Subgroups — #23757669

Order 10048.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19097771	C₂	1	1	yes	maximal, characteristic
2	#19097702	C₂	1	4	yes	maximal
3	C₄×#11929434	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₃₁₄	—	2	1	no	—
6	C₂×#11929434	C₂²	1	2	yes	—
7	C₄⋊Dic₁₅₇	—	2	1	no	—
8	C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
9	C₂×Q₈×C₁₅₇	—	2	1	no	—
10	C₄⋊C₄×C₁₅₇	—	2	1	no	—
11	C₄⋊C₄×C₁₅₇	C₂²	1	2	yes	—
12	C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
13	Dic₃₁₄	—	4	2	no	—
14	C₂×Dic₁₅₇	—	2	2	no	—
15	#11929434	Q₈	1	4	yes	complemented
16	Q₈×C₁₅₇	—	4	2	no	—
17	C₂×C₄×C₁₅₇	—	2	2	no	—
18	C₂×C₄×C₁₅₇	D₄	1	1	yes	characteristic
19	C₂×C₄×C₁₅₇	D₄	1	1	yes	characteristic
20	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
21	C₂²×C₁₅₇	C₂×D₄	1	1	yes	characteristic
22	C₄×C₁₅₇	—	4	2	no	—
23	C₄×C₁₅₇	SD₁₆	1	2	yes	—
24	C₄×C₁₅₇	SD₁₆	1	2	yes	—
25	C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
26	Dic₁₅₇	—	4	2	no	—
27	C₂×C₁₅₇	C₄⋊Q₈	1	1	yes	characteristic
28	C₂×C₁₅₇	C₂×SD₁₆	1	1	yes	characteristic
29	C₂×C₁₅₇	C₂×SD₁₆	1	1	yes	characteristic
30	C₁₅₇	#498	1	1	yes	characteristic, complemented, Sylow
31	#498	—	157	1	no	maximal, Sylow
32	C₄⋊Q₈	—	157	1	no	—
33	C₄⋊Q₈	D₁₅₇	1	1	yes	characteristic
34	#106	—	157	4	no	—
35	C₄×C₈	—	157	1	no	—
36	C₂×Q₈	—	2	1	no	—
37	C₂×Q₈	—	314	1	no	—
38	C₂×C₈	—	157	2	no	—
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	—	314	1	no	—
41	C₄⋊C₄	—	157	2	no	—
42	C₄⋊C₄	C₂×D₁₅₇	1	2	yes	—
43	C₄²	C₂×D₁₅₇	1	1	yes	characteristic
44	Q₈	—	4	2	no	—
45	Q₈	—	628	2	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	314	2	no	—
48	C₂×C₄	C₂²⋊D₁₅₇	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊D₁₅₇	1	1	yes	characteristic
50	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
51	C₈	—	157	4	no	—
52	C₂²	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	—	628	2	no	—
55	C₄	#14578941	1	2	yes	—
56	C₄	#14578942	1	2	yes	—
57	C₄	Q₈×D₁₅₇	1	2	yes	—
58	C₂	#19097769	1	1	yes	characteristic
59	C₂	C₂×#14578941	1	1	yes	characteristic
60	C₂	C₂×#14578942	1	1	yes	characteristic

	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, prime radical	C₂²×C₁₅₇	C₂×D₄
hypercenter	C₄⋊Q₈	D₁₅₇
hypocenter	C₁₅₇	#498
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
15. #11929434	Q₈	semidirect
30. C₁₅₇	#498	semidirect