Subgroups — #34102022

Order 20096.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757390	C₂	1	2	yes	maximal, complemented
2	#23757103	C₂	1	1	yes	maximal, characteristic, complemented
3	#23757080	C₂	1	2	yes	maximal, complemented
4	#23757048	C₂	1	1	yes	maximal, characteristic, complemented
5	#508×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	#19097778	—	2	2	no	—
7	C₂×#14578929	C₂²	1	1	yes	characteristic
8	#19097707	C₂²	1	2	yes	—
9	#19097698	—	2	2	no	—
10	C₈×Dic₁₅₇	C₂²	1	1	yes	characteristic
11	#135×C₁₅₇	C₂²	1	2	yes	—
12	Q₃₂×C₁₅₇	C₄	1	1	yes	characteristic, complemented
13	SD₃₂×C₁₅₇	C₄	1	2	yes	complemented
14	D₁₆×C₁₅₇	C₄	1	1	yes	characteristic, complemented
15	C₂×C₁₆×C₁₅₇	C₂²	1	1	yes	characteristic
16	#14578938	—	2	2	no	—
17	#14578938	—	4	2	no	—
18	C₂×#11929434	—	4	2	no	—
19	C₂×#11929434	D₄	1	1	yes	characteristic
20	C₄×Dic₁₅₇	D₄	1	1	yes	characteristic
21	#14578929	—	2	1	no	—
22	#41×C₁₅₇	—	2	2	no	—
23	Q₁₆×C₁₅₇	C₂×C₄	1	2	yes	—
24	SD₁₆×C₁₅₇	—	2	2	no	—
25	D₈×C₁₅₇	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
27	C₁₆×C₁₅₇	C₂×C₄	1	1	yes	characteristic
28	C₁₆×C₁₅₇	C₂×C₄	1	1	yes	characteristic
29	C₂×Dic₁₅₇	—	2	1	no	—
30	#11929434	—	4	2	no	—
31	#11929434	D₈	1	2	yes	—
32	Q₈×C₁₅₇	—	2	2	no	—
33	D₄×C₁₅₇	—	2	2	no	—
34	D₄×C₁₅₇	—	4	2	no	—
35	C₂×C₄×C₁₅₇	—	4	2	no	—
36	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
37	C₈×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
38	C₈×C₁₅₇	#41	1	1	yes	characteristic
39	C₂²×C₁₅₇	—	4	2	no	—
40	C₂²×C₁₅₇	#135	1	1	yes	characteristic
41	C₄×C₁₅₇	—	4	2	no	—
42	C₄×C₁₅₇	C₄×D₄	1	1	yes	characteristic
43	C₄×C₁₅₇	C₂×D₈	1	1	yes	characteristic
44	Dic₁₅₇	—	4	1	no	—
45	C₂×C₁₅₇	—	2	1	no	—
46	C₂×C₁₅₇	—	8	2	no	—
47	C₂×C₁₅₇	C₄×D₈	1	1	yes	characteristic
48	C₁₅₇	#2179	1	1	yes	characteristic, complemented, Sylow
49	#2179	—	157	1	no	maximal, Sylow
50	#508	D₁₅₇	1	1	yes	characteristic
51	#443	—	157	2	no	—
52	#368	—	157	1	no	—
53	#359	—	157	2	no	—
54	#346	—	157	1	no	—
55	#135	C₂×D₁₅₇	1	2	yes	—
56	#131	—	314	2	no	—
57	Q₃₂	Dic₁₅₇	1	1	yes	characteristic, complemented
58	SD₃₂	Dic₁₅₇	1	2	yes	complemented
59	D₁₆	Dic₁₅₇	1	1	yes	characteristic, complemented
60	C₂×C₁₆	—	157	1	no	—
61	C₂×C₁₆	C₂×D₁₅₇	1	1	yes	characteristic
62	#108	—	157	2	no	—
63	#104	—	314	2	no	—
64	C₄×C₈	—	157	1	no	—
65	#41	—	2	2	no	—
66	Q₁₆	C₂×Dic₁₅₇	1	2	yes	—
67	SD₁₆	—	2	2	no	—
68	D₈	C₂×Dic₁₅₇	1	2	yes	—
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	C₂²×D₁₅₇	1	1	yes	characteristic
74	C₄²	—	157	1	no	—
75	C₁₆	—	314	1	no	—
76	C₁₆	C₂×Dic₁₅₇	1	1	yes	characteristic
77	C₁₆	C₂×Dic₁₅₇	1	1	yes	characteristic
78	Q₈	—	2	2	no	—
79	D₄	—	2	2	no	—
80	D₄	—	4	2	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	314	1	no	—
83	C₂×C₄	D₄×D₁₅₇	1	1	yes	characteristic
84	C₈	—	628	2	no	—
85	C₈	—	157	2	no	—
86	C₈	C₂²×Dic₁₅₇	1	1	yes	characteristic
87	C₈	#14578953	1	1	yes	characteristic
88	C₂²	—	4	2	no	—
89	C₂²	#19097790	1	1	yes	characteristic
90	C₄	—	4	2	no	—
91	C₄	—	628	1	no	—
92	C₄	D₄×Dic₁₅₇	1	1	yes	characteristic
93	C₄	D₈×D₁₅₇	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	8	2	no	—
96	C₂	D₈×Dic₁₅₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₈×D₁₅₇
commutator	C₈×C₁₅₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₅₇
Fitting, prime radical	#508×C₁₅₇	C₂
socle	C₂×C₁₅₇	C₄×D₈
cosocle	C₂×C₈×C₁₅₇	C₂³
hypercenter	#508	D₁₅₇
hypocenter	C₁₅₇	#2179
LCS	G ▷ C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #508

Complements

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

N	H ≅ G/N	type
1. #23757390	C₂	semidirect
2. #23757103	C₂	semidirect
3. #23757080	C₂	semidirect
4. #23757048	C₂	semidirect
12. Q₃₂×C₁₅₇	C₄	semidirect
13. SD₃₂×C₁₅₇	C₄	semidirect
14. D₁₆×C₁₅₇	C₄	semidirect
48. C₁₅₇	#2179	semidirect
57. Q₃₂	Dic₁₅₇	semidirect
58. SD₃₂	Dic₁₅₇	semidirect
59. D₁₆	Dic₁₅₇	semidirect

Maximal subgroups

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

subgroup	index
#23757390	2
#23757103	2
#23757080	2
#23757048	2
#508×C₁₅₇	2
#2179	157