Group #34097985 — subgroups

Subgroups — #34097985

Order 20096.

160 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11929434	C₂	1	1	yes	maximal, characteristic
2	C₄×#14578938	C₂	1	1	yes	maximal, characteristic
3	#431×C₁₅₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14578938	C₄	1	2	yes	—
5	C₂²×#11929434	C₄	1	2	yes	—
6	C₄⋊#11929434	—	2	1	no	—
7	C₄⋊#11929434	—	2	1	no	—
8	#19097682	—	2	1	no	—
9	C₄×#11929434	—	2	1	no	—
10	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₁₅₇	—	2	1	no	—
12	C₂²⋊C₈×C₁₅₇	C₄	1	2	yes	—
13	C₄×C₈×C₁₅₇	—	2	1	no	—
14	#14578938	—	2	2	no	—
15	#14578938	C₈	1	4	yes	complemented
16	C₂×#11929434	—	2	2	no	—
17	C₂×#11929434	—	2	2	no	—
18	C₂×#11929434	—	4	2	no	—
19	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₅₇	—	2	2	no	—
23	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
24	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
25	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
26	C₄²×C₁₅₇	Q₈	1	1	yes	characteristic
27	#11929434	—	2	4	no	—
28	#11929434	—	4	2	no	—
29	C₂³×C₁₅₇	C₄²	1	1	yes	characteristic
30	C₂×C₄×C₁₅₇	—	2	1	no	—
31	C₂×C₄×C₁₅₇	—	2	1	no	—
32	C₂×C₄×C₁₅₇	—	4	1	no	—
33	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₅₇	C₂×C₈	1	2	yes	—
40	C₂×C₄×C₁₅₇	M₁₆	1	2	yes	—
41	C₈×C₁₅₇	—	4	2	no	—
42	C₂²×C₁₅₇	—	2	1	no	—
43	C₂²×C₁₅₇	—	2	1	no	—
44	C₂²×C₁₅₇	#95	1	1	yes	characteristic
45	C₂²×C₁₅₇	C₄×C₈	1	1	yes	characteristic
46	C₂²×C₁₅₇	#97	1	1	yes	characteristic
47	C₄×C₁₅₇	—	2	1	no	—
48	C₄×C₁₅₇	—	2	1	no	—
49	C₄×C₁₅₇	—	2	1	no	—
50	C₄×C₁₅₇	C₂²⋊C₈	1	2	yes	—
51	C₄×C₁₅₇	#100	1	1	yes	characteristic
52	C₄×C₁₅₇	#101	1	1	yes	characteristic
53	C₄×C₁₅₇	C₄⋊C₈	1	2	yes	—
54	C₂×C₁₅₇	—	2	1	no	—
55	C₂×C₁₅₇	—	2	1	no	—
56	C₂×C₁₅₇	#336	1	1	yes	characteristic
57	C₂×C₁₅₇	#343	1	1	yes	characteristic
58	C₂×C₁₅₇	#344	1	1	yes	characteristic
59	C₁₅₇	#1283	1	1	yes	characteristic, complemented, Sylow
60	#1283	—	157	1	no	maximal, Sylow
61	#431	D₁₅₇	1	1	yes	characteristic
62	C₂×C₄⋊C₈	—	157	1	no	—
63	C₄×M₁₆	—	157	1	no	—
64	C₂×M₁₆	—	157	2	no	—
65	C₂²×C₈	—	157	2	no	—
66	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
67	C₄⋊C₈	—	2	1	no	—
68	C₄⋊C₈	—	314	1	no	—
69	C₄⋊C₈	—	314	1	no	—
70	C₂²⋊C₈	Dic₁₅₇	1	2	yes	—
71	#97	—	314	1	no	—
72	C₄×C₈	—	2	1	no	—
73	C₄×C₈	—	314	1	no	—
74	C₂²×C₄	C₄×D₁₅₇	1	1	yes	characteristic
75	C₂²×C₄	C₄×D₁₅₇	1	1	yes	characteristic
76	C₂²×C₄	C₂×Dic₁₅₇	1	1	yes	characteristic
77	M₁₆	—	314	2	no	—
78	M₁₆	—	157	4	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	314	2	no	—
81	C₂×C₈	—	314	2	no	—
82	C₂×C₈	—	628	2	no	—
83	C₄²	C₂²⋊D₁₅₇	1	1	yes	characteristic
84	C₄²	C₂²⋊D₁₅₇	1	1	yes	characteristic
85	C₄²	D₆₂₈	1	1	yes	characteristic
86	C₄²	Dic₃₁₄	1	1	yes	characteristic
87	C₂³	C₄×Dic₁₅₇	1	1	yes	characteristic
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	4	1	no	—
91	C₂×C₄	#14578931	1	1	yes	characteristic
92	C₂×C₄	#14578931	1	1	yes	characteristic
93	C₂×C₄	C₂²⋊Dic₁₅₇	1	1	yes	characteristic
94	C₂×C₄	#14578933	1	1	yes	characteristic
95	C₂×C₄	#14578933	1	1	yes	characteristic
96	C₂×C₄	C₄⋊Dic₁₅₇	1	1	yes	characteristic
97	C₂×C₄	C₈×D₁₅₇	1	2	yes	—
98	C₂×C₄	#14578937	1	2	yes	—
99	C₈	—	4	2	no	—
100	C₈	—	314	4	no	—
101	C₈	—	628	2	no	—
102	C₂²	—	2	1	no	—
103	C₂²	—	2	1	no	—
104	C₂²	#19097678	1	1	yes	characteristic
105	C₂²	C₈×Dic₁₅₇	1	1	yes	characteristic
106	C₂²	#19097681	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	#19097683	1	2	yes	—
111	C₄	#19097687	1	1	yes	characteristic
112	C₄	#19097689	1	1	yes	characteristic
113	C₄	#19097700	1	2	yes	—
114	C₂	—	2	1	no	—
115	C₂	—	2	1	no	—
116	C₂	#23757027	1	1	yes	characteristic
117	C₂	#23757041	1	1	yes	characteristic
118	C₂	#23757043	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11929434

C₂

yes

maximal, characteristic

C₄×#14578938

C₂

yes

maximal, characteristic

#431×C₁₅₇

C₂

yes

maximal, characteristic

C₂×#14578938

C₄

yes

—

C₂²×#11929434

C₄

yes

—

—

—

—

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₅₇

—

C₂²⋊C₈×C₁₅₇

C₄

yes

—

C₄×C₈×C₁₅₇

—

#14578938

—

#14578938

C₈

yes

—

—

—

C₂²×C₄×C₁₅₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₅₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₅₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₅₇

—

C₄²×C₁₅₇

D₄

yes

characteristic

C₄²×C₁₅₇

D₄

yes

characteristic

C₄²×C₁₅₇

D₄

yes

characteristic

C₄²×C₁₅₇

Q₈

yes

characteristic

#11929434

—

#11929434

—

C₂³×C₁₅₇

C₄²

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×C₈

yes

—

C₂×C₄×C₁₅₇

M₁₆

yes

—

C₈×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

#95

yes

characteristic

C₂²×C₁₅₇

C₄×C₈

yes

characteristic

C₂²×C₁₅₇

#97

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂²⋊C₈

yes

—

C₄×C₁₅₇

yes

C₄×C₁₅₇

yes

C₄×C₁₅₇

C₄⋊C₈

yes

—

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₁₅₇

yes

characteristic, complemented, Sylow

#1283

—

157

maximal, Sylow

#431

D₁₅₇

yes

characteristic

C₂×C₄⋊C₈

—

157

—

C₄×M₁₆

—

157

—

C₂×M₁₆

—

157

—

C₂²×C₈

—

157

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

314

—

C₄⋊C₈

—

314

—

C₂²⋊C₈

Dic₁₅₇

yes

—

#97

—

314

—

C₄×C₈

—

C₄×C₈

—

314

—

C₂²×C₄

C₄×D₁₅₇

yes

characteristic

C₂²×C₄

C₄×D₁₅₇

yes

characteristic

C₂²×C₄

C₂×Dic₁₅₇

yes

characteristic

M₁₆

—

314

—

M₁₆

—

157

—

C₂×C₈

—

C₂×C₈

—

314

—

C₂×C₈

—

314

—

C₂×C₈

—

628

—

C₄²

C₂²⋊D₁₅₇

yes

characteristic

C₄²

C₂²⋊D₁₅₇

yes

characteristic

C₄²

D₆₂₈

yes

characteristic

C₄²

Dic₃₁₄

yes

characteristic

C₂³

C₄×Dic₁₅₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊Dic₁₅₇

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₄⋊Dic₁₅₇

yes

characteristic

C₂×C₄

C₈×D₁₅₇

yes

—

C₂×C₄

#14578937

yes

—

C₈

—

100

C₈

—

314

—

101

C₈

—

628

—

102

C₂²

—

103

C₂²

—

104

C₂²

#19097678

yes

characteristic

105

C₂²

C₈×Dic₁₅₇

yes

characteristic

106

C₂²

#19097681

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

#19097683

yes

—

111

C₄

#19097687

yes

characteristic

112

C₄

#19097689

yes

characteristic

113

C₄

#19097700

yes

—

114

C₂

—

115

C₂

—

116

C₂

#23757027

yes

characteristic

117

C₂

#23757041

yes

characteristic

118

C₂

#23757043

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14578931
commutator	C₂²×C₁₅₇	C₄×C₈
Frattini	C₂×C₄²	C₂×D₁₅₇
Fitting	#431×C₁₅₇	C₂
socle	C₂²×C₁₅₇	#95
prime radical	C₂³×C₁₅₇	C₄²
cosocle	C₂×C₄²×C₁₅₇	C₂²
hypercenter	#431	D₁₅₇
hypocenter	C₁₅₇	#1283
LCS	G ▷ C₂²×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

#14578931

commutator

C₂²×C₁₅₇

C₄×C₈

Frattini

C₂×C₄²

C₂×D₁₅₇

Fitting

#431×C₁₅₇

C₂

socle

C₂²×C₁₅₇

#95

prime radical

C₂³×C₁₅₇

C₄²

cosocle

C₂×C₄²×C₁₅₇

C₂²

D₁₅₇

C₁₅₇

G ▷ C₂²×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
15. #14578938	C₈	semidirect
59. C₁₅₇	#1283	semidirect

H ≅ G/N

type

15. #14578938

C₈

semidirect

59. C₁₅₇

#1283

semidirect

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

subgroup	index
C₂×C₄⋊#11929434	2
C₄×#14578938	2
#431×C₁₅₇	2
#1283	157

subgroup

index

C₂×C₄⋊#11929434

C₄×#14578938

#431×C₁₅₇

#1283

157

Subgroups — #34097985

Conjugacy classes of subgroups

Complements

Maximal subgroups