Subgroups — #37554817

Order 23488.

206 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26044260	C₂	1	4	yes	maximal, complemented
2	#26044252	C₂	1	4	yes	maximal, complemented
3	#26044240	C₂	1	4	yes	maximal
4	#26044220	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₄×C₃₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₃₄	C₂²	1	4	yes	—
7	C₄⋊Dic₃₆₇	C₂²	1	2	yes	—
8	C₄⋊Dic₃₆₇	C₂²	1	2	yes	—
9	C₄⋊Dic₃₆₇	C₂²	1	4	yes	—
10	Dic₃₆₇⋊C₄	C₂²	1	8	yes	—
11	C₂²⋊Dic₃₆₇	C₂²	1	4	yes	—
12	C₄×Dic₃₆₇	C₂²	1	4	yes	—
13	C₂²×C₄×C₃₆₇	C₂²	1	2	yes	—
14	C₂²×C₄×C₃₆₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₃₆₇	C₂²	1	4	yes	—
16	Dic₇₃₄	—	4	4	no	—
17	C₂×Dic₃₆₇	C₂³	1	4	yes	—
18	C₂×Dic₃₆₇	C₂³	1	4	yes	—
19	C₂³×C₃₆₇	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₃₆₇	—	4	2	no	—
21	C₂×C₄×C₃₆₇	Q₈	1	4	yes	complemented
22	C₂×C₄×C₃₆₇	C₂³	1	4	yes	—
23	C₂×C₄×C₃₆₇	C₂³	1	2	yes	—
24	C₂²×C₃₆₇	—	2	1	no	—
25	C₂²×C₃₆₇	—	2	1	no	—
26	C₂²×C₃₆₇	C₂×Q₈	1	2	yes	—
27	C₂²×C₃₆₇	C₂⁴	1	1	yes	characteristic
28	C₄×C₃₆₇	—	2	4	no	—
29	C₄×C₃₆₇	C₂×Q₈	1	4	yes	—
30	Dic₃₆₇	—	2	4	no	—
31	Dic₃₆₇	—	2	4	no	—
32	C₂×C₃₆₇	—	2	2	no	—
33	C₂×C₃₆₇	C₂²×Q₈	1	1	yes	characteristic
34	C₂×C₃₆₇	H(5,𝔽₂)	1	1	yes	characteristic
35	C₂×C₃₆₇	2₋¹⁺⁴	1	1	yes	characteristic
36	C₃₆₇	#544	1	1	yes	characteristic, complemented, Sylow
37	#544	—	367	1	no	maximal, Sylow
38	C₄⋊Q₈	—	367	4	no	—
39	#125	—	367	4	no	—
40	C₂²⋊Q₈	—	367	4	no	—
41	#117	—	367	2	no	—
42	C₂×C₄⋊C₄	D₃₆₇	1	1	yes	characteristic
43	C₂×Q₈	—	367	4	no	—
44	C₂²×C₄	C₂×D₃₆₇	1	2	yes	—
45	C₂²×C₄	C₂×D₃₆₇	1	1	yes	characteristic
46	C₄⋊C₄	—	367	2	no	—
47	C₄⋊C₄	—	367	2	no	—
48	C₄⋊C₄	—	367	8	no	—
49	C₄⋊C₄	—	367	4	no	—
50	C₄⋊C₄	C₂×D₃₆₇	1	4	yes	—
51	C₂²⋊C₄	—	367	4	no	—
52	C₄²	—	367	4	no	—
53	C₂³	C₂²×D₃₆₇	1	1	yes	characteristic
54	Q₈	—	1468	4	no	—
55	C₂×C₄	—	4	2	no	—
56	C₂×C₄	—	367	4	no	—
57	C₂×C₄	—	367	4	no	—
58	C₂×C₄	Dic₇₃₄	1	4	yes	complemented
59	C₂×C₄	C₂²×D₃₆₇	1	4	yes	—
60	C₂×C₄	C₂²×D₃₆₇	1	2	yes	—
61	C₂²	—	2	1	no	—
62	C₂²	—	2	1	no	—
63	C₂²	C₂×Dic₇₃₄	1	2	yes	—
64	C₂²	C₂³×D₃₆₇	1	1	yes	characteristic
65	C₄	—	2	4	no	—
66	C₄	—	734	4	no	—
67	C₄	—	734	4	no	—
68	C₄	C₂×Dic₇₃₄	1	4	yes	—
69	C₂	—	2	2	no	—
70	C₂	C₂²×Dic₇₃₄	1	1	yes	characteristic
71	C₂	#26044306	1	1	yes	characteristic
72	C₂	#26044309	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₆₇
commutator, socle, cosocle	C₂²×C₃₆₇	C₂⁴
Fitting	C₂×C₄⋊C₄×C₃₆₇	C₂
prime radical	C₂³×C₃₆₇	C₂³
hypercenter	C₂×C₄⋊C₄	D₃₆₇
hypocenter	C₃₆₇	#544
LCS	G ▷ C₂²×C₃₆₇ ▷ C₃₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. #26044260	C₂	semidirect
2. #26044252	C₂	semidirect
21. C₂×C₄×C₃₆₇	Q₈	semidirect
36. C₃₆₇	#544	semidirect
58. C₂×C₄	Dic₇₃₄	semidirect

Maximal subgroups

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

subgroup	index
#26044260	2
#26044252	2
#26044240	2
#26044220	2
C₂×C₄⋊C₄×C₃₆₇	2
#544	367