Subgroups — #27150817

Order 12416.

396 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11715895	C₂	1	8	yes	maximal, complemented
2	#20260788	C₂	1	6	yes	maximal
3	C₂×C₄×Q₈×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11715895	C₂²	1	12	yes	—
5	C₂²⋊#11715895	C₂²	1	4	yes	—
6	C₄×#11715895	C₂²	1	12	yes	—
7	C₂²×Q₈×C₉₇	C₄	1	1	yes	characteristic
8	C₄×Q₈×C₉₇	C₄	1	4	yes	—
9	C₄×Q₈×C₉₇	C₂²	1	4	yes	—
10	C₂×C₄⋊C₄×C₉₇	C₄	1	3	yes	—
11	C₂×C₄²×C₉₇	C₂²	1	3	yes	—
12	C₂×#11715895	C₂³	1	8	yes	—
13	C₂×Q₈×C₉₇	C₈	1	8	yes	complemented
14	C₂×Q₈×C₉₇	C₂×C₄	1	4	yes	—
15	C₂²×C₄×C₉₇	C₂×C₄	1	3	yes	—
16	C₂²×C₄×C₉₇	C₂×C₄	1	3	yes	—
17	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
18	C₄⋊C₄×C₉₇	C₂×C₄	1	12	yes	—
19	C₄²×C₉₇	C₂×C₄	1	6	yes	—
20	C₄²×C₉₇	C₂³	1	6	yes	—
21	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
22	Q₈×C₉₇	C₂×C₈	1	16	yes	—
23	C₂×C₄×C₉₇	—	2	2	no	—
24	C₂×C₄×C₉₇	—	4	3	no	—
25	C₂×C₄×C₉₇	C₂×C₈	1	12	yes	—
26	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₉₇	C₂²×C₄	1	6	yes	—
28	C₂×C₄×C₉₇	C₂²×C₄	1	6	yes	—
29	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
30	#11715895	—	2	8	no	—
31	C₂²×C₉₇	—	2	1	no	—
32	C₂²×C₉₇	—	2	1	no	—
33	C₂²×C₉₇	C₂²×C₈	1	2	yes	—
34	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
35	C₄×C₉₇	—	2	1	no	—
36	C₄×C₉₇	—	2	6	no	—
37	C₄×C₉₇	C₂²×C₈	1	12	yes	—
38	C₄×C₉₇	2₋¹⁺⁴	1	2	yes	—
39	C₂×C₉₇	—	2	2	no	—
40	C₂×C₉₇	#519	1	1	yes	characteristic
41	C₂×C₉₇	C₂³×C₈	1	1	yes	characteristic
42	C₂×C₉₇	#568	1	1	yes	characteristic
43	#2982	—	97	1	no	maximal, Sylow
44	C₉₇	#2982	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₄×Q₈	D₉₇	1	1	yes	characteristic
46	C₈×Q₈	—	97	8	no	—
47	#431	—	97	6	no	—
48	C₂²×Q₈	Dic₉₇	1	1	yes	characteristic
49	C₄×Q₈	Dic₉₇	1	4	yes	—
50	C₄×Q₈	C₂×D₉₇	1	4	yes	—
51	C₂×C₄⋊C₄	Dic₉₇	1	3	yes	—
52	C₂×C₄²	C₂×D₉₇	1	3	yes	—
53	C₄⋊C₈	—	97	12	no	—
54	C₂²⋊C₈	—	97	4	no	—
55	C₄×C₈	—	97	12	no	—
56	C₂×Q₈	#11715895	1	8	yes	complemented
57	C₂×Q₈	C₂×Dic₉₇	1	4	yes	—
58	C₂²×C₄	C₂×Dic₉₇	1	3	yes	—
59	C₂²×C₄	C₂×Dic₉₇	1	3	yes	—
60	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
61	C₂×C₈	—	97	8	no	—
62	C₄⋊C₄	C₂×Dic₉₇	1	12	yes	—
63	C₄²	C₂×Dic₉₇	1	6	yes	—
64	C₄²	C₂²×D₉₇	1	6	yes	—
65	C₂³	C₂²×Dic₉₇	1	1	yes	characteristic
66	Q₈	C₂×#11715895	1	16	yes	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	4	3	no	—
69	C₂×C₄	C₂×#11715895	1	12	yes	—
70	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
71	C₂×C₄	C₂²×Dic₉₇	1	6	yes	—
72	C₂×C₄	C₂²×Dic₉₇	1	6	yes	—
73	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
74	C₈	—	194	8	no	—
75	C₂²	—	2	1	no	—
76	C₂²	—	2	1	no	—
77	C₂²	C₂²×#11715895	1	2	yes	—
78	C₂²	C₂³×Dic₉₇	1	1	yes	characteristic
79	C₄	—	2	1	no	—
80	C₄	—	2	6	no	—
81	C₄	C₂²×#11715895	1	12	yes	—
82	C₄	#15690858	1	2	yes	—
83	C₂	—	2	2	no	—
84	C₂	#20261213	1	1	yes	characteristic
85	C₂	C₂³×#11715895	1	1	yes	characteristic
86	C₂	#20261502	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₉₇
commutator	C₂×C₉₇	C₂³×C₈
Fitting	C₂×C₄×Q₈×C₉₇	C₂
socle	C₂²×C₉₇	C₂³×C₄
prime radical	C₂³×C₉₇	C₂²×C₄
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	C₂×C₄×Q₈	D₉₇
hypocenter	C₉₇	#2982
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×#11715895	C₂	semidirect
13. C₂×Q₈×C₉₇	C₈	semidirect
44. C₉₇	#2982	semidirect
56. C₂×Q₈	#11715895	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#11715895	2
#20260788	2
C₂×C₄×Q₈×C₉₇	2
#2982	97