Subgroups — #37554238

Order 23488.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×D₃₆₇	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₃₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₂×D₃₆₇)⋊C₈	C₂	1	2	yes	maximal
4	C₈×Dic₃₆₇	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₈×C₃₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₃₆₇	C₄	1	1	yes	characteristic
7	C₂×C₄×D₃₆₇	C₄	1	1	yes	characteristic
8	C₂×C₄×D₃₆₇	C₂²	1	1	yes	characteristic
9	C₂×C₃₆₇⋊C₈	C₂²	1	2	yes	—
10	C₄×Dic₃₆₇	C₄	1	1	yes	characteristic
11	C₄×Dic₃₆₇	C₄	1	1	yes	characteristic
12	C₄×Dic₃₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₈×C₃₆₇	C₂²	1	2	yes	—
14	C₄²×C₃₆₇	C₂²	1	1	yes	characteristic
15	C₂²×D₃₆₇	C₂×C₄	1	1	yes	characteristic
16	C₂×Dic₃₆₇	C₂×C₄	1	1	yes	characteristic
17	C₂×Dic₃₆₇	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₃₆₇	C₂×C₄	1	1	yes	characteristic
19	C₄×D₃₆₇	—	2	1	no	—
20	C₄×D₃₆₇	—	2	1	no	—
21	C₄×D₃₆₇	—	2	2	no	—
22	C₄×D₃₆₇	C₈	1	4	yes	complemented
23	C₃₆₇⋊C₈	—	2	2	no	—
24	C₂×C₄×C₃₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₃₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₃₆₇	C₂³	1	1	yes	characteristic
27	C₈×C₃₆₇	—	2	2	no	—
28	C₂²×C₃₆₇	C₂²×C₄	1	1	yes	characteristic
29	C₂×D₃₆₇	—	2	1	no	—
30	C₂×D₃₆₇	—	2	1	no	—
31	C₂×D₃₆₇	C₂×C₈	1	2	yes	—
32	C₄×C₃₆₇	—	2	1	no	—
33	C₄×C₃₆₇	C₂×C₈	1	2	yes	—
34	C₄×C₃₆₇	#41	1	1	yes	characteristic
35	C₄×C₃₆₇	#41	1	1	yes	characteristic
36	Dic₃₆₇	—	2	1	no	—
37	Dic₃₆₇	C₂×C₈	1	2	yes	—
38	Dic₃₆₇	M₁₆	1	2	yes	—
39	C₂×C₃₆₇	#117	1	1	yes	characteristic
40	C₂×C₃₆₇	C₂²×C₈	1	1	yes	characteristic
41	C₂×C₃₆₇	C₂×M₁₆	1	1	yes	characteristic
42	D₃₆₇	—	2	2	no	—
43	C₃₆₇	#431	1	1	yes	characteristic, complemented, Sylow
44	#431	—	367	1	no	maximal, Sylow
45	C₂×C₄²	—	367	1	no	—
46	C₄⋊C₈	—	367	1	no	—
47	C₄⋊C₈	D₃₆₇	1	1	yes	characteristic, complemented
48	C₂²⋊C₈	—	367	2	no	—
49	C₄×C₈	—	367	2	no	—
50	C₂²×C₄	—	367	1	no	—
51	C₂²×C₄	—	367	1	no	—
52	C₂²×C₄	—	367	1	no	—
53	C₂×C₈	—	367	2	no	—
54	C₂×C₈	C₂×D₃₆₇	1	2	yes	—
55	C₄²	—	367	1	no	—
56	C₄²	—	367	1	no	—
57	C₄²	—	367	1	no	—
58	C₄²	C₂×D₃₆₇	1	1	yes	characteristic
59	C₂³	—	367	1	no	—
60	C₂×C₄	—	367	1	no	—
61	C₂×C₄	—	367	1	no	—
62	C₂×C₄	—	367	4	no	—
63	C₂×C₄	—	367	1	no	—
64	C₂×C₄	—	734	2	no	—
65	C₂×C₄	—	734	1	no	—
66	C₂×C₄	—	734	1	no	—
67	C₂×C₄	C₄×D₃₆₇	1	1	yes	characteristic
68	C₂×C₄	C₄×D₃₆₇	1	1	yes	characteristic
69	C₂×C₄	C₂²×D₃₆₇	1	1	yes	characteristic
70	C₈	—	2	2	no	—
71	C₈	—	734	2	no	—
72	C₂²	—	367	2	no	—
73	C₂²	—	734	1	no	—
74	C₂²	—	734	1	no	—
75	C₂²	C₂×C₄×D₃₆₇	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	367	2	no	—
78	C₄	—	367	2	no	—
79	C₄	—	734	1	no	—
80	C₄	C₈×D₃₆₇	1	2	yes	—
81	C₄	#19154576	1	1	yes	characteristic
82	C₄	#19154578	1	1	yes	characteristic
83	C₂	—	734	2	no	—
84	C₂	#26044219	1	1	yes	characteristic
85	C₂	C₂×C₈×D₃₆₇	1	1	yes	characteristic
86	C₂	M₁₆×D₃₆₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃₆₇
commutator	C₂×C₃₆₇	C₂²×C₈
Fitting	C₄⋊C₈×C₃₆₇	C₂
socle	C₂²×C₃₆₇	C₂²×C₄
prime radical	C₂²×D₃₆₇	C₂×C₄
cosocle	C₂×C₄×C₃₆₇	C₂³
hypercenter	C₄⋊C₈	D₃₆₇
hypocenter	C₃₆₇	#431
LCS	G ▷ C₂×C₃₆₇ ▷ C₃₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₄⋊(C₃₆₇⋊C₈)	C₂	semidirect
4. C₈×Dic₃₆₇	C₂	semidirect
5. C₄⋊C₈×C₃₆₇	C₂	semidirect
22. C₄×D₃₆₇	C₈	semidirect
43. C₃₆₇	#431	semidirect
47. C₄⋊C₈	D₃₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₄²×D₃₆₇	2
C₄⋊(C₃₆₇⋊C₈)	2
(C₂×D₃₆₇)⋊C₈	2
C₈×Dic₃₆₇	2
C₄⋊C₈×C₃₆₇	2
#431	367