Subgroups — #23757084

Order 10048.

58 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	#19097703	C₂	1	1	yes	maximal, characteristic, complemented
3	M₃₂×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#14578954	—	2	1	no	—
5	Dic₆₂₈	C₄	1	1	yes	characteristic, complemented
6	#14578943	—	2	1	no	—
7	D₁₂₅₆	C₄	1	1	yes	characteristic, complemented
8	C₄⋊Dic₁₅₇	—	2	1	no	—
9	C₂×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
10	C₁₆×C₁₅₇	—	2	1	no	—
11	Dic₃₁₄	—	2	1	no	—
12	D₆₂₈	—	2	1	no	—
13	C₂²⋊D₁₅₇	—	4	1	no	—
14	C₂×Dic₁₅₇	—	4	1	no	—
15	C₄×D₁₅₇	—	4	1	no	—
16	C₂×C₄×C₁₅₇	D₄	1	1	yes	characteristic
17	C₈×C₁₅₇	C₂×C₄	1	1	yes	characteristic
18	C₈×C₁₅₇	D₄	1	1	yes	characteristic
19	C₂²×C₁₅₇	D₈	1	1	yes	characteristic
20	C₂×D₁₅₇	—	4	1	no	—
21	C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₁₅₇	SD₁₆	1	1	yes	characteristic
23	Dic₁₅₇	—	4	1	no	—
24	Dic₁₅₇	—	8	1	no	—
25	C₂×C₁₅₇	—	2	1	no	—
26	C₂×C₁₅₇	#102	1	1	yes	characteristic
27	D₁₅₇	—	8	1	no	—
28	C₁₅₇	#360	1	1	yes	characteristic, complemented, Sylow
29	#360	—	157	1	no	maximal, Sylow
30	#135	—	157	1	no	—
31	M₃₂	D₁₅₇	1	1	yes	characteristic, complemented
32	#106	—	157	1	no	—
33	#41	—	314	1	no	—
34	Q₁₆	—	157	1	no	—
35	SD₁₆	—	314	1	no	—
36	D₈	—	157	1	no	—
37	C₂×C₈	C₂×D₁₅₇	1	1	yes	characteristic
38	C₄⋊C₄	—	314	1	no	—
39	C₁₆	—	2	1	no	—
40	Q₈	—	314	1	no	—
41	D₄	—	314	1	no	—
42	D₄	—	628	1	no	—
43	C₂×C₄	—	628	1	no	—
44	C₂×C₄	—	628	1	no	—
45	C₂×C₄	D₆₂₈	1	1	yes	characteristic
46	C₈	C₄×D₁₅₇	1	1	yes	characteristic
47	C₈	C₂²⋊D₁₅₇	1	1	yes	characteristic
48	C₂²	—	628	1	no	—
49	C₂²	D₁₂₅₆	1	1	yes	characteristic
50	C₄	—	628	1	no	—
51	C₄	—	1256	1	no	—
52	C₄	#14578931	1	1	yes	characteristic
53	C₄	#14578943	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	1256	1	no	—
56	C₂	#19097693	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#19097693
commutator	C₈×C₁₅₇	C₂×C₄
Frattini	C₂×C₈	C₂×D₁₅₇
Fitting	M₃₂×C₁₅₇	C₂
socle	C₂×C₁₅₇	#102
prime radical	#19097793	C₂
cosocle	C₂×C₈×C₁₅₇	C₂²
hypercenter	M₃₂	D₁₅₇
hypocenter	C₁₅₇	#360
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. #19097703	C₂	semidirect
3. M₃₂×C₁₅₇	C₂	semidirect
5. Dic₆₂₈	C₄	semidirect
7. D₁₂₅₆	C₄	semidirect
28. C₁₅₇	#360	semidirect
31. M₃₂	D₁₅₇	semidirect