Subgroups — #23757706

Order 10048.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19097794	C₂	1	2	yes	maximal, complemented
2	#19097714	C₂	1	2	yes	maximal, complemented
3	D₂₅₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₆×D₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₃₂×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#14578955	—	2	2	no	—
7	#14578941	—	2	2	no	—
8	D₁₂₅₆	C₂²	1	2	yes	—
9	C₈×D₁₅₇	C₂²	1	1	yes	characteristic
10	#14578929	C₂²	1	1	yes	characteristic
11	Q₁₆×C₁₅₇	C₂²	1	2	yes	—
12	C₁₆×C₁₅₇	C₂²	1	1	yes	characteristic
13	D₆₂₈	—	2	2	no	—
14	D₆₂₈	—	4	2	no	—
15	C₄×D₁₅₇	—	4	2	no	—
16	C₄×D₁₅₇	D₄	1	1	yes	characteristic
17	#11929434	D₄	1	1	yes	characteristic
18	Q₈×C₁₅₇	—	2	2	no	—
19	C₈×C₁₅₇	C₂³	1	1	yes	characteristic
20	C₂×D₁₅₇	—	4	2	no	—
21	C₂×D₁₅₇	D₈	1	1	yes	characteristic
22	C₄×C₁₅₇	—	4	2	no	—
23	C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
24	Dic₁₅₇	D₈	1	1	yes	characteristic
25	C₂×C₁₅₇	C₂×D₈	1	1	yes	characteristic
26	D₁₅₇	—	2	1	no	—
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	D₁₅₇	1	1	yes	characteristic, complemented
32	SD₃₂	—	157	2	no	—
33	D₁₆	—	157	1	no	—
34	C₂×C₁₆	—	157	1	no	—
35	#41	—	314	2	no	—
36	Q₁₆	C₂×D₁₅₇	1	2	yes	—
37	SD₁₆	—	314	2	no	—
38	D₈	—	157	2	no	—
39	C₂×C₈	—	157	1	no	—
40	C₁₆	—	157	1	no	—
41	C₁₆	C₂×D₁₅₇	1	1	yes	characteristic
42	Q₈	—	2	2	no	—
43	D₄	—	314	2	no	—
44	D₄	—	628	2	no	—
45	C₂×C₄	—	628	2	no	—
46	C₂×C₄	—	157	1	no	—
47	C₈	—	157	1	no	—
48	C₈	C₂²×D₁₅₇	1	1	yes	characteristic
49	C₂²	—	628	2	no	—
50	C₂²	—	157	1	no	—
51	C₄	—	4	2	no	—
52	C₄	—	157	1	no	—
53	C₄	D₄×D₁₅₇	1	1	yes	characteristic
54	C₂	—	314	1	no	—
55	C₂	—	1256	2	no	—
56	C₂	D₈×D₁₅₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₁₅₇
commutator, cosocle	C₈×C₁₅₇	C₂³
Frattini	C₈	C₂²×D₁₅₇
Fitting	Q₃₂×C₁₅₇	C₂
socle	C₂×C₁₅₇	C₂×D₈
hypercenter	Q₃₂	D₁₅₇
hypocenter	C₁₅₇	#508
LCS	G ▷ C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
1. #19097794	C₂	semidirect
2. #19097714	C₂	semidirect
3. D₂₅₁₂	C₂	semidirect
4. C₁₆×D₁₅₇	C₂	semidirect
5. Q₃₂×C₁₅₇	C₂	semidirect
28. C₁₅₇	#508	semidirect
31. Q₃₂	D₁₅₇	semidirect