Subgroups — #37554213

Order 23488.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19154561	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₃₆₇⋊C₈)	C₂	1	4	yes	maximal, complemented
4	#117×C₃₆₇	C₂	1	1	yes	maximal, characteristic
5	#19154561	—	2	2	no	—
6	C₂×C₃₆₇⋊C₈	—	2	2	no	—
7	C₂×C₃₆₇⋊C₈	C₂²	1	2	yes	—
8	C₂×C₃₆₇⋊C₈	C₂²	1	2	yes	—
9	C₂²×C₄×C₃₆₇	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₃₆₇	C₄	1	2	yes	—
11	C₂²⋊C₄×C₃₆₇	C₄	1	2	yes	—
12	C₄²×C₃₆₇	C₂²	1	2	yes	—
13	C₃₆₇⋊C₈	—	2	2	no	—
14	C₃₆₇⋊C₈	—	2	2	no	—
15	C₂³×C₃₆₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₃₆₇	C₂×C₄	1	4	yes	—
17	C₂×C₄×C₃₆₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₃₆₇	D₄	1	2	yes	—
19	C₂×C₄×C₃₆₇	Q₈	1	2	yes	—
20	C₂×C₄×C₃₆₇	C₂³	1	1	yes	characteristic
21	C₂²×C₃₆₇	—	2	2	no	—
22	C₂²×C₃₆₇	C₄⋊C₄	1	2	yes	—
23	C₂²×C₃₆₇	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₃₆₇	—	2	4	no	—
25	C₄×C₃₆₇	C₄⋊C₄	1	2	yes	—
26	C₄×C₃₆₇	C₂×D₄	1	1	yes	characteristic
27	C₄×C₃₆₇	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₃₆₇	—	2	2	no	—
29	C₂×C₃₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₃₆₇	#131	1	2	yes	—
31	C₃₆₇	#424	1	1	yes	characteristic, complemented, Sylow
32	#424	—	367	1	no	maximal, Sylow
33	C₂×M₁₆	—	367	1	no	—
34	C₂²×C₈	—	367	1	no	—
35	#117	D₃₆₇	1	1	yes	characteristic
36	C₄⋊C₈	—	367	4	no	—
37	C₂²×C₄	C₂×D₃₆₇	1	1	yes	characteristic
38	M₁₆	—	734	2	no	—
39	C₂×C₈	—	367	2	no	—
40	C₂×C₈	—	367	2	no	—
41	C₂×C₈	—	734	2	no	—
42	C₄⋊C₄	Dic₃₆₇	1	2	yes	—
43	C₂²⋊C₄	Dic₃₆₇	1	2	yes	—
44	C₄²	C₂×D₃₆₇	1	2	yes	—
45	C₂³	C₂×Dic₃₆₇	1	1	yes	characteristic
46	C₂×C₄	C₂×Dic₃₆₇	1	4	yes	—
47	C₂×C₄	C₂×Dic₃₆₇	1	1	yes	characteristic
48	C₂×C₄	D₁₄₆₈	1	2	yes	—
49	C₂×C₄	Dic₇₃₄	1	2	yes	—
50	C₂×C₄	C₂²×D₃₆₇	1	1	yes	characteristic
51	C₈	—	734	2	no	—
52	C₈	—	734	2	no	—
53	C₂²	—	2	2	no	—
54	C₂²	C₄⋊Dic₃₆₇	1	2	yes	—
55	C₂²	C₂²×Dic₃₆₇	1	1	yes	characteristic
56	C₄	—	2	4	no	—
57	C₄	C₄⋊Dic₃₆₇	1	2	yes	—
58	C₄	C₂×D₁₄₆₈	1	1	yes	characteristic
59	C₄	C₂×Dic₇₃₄	1	1	yes	characteristic
60	C₂	—	2	2	no	—
61	C₂	C₂×C₄⋊Dic₃₆₇	1	1	yes	characteristic
62	C₂	#26044268	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃₆₇
commutator, socle	C₂²×C₃₆₇	C₂²×C₄
Fitting	#117×C₃₆₇	C₂
prime radical	C₂³×C₃₆₇	C₂×C₄
cosocle	C₂×C₄×C₃₆₇	C₂³
hypercenter	#117	D₃₆₇
hypocenter	C₃₆₇	#424
LCS	G ▷ C₂²×C₃₆₇ ▷ C₃₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
3. C₄⋊(C₃₆₇⋊C₈)	C₂	semidirect
31. C₃₆₇	#424	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19154561	2
C₂²×C₃₆₇⋊C₈	2
C₄⋊(C₃₆₇⋊C₈)	2
#117×C₃₆₇	2
#424	367