Subgroups — #34101994

Order 20096.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₂₅₆	C₂	1	1	yes	maximal, characteristic
2	#23757367	C₂	1	2	yes	maximal
3	#23757076	C₂	1	2	yes	maximal
4	C₁₆×Dic₁₅₇	C₂	1	1	yes	maximal, characteristic
5	#366×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₆₂₈	C₂²	1	2	yes	—
7	#19097736	—	2	2	no	—
8	Dic₁₂₅₆	C₄	1	4	yes	complemented
9	C₂×#14578929	C₂²	1	1	yes	characteristic
10	#19097694	—	2	2	no	—
11	C₈×Dic₁₅₇	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₁₅₇	C₂²	1	1	yes	characteristic
13	#107×C₁₅₇	C₂²	1	2	yes	—
14	C₂×Dic₃₁₄	—	2	2	no	—
15	Dic₆₂₈	—	2	2	no	—
16	Dic₆₂₈	C₂×C₄	1	4	yes	—
17	C₂×#11929434	D₄	1	1	yes	characteristic
18	#14578933	—	4	2	no	—
19	C₄×Dic₁₅₇	—	4	2	no	—
20	C₄×Dic₁₅₇	D₄	1	1	yes	characteristic
21	#14578929	—	2	1	no	—
22	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₁₅₇	—	2	2	no	—
24	C₁₆×C₁₅₇	C₂×C₄	1	2	yes	—
25	Dic₃₁₄	—	2	4	no	—
26	Dic₃₁₄	—	4	2	no	—
27	C₂×Dic₁₅₇	—	4	2	no	—
28	C₂×Dic₁₅₇	D₈	1	1	yes	characteristic
29	C₂×Dic₁₅₇	D₈	1	1	yes	characteristic
30	#11929434	—	2	1	no	—
31	C₂×C₄×C₁₅₇	—	4	2	no	—
32	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
33	C₈×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
34	C₈×C₁₅₇	#41	1	1	yes	characteristic
35	C₂²×C₁₅₇	C₂×D₈	1	1	yes	characteristic
36	C₄×C₁₅₇	—	8	2	no	—
37	C₄×C₁₅₇	C₄×D₄	1	1	yes	characteristic
38	C₄×C₁₅₇	#135	1	1	yes	characteristic
39	Dic₁₅₇	—	2	1	no	—
40	Dic₁₅₇	—	4	4	no	—
41	Dic₁₅₇	Q₃₂	1	2	yes	complemented
42	C₂×C₁₅₇	C₄×D₈	1	1	yes	characteristic
43	C₂×C₁₅₇	C₂×Q₃₂	1	1	yes	characteristic
44	C₂×C₁₅₇	#508	1	1	yes	characteristic
45	C₁₅₇	C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
46	C₄×Q₃₂	—	157	1	no	maximal, Sylow
47	C₂×Q₃₂	—	157	1	no	—
48	C₄×Q₁₆	—	157	2	no	—
49	#366	D₁₅₇	1	1	yes	characteristic
50	#358	—	157	2	no	—
51	C₄×C₁₆	—	157	1	no	—
52	C₂×Q₁₆	—	157	2	no	—
53	C₄×Q₈	—	314	2	no	—
54	Q₃₂	—	157	4	no	—
55	C₂×C₁₆	—	157	1	no	—
56	C₂×C₁₆	C₂×D₁₅₇	1	1	yes	characteristic
57	#107	C₂×D₁₅₇	1	2	yes	—
58	#103	—	314	2	no	—
59	C₄×C₈	—	157	1	no	—
60	C₂×Q₈	—	314	2	no	—
61	Q₁₆	—	314	2	no	—
62	Q₁₆	—	157	4	no	—
63	C₂×C₈	—	157	1	no	—
64	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	628	2	no	—
67	C₄²	—	628	2	no	—
68	C₄²	—	157	1	no	—
69	C₁₆	—	314	1	no	—
70	C₁₆	C₄×D₁₅₇	1	2	yes	—
71	Q₈	—	314	4	no	—
72	Q₈	—	628	2	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	—	628	2	no	—
75	C₂×C₄	—	157	1	no	—
76	C₂×C₄	—	157	1	no	—
77	C₂×C₄	D₄×D₁₅₇	1	1	yes	characteristic
78	C₈	—	314	1	no	—
79	C₈	C₂×C₄×D₁₅₇	1	1	yes	characteristic
80	C₈	#14578955	1	1	yes	characteristic
81	C₂²	D₈×D₁₅₇	1	1	yes	characteristic
82	C₄	—	8	2	no	—
83	C₄	—	314	1	no	—
84	C₄	—	628	4	no	—
85	C₄	—	157	2	no	—
86	C₄	#19097733	1	1	yes	characteristic
87	C₄	#19097794	1	1	yes	characteristic
88	C₂	#23757355	1	1	yes	characteristic
89	C₂	Q₃₂×D₁₅₇	1	1	yes	characteristic
90	C₂	#23757702	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₅₇
commutator	C₈×C₁₅₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₅₇
Fitting	#366×C₁₅₇	C₂
socle, prime radical	C₂²×C₁₅₇	C₂×D₈
cosocle	C₂×C₈×C₁₅₇	C₂³
hypercenter	#366	D₁₅₇
hypocenter	C₁₅₇	C₄×Q₃₂
LCS	G ▷ C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

Complements

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

N	H ≅ G/N	type
8. Dic₁₂₅₆	C₄	semidirect
41. Dic₁₅₇	Q₃₂	semidirect
45. C₁₅₇	C₄×Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₂₅₆	2
#23757367	2
#23757076	2
C₁₆×Dic₁₅₇	2
#366×C₁₅₇	2
C₄×Q₃₂	157