Subgroups — #34099734

Order 20096.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757669	C₂	1	2	yes	maximal
2	#23757653	C₂	1	1	yes	maximal, characteristic
3	#23757623	C₂	1	2	yes	maximal
4	C₈×#11929434	C₂	1	1	yes	maximal, characteristic
5	#500×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₆₂₈	—	2	2	no	—
7	#19097771	C₂²	1	2	yes	—
8	#19097702	—	2	4	no	—
9	#19097695	—	2	4	no	—
10	C₄×#11929434	C₂²	1	1	yes	characteristic
11	C₄×#11929434	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₁₅₇	C₂²	1	2	yes	—
13	#107×C₁₅₇	—	2	2	no	—
14	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₁₄	—	2	2	no	—
16	Dic₆₂₈	—	4	4	no	—
17	C₂×#11929434	D₄	1	2	yes	—
18	C₂×#11929434	D₄	1	2	yes	—
19	C₄⋊Dic₁₅₇	—	2	2	no	—
20	C₄⋊Dic₁₅₇	—	2	2	no	—
21	C₂×Q₈×C₁₅₇	—	2	2	no	—
22	C₂×C₈×C₁₅₇	D₄	1	2	yes	—
23	C₄⋊C₄×C₁₅₇	—	2	2	no	—
24	C₄⋊C₄×C₁₅₇	—	2	2	no	—
25	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
26	Dic₃₁₄	—	4	4	no	—
27	C₂×Dic₁₅₇	—	4	2	no	—
28	#11929434	—	2	2	no	—
29	#11929434	Q₁₆	1	4	yes	complemented
30	Q₈×C₁₅₇	—	4	4	no	—
31	C₂×C₄×C₁₅₇	—	4	2	no	—
32	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
35	C₈×C₁₅₇	SD₁₆	1	4	yes	—
36	C₂²×C₁₅₇	#127	1	1	yes	characteristic
37	C₄×C₁₅₇	—	8	2	no	—
38	C₄×C₁₅₇	C₂×SD₁₆	1	2	yes	—
39	C₄×C₁₅₇	C₂×Q₁₆	1	2	yes	—
40	C₄×C₁₅₇	#135	1	2	yes	—
41	Dic₁₅₇	—	8	2	no	—
42	C₂×C₁₅₇	#492	1	1	yes	characteristic
43	C₂×C₁₅₇	#494	1	1	yes	characteristic
44	C₂×C₁₅₇	#495	1	1	yes	characteristic
45	C₁₅₇	#1707	1	1	yes	characteristic, complemented, Sylow
46	#1707	—	157	1	no	maximal, Sylow
47	#500	D₁₅₇	1	1	yes	characteristic
48	#498	—	157	2	no	—
49	#494	—	157	1	no	—
50	#487	—	157	2	no	—
51	C₈²	—	157	1	no	—
52	C₂×Q₁₆	—	314	2	no	—
53	C₄⋊Q₈	—	157	2	no	—
54	C₄⋊Q₈	C₂×D₁₅₇	1	2	yes	—
55	#107	—	2	2	no	—
56	#106	—	314	4	no	—
57	#103	—	314	4	no	—
58	C₄×C₈	—	157	1	no	—
59	C₄×C₈	—	157	1	no	—
60	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic
61	C₂×Q₈	—	2	2	no	—
62	C₂×Q₈	—	314	2	no	—
63	Q₁₆	—	628	4	no	—
64	C₂×C₈	—	157	2	no	—
65	C₂×C₈	—	157	2	no	—
66	C₂×C₈	C₂²⋊D₁₅₇	1	2	yes	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	2	2	no	—
69	C₄⋊C₄	—	314	2	no	—
70	C₄⋊C₄	—	314	2	no	—
71	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
72	Q₈	—	4	4	no	—
73	Q₈	—	628	4	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	628	2	no	—
76	C₂×C₄	D₄×D₁₅₇	1	1	yes	characteristic
77	C₂×C₄	D₄×D₁₅₇	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
79	C₈	—	314	2	no	—
80	C₈	—	157	4	no	—
81	C₈	#14578941	1	4	yes	—
82	C₂²	#19097767	1	1	yes	characteristic
83	C₄	—	8	2	no	—
84	C₄	—	1256	2	no	—
85	C₄	C₂×#14578941	1	2	yes	—
86	C₄	Q₁₆×D₁₅₇	1	2	yes	—
87	C₄	#19097790	1	2	yes	—
88	C₂	#23757647	1	1	yes	characteristic
89	C₂	#23757651	1	1	yes	characteristic
90	C₂	#23757656	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19097767
commutator, cosocle	C₄²×C₁₅₇	C₂³
Frattini	C₄²	C₂²×D₁₅₇
Fitting	#500×C₁₅₇	C₂
socle, prime radical	C₂²×C₁₅₇	#127
hypercenter	#500	D₁₅₇
hypocenter	C₁₅₇	#1707
LCS	G ▷ C₄²×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

Complements

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

N	H ≅ G/N	type
29. #11929434	Q₁₆	semidirect
45. C₁₅₇	#1707	semidirect

Maximal subgroups

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

subgroup	index
#23757669	2
#23757653	2
#23757623	2
C₈×#11929434	2
#500×C₁₅₇	2
#1707	157