Group #34101278 — subgroups

Subgroups — #34101278

Order 20096.

290 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 91 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×#19097771	C₂	1	1	yes	maximal, characteristic
2	C₂×#19097696	C₂	1	2	yes	maximal
3	#23757179	C₂	1	1	yes	maximal, characteristic
4	#23757036	C₂	1	2	yes	maximal
5	#336×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₁₄	—	2	1	no	—
7	C₂²×Dic₃₁₄	C₂²	1	1	yes	characteristic
8	#19097771	—	2	4	no	—
9	C₂×C₄⋊Dic₁₅₇	—	2	1	no	—
10	C₂×C₄⋊Dic₁₅₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
12	#19097696	—	2	4	no	—
13	#19097678	—	2	1	no	—
14	#19097678	—	2	1	no	—
15	C₂²×C₈×C₁₅₇	C₂²	1	2	yes	—
16	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₁₄	—	2	4	no	—
18	C₂×Dic₃₁₄	—	2	2	no	—
19	C₂×Dic₃₁₄	—	2	1	no	—
20	C₂×Dic₃₁₄	—	2	2	no	—
21	C₂×Dic₃₁₄	—	4	2	no	—
22	C₂×Dic₃₁₄	—	4	2	no	—
23	C₂×Dic₃₁₄	Q₈	1	2	yes	complemented
24	C₂²×Dic₁₅₇	—	2	1	no	—
25	C₂²×Dic₁₅₇	—	2	2	no	—
26	C₂²×Dic₁₅₇	—	2	1	no	—
27	C₄⋊Dic₁₅₇	—	2	4	no	—
28	C₄⋊Dic₁₅₇	—	2	2	no	—
29	C₄⋊Dic₁₅₇	—	2	2	no	—
30	C₄⋊Dic₁₅₇	D₄	1	4	yes	—
31	C₂²×C₄×C₁₅₇	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₅₇	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₁₅₇	—	2	2	no	—
35	C₂×C₈×C₁₅₇	—	2	2	no	—
36	C₂×C₈×C₁₅₇	—	2	2	no	—
37	C₄²×C₁₅₇	—	2	2	no	—
38	Dic₃₁₄	—	2	4	no	—
39	Dic₃₁₄	—	4	8	no	—
40	Dic₃₁₄	—	4	2	no	—
41	C₂×Dic₁₅₇	—	2	4	no	—
42	C₂×Dic₁₅₇	—	2	2	no	—
43	C₂×Dic₁₅₇	—	4	1	no	—
44	C₂×Dic₁₅₇	—	4	1	no	—
45	C₂×Dic₁₅₇	—	4	2	no	—
46	C₂×Dic₁₅₇	—	4	2	no	—
47	C₂×Dic₁₅₇	—	4	1	no	—
48	C₂×Dic₁₅₇	—	4	1	no	—
49	C₂×Dic₁₅₇	—	4	1	no	—
50	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₅₇	—	2	1	no	—
52	C₂×C₄×C₁₅₇	—	2	1	no	—
53	C₂×C₄×C₁₅₇	—	2	1	no	—
54	C₂×C₄×C₁₅₇	—	2	1	no	—
55	C₂×C₄×C₁₅₇	SD₁₆	1	2	yes	—
56	C₂×C₄×C₁₅₇	Q₁₆	1	2	yes	—
57	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₄×C₁₅₇	#41	1	1	yes	characteristic
60	C₂×C₄×C₁₅₇	#41	1	2	yes	—
61	C₈×C₁₅₇	—	4	2	no	—
62	C₂²×C₁₅₇	#120	1	1	yes	characteristic
63	C₂²×C₁₅₇	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂²×C₁₅₇	#124	1	1	yes	characteristic
65	C₂²×C₁₅₇	C₂×SD₁₆	1	1	yes	characteristic
66	C₂²×C₁₅₇	C₂×Q₁₆	1	1	yes	characteristic
67	C₂²×C₁₅₇	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₂²×C₁₅₇	#137	1	1	yes	characteristic
69	C₄×C₁₅₇	—	2	2	no	—
70	C₄×C₁₅₇	—	2	2	no	—
71	C₄×C₁₅₇	C₂²⋊Q₈	1	2	yes	—
72	C₄×C₁₅₇	#124	1	2	yes	—
73	Dic₁₅₇	—	4	2	no	—
74	Dic₁₅₇	—	4	4	no	—
75	Dic₁₅₇	—	8	1	no	—
76	C₂×C₁₅₇	#393	1	1	yes	characteristic
77	C₂×C₁₅₇	#450	1	1	yes	characteristic
78	C₂×C₁₅₇	#451	1	1	yes	characteristic
79	C₂×C₁₅₇	#475	1	1	yes	characteristic
80	C₂×C₁₅₇	#477	1	1	yes	characteristic
81	C₂×C₁₅₇	#487	1	1	yes	characteristic
82	C₂×C₁₅₇	#489	1	1	yes	characteristic
83	C₁₅₇	#2024	1	1	yes	characteristic, complemented, Sylow
84	#2024	—	157	1	no	maximal, Sylow
85	C₂×C₄⋊Q₈	—	157	1	no	—
86	C₂×#103	—	157	2	no	—
87	#391	—	157	1	no	—
88	#340	—	157	2	no	—
89	#336	D₁₅₇	1	1	yes	characteristic
90	C₂²×Q₈	—	314	1	no	—
91	C₂²×Q₈	—	157	1	no	—
92	C₂²×C₈	C₂×D₁₅₇	1	2	yes	—
93	C₄⋊Q₈	—	314	4	no	—
94	C₂×C₄⋊C₄	—	314	1	no	—
95	C₂×C₄⋊C₄	—	157	1	no	—
96	C₂×C₄⋊C₄	—	157	2	no	—
97	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
98	#103	—	314	4	no	—
99	#95	—	314	1	no	—
100	#95	—	314	1	no	—
101	C₂×Q₈	—	314	4	no	—
102	C₂×Q₈	—	314	1	no	—
103	C₂×Q₈	—	314	2	no	—
104	C₂×Q₈	—	314	2	no	—
105	C₂×Q₈	—	628	2	no	—
106	C₂×Q₈	—	628	2	no	—
107	C₂×Q₈	—	157	2	no	—
108	C₂²×C₄	—	314	1	no	—
109	C₂²×C₄	—	314	2	no	—
110	C₂²×C₄	—	314	1	no	—
111	C₂²×C₄	D₆₂₈	1	1	yes	characteristic
112	C₂²×C₄	D₆₂₈	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₄⋊C₄	—	314	4	no	—
118	C₄⋊C₄	—	314	2	no	—
119	C₄⋊C₄	—	314	2	no	—
120	C₄⋊C₄	—	157	4	no	—
121	C₄²	—	2	2	no	—
122	C₂³	C₂×D₆₂₈	1	1	yes	characteristic
123	Q₈	—	314	4	no	—
124	Q₈	—	628	8	no	—
125	Q₈	—	628	2	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	314	4	no	—
131	C₂×C₄	—	314	2	no	—
132	C₂×C₄	—	628	1	no	—
133	C₂×C₄	—	628	1	no	—
134	C₂×C₄	—	628	2	no	—
135	C₂×C₄	—	628	2	no	—
136	C₂×C₄	—	628	1	no	—
137	C₂×C₄	—	628	1	no	—
138	C₂×C₄	—	628	1	no	—
139	C₂×C₄	#14578943	1	2	yes	—
140	C₂×C₄	Dic₆₂₈	1	2	yes	—
141	C₂×C₄	D₄×D₁₅₇	1	2	yes	—
142	C₂×C₄	Q₈×D₁₅₇	1	1	yes	characteristic
143	C₂×C₄	#14578953	1	1	yes	characteristic
144	C₂×C₄	#14578954	1	2	yes	—
145	C₈	—	4	2	no	—
146	C₂²	#19097741	1	1	yes	characteristic
147	C₂²	#19097751	1	1	yes	characteristic
148	C₂²	#19097759	1	1	yes	characteristic
149	C₂²	C₂×#14578943	1	1	yes	characteristic
150	C₂²	C₂×Dic₆₂₈	1	1	yes	characteristic
151	C₂²	#19097799	1	1	yes	characteristic
152	C₂²	#19097804	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	628	4	no	—
156	C₄	—	628	2	no	—
157	C₄	—	1256	1	no	—
158	C₄	#19097749	1	2	yes	—
159	C₄	#19097757	1	2	yes	—
160	C₂	#23757186	1	1	yes	characteristic
161	C₂	#23757426	1	1	yes	characteristic
162	C₂	#23757431	1	1	yes	characteristic
163	C₂	#23757559	1	1	yes	characteristic
164	C₂	#23757569	1	1	yes	characteristic
165	C₂	#23757624	1	1	yes	characteristic
166	C₂	#23757633	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19097771

C₂

yes

maximal, characteristic

C₂×#19097696

C₂

yes

maximal

#23757179

C₂

yes

maximal, characteristic

#23757036

C₂

yes

maximal

#336×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×Dic₃₁₄

—

C₂²×Dic₃₁₄

C₂²

yes

characteristic

#19097771

—

C₂×C₄⋊Dic₁₅₇

—

C₂×C₄⋊Dic₁₅₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₅₇

C₂²

yes

—

#19097696

—

#19097678

—

#19097678

—

C₂²×C₈×C₁₅₇

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

Q₈

yes

complemented

C₂²×Dic₁₅₇

—

C₂²×Dic₁₅₇

—

C₂²×Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

D₄

yes

—

C₂²×C₄×C₁₅₇

D₄

yes

characteristic

C₂²×C₄×C₁₅₇

D₄

yes

characteristic

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₄²×C₁₅₇

—

Dic₃₁₄

—

Dic₃₁₄

—

Dic₃₁₄

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂³×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

SD₁₆

yes

—

C₂×C₄×C₁₅₇

Q₁₆

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×Q₈

yes

C₂×C₄×C₁₅₇

yes

C₂×C₄×C₁₅₇

yes

—

C₈×C₁₅₇

—

C₂²×C₁₅₇

#120

yes

characteristic

C₂²×C₁₅₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₅₇

#124

yes

characteristic

C₂²×C₁₅₇

C₂×SD₁₆

yes

characteristic

C₂²×C₁₅₇

C₂×Q₁₆

yes

C₂²×C₁₅₇

yes

C₂²×C₁₅₇

yes

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂²⋊Q₈

yes

—

C₄×C₁₅₇

#124

yes

—

Dic₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₁₅₇

yes

characteristic, complemented, Sylow

#2024

—

157

maximal, Sylow

C₂×C₄⋊Q₈

—

157

—

C₂×#103

—

157

—

#391

—

157

—

#340

—

157

—

#336

D₁₅₇

yes

characteristic

C₂²×Q₈

—

314

—

C₂²×Q₈

—

157

—

C₂²×C₈

C₂×D₁₅₇

yes

—

C₄⋊Q₈

—

314

—

C₂×C₄⋊C₄

—

314

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

#103

—

314

—

#95

—

314

—

100

#95

—

314

—

101

C₂×Q₈

—

314

—

102

C₂×Q₈

—

314

—

103

C₂×Q₈

—

314

—

104

C₂×Q₈

—

314

—

105

C₂×Q₈

—

628

—

106

C₂×Q₈

—

628

—

107

C₂×Q₈

—

157

—

108

C₂²×C₄

—

314

—

109

C₂²×C₄

—

314

—

110

C₂²×C₄

—

314

—

111

C₂²×C₄

D₆₂₈

yes

characteristic

112

C₂²×C₄

D₆₂₈

yes

characteristic

113

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₄⋊C₄

—

314

—

118

C₄⋊C₄

—

314

—

119

C₄⋊C₄

—

314

—

120

C₄⋊C₄

—

157

—

121

C₄²

—

122

C₂³

C₂×D₆₂₈

yes

characteristic

123

Q₈

—

314

—

124

Q₈

—

628

—

125

Q₈

—

628

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

314

—

131

C₂×C₄

—

314

—

132

C₂×C₄

—

628

—

133

C₂×C₄

—

628

—

134

C₂×C₄

—

628

—

135

C₂×C₄

—

628

—

136

C₂×C₄

—

628

—

137

C₂×C₄

—

628

—

138

C₂×C₄

—

628

—

139

C₂×C₄

#14578943

yes

—

140

C₂×C₄

Dic₆₂₈

yes

—

141

C₂×C₄

D₄×D₁₅₇

yes

—

142

C₂×C₄

Q₈×D₁₅₇

yes

characteristic

143

C₂×C₄

#14578953

yes

characteristic

144

C₂×C₄

#14578954

yes

—

145

C₈

—

146

C₂²

#19097741

yes

characteristic

147

C₂²

#19097751

yes

characteristic

148

C₂²

#19097759

yes

characteristic

149

C₂²

C₂×#14578943

yes

characteristic

150

C₂²

C₂×Dic₆₂₈

yes

characteristic

151

C₂²

#19097799

yes

characteristic

152

C₂²

#19097804

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

628

—

156

C₄

—

628

—

157

C₄

—

1256

—

158

C₄

#19097749

yes

—

159

C₄

#19097757

yes

—

160

C₂

#23757186

yes

characteristic

161

C₂

#23757426

yes

characteristic

162

C₂

#23757431

yes

characteristic

163

C₂

#23757559

yes

characteristic

164

C₂

#23757569

yes

characteristic

165

C₂

#23757624

yes

characteristic

166

C₂

#23757633

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₆₂₈
commutator, cosocle	C₂²×C₄×C₁₅₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	#336×C₁₅₇	C₂
socle, prime radical	C₂³×C₁₅₇	C₂×D₄
hypercenter	#336	D₁₅₇
hypocenter	C₁₅₇	#2024
LCS	G ▷ C₂²×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₆₂₈

commutator, cosocle

C₂²×C₄×C₁₅₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

#336×C₁₅₇

C₂

socle, prime radical

C₂³×C₁₅₇

C₂×D₄

D₁₅₇

C₁₅₇

G ▷ C₂²×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
23. C₂×Dic₃₁₄	Q₈	semidirect
83. C₁₅₇	#2024	semidirect

H ≅ G/N

type

23. C₂×Dic₃₁₄

Q₈

semidirect

83. C₁₅₇

#2024

semidirect

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

subgroup	index
C₂×#19097771	2
C₂×#19097696	2
#23757179	2
#23757036	2
#336×C₁₅₇	2
#2024	157

subgroup

index

#336×C₁₅₇

157

Subgroups — #34101278

Conjugacy classes of subgroups

Complements

Maximal subgroups