Subgroups — #23757075

Order 10048.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₂₅₆	C₂	1	1	yes	maximal, characteristic
2	#19097705	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₆×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₆₂₈	—	2	1	no	—
5	D₁₂₅₆	—	2	1	no	—
6	D₁₂₅₆	C₄	1	2	yes	complemented
7	C₄⋊Dic₁₅₇	—	2	1	no	—
8	C₂×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
9	C₁₆×C₁₅₇	—	2	1	no	—
10	C₂²×D₁₅₇	—	4	1	no	—
11	D₆₂₈	—	2	2	no	—
12	D₆₂₈	—	4	1	no	—
13	C₂×Dic₁₅₇	—	4	1	no	—
14	C₂×C₄×C₁₅₇	D₄	1	1	yes	characteristic
15	C₈×C₁₅₇	C₂×C₄	1	1	yes	characteristic
16	C₈×C₁₅₇	D₄	1	1	yes	characteristic
17	C₂²×C₁₅₇	D₈	1	1	yes	characteristic
18	C₂×D₁₅₇	—	4	2	no	—
19	C₂×D₁₅₇	—	8	1	no	—
20	C₂×D₁₅₇	—	8	1	no	—
21	C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₁₅₇	SD₁₆	1	1	yes	characteristic
23	Dic₁₅₇	—	8	1	no	—
24	C₂×C₁₅₇	#102	1	1	yes	characteristic
25	C₂×C₁₅₇	D₁₆	1	1	yes	characteristic
26	C₂×C₁₅₇	SD₃₂	1	1	yes	characteristic
27	D₁₅₇	—	8	2	no	—
28	C₁₅₇	#357	1	1	yes	characteristic, complemented, Sylow
29	#357	—	157	1	no	maximal, Sylow
30	C₂×D₈	—	157	1	no	—
31	C₂×C₁₆	D₁₅₇	1	1	yes	characteristic, complemented
32	#107	—	157	1	no	—
33	C₂×D₄	—	314	1	no	—
34	D₈	—	314	1	no	—
35	D₈	—	157	2	no	—
36	C₂×C₈	C₂×D₁₅₇	1	1	yes	characteristic
37	C₄⋊C₄	—	314	1	no	—
38	C₁₆	—	2	1	no	—
39	C₂³	—	628	1	no	—
40	D₄	—	314	2	no	—
41	D₄	—	628	1	no	—
42	C₂×C₄	—	628	1	no	—
43	C₂×C₄	D₆₂₈	1	1	yes	characteristic
44	C₈	C₄×D₁₅₇	1	1	yes	characteristic
45	C₈	C₂²⋊D₁₅₇	1	1	yes	characteristic
46	C₂²	—	628	2	no	—
47	C₂²	—	1256	1	no	—
48	C₂²	—	1256	1	no	—
49	C₂²	D₁₂₅₆	1	1	yes	characteristic
50	C₄	—	1256	1	no	—
51	C₄	#14578931	1	1	yes	characteristic
52	C₄	#14578943	1	1	yes	characteristic
53	C₂	—	1256	2	no	—
54	C₂	#19097693	1	1	yes	characteristic
55	C₂	D₂₅₁₂	1	1	yes	characteristic
56	C₂	#19097716	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₂₅₆
commutator	C₈×C₁₅₇	C₂×C₄
Frattini	C₂×C₈	C₂×D₁₅₇
Fitting	C₂×C₁₆×C₁₅₇	C₂
socle	C₂²×C₁₅₇	D₈
prime radical	C₂×D₁₂₅₆	C₂
cosocle	C₂×C₈×C₁₅₇	C₂²
hypercenter	C₂×C₁₆	D₁₅₇
hypocenter	C₁₅₇	#357
LCS	G ▷ C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
2. #19097705	C₂	semidirect
3. C₂×C₁₆×C₁₅₇	C₂	semidirect
6. D₁₂₅₆	C₄	semidirect
28. C₁₅₇	#357	semidirect
31. C₂×C₁₆	D₁₅₇	semidirect