Group #34079564 — subgroups

Subgroups — #34079564

Order 19328.

552 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 145 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×#18012768	C₂	1	4	yes	maximal, complemented
2	#23752760	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Q₈×D₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#23752603	C₂	1	4	yes	maximal, complemented
5	#23752592	C₂	1	1	yes	maximal, characteristic, complemented
6	#23752591	C₂	1	1	yes	maximal, characteristic, complemented
7	#23752584	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#14558987	C₂	1	1	yes	maximal, characteristic, complemented
9	#492×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14559003	C₂²	1	2	yes	—
11	C₂×Q₈×D₁₅₁	C₂²	1	2	yes	—
12	#18012768	—	2	16	no	—
13	C₂×Dic₆₀₄	C₂²	1	4	yes	—
14	C₂×#14558994	C₂²	1	4	yes	—
15	C₂×#14558992	C₂²	1	4	yes	—
16	C₂×#14558987	C₂²	1	2	yes	—
17	#18012739	C₂²	1	1	yes	characteristic
18	#18012739	C₂²	1	1	yes	characteristic
19	#18012738	—	2	2	no	—
20	#18012737	C₂²	1	2	yes	—
21	#18012726	C₂²	1	2	yes	—
22	#18012710	—	2	2	no	—
23	D₄×Dic₁₅₁	—	2	2	no	—
24	C₄⋊C₄×D₁₅₁	—	2	2	no	—
25	C₄²×D₁₅₁	C₂²	1	1	yes	characteristic
26	#18012649	C₂²	1	2	yes	—
27	C₄×C₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×SD₁₆×C₁₅₁	C₂²	1	4	yes	—
29	C₄⋊Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
30	#127×C₁₅₁	C₂²	1	1	yes	characteristic
31	C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
32	#14559003	—	2	8	no	—
33	C₂×Dic₃₀₂	—	2	2	no	—
34	C₂×Dic₃₀₂	C₂³	1	2	yes	—
35	C₂×Dic₃₀₂	C₂³	1	2	yes	—
36	Q₈×D₁₅₁	—	2	8	no	—
37	C₂×C₂²⋊D₁₅₁	—	2	2	no	—
38	C₂²×Dic₁₅₁	—	2	2	no	—
39	C₂×C₄×D₁₅₁	—	2	2	no	—
40	C₂×C₄×D₁₅₁	D₄	1	2	yes	—
41	C₂×C₄×D₁₅₁	C₂³	1	1	yes	characteristic
42	Dic₆₀₄	—	2	8	no	—
43	#14558994	—	2	8	no	—
44	#14558992	—	2	8	no	—
45	#14558987	D₄	1	8	yes	complemented
46	C₂×C₁₅₁⋊C₈	C₂³	1	2	yes	—
47	C₄⋊Dic₁₅₁	—	2	2	no	—
48	C₄⋊Dic₁₅₁	—	2	2	no	—
49	Dic₁₅₁⋊C₄	—	2	4	no	—
50	C₂²⋊Dic₁₅₁	—	2	4	no	—
51	C₄×Dic₁₅₁	D₄	1	2	yes	—
52	C₄×Dic₁₅₁	C₂³	1	1	yes	characteristic
53	C₂×Q₈×C₁₅₁	C₂³	1	2	yes	—
54	C₂×D₄×C₁₅₁	—	2	2	no	—
55	C₂×D₄×C₁₅₁	C₂³	1	2	yes	—
56	SD₁₆×C₁₅₁	—	2	8	no	—
57	C₂×C₈×C₁₅₁	C₂³	1	2	yes	—
58	C₄⋊C₄×C₁₅₁	—	2	2	no	—
59	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
60	C₂²×D₁₅₁	C₂×D₄	1	1	yes	characteristic
61	Dic₃₀₂	—	2	4	no	—
62	Dic₃₀₂	—	2	4	no	—
63	Dic₃₀₂	—	4	4	no	—
64	C₂²⋊D₁₅₁	—	4	4	no	—
65	C₂×Dic₁₅₁	—	2	2	no	—
66	C₂×Dic₁₅₁	—	2	2	no	—
67	C₂×Dic₁₅₁	—	4	4	no	—
68	C₂×Dic₁₅₁	C₂×D₄	1	2	yes	—
69	C₂×Dic₁₅₁	C₂×D₄	1	1	yes	characteristic
70	C₄×D₁₅₁	—	2	4	no	—
71	C₄×D₁₅₁	—	4	4	no	—
72	C₄×D₁₅₁	C₂×D₄	1	4	yes	—
73	C₁₅₁⋊C₈	C₂×D₄	1	4	yes	—
74	C₂³×C₁₅₁	—	2	2	no	—
75	Q₈×C₁₅₁	—	2	4	no	—
76	D₄×C₁₅₁	—	2	4	no	—
77	D₄×C₁₅₁	—	4	4	no	—
78	C₂×C₄×C₁₅₁	—	2	2	no	—
79	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
81	C₈×C₁₅₁	C₂×D₄	1	4	yes	—
82	C₂²×C₁₅₁	—	4	2	no	—
83	C₂²×C₁₅₁	—	4	4	no	—
84	C₂²×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
85	C₂×D₁₅₁	—	2	2	no	—
86	C₂×D₁₅₁	#127	1	2	yes	—
87	C₄×C₁₅₁	—	4	2	no	—
88	C₄×C₁₅₁	#137	1	4	yes	complemented
89	C₄×C₁₅₁	C₂²×D₄	1	2	yes	—
90	Dic₁₅₁	—	2	2	no	—
91	Dic₁₅₁	—	4	2	no	—
92	Dic₁₅₁	—	4	2	no	—
93	Dic₁₅₁	#127	1	2	yes	—
94	C₂×C₁₅₁	—	8	2	no	—
95	C₂×C₁₅₁	C₂×#127	1	1	yes	characteristic
96	C₂×C₁₅₁	C₂×#137	1	2	yes	—
97	D₁₅₁	—	2	2	no	—
98	C₁₅₁	#3152	1	1	yes	characteristic, complemented, Sylow
99	#3152	—	151	1	no	maximal, Sylow
100	C₂×#137	—	151	4	no	—
101	#532	—	151	1	no	—
102	C₂×C₄⋊Q₈	—	151	1	no	—
103	#497	—	151	4	no	—
104	#494	—	151	1	no	—
105	#494	—	151	1	no	—
106	#492	—	151	1	no	—
107	#492	D₁₅₁	1	1	yes	characteristic, complemented
108	C₄×M₁₆	—	151	1	no	—
109	C₂×#41	—	151	2	no	—
110	C₂²×Q₈	—	151	2	no	—
111	#137	—	302	16	no	—
112	C₂×Q₁₆	—	151	4	no	—
113	C₂×Q₁₆	—	151	4	no	—
114	C₂×SD₁₆	—	151	4	no	—
115	C₂×SD₁₆	C₂×D₁₅₁	1	4	yes	—
116	C₂×M₁₆	—	151	2	no	—
117	C₄⋊Q₈	—	302	2	no	—
118	C₄⋊Q₈	—	151	2	no	—
119	C₄⋊Q₈	—	151	1	no	—
120	C₄⋊Q₈	—	151	1	no	—
121	C₄⋊Q₈	C₂×D₁₅₁	1	1	yes	characteristic
122	#127	C₂×D₁₅₁	1	1	yes	characteristic
123	#124	—	151	2	no	—
124	#121	—	302	2	no	—
125	C₄×D₄	—	302	2	no	—
126	C₂×C₄⋊C₄	—	302	2	no	—
127	C₂×C₄²	—	151	1	no	—
128	#97	—	151	2	no	—
129	C₄×C₈	—	151	1	no	—
130	C₄×C₈	C₂×D₁₅₁	1	1	yes	characteristic
131	#41	—	302	8	no	—
132	C₂×Q₈	—	302	2	no	—
133	C₂×Q₈	—	302	8	no	—
134	C₂×Q₈	—	151	2	no	—
135	C₂×Q₈	—	151	2	no	—
136	C₂×Q₈	C₂²×D₁₅₁	1	2	yes	—
137	C₂×D₄	—	2	2	no	—
138	C₂×D₄	—	302	2	no	—
139	C₂×D₄	C₂²×D₁₅₁	1	2	yes	—
140	C₂²×C₄	—	302	2	no	—
141	C₂²×C₄	—	302	2	no	—
142	C₂²×C₄	—	151	2	no	—
143	C₂²×C₄	—	151	1	no	—
144	Q₁₆	—	302	8	no	—
145	Q₁₆	—	302	8	no	—
146	SD₁₆	—	2	8	no	—
147	SD₁₆	—	302	8	no	—
148	M₁₆	—	151	8	no	—
149	C₂×C₈	—	151	2	no	—
150	C₂×C₈	C₂²×D₁₅₁	1	2	yes	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	302	4	no	—
153	C₄⋊C₄	—	302	2	no	—
154	C₄⋊C₄	—	302	2	no	—
155	C₂²⋊C₄	—	302	4	no	—
156	C₄²	—	151	2	no	—
157	C₄²	—	151	1	no	—
158	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
159	C₂³	—	2	2	no	—
160	C₂³	—	151	1	no	—
161	Q₈	—	2	4	no	—
162	Q₈	—	302	4	no	—
163	Q₈	—	302	4	no	—
164	Q₈	—	604	4	no	—
165	D₄	—	2	4	no	—
166	D₄	—	4	4	no	—
167	D₄	—	604	4	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	302	4	no	—
170	C₂×C₄	—	302	2	no	—
171	C₂×C₄	—	302	2	no	—
172	C₂×C₄	—	604	4	no	—
173	C₂×C₄	—	604	4	no	—
174	C₂×C₄	—	151	2	no	—
175	C₂×C₄	—	151	1	no	—
176	C₂×C₄	—	151	4	no	—
177	C₂×C₄	D₄×D₁₅₁	1	2	yes	—
178	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
179	C₈	—	151	4	no	—
180	C₈	D₄×D₁₅₁	1	4	yes	—
181	C₂²	—	4	4	no	—
182	C₂²	—	4	2	no	—
183	C₂²	—	302	2	no	—
184	C₂²	—	151	2	no	—
185	C₂²	C₂×D₄×D₁₅₁	1	1	yes	characteristic
186	C₄	—	4	2	no	—
187	C₄	—	302	2	no	—
188	C₄	—	604	2	no	—
189	C₄	—	604	2	no	—
190	C₄	—	151	2	no	—
191	C₄	#18012768	1	4	yes	complemented
192	C₄	C₂×D₄×D₁₅₁	1	2	yes	—
193	C₂	—	8	2	no	—
194	C₂	—	302	2	no	—
195	C₂	#127×D₁₅₁	1	1	yes	characteristic
196	C₂	C₂×#18012768	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18012768

C₂

yes

maximal, complemented

#23752760

C₂

yes

maximal, characteristic

C₄⋊Q₈×D₁₅₁

C₂

yes

maximal, characteristic, complemented

#23752603

C₂

yes

maximal, complemented

#23752592

C₂

yes

maximal, characteristic, complemented

#23752591

C₂

yes

maximal, characteristic, complemented

#23752584

C₂

yes

maximal, characteristic, complemented

C₄×#14558987

C₂

yes

maximal, characteristic, complemented

#492×C₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂×#14559003

C₂²

yes

—

C₂×Q₈×D₁₅₁

C₂²

yes

—

#18012768

—

C₂×Dic₆₀₄

C₂²

yes

—

C₂×#14558994

C₂²

yes

—

C₂×#14558992

C₂²

yes

—

C₂×#14558987

C₂²

yes

—

#18012739

C₂²

yes

characteristic

#18012739

C₂²

yes

characteristic

#18012738

—

#18012737

C₂²

yes

—

#18012726

C₂²

yes

—

#18012710

—

D₄×Dic₁₅₁

—

C₄⋊C₄×D₁₅₁

—

C₄²×D₁₅₁

C₂²

yes

characteristic

#18012649

C₂²

yes

—

C₄×C₁₅₁⋊C₈

C₂²

yes

characteristic

C₂×SD₁₆×C₁₅₁

C₂²

yes

—

C₄⋊Q₈×C₁₅₁

C₂²

yes

characteristic

#127×C₁₅₁

C₂²

yes

characteristic

C₄×C₈×C₁₅₁

C₂²

yes

characteristic

#14559003

—

C₂×Dic₃₀₂

—

C₂×Dic₃₀₂

C₂³

yes

—

C₂×Dic₃₀₂

C₂³

yes

—

Q₈×D₁₅₁

—

C₂×C₂²⋊D₁₅₁

—

C₂²×Dic₁₅₁

—

C₂×C₄×D₁₅₁

—

C₂×C₄×D₁₅₁

D₄

yes

—

C₂×C₄×D₁₅₁

C₂³

yes

characteristic

Dic₆₀₄

—

#14558994

—

#14558992

—

#14558987

D₄

yes

complemented

C₂×C₁₅₁⋊C₈

C₂³

yes

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

—

Dic₁₅₁⋊C₄

—

C₂²⋊Dic₁₅₁

—

C₄×Dic₁₅₁

D₄

yes

—

C₄×Dic₁₅₁

C₂³

yes

characteristic

C₂×Q₈×C₁₅₁

C₂³

yes

—

C₂×D₄×C₁₅₁

—

C₂×D₄×C₁₅₁

C₂³

yes

—

SD₁₆×C₁₅₁

—

C₂×C₈×C₁₅₁

C₂³

yes

—

C₄⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

C₂³

yes

characteristic

C₂²×D₁₅₁

C₂×D₄

yes

characteristic

Dic₃₀₂

—

Dic₃₀₂

—

Dic₃₀₂

—

C₂²⋊D₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

C₂×D₄

yes

—

C₂×Dic₁₅₁

C₂×D₄

yes

characteristic

C₄×D₁₅₁

—

C₄×D₁₅₁

—

C₄×D₁₅₁

C₂×D₄

yes

—

C₁₅₁⋊C₈

C₂×D₄

yes

—

C₂³×C₁₅₁

—

Q₈×C₁₅₁

—

D₄×C₁₅₁

—

D₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

—

C₂×C₄×C₁₅₁

C₂⁴

yes

characteristic

C₈×C₁₅₁

C₂×D₄

yes

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

C₂²×D₄

yes

characteristic

C₂×D₁₅₁

—

C₂×D₁₅₁

#127

yes

—

C₄×C₁₅₁

—

C₄×C₁₅₁

#137

yes

complemented

C₄×C₁₅₁

C₂²×D₄

yes

—

Dic₁₅₁

—

Dic₁₅₁

—

Dic₁₅₁

—

Dic₁₅₁

#127

yes

—

C₂×C₁₅₁

—

C₂×C₁₅₁

C₂×#127

yes

characteristic

C₂×C₁₅₁

C₂×#137

yes

—

D₁₅₁

—

C₁₅₁

#3152

yes

characteristic, complemented, Sylow

#3152

—

151

maximal, Sylow

100

C₂×#137

—

151

—

101

#532

—

151

—

102

C₂×C₄⋊Q₈

—

151

—

103

#497

—

151

—

104

#494

—

151

—

105

#494

—

151

—

106

#492

—

151

—

107

#492

D₁₅₁

yes

characteristic, complemented

108

C₄×M₁₆

—

151

—

109

C₂×#41

—

151

—

110

C₂²×Q₈

—

151

—

111

#137

—

302

—

112

C₂×Q₁₆

—

151

—

113

C₂×Q₁₆

—

151

—

114

C₂×SD₁₆

—

151

—

115

C₂×SD₁₆

C₂×D₁₅₁

yes

—

116

C₂×M₁₆

—

151

—

117

C₄⋊Q₈

—

302

—

118

C₄⋊Q₈

—

151

—

119

C₄⋊Q₈

—

151

—

120

C₄⋊Q₈

—

151

—

121

C₄⋊Q₈

C₂×D₁₅₁

yes

characteristic

122

#127

C₂×D₁₅₁

yes

characteristic

123

#124

—

151

—

124

#121

—

302

—

125

C₄×D₄

—

302

—

126

C₂×C₄⋊C₄

—

302

—

127

C₂×C₄²

—

151

—

128

#97

—

151

—

129

C₄×C₈

—

151

—

130

C₄×C₈

C₂×D₁₅₁

yes

characteristic

131

#41

—

302

—

132

C₂×Q₈

—

302

—

133

C₂×Q₈

—

302

—

134

C₂×Q₈

—

151

—

135

C₂×Q₈

—

151

—

136

C₂×Q₈

C₂²×D₁₅₁

yes

—

137

C₂×D₄

—

138

C₂×D₄

—

302

—

139

C₂×D₄

C₂²×D₁₅₁

yes

—

140

C₂²×C₄

—

302

—

141

C₂²×C₄

—

302

—

142

C₂²×C₄

—

151

—

143

C₂²×C₄

—

151

—

144

Q₁₆

—

302

—

145

Q₁₆

—

302

—

146

SD₁₆

—

147

SD₁₆

—

302

—

148

M₁₆

—

151

—

149

C₂×C₈

—

151

—

150

C₂×C₈

C₂²×D₁₅₁

yes

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

302

—

153

C₄⋊C₄

—

302

—

154

C₄⋊C₄

—

302

—

155

C₂²⋊C₄

—

302

—

156

C₄²

—

151

—

157

C₄²

—

151

—

158

C₄²

C₂²×D₁₅₁

yes

characteristic

159

C₂³

—

160

C₂³

—

151

—

161

Q₈

—

162

Q₈

—

302

—

163

Q₈

—

302

—

164

Q₈

—

604

—

165

D₄

—

166

D₄

—

167

D₄

—

604

—

168

C₂×C₄

—

169

C₂×C₄

—

302

—

170

C₂×C₄

—

302

—

171

C₂×C₄

—

302

—

172

C₂×C₄

—

604

—

173

C₂×C₄

—

604

—

174

C₂×C₄

—

151

—

175

C₂×C₄

—

151

—

176

C₂×C₄

—

151

—

177

C₂×C₄

D₄×D₁₅₁

yes

—

178

C₂×C₄

C₂³×D₁₅₁

yes

characteristic

179

C₈

—

151

—

180

C₈

D₄×D₁₅₁

yes

—

181

C₂²

—

182

C₂²

—

183

C₂²

—

302

—

184

C₂²

—

151

—

185

C₂²

C₂×D₄×D₁₅₁

yes

characteristic

186

C₄

—

187

C₄

—

302

—

188

C₄

—

604

—

189

C₄

—

604

—

190

C₄

—

151

—

191

C₄

#18012768

yes

complemented

192

C₄

C₂×D₄×D₁₅₁

yes

—

193

C₂

—

194

C₂

—

302

—

195

C₂

#127×D₁₅₁

yes

characteristic

196

C₂

C₂×#18012768

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₅₁
commutator, cosocle	C₂×C₄×C₁₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₁
Fitting	#492×C₁₅₁	C₂
socle	C₂²×C₁₅₁	C₂²×D₄
prime radical	#23752760	C₂
hypercenter	#492	D₁₅₁
hypocenter	C₁₅₁	#3152
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

C₂×D₄×D₁₅₁

commutator, cosocle

C₂×C₄×C₁₅₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₅₁

Fitting

#492×C₁₅₁

C₂

socle

C₂²×C₁₅₁

C₂²×D₄

C₂

D₁₅₁

C₁₅₁

G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. C₂×#18012768	C₂	semidirect
3. C₄⋊Q₈×D₁₅₁	C₂	semidirect
4. #23752603	C₂	semidirect
5. #23752592	C₂	semidirect
6. #23752591	C₂	semidirect
7. #23752584	C₂	semidirect
8. C₄×#14558987	C₂	semidirect
9. #492×C₁₅₁	C₂	semidirect
45. #14558987	D₄	semidirect
88. C₄×C₁₅₁	#137	semidirect
98. C₁₅₁	#3152	semidirect
107. #492	D₁₅₁	semidirect
191. C₄	#18012768	semidirect

H ≅ G/N

type

1. C₂×#18012768

C₂

semidirect

3. C₄⋊Q₈×D₁₅₁

C₂

semidirect

4. #23752603

C₂

semidirect

5. #23752592

C₂

semidirect

6. #23752591

C₂

semidirect

7. #23752584

C₂

semidirect

8. C₄×#14558987

C₂

semidirect

9. #492×C₁₅₁

C₂

semidirect

45. #14558987

D₄

semidirect

88. C₄×C₁₅₁

#137

semidirect

98. C₁₅₁

#3152

semidirect

107. #492

D₁₅₁

semidirect

191. C₄

#18012768

semidirect

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

subgroup	index
C₂×#18012768	2
#23752760	2
C₄⋊Q₈×D₁₅₁	2
#23752603	2
#23752592	2
#23752591	2
#23752584	2
C₄×#14558987	2
#492×C₁₅₁	2
#3152	151

subgroup

index

C₄⋊Q₈×D₁₅₁

#492×C₁₅₁

151

Subgroups — #34079564

Conjugacy classes of subgroups

Complements

Maximal subgroups