Subgroups — #37554221

Order 23488.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₃₆₇	C₂	1	1	yes	maximal, characteristic
2	#26044219	C₂	1	2	yes	maximal
3	#26044194	C₂	1	2	yes	maximal, complemented
4	#26044193	C₂	1	1	yes	maximal, characteristic, complemented
5	#106×C₃₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₃₆₇	C₂²	1	1	yes	characteristic
7	C₂×C₃₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₈×D₃₆₇	C₄	1	4	yes	complemented
9	C₄⋊Dic₃₆₇	C₂²	1	2	yes	—
10	(C₂×D₃₆₇)⋊C₄	—	2	2	no	—
11	C₄×Dic₃₆₇	—	2	2	no	—
12	C₂×C₈×C₃₆₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₃₆₇	C₂²	1	2	yes	—
14	C₂²×D₃₆₇	D₄	1	1	yes	characteristic
15	C₂×Dic₃₆₇	—	2	2	no	—
16	C₂×Dic₃₆₇	D₄	1	1	yes	characteristic
17	C₄×D₃₆₇	C₂×C₄	1	2	yes	—
18	C₄×D₃₆₇	Q₈	1	2	yes	—
19	C₃₆₇⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₃₆₇	—	2	2	no	—
21	C₂×C₄×C₃₆₇	C₂³	1	1	yes	characteristic
22	C₈×C₃₆₇	C₂×C₄	1	2	yes	—
23	C₂²×C₃₆₇	C₂×D₄	1	1	yes	characteristic
24	C₂×D₃₆₇	—	2	2	no	—
25	C₂×D₃₆₇	C₄⋊C₄	1	2	yes	—
26	C₄×C₃₆₇	—	4	2	no	—
27	C₄×C₃₆₇	C₂²×C₄	1	1	yes	characteristic
28	C₄×C₃₆₇	C₂×Q₈	1	1	yes	characteristic
29	Dic₃₆₇	—	4	2	no	—
30	Dic₃₆₇	C₄⋊C₄	1	2	yes	—
31	C₂×C₃₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₃₆₇	#135	1	2	yes	—
33	D₃₆₇	—	2	2	no	—
34	C₃₆₇	#427	1	1	yes	characteristic, complemented, Sylow
35	#427	—	367	1	no	maximal, Sylow
36	C₂²×C₈	—	367	1	no	—
37	#117	—	367	2	no	—
38	#107	—	367	2	no	—
39	#106	—	367	1	no	—
40	#106	D₃₆₇	1	1	yes	characteristic, complemented
41	C₂²×C₄	—	367	1	no	—
42	C₂×C₈	—	367	1	no	—
43	C₂×C₈	—	367	4	no	—
44	C₂×C₈	C₂×D₃₆₇	1	1	yes	characteristic
45	C₄⋊C₄	—	367	2	no	—
46	C₄⋊C₄	C₂×D₃₆₇	1	2	yes	—
47	C₂²⋊C₄	—	734	2	no	—
48	C₄²	—	734	2	no	—
49	C₂³	—	367	1	no	—
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	—	367	1	no	—
52	C₂×C₄	—	367	2	no	—
53	C₂×C₄	—	367	2	no	—
54	C₂×C₄	—	734	2	no	—
55	C₂×C₄	C₂²×D₃₆₇	1	1	yes	characteristic
56	C₈	—	367	2	no	—
57	C₈	C₄×D₃₆₇	1	2	yes	—
58	C₂²	—	367	2	no	—
59	C₂²	—	734	2	no	—
60	C₂²	D₄×D₃₆₇	1	1	yes	characteristic
61	C₄	—	4	2	no	—
62	C₄	—	367	2	no	—
63	C₄	—	1468	2	no	—
64	C₄	C₂×C₄×D₃₆₇	1	1	yes	characteristic
65	C₄	Q₈×D₃₆₇	1	1	yes	characteristic
66	C₂	—	734	2	no	—
67	C₂	C₄⋊C₄×D₃₆₇	1	1	yes	characteristic
68	C₂	#26044282	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₃₆₇
commutator	C₄×C₃₆₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃₆₇
Fitting	#106×C₃₆₇	C₂
socle	C₂²×C₃₆₇	C₂×D₄
prime radical	C₂²×D₃₆₇	D₄
cosocle	C₂×C₄×C₃₆₇	C₂³
hypercenter	#106	D₃₆₇
hypocenter	C₃₆₇	#427
LCS	G ▷ C₄×C₃₆₇ ▷ C₂×C₃₆₇ ▷ C₃₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
3. #26044194	C₂	semidirect
4. #26044193	C₂	semidirect
5. #106×C₃₆₇	C₂	semidirect
8. C₈×D₃₆₇	C₄	semidirect
34. C₃₆₇	#427	semidirect
40. #106	D₃₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₃₆₇	2
#26044219	2
#26044194	2
#26044193	2
#106×C₃₆₇	2
#427	367