Subgroups — #23758078

Order 10048.

268 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19097817	C₂	1	2	yes	maximal, complemented
2	#19097799	C₂	1	6	yes	maximal, complemented
3	#19097793	C₂	1	3	yes	maximal, complemented
4	C₂×D₁₂₅₆	C₂	1	3	yes	maximal, complemented
5	#131×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#14578955	—	2	2	no	—
7	#14578954	C₂²	1	6	yes	complemented
8	C₂×D₆₂₈	C₂²	1	6	yes	complemented
9	D₄×D₁₅₇	—	2	6	no	—
10	Dic₆₂₈	C₂²	1	1	yes	characteristic, complemented
11	#14578943	C₂²	1	6	yes	complemented
12	D₁₂₅₆	C₂²	1	3	yes	complemented
13	D₁₂₅₆	C₂²	1	6	yes	complemented
14	#41×C₁₅₇	C₂²	1	1	yes	characteristic
15	M₁₆×C₁₅₇	C₂²	1	3	yes	complemented
16	C₂×C₈×C₁₅₇	C₂²	1	3	yes	complemented
17	C₂²×D₁₅₇	—	2	6	no	—
18	Dic₃₁₄	C₂³	1	2	yes	—
19	D₆₂₈	—	2	6	no	—
20	D₆₂₈	C₂³	1	6	yes	—
21	C₂²⋊D₁₅₇	—	2	6	no	—
22	C₄×D₁₅₇	—	2	6	no	—
23	Q₈×C₁₅₇	D₄	1	1	yes	characteristic
24	D₄×C₁₅₇	D₄	1	3	yes	—
25	C₂×C₄×C₁₅₇	C₂³	1	3	yes	—
26	C₈×C₁₅₇	C₂³	1	1	yes	characteristic
27	C₈×C₁₅₇	C₂³	1	3	yes	—
28	C₂²×C₁₅₇	C₂×D₄	1	3	yes	—
29	C₂×D₁₅₇	—	2	6	no	—
30	C₂×D₁₅₇	—	8	6	no	—
31	C₄×C₁₅₇	C₂×D₄	1	3	yes	—
32	C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
33	Dic₁₅₇	—	2	2	no	—
34	C₂×C₁₅₇	—	2	3	no	—
35	C₂×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
36	D₁₅₇	—	4	6	no	—
37	C₁₅₇	#576	1	1	yes	characteristic, complemented, Sylow
38	#576	—	157	1	no	maximal, Sylow
39	H(5,𝔽₂)	—	157	2	no	—
40	AGL(1,ℤ/8)	—	157	6	no	—
41	#135	—	157	3	no	—
42	C₂×D₈	—	157	3	no	—
43	#131	D₁₅₇	1	1	yes	characteristic, complemented
44	#41	—	314	2	no	—
45	#41	—	157	6	no	—
46	#41	C₂×D₁₅₇	1	1	yes	characteristic
47	C₂×D₄	—	314	6	no	—
48	C₂×D₄	—	157	6	no	—
49	Q₁₆	—	157	1	no	—
50	SD₁₆	—	157	6	no	—
51	D₈	—	157	3	no	—
52	D₈	—	157	6	no	—
53	M₁₆	C₂×D₁₅₇	1	3	yes	complemented
54	C₂×C₈	C₂×D₁₅₇	1	3	yes	complemented
55	C₂³	—	314	6	no	—
56	Q₈	—	157	2	no	—
57	Q₈	D₆₂₈	1	1	yes	characteristic
58	D₄	—	314	6	no	—
59	D₄	—	314	6	no	—
60	D₄	—	157	6	no	—
61	D₄	D₆₂₈	1	3	yes	—
62	C₂×C₄	—	314	6	no	—
63	C₂×C₄	C₂²×D₁₅₇	1	3	yes	—
64	C₈	C₂²×D₁₅₇	1	1	yes	characteristic
65	C₈	C₂²×D₁₅₇	1	3	yes	—
66	C₂²	—	314	6	no	—
67	C₂²	—	1256	6	no	—
68	C₂²	C₂×D₆₂₈	1	3	yes	—
69	C₄	—	314	2	no	—
70	C₄	C₂×D₆₂₈	1	3	yes	—
71	C₄	C₂³×D₁₅₇	1	1	yes	characteristic
72	C₂	—	2	3	no	—
73	C₂	—	628	6	no	—
74	C₂	C₂²×D₆₂₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₆₂₈
commutator, cosocle	C₄×C₁₅₇	C₂⁴
Frattini	C₄	C₂³×D₁₅₇
Fitting	#131×C₁₅₇	C₂
socle	C₂×C₁₅₇	C₂²×D₄
hypercenter	#131	D₁₅₇
hypocenter	C₁₅₇	#576
LCS	G ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #19097817	C₂	semidirect
2. #19097799	C₂	semidirect
3. #19097793	C₂	semidirect
4. C₂×D₁₂₅₆	C₂	semidirect
5. #131×C₁₅₇	C₂	semidirect
7. #14578954	C₂²	semidirect
8. C₂×D₆₂₈	C₂²	semidirect
10. Dic₆₂₈	C₂²	semidirect
11. #14578943	C₂²	semidirect
12. D₁₂₅₆	C₂²	semidirect
13. D₁₂₅₆	C₂²	semidirect
15. M₁₆×C₁₅₇	C₂²	semidirect
16. C₂×C₈×C₁₅₇	C₂²	semidirect
37. C₁₅₇	#576	semidirect
43. #131	D₁₅₇	semidirect
53. M₁₆	C₂×D₁₅₇	semidirect
54. C₂×C₈	C₂×D₁₅₇	semidirect