Group #31916001 — subgroups

Subgroups — #31916001

Order 18176.

316 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 115 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	#22597272	C₂	1	1	yes	maximal, characteristic
2	#22590106	C₂	1	1	yes	maximal, characteristic
3	#22586427	C₂	1	2	yes	maximal
4	#22586421	C₂	1	2	yes	maximal
5	C₇₁×#2110	C₂	1	1	yes	maximal, characteristic
6	#17986552	C₄	1	1	yes	characteristic
7	C₂×#14556068	C₂²	1	2	yes	—
8	#17986127	—	2	2	no	—
9	C₈×Dic₁₄₂	—	2	2	no	—
10	#17986027	C₄	1	1	yes	characteristic
11	#17986026	C₄	1	1	yes	characteristic
12	#17986022	C₄	1	1	yes	characteristic
13	#17985954	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
15	#17985820	—	2	2	no	—
16	C₂×M₃₂×C₇₁	C₂²	1	2	yes	—
17	#405×C₇₁	C₂²	1	1	yes	characteristic
18	#347×C₇₁	—	2	2	no	—
19	C₂×#11765849	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
21	#14556127	—	2	1	no	—
22	#14556119	—	2	1	no	—
23	#14556105	—	2	2	no	—
24	C₄×Dic₁₄₂	—	2	2	no	—
25	#14556087	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
27	#14556068	—	2	2	no	—
28	#14556068	—	2	2	no	—
29	C₂×C₇₁⋊C₁₆	—	2	2	no	—
30	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
31	Dic₇₁⋊C₈	—	2	2	no	—
32	Dic₇₁⋊C₈	—	2	2	no	—
33	#14556038	—	2	1	no	—
34	C₈×Dic₇₁	—	2	1	no	—
35	C₈×Dic₇₁	—	2	1	no	—
36	C₈×Dic₇₁	—	2	1	no	—
37	C₂×M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
39	#117×C₇₁	C₂×C₄	1	1	yes	characteristic
40	M₃₂×C₇₁	—	2	2	no	—
41	M₃₂×C₇₁	Q₈	1	4	yes	complemented
42	C₂×C₁₆×C₇₁	—	2	2	no	—
43	#97×C₇₁	D₄	1	2	yes	—
44	C₄×C₈×C₇₁	D₄	1	2	yes	—
45	C₂×Dic₁₄₂	—	2	2	no	—
46	C₂²×Dic₇₁	—	2	1	no	—
47	#11765849	—	2	2	no	—
48	C₂×C₇₁⋊C₈	—	2	1	no	—
49	C₂×C₇₁⋊C₈	—	2	1	no	—
50	C₂×C₇₁⋊C₈	—	2	2	no	—
51	C₄⋊Dic₇₁	—	2	2	no	—
52	C₄⋊Dic₇₁	—	2	1	no	—
53	Dic₇₁⋊C₄	—	2	2	no	—
54	Dic₇₁⋊C₄	—	2	2	no	—
55	C₂²⋊Dic₇₁	M₁₆	1	2	yes	—
56	C₄×Dic₇₁	—	2	1	no	—
57	C₄×Dic₇₁	—	2	1	no	—
58	C₄×Dic₇₁	M₁₆	1	1	yes	characteristic
59	C₄×Dic₇₁	M₁₆	1	1	yes	characteristic
60	C₇₁⋊C₁₆	—	4	2	no	—
61	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
62	M₁₆×C₇₁	—	2	2	no	—
63	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
67	C₂×C₈×C₇₁	#41	1	2	yes	—
68	C₄⋊C₄×C₇₁	—	2	1	no	—
69	C₂²⋊C₄×C₇₁	—	2	1	no	—
70	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
71	C₁₆×C₇₁	—	2	4	no	—
72	C₂³×C₇₁	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₇₁	—	2	2	no	—
74	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
76	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₇₁	#131	1	2	yes	—
78	C₂×Dic₇₁	—	2	1	no	—
79	C₂×Dic₇₁	—	2	1	no	—
80	C₂×Dic₇₁	—	2	2	no	—
81	C₂×Dic₇₁	—	4	2	no	—
82	Dic₁₄₂	—	4	4	no	—
83	C₈×C₇₁	—	2	2	no	—
84	C₈×C₇₁	C₂²⋊Q₈	1	2	yes	—
85	C₈×C₇₁	#124	1	1	yes	characteristic
86	C₈×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
87	C₇₁⋊C₈	—	4	1	no	—
88	C₇₁⋊C₈	—	4	1	no	—
89	C₂²×C₇₁	—	4	1	no	—
90	C₂²×C₇₁	#407	1	1	yes	characteristic
91	C₂²×C₇₁	#446	1	2	yes	—
92	C₄×C₇₁	—	4	2	no	—
93	C₄×C₇₁	#391	1	1	yes	characteristic
94	C₄×C₇₁	#408	1	1	yes	characteristic
95	C₄×C₇₁	#446	1	2	yes	—
96	Dic₇₁	—	2	2	no	—
97	Dic₇₁	—	4	2	no	—
98	Dic₇₁	—	4	1	no	—
99	#13270	—	71	1	no	maximal, Sylow
100	C₂×C₇₁	—	2	1	no	—
101	C₂×C₇₁	—	2	2	no	—
102	C₂×C₇₁	#1985	1	1	yes	characteristic
103	#2960	—	71	1	no	—
104	#2153	—	71	1	no	—
105	#2110	D₇₁	1	1	yes	characteristic
106	#1378	—	71	2	no	—
107	#1376	—	71	2	no	—
108	C₇₁	#13270	1	1	yes	characteristic, complemented, Sylow
109	#533	—	71	1	no	—
110	C₂×M₃₂	—	71	2	no	—
111	C₂×M₃₂	C₂×D₇₁	1	2	yes	—
112	#446	—	142	2	no	—
113	C₈×Q₈	—	142	2	no	—
114	#424	—	71	1	no	—
115	#424	—	71	1	no	—
116	#423	—	71	1	no	—
117	#405	—	71	1	no	—
118	#405	C₂×D₇₁	1	1	yes	characteristic
119	C₂×C₄×C₈	—	71	1	no	—
120	#364	—	142	2	no	—
121	#347	—	2	2	no	—
122	C₂×M₁₆	—	71	1	no	—
123	C₂×M₁₆	C₄×D₇₁	1	1	yes	characteristic
124	C₂²×C₈	—	71	1	no	—
125	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
126	C₄⋊Q₈	—	142	1	no	—
127	#125	—	142	1	no	—
128	C₂²⋊Q₈	—	142	2	no	—
129	C₄×Q₈	—	142	2	no	—
130	#117	—	71	1	no	—
131	#117	C₄×D₇₁	1	1	yes	characteristic
132	C₂×C₄²	—	71	1	no	—
133	M₃₂	—	2	2	no	—
134	M₃₂	—	142	2	no	—
135	M₃₂	—	142	2	no	—
136	M₃₂	Dic₁₄₂	1	4	yes	complemented
137	C₂×C₁₆	—	2	2	no	—
138	C₂×C₁₆	—	142	2	no	—
139	C₄⋊C₈	—	142	2	no	—
140	C₄⋊C₈	—	142	2	no	—
141	C₄⋊C₈	—	142	2	no	—
142	#97	—	142	1	no	—
143	#97	C₂²⋊D₇₁	1	2	yes	—
144	C₄×C₈	—	142	1	no	—
145	C₄×C₈	—	142	1	no	—
146	C₄×C₈	—	142	1	no	—
147	C₄×C₈	D₂₈₄	1	2	yes	—
148	C₂×Q₈	—	142	2	no	—
149	C₂²×C₄	—	142	1	no	—
150	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
151	M₁₆	—	2	2	no	—
152	M₁₆	—	142	2	no	—
153	C₂×C₈	—	142	1	no	—
154	C₂×C₈	—	142	1	no	—
155	C₂×C₈	—	142	2	no	—
156	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
157	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
158	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
159	C₂×C₈	C₂×Dic₁₄₂	1	2	yes	—
160	C₂×C₈	#11765865	1	2	yes	—
161	C₄⋊C₄	—	2	1	no	—
162	C₄⋊C₄	—	142	2	no	—
163	C₄⋊C₄	—	142	2	no	—
164	C₄⋊C₄	—	142	2	no	—
165	C₄⋊C₄	—	142	1	no	—
166	C₂²⋊C₄	—	2	1	no	—
167	C₂²⋊C₄	—	71	2	no	—
168	C₄²	—	71	1	no	—
169	C₄²	—	71	1	no	—
170	C₄²	—	142	1	no	—
171	C₄²	—	142	1	no	—
172	C₄²	(C₂×D₇₁)⋊C₄	1	2	yes	—
173	C₁₆	—	2	4	no	—
174	C₁₆	—	284	2	no	—
175	C₂³	M₁₆×D₇₁	1	1	yes	characteristic
176	Q₈	—	284	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	284	2	no	—
179	C₂×C₄	—	142	1	no	—
180	C₂×C₄	—	142	2	no	—
181	C₂×C₄	—	142	1	no	—
182	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
183	C₂×C₄	C₄×Dic₁₄₂	1	2	yes	—
184	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
185	C₂×C₄	#14556136	1	2	yes	—
186	C₈	—	2	2	no	—
187	C₈	—	284	1	no	—
188	C₈	—	284	1	no	—
189	C₈	#14556107	1	2	yes	—
190	C₈	#14556116	1	1	yes	characteristic
191	C₈	#14556128	1	1	yes	characteristic
192	C₂²	—	4	1	no	—
193	C₂²	#17985962	1	1	yes	characteristic
194	C₂²	#17986127	1	2	yes	—
195	C₄	—	4	2	no	—
196	C₄	—	284	1	no	—
197	C₄	—	284	2	no	—
198	C₄	—	142	2	no	—
199	C₄	#17985905	1	1	yes	characteristic
200	C₄	#17985967	1	1	yes	characteristic
201	C₄	#17986127	1	2	yes	—
202	C₂	—	2	1	no	—
203	C₂	—	2	2	no	—
204	C₂	#22589280	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22597272

C₂

yes

maximal, characteristic

#22590106

C₂

yes

maximal, characteristic

#22586427

C₂

yes

maximal

#22586421

C₂

yes

maximal

C₇₁×#2110

C₂

yes

maximal, characteristic

#17986552

C₄

yes

characteristic

C₂×#14556068

C₂²

yes

—

#17986127

—

C₈×Dic₁₄₂

—

#17986027

C₄

yes

characteristic

#17986026

C₄

yes

characteristic

#17986022

C₄

yes

characteristic

#17985954

C₂²

yes

characteristic

C₂×C₈×Dic₇₁

C₂²

yes

characteristic

#17985820

—

C₂×M₃₂×C₇₁

C₂²

yes

—

#405×C₇₁

C₂²

yes

characteristic

#347×C₇₁

—

C₂×#11765849

C₂×C₄

yes

characteristic

C₂²×C₇₁⋊C₈

C₂×C₄

yes

—

—

—

C₄×Dic₁₄₂

—

#14556087

C₂×C₄

yes

characteristic

C₂×C₄×Dic₇₁

C₂×C₄

yes

characteristic

#14556068

—

#14556068

—

C₂×C₇₁⋊C₁₆

—

C₄⋊(C₇₁⋊C₈)

—

Dic₇₁⋊C₈

—

Dic₇₁⋊C₈

—

#14556038

—

C₈×Dic₇₁

—

C₈×Dic₇₁

—

C₈×Dic₇₁

—

C₂×M₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

#117×C₇₁

C₂×C₄

yes

characteristic

M₃₂×C₇₁

—

M₃₂×C₇₁

Q₈

yes

complemented

C₂×C₁₆×C₇₁

—

#97×C₇₁

D₄

yes

—

C₄×C₈×C₇₁

D₄

yes

—

C₂×Dic₁₄₂

—

C₂²×Dic₇₁

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

—

C₂²⋊Dic₇₁

M₁₆

yes

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

M₁₆

yes

characteristic

C₄×Dic₇₁

M₁₆

yes

characteristic

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

M₁₆×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

—

C₂×C₈×C₇₁

#41

yes

—

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

C₂²⋊C₄

yes

—

C₁₆×C₇₁

—

C₂³×C₇₁

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄×Q₈

yes

—

C₂×C₄×C₇₁

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₁

#131

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

C₂²⋊Q₈

yes

—

C₈×C₇₁

#124

yes

characteristic

C₈×C₇₁

C₄⋊Q₈

yes

characteristic

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

#407

yes

characteristic

C₂²×C₇₁

#446

yes

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

#13270

—

maximal, Sylow

100

C₂×C₇₁

—

101

C₂×C₇₁

—

102

C₂×C₇₁

#1985

yes

characteristic

103

#2960

—

104

#2153

—

105

#2110

D₇₁

yes

characteristic

106

#1378

—

107

#1376

—

108

C₇₁

#13270

yes

characteristic, complemented, Sylow

109

#533

—

110

C₂×M₃₂

—

111

C₂×M₃₂

C₂×D₇₁

yes

—

112

#446

—

142

—

113

C₈×Q₈

—

142

—

114

#424

—

115

#424

—

116

#423

—

117

#405

—

118

#405

C₂×D₇₁

yes

characteristic

119

C₂×C₄×C₈

—

120

#364

—

142

—

121

#347

—

122

C₂×M₁₆

—

123

C₂×M₁₆

C₄×D₇₁

yes

characteristic

124

C₂²×C₈

—

125

C₂²×C₈

C₂²×D₇₁

yes

characteristic

126

C₄⋊Q₈

—

142

—

127

#125

—

142

—

128

C₂²⋊Q₈

—

142

—

129

C₄×Q₈

—

142

—

130

#117

—

131

#117

C₄×D₇₁

yes

characteristic

132

C₂×C₄²

—

133

M₃₂

—

134

M₃₂

—

142

—

135

M₃₂

—

142

—

136

M₃₂

Dic₁₄₂

yes

complemented

137

C₂×C₁₆

—

138

C₂×C₁₆

—

142

—

139

C₄⋊C₈

—

142

—

140

C₄⋊C₈

—

142

—

141

C₄⋊C₈

—

142

—

142

#97

—

142

—

143

#97

C₂²⋊D₇₁

yes

—

144

C₄×C₈

—

142

—

145

C₄×C₈

—

142

—

146

C₄×C₈

—

142

—

147

C₄×C₈

D₂₈₄

yes

—

148

C₂×Q₈

—

142

—

149

C₂²×C₄

—

142

—

150

C₂²×C₄

C₂×C₄×D₇₁

yes

characteristic

151

M₁₆

—

152

M₁₆

—

142

—

153

C₂×C₈

—

142

—

154

C₂×C₈

—

142

—

155

C₂×C₈

—

142

—

156

C₂×C₈

(C₂×D₇₁)⋊C₄

yes

—

157

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

158

C₂×C₈

C₂×D₂₈₄

yes

characteristic

159

C₂×C₈

C₂×Dic₁₄₂

yes

—

160

C₂×C₈

#11765865

yes

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

142

—

163

C₄⋊C₄

—

142

—

164

C₄⋊C₄

—

142

—

165

C₄⋊C₄

—

142

—

166

C₂²⋊C₄

—

167

C₂²⋊C₄

—

168

C₄²

—

169

C₄²

—

170

C₄²

—

142

—

171

C₄²

—

142

—

172

C₄²

(C₂×D₇₁)⋊C₄

yes

—

173

C₁₆

—

174

C₁₆

—

284

—

175

C₂³

M₁₆×D₇₁

yes

characteristic

176

Q₈

—

284

—

177

C₂×C₄

—

178

C₂×C₄

—

284

—

179

C₂×C₄

—

142

—

180

C₂×C₄

—

142

—

181

C₂×C₄

—

142

—

182

C₂×C₄

C₂×(C₂×D₇₁)⋊C₄

yes

characteristic

183

C₂×C₄

C₄×Dic₁₄₂

yes

—

184

C₂×C₄

M₁₆×D₇₁

yes

characteristic

185

C₂×C₄

#14556136

yes

—

186

C₈

—

187

C₈

—

284

—

188

C₈

—

284

—

189

C₈

#14556107

yes

—

190

C₈

#14556116

yes

characteristic

191

C₈

#14556128

yes

characteristic

192

C₂²

—

193

C₂²

#17985962

yes

characteristic

194

C₂²

#17986127

yes

—

195

C₄

—

196

C₄

—

284

—

197

C₄

—

284

—

198

C₄

—

142

—

199

C₄

#17985905

yes

characteristic

200

C₄

#17985967

yes

characteristic

201

C₄

#17986127

yes

—

202

C₂

—

203

C₂

—

204

C₂

#22589280

yes

characteristic

	subgroup	G/H
center	C₈	#14556116
commutator	C₂²×C₄×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#2110	C₂
socle	C₂×C₇₁	#1985
prime radical	C₂³×C₇₁	C₂×M₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#2110	D₇₁
hypocenter	C₇₁	#13270
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2110

subgroup

G/H

center

C₈

#14556116

commutator

C₂²×C₄×C₇₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₇₁×#2110

C₂

socle

C₂×C₇₁

#1985

prime radical

C₂³×C₇₁

C₂×M₁₆

cosocle

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2110

N	H ≅ G/N	type
41. M₃₂×C₇₁	Q₈	semidirect
108. C₇₁	#13270	semidirect
136. M₃₂	Dic₁₄₂	semidirect

H ≅ G/N

type

41. M₃₂×C₇₁

Q₈

semidirect

108. C₇₁

#13270

semidirect

136. M₃₂

Dic₁₄₂

semidirect

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

subgroup	index
#22597272	2
#22590106	2
#22586427	2
#22586421	2
C₇₁×#2110	2
#13270	71

subgroup

index

C₇₁×#2110

Subgroups — #31916001

Conjugacy classes of subgroups

Complements

Maximal subgroups