Subgroups — #37554799

Order 23488.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26044261	C₂	1	1	yes	maximal, characteristic, complemented
2	#26044256	C₂	1	4	yes	maximal
3	#26044252	C₂	1	4	yes	maximal, complemented
4	#26044241	C₂	1	2	yes	maximal
5	#26044226	C₂	1	2	yes	maximal, complemented
6	#26044218	C₂	1	1	yes	maximal, characteristic
7	#125×C₃₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₇₃₄	C₂²	1	2	yes	—
9	C₂×C₄×D₃₆₇	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₃₆₇	C₂²	1	4	yes	—
11	C₄⋊Dic₃₆₇	C₂²	1	4	yes	—
12	Dic₃₆₇⋊C₄	C₂²	1	2	yes	—
13	Dic₃₆₇⋊C₄	C₂²	1	4	yes	—
14	(C₂×D₃₆₇)⋊C₄	C₂²	1	2	yes	—
15	(C₂×D₃₆₇)⋊C₄	C₂²	1	4	yes	—
16	C₄×Dic₃₆₇	C₂²	1	1	yes	characteristic
17	C₄×Dic₃₆₇	C₂²	1	4	yes	—
18	C₄⋊C₄×C₃₆₇	C₂²	1	2	yes	—
19	C₄⋊C₄×C₃₆₇	C₂²	1	4	yes	—
20	C₄²×C₃₆₇	C₂²	1	1	yes	characteristic
21	C₂²×D₃₆₇	C₂³	1	1	yes	characteristic
22	Dic₇₃₄	—	2	4	no	—
23	C₂×Dic₃₆₇	C₂³	1	2	yes	—
24	C₂×Dic₃₆₇	C₂³	1	4	yes	—
25	C₂×Dic₃₆₇	C₂³	1	1	yes	characteristic
26	C₄×D₃₆₇	—	2	2	no	—
27	C₂×C₄×C₃₆₇	C₂³	1	2	yes	—
28	C₂×C₄×C₃₆₇	C₂³	1	4	yes	—
29	C₂×C₄×C₃₆₇	C₂³	1	1	yes	characteristic
30	C₂²×C₃₆₇	C₂⁴	1	1	yes	characteristic
31	C₂×D₃₆₇	—	2	1	no	—
32	C₂×D₃₆₇	—	2	2	no	—
33	C₄×C₃₆₇	—	2	2	no	—
34	C₄×C₃₆₇	—	2	4	no	—
35	C₄×C₃₆₇	#41	1	2	yes	—
36	Dic₃₆₇	—	2	1	no	—
37	Dic₃₆₇	—	2	4	no	—
38	Dic₃₆₇	—	2	2	no	—
39	C₂×C₃₆₇	C₂×#41	1	1	yes	characteristic
40	C₂×C₃₆₇	2₋¹⁺⁴	1	2	yes	—
41	D₃₆₇	—	4	1	no	—
42	C₃₆₇	#541	1	1	yes	characteristic, complemented, Sylow
43	#541	—	367	1	no	maximal, Sylow
44	C₄⋊Q₈	—	367	1	no	—
45	#126	—	367	4	no	—
46	#125	—	367	4	no	—
47	#125	D₃₆₇	1	1	yes	characteristic, complemented
48	C₂²⋊Q₈	—	367	2	no	—
49	C₄×Q₈	—	367	2	no	—
50	#117	—	367	1	no	—
51	C₂×Q₈	—	367	2	no	—
52	C₂²×C₄	—	367	1	no	—
53	C₄⋊C₄	—	367	2	no	—
54	C₄⋊C₄	—	367	4	no	—
55	C₄⋊C₄	—	367	4	no	—
56	C₄⋊C₄	—	367	4	no	—
57	C₄⋊C₄	C₂×D₃₆₇	1	2	yes	—
58	C₄⋊C₄	C₂×D₃₆₇	1	4	yes	—
59	C₂²⋊C₄	—	367	2	no	—
60	C₂²⋊C₄	—	367	4	no	—
61	C₄²	—	367	1	no	—
62	C₄²	—	367	4	no	—
63	C₄²	C₂×D₃₆₇	1	1	yes	characteristic
64	C₂³	—	367	1	no	—
65	Q₈	—	734	4	no	—
66	C₂×C₄	—	367	2	no	—
67	C₂×C₄	—	367	4	no	—
68	C₂×C₄	—	367	1	no	—
69	C₂×C₄	—	734	2	no	—
70	C₂×C₄	C₂²×D₃₆₇	1	4	yes	—
71	C₂×C₄	C₂²×D₃₆₇	1	2	yes	—
72	C₂×C₄	C₂²×D₃₆₇	1	1	yes	characteristic
73	C₂²	—	734	1	no	—
74	C₂²	—	734	2	no	—
75	C₂²	C₂³×D₃₆₇	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	—	2	2	no	—
78	C₄	—	734	1	no	—
79	C₄	—	734	4	no	—
80	C₄	—	734	2	no	—
81	C₄	#19154578	1	2	yes	—
82	C₂	—	1468	1	no	—
83	C₂	C₂×#19154578	1	1	yes	characteristic
84	C₂	#26044308	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₆₇
commutator, socle, cosocle	C₂²×C₃₆₇	C₂⁴
Fitting	#125×C₃₆₇	C₂
prime radical	C₂²×D₃₆₇	C₂³
hypercenter	#125	D₃₆₇
hypocenter	C₃₆₇	#541
LCS	G ▷ C₂²×C₃₆₇ ▷ C₃₆₇
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. #26044261	C₂	semidirect
3. #26044252	C₂	semidirect
5. #26044226	C₂	semidirect
7. #125×C₃₆₇	C₂	semidirect
42. C₃₆₇	#541	semidirect
47. #125	D₃₆₇	semidirect

Maximal subgroups

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

subgroup	index
#26044261	2
#26044256	2
#26044252	2
#26044241	2
#26044226	2
#26044218	2
#125×C₃₆₇	2
#541	367