Subgroups — #34115286

Order 20096.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₅₇⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	#23758199	C₂	1	1	yes	maximal, characteristic
3	Q₃₂×D₁₅₇	C₂	1	1	yes	maximal, characteristic
4	#19097840	—	2	1	no	—
5	#19097821	—	2	1	no	—
6	Q₁₆×D₁₅₇	—	2	1	no	—
7	Dic₁₂₅₆	C₄	1	1	yes	characteristic, complemented
8	#19097717	—	2	1	no	—
9	C₁₆×D₁₅₇	C₂²	1	1	yes	characteristic
10	Q₃₂×C₁₅₇	C₄	1	1	yes	characteristic, complemented
11	C₄⋊#10597845	—	4	1	no	—
12	Q₈×D₁₅₇	—	4	1	no	—
13	Dic₆₂₈	—	2	1	no	—
14	#14578944	—	4	1	no	—
15	C₈×D₁₅₇	D₄	1	1	yes	characteristic
16	#14578929	D₄	1	1	yes	characteristic
17	Q₁₆×C₁₅₇	—	2	1	no	—
18	C₁₆×C₁₅₇	C₂×C₄	1	1	yes	characteristic
19	C₂×#10597845	—	8	1	no	—
20	Dic₃₁₄	—	4	1	no	—
21	Dic₃₁₄	—	8	1	no	—
22	C₄×D₁₅₇	—	8	1	no	—
23	C₄×D₁₅₇	D₈	1	1	yes	characteristic
24	#11929434	SD₁₆	1	1	yes	characteristic
25	Q₈×C₁₅₇	—	4	1	no	—
26	C₈×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×D₁₅₇	D₁₆	1	1	yes	characteristic
28	#10597845	—	16	1	no	—
29	C₄×C₁₅₇	—	8	1	no	—
30	C₄×C₁₅₇	#102	1	1	yes	characteristic
31	Dic₁₅₇	—	8	1	no	—
32	Dic₁₅₇	SD₃₂	1	1	yes	characteristic
33	C₂×C₁₅₇	#357	1	1	yes	characteristic
34	D₁₅₇	SD₆₄	1	1	yes	characteristic
35	D₁₅₇	Q₆₄	1	1	yes	characteristic
36	C₁₅₇	#1416	1	1	yes	characteristic, complemented, Sylow
37	#1416	—	157	1	no	maximal, Sylow
38	C₂×Q₃₂	—	157	1	no	—
39	C₂×C₃₂	—	157	1	no	—
40	#366	—	157	1	no	—
41	C₂×Q₁₆	—	314	1	no	—
42	Q₃₂	—	314	1	no	—
43	Q₃₂	—	157	1	no	—
44	Q₃₂	#10597845	1	1	yes	characteristic, complemented
45	C₂×C₁₆	—	157	1	no	—
46	#107	—	314	1	no	—
47	C₃₂	—	314	1	no	—
48	C₂×Q₈	—	628	1	no	—
49	Q₁₆	—	2	1	no	—
50	Q₁₆	—	314	1	no	—
51	Q₁₆	—	628	1	no	—
52	C₂×C₈	—	157	1	no	—
53	C₄⋊C₄	—	628	1	no	—
54	C₁₆	—	157	1	no	—
55	C₁₆	C₂×#10597845	1	1	yes	characteristic
56	Q₈	—	4	1	no	—
57	Q₈	—	628	1	no	—
58	Q₈	—	1256	1	no	—
59	C₂×C₄	—	157	1	no	—
60	C₂×C₄	—	1256	1	no	—
61	C₂×C₄	—	1256	1	no	—
62	C₈	—	157	1	no	—
63	C₈	C₂²⋊#10597845	1	1	yes	characteristic
64	C₂²	—	157	1	no	—
65	C₄	—	8	1	no	—
66	C₄	—	157	1	no	—
67	C₄	—	2512	1	no	—
68	C₄	—	1256	1	no	—
69	C₄	#19097833	1	1	yes	characteristic
70	C₂	—	157	1	no	—
71	C₂	—	157	1	no	—
72	C₂	#23758185	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23758185
commutator	C₁₆×C₁₅₇	C₂×C₄
Frattini	C₁₆	C₂×#10597845
Fitting	Q₃₂×C₁₅₇	C₄
socle	C₂×C₁₅₇	#357
prime radical	C₂×D₁₅₇	D₁₆
cosocle	C₁₆×D₁₅₇	C₂²
hypercenter	Q₃₂	#10597845
hypocenter	C₁₅₇	#1416
LCS	G ▷ C₁₆×C₁₅₇ ▷ C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
7. Dic₁₂₅₆	C₄	semidirect
10. Q₃₂×C₁₅₇	C₄	semidirect
36. C₁₅₇	#1416	semidirect
44. Q₃₂	#10597845	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
D₁₅₇⋊C₃₂	2
#23758199	2
Q₃₂×D₁₅₇	2
#1416	157