Subgroups — #23757665

Order 10048.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14578944	C₂	1	2	yes	maximal, complemented
2	C₂×#14578941	C₂	1	2	yes	maximal
3	#19097759	C₂	1	1	yes	maximal, characteristic
4	#19097682	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊Q₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₃₁₄	C₂²	1	1	yes	characteristic
7	C₂×D₆₂₈	C₂²	1	1	yes	characteristic
8	#14578944	—	2	4	no	—
9	#14578941	—	2	4	no	—
10	C₂×#11929434	C₂²	1	2	yes	—
11	#14578931	—	2	2	no	—
12	C₂×Q₈×C₁₅₇	C₂²	1	2	yes	—
13	C₄⋊C₄×C₁₅₇	—	2	2	no	—
14	C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
15	C₂²×D₁₅₇	—	2	1	no	—
16	Dic₃₁₄	—	2	2	no	—
17	D₆₂₈	—	2	2	no	—
18	C₂×Dic₁₅₇	—	2	1	no	—
19	#11929434	D₄	1	4	yes	—
20	Q₈×C₁₅₇	—	2	2	no	—
21	Q₈×C₁₅₇	—	2	2	no	—
22	C₂×C₄×C₁₅₇	—	2	2	no	—
23	C₂×C₄×C₁₅₇	D₄	1	2	yes	—
24	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
25	C₂²×C₁₅₇	C₂×D₄	1	1	yes	characteristic
26	C₂×D₁₅₇	—	4	1	no	—
27	C₂×D₁₅₇	—	4	2	no	—
28	C₄×C₁₅₇	—	2	2	no	—
29	C₄×C₁₅₇	—	4	2	no	—
30	C₄×C₁₅₇	C₂×D₄	1	2	yes	—
31	Dic₁₅₇	—	4	1	no	—
32	C₂×C₁₅₇	#127	1	1	yes	characteristic
33	C₂×C₁₅₇	#137	1	2	yes	—
34	D₁₅₇	—	8	1	no	—
35	C₁₅₇	#497	1	1	yes	characteristic, complemented, Sylow
36	#497	—	157	1	no	maximal, Sylow
37	C₂×Q₁₆	—	157	2	no	—
38	C₂×SD₁₆	—	157	2	no	—
39	C₄⋊Q₈	D₁₅₇	1	1	yes	characteristic, complemented
40	#124	—	157	1	no	—
41	#97	—	157	1	no	—
42	C₂×Q₈	—	157	1	no	—
43	C₂×Q₈	C₂×D₁₅₇	1	2	yes	—
44	C₂×D₄	—	157	1	no	—
45	Q₁₆	—	314	4	no	—
46	SD₁₆	—	314	4	no	—
47	C₂×C₈	—	157	2	no	—
48	C₄⋊C₄	—	2	2	no	—
49	C₂²⋊C₄	—	314	2	no	—
50	C₄²	C₂×D₁₅₇	1	1	yes	characteristic
51	C₂³	—	314	1	no	—
52	Q₈	—	2	2	no	—
53	Q₈	—	2	2	no	—
54	Q₈	—	314	2	no	—
55	D₄	—	314	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	314	1	no	—
58	C₂×C₄	C₂²⋊D₁₅₇	1	2	yes	—
59	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
60	C₈	—	157	4	no	—
61	C₂²	—	628	1	no	—
62	C₂²	—	628	2	no	—
63	C₂²	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	4	2	no	—
66	C₄	—	628	1	no	—
67	C₄	D₄×D₁₅₇	1	2	yes	—
68	C₂	—	1256	1	no	—
69	C₂	#19097767	1	1	yes	characteristic
70	C₂	#19097802	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₅₇
commutator, cosocle	C₂×C₄×C₁₅₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₅₇
Fitting	C₄⋊Q₈×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂×D₄
prime radical	C₂×D₆₂₈	C₂²
hypercenter	C₄⋊Q₈	D₁₅₇
hypocenter	C₁₅₇	#497
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#14578944	C₂	semidirect
4. #19097682	C₂	semidirect
5. C₄⋊Q₈×C₁₅₇	C₂	semidirect
35. C₁₅₇	#497	semidirect
39. C₄⋊Q₈	D₁₅₇	semidirect