Subgroups — #37554479

Order 23488.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26044260	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₃₆₇	C₂	1	1	yes	maximal, characteristic
3	#26044195	C₂	1	1	yes	maximal, characteristic
4	#26044194	C₂	1	1	yes	maximal, characteristic
5	Dic₃₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#26044185	C₂	1	1	yes	maximal, characteristic
7	#103×C₃₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₇₃₄	—	2	1	no	—
9	C₂×C₃₆₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₃₆₇	—	2	1	no	—
11	C₄⋊Dic₃₆₇	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₃₆₇	C₂²	1	1	yes	characteristic
13	Dic₃₆₇⋊C₄	—	2	1	no	—
14	C₄×Dic₃₆₇	—	2	1	no	—
15	C₄×Dic₃₆₇	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₃₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₃₆₇	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₃₆₇	C₂²	1	1	yes	characteristic
19	Dic₇₃₄	—	4	2	no	—
20	C₂×Dic₃₆₇	—	2	1	no	—
21	C₂×Dic₃₆₇	—	2	1	no	—
22	C₂×Dic₃₆₇	D₄	1	1	yes	characteristic
23	C₂×Dic₃₆₇	D₄	1	1	yes	characteristic
24	C₃₆₇⋊C₈	—	2	1	no	—
25	Q₈×C₃₆₇	—	2	1	no	—
26	Q₈×C₃₆₇	Q₈	1	2	yes	complemented
27	C₂×C₄×C₃₆₇	—	2	1	no	—
28	C₂×C₄×C₃₆₇	—	2	1	no	—
29	C₂×C₄×C₃₆₇	C₂³	1	1	yes	characteristic
30	C₈×C₃₆₇	—	2	1	no	—
31	C₂²×C₃₆₇	C₂×D₄	1	1	yes	characteristic
32	C₄×C₃₆₇	—	2	2	no	—
33	C₄×C₃₆₇	—	4	1	no	—
34	C₄×C₃₆₇	C₂×Q₈	1	1	yes	characteristic
35	C₄×C₃₆₇	#41	1	1	yes	characteristic
36	Dic₃₆₇	—	2	1	no	—
37	Dic₃₆₇	—	4	1	no	—
38	Dic₃₆₇	—	4	1	no	—
39	Dic₃₆₇	Q₁₆	1	2	yes	—
40	C₂×C₃₆₇	C₂²⋊Q₈	1	1	yes	characteristic
41	C₂×C₃₆₇	C₂×Q₁₆	1	1	yes	characteristic
42	C₂×C₃₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
43	C₃₆₇	#477	1	1	yes	characteristic, complemented, Sylow
44	#477	—	367	1	no	maximal, Sylow
45	C₄⋊Q₈	—	367	1	no	—
46	C₄×Q₈	—	367	1	no	—
47	#107	—	367	1	no	—
48	#107	—	367	1	no	—
49	C₄⋊C₈	—	367	1	no	—
50	#103	—	367	1	no	—
51	#103	D₃₆₇	1	1	yes	characteristic
52	C₂×Q₈	—	734	1	no	—
53	C₂×Q₈	C₂×D₃₆₇	1	1	yes	characteristic
54	C₂×C₈	—	367	1	no	—
55	C₂×C₈	C₂×D₃₆₇	1	1	yes	characteristic
56	C₄⋊C₄	—	367	1	no	—
57	C₄⋊C₄	—	367	1	no	—
58	C₄⋊C₄	—	734	1	no	—
59	C₄⋊C₄	—	734	1	no	—
60	C₄⋊C₄	C₂×D₃₆₇	1	1	yes	characteristic
61	C₄²	—	367	1	no	—
62	C₄²	—	734	1	no	—
63	Q₈	—	2	1	no	—
64	Q₈	—	1468	2	no	—
65	Q₈	Dic₇₃₄	1	2	yes	complemented
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	367	1	no	—
69	C₂×C₄	—	367	1	no	—
70	C₂×C₄	—	734	1	no	—
71	C₂×C₄	—	734	1	no	—
72	C₂×C₄	C₂²×D₃₆₇	1	1	yes	characteristic
73	C₈	—	2	1	no	—
74	C₈	—	734	1	no	—
75	C₂²	D₄×D₃₆₇	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	—	4	1	no	—
78	C₄	—	367	2	no	—
79	C₄	—	1468	1	no	—
80	C₄	—	1468	1	no	—
81	C₄	—	734	1	no	—
82	C₄	C₂×Dic₇₃₄	1	1	yes	characteristic
83	C₄	#19154576	1	1	yes	characteristic
84	C₂	#26044238	1	1	yes	characteristic
85	C₂	Q₁₆×D₃₆₇	1	1	yes	characteristic
86	C₂	#26044288	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₃₆₇
commutator, cosocle	C₂×C₄×C₃₆₇	C₂³
Frattini	C₂×C₄	C₂²×D₃₆₇
Fitting	#103×C₃₆₇	C₂
socle, prime radical	C₂²×C₃₆₇	C₂×D₄
hypercenter	#103	D₃₆₇
hypocenter	C₃₆₇	#477
LCS	G ▷ C₂×C₄×C₃₆₇ ▷ C₂×C₃₆₇ ▷ C₃₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
26. Q₈×C₃₆₇	Q₈	semidirect
43. C₃₆₇	#477	semidirect
65. Q₈	Dic₇₃₄	semidirect

Maximal subgroups

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

subgroup	index
#26044260	2
Q₈×Dic₃₆₇	2
#26044195	2
#26044194	2
Dic₃₆₇⋊C₈	2
#26044185	2
#103×C₃₆₇	2
#477	367