Subgroups — #27156866

Order 12416.

98 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	#20261129	C₂	1	1	yes	maximal, characteristic
4	#15690871	—	2	1	no	—
5	#15690869	—	2	1	no	—
6	C₂×Dic₃₈₈	—	2	1	no	—
7	C₂×#13095500	—	2	1	no	—
8	#15690810	C₄	1	1	yes	characteristic, complemented
9	#15690798	C₂²	1	1	yes	characteristic
10	#97×C₉₇	C₄	1	1	yes	characteristic, complemented
11	#13095520	—	4	1	no	—
12	C₂²⋊#87604	—	4	1	no	—
13	C₂×Dic₁₉₄	—	2	1	no	—
14	C₂×Dic₁₉₄	D₄	1	1	yes	characteristic
15	C₂×D₃₈₈	D₄	1	1	yes	characteristic
16	Dic₃₈₈	—	4	2	no	—
17	#13095500	—	4	2	no	—
18	C₄⋊Dic₉₇	—	4	1	no	—
19	#13095488	—	4	1	no	—
20	C₂×C₈×C₉₇	—	2	1	no	—
21	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×D₉₇	—	2	1	no	—
23	C₂×#87604	—	8	1	no	—
24	C₂×C₄×C₉₇	—	2	1	no	—
25	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×Dic₉₇	—	2	1	no	—
27	C₂×Dic₉₇	—	4	1	no	—
28	D₃₈₈	—	4	1	no	—
29	Dic₁₉₄	—	4	1	no	—
30	Dic₁₉₄	—	4	1	no	—
31	Dic₁₉₄	—	4	1	no	—
32	#11715897	—	8	1	no	—
33	C₈×C₉₇	—	2	2	no	—
34	C₂²×C₉₇	#99	1	1	yes	characteristic
35	C₂×D₉₇	—	4	1	no	—
36	C₂×D₉₇	—	8	1	no	—
37	#87604	—	16	1	no	—
38	C₄×C₉₇	—	2	1	no	—
39	C₄×C₉₇	—	4	1	no	—
40	Dic₉₇	—	4	1	no	—
41	Dic₉₇	—	8	1	no	—
42	C₂×C₉₇	—	2	1	no	—
43	C₂×C₉₇	#353	1	1	yes	characteristic
44	D₉₇	—	8	1	no	—
45	#1407	—	97	1	no	maximal, Sylow
46	C₉₇	#1407	1	1	yes	characteristic, complemented, Sylow
47	#497	—	97	1	no	—
48	#355	—	97	1	no	—
49	#354	—	97	1	no	—
50	C₂×Q₁₆	—	194	1	no	—
51	C₂×SD₁₆	—	194	1	no	—
52	C₄⋊Q₈	—	97	1	no	—
53	#124	—	97	1	no	—
54	#101	—	194	1	no	—
55	#99	—	194	1	no	—
56	#97	#87604	1	1	yes	characteristic, complemented
57	C₂×Q₈	—	97	1	no	—
58	C₂×Q₈	—	194	1	no	—
59	C₂×D₄	—	97	1	no	—
60	Q₁₆	—	388	2	no	—
61	SD₁₆	—	388	2	no	—
62	M₁₆	—	388	1	no	—
63	C₂×C₈	—	2	1	no	—
64	C₄⋊C₄	—	388	1	no	—
65	C₂²⋊C₄	—	388	1	no	—
66	C₂²⋊C₄	—	388	1	no	—
67	C₄²	C₂×#87604	1	1	yes	characteristic
68	C₂³	—	194	1	no	—
69	Q₈	—	388	1	no	—
70	Q₈	—	388	1	no	—
71	Q₈	—	388	1	no	—
72	D₄	—	388	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	776	1	no	—
75	C₂×C₄	—	388	1	no	—
76	C₂×C₄	—	194	1	no	—
77	C₂×C₄	C₂²⋊#87604	1	1	yes	characteristic
78	C₈	—	2	2	no	—
79	C₈	—	776	1	no	—
80	C₂²	—	776	1	no	—
81	C₂²	—	388	1	no	—
82	C₂²	#15690869	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	4	1	no	—
85	C₄	—	776	1	no	—
86	C₄	—	1552	1	no	—
87	C₄	—	388	1	no	—
88	C₂	—	2	1	no	—
89	C₂	—	776	1	no	—
90	C₂	#20261640	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20261640
commutator	C₄²×C₉₇	C₂×C₄
Frattini	C₄²	C₂×#87604
Fitting	#97×C₉₇	C₄
socle	C₂×C₉₇	#353
prime radical	C₂×D₃₈₈	D₄
cosocle	#15690798	C₂²
hypercenter	#97	#87604
hypocenter	C₉₇	#1407
LCS	G ▷ C₄²×C₉₇ ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
8. #15690810	C₄	semidirect
10. #97×C₉₇	C₄	semidirect
46. C₉₇	#1407	semidirect
56. #97	#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
#20261129	2
#1407	97