Subgroups — #27156882

Order 12416.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261644	C₂	1	1	yes	maximal, characteristic
2	#20261642	C₂	1	1	yes	maximal, characteristic
3	#20261092	C₂	1	1	yes	maximal, characteristic
4	#15690871	—	2	1	no	—
5	#15690869	—	2	1	no	—
6	#15690798	C₂²	1	1	yes	characteristic
7	#15690733	—	2	1	no	—
8	#15690730	—	2	1	no	—
9	C₄×#11715895	C₄	1	1	yes	characteristic, complemented
10	#125×C₉₇	C₄	1	1	yes	characteristic, complemented
11	#13095520	—	4	1	no	—
12	C₂²⋊#87604	—	4	1	no	—
13	C₂×Dic₁₉₄	D₄	1	1	yes	characteristic
14	C₂×D₃₈₈	D₄	1	1	yes	characteristic
15	C₂×#11715895	—	2	1	no	—
16	#13095488	—	4	1	no	—
17	C₄⋊C₄×C₉₇	—	2	1	no	—
18	C₄⋊C₄×C₉₇	—	4	1	no	—
19	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×D₉₇	—	2	1	no	—
21	C₂×#87604	—	8	1	no	—
22	C₂×C₄×C₉₇	—	2	1	no	—
23	C₂×C₄×C₉₇	—	4	1	no	—
24	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×Dic₉₇	—	2	1	no	—
26	D₃₈₈	—	4	1	no	—
27	Dic₁₉₄	—	4	1	no	—
28	#11715897	—	8	1	no	—
29	#11715895	—	4	1	no	—
30	C₂²×C₉₇	#99	1	1	yes	characteristic
31	C₂×D₉₇	—	4	1	no	—
32	C₂×D₉₇	—	8	1	no	—
33	#87604	—	16	1	no	—
34	C₄×C₉₇	—	2	1	no	—
35	C₄×C₉₇	—	4	1	no	—
36	C₄×C₉₇	—	8	1	no	—
37	Dic₉₇	—	4	1	no	—
38	C₂×C₉₇	—	2	1	no	—
39	C₂×C₉₇	#354	1	1	yes	characteristic
40	D₉₇	—	8	1	no	—
41	#1414	—	97	1	no	maximal, Sylow
42	C₉₇	#1414	1	1	yes	characteristic, complemented, Sylow
43	#488	—	97	1	no	—
44	#355	—	97	1	no	—
45	#354	—	97	1	no	—
46	#125	#87604	1	1	yes	characteristic, complemented
47	#124	—	97	1	no	—
48	#103	—	194	1	no	—
49	#102	—	194	1	no	—
50	#101	—	194	1	no	—
51	#99	—	194	1	no	—
52	C₄×C₈	—	97	1	no	—
53	C₂×Q₈	—	97	1	no	—
54	C₂×D₄	—	97	1	no	—
55	M₁₆	—	388	1	no	—
56	C₂×C₈	—	194	1	no	—
57	C₄⋊C₄	—	2	1	no	—
58	C₄⋊C₄	—	4	1	no	—
59	C₂²⋊C₄	—	388	1	no	—
60	C₂²⋊C₄	—	388	1	no	—
61	C₄²	C₂×#87604	1	1	yes	characteristic
62	C₂³	—	194	1	no	—
63	Q₈	—	388	1	no	—
64	D₄	—	388	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	4	1	no	—
67	C₂×C₄	—	776	1	no	—
68	C₂×C₄	—	194	1	no	—
69	C₂×C₄	C₂²⋊#87604	1	1	yes	characteristic
70	C₈	—	776	1	no	—
71	C₈	—	388	1	no	—
72	C₂²	—	776	1	no	—
73	C₂²	—	388	1	no	—
74	C₂²	#15690868	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	8	1	no	—
78	C₄	—	1552	1	no	—
79	C₄	—	388	1	no	—
80	C₂	—	2	1	no	—
81	C₂	—	776	1	no	—
82	C₂	#20261641	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20261641
commutator	C₄²×C₉₇	C₂×C₄
Frattini	C₄²	C₂×#87604
Fitting	#125×C₉₇	C₄
socle	C₂×C₉₇	#354
prime radical	C₂×D₃₈₈	D₄
cosocle	#15690798	C₂²
hypercenter	#125	#87604
hypocenter	C₉₇	#1414
LCS	G ▷ C₄²×C₉₇ ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
9. C₄×#11715895	C₄	semidirect
10. #125×C₉₇	C₄	semidirect
42. C₉₇	#1414	semidirect
46. #125	#87604	semidirect

Maximal subgroups

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

subgroup	index
#20261644	2
#20261642	2
#20261092	2
#1414	97