Subgroups — #23757716

Order 10048.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19097793	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₆₂₈	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₁₂₅₆	C₂	1	2	yes	maximal, complemented
4	#19097716	C₂	1	2	yes	maximal, complemented
5	M₃₂×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#14578954	—	2	1	no	—
7	C₂×Dic₃₁₄	—	2	1	no	—
8	Dic₆₂₈	—	2	1	no	—
9	Dic₆₂₈	C₂²	1	2	yes	—
10	Dic₆₂₈	C₂²	1	1	yes	characteristic
11	#14578943	—	2	1	no	—
12	D₁₂₅₆	C₂²	1	1	yes	characteristic
13	C₂×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
14	C₁₆×C₁₅₇	C₂²	1	2	yes	—
15	Dic₃₁₄	—	2	2	no	—
16	Dic₃₁₄	—	2	1	no	—
17	Dic₃₁₄	—	4	1	no	—
18	D₆₂₈	—	2	1	no	—
19	C₂²⋊D₁₅₇	—	4	1	no	—
20	C₂×Dic₁₅₇	—	4	1	no	—
21	C₄×D₁₅₇	—	4	1	no	—
22	C₂×C₄×C₁₅₇	D₄	1	1	yes	characteristic
23	C₈×C₁₅₇	D₄	1	1	yes	characteristic
24	C₈×C₁₅₇	C₂³	1	1	yes	characteristic
25	C₂²×C₁₅₇	D₈	1	1	yes	characteristic
26	C₂×D₁₅₇	—	4	1	no	—
27	C₄×C₁₅₇	D₈	1	1	yes	characteristic
28	C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
29	Dic₁₅₇	—	4	1	no	—
30	Dic₁₅₇	—	4	2	no	—
31	C₂×C₁₅₇	—	2	1	no	—
32	C₂×C₁₅₇	C₂×D₈	1	1	yes	characteristic
33	D₁₅₇	—	8	1	no	—
34	C₁₅₇	#510	1	1	yes	characteristic, complemented, Sylow
35	#510	—	157	1	no	maximal, Sylow
36	#135	—	157	1	no	—
37	C₂×Q₁₆	—	157	1	no	—
38	Q₃₂	—	157	2	no	—
39	SD₃₂	—	157	2	no	—
40	M₃₂	D₁₅₇	1	1	yes	characteristic, complemented
41	#41	—	314	1	no	—
42	C₂×Q₈	—	314	1	no	—
43	Q₁₆	—	314	1	no	—
44	Q₁₆	—	157	2	no	—
45	Q₁₆	—	157	1	no	—
46	SD₁₆	—	314	1	no	—
47	D₈	—	157	1	no	—
48	C₂×C₈	C₂×D₁₅₇	1	1	yes	characteristic
49	C₁₆	C₂×D₁₅₇	1	2	yes	—
50	Q₈	—	314	1	no	—
51	Q₈	—	314	2	no	—
52	Q₈	—	628	1	no	—
53	D₄	—	314	1	no	—
54	D₄	—	628	1	no	—
55	C₂×C₄	—	628	1	no	—
56	C₂×C₄	—	628	1	no	—
57	C₂×C₄	D₆₂₈	1	1	yes	characteristic
58	C₈	D₆₂₈	1	1	yes	characteristic
59	C₈	C₂²×D₁₅₇	1	1	yes	characteristic
60	C₂²	—	628	1	no	—
61	C₂²	D₁₂₅₆	1	1	yes	characteristic
62	C₄	—	628	1	no	—
63	C₄	—	628	2	no	—
64	C₄	D₁₂₅₆	1	1	yes	characteristic
65	C₄	C₂×D₆₂₈	1	1	yes	characteristic
66	C₂	—	2	1	no	—
67	C₂	—	1256	1	no	—
68	C₂	C₂×D₁₂₅₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₁₂₅₆
commutator, cosocle	C₈×C₁₅₇	C₂³
Frattini	C₈	C₂²×D₁₅₇
Fitting	M₃₂×C₁₅₇	C₂
socle	C₂×C₁₅₇	C₂×D₈
prime radical	#19097793	C₂
hypercenter	M₃₂	D₁₅₇
hypocenter	C₁₅₇	#510
LCS	G ▷ C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₆₂₈	C₂	semidirect
3. Dic₁₂₅₆	C₂	semidirect
4. #19097716	C₂	semidirect
5. M₃₂×C₁₅₇	C₂	semidirect
34. C₁₅₇	#510	semidirect
40. M₃₂	D₁₅₇	semidirect