Subgroups — #27144045

Order 12416.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261167	C₂	1	2	yes	maximal, complemented
2	Dic₁₅₅₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#20260574	C₂	1	2	yes	maximal, complemented
4	D₃₁₀₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₂×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#15690832	—	2	2	no	—
7	Dic₇₇₆	C₂²	1	2	yes	—
8	#15690755	—	2	2	no	—
9	D₁₅₅₂	C₂²	1	2	yes	—
10	C₂×C₁₆×C₉₇	C₂²	1	1	yes	characteristic
11	C₃₂×C₉₇	C₂²	1	1	yes	characteristic
12	C₃₂×C₉₇	C₂²	1	1	yes	characteristic
13	#13095511	—	4	2	no	—
14	Dic₃₈₈	—	2	2	no	—
15	#13095500	—	4	2	no	—
16	D₇₇₆	—	2	2	no	—
17	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
18	C₁₆×C₉₇	D₄	1	1	yes	characteristic
19	C₁₆×C₉₇	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
21	C₂²⋊D₉₇	—	8	2	no	—
22	D₃₈₈	—	4	2	no	—
23	C₄×D₉₇	—	8	2	no	—
24	Dic₁₉₄	—	4	2	no	—
25	C₈×C₉₇	D₈	1	1	yes	characteristic
26	C₈×C₉₇	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₉₇	D₁₆	1	1	yes	characteristic
28	C₂×D₉₇	—	8	2	no	—
29	C₄×C₉₇	D₁₆	1	1	yes	characteristic
30	C₄×C₉₇	C₂×D₈	1	1	yes	characteristic
31	Dic₉₇	—	8	2	no	—
32	C₂×C₉₇	—	2	1	no	—
33	C₂×C₉₇	C₂×D₁₆	1	1	yes	characteristic
34	D₉₇	—	16	2	no	—
35	#2262	—	97	1	no	maximal, Sylow
36	C₉₇	#2262	1	1	yes	characteristic, complemented, Sylow
37	#508	—	97	2	no	—
38	Q₆₄	—	97	1	no	—
39	SD₆₄	—	97	2	no	—
40	D₃₂	—	97	1	no	—
41	C₂×C₃₂	D₉₇	1	1	yes	characteristic, complemented
42	#135	—	194	2	no	—
43	Q₃₂	—	97	2	no	—
44	SD₃₂	—	194	2	no	—
45	D₁₆	—	97	2	no	—
46	C₂×C₁₆	C₂×D₉₇	1	1	yes	characteristic
47	C₃₂	C₂×D₉₇	1	1	yes	characteristic
48	C₃₂	C₂×D₉₇	1	1	yes	characteristic
49	#41	—	388	2	no	—
50	Q₁₆	—	194	2	no	—
51	SD₁₆	—	388	2	no	—
52	D₈	—	194	2	no	—
53	C₂×C₈	D₃₈₈	1	1	yes	characteristic
54	C₁₆	D₃₈₈	1	1	yes	characteristic
55	C₁₆	C₂²×D₉₇	1	1	yes	characteristic
56	Q₈	—	388	2	no	—
57	D₄	—	776	2	no	—
58	D₄	—	388	2	no	—
59	C₂×C₄	—	776	2	no	—
60	C₂×C₄	D₇₇₆	1	1	yes	characteristic
61	C₈	D₇₇₆	1	1	yes	characteristic
62	C₈	C₂×D₃₈₈	1	1	yes	characteristic
63	C₂²	—	776	2	no	—
64	C₂²	D₁₅₅₂	1	1	yes	characteristic
65	C₄	—	776	2	no	—
66	C₄	D₁₅₅₂	1	1	yes	characteristic
67	C₄	C₂×D₇₇₆	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	1552	2	no	—
70	C₂	C₂×D₁₅₅₂	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₁₅₅₂
commutator, cosocle	C₁₆×C₉₇	C₂³
Frattini	C₁₆	C₂²×D₉₇
Fitting	C₂×C₃₂×C₉₇	C₂
socle	C₂×C₉₇	C₂×D₁₆
hypercenter	C₂×C₃₂	D₉₇
hypocenter	C₉₇	#2262
LCS	G ▷ C₁₆×C₉₇ ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
1. #20261167	C₂	semidirect
2. Dic₁₅₅₂	C₂	semidirect
3. #20260574	C₂	semidirect
4. D₃₁₀₄	C₂	semidirect
5. C₂×C₃₂×C₉₇	C₂	semidirect
36. C₉₇	#2262	semidirect
41. C₂×C₃₂	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
Dic₁₅₅₂	2
#20260574	2
D₃₁₀₄	2
C₂×C₃₂×C₉₇	2
#2262	97