Subgroups — #37554453

Order 23488.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26044287	C₂	1	2	yes	maximal, complemented
2	#26044267	C₂	1	1	yes	maximal, characteristic, complemented
3	#26044197	C₂	1	1	yes	maximal, characteristic, complemented
4	#26044178	C₂	1	2	yes	maximal, complemented
5	C₂×D₈×C₃₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#19154577	C₂²	1	1	yes	characteristic
7	#19154565	—	2	2	no	—
8	#19154562	—	2	2	no	—
9	#19154561	C₂²	1	1	yes	characteristic
10	#19154561	C₂²	1	2	yes	—
11	#19154560	—	2	1	no	—
12	C₈×D₃₆₇	—	2	1	no	—
13	C₂×D₄×C₃₆₇	C₂²	1	2	yes	—
14	D₈×C₃₆₇	—	2	1	no	—
15	D₈×C₃₆₇	—	2	1	no	—
16	C₂×C₈×C₃₆₇	C₂²	1	1	yes	characteristic
17	Dic₇₃₄	D₄	1	1	yes	characteristic
18	D₁₄₆₈	D₄	1	1	yes	characteristic
19	C₂²⋊D₃₆₇	—	2	1	no	—
20	C₄×D₃₆₇	—	2	1	no	—
21	C₃₆₇⋊C₈	—	2	1	no	—
22	C₃₆₇⋊C₈	—	2	2	no	—
23	C₂³×C₃₆₇	—	2	2	no	—
24	D₄×C₃₆₇	—	2	2	no	—
25	D₄×C₃₆₇	—	2	2	no	—
26	C₂×C₄×C₃₆₇	C₂³	1	1	yes	characteristic
27	C₈×C₃₆₇	D₄	1	1	yes	characteristic
28	C₈×C₃₆₇	D₄	1	1	yes	characteristic
29	C₂²×C₃₆₇	—	4	2	no	—
30	C₂²×C₃₆₇	—	8	2	no	—
31	C₂²×C₃₆₇	#41	1	1	yes	characteristic
32	C₂×D₃₆₇	—	2	1	no	—
33	C₄×C₃₆₇	C₂×D₄	1	1	yes	characteristic
34	C₄×C₃₆₇	C₂×D₄	1	1	yes	characteristic
35	Dic₃₆₇	—	2	1	no	—
36	C₂×C₃₆₇	—	2	1	no	—
37	C₂×C₃₆₇	—	8	2	no	—
38	C₂×C₃₆₇	#121	1	1	yes	characteristic
39	D₃₆₇	—	4	1	no	—
40	C₃₆₇	#472	1	1	yes	characteristic, complemented, Sylow
41	#472	—	367	1	no	maximal, Sylow
42	AGL(1,ℤ/8)	—	367	2	no	—
43	C₂×D₈	D₃₆₇	1	1	yes	characteristic, complemented
44	#131	—	367	1	no	—
45	#108	—	367	1	no	—
46	#100	—	367	2	no	—
47	#41	—	367	1	no	—
48	C₂×D₄	C₂×D₃₆₇	1	2	yes	—
49	SD₁₆	—	734	2	no	—
50	D₈	—	2	1	no	—
51	D₈	—	2	1	no	—
52	D₈	—	734	2	no	—
53	M₁₆	—	367	1	no	—
54	M₁₆	—	367	2	no	—
55	M₁₆	—	734	1	no	—
56	C₂×C₈	—	734	1	no	—
57	C₂×C₈	C₂×D₃₆₇	1	1	yes	characteristic
58	C₂³	—	2	2	no	—
59	Q₈	—	367	1	no	—
60	D₄	—	2	2	no	—
61	D₄	—	2	2	no	—
62	D₄	—	367	1	no	—
63	D₄	—	734	1	no	—
64	C₂×C₄	—	734	1	no	—
65	C₂×C₄	C₂²×D₃₆₇	1	1	yes	characteristic
66	C₈	—	734	2	no	—
67	C₈	—	734	1	no	—
68	C₈	C₂²⋊D₃₆₇	1	1	yes	characteristic
69	C₈	C₂²⋊D₃₆₇	1	1	yes	characteristic
70	C₂²	—	4	2	no	—
71	C₂²	—	8	2	no	—
72	C₂²	—	734	1	no	—
73	C₂²	#19154576	1	1	yes	characteristic
74	C₄	—	734	1	no	—
75	C₄	D₄×D₃₆₇	1	1	yes	characteristic
76	C₄	C₂×C₂²⋊D₃₆₇	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	—	8	2	no	—
79	C₂	—	1468	1	no	—
80	C₂	#26044232	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26044232
commutator, cosocle	C₂×C₄×C₃₆₇	C₂³
Frattini	C₂×C₄	C₂²×D₃₆₇
Fitting	C₂×D₈×C₃₆₇	C₂
socle	C₂×C₃₆₇	#121
hypercenter	C₂×D₈	D₃₆₇
hypocenter	C₃₆₇	#472
LCS	G ▷ C₂×C₄×C₃₆₇ ▷ C₂×C₃₆₇ ▷ C₃₆₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. #26044287	C₂	semidirect
2. #26044267	C₂	semidirect
3. #26044197	C₂	semidirect
4. #26044178	C₂	semidirect
5. C₂×D₈×C₃₆₇	C₂	semidirect
40. C₃₆₇	#472	semidirect
43. C₂×D₈	D₃₆₇	semidirect

Maximal subgroups

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

subgroup	index
#26044287	2
#26044267	2
#26044197	2
#26044178	2
C₂×D₈×C₃₆₇	2
#472	367