Subgroup structure of groups of order 32

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This article gives specific information, namely, subgroup structure, about a family of groups, namely: groups of order 32.
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Numerical information on counts of subgroups by order

We note the following:

General assertion Implication for the counts in this case
Congruence condition on number of subgroups of given prime power order states that the number of subgroups of order equal to a given prime power dividing the order of the group is congruent to 1 modulo the prime.
Further, if the whole group itself has prime power order, the number of normal subgroups of order equal to a given prime power dividing the order is also congruent to 1 modulo the prime.
Here, the prime is 2, and congruent to 1 mod 2 is the same as saying odd, so we get that each of the following counts is odd: (number of subgroups of order 2), (number of normal subgroups of order 2), (number of subgroups of order 4), (number of normal subgroups of order 4), (number of subgroups of order 8), (number of normal subgroups of order 8), (number of subgroups of order 16), (number of normal subgroups of order 16).
Index two implies normal (more general prime version: subgroup of index equal to least prime divisor of group order is normal) All the subgroups of order 16 (equivalently, index 2) are normal. Thus, (number of subgroups of order 16) = (number of normal subgroups of order 16).
In a group of prime power order, say p^n, the maximal subgroups are all normal, and are precisely the subgroups of index p. They correspond to maximal subgroups in the Frattini quotient via the fourth isomorphism theorem. The Frattini quotient is an elementary abelian group of order p^r where r is the minimum size of generating set for the group, and the number of maximal subgroups turns out to be (p^r - 1)/(p - 1). Note that 1 \le r \le n. The number of maximal subgroups is (2^r - 1)/(2 - 1) = 2^r - 1. Thus, the only possibilities for the number of subgroups of order 16 in a group of order 32 are 1, 3, 7, 15, and 31.
In a group of prime power order p^n, the normal subgroups of prime order are precisely the subgroups of prime order inside the socle, which is the first omega subgroup of the center, and is elementary abelian of order p^s where s is the rank of the center. (See minimal normal implies central in nilpotent). The number of normal subgroups of prime order is thus (p^s - 1)/(p - 1), where 1 \le s \le n. For a non-abelian group, s \le n - 2. In our case, the number of normal subgroups of order 2 is (2^s - 1)/(2 - 1) = 2^s - 1, which must be one of the numbers 1,3,7,15,31. For a non-abelian group, the socle cannot have order 16 or 32, so the number of normal subgroups of order 2 is one of the numbers 1,3,7.
Subgroup lattice and quotient lattice of finite abelian group are isomorphic as well as abelian implies every subgroup is normal guarantees that, in a finite abelian group, the counts for subgroups coincide with the counts for normal subgroups, and also that the number of subgroups of a given order equals the number of subgroups of order equal to the order of the whole group divided by that order. For abelian groups of order 32, we have: (number of subgroups of order 2) = (number of normal subgroups of order 2) = (number of subgroups of order 16) = (number of normal subgroups of order 16). Separately, (number of subgroups of order 4) = (number of normal subgroups of order 4) = (number of subgroups of order 8) = (number of normal subgroups of order 8).
Group GAP ID (2nd part) Hall-Senior number Hall-Senior symbol Nilpotency class Number of subgroups of order 2 Number of normal subgroups of order 2 Number of subgroups of order 4 Number of normal subgroups of order 4 Number of subgroups of order 8 Number of normal subgroups of order 8 Number of subgroups of order 16 Number of normal subgroups of order 16
Cyclic group:Z32 1 7 (5) 1 1 1 1 1 1 1 1 1
SmallGroup(32,2) 2 18 \Gamma_2h 2 7 7 19 7 19 7 3 3
Direct product of Z8 and Z4 3 5 (32) 1 3 3 7 7 7 7 3 3
Semidirect product of Z8 and Z4 of M-type 4 19 \Gamma_2i 2 3 3 7 3 7 7 3 3
SmallGroup(32,5) 5 20 \Gamma_2j_1 2 7 3 11 5 11 3 3 3
Faithful semidirect product of E8 and Z4 6 46 \Gamma_7a_1 3 11 1 23 3 11 3 3 3
SmallGroup(32,7) 7 47 \Gamma_7a_2 3 11 1 15 3 11 3 3 3
SmallGroup(32,8) 8 48 \Gamma_7a_3 3 3 1 7 3 11 3 3 3
SmallGroup(32,9) 9 27 \Gamma_3c_1 3 11 3 19 3 11 3 3 3
SmallGroup(32,10) 10 28 \Gamma_3c_2 3 3 3 11 3 11 3 3 3
Wreath product of Z4 and Z2 11 31 \Gamma_3e 3 7 1 11 3 11 3 3 3
SmallGroup(32,12) 12 21 \Gamma_2j_2 2 3 3 7 5 7 3 3 3
Semidirect product of Z8 and Z4 of semidihedral type 13 30 \Gamma_3d_2 3 3 3 11 3 7 3 3 3
Semidirect product of Z8 and Z4 of dihedral type 14 29 \Gamma_3d_1 3 3 3 11 3 7 3 3 3
SmallGroup(32,15) 15 32 \Gamma_3f 3 3 1 3 3 7 3 3 3
Direct product of Z16 and Z2 16 6 (41) 1 3 3 3 3 3 3 3 3
M32 17 22 \Gamma_2k 2 3 1 3 3 3 3 3 3
Dihedral group:D32 18 49 \Gamma_8a_1 4 17 1 9 1 5 1 3 3
Semidihedral group:SD32 19 50 \Gamma_8a_2 4 9 1 9 1 5 1 3 3
Generalized quaternion group:Q32 20 51 \Gamma_8a_3 4 1 1 9 1 5 1 3 3
Direct product of Z4 and Z4 and Z2 21 3 (2^21) 1 7 7 19 19 19 19 7 7
Direct product of SmallGroup(16,3) and Z2 22 11 \Gamma_2c_1 2 15 7 43 11 27 11 7 7
Direct product of SmallGroup(16,4) and Z2 23 12 \Gamma_2c_2 2 7 7 19 11 19 11 7 7
SmallGroup(32,24) 24 16 \Gamma_2f 2 7 3 19 7 11 11 7 7
Direct product of D8 and Z4 25 14 \Gamma_2e_1 2 11 3 23 9 19 11 7 7
Direct product of Q8 and Z4 26 15 \Gamma_2e_2 2 3 3 15 9 11 11 7 7
SmallGroup(32,27) 27 33 \Gamma_4a_1 2 19 3 47 7 31 7 7 7
SmallGroup(32,28) 28 36 \Gamma_4b_1 2 15 3 27 5 23 7 7 7
SmallGroup(32,29) 29 37 \Gamma_4b_2 2 7 3 19 5 15 7 7 7
SmallGroup(32,30) 30 38 \Gamma_4c_1 2 11 3 23 3 15 7 7 7
SmallGroup(32,31) 31 39 \Gamma_4c_2 2 11 3 23 3 15 7 7 7
SmallGroup(32,32) 32 40 \Gamma_4c_3 2 3 3 15 3 7 7 7 7
SmallGroup(32,33) 33 41 \Gamma_4d 2 7 3 19 1 7 7 7 7
Generalized dihedral group for direct product of Z4 and Z4 34 34 \Gamma_4a_2 2 19 3 31 7 31 7 7 7
SmallGroup(32,35) 35 35 \Gamma_4a_3 2 3 3 15 7 15 7 7 7
Direct product of Z8 and V4 36 4 (31^2) 1 7 7 11 11 11 11 7 7
Direct product of M16 and Z2 37 13 \Gamma_2d 2 7 3 11 7 11 11 7 7
Central product of D8 and Z8 38 17 \Gamma_2g 2 7 1 7 7 11 11 7 7
Direct product of D16 and Z2 39 23 \Gamma_3a_1 3 19 3 27 3 15 7 7 7
Direct product of SD16 and Z2 40 24 \Gamma_3a_2 3 11 3 19 3 15 7 7 7
Direct product of Q16 and Z2 41 25 \Gamma_3a_3 3 3 3 11 3 15 7 7 7
Central product of D16 and Z4 42 26 \Gamma_3b 3 11 1 11 3 15 7 7 7
Holomorph of Z8 43 44 \Gamma_6a_1 3 15 1 19 3 15 7 7 7
SmallGroup(32,44) 44 45 \Gamma_6a_2 3 7 1 11 3 15 7 7 7
Direct product of E8 and Z4 45 2 (21^3) 1 15 15 43 43 43 43 15 15
Direct product of D8 and V4 46 8 \Gamma_2a_1 2 23 7 67 19 51 35 15 15
Direct product of Q8 and V4 47 9 \Gamma_2a_2 2 7 7 19 19 35 35 15 15
Direct product of SmallGroup(16,13) and Z2 48 10 \Gamma_2b 2 15 3 27 15 35 35 15 15
Inner holomorph of D8 49 42 \Gamma_5a_1 2 19 1 39 15 35 35 15 15
Central product of D8 and Q8 50 43 \Gamma_5a_2 2 11 1 15 15 35 35 15 15
Elementary abelian group:E32 51 1 (1^5) 1 31 31 155 155 155 155 31 31
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Abelian subgroups

Further information: abelian subgroup structure of groups of order 32

Counts of abelian subgroups and abelian normal subgroups

Column (in counts table) about which assertion is being made Fact being asserted about the column Explanation/general version
number of abelian subgroups of order 2 odd congruence condition on number of subgroups of given prime power order, along with the observation that any group of order 2 is abelian.
Another way of saying this is that the singleton set cyclic group:Z2 is a collection of groups satisfying a universal congruence condition.
number of abelian normal subgroups of order 2 odd -
number of abelian subgroups of order 4
number of abelian normal subgroups of order 4
odd
odd
congruence condition on number of subgroups of given prime power order, along with the observation that any group of order 4 is abelian (see groups of order 4)
Another way of putting this is that the collection of abelian groups of order 4 is a collection of groups satisfying a universal congruence condition.
number of abelian subgroups of order 8
number of abelian normal subgroups of order 8
odd
odd
existence of abelian normal subgroups of small prime power order guarantees first the existence of an abelian normal subgroup of order 8. Congruence condition on number of abelian subgroups of prime-cube order then shows that the total count of abelian normal subgroups is odd.
number of abelian subgroups of order 16 = number of abelian normal subgroups of order 16 odd if nonzero. In a non-abelian group, it is either 0, 1, or 3. index two implies normal, so the abelian subgroups of order 16 are normal. Further, by congruence condition on number of abelian subgroups of prime index, this number is odd.
number of abelian normal subgroups of order 2 one of the numbers 1,3,7,15,31. For a non-abelian group, one of the numbers 1,3,7. The abelian normal subgroups of order 2 are the same as the normal subgroups of order 2, which are precisely the subgroups of order 2 contained in the socle, which in this case is the first omega subgroup of the center. It is thus elementary abelian of order 2^s where s is the rank of the center, so its number of subgroups of order 2 is 2^s - 1.

Two additional points:

  • For the abelian groups: note that abelian implies every subgroup is normal and also that subgroup lattice and quotient lattice of finite abelian group are isomorphic. Thus, when the whole group is abelian, we have: number of abelian subgroups of order 2 = number of abelian normal subgroups of order 2 = number of abelian subgroups of order 16 = number of abelian normal subgroups of order 16. Separately, we have number of abelian subgroups of order 4 = number of abelian normal subgroups of order 4 = number of abelian subgroups of order 8 = number of abelian normal subgroups of order 8.
  • The "number of abelian normal subgroups" columns depend only on the Hall-Senior genus, i.e., two groups with the same Hall-Senior genus have the same "number of abelian normal subgroups" of each order. The Hall-Senior genus is the part of the Hall-Senior symbol excluding the very final subscript, so for instance 32\Gamma_2a_1 and 32\Gamma_2a_2 both belong to the Hall-Senior genus 32\Gamma_2a and hence have the same number of abelian normal subgroups of each order.
Group GAP ID (2nd part) Hall-Senior number Hall-Senior symbol Nilpotency class Number of abelian subgroups of order 2 Number of abelian normal subgroups of order 2 Number of abelian subgroups of order 4 Number of abelian normal subgroups of order 4 Number of abelian subgroups of order 8 Number of abelian normal subgroups of order 8 Number of abelian subgroups of order 16 Number of abelian normal subgroups of order 16
Cyclic group:Z32 1 7 (5) 1 1 1 1 1 1 1 1 1
SmallGroup(32,2) 2 18 \Gamma_2h 2 7 7 19 7 19 7 3 3
Direct product of Z8 and Z4 3 5 (32) 1 3 3 7 7 7 7 3 3
Semidirect product of Z8 and Z4 of M-type 4 19 \Gamma_2i 2 3 3 7 3 7 7 3 3
SmallGroup(32,5) 5 20 \Gamma_2j_1 2 7 3 11 5 11 3 3 3
Faithful semidirect product of E8 and Z4 6 46 \Gamma_7a_1 3 11 1 23 3 7 3 0 0
SmallGroup(32,7) 7 47 \Gamma_7a_2 3 11 1 15 3 7 3 0 0
SmallGroup(32,8) 8 48 \Gamma_7a_3 3 3 1 7 3 7 3 0 0
SmallGroup(32,9) 9 27 \Gamma_3c_1 3 11 3 19 3 7 1 1 1
SmallGroup(32,10) 10 28 \Gamma_3c_2 3 3 3 11 3 7 1 1 1
Wreath product of Z4 and Z2 11 31 \Gamma_3e 3 7 1 11 3 7 1 1 1
SmallGroup(32,12) 12 21 \Gamma_2j_2 2 3 3 7 5 7 3 3 3
Semidirect product of Z8 and Z4 of semidihedral type 13 3 3 3 11 3 7 3 1 1
Semidirect product of Z8 and Z4 of dihedral type 14 3 3 3 11 3 7 3 1 1
SmallGroup(32,15) 15 32 \Gamma_3f 3 3 1 3 3 7 3 1 1
Direct product of Z16 and Z2 16 6 (41) 1 3 3 3 3 3 3 3 3
M32 17 22 \Gamma_2k 2 3 1 3 3 3 3 3 3
Dihedral group:D32 18 49 \Gamma_8a_1 4 17 1 9 1 1 1 1 1
Semidihedral group:SD32 19 50 \Gamma_8a_2 4 9 1 9 1 1 1 1 1
Generalized quaternion group:Q32 20 51 \Gamma_8a_3 4 1 1 9 1 1 1 1 1
Direct product of Z4 and Z4 and Z2 21 3 (2^21) 1 7 7 19 19 19 19 7 7
Direct product of SmallGroup(16,3) and Z2 22 11 \Gamma_2c_1 2 15 7 43 11 27 11 3 3
Direct product of SmallGroup(16,4) and Z2 23 12 \Gamma_2c_2 2 7 7 19 11 19 11 3 3
SmallGroup(32,24) 24 16 \Gamma_2f 2 7 3 19 7 11 11 3 3
Direct product of D8 and Z4 25 14 \Gamma_2e_1 2 11 3 23 9 15 7 3 3
Direct product of Q8 and Z4 26 15 \Gamma_2e_2 2 3 3 15 9 7 7 3 3
SmallGroup(32,27) 27 33 \Gamma_4a_1 2 19 3 47 7 19 7 1 1
SmallGroup(32,28) 28 36 \Gamma_4b_1 2 15 3 27 5 11 7 1 1
SmallGroup(32,29) 29 37 \Gamma_4b_2 2 7 3 19 5 11 7 1 1
SmallGroup(32,30) 30 38 \Gamma_4c_1 2 11 3 23 3 11 7 1 1
SmallGroup(32,31) 31 39 \Gamma_4c_2 2 11 3 23 3 7 7 1 1
SmallGroup(32,32) 32 40 \Gamma_4c_3 2 3 3 15 3 7 7 1 1
SmallGroup(32,33) 33 41 \Gamma_4d 2 7 3 19 1 7 7 1 1
Generalized dihedral group for direct product of Z4 and Z4 34 34 \Gamma_4a_2 2 19 3 31 7 7 7 1 1
SmallGroup(32,35) 35 35 \Gamma_4a_3 2 3 3 15 7 7 7 1 1
Direct product of Z8 and V4 36 4 (31^2) 1 7 7 11 11 11 11 7 7
Direct product of M16 and Z2 37 13 \Gamma_2d 2 7 3 11 7 11 11 3 3
Central product of D8 and Z8 38 17 \Gamma_2g 2 7 1 7 7 7 7 3 3
Direct product of D16 and Z2 39 23 \Gamma_3a_1 3 19 3 27 3 7 3 1 1
Direct product of SD16 and Z2 40 24 \Gamma_3a_2 3 11 3 19 3 7 3 1 1
Direct product of Q16 and Z2 41 25 \Gamma_3a_3 3 3 3 11 3 7 3 1 1
Central product of D16 and Z4 42 26 \Gamma_3b 3 11 1 11 3 7 3 1 1
Holomorph of Z8 43 44 \Gamma_6a_1 3 15 1 19 3 7 3 0 0
SmallGroup(32,44) 44 45 \Gamma_6a_2 3 7 1 11 3 7 3 0 0
Direct product of E8 and Z4 45 2 (21^3) 1 15 15 43 43 43 43 15 15
Direct product of D8 and V4 46 8 \Gamma_2a_1 2 23 7 67 19 35 19 3 3
Direct product of Q8 and V4 47 9 \Gamma_2a_2 2 7 7 19 19 19 19 3 3
Direct product of SmallGroup(16,13) and Z2 48 10 \Gamma_2b 2 15 3 27 15 19 19 3 3
Inner holomorph of D8 49 42 \Gamma_5a_1 2 19 1 39 15 15 15 0 0
Central product of D8 and Q8 50 43 \Gamma_5a_2 2 11 1 15 15 15 15 0 0
Elementary abelian group:E32 51 1 (1^5) 1 31 31 155 155 155 155 31 31
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