Subgroups — #27139923

Order 12416.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260569	C₂	1	1	yes	maximal, characteristic, complemented
2	#20260563	C₂	1	1	yes	maximal, characteristic, complemented
3	#508×C₉₇	C₂	1	1	yes	maximal, characteristic
4	#15690744	—	2	1	no	—
5	#15690716	—	2	1	no	—
6	#135×C₉₇	—	2	1	no	—
7	Q₃₂×C₉₇	C₄	1	1	yes	characteristic, complemented
8	SD₃₂×C₉₇	—	2	1	no	—
9	D₁₆×C₉₇	C₄	1	1	yes	characteristic, complemented
10	C₂×C₁₆×C₉₇	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₉₇	—	4	1	no	—
12	#41×C₉₇	—	4	1	no	—
13	Q₁₆×C₉₇	—	2	1	no	—
14	SD₁₆×C₉₇	—	4	1	no	—
15	D₈×C₉₇	—	2	1	no	—
16	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
17	C₁₆×C₉₇	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₉₇	D₄	1	1	yes	characteristic
19	Q₈×C₉₇	—	4	1	no	—
20	D₄×C₉₇	—	4	1	no	—
21	D₄×C₉₇	—	8	1	no	—
22	C₂×C₄×C₉₇	—	8	1	no	—
23	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
24	C₂×Dic₉₇	—	8	1	no	—
25	C₈×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₉₇	SD₁₆	1	1	yes	characteristic
27	C₂²×C₉₇	—	8	1	no	—
28	C₂²×C₉₇	D₁₆	1	1	yes	characteristic
29	C₄×C₉₇	—	8	1	no	—
30	C₄×C₉₇	#102	1	1	yes	characteristic
31	C₄×C₉₇	SD₃₂	1	1	yes	characteristic
32	Dic₉₇	—	16	1	no	—
33	C₂×C₉₇	—	2	1	no	—
34	C₂×C₉₇	—	16	1	no	—
35	C₂×C₉₇	#357	1	1	yes	characteristic
36	#1418	—	97	1	no	maximal, Sylow
37	C₉₇	#1418	1	1	yes	characteristic, complemented, Sylow
38	#508	D₉₇	1	1	yes	characteristic
39	M₆₄	—	97	1	no	—
40	#367	—	97	1	no	—
41	#135	—	2	1	no	—
42	Q₃₂	Dic₉₇	1	1	yes	characteristic, complemented
43	SD₃₂	—	2	1	no	—
44	D₁₆	Dic₉₇	1	1	yes	characteristic, complemented
45	C₂×C₁₆	C₂×D₉₇	1	1	yes	characteristic
46	#107	—	194	1	no	—
47	C₃₂	—	194	1	no	—
48	#41	—	4	1	no	—
49	Q₁₆	—	2	1	no	—
50	SD₁₆	—	4	1	no	—
51	D₈	—	2	1	no	—
52	C₂×C₈	C₂²⋊D₉₇	1	1	yes	characteristic
53	C₄⋊C₄	—	388	1	no	—
54	C₁₆	C₂×Dic₉₇	1	1	yes	characteristic
55	C₁₆	C₂²⋊D₉₇	1	1	yes	characteristic
56	Q₈	—	4	1	no	—
57	D₄	—	4	1	no	—
58	D₄	—	8	1	no	—
59	C₂×C₄	—	8	1	no	—
60	C₂×C₄	—	776	1	no	—
61	C₂×C₄	#13095496	1	1	yes	characteristic
62	C₈	C₂²⋊Dic₉₇	1	1	yes	characteristic
63	C₈	#13095499	1	1	yes	characteristic
64	C₂²	—	8	1	no	—
65	C₂²	#15690751	1	1	yes	characteristic
66	C₄	—	8	1	no	—
67	C₄	—	1552	1	no	—
68	C₄	#15690731	1	1	yes	characteristic
69	C₄	#15690754	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	16	1	no	—
72	C₂	#20260536	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20260536
commutator	C₁₆×C₉₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₉₇
Fitting, prime radical	#508×C₉₇	C₂
socle	C₂×C₉₇	#357
cosocle	C₂×C₁₆×C₉₇	C₂²
hypercenter	#508	D₉₇
hypocenter	C₉₇	#1418
LCS	G ▷ C₁₆×C₉₇ ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #508

Complements

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

N	H ≅ G/N	type
1. #20260569	C₂	semidirect
2. #20260563	C₂	semidirect
7. Q₃₂×C₉₇	C₄	semidirect
9. D₁₆×C₉₇	C₄	semidirect
37. C₉₇	#1418	semidirect
42. Q₃₂	Dic₉₇	semidirect
44. D₁₆	Dic₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20260569	2
#20260563	2
#508×C₉₇	2
#1418	97