Subgroup structure of groups of order 81

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This article gives specific information, namely, subgroup structure, about a family of groups, namely: groups of order 81.
View subgroup structure of group families | View subgroup structure of groups of a particular order |View other specific information about groups of order 81
FACTS TO CHECK AGAINST FOR SUBGROUP STRUCTURE: (group of prime power order)
Lagrange's theorem (order of subgroup times index of subgroup equals order of whole group, so all subgroups have prime power orders)|order of quotient group divides order of group (and equals index of corresponding normal subgroup, so all quotients have prime power orders)
prime power order implies not centerless | prime power order implies nilpotent | prime power order implies center is normality-large
size of conjugacy class of subgroups divides index of center
congruence condition on number of subgroups of given prime power order: The total number of subgroups of any fixed prime power order is congruent to 1 mod the prime.

Numerical information on counts of subgroups by order

Number of subgroups per order

Due to congruence condition on number of subgroups of given prime power order, all the counts of subgroups, as well as of normal subgroups, are congruent to 1 modulo 3. Further, note the following:

  • For an abelian group, the number of subgroups of a given order equals the number of normal subgroups. Moreover, because subgroup lattice and quotient lattice of finite abelian group are isomorphic, we get that (number of subgroups of order 3) = (number of normal subgroups of order 3) = (number of subgroups of order 27) = (number of normal subgroups of order 27), and separately, (number of subgroups of order 9) = (number of normal subgroups of order 9).
  • Since prime power order implies nilpotent, and nilpotent implies every maximal subgroup is normal, we have, in all cases, that the number of subgroups of order 27 = number of normal subgroups of order 27.
  • The subgroups of order 27 (all of which are normal) correspond to the maximal subgroups of the Frattini quotient, which is an elementary abelian group of order 3^r, 1 \le r \le 4 where r is the minimum size of generating set for the group. The number of such subgroups is (3^r - 1)/(3 - 1) = 3^{r-1} + 3^{r-2} + \dots + 3 + 1, and hence must be one of the numbers 1,4,13,40.
  • The normal subgroups of order 3 correspond precisely to the subgroups of order 3 in the socle, which is an elementary abelian subgroup defined as \Omega_1 of the center. If the socle has order 3^s, 1 \le s \le 4, the number of normal subgroups of order 3 is (3^s - 1)/(3 - 1) = 3^{s-1} + 3^{s-2} + \dots + 3 + 1. See minimal normal implies central in nilpotent. Thus, the count of normal subgroups of order 3 must be 1, 4, 13, or 40. Moroever, for a non-abelian group, the socle can have order either 3 or 9 (cannot have order 27 or 81) so the number of normal subgroups is either 1 or 3.
Common name for group Second part of GAP ID (GAP ID is (p^4, second part)) Nilpotency class Number of subgroups of order 3 Number of normal subgroups of order 3 Number of subgroups of order 9 Number of normal subgroups of order 9 Number of subgroups of order 27 Number of normal subgroups of order 27
Cyclic group:Z81 1 1 1 1 1 1 1 1
Direct product of Z9 and Z9 2 1 4 4 13 13 4 4
SmallGroup(81,3) 3 2 13 4 22 4 4 4
Nontrivial semidirect product of Z9 and Z9 4 2 4 4 13 4 4 4
Direct product of Z27 and Z3 5 1 4 4 4 4 4 4
M81 6 2 4 1 4 4 4 4
Wreath product of Z3 and Z3 7 3 22 1 22 1 4 4
SmallGroup(81,8) 8 3 13 1 13 1 4 4
SmallGroup(81,9) 9 3 31 1 13 1 4 4
SmallGroup(81,10) 10 3 4 1 13 1 4 4
Direct product of Z9 and E9 11 1 13 13 22 22 13 13
Direct product of prime-cube order group:U(3,3) and Z3 12 2 40 4 49 13 13 13
Direct product of semidirect product of Z9 and Z3 and Z3 13 2 13 4 22 13 13 13
Central product of prime-cube order group:U(3,3) and Z9 14 2 13 1 13 13 13 13
Elementary abelian group:E81 15 1 40 40 130 130 40 40

Abelian subgroups

Counts of abelian subgroups and abelian normal subgroups

Note the following:

The upshot is that all counts in the table below are odd.

Common name for group Second part of GAP ID (GAP ID is (p^4, second part)) Nilpotency class Number of abelian subgroups of order 3 Number of abelian normal subgroups of order 3 Number of abelian subgroups of order 9 Number of abelian normal subgroups of order 9 Number of abelian subgroups of order 27 Number of abelian normal subgroups of order 27
Cyclic group:Z81 1 1 1 1 1 1 1 1
Direct product of Z9 and Z9 2 1 4 4 13 13 4 4
SmallGroup(81,3) 3 2 13 4 22 4 4 4
Nontrivial semidirect product of Z9 and Z9 4 2 4 4 13 4 4 4
Direct product of Z27 and Z3 5 1 4 4 4 4 4 4
M81 6 2 4 1 4 4 4 4
Wreath product of Z3 and Z3 7 3 22 1 22 1 1 1
SmallGroup(81,8) 8 3 13 1 13 1 1 1
SmallGroup(81,9) 9 3 31 1 13 1 1 1
SmallGroup(81,10) 10 3 4 1 13 1 1 1
Direct product of Z9 and E9 11 1 13 13 22 22 13 13
Direct product of prime-cube order group:U(3,3) and Z3 12 2 40 4 49 13 4 4
Direct product of semidirect product of Z9 and Z3 and Z3 13 2 13 4 22 13 4 4
Central product of prime-cube order group:U(3,3) and Z9 14 2 13 1 13 13 4 4
Elementary abelian group:E81 15 1 40 40 130 130 40 40