Subgroups — #23757023

Order 10048.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#11929434	C₂	1	1	yes	maximal, characteristic
2	C₄×#11929434	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#11929434	—	2	1	no	—
5	C₂×#11929434	C₄	1	1	yes	characteristic
6	C₂×#11929434	C₄	1	1	yes	characteristic
7	C₂×C₈×C₁₅₇	—	2	1	no	—
8	C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
9	#11929434	—	2	1	no	—
10	#11929434	—	4	1	no	—
11	#11929434	C₈	1	2	yes	complemented
12	C₂×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₁₅₇	D₄	1	1	yes	characteristic
14	C₂×C₄×C₁₅₇	Q₈	1	1	yes	characteristic
15	C₈×C₁₅₇	—	4	1	no	—
16	C₂²×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
17	C₄×C₁₅₇	—	2	1	no	—
18	C₄×C₁₅₇	C₂×C₈	1	1	yes	characteristic
19	C₄×C₁₅₇	M₁₆	1	1	yes	characteristic
20	C₄×C₁₅₇	SD₁₆	1	1	yes	characteristic
21	C₄×C₁₅₇	SD₁₆	1	1	yes	characteristic
22	C₂×C₁₅₇	C₄⋊C₈	1	1	yes	characteristic
23	C₂×C₁₅₇	#106	1	1	yes	characteristic
24	C₂×C₁₅₇	#108	1	1	yes	characteristic
25	C₁₅₇	#334	1	1	yes	characteristic, complemented, Sylow
26	#334	—	157	1	no	maximal, Sylow
27	C₄⋊C₈	—	157	1	no	—
28	C₄⋊C₈	D₁₅₇	1	1	yes	characteristic
29	C₄×C₈	—	157	1	no	—
30	C₂×C₈	—	2	1	no	—
31	C₂×C₈	—	314	1	no	—
32	C₂×C₈	—	157	1	no	—
33	C₂×C₈	—	157	1	no	—
34	C₄²	C₂×D₁₅₇	1	1	yes	characteristic
35	C₂×C₄	C₄×D₁₅₇	1	1	yes	characteristic
36	C₂×C₄	C₂²⋊D₁₅₇	1	1	yes	characteristic
37	C₂×C₄	Dic₃₁₄	1	1	yes	characteristic
38	C₈	—	4	1	no	—
39	C₈	—	314	1	no	—
40	C₈	—	628	1	no	—
41	C₈	—	157	2	no	—
42	C₂²	#14578933	1	1	yes	characteristic
43	C₄	—	2	1	no	—
44	C₄	C₈×D₁₅₇	1	1	yes	characteristic
45	C₄	#14578937	1	1	yes	characteristic
46	C₄	#14578941	1	1	yes	characteristic
47	C₄	#14578942	1	1	yes	characteristic
48	C₂	#19097700	1	1	yes	characteristic
49	C₂	#19097702	1	1	yes	characteristic
50	C₂	#19097706	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₁₅₇
commutator	C₄×C₁₅₇	C₂×C₈
Frattini	C₄²	C₂×D₁₅₇
Fitting	C₄⋊C₈×C₁₅₇	C₂
socle, prime radical	C₂²×C₁₅₇	C₄⋊C₄
cosocle	C₄²×C₁₅₇	C₂²
hypercenter	C₄⋊C₈	D₁₅₇
hypocenter	C₁₅₇	#334
LCS	G ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
11. #11929434	C₈	semidirect
25. C₁₅₇	#334	semidirect