Subgroups — #23757705

Order 10048.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19097793	C₂	1	2	yes	maximal, complemented
2	Dic₁₂₅₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#19097716	C₂	1	2	yes	maximal, complemented
4	D₂₅₁₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₆×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#14578954	—	2	2	no	—
7	Dic₆₂₈	C₂²	1	2	yes	—
8	#14578943	—	2	2	no	—
9	D₁₂₅₆	C₂²	1	2	yes	—
10	C₂×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
11	C₁₆×C₁₅₇	C₂²	1	1	yes	characteristic
12	C₁₆×C₁₅₇	C₂²	1	1	yes	characteristic
13	Dic₃₁₄	—	2	2	no	—
14	D₆₂₈	—	2	2	no	—
15	C₂²⋊D₁₅₇	—	4	2	no	—
16	C₄×D₁₅₇	—	4	2	no	—
17	C₂×C₄×C₁₅₇	D₄	1	1	yes	characteristic
18	C₈×C₁₅₇	D₄	1	1	yes	characteristic
19	C₈×C₁₅₇	C₂³	1	1	yes	characteristic
20	C₂²×C₁₅₇	D₈	1	1	yes	characteristic
21	C₂×D₁₅₇	—	4	2	no	—
22	C₄×C₁₅₇	D₈	1	1	yes	characteristic
23	C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
24	Dic₁₅₇	—	4	2	no	—
25	C₂×C₁₅₇	—	2	1	no	—
26	C₂×C₁₅₇	C₂×D₈	1	1	yes	characteristic
27	D₁₅₇	—	8	2	no	—
28	C₁₅₇	#508	1	1	yes	characteristic, complemented, Sylow
29	#508	—	157	1	no	maximal, Sylow
30	#135	—	157	2	no	—
31	Q₃₂	—	157	1	no	—
32	SD₃₂	—	157	2	no	—
33	D₁₆	—	157	1	no	—
34	C₂×C₁₆	D₁₅₇	1	1	yes	characteristic, complemented
35	#41	—	314	2	no	—
36	Q₁₆	—	157	2	no	—
37	SD₁₆	—	314	2	no	—
38	D₈	—	157	2	no	—
39	C₂×C₈	C₂×D₁₅₇	1	1	yes	characteristic
40	C₁₆	C₂×D₁₅₇	1	1	yes	characteristic
41	C₁₆	C₂×D₁₅₇	1	1	yes	characteristic
42	Q₈	—	314	2	no	—
43	D₄	—	314	2	no	—
44	D₄	—	628	2	no	—
45	C₂×C₄	—	628	2	no	—
46	C₂×C₄	D₆₂₈	1	1	yes	characteristic
47	C₈	D₆₂₈	1	1	yes	characteristic
48	C₈	C₂²×D₁₅₇	1	1	yes	characteristic
49	C₂²	—	628	2	no	—
50	C₂²	D₁₂₅₆	1	1	yes	characteristic
51	C₄	—	628	2	no	—
52	C₄	D₁₂₅₆	1	1	yes	characteristic
53	C₄	C₂×D₆₂₈	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	1256	2	no	—
56	C₂	C₂×D₁₂₅₆	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₁₂₅₆
commutator, cosocle	C₈×C₁₅₇	C₂³
Frattini	C₈	C₂²×D₁₅₇
Fitting	C₂×C₁₆×C₁₅₇	C₂
socle	C₂×C₁₅₇	C₂×D₈
hypercenter	C₂×C₁₆	D₁₅₇
hypocenter	C₁₅₇	#508
LCS	G ▷ C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
1. #19097793	C₂	semidirect
2. Dic₁₂₅₆	C₂	semidirect
3. #19097716	C₂	semidirect
4. D₂₅₁₂	C₂	semidirect
5. C₂×C₁₆×C₁₅₇	C₂	semidirect
28. C₁₅₇	#508	semidirect
34. C₂×C₁₆	D₁₅₇	semidirect