Subgroups — #34100250

Order 20096.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19097690	C₂	1	2	yes	maximal
2	#23757043	C₂	1	4	yes	maximal
3	#519×C₁₅₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14578938	C₂²	1	4	yes	—
5	#19097690	—	2	4	no	—
6	C₂²×Q₈×C₁₅₇	C₂²	1	1	yes	characteristic
7	C₄×Q₈×C₁₅₇	—	2	4	no	—
8	#117×C₁₅₇	C₄	1	4	yes	—
9	#117×C₁₅₇	C₂²	1	2	yes	—
10	#14578938	—	2	4	no	—
11	#14578938	—	2	4	no	—
12	C₂×#11929434	—	2	4	no	—
13	C₂×Q₈×C₁₅₇	—	2	4	no	—
14	C₂×Q₈×C₁₅₇	D₄	1	4	yes	—
15	C₂²×C₄×C₁₅₇	C₂×C₄	1	2	yes	—
16	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
17	C₄⋊C₄×C₁₅₇	—	2	4	no	—
18	C₄⋊C₄×C₁₅₇	—	2	2	no	—
19	C₂²⋊C₄×C₁₅₇	C₂×C₄	1	4	yes	—
20	C₄²×C₁₅₇	—	2	4	no	—
21	C₄²×C₁₅₇	—	2	2	no	—
22	#11929434	—	4	4	no	—
23	C₂³×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
24	Q₈×C₁₅₇	—	2	8	no	—
25	C₂×C₄×C₁₅₇	—	2	4	no	—
26	C₂×C₄×C₁₅₇	—	2	4	no	—
27	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	4	yes	—
28	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
29	C₂×C₄×C₁₅₇	#41	1	4	yes	—
30	C₂²×C₁₅₇	—	4	1	no	—
31	C₂²×C₁₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₁₅₇	#117	1	2	yes	—
33	C₄×C₁₅₇	—	2	4	no	—
34	C₄×C₁₅₇	—	4	4	no	—
35	C₄×C₁₅₇	#123	1	4	yes	—
36	C₂×C₁₅₇	—	2	1	no	—
37	C₂×C₁₅₇	—	2	2	no	—
38	C₂×C₁₅₇	#381	1	1	yes	characteristic
39	C₁₅₇	#1830	1	1	yes	characteristic, complemented, Sylow
40	#1830	—	157	1	no	maximal, Sylow
41	#519	D₁₅₇	1	1	yes	characteristic
42	C₂×#101	—	157	2	no	—
43	#344	—	157	4	no	—
44	C₂²×Q₈	C₂×D₁₅₇	1	1	yes	characteristic
45	C₂×M₁₆	—	157	4	no	—
46	C₄×Q₈	—	2	4	no	—
47	#117	Dic₁₅₇	1	4	yes	—
48	#117	C₂×D₁₅₇	1	2	yes	—
49	#101	—	314	4	no	—
50	C₂×Q₈	—	2	4	no	—
51	C₂×Q₈	C₂²⋊D₁₅₇	1	4	yes	—
52	C₂²×C₄	C₂×Dic₁₅₇	1	2	yes	—
53	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
54	M₁₆	—	314	4	no	—
55	M₁₆	—	314	4	no	—
56	C₂×C₈	—	314	4	no	—
57	C₄⋊C₄	—	2	2	no	—
58	C₄⋊C₄	—	2	4	no	—
59	C₂²⋊C₄	C₂×Dic₁₅₇	1	4	yes	—
60	C₄²	—	2	2	no	—
61	C₄²	—	2	4	no	—
62	C₂³	C₂²×Dic₁₅₇	1	1	yes	characteristic
63	Q₈	—	2	8	no	—
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	C₂²⋊Dic₁₅₇	1	4	yes	—
67	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	2	yes	—
68	C₂×C₄	#14578954	1	4	yes	—
69	C₈	—	628	4	no	—
70	C₂²	—	4	1	no	—
71	C₂²	C₂×C₂²⋊Dic₁₅₇	1	1	yes	characteristic
72	C₂²	#19097730	1	2	yes	—
73	C₄	—	2	4	no	—
74	C₄	—	4	4	no	—
75	C₄	#19097754	1	4	yes	—
76	C₂	—	2	1	no	—
77	C₂	—	2	2	no	—
78	C₂	#23757136	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23757136
commutator, prime radical	C₂³×C₁₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	#519×C₁₅₇	C₂
socle	C₂×C₁₅₇	#381
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	#519	D₁₅₇
hypocenter	C₁₅₇	#1830
LCS	G ▷ C₂³×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #519

Complements

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

N	H ≅ G/N	type
39. C₁₅₇	#1830	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19097690	2
#23757043	2
#519×C₁₅₇	2
#1830	157