Subgroups — #24923705

Order 10688.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128854	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128776	C₂	1	2	yes	maximal, complemented
3	#19128773	C₂	1	2	yes	maximal, complemented
4	#19128772	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₈×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#14581950	—	2	1	no	—
7	Dic₆₆₈	C₂²	1	1	yes	characteristic, complemented
8	#14581939	—	2	1	no	—
9	D₁₃₃₆	C₂²	1	1	yes	characteristic, complemented
10	C₁₆₇⋊C₁₆	C₂²	1	2	yes	complemented
11	C₂×D₄×C₁₆₇	—	2	1	no	—
12	D₈×C₁₆₇	—	2	1	no	—
13	D₈×C₁₆₇	C₂²	1	2	yes	—
14	C₂×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
15	Dic₃₃₄	—	2	1	no	—
16	D₆₆₈	—	2	1	no	—
17	C₂²⋊D₁₆₇	—	4	1	no	—
18	C₄×D₁₆₇	—	4	1	no	—
19	C₂³×C₁₆₇	—	4	1	no	—
20	D₄×C₁₆₇	—	2	2	no	—
21	D₄×C₁₆₇	—	4	1	no	—
22	C₂×C₄×C₁₆₇	D₄	1	1	yes	characteristic
23	C₈×C₁₆₇	D₄	1	1	yes	characteristic
24	C₈×C₁₆₇	C₂³	1	1	yes	characteristic
25	C₂²×C₁₆₇	—	4	2	no	—
26	C₂²×C₁₆₇	—	8	2	no	—
27	C₂²×C₁₆₇	D₈	1	1	yes	characteristic
28	C₂×D₁₆₇	—	4	1	no	—
29	C₄×C₁₆₇	D₈	1	1	yes	characteristic
30	C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
31	Dic₁₆₇	—	4	1	no	—
32	C₂×C₁₆₇	—	2	1	no	—
33	C₂×C₁₆₇	—	8	2	no	—
34	C₂×C₁₆₇	C₂×D₈	1	1	yes	characteristic
35	D₁₆₇	—	8	1	no	—
36	C₁₆₇	#509	1	1	yes	characteristic, complemented, Sylow
37	#509	—	167	1	no	maximal, Sylow
38	#135	—	167	1	no	—
39	C₂×D₈	D₁₆₇	1	1	yes	characteristic, complemented
40	SD₃₂	—	167	2	no	—
41	D₁₆	—	167	2	no	—
42	M₃₂	—	167	1	no	—
43	#41	—	334	1	no	—
44	C₂×D₄	—	2	1	no	—
45	Q₁₆	—	167	1	no	—
46	SD₁₆	—	334	1	no	—
47	D₈	—	2	1	no	—
48	D₈	—	167	1	no	—
49	D₈	C₂×D₁₆₇	1	2	yes	—
50	C₂×C₈	C₂×D₁₆₇	1	1	yes	characteristic
51	C₁₆	—	167	2	no	—
52	C₂³	—	4	1	no	—
53	Q₈	—	334	1	no	—
54	D₄	—	2	2	no	—
55	D₄	—	4	1	no	—
56	D₄	—	334	1	no	—
57	D₄	—	668	1	no	—
58	C₂×C₄	—	668	1	no	—
59	C₂×C₄	C₂²⋊D₁₆₇	1	1	yes	characteristic
60	C₈	C₂²⋊D₁₆₇	1	1	yes	characteristic
61	C₈	C₂²×D₁₆₇	1	1	yes	characteristic
62	C₂²	—	4	2	no	—
63	C₂²	—	8	2	no	—
64	C₂²	—	668	1	no	—
65	C₂²	#14581935	1	1	yes	characteristic
66	C₄	—	668	1	no	—
67	C₄	#14581935	1	1	yes	characteristic
68	C₄	C₂×C₂²⋊D₁₆₇	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	—	8	2	no	—
71	C₂	—	1336	1	no	—
72	C₂	C₂×#14581935	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#14581935
commutator, cosocle	C₈×C₁₆₇	C₂³
Frattini	C₈	C₂²×D₁₆₇
Fitting	C₂×D₈×C₁₆₇	C₂
socle	C₂×C₁₆₇	C₂×D₈
hypercenter	C₂×D₈	D₁₆₇
hypocenter	C₁₆₇	#509
LCS	G ▷ C₈×C₁₆₇ ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. #19128854	C₂	semidirect
2. #19128776	C₂	semidirect
3. #19128773	C₂	semidirect
4. #19128772	C₂	semidirect
5. C₂×D₈×C₁₆₇	C₂	semidirect
7. Dic₆₆₈	C₂²	semidirect
9. D₁₃₃₆	C₂²	semidirect
10. C₁₆₇⋊C₁₆	C₂²	semidirect
36. C₁₆₇	#509	semidirect
39. C₂×D₈	D₁₆₇	semidirect