Subgroups — #37554478

Order 23488.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₇₃₄	C₂	1	1	yes	maximal, characteristic
2	#26044194	C₂	1	2	yes	maximal
3	C₄⋊(C₃₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#26044184	C₂	1	2	yes	maximal
5	C₄⋊Q₈×C₃₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₃₄	C₂²	1	1	yes	characteristic
7	C₂×C₃₆₇⋊C₈	C₂²	1	2	yes	—
8	C₄⋊Dic₃₆₇	C₂²	1	1	yes	characteristic
9	Dic₃₆₇⋊C₄	—	2	1	no	—
10	C₄×Dic₃₆₇	—	2	1	no	—
11	C₂×Q₈×C₃₆₇	—	2	1	no	—
12	C₄⋊C₄×C₃₆₇	—	2	1	no	—
13	C₄⋊C₄×C₃₆₇	C₂²	1	2	yes	—
14	C₄²×C₃₆₇	C₂²	1	1	yes	characteristic
15	Dic₇₃₄	—	2	1	no	—
16	Dic₇₃₄	Q₈	1	2	yes	complemented
17	C₂×Dic₃₆₇	—	2	1	no	—
18	C₂×Dic₃₆₇	—	2	1	no	—
19	C₃₆₇⋊C₈	—	2	2	no	—
20	Q₈×C₃₆₇	—	4	2	no	—
21	C₂×C₄×C₃₆₇	—	2	2	no	—
22	C₂×C₄×C₃₆₇	D₄	1	1	yes	characteristic
23	C₂×C₄×C₃₆₇	D₄	1	1	yes	characteristic
24	C₂×C₄×C₃₆₇	C₂³	1	1	yes	characteristic
25	C₂²×C₃₆₇	C₂×D₄	1	1	yes	characteristic
26	C₄×C₃₆₇	—	2	1	no	—
27	C₄×C₃₆₇	—	4	2	no	—
28	C₄×C₃₆₇	Q₁₆	1	2	yes	—
29	C₄×C₃₆₇	C₂×Q₈	1	1	yes	characteristic
30	C₄×C₃₆₇	#41	1	1	yes	characteristic
31	Dic₃₆₇	—	2	2	no	—
32	Dic₃₆₇	—	4	1	no	—
33	C₂×C₃₆₇	C₂²⋊Q₈	1	1	yes	characteristic
34	C₂×C₃₆₇	C₂×Q₁₆	1	1	yes	characteristic
35	C₂×C₃₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
36	C₃₆₇	#477	1	1	yes	characteristic, complemented, Sylow
37	#477	—	367	1	no	maximal, Sylow
38	C₄⋊Q₈	D₃₆₇	1	1	yes	characteristic
39	C₄×Q₈	—	367	1	no	—
40	#107	—	367	2	no	—
41	C₄⋊C₈	—	367	1	no	—
42	#103	—	367	2	no	—
43	C₂×Q₈	—	2	1	no	—
44	C₂×Q₈	—	367	1	no	—
45	C₂×C₈	—	367	2	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	—	367	1	no	—
48	C₄⋊C₄	—	734	1	no	—
49	C₄⋊C₄	C₂×D₃₆₇	1	2	yes	—
50	C₄²	—	734	1	no	—
51	C₄²	C₂×D₃₆₇	1	1	yes	characteristic
52	Q₈	—	4	2	no	—
53	Q₈	—	367	2	no	—
54	Q₈	—	734	1	no	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	734	1	no	—
57	C₂×C₄	—	734	1	no	—
58	C₂×C₄	C₂²⋊D₃₆₇	1	1	yes	characteristic
59	C₂×C₄	C₂²⋊D₃₆₇	1	1	yes	characteristic
60	C₂×C₄	C₂²×D₃₆₇	1	1	yes	characteristic
61	C₈	—	734	2	no	—
62	C₂²	C₂×C₂²⋊D₃₆₇	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	—	4	2	no	—
65	C₄	—	1468	1	no	—
66	C₄	—	734	2	no	—
67	C₄	#19154567	1	2	yes	—
68	C₄	Q₈×D₃₆₇	1	1	yes	characteristic
69	C₄	#19154578	1	1	yes	characteristic
70	C₂	#26044237	1	1	yes	characteristic
71	C₂	C₂×#19154567	1	1	yes	characteristic
72	C₂	#26044287	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₃₆₇
commutator, cosocle	C₂×C₄×C₃₆₇	C₂³
Frattini	C₂×C₄	C₂²×D₃₆₇
Fitting	C₄⋊Q₈×C₃₆₇	C₂
socle, prime radical	C₂²×C₃₆₇	C₂×D₄
hypercenter	C₄⋊Q₈	D₃₆₇
hypocenter	C₃₆₇	#477
LCS	G ▷ C₂×C₄×C₃₆₇ ▷ C₂×C₃₆₇ ▷ C₃₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
16. Dic₇₃₄	Q₈	semidirect
36. C₃₆₇	#477	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₇₃₄	2
#26044194	2
C₄⋊(C₃₆₇⋊C₈)	2
#26044184	2
C₄⋊Q₈×C₃₆₇	2
#477	367