Subgroups — #34116539

Order 20096.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23758317	C₂	1	2	yes	maximal, complemented
2	#23758202	C₂	1	1	yes	maximal, characteristic, complemented
3	#23758188	C₂	1	2	yes	maximal, complemented
4	#23758160	C₂	1	1	yes	maximal, characteristic, complemented
5	#23757706	C₂	1	1	yes	maximal, characteristic
6	#19097859	—	2	2	no	—
7	D₁₅₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	#19097842	C₂²	1	2	yes	—
9	#19097836	—	2	2	no	—
10	C₈×#10597845	C₂²	1	1	yes	characteristic
11	#19097794	C₂²	1	2	yes	—
12	#19097714	C₄	1	2	yes	complemented
13	D₂₅₁₂	C₄	1	1	yes	characteristic, complemented
14	C₁₆×D₁₅₇	C₂²	1	1	yes	characteristic
15	Q₃₂×C₁₅₇	C₄	1	1	yes	characteristic, complemented
16	#14578964	—	2	2	no	—
17	#14578964	—	4	2	no	—
18	D₁₅₇⋊C₈	—	4	2	no	—
19	D₁₅₇⋊C₈	D₄	1	1	yes	characteristic
20	C₄×#10597845	D₄	1	1	yes	characteristic
21	#14578957	—	2	1	no	—
22	#14578955	—	2	2	no	—
23	#14578941	—	2	2	no	—
24	D₁₂₅₆	C₂×C₄	1	2	yes	—
25	C₈×D₁₅₇	C₂³	1	1	yes	characteristic
26	#14578929	C₂×C₄	1	1	yes	characteristic
27	Q₁₆×C₁₅₇	C₂×C₄	1	2	yes	—
28	C₁₆×C₁₅₇	C₂×C₄	1	1	yes	characteristic
29	C₂×#10597845	—	2	1	no	—
30	#11929441	—	4	2	no	—
31	#11929441	D₈	1	2	yes	—
32	D₆₂₈	—	2	2	no	—
33	D₆₂₈	—	4	2	no	—
34	C₄×D₁₅₇	—	4	2	no	—
35	C₄×D₁₅₇	C₂×D₄	1	1	yes	characteristic
36	#11929434	#41	1	1	yes	characteristic
37	Q₈×C₁₅₇	—	2	2	no	—
38	C₈×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×D₁₅₇	—	4	2	no	—
40	C₂×D₁₅₇	#135	1	1	yes	characteristic
41	#10597845	—	4	1	no	—
42	C₄×C₁₅₇	—	4	2	no	—
43	C₄×C₁₅₇	C₄×D₄	1	1	yes	characteristic
44	Dic₁₅₇	C₂×D₈	1	1	yes	characteristic
45	C₂×C₁₅₇	C₄×D₈	1	1	yes	characteristic
46	D₁₅₇	—	2	1	no	—
47	D₁₅₇	—	8	2	no	—
48	C₁₅₇	#2179	1	1	yes	characteristic, complemented, Sylow
49	#2179	—	157	1	no	maximal, Sylow
50	#508	—	157	1	no	—
51	#443	—	157	2	no	—
52	#368	—	157	1	no	—
53	#359	—	157	2	no	—
54	#346	—	157	1	no	—
55	#135	—	157	2	no	—
56	#131	—	314	2	no	—
57	Q₃₂	#10597845	1	1	yes	characteristic, complemented
58	SD₃₂	—	157	2	no	—
59	D₁₆	—	157	1	no	—
60	C₂×C₁₆	—	157	1	no	—
61	C₂×C₁₆	—	157	1	no	—
62	#108	—	157	2	no	—
63	#104	—	314	2	no	—
64	C₄×C₈	—	157	1	no	—
65	#41	—	314	2	no	—
66	Q₁₆	C₂×#10597845	1	2	yes	—
67	SD₁₆	—	314	2	no	—
68	D₈	—	157	2	no	—
69	M₁₆	—	314	2	no	—
70	M₁₆	—	628	2	no	—
71	C₂×C₈	—	628	2	no	—
72	C₂×C₈	—	157	1	no	—
73	C₂×C₈	—	157	1	no	—
74	C₄²	—	157	1	no	—
75	C₁₆	—	314	1	no	—
76	C₁₆	—	157	1	no	—
77	C₁₆	C₂×#10597845	1	1	yes	characteristic
78	Q₈	—	2	2	no	—
79	D₄	—	314	2	no	—
80	D₄	—	628	2	no	—
81	C₂×C₄	—	314	1	no	—
82	C₂×C₄	—	628	2	no	—
83	C₂×C₄	—	157	1	no	—
84	C₈	—	628	2	no	—
85	C₈	—	157	2	no	—
86	C₈	—	157	1	no	—
87	C₈	C₂²×#10597845	1	1	yes	characteristic
88	C₂²	—	628	2	no	—
89	C₂²	—	157	1	no	—
90	C₄	—	4	2	no	—
91	C₄	—	628	1	no	—
92	C₄	—	157	1	no	—
93	C₄	D₄×#10597845	1	1	yes	characteristic
94	C₂	—	314	1	no	—
95	C₂	—	1256	2	no	—
96	C₂	D₈×#10597845	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×#10597845
commutator	C₈×C₁₅₇	C₂²×C₄
Frattini	C₈	C₂²×#10597845
Fitting	Q₃₂×C₁₅₇	C₄
socle	C₂×C₁₅₇	C₄×D₈
prime radical	#23757706	C₂
cosocle	C₈×D₁₅₇	C₂³
hypercenter	Q₃₂	#10597845
hypocenter	C₁₅₇	#2179
LCS	G ▷ C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
1. #23758317	C₂	semidirect
2. #23758202	C₂	semidirect
3. #23758188	C₂	semidirect
4. #23758160	C₂	semidirect
12. #19097714	C₄	semidirect
13. D₂₅₁₂	C₄	semidirect
15. Q₃₂×C₁₅₇	C₄	semidirect
48. C₁₅₇	#2179	semidirect
57. Q₃₂	#10597845	semidirect

Maximal subgroups

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

subgroup	index
#23758317	2
#23758202	2
#23758188	2
#23758160	2
#23757706	2
#2179	157