Group #34109077 — subgroups

Subgroups — #34109077

Order 20096.

348 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 221 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	#23757404	C₂	1	8	yes	maximal, complemented
2	#23757300	C₂	1	2	yes	maximal
3	#23757297	C₂	1	1	yes	maximal, characteristic
4	#23757229	C₂	1	2	yes	maximal
5	C₂×#19097682	C₂	1	1	yes	maximal, characteristic
6	#533×C₁₅₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#14578938	C₂²	1	2	yes	—
8	C₂²×#11929434	C₂²	1	1	yes	characteristic
9	C₂²×#11929434	C₂²	1	1	yes	characteristic
10	C₄⋊#11929434	C₂²	1	4	yes	—
11	C₄⋊#11929434	C₂²	1	8	yes	—
12	#19097682	C₂²	1	4	yes	—
13	#19097682	C₂²	1	4	yes	—
14	C₄×#11929434	C₂²	1	4	yes	—
15	C₄⋊Q₈×C₁₅₇	C₄	1	2	yes	—
16	#125×C₁₅₇	C₄	1	2	yes	—
17	C₂²⋊Q₈×C₁₅₇	C₄	1	4	yes	—
18	C₄×Q₈×C₁₅₇	C₂²	1	4	yes	—
19	#117×C₁₅₇	C₂²	1	2	yes	—
20	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
21	#14578938	—	2	4	no	—
22	C₂×#11929434	—	2	2	no	—
23	C₂×#11929434	Q₈	1	4	yes	complemented
24	C₂×#11929434	C₂³	1	4	yes	—
25	C₂×#11929434	C₂³	1	2	yes	—
26	C₂×#11929434	C₂³	1	2	yes	—
27	C₂×Q₈×C₁₅₇	C₂×C₄	1	4	yes	—
28	C₂²×C₄×C₁₅₇	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₁₅₇	C₂×C₄	1	4	yes	—
31	C₄⋊C₄×C₁₅₇	C₂×C₄	1	8	yes	—
32	C₄⋊C₄×C₁₅₇	C₂×C₄	1	4	yes	—
33	C₂²⋊C₄×C₁₅₇	C₂×C₄	1	4	yes	—
34	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₁₅₇	C₂³	1	2	yes	—
37	C₄²×C₁₅₇	C₂³	1	4	yes	—
38	#11929434	—	2	2	no	—
39	#11929434	—	2	4	no	—
40	#11929434	C₂×Q₈	1	4	yes	—
41	C₂³×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
42	Q₈×C₁₅₇	—	4	4	no	—
43	C₂×C₄×C₁₅₇	—	4	2	no	—
44	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
45	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₁₅₇	C₂²×C₄	1	8	yes	—
47	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₁₅₇	#41	1	2	yes	—
50	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
51	C₂²×C₁₅₇	—	2	2	no	—
52	C₂²×C₁₅₇	C₄×Q₈	1	2	yes	—
53	C₂²×C₁₅₇	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₁₅₇	—	2	2	no	—
55	C₄×C₁₅₇	—	2	2	no	—
56	C₄×C₁₅₇	—	2	8	no	—
57	C₄×C₁₅₇	C₄×Q₈	1	2	yes	—
58	C₄×C₁₅₇	C₂²×Q₈	1	1	yes	characteristic
59	C₄×C₁₅₇	C₂×#41	1	1	yes	characteristic
60	C₂×C₁₅₇	—	2	2	no	—
61	C₂×C₁₅₇	C₂×C₄×Q₈	1	1	yes	characteristic
62	C₂×C₁₅₇	#568	1	2	yes	—
63	C₁₅₇	#2961	1	1	yes	characteristic, complemented, Sylow
64	#2961	—	157	1	no	maximal, Sylow
65	#533	D₁₅₇	1	1	yes	characteristic
66	#446	—	157	8	no	—
67	#424	—	157	2	no	—
68	#423	—	157	1	no	—
69	#405	—	157	2	no	—
70	C₂×#97	—	157	1	no	—
71	C₂×M₁₆	—	157	2	no	—
72	C₂²×C₈	—	157	1	no	—
73	C₂²×C₈	—	157	1	no	—
74	C₄⋊Q₈	Dic₁₅₇	1	2	yes	—
75	#125	Dic₁₅₇	1	2	yes	—
76	C₂²⋊Q₈	Dic₁₅₇	1	4	yes	—
77	C₄×Q₈	C₂×D₁₅₇	1	4	yes	—
78	#117	C₂×D₁₅₇	1	2	yes	—
79	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
80	C₄⋊C₈	—	157	4	no	—
81	C₄⋊C₈	—	157	8	no	—
82	#97	—	157	4	no	—
83	#97	—	157	4	no	—
84	C₄×C₈	—	157	4	no	—
85	C₂×Q₈	C₂×Dic₁₅₇	1	4	yes	—
86	C₂²×C₄	C₂×Dic₁₅₇	1	2	yes	—
87	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
88	M₁₆	—	314	4	no	—
89	C₂×C₈	—	314	2	no	—
90	C₂×C₈	—	157	4	no	—
91	C₂×C₈	—	157	2	no	—
92	C₂×C₈	—	157	4	no	—
93	C₂×C₈	—	157	2	no	—
94	C₄⋊C₄	C₂×Dic₁₅₇	1	4	yes	—
95	C₄⋊C₄	C₂×Dic₁₅₇	1	8	yes	—
96	C₄⋊C₄	C₂×Dic₁₅₇	1	4	yes	—
97	C₂²⋊C₄	C₂×Dic₁₅₇	1	4	yes	—
98	C₄²	C₂×Dic₁₅₇	1	1	yes	characteristic
99	C₄²	C₂×Dic₁₅₇	1	1	yes	characteristic
100	C₄²	C₂²×D₁₅₇	1	4	yes	—
101	C₄²	C₂²×D₁₅₇	1	2	yes	—
102	C₂³	C₂²×Dic₁₅₇	1	1	yes	characteristic
103	Q₈	—	4	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	C₂²×Dic₁₅₇	1	2	yes	—
106	C₂×C₄	C₂²×Dic₁₅₇	1	2	yes	—
107	C₂×C₄	C₂²×Dic₁₅₇	1	8	yes	—
108	C₂×C₄	C₂²×Dic₁₅₇	1	1	yes	characteristic
109	C₂×C₄	Q₈×D₁₅₇	1	2	yes	—
110	C₂×C₄	#14578955	1	2	yes	—
111	C₂×C₄	C₂³×D₁₅₇	1	1	yes	characteristic
112	C₈	—	314	2	no	—
113	C₈	—	314	4	no	—
114	C₈	—	157	4	no	—
115	C₂²	—	2	2	no	—
116	C₂²	Q₈×Dic₁₅₇	1	2	yes	—
117	C₂²	C₂³×Dic₁₅₇	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	2	8	no	—
121	C₄	Q₈×Dic₁₅₇	1	2	yes	—
122	C₄	C₂×Q₈×D₁₅₇	1	1	yes	characteristic
123	C₄	C₂×#14578955	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	C₂×Q₈×Dic₁₅₇	1	1	yes	characteristic
126	C₂	#23758040	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23757404

C₂

yes

maximal, complemented

#23757300

C₂

yes

maximal

#23757297

C₂

yes

maximal, characteristic

#23757229

C₂

yes

maximal

C₂×#19097682

C₂

yes

maximal, characteristic

#533×C₁₅₇

C₂

yes

maximal, characteristic

C₂×#14578938

C₂²

yes

—

C₂²×#11929434

C₂²

yes

characteristic

C₂²×#11929434

C₂²

yes

characteristic

C₄⋊#11929434

C₂²

yes

—

C₄⋊#11929434

C₂²

yes

—

#19097682

C₂²

yes

—

#19097682

C₂²

yes

—

C₄×#11929434

C₂²

yes

—

C₄⋊Q₈×C₁₅₇

C₄

yes

—

#125×C₁₅₇

C₄

yes

—

C₂²⋊Q₈×C₁₅₇

C₄

yes

—

C₄×Q₈×C₁₅₇

C₂²

yes

—

#117×C₁₅₇

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

—

—

Q₈

yes

complemented

C₂×#11929434

C₂³

yes

—

C₂×#11929434

C₂³

yes

—

C₂×#11929434

C₂³

yes

—

C₂×Q₈×C₁₅₇

C₂×C₄

yes

—

C₂²×C₄×C₁₅₇

C₂×C₄

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₅₇

C₂×C₄

yes

—

C₄⋊C₄×C₁₅₇

C₂×C₄

yes

—

C₄⋊C₄×C₁₅₇

C₂×C₄

yes

—

C₂²⋊C₄×C₁₅₇

C₂×C₄

yes

—

C₄²×C₁₅₇

C₂×C₄

yes

characteristic

C₄²×C₁₅₇

C₂×C₄

yes

characteristic

C₄²×C₁₅₇

C₂³

yes

—

C₄²×C₁₅₇

C₂³

yes

—

#11929434

—

#11929434

—

#11929434

C₂×Q₈

yes

—

C₂³×C₁₅₇

C₂²×C₄

yes

characteristic

Q₈×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×Q₈

yes

—

C₂×C₄×C₁₅₇

#41

yes

—

C₂×C₄×C₁₅₇

C₂⁴

yes

characteristic

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₄×Q₈

yes

—

C₂²×C₁₅₇

C₂³×C₄

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₄×Q₈

yes

—

C₄×C₁₅₇

C₂²×Q₈

yes

characteristic

C₄×C₁₅₇

C₂×#41

yes

characteristic

C₂×C₁₅₇

—

C₂×C₁₅₇

C₂×C₄×Q₈

yes

characteristic

C₂×C₁₅₇

#568

yes

—

C₁₅₇

#2961

yes

characteristic, complemented, Sylow

#2961

—

157

maximal, Sylow

#533

D₁₅₇

yes

characteristic

#446

—

157

—

#424

—

157

—

#423

—

157

—

#405

—

157

—

C₂×#97

—

157

—

C₂×M₁₆

—

157

—

C₂²×C₈

—

157

—

C₂²×C₈

—

157

—

C₄⋊Q₈

Dic₁₅₇

yes

—

#125

Dic₁₅₇

yes

—

C₂²⋊Q₈

Dic₁₅₇

yes

—

C₄×Q₈

C₂×D₁₅₇

yes

—

#117

C₂×D₁₅₇

yes

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

C₄⋊C₈

—

157

—

C₄⋊C₈

—

157

—

#97

—

157

—

#97

—

157

—

C₄×C₈

—

157

—

C₂×Q₈

C₂×Dic₁₅₇

yes

—

C₂²×C₄

C₂×Dic₁₅₇

yes

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

M₁₆

—

314

—

C₂×C₈

—

314

—

C₂×C₈

—

157

—

C₂×C₈

—

157

—

C₂×C₈

—

157

—

C₂×C₈

—

157

—

C₄⋊C₄

C₂×Dic₁₅₇

yes

—

C₄⋊C₄

C₂×Dic₁₅₇

yes

—

C₄⋊C₄

C₂×Dic₁₅₇

yes

—

C₂²⋊C₄

C₂×Dic₁₅₇

yes

—

C₄²

C₂×Dic₁₅₇

yes

characteristic

C₄²

C₂×Dic₁₅₇

yes

characteristic

100

C₄²

C₂²×D₁₅₇

yes

—

101

C₄²

C₂²×D₁₅₇

yes

—

102

C₂³

C₂²×Dic₁₅₇

yes

characteristic

103

Q₈

—

104

C₂×C₄

—

105

C₂×C₄

C₂²×Dic₁₅₇

yes

—

106

C₂×C₄

C₂²×Dic₁₅₇

yes

—

107

C₂×C₄

C₂²×Dic₁₅₇

yes

—

108

C₂×C₄

C₂²×Dic₁₅₇

yes

characteristic

109

C₂×C₄

Q₈×D₁₅₇

yes

—

110

C₂×C₄

#14578955

yes

—

111

C₂×C₄

C₂³×D₁₅₇

yes

characteristic

112

C₈

—

314

—

113

C₈

—

314

—

114

C₈

—

157

—

115

C₂²

—

116

C₂²

Q₈×Dic₁₅₇

yes

—

117

C₂²

C₂³×Dic₁₅₇

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

Q₈×Dic₁₅₇

yes

—

122

C₄

C₂×Q₈×D₁₅₇

yes

characteristic

123

C₄

C₂×#14578955

yes

characteristic

124

C₂

—

125

C₂

C₂×Q₈×Dic₁₅₇

yes

characteristic

126

C₂

#23758040

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₅₇
commutator, socle	C₂²×C₁₅₇	C₂³×C₄
Fitting	#533×C₁₅₇	C₂
prime radical	C₂³×C₁₅₇	C₂²×C₄
cosocle	C₂×C₄×C₁₅₇	C₂⁴
hypercenter	#533	D₁₅₇
hypocenter	C₁₅₇	#2961
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ #533

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₅₇

commutator, socle

C₂²×C₁₅₇

C₂³×C₄

Fitting

#533×C₁₅₇

C₂

prime radical

C₂³×C₁₅₇

C₂²×C₄

cosocle

C₂×C₄×C₁₅₇

C₂⁴

D₁₅₇

C₁₅₇

G ▷ C₂²×C₁₅₇ ▷ C₁₅₇

C₁ ◁ C₂×C₄ ◁ #533

N	H ≅ G/N	type
1. #23757404	C₂	semidirect
23. C₂×#11929434	Q₈	semidirect
63. C₁₅₇	#2961	semidirect

H ≅ G/N

type

1. #23757404

C₂

semidirect

23. C₂×#11929434

Q₈

semidirect

63. C₁₅₇

#2961

semidirect

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

subgroup	index
#23757404	2
#23757300	2
#23757297	2
#23757229	2
C₂×#19097682	2
#533×C₁₅₇	2
#2961	157

subgroup

index

#533×C₁₅₇

157

Subgroups — #34109077

Conjugacy classes of subgroups

Complements

Maximal subgroups