Subgroups — #27144051

Order 12416.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261167	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₁₅₅₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#20260574	C₂	1	2	yes	maximal, complemented
4	D₃₁₀₄	C₂	1	2	yes	maximal, complemented
5	M₆₄×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#15690832	—	2	1	no	—
7	C₂×D₇₇₆	—	2	1	no	—
8	Dic₇₇₆	C₂²	1	1	yes	characteristic, complemented
9	#15690755	—	2	1	no	—
10	D₁₅₅₂	—	2	1	no	—
11	D₁₅₅₂	C₂²	1	2	yes	—
12	D₁₅₅₂	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₁₆×C₉₇	C₂²	1	1	yes	characteristic
14	C₃₂×C₉₇	C₂²	1	2	yes	complemented
15	#13095511	—	4	1	no	—
16	C₂×D₃₈₈	—	4	1	no	—
17	Dic₃₈₈	—	2	1	no	—
18	#13095500	—	4	1	no	—
19	D₇₇₆	—	2	2	no	—
20	D₇₇₆	—	2	1	no	—
21	D₇₇₆	—	4	1	no	—
22	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
23	C₁₆×C₉₇	D₄	1	1	yes	characteristic
24	C₁₆×C₉₇	C₂³	1	1	yes	characteristic
25	C₂²×D₉₇	—	8	1	no	—
26	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
27	C₂²⋊D₉₇	—	8	1	no	—
28	D₃₈₈	—	4	2	no	—
29	D₃₈₈	—	4	1	no	—
30	D₃₈₈	—	8	1	no	—
31	C₄×D₉₇	—	8	1	no	—
32	Dic₁₉₄	—	4	1	no	—
33	C₈×C₉₇	D₈	1	1	yes	characteristic
34	C₈×C₉₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₉₇	D₁₆	1	1	yes	characteristic
36	C₂×D₉₇	—	8	1	no	—
37	C₂×D₉₇	—	8	2	no	—
38	C₂×D₉₇	—	16	2	no	—
39	C₄×C₉₇	D₁₆	1	1	yes	characteristic
40	C₄×C₉₇	C₂×D₈	1	1	yes	characteristic
41	Dic₉₇	—	8	1	no	—
42	C₂×C₉₇	—	2	1	no	—
43	C₂×C₉₇	C₂×D₁₆	1	1	yes	characteristic
44	D₉₇	—	16	1	no	—
45	D₉₇	—	16	2	no	—
46	#2263	—	97	1	no	maximal, Sylow
47	C₉₇	#2263	1	1	yes	characteristic, complemented, Sylow
48	#508	—	97	1	no	—
49	C₂×D₁₆	—	97	1	no	—
50	SD₆₄	—	97	2	no	—
51	D₃₂	—	97	2	no	—
52	M₆₄	D₉₇	1	1	yes	characteristic, complemented
53	#135	—	194	1	no	—
54	C₂×D₈	—	194	1	no	—
55	Q₃₂	—	97	1	no	—
56	SD₃₂	—	194	1	no	—
57	D₁₆	—	97	2	no	—
58	D₁₆	—	97	1	no	—
59	D₁₆	—	194	1	no	—
60	C₂×C₁₆	C₂×D₉₇	1	1	yes	characteristic
61	C₃₂	C₂×D₉₇	1	2	yes	complemented
62	#41	—	388	1	no	—
63	C₂×D₄	—	388	1	no	—
64	Q₁₆	—	194	1	no	—
65	SD₁₆	—	388	1	no	—
66	D₈	—	388	1	no	—
67	D₈	—	194	2	no	—
68	D₈	—	194	1	no	—
69	C₂×C₈	D₃₈₈	1	1	yes	characteristic
70	C₁₆	D₃₈₈	1	1	yes	characteristic
71	C₁₆	C₂²×D₉₇	1	1	yes	characteristic
72	C₂³	—	776	1	no	—
73	Q₈	—	388	1	no	—
74	D₄	—	776	1	no	—
75	D₄	—	776	1	no	—
76	D₄	—	388	2	no	—
77	D₄	—	388	1	no	—
78	C₂×C₄	—	776	1	no	—
79	C₂×C₄	D₇₇₆	1	1	yes	characteristic
80	C₈	D₇₇₆	1	1	yes	characteristic
81	C₈	C₂×D₃₈₈	1	1	yes	characteristic
82	C₂²	—	776	1	no	—
83	C₂²	—	776	2	no	—
84	C₂²	—	1552	2	no	—
85	C₂²	D₁₅₅₂	1	1	yes	characteristic
86	C₄	—	776	1	no	—
87	C₄	D₁₅₅₂	1	1	yes	characteristic
88	C₄	C₂×D₇₇₆	1	1	yes	characteristic
89	C₂	—	2	1	no	—
90	C₂	—	1552	2	no	—
91	C₂	—	1552	1	no	—
92	C₂	C₂×D₁₅₅₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₁₅₅₂
commutator, cosocle	C₁₆×C₉₇	C₂³
Frattini	C₁₆	C₂²×D₉₇
Fitting	M₆₄×C₉₇	C₂
socle	C₂×C₉₇	C₂×D₁₆
hypercenter	M₆₄	D₉₇
hypocenter	C₉₇	#2263
LCS	G ▷ C₁₆×C₉₇ ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
1. #20261167	C₂	semidirect
2. C₂×D₁₅₅₂	C₂	semidirect
3. #20260574	C₂	semidirect
4. D₃₁₀₄	C₂	semidirect
5. M₆₄×C₉₇	C₂	semidirect
8. Dic₇₇₆	C₂²	semidirect
12. D₁₅₅₂	C₂²	semidirect
14. C₃₂×C₉₇	C₂²	semidirect
47. C₉₇	#2263	semidirect
52. M₆₄	D₉₇	semidirect
61. C₃₂	C₂×D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261167	2
C₂×D₁₅₅₂	2
#20260574	2
D₃₁₀₄	2
M₆₄×C₉₇	2
#2263	97