Subgroups — C₂⁴×#11715897 #27158719

Order 12416.

1976 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11715897	C₂	1	30	yes	maximal, direct product factor
2	C₂⁴×Dic₉₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11715897	C₂²	1	140	yes	direct product factor
4	C₂³×Dic₉₇	C₄	1	15	yes	—
5	C₂³×Dic₉₇	C₂²	1	15	yes	—
6	C₂⁵×C₉₇	C₄	1	1	yes	characteristic
7	C₂×#11715897	C₂³	1	120	yes	direct product factor
8	C₂²×Dic₉₇	C₂×C₄	1	105	yes	—
9	C₂²×Dic₉₇	C₂³	1	35	yes	—
10	C₂⁴×C₉₇	C₈	1	16	yes	complemented
11	C₂⁴×C₉₇	C₂×C₄	1	15	yes	—
12	C₂³×C₉₇	C₂×C₈	1	120	yes	complemented
13	C₂³×C₉₇	C₂²×C₄	1	35	yes	—
14	C₂×Dic₉₇	C₂²×C₄	1	105	yes	—
15	C₂×Dic₉₇	C₂⁴	1	15	yes	—
16	#11715897	C₂⁴	1	16	yes	direct product factor
17	C₂²×C₉₇	C₂²×C₈	1	140	yes	complemented
18	C₂²×C₉₇	C₂³×C₄	1	15	yes	—
19	Dic₉₇	C₂³×C₄	1	15	yes	—
20	Dic₉₇	C₂⁵	1	1	yes	characteristic
21	C₂×C₉₇	C₂³×C₈	1	30	yes	complemented
22	C₂×C₉₇	C₂⁴×C₄	1	1	yes	characteristic
23	C₂⁴×C₈	—	97	1	no	maximal, Sylow
24	C₉₇	C₂⁴×C₈	1	1	yes	characteristic, complemented, Sylow
25	C₂⁴×C₄	—	97	1	no	—
26	C₂³×C₈	—	97	30	no	—
27	C₂⁵	#87604	1	1	yes	characteristic
28	C₂³×C₄	—	97	15	no	—
29	C₂³×C₄	—	97	15	no	—
30	C₂²×C₈	—	97	140	no	—
31	C₂⁴	#11715897	1	16	yes	direct product factor
32	C₂⁴	C₂×#87604	1	15	yes	—
33	C₂²×C₄	—	97	105	no	—
34	C₂²×C₄	—	97	35	no	—
35	C₂×C₈	—	97	120	no	—
36	C₂³	C₂×#11715897	1	120	yes	direct product factor
37	C₂³	C₂²×#87604	1	35	yes	—
38	C₂×C₄	—	97	15	no	—
39	C₂×C₄	—	97	105	no	—
40	C₈	—	97	16	no	—
41	C₂²	C₂²×#11715897	1	140	yes	direct product factor
42	C₂²	C₂³×#87604	1	15	yes	—
43	C₄	—	97	1	no	—
44	C₄	—	97	15	no	—
45	C₂	C₂³×#11715897	1	30	yes	direct product factor
46	C₂	C₂⁴×#87604	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂⁵	#87604
commutator, hypocenter	C₉₇	C₂⁴×C₈
Frattini	C₂	C₂⁴×#87604
Fitting, socle, prime radical	C₂⁵×C₉₇	C₄
cosocle	Dic₉₇	C₂⁵
LCS	G ▷ C₉₇
UCS	C₁ ◁ C₂⁵

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂³×#11715897	C₂	direct (N × H)
3. C₂²×#11715897	C₂²	direct (N × H)
7. C₂×#11715897	C₂³	direct (N × H)
10. C₂⁴×C₉₇	C₈	semidirect
12. C₂³×C₉₇	C₂×C₈	semidirect
16. #11715897	C₂⁴	direct (N × H)
17. C₂²×C₉₇	C₂²×C₈	semidirect
21. C₂×C₉₇	C₂³×C₈	semidirect
24. C₉₇	C₂⁴×C₈	semidirect
31. C₂⁴	#11715897	direct (N × H)
36. C₂³	C₂×#11715897	direct (N × H)
41. C₂²	C₂²×#11715897	direct (N × H)
45. C₂	C₂³×#11715897	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×#11715897	2
C₂⁴×Dic₉₇	2
C₂⁴×C₈	97

Subgroups — C24×#11715897 #27158719

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×#11715897 #27158719