Group #34109070 — subgroups

Subgroups — #34109070

Order 20096.

364 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 197 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	#23757830	C₂	1	1	yes	maximal, characteristic
2	#23757403	C₂	1	4	yes	maximal, complemented
3	C₈×Dic₃₁₄	C₂	1	4	yes	maximal, complemented
4	#23757299	C₂	1	2	yes	maximal
5	#23757297	C₂	1	1	yes	maximal, characteristic
6	#23757228	C₂	1	2	yes	maximal
7	C₂×C₄×C₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#14578938	C₂²	1	2	yes	—
9	#19097771	C₄	1	2	yes	—
10	#19097763	C₄	1	2	yes	—
11	#19097750	C₄	1	4	yes	—
12	C₄×Dic₃₁₄	C₂²	1	4	yes	—
13	#19097730	C₂²	1	2	yes	—
14	C₄⋊#11929434	C₂²	1	4	yes	—
15	#19097700	C₂²	1	8	yes	—
16	#19097681	C₂²	1	4	yes	—
17	C₈×Dic₁₅₇	C₂²	1	4	yes	—
18	C₂²×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₁₅₇	C₂²	1	2	yes	—
22	C₄×C₈×C₁₅₇	C₂²	1	2	yes	—
23	C₂×Dic₃₁₄	C₂×C₄	1	4	yes	—
24	#14578938	—	2	4	no	—
25	C₂×#11929434	C₂³	1	4	yes	—
26	C₄⋊Dic₁₅₇	C₂×C₄	1	4	yes	—
27	C₄⋊Dic₁₅₇	C₂×C₄	1	4	yes	—
28	#14578933	C₂×C₄	1	8	yes	—
29	C₂²⋊Dic₁₅₇	C₂×C₄	1	4	yes	—
30	C₄×Dic₁₅₇	C₂³	1	4	yes	—
31	C₂²×C₄×C₁₅₇	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₁₅₇	—	2	2	no	—
34	C₂×C₈×C₁₅₇	Q₈	1	4	yes	complemented
35	C₂×C₈×C₁₅₇	C₂³	1	2	yes	—
36	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
38	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
40	C₄²×C₁₅₇	C₂³	1	2	yes	—
41	Dic₃₁₄	—	2	8	no	—
42	C₂×Dic₁₅₇	C₂²×C₄	1	8	yes	—
43	#11929434	—	2	4	no	—
44	C₂³×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₅₇	—	2	4	no	—
46	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
50	C₂×C₄×C₁₅₇	#41	1	2	yes	—
51	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
52	C₈×C₁₅₇	—	2	2	no	—
53	C₈×C₁₅₇	C₂×Q₈	1	2	yes	—
54	C₈×C₁₅₇	C₂×Q₈	1	2	yes	—
55	C₂²×C₁₅₇	—	2	2	no	—
56	C₂²×C₁₅₇	C₄×Q₈	1	2	yes	—
57	C₂²×C₁₅₇	C₂³×C₄	1	1	yes	characteristic
58	C₄×C₁₅₇	—	2	2	no	—
59	C₄×C₁₅₇	C₄×Q₈	1	2	yes	—
60	C₄×C₁₅₇	#131	1	4	yes	—
61	C₄×C₁₅₇	C₂²×Q₈	1	1	yes	characteristic
62	C₄×C₁₅₇	C₂×#41	1	1	yes	characteristic
63	Dic₁₅₇	—	2	8	no	—
64	C₂×C₁₅₇	—	2	2	no	—
65	C₂×C₁₅₇	C₂×C₄×Q₈	1	1	yes	characteristic
66	C₂×C₁₅₇	C₂×#131	1	2	yes	—
67	C₁₅₇	#2960	1	1	yes	characteristic, complemented, Sylow
68	#2960	—	157	1	no	maximal, Sylow
69	#533	—	157	1	no	—
70	#446	—	157	4	no	—
71	C₈×Q₈	—	157	4	no	—
72	#424	—	157	2	no	—
73	#423	—	157	1	no	—
74	#405	—	157	2	no	—
75	C₂×C₄×C₈	D₁₅₇	1	1	yes	characteristic
76	C₂×M₁₆	—	157	2	no	—
77	C₂²×C₈	C₂×D₁₅₇	1	1	yes	characteristic
78	C₂²×C₈	C₂×D₁₅₇	1	1	yes	characteristic
79	C₄⋊Q₈	—	157	2	no	—
80	#125	—	157	2	no	—
81	C₂²⋊Q₈	—	157	4	no	—
82	C₄×Q₈	—	157	4	no	—
83	#117	—	157	2	no	—
84	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
85	C₄⋊C₈	—	157	8	no	—
86	C₄⋊C₈	—	157	4	no	—
87	#97	—	157	4	no	—
88	C₄×C₈	—	157	4	no	—
89	C₄×C₈	C₂×D₁₅₇	1	2	yes	—
90	C₄×C₈	C₂×D₁₅₇	1	2	yes	—
91	C₂×Q₈	—	157	4	no	—
92	C₂²×C₄	C₄×D₁₅₇	1	2	yes	—
93	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
94	M₁₆	—	314	4	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	157	4	no	—
97	C₂×C₈	Dic₃₁₄	1	4	yes	complemented
98	C₂×C₈	C₂²×D₁₅₇	1	2	yes	—
99	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
100	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
101	C₄⋊C₄	—	157	4	no	—
102	C₄⋊C₄	—	157	8	no	—
103	C₄⋊C₄	—	157	4	no	—
104	C₂²⋊C₄	—	157	4	no	—
105	C₄²	—	157	4	no	—
106	C₄²	C₄×D₁₅₇	1	1	yes	characteristic
107	C₄²	C₄×D₁₅₇	1	1	yes	characteristic
108	C₄²	C₂²×D₁₅₇	1	2	yes	—
109	C₂³	C₂×C₄×D₁₅₇	1	1	yes	characteristic
110	Q₈	—	314	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	157	8	no	—
113	C₂×C₄	C₂×C₄×D₁₅₇	1	2	yes	—
114	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
115	C₂×C₄	C₂×C₄×D₁₅₇	1	2	yes	—
116	C₂×C₄	C₂×Dic₃₁₄	1	2	yes	—
117	C₂×C₄	#14578954	1	2	yes	—
118	C₂×C₄	C₂³×D₁₅₇	1	1	yes	characteristic
119	C₈	—	2	2	no	—
120	C₈	—	314	4	no	—
121	C₈	C₂×Dic₃₁₄	1	2	yes	—
122	C₈	C₂×Dic₃₁₄	1	2	yes	—
123	C₂²	—	2	2	no	—
124	C₂²	C₄×Dic₃₁₄	1	2	yes	—
125	C₂²	C₂²×C₄×D₁₅₇	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	314	8	no	—
128	C₄	C₄×Dic₃₁₄	1	2	yes	—
129	C₄	#19097777	1	4	yes	—
130	C₄	C₂²×Dic₃₁₄	1	1	yes	characteristic
131	C₄	C₂×#14578954	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×C₄×Dic₃₁₄	1	1	yes	characteristic
134	C₂	C₂×#19097777	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23757830

C₂

yes

maximal, characteristic

#23757403

C₂

yes

maximal, complemented

C₈×Dic₃₁₄

C₂

yes

maximal, complemented

#23757299

C₂

yes

maximal

#23757297

C₂

yes

maximal, characteristic

#23757228

C₂

yes

maximal

C₂×C₄×C₈×C₁₅₇

C₂

yes

maximal, characteristic

C₂×#14578938

C₂²

yes

—

#19097771

C₄

yes

—

#19097763

C₄

yes

—

#19097750

C₄

yes

—

C₄×Dic₃₁₄

C₂²

yes

—

#19097730

C₂²

yes

—

C₄⋊#11929434

C₂²

yes

—

#19097700

C₂²

yes

—

#19097681

C₂²

yes

—

C₈×Dic₁₅₇

C₂²

yes

—

C₂²×C₈×C₁₅₇

C₂²

yes

characteristic

C₂²×C₈×C₁₅₇

C₂²

yes

characteristic

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

C₄×C₈×C₁₅₇

C₂²

yes

—

C₄×C₈×C₁₅₇

C₂²

yes

—

C₂×Dic₃₁₄

C₂×C₄

yes

—

#14578938

—

C₂×#11929434

C₂³

yes

—

C₄⋊Dic₁₅₇

C₂×C₄

yes

—

C₄⋊Dic₁₅₇

C₂×C₄

yes

—

#14578933

C₂×C₄

yes

—

C₂²⋊Dic₁₅₇

C₂×C₄

yes

—

C₄×Dic₁₅₇

C₂³

yes

—

C₂²×C₄×C₁₅₇

C₂×C₄

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

Q₈

yes

complemented

C₂×C₈×C₁₅₇

C₂³

yes

—

C₂×C₈×C₁₅₇

C₂³

yes

characteristic

C₂×C₈×C₁₅₇

C₂³

yes

characteristic

C₄²×C₁₅₇

C₂×C₄

yes

characteristic

C₄²×C₁₅₇

C₂×C₄

yes

characteristic

C₄²×C₁₅₇

C₂³

yes

—

Dic₃₁₄

—

C₂×Dic₁₅₇

C₂²×C₄

yes

—

#11929434

—

C₂³×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂²×C₄

yes

—

C₂×C₄×C₁₅₇

C₂×Q₈

yes

—

C₂×C₄×C₁₅₇

#41

yes

—

C₂×C₄×C₁₅₇

C₂⁴

yes

characteristic

C₈×C₁₅₇

—

C₈×C₁₅₇

C₂×Q₈

yes

—

C₈×C₁₅₇

C₂×Q₈

yes

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₄×Q₈

yes

—

C₂²×C₁₅₇

C₂³×C₄

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

C₄×Q₈

yes

—

C₄×C₁₅₇

#131

yes

—

C₄×C₁₅₇

C₂²×Q₈

yes

characteristic

C₄×C₁₅₇

C₂×#41

yes

characteristic

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

C₂×C₄×Q₈

yes

characteristic

C₂×C₁₅₇

C₂×#131

yes

—

C₁₅₇

#2960

yes

characteristic, complemented, Sylow

#2960

—

157

maximal, Sylow

#533

—

157

—

#446

—

157

—

C₈×Q₈

—

157

—

#424

—

157

—

#423

—

157

—

#405

—

157

—

C₂×C₄×C₈

D₁₅₇

yes

characteristic

C₂×M₁₆

—

157

—

C₂²×C₈

C₂×D₁₅₇

yes

characteristic

C₂²×C₈

C₂×D₁₅₇

yes

characteristic

C₄⋊Q₈

—

157

—

#125

—

157

—

C₂²⋊Q₈

—

157

—

C₄×Q₈

—

157

—

#117

—

157

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

C₄⋊C₈

—

157

—

C₄⋊C₈

—

157

—

#97

—

157

—

C₄×C₈

—

157

—

C₄×C₈

C₂×D₁₅₇

yes

—

C₄×C₈

C₂×D₁₅₇

yes

—

C₂×Q₈

—

157

—

C₂²×C₄

C₄×D₁₅₇

yes

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

M₁₆

—

314

—

C₂×C₈

—

C₂×C₈

—

157

—

C₂×C₈

Dic₃₁₄

yes

complemented

C₂×C₈

C₂²×D₁₅₇

yes

—

C₂×C₈

C₂²×D₁₅₇

yes

characteristic

100

C₂×C₈

C₂²×D₁₅₇

yes

characteristic

101

C₄⋊C₄

—

157

—

102

C₄⋊C₄

—

157

—

103

C₄⋊C₄

—

157

—

104

C₂²⋊C₄

—

157

—

105

C₄²

—

157

—

106

C₄²

C₄×D₁₅₇

yes

characteristic

107

C₄²

C₄×D₁₅₇

yes

characteristic

108

C₄²

C₂²×D₁₅₇

yes

—

109

C₂³

C₂×C₄×D₁₅₇

yes

characteristic

110

Q₈

—

314

—

111

C₂×C₄

—

112

C₂×C₄

—

157

—

113

C₂×C₄

C₂×C₄×D₁₅₇

yes

—

114

C₂×C₄

C₂×C₄×D₁₅₇

yes

characteristic

115

C₂×C₄

C₂×C₄×D₁₅₇

yes

—

116

C₂×C₄

C₂×Dic₃₁₄

yes

—

117

C₂×C₄

#14578954

yes

—

118

C₂×C₄

C₂³×D₁₅₇

yes

characteristic

119

C₈

—

120

C₈

—

314

—

121

C₈

C₂×Dic₃₁₄

yes

—

122

C₈

C₂×Dic₃₁₄

yes

—

123

C₂²

—

124

C₂²

C₄×Dic₃₁₄

yes

—

125

C₂²

C₂²×C₄×D₁₅₇

yes

characteristic

126

C₄

—

127

C₄

—

314

—

128

C₄

C₄×Dic₃₁₄

yes

—

129

C₄

#19097777

yes

—

130

C₄

C₂²×Dic₃₁₄

yes

characteristic

131

C₄

C₂×#14578954

yes

characteristic

132

C₂

—

133

C₂

C₂×C₄×Dic₃₁₄

yes

characteristic

134

C₂

C₂×#19097777

yes

—

	subgroup	G/H
center	C₂×C₈	C₂²×D₁₅₇
commutator, socle	C₂²×C₁₅₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₅₇
Fitting	C₂×C₄×C₈×C₁₅₇	C₂
prime radical	C₂³×C₁₅₇	C₂²×C₄
cosocle	C₂×C₄×C₁₅₇	C₂⁴
hypercenter	C₂×C₄×C₈	D₁₅₇
hypocenter	C₁₅₇	#2960
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₈

C₂²×D₁₅₇

commutator, socle

C₂²×C₁₅₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₁₅₇

Fitting

C₂×C₄×C₈×C₁₅₇

C₂

prime radical

C₂³×C₁₅₇

C₂²×C₄

cosocle

C₂×C₄×C₁₅₇

C₂⁴

hypercenter

C₂×C₄×C₈

D₁₅₇

C₁₅₇

G ▷ C₂²×C₁₅₇ ▷ C₁₅₇

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #23757403	C₂	semidirect
3. C₈×Dic₃₁₄	C₂	semidirect
34. C₂×C₈×C₁₅₇	Q₈	semidirect
67. C₁₅₇	#2960	semidirect
97. C₂×C₈	Dic₃₁₄	semidirect

H ≅ G/N

type

2. #23757403

C₂

semidirect

3. C₈×Dic₃₁₄

C₂

semidirect

34. C₂×C₈×C₁₅₇

Q₈

semidirect

67. C₁₅₇

#2960

semidirect

97. C₂×C₈

Dic₃₁₄

semidirect

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

subgroup	index
#23757830	2
#23757403	2
C₈×Dic₃₁₄	2
#23757299	2
#23757297	2
#23757228	2
C₂×C₄×C₈×C₁₅₇	2
#2960	157

subgroup

index

C₈×Dic₃₁₄

C₂×C₄×C₈×C₁₅₇

157

Subgroups — #34109070

Conjugacy classes of subgroups

Complements

Maximal subgroups