Subgroups — #37554096

Order 23488.

322 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₄₆₈	C₂	1	3	yes	maximal, complemented
2	C₂×(C₂×D₃₆₇)⋊C₄	C₂	1	3	yes	maximal, complemented
3	#95×C₃₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₃₆₇	C₂²	1	3	yes	—
5	C₂×D₁₄₆₈	—	2	12	no	—
6	C₂×D₁₄₆₈	—	2	6	no	—
7	C₂²×Dic₃₆₇	C₂²	1	1	yes	characteristic
8	(C₂×D₃₆₇)⋊C₄	—	2	6	no	—
9	C₂²×C₄×C₃₆₇	C₂²	1	3	yes	—
10	C₂²×D₃₆₇	—	2	3	no	—
11	C₂²×D₃₆₇	—	2	6	no	—
12	C₂²×D₃₆₇	—	2	6	no	—
13	C₂²×D₃₆₇	—	2	3	no	—
14	C₂²×D₃₆₇	D₄	1	6	yes	complemented
15	D₁₄₆₈	—	2	24	no	—
16	C₂×Dic₃₆₇	—	2	3	no	—
17	C₂³×C₃₆₇	C₂³	1	1	yes	characteristic
18	C₂×C₄×C₃₆₇	—	2	6	no	—
19	C₂×C₄×C₃₆₇	D₄	1	6	yes	complemented
20	C₂²×C₃₆₇	C₂×D₄	1	3	yes	—
21	C₂²×C₃₆₇	C₂×D₄	1	3	yes	—
22	C₂²×C₃₆₇	#41	1	1	yes	characteristic
23	C₂×D₃₆₇	—	2	12	no	—
24	C₂×D₃₆₇	—	2	6	no	—
25	C₂×D₃₆₇	—	4	3	no	—
26	C₂×D₃₆₇	—	4	6	no	—
27	C₂×D₃₆₇	—	4	3	no	—
28	C₄×C₃₆₇	—	2	6	no	—
29	Dic₃₆₇	—	4	1	no	—
30	C₂×C₃₆₇	#120	1	3	yes	—
31	C₂×C₃₆₇	#121	1	3	yes	—
32	C₂×C₃₆₇	#127	1	1	yes	characteristic
33	D₃₆₇	—	4	6	no	—
34	C₃₆₇	#392	1	1	yes	characteristic, complemented, Sylow
35	#392	—	367	1	no	maximal, Sylow
36	C₂²×D₄	—	367	3	no	—
37	C₂×C₂²⋊C₄	—	367	3	no	—
38	#95	D₃₆₇	1	1	yes	characteristic, complemented
39	C₂⁴	—	367	3	no	—
40	C₂×D₄	—	734	12	no	—
41	C₂×D₄	—	734	6	no	—
42	C₂²×C₄	—	367	1	no	—
43	C₂²×C₄	C₂×D₃₆₇	1	3	yes	—
44	C₂²⋊C₄	—	734	6	no	—
45	C₂³	—	367	6	no	—
46	C₂³	—	734	6	no	—
47	C₂³	—	734	3	no	—
48	C₂³	—	734	6	no	—
49	C₂³	—	734	3	no	—
50	C₂³	C₂²×D₃₆₇	1	1	yes	characteristic
51	D₄	—	734	24	no	—
52	C₂×C₄	—	2	6	no	—
53	C₂×C₄	—	734	3	no	—
54	C₂×C₄	D₁₄₆₈	1	6	yes	complemented
55	C₂²	—	1468	3	no	—
56	C₂²	—	1468	6	no	—
57	C₂²	—	1468	3	no	—
58	C₂²	—	734	12	no	—
59	C₂²	—	734	6	no	—
60	C₂²	D₄×D₃₆₇	1	3	yes	—
61	C₂²	C₂×D₁₄₆₈	1	3	yes	—
62	C₂²	#19154578	1	1	yes	characteristic
63	C₄	—	2	6	no	—
64	C₄	—	1468	1	no	—
65	C₂	—	1468	6	no	—
66	C₂	#26044231	1	3	yes	—
67	C₂	#26044236	1	3	yes	—
68	C₂	#26044258	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₃₆₇
commutator, socle, cosocle	C₂³×C₃₆₇	C₂³
Fitting	#95×C₃₆₇	C₂
hypercenter	#95	D₃₆₇
hypocenter	C₃₆₇	#392
LCS	G ▷ C₂³×C₃₆₇ ▷ C₃₆₇
UCS	C₁ ◁ C₂³ ◁ #95

Complements

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

N	H ≅ G/N	type
1. C₂²×D₁₄₆₈	C₂	semidirect
2. C₂×(C₂×D₃₆₇)⋊C₄	C₂	semidirect
3. #95×C₃₆₇	C₂	semidirect
14. C₂²×D₃₆₇	D₄	semidirect
19. C₂×C₄×C₃₆₇	D₄	semidirect
34. C₃₆₇	#392	semidirect
38. #95	D₃₆₇	semidirect
54. C₂×C₄	D₁₄₆₈	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₁₄₆₈	2
C₂×(C₂×D₃₆₇)⋊C₄	2
#95×C₃₆₇	2
#392	367