Abelian subgroup structure of groups of order 32

This article gives specific information, namely, abelian subgroup structure, about a family of groups, namely: groups of order 32.
View abelian subgroup structure of group families | View abelian subgroup structure of groups of a particular order |View other specific information about groups of order 32

Summary on existence, congruence, and replacement

Order of subgroup	Existence of abelian subgroup of that order guaranteed?	Existence of abelian normal subgroup of that order guaranteed?	Abelian-to-normal replacement for that order guaranteed?	Number of abelian subgroups of that order always odd if nonzero?
1	Yes	Yes	Yes	Yes
2	Yes	Yes	Yes	Yes
4	Yes	Yes	Yes	Yes
8	Yes	Yes	Yes	Yes
16	No	No	Yes	Yes
32	No	No	Yes	Yes

Abelian normal subgroups

FACTS TO CHECK AGAINST for abelian normal subgroup of group of prime power order:
EXISTENCE: existence of abelian normal subgroups of small prime power order
CONGRUENCE, IMPLIES ABELIAN-TO-NORMAL REPLACEMENT: congruence condition on number of subgroups of given prime power order (covers cases of $p$ and $p^{2}$) | congruence condition on number of abelian subgroups of prime-cube order | congruence condition on number of abelian subgroups of prime-fourth order | Jonah-Konvisser congruence condition on number of abelian subgroups of small prime power order | congruence condition on number of abelian subgroups of prime index | congruence condition on number of abelian subgroups of prime-square index for odd prime |abelian-to-normal replacement theorem for prime-square index | abelian-to-normal replacement theorem for prime-cube index for odd prime
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RELATION WITH REPRESENTATION THEORY: degree of irreducible representation divides index of abelian normal subgroup

Abelian normal subgroups of order 16

Note that index two implies normal, so the abelian subgroups of order 16 are precisely the same as the abelian normal subgroups of order 16.

• Existence is not guaranteed!: It is possible for there to be no abelian normal subgroup of order 16.
• Count: The congruence condition on number of abelian subgroups of prime index tells us that the number of abelian normal subgroups of order 16, if nonzero, is odd. However, we can be more specific. In a non-abelian group of order 32, the number of abelian normal subgroups is 0, 1, or 3. The "3" case arises if and only if the inner automorphism group is a Klein four-group, which occurs only for the Hall-Senior family (isoclinism class) $\Gamma _{2}$.

For an abelian group of order 32 and rank $r$ (i.e., $r$ is the minimum size of generating set), the number of abelian normal subgroups of order 16 is $2^{r}-1$, which could be 1, 3, 7, 15, or 31.

Group	GAP ID (2nd part)	Hall-Senior number	Hall-Senior symbol	Nilpotency class	Minimum size of generating set	Number of subgroups isomorphic to elementary abelian group:E16	Number of subgroups isomorphic to direct product of Z4 and V4	Number of subgroups isomorphic to direct product of Z4 and Z4	Number of subgroups isomorphic to direct product of Z8 and Z2	Number of subgroups isomorphic to cyclic group:Z16	Total number of abelian (normal) subgroups of order 16
Cyclic group:Z32	1	7	$(5)$	1	1	0	0	0	0	1	1
SmallGroup(32,2)	2	18	$\Gamma _{2}h$	2	2	0	3	0	0	0	3
Direct product of Z8 and Z4	3	5	$(32)$	1	2	0	0	1	2	0	3
Semidirect product of Z8 and Z4 of M-type	4	19	$\Gamma _{2}i$	2	2	0	0	1	2	0	3
SmallGroup(32,5)	5	20	$\Gamma _{2}j_{1}$	2	2	0	1	0	2	0	3
Faithful semidirect product of E8 and Z4	6	46	$\Gamma _{7}a_{1}$	3	2	0	0	0	0	0	0
SmallGroup(32,7)	7	47	$\Gamma _{7}a_{2}$	3	2	0	0	0	0	0	0
SmallGroup(32,8)	8	48	$\Gamma _{7}a_{3}$	3	2	0	0	0	0	0	0
SmallGroup(32,9)	9	27	$\Gamma _{3}c_{1}$	3	2	0	0	0	1	0	1
SmallGroup(32,10)	10	28	$\Gamma _{3}c_{2}$	3	2	0	0	0	1	0	1
Wreath product of Z4 and Z2	11	31	$\Gamma _{3}e$	3	2	0	0	1	0	0	1
SmallGroup(32,12)	12	21	$\Gamma _{2}j_{2}$	2	2	0	0	1	2	0	3
Semidirect product of Z8 and Z4 of semidihedral type	13			3	2	0	0	0	1	0	1
Semidirect product of Z8 and Z4 of dihedral type	14			3	2	0	0	0	1	0	1
SmallGroup(32,15)	15	32	$\Gamma _{3}f$	3	2	0	0	0	1	0	1
Direct product of Z16 and Z2	16	6	$(41)$	1	2	0	0	0	1	2	3
M32	17	22	$\Gamma _{2}k$	2	2	0	0	0	1	2	3
Dihedral group:D32	18	49	$\Gamma _{8}a_{1}$	4	2	0	0	0	0	1	1
Semidihedral group:SD32	19	50	$\Gamma _{8}a_{2}$	4	2	0	0	0	0	1	1
Generalized quaternion group:Q32	20	51	$\Gamma _{8}a_{3}$	4	2	0	0	0	0	1	1
Direct product of Z4 and Z4 and Z2	21	3	$(2^{2}1)$	1	3	0	3	4	0	0	7
Direct product of SmallGroup(16,3) and Z2	22	11	$\Gamma _{2}c_{1}$	2	3	1	2	0	0	0	3
Direct product of SmallGroup(16,4) and Z2	23	12	$\Gamma _{2}c_{2}$	2	3	0	3	0	0	0	3
SmallGroup(32,24)	24	16	$\Gamma _{2}f$	2	3	0	1	2	0	0	3
Direct product of D8 and Z4	25	14	$\Gamma _{2}e_{1}$	2	3	0	2	1	0	0	3
Direct product of Q8 and Z4	26	15	$\Gamma _{2}e_{2}$	2	3	0	0	3	0	0	3
SmallGroup(32,27)	27	33	$\Gamma _{4}a_{1}$	2	3	1	0	0	0	0	1
SmallGroup(32,28)	28	36	$\Gamma _{4}b_{1}$	2	3	0	1	0	0	0	1
SmallGroup(32,29)	29	37	$\Gamma _{4}b_{2}$	2	3	0	1	0	0	0	1
SmallGroup(32,30)	30	38	$\Gamma _{4}c_{1}$	2	3	0	0	1	0	0	1
SmallGroup(32,31)	31	39	$\Gamma _{4}c_{2}$	2	3	0	0	1	0	0	1
SmallGroup(32,32)	32	40	$\Gamma _{4}c_{3}$	2	3	0	0	1	0	0	1
SmallGroup(32,33)	33	41	$\Gamma _{4}d$	2	3	0	0	1	0	0	1
Generalized dihedral group for direct product of Z4 and Z4	34	34	$\Gamma _{4}a_{2}$	2	3	0	0	1	0	0	1
SmallGroup(32,35)	35	35	$\Gamma _{4}a_{3}$	2	3	0	0	1	0	0	1
Direct product of Z8 and V4	36	4	$(31^{2})$	1	3	0	1	0	6	0	7
Direct product of M16 and Z2	37	13	$\Gamma _{2}d$	2	3	0	1	0	2	0	3
Central product of D8 and Z8	38	17	$\Gamma _{2}g$	2	3	0	0	0	3	0	3
Direct product of D16 and Z2	39	23	$\Gamma _{3}a_{1}$	3	3	0	0	0	1	0	1
Direct product of SD16 and Z2	40	24	$\Gamma _{3}a_{2}$	3	3	0	0	0	1	0	1
Direct product of Q16 and Z2	41	25	$\Gamma _{3}a_{3}$	3	3	0	0	0	1	0	1
Central product of D16 and Z4	42	26	$\Gamma _{3}b$	3	3	0	0	0	1	0	1
Holomorph of Z8	43	44	$\Gamma _{6}a_{1}$	3	3	0	0	0	0	0	0
SmallGroup(32,44)	44	45	$\Gamma _{6}a_{2}$	3	3	0	0	0	0	0	0
Direct product of E8 and Z4	45	2	$(21^{3})$	1	4	1	14	0	0	0	15
Direct product of D8 and V4	46	8	$\Gamma _{2}a_{1}$	2	4	2	1	0	0	0	3
Direct product of Q8 and V4	47	9	$\Gamma _{2}a_{2}$	2	4	0	3	0	0	0	3
Direct product of SmallGroup(16,13) and Z2	48	10	$\Gamma _{2}b$	2	4	0	3	0	0	0	3
Inner holomorph of D8	49	42	$\Gamma _{5}a_{1}$	2	4	0	0	0	0	0	0
Central product of D8 and Q8	50	43	$\Gamma _{5}a_{2}$	2	4	0	0	0	0	0	0
Elementary abelian group:E32	51	1	$(1^{5})$	1	5	31	0	0	0	0	31

We now construct a table derived from the above, that lists the total number of abelian normal subgroups of order eight and exponent bounded by some specific number (2, 4, 8, or 16). Note that the exponent dividing 2 count may be a nonzero even number (specifically, it can be 2), but the exponent dividing 4, exponent dividing 8, and exponent dividing 16 counts are all either zero or odd. As noted earlier, for non-abelian groups, these counts are either 0, 1, or 3.

Group	GAP ID (2nd part)	Hall-Senior number	Hall-Senior symbol	Nilpotency class	Minimum size of generating set	Number of subgroups of order 16, exponent dividing 2	Number of subgroups of order 16, exponent dividing 4	Number of subgroups of order 16, exponent dividing 8	Number of subgroups of order 16, exponent dividing 16
Cyclic group:Z32	1	7	$(5)$	1	1	0	0	0	1
SmallGroup(32,2)	2	18	$\Gamma _{2}h$	2	2	0	3	3	3
Direct product of Z8 and Z4	3	5	$(32)$	1	2	0	1	3	3
Semidirect product of Z8 and Z4 of M-type	4	19	$\Gamma _{2}i$	2	2	0	1	3	3
SmallGroup(32,5)	5	20	$\Gamma _{2}j_{1}$	2	2	0	1	3	3
Faithful semidirect product of E8 and Z4	6	46	$\Gamma _{7}a_{1}$	3	2	0	0	0	0
SmallGroup(32,7)	7	47	$\Gamma _{7}a_{2}$	3	2	0	0	0	0
SmallGroup(32,8)	8	48	$\Gamma _{7}a_{3}$	3	2	0	0	0	0
SmallGroup(32,9)	9	27	$\Gamma _{3}c_{1}$	3	2	0	0	1	1
SmallGroup(32,10)	10	28	$\Gamma _{3}c_{2}$	3	2	0	0	1	1
Wreath product of Z4 and Z2	11	31	$\Gamma _{3}e$	3	2	0	1	1	1
SmallGroup(32,12)	12	21	$\Gamma _{2}j_{2}$	2	2	0	1	3	3
Semidirect product of Z8 and Z4 of semidihedral type	13			3	2	0	0	1	1
Semidirect product of Z8 and Z4 of dihedral type	14			3	2	0	0	1	1
SmallGroup(32,15)	15	32	$\Gamma _{3}f$	3	2	0	0	1	1
Direct product of Z16 and Z2	16	6	$(41)$	1	2	0	0	1	3
M32	17	22	$\Gamma _{2}k$	2	2	0	0	1	3
Dihedral group:D32	18	49	$\Gamma _{8}a_{1}$	4	2	0	0	0	1
Semidihedral group:SD32	19	50	$\Gamma _{8}a_{2}$	4	2	0	0	0	1
Generalized quaternion group:Q32	20	51	$\Gamma _{8}a_{3}$	4	2	0	0	0	1
Direct product of Z4 and Z4 and Z2	21	3	$(2^{2}1)$	1	3	0	7	7	7
Direct product of SmallGroup(16,3) and Z2	22	11	$\Gamma _{2}c_{1}$	2	3	1	3	3	3
Direct product of SmallGroup(16,4) and Z2	23	12	$\Gamma _{2}c_{2}$	2	3	0	3	3	3
SmallGroup(32,24)	24	16	$\Gamma _{2}f$	2	3	0	1	3	3
Direct product of D8 and Z4	25	14	$\Gamma _{2}e_{1}$	2	3	0	3	3	3
Direct product of Q8 and Z4	26	15	$\Gamma _{2}e_{2}$	2	3	0	3	3	3
SmallGroup(32,27)	27	33	$\Gamma _{4}a_{1}$	2	3	1	1	1	1
SmallGroup(32,28)	28	36	$\Gamma _{4}b_{1}$	2	3	0	1	1	1
SmallGroup(32,29)	29	37	$\Gamma _{4}b_{2}$	2	3	0	1	1	1
SmallGroup(32,30)	30	38	$\Gamma _{4}c_{1}$	2	3	0	1	1	1
SmallGroup(32,31)	31	39	$\Gamma _{4}c_{2}$	2	3	0	1	1	1
SmallGroup(32,32)	32	40	$\Gamma _{4}c_{3}$	2	3	0	1	1	1
SmallGroup(32,33)	33	41	$\Gamma _{4}d$	2	3	0	1	1	1
Generalized dihedral group for direct product of Z4 and Z4	34	34	$\Gamma _{4}a_{2}$	2	3	0	1	1	1
SmallGroup(32,35)	35	35	$\Gamma _{4}a_{3}$	2	3	0	1	1	1
Direct product of Z8 and V4	36	4	$(31^{2})$	1	3	0	1	7	7
Direct product of M16 and Z2	37	13	$\Gamma _{2}d$	2	3	0	1	3	3
Central product of D8 and Z8	38	17	$\Gamma _{2}g$	2	3	0	0	3	3
Direct product of D16 and Z2	39	23	$\Gamma _{3}a_{1}$	3	3	0	0	1	1
Direct product of SD16 and Z2	40	24	$\Gamma _{3}a_{2}$	3	3	0	0	1	1
Direct product of Q16 and Z2	41	25	$\Gamma _{3}a_{3}$	3	3	0	0	1	1
Central product of D16 and Z4	42	26	$\Gamma _{3}b$	3	3	0	0	1	1
Holomorph of Z8	43	44	$\Gamma _{6}a_{1}$	3	3	0	0	0	0
SmallGroup(32,44)	44	45	$\Gamma _{6}a_{2}$	3	3	0	0	0	0
Direct product of E8 and Z4	45	2	$(21^{3})$	1	4	1	15	15	15
Direct product of D8 and V4	46	8	$\Gamma _{2}a_{1}$	2	4	2	3	3	3
Direct product of Q8 and V4	47	9	$\Gamma _{2}a_{2}$	2	4	0	3	3	3
Direct product of SmallGroup(16,13) and Z2	48	10	$\Gamma _{2}b$	2	4	0	3	3	3
Inner holomorph of D8	49	42	$\Gamma _{5}a_{1}$	2	4	0	0	0	0
Central product of D8 and Q8	50	43	$\Gamma _{5}a_{2}$	2	4	0	0	0	0
Elementary abelian group:E32	51	1	$(1^{5})$	1	5	31	31	31	31

Abelian normal subgroups of order 8

• Existence: There exist abelian normal subgroups of order 8 by the existence of abelian normal subgroups of small prime power order. In fact, any subgroup that is maximal among abelian normal subgroups must have orer at least 8.
• Count: The congruence condition on number of abelian subgroups of prime-cube order tells us that the number of abelian normal subgroups of order 8 is odd.

Abelian normal subgroups of order 4

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Abelian normal subgroups of order 2

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Abelian characteristic subgroups

We make cases based on the nilpotency class, isoclinism class (i.e., Hall-Senior family) and other aspects of the group structure:

• For an abelian group of order 32, the whole group is the unique subgroup of itself that is maximal among abelian characteristic subgroups.