Subgroups — #27156752

Order 12416.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261656	C₂	1	1	yes	maximal, characteristic
2	#20261597	C₂	1	1	yes	maximal, characteristic
3	#20261136	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095514	—	2	1	no	—
5	C₄⋊#11715897	—	2	1	no	—
6	#15690801	—	2	1	no	—
7	#15690744	C₄	1	1	yes	characteristic
8	#15690741	—	2	1	no	—
9	C₈×Dic₉₇	C₂²	1	1	yes	characteristic
10	#107×C₉₇	C₄	1	1	yes	characteristic
11	C₂×#11715897	—	4	1	no	—
12	#13095514	—	4	1	no	—
13	C₂×#11715895	D₄	1	1	yes	characteristic
14	C₄⋊Dic₉₇	—	2	1	no	—
15	C₄⋊Dic₉₇	—	4	1	no	—
16	#13095490	—	4	1	no	—
17	C₄×Dic₉₇	D₄	1	1	yes	characteristic
18	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
19	C₄⋊C₄×C₉₇	—	2	1	no	—
20	C₂×C₄×C₉₇	—	4	1	no	—
21	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×Dic₉₇	—	4	1	no	—
23	C₂×Dic₉₇	D₈	1	1	yes	characteristic
24	C₂×Dic₉₇	SD₁₆	1	1	yes	characteristic
25	#11715897	—	8	1	no	—
26	C₈×C₉₇	—	2	1	no	—
27	#11715895	SD₁₆	1	1	yes	characteristic
28	#11715895	Q₁₆	1	1	yes	characteristic
29	C₂²×C₉₇	#102	1	1	yes	characteristic
30	C₄×C₉₇	—	8	1	no	—
31	C₄×C₉₇	#101	1	1	yes	characteristic
32	C₄×C₉₇	#103	1	1	yes	characteristic
33	Dic₉₇	—	2	1	no	—
34	Dic₉₇	—	2	1	no	—
35	Dic₉₇	—	8	1	no	—
36	C₂×C₉₇	#332	1	1	yes	characteristic
37	C₂×C₉₇	#359	1	1	yes	characteristic
38	C₂×C₉₇	#360	1	1	yes	characteristic
39	#1363	—	97	1	no	maximal, Sylow
40	C₉₇	#1363	1	1	yes	characteristic, complemented, Sylow
41	#499	—	97	1	no	—
42	C₄⋊C₁₆	—	97	1	no	—
43	#334	—	97	1	no	—
44	#125	—	194	1	no	—
45	C₂×C₁₆	—	194	1	no	—
46	#107	—	97	1	no	—
47	#107	#87604	1	1	yes	characteristic
48	#106	—	194	1	no	—
49	C₄⋊C₈	—	194	1	no	—
50	C₄×C₈	—	97	1	no	—
51	C₂×C₈	—	97	1	no	—
52	C₂×C₈	—	388	1	no	—
53	C₂×C₈	C₂×#87604	1	1	yes	characteristic
54	C₄⋊C₄	—	2	1	no	—
55	C₄⋊C₄	—	388	1	no	—
56	C₄⋊C₄	—	388	1	no	—
57	C₄⋊C₄	—	194	1	no	—
58	C₄²	—	97	1	no	—
59	C₁₆	—	388	1	no	—
60	C₂×C₄	—	4	1	no	—
61	C₂×C₄	—	97	1	no	—
62	C₂×C₄	—	97	1	no	—
63	C₂×C₄	—	388	1	no	—
64	C₂×C₄	C₂²⋊#87604	1	1	yes	characteristic
65	C₈	—	2	1	no	—
66	C₈	—	776	1	no	—
67	C₈	—	97	1	no	—
68	C₈	—	97	1	no	—
69	C₂²	#15690872	1	1	yes	characteristic
70	C₄	—	8	1	no	—
71	C₄	—	776	1	no	—
72	C₄	—	194	1	no	—
73	C₄	—	194	1	no	—
74	C₄	#15690871	1	1	yes	characteristic
75	C₄	#15690873	1	1	yes	characteristic
76	C₂	#20261594	1	1	yes	characteristic
77	C₂	#20261649	1	1	yes	characteristic
78	C₂	#20261652	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15690872
commutator	C₂×C₈×C₉₇	C₂×C₄
Frattini	C₂×C₈	C₂×#87604
Fitting	#107×C₉₇	C₄
socle, prime radical	C₂²×C₉₇	#102
cosocle	C₈×Dic₉₇	C₂²
hypercenter	#107	#87604
hypocenter	C₉₇	#1363
LCS	G ▷ C₂×C₈×C₉₇ ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #107

Complements

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

N	H ≅ G/N	type
40. C₉₇	#1363	semidirect

Maximal subgroups

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

subgroup	index
#20261656	2
#20261597	2
#20261136	2
#1363	97