Subgroups — #37554386

Order 23488.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19154564	C₂	1	2	yes	maximal, complemented
2	#26044258	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₃₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#26044181	C₂	1	2	yes	maximal, complemented
5	C₄×Q₈×C₃₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₄₆₈	—	2	1	no	—
7	C₂×D₁₄₆₈	—	2	1	no	—
8	C₂×D₁₄₆₈	C₂²	1	2	yes	—
9	#19154564	—	2	4	no	—
10	C₂×C₃₆₇⋊C₈	C₂²	1	2	yes	—
11	C₂×Q₈×C₃₆₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₃₆₇	—	2	1	no	—
13	C₄⋊C₄×C₃₆₇	C₂²	1	1	yes	characteristic
14	C₄²×C₃₆₇	—	2	1	no	—
15	C₄²×C₃₆₇	C₂²	1	1	yes	characteristic
16	C₂²×D₃₆₇	—	2	2	no	—
17	D₁₄₆₈	—	2	2	no	—
18	D₁₄₆₈	—	2	2	no	—
19	D₁₄₆₈	—	4	2	no	—
20	D₁₄₆₈	—	4	2	no	—
21	C₃₆₇⋊C₈	—	2	2	no	—
22	Q₈×C₃₆₇	—	2	1	no	—
23	Q₈×C₃₆₇	D₄	1	2	yes	complemented
24	C₂×C₄×C₃₆₇	—	2	1	no	—
25	C₂×C₄×C₃₆₇	—	2	1	no	—
26	C₂×C₄×C₃₆₇	D₄	1	1	yes	characteristic
27	C₂×C₄×C₃₆₇	D₄	1	1	yes	characteristic
28	C₂×C₄×C₃₆₇	C₂³	1	1	yes	characteristic
29	C₂²×C₃₆₇	C₂×D₄	1	1	yes	characteristic
30	C₂×D₃₆₇	—	4	2	no	—
31	C₂×D₃₆₇	—	4	2	no	—
32	C₂×D₃₆₇	—	4	2	no	—
33	C₄×C₃₆₇	—	2	2	no	—
34	C₄×C₃₆₇	—	2	1	no	—
35	C₄×C₃₆₇	—	4	1	no	—
36	C₄×C₃₆₇	SD₁₆	1	2	yes	complemented
37	C₄×C₃₆₇	C₂×D₄	1	1	yes	characteristic
38	C₄×C₃₆₇	#41	1	1	yes	characteristic
39	C₂×C₃₆₇	#121	1	1	yes	characteristic
40	C₂×C₃₆₇	C₂×SD₁₆	1	1	yes	characteristic
41	C₂×C₃₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
42	D₃₆₇	—	8	2	no	—
43	C₃₆₇	#460	1	1	yes	characteristic, complemented, Sylow
44	#460	—	367	1	no	maximal, Sylow
45	C₂×SD₁₆	—	367	2	no	—
46	#127	—	367	1	no	—
47	C₄×Q₈	D₃₆₇	1	1	yes	characteristic, complemented
48	C₄⋊C₈	—	367	1	no	—
49	#102	—	367	2	no	—
50	C₂×Q₈	C₂×D₃₆₇	1	1	yes	characteristic
51	C₂×D₄	—	367	2	no	—
52	C₂×D₄	—	734	1	no	—
53	C₂×D₄	—	734	1	no	—
54	SD₁₆	—	734	4	no	—
55	C₂×C₈	—	367	2	no	—
56	C₄⋊C₄	—	2	1	no	—
57	C₄⋊C₄	C₂×D₃₆₇	1	1	yes	characteristic
58	C₄²	—	2	1	no	—
59	C₄²	C₂×D₃₆₇	1	1	yes	characteristic
60	C₂³	—	734	2	no	—
61	Q₈	—	2	1	no	—
62	Q₈	D₁₄₆₈	1	2	yes	complemented
63	D₄	—	1468	2	no	—
64	D₄	—	1468	2	no	—
65	D₄	—	734	2	no	—
66	D₄	—	734	2	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	C₂²⋊D₃₆₇	1	1	yes	characteristic
70	C₂×C₄	C₂²⋊D₃₆₇	1	1	yes	characteristic
71	C₂×C₄	C₂²×D₃₆₇	1	1	yes	characteristic
72	C₈	—	734	2	no	—
73	C₂²	—	1468	2	no	—
74	C₂²	—	1468	2	no	—
75	C₂²	—	1468	2	no	—
76	C₂²	C₂×C₂²⋊D₃₆₇	1	1	yes	characteristic
77	C₄	—	2	2	no	—
78	C₄	—	2	1	no	—
79	C₄	—	4	1	no	—
80	C₄	#19154564	1	2	yes	complemented
81	C₄	C₂×D₁₄₆₈	1	1	yes	characteristic
82	C₄	#19154577	1	1	yes	characteristic
83	C₂	—	2936	2	no	—
84	C₂	#26044235	1	1	yes	characteristic
85	C₂	C₂×#19154564	1	1	yes	characteristic
86	C₂	#26044286	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₃₆₇
commutator, cosocle	C₂×C₄×C₃₆₇	C₂³
Frattini	C₂×C₄	C₂²×D₃₆₇
Fitting	C₄×Q₈×C₃₆₇	C₂
socle	C₂²×C₃₆₇	C₂×D₄
prime radical	#26044258	C₂
hypercenter	C₄×Q₈	D₃₆₇
hypocenter	C₃₆₇	#460
LCS	G ▷ C₂×C₄×C₃₆₇ ▷ C₂×C₃₆₇ ▷ C₃₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#19154564	C₂	semidirect
3. C₄⋊(C₃₆₇⋊C₈)	C₂	semidirect
4. #26044181	C₂	semidirect
5. C₄×Q₈×C₃₆₇	C₂	semidirect
23. Q₈×C₃₆₇	D₄	semidirect
36. C₄×C₃₆₇	SD₁₆	semidirect
43. C₃₆₇	#460	semidirect
47. C₄×Q₈	D₃₆₇	semidirect
62. Q₈	D₁₄₆₈	semidirect
80. C₄	#19154564	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19154564	2
#26044258	2
C₄⋊(C₃₆₇⋊C₈)	2
#26044181	2
C₄×Q₈×C₃₆₇	2
#460	367