Subgroups — #23758186

Order 10048.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₅₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#19097840	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×D₁₅₇	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#10597845	—	2	1	no	—
5	#14578957	—	2	1	no	—
6	Q₈×D₁₅₇	—	2	1	no	—
7	Dic₆₂₈	C₄	1	1	yes	characteristic, complemented
8	#14578944	—	2	1	no	—
9	C₈×D₁₅₇	C₂²	1	1	yes	characteristic
10	Q₁₆×C₁₅₇	C₄	1	1	yes	characteristic, complemented
11	C₂×#10597845	—	4	1	no	—
12	Dic₃₁₄	—	2	1	no	—
13	Dic₃₁₄	—	4	1	no	—
14	C₄×D₁₅₇	—	4	1	no	—
15	C₄×D₁₅₇	D₄	1	1	yes	characteristic
16	#11929434	D₄	1	1	yes	characteristic
17	Q₈×C₁₅₇	—	2	1	no	—
18	C₈×C₁₅₇	C₂×C₄	1	1	yes	characteristic
19	C₂×D₁₅₇	D₈	1	1	yes	characteristic
20	#10597845	—	8	1	no	—
21	C₄×C₁₅₇	—	4	1	no	—
22	C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
23	Dic₁₅₇	—	4	1	no	—
24	Dic₁₅₇	SD₁₆	1	1	yes	characteristic
25	C₂×C₁₅₇	#102	1	1	yes	characteristic
26	D₁₅₇	SD₃₂	1	1	yes	characteristic
27	D₁₅₇	Q₃₂	1	1	yes	characteristic
28	C₁₅₇	#358	1	1	yes	characteristic, complemented, Sylow
29	#358	—	157	1	no	maximal, Sylow
30	C₂×Q₁₆	—	157	1	no	—
31	C₂×C₁₆	—	157	1	no	—
32	#107	—	157	1	no	—
33	C₂×Q₈	—	314	1	no	—
34	Q₁₆	—	314	1	no	—
35	Q₁₆	—	157	1	no	—
36	Q₁₆	#10597845	1	1	yes	characteristic, complemented
37	C₂×C₈	—	157	1	no	—
38	C₄⋊C₄	—	314	1	no	—
39	C₁₆	—	314	1	no	—
40	Q₈	—	2	1	no	—
41	Q₈	—	314	1	no	—
42	Q₈	—	628	1	no	—
43	C₂×C₄	—	628	1	no	—
44	C₂×C₄	—	628	1	no	—
45	C₂×C₄	—	157	1	no	—
46	C₈	—	157	1	no	—
47	C₈	C₂×#10597845	1	1	yes	characteristic
48	C₂²	—	157	1	no	—
49	C₄	—	4	1	no	—
50	C₄	—	628	1	no	—
51	C₄	—	157	1	no	—
52	C₄	—	1256	1	no	—
53	C₄	C₂²⋊#10597845	1	1	yes	characteristic
54	C₂	—	157	1	no	—
55	C₂	—	157	1	no	—
56	C₂	#19097833	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#19097833
commutator	C₈×C₁₅₇	C₂×C₄
Frattini	C₈	C₂×#10597845
Fitting	Q₁₆×C₁₅₇	C₄
socle	C₂×C₁₅₇	#102
prime radical	C₂×D₁₅₇	D₈
cosocle	C₈×D₁₅₇	C₂²
hypercenter	Q₁₆	#10597845
hypocenter	C₁₅₇	#358
LCS	G ▷ C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
7. Dic₆₂₈	C₄	semidirect
10. Q₁₆×C₁₅₇	C₄	semidirect
28. C₁₅₇	#358	semidirect
36. Q₁₆	#10597845	semidirect