Subgroups — #27141350

Order 12416.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261112	C₂	1	1	yes	maximal, characteristic, complemented
2	#20261111	C₂	1	4	yes	maximal, complemented
3	C₈×#11715895	C₂	1	1	yes	maximal, characteristic, complemented
4	#493×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₇₇₆	—	2	2	no	—
6	C₂×#13095496	—	2	8	no	—
7	#15690807	C₂²	1	2	yes	—
8	C₄×#11715895	C₂²	1	2	yes	—
9	C₂×D₈×C₉₇	—	2	2	no	—
10	#127×C₉₇	C₂²	1	2	yes	—
11	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
12	C₂×D₃₈₈	—	2	4	no	—
13	C₂×D₃₈₈	—	2	2	no	—
14	D₇₇₆	—	4	4	no	—
15	#13095496	—	4	16	no	—
16	C₂×#11715895	D₄	1	4	yes	—
17	C₂×D₄×C₉₇	—	2	4	no	—
18	C₂×D₄×C₉₇	—	2	2	no	—
19	D₈×C₉₇	—	4	4	no	—
20	C₂×C₈×C₉₇	D₄	1	2	yes	—
21	C₄²×C₉₇	C₂³	1	1	yes	characteristic
22	C₂³×C₉₇	—	4	2	no	—
23	C₂²×D₉₇	—	4	2	no	—
24	D₄×C₉₇	—	4	4	no	—
25	D₄×C₉₇	—	4	8	no	—
26	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
27	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
28	D₃₈₈	—	4	8	no	—
29	D₃₈₈	—	4	4	no	—
30	C₈×C₉₇	D₈	1	4	yes	complemented
31	#11715895	D₈	1	8	yes	complemented
32	C₂²×C₉₇	—	8	4	no	—
33	C₂²×C₉₇	—	8	2	no	—
34	C₂²×C₉₇	#127	1	1	yes	characteristic
35	C₂×D₉₇	—	8	4	no	—
36	C₂×D₉₇	—	8	2	no	—
37	C₄×C₉₇	C₂×D₈	1	4	yes	—
38	C₄×C₉₇	C₂×D₈	1	2	yes	—
39	C₂×C₉₇	—	16	2	no	—
40	C₂×C₉₇	#493	1	2	yes	—
41	C₂×C₉₇	#493	1	1	yes	characteristic
42	D₉₇	—	16	2	no	—
43	#1712	—	97	1	no	maximal, Sylow
44	C₉₇	#1712	1	1	yes	characteristic, complemented, Sylow
45	#493	—	97	4	no	—
46	#493	—	97	1	no	—
47	#493	D₉₇	1	1	yes	characteristic, complemented
48	C₈²	—	97	1	no	—
49	C₂×D₈	—	2	2	no	—
50	C₂×D₈	—	194	8	no	—
51	C₂×D₈	—	194	2	no	—
52	#127	—	97	2	no	—
53	#127	C₂×D₉₇	1	2	yes	—
54	C₄×C₈	—	97	2	no	—
55	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
56	C₂×D₄	—	2	4	no	—
57	C₂×D₄	—	2	2	no	—
58	C₂×D₄	—	194	4	no	—
59	C₂×D₄	—	194	2	no	—
60	D₈	—	4	4	no	—
61	D₈	—	388	16	no	—
62	D₈	—	388	4	no	—
63	C₂×C₈	—	97	4	no	—
64	C₂×C₈	C₂²⋊D₉₇	1	2	yes	—
65	C₄²	C₂²×D₉₇	1	1	yes	characteristic
66	C₂³	—	4	2	no	—
67	C₂³	—	388	2	no	—
68	D₄	—	4	8	no	—
69	D₄	—	4	4	no	—
70	D₄	—	388	8	no	—
71	D₄	—	388	4	no	—
72	C₂×C₄	D₄×D₉₇	1	2	yes	—
73	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
74	C₈	—	97	8	no	—
75	C₈	#13095496	1	4	yes	complemented
76	C₂²	—	8	4	no	—
77	C₂²	—	8	2	no	—
78	C₂²	—	776	4	no	—
79	C₂²	—	776	2	no	—
80	C₂²	#15690806	1	1	yes	characteristic
81	C₄	D₈×D₉₇	1	4	yes	—
82	C₄	C₂×#13095496	1	2	yes	—
83	C₂	—	16	2	no	—
84	C₂	—	1552	2	no	—
85	C₂	#20261110	1	2	yes	—
86	C₂	#20261111	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15690806
commutator, cosocle	C₄²×C₉₇	C₂³
Frattini	C₄²	C₂²×D₉₇
Fitting	#493×C₉₇	C₂
socle	C₂²×C₉₇	#127
hypercenter	#493	D₉₇
hypocenter	C₉₇	#1712
LCS	G ▷ C₄²×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

Complements

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

N	H ≅ G/N	type
1. #20261112	C₂	semidirect
2. #20261111	C₂	semidirect
3. C₈×#11715895	C₂	semidirect
4. #493×C₉₇	C₂	semidirect
30. C₈×C₉₇	D₈	semidirect
31. #11715895	D₈	semidirect
44. C₉₇	#1712	semidirect
47. #493	D₉₇	semidirect
75. C₈	#13095496	semidirect

Maximal subgroups

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

subgroup	index
#20261112	2
#20261111	2
C₈×#11715895	2
#493×C₉₇	2
#1712	97