Subgroups — #34102429

Order 20096.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757689	C₂	1	1	yes	maximal, characteristic
2	#23757107	C₂	1	1	yes	maximal, characteristic, complemented
3	#23757106	C₂	1	2	yes	maximal, complemented
4	C₃₂×D₁₅₇	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₂×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#19097777	C₄	1	1	yes	characteristic
7	#19097711	C₄	1	1	yes	characteristic
8	#19097710	C₄	1	2	yes	—
9	C₁₆×D₁₅₇	C₂²	1	2	yes	—
10	#19097677	C₂²	1	2	yes	—
11	C₂×C₁₆×C₁₅₇	C₂²	1	1	yes	characteristic
12	C₃₂×C₁₅₇	C₂²	1	1	yes	characteristic
13	C₃₂×C₁₅₇	C₂²	1	1	yes	characteristic
14	#14578954	C₈	1	1	yes	characteristic
15	#14578938	C₈	1	1	yes	characteristic
16	#14578937	C₈	1	2	yes	—
17	C₈×D₁₅₇	C₂×C₄	1	2	yes	—
18	#14578929	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₁₅₇	C₂×C₄	1	1	yes	characteristic
20	C₁₆×C₁₅₇	C₂×C₄	1	1	yes	characteristic
21	C₁₆×C₁₅₇	C₂³	1	1	yes	characteristic
22	Dic₃₁₄	C₁₆	1	1	yes	characteristic
23	D₆₂₈	C₁₆	1	1	yes	characteristic
24	C₂²⋊D₁₅₇	C₁₆	1	2	yes	—
25	C₄×D₁₅₇	C₂×C₈	1	2	yes	—
26	#11929434	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₁₅₇	C₂×C₈	1	1	yes	characteristic
28	C₈×C₁₅₇	C₂×C₈	1	1	yes	characteristic
29	C₈×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₁₅₇	C₂×C₁₆	1	1	yes	characteristic
31	C₂×D₁₅₇	C₂×C₁₆	1	2	yes	—
32	C₄×C₁₅₇	C₂×C₁₆	1	1	yes	characteristic
33	C₄×C₁₅₇	C₂²×C₈	1	1	yes	characteristic
34	Dic₁₅₇	C₂×C₁₆	1	2	yes	—
35	C₂×C₁₅₇	—	2	1	no	—
36	C₂×C₁₅₇	C₂²×C₁₆	1	1	yes	characteristic
37	D₁₅₇	—	2	2	no	—
38	C₁₅₇	#2258	1	1	yes	characteristic, complemented, Sylow
39	#2258	—	157	1	no	maximal, Sylow
40	#504	—	157	1	no	—
41	M₆₄	—	157	1	no	—
42	M₆₄	—	157	2	no	—
43	C₂×C₃₂	—	157	2	no	—
44	C₂×C₃₂	D₁₅₇	1	1	yes	characteristic, complemented
45	#131	—	157	1	no	—
46	M₃₂	—	157	1	no	—
47	M₃₂	—	157	2	no	—
48	C₂×C₁₆	—	157	2	no	—
49	C₂×C₁₆	C₂×D₁₅₇	1	1	yes	characteristic
50	C₃₂	—	157	2	no	—
51	C₃₂	C₂×D₁₅₇	1	1	yes	characteristic
52	C₃₂	C₂×D₁₅₇	1	1	yes	characteristic
53	#41	—	157	1	no	—
54	M₁₆	—	157	1	no	—
55	M₁₆	—	157	2	no	—
56	C₂×C₈	—	157	2	no	—
57	C₂×C₈	C₄×D₁₅₇	1	1	yes	characteristic
58	C₁₆	—	157	2	no	—
59	C₁₆	C₄×D₁₅₇	1	1	yes	characteristic
60	C₁₆	C₂²×D₁₅₇	1	1	yes	characteristic
61	Q₈	—	157	1	no	—
62	D₄	—	157	2	no	—
63	D₄	—	157	1	no	—
64	C₂×C₄	—	157	2	no	—
65	C₂×C₄	C₈×D₁₅₇	1	1	yes	characteristic
66	C₈	—	157	2	no	—
67	C₈	C₈×D₁₅₇	1	1	yes	characteristic
68	C₈	C₂×C₄×D₁₅₇	1	1	yes	characteristic
69	C₂²	—	157	2	no	—
70	C₂²	C₁₆×D₁₅₇	1	1	yes	characteristic
71	C₄	—	157	2	no	—
72	C₄	C₁₆×D₁₅₇	1	1	yes	characteristic
73	C₄	C₂×C₈×D₁₅₇	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	—	314	2	no	—
76	C₂	C₂×C₁₆×D₁₅₇	1	1	yes	characteristic

	subgroup	G/H
center	C₃₂	C₂×D₁₅₇
commutator, socle	C₂×C₁₅₇	C₂²×C₁₆
Frattini	C₁₆	C₂²×D₁₅₇
Fitting	C₂×C₃₂×C₁₅₇	C₂
prime radical	#14578954	C₈
cosocle	C₁₆×C₁₅₇	C₂³
hypercenter	C₂×C₃₂	D₁₅₇
hypocenter	C₁₅₇	#2258
LCS	G ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₃₂ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
2. #23757107	C₂	semidirect
3. #23757106	C₂	semidirect
4. C₃₂×D₁₅₇	C₂	semidirect
5. C₂×C₃₂×C₁₅₇	C₂	semidirect
38. C₁₅₇	#2258	semidirect
44. C₂×C₃₂	D₁₅₇	semidirect

Maximal subgroups

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

subgroup	index
#23757689	2
#23757107	2
#23757106	2
C₃₂×D₁₅₇	2
C₂×C₃₂×C₁₅₇	2
#2258	157