Group #31913820 — subgroups

Subgroups — #31913820

Order 18176.

234 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 95 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	#22590105	C₂	1	1	yes	maximal, characteristic
2	#22590100	C₂	1	1	yes	maximal, characteristic
3	#22588095	C₂	1	1	yes	maximal, characteristic
4	#22586430	C₂	1	1	yes	maximal, characteristic
5	#22586429	C₂	1	1	yes	maximal, characteristic
6	#22586427	C₂	1	1	yes	maximal, characteristic
7	C₇₁×#1378	C₂	1	1	yes	maximal, characteristic
8	C₂×#14556068	C₂²	1	1	yes	characteristic
9	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
10	#17985954	C₂²	1	1	yes	characteristic
11	#17985954	C₂²	1	1	yes	characteristic
12	#17985819	—	2	2	no	—
13	Dic₇₁⋊C₁₆	—	2	2	no	—
14	C₇₁⋊#338	C₄	1	1	yes	characteristic
15	#17985756	C₄	1	1	yes	characteristic
16	#17985755	C₄	1	1	yes	characteristic
17	#17985755	C₄	1	1	yes	characteristic
18	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
20	#405×C₇₁	C₂²	1	1	yes	characteristic
21	C₂×#11765849	C₂×C₄	1	1	yes	characteristic
22	C₂×#11765849	C₂×C₄	1	1	yes	characteristic
23	#14556087	C₂×C₄	1	1	yes	characteristic
24	#14556087	C₂×C₄	1	1	yes	characteristic
25	#14556068	—	2	1	no	—
26	#14556068	—	2	1	no	—
27	C₂×C₇₁⋊C₁₆	—	2	1	no	—
28	C₂×C₇₁⋊C₁₆	—	2	1	no	—
29	C₂×C₇₁⋊C₁₆	—	2	1	no	—
30	C₂×C₇₁⋊C₁₆	—	2	1	no	—
31	#14556038	—	2	1	no	—
32	#14556038	Q₈	1	2	yes	—
33	C₈×Dic₇₁	—	2	1	no	—
34	C₈×Dic₇₁	D₄	1	2	yes	—
35	C₂×M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
37	#117×C₇₁	C₂×C₄	1	1	yes	characteristic
38	M₃₂×C₇₁	—	2	1	no	—
39	M₃₂×C₇₁	—	2	1	no	—
40	M₃₂×C₇₁	—	2	1	no	—
41	M₃₂×C₇₁	—	2	1	no	—
42	C₂×C₁₆×C₇₁	—	2	1	no	—
43	C₂×C₁₆×C₇₁	—	2	1	no	—
44	#97×C₇₁	—	2	1	no	—
45	C₄×C₈×C₇₁	—	2	1	no	—
46	#11765849	—	2	1	no	—
47	#11765849	—	2	1	no	—
48	#11765849	—	2	1	no	—
49	#11765849	—	2	1	no	—
50	C₂×C₇₁⋊C₈	—	2	1	no	—
51	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
52	C₄⋊Dic₇₁	—	2	1	no	—
53	C₄⋊Dic₇₁	—	2	1	no	—
54	C₂²⋊Dic₇₁	—	2	1	no	—
55	C₂²⋊Dic₇₁	—	2	1	no	—
56	C₄×Dic₇₁	—	2	1	no	—
57	C₄×Dic₇₁	C₄⋊C₄	1	2	yes	—
58	C₇₁⋊C₁₆	—	4	1	no	—
59	C₇₁⋊C₁₆	—	4	1	no	—
60	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
61	M₁₆×C₇₁	—	2	1	no	—
62	M₁₆×C₇₁	—	2	1	no	—
63	C₂×C₈×C₇₁	—	2	1	no	—
64	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
67	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
68	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
69	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
70	C₄⋊C₄×C₇₁	—	2	1	no	—
71	C₂²⋊C₄×C₇₁	—	2	1	no	—
72	C₄²×C₇₁	—	2	1	no	—
73	C₁₆×C₇₁	—	4	1	no	—
74	C₁₆×C₇₁	—	4	1	no	—
75	C₂³×C₇₁	#131	1	1	yes	characteristic
76	C₂×C₄×C₇₁	—	4	1	no	—
77	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
79	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
80	C₂×C₄×C₇₁	#131	1	1	yes	characteristic
81	C₂×C₄×C₇₁	#131	1	1	yes	characteristic
82	C₂×C₄×C₇₁	#131	1	1	yes	characteristic
83	C₂×Dic₇₁	—	2	2	no	—
84	C₂×Dic₇₁	—	4	1	no	—
85	C₈×C₇₁	—	4	1	no	—
86	C₈×C₇₁	#124	1	1	yes	characteristic
87	C₈×C₇₁	#124	1	1	yes	characteristic
88	C₈×C₇₁	#125	1	1	yes	characteristic
89	C₈×C₇₁	#125	1	1	yes	characteristic
90	C₇₁⋊C₈	—	2	2	no	—
91	C₇₁⋊C₈	—	4	1	no	—
92	C₂²×C₇₁	—	4	1	no	—
93	C₂²×C₇₁	#424	1	1	yes	characteristic
94	C₂²×C₇₁	#433	1	1	yes	characteristic
95	C₂²×C₇₁	#433	1	1	yes	characteristic
96	C₄×C₇₁	—	8	1	no	—
97	C₄×C₇₁	#382	1	1	yes	characteristic
98	C₄×C₇₁	#424	1	1	yes	characteristic
99	C₄×C₇₁	#433	1	1	yes	characteristic
100	C₄×C₇₁	#433	1	1	yes	characteristic
101	Dic₇₁	—	4	2	no	—
102	Dic₇₁	—	8	1	no	—
103	#12885	—	71	1	no	maximal, Sylow
104	C₂×C₇₁	—	2	1	no	—
105	C₂×C₇₁	—	2	1	no	—
106	C₂×C₇₁	—	2	1	no	—
107	C₂×C₇₁	#1832	1	1	yes	characteristic
108	#2153	—	71	1	no	—
109	#2151	—	71	1	no	—
110	#1747	—	71	1	no	—
111	#1379	—	71	1	no	—
112	#1379	—	71	1	no	—
113	#1378	—	71	1	no	—
114	#1378	D₇₁	1	1	yes	characteristic
115	C₇₁	#12885	1	1	yes	characteristic, complemented, Sylow
116	C₂×M₃₂	—	71	1	no	—
117	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
118	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
119	C₂²×C₁₆	—	71	1	no	—
120	#405	—	71	1	no	—
121	#405	—	71	1	no	—
122	#405	C₂×D₇₁	1	1	yes	characteristic
123	#364	—	142	2	no	—
124	C₄⋊C₁₆	—	142	2	no	—
125	#338	—	71	1	no	—
126	H(3,ℤ/4)	—	71	1	no	—
127	H(3,ℤ/4)	—	71	1	no	—
128	H(3,ℤ/4)	—	71	1	no	—
129	C₂×M₁₆	—	71	1	no	—
130	C₂×M₁₆	—	71	1	no	—
131	C₂×M₁₆	C₄×D₇₁	1	1	yes	characteristic
132	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
133	#117	—	71	1	no	—
134	#117	—	71	1	no	—
135	#117	C₄×D₇₁	1	1	yes	characteristic
136	M₃₂	—	2	1	no	—
137	M₃₂	—	2	1	no	—
138	M₃₂	—	2	1	no	—
139	M₃₂	—	2	1	no	—
140	M₃₂	—	142	1	no	—
141	M₃₂	—	142	1	no	—
142	C₂×C₁₆	—	2	1	no	—
143	C₂×C₁₆	—	2	1	no	—
144	C₂×C₁₆	—	142	1	no	—
145	C₂×C₁₆	—	142	1	no	—
146	C₂×C₁₆	—	142	1	no	—
147	C₂×C₁₆	—	142	1	no	—
148	#97	—	2	1	no	—
149	#97	—	71	2	no	—
150	#97	—	142	1	no	—
151	C₄×C₈	—	2	1	no	—
152	C₄×C₈	—	71	2	no	—
153	C₄×C₈	—	142	1	no	—
154	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
155	M₁₆	—	2	1	no	—
156	M₁₆	—	2	1	no	—
157	M₁₆	—	142	1	no	—
158	M₁₆	—	142	1	no	—
159	M₁₆	—	142	1	no	—
160	M₁₆	—	142	1	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	71	2	no	—
163	C₂×C₈	—	142	1	no	—
164	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
165	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
166	C₂×C₈	#11765864	1	1	yes	characteristic
167	C₂×C₈	#11765864	1	1	yes	characteristic
168	C₂×C₈	#11765865	1	1	yes	characteristic
169	C₂×C₈	#11765865	1	1	yes	characteristic
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	142	1	no	—
172	C₄⋊C₄	—	142	1	no	—
173	C₂²⋊C₄	—	2	1	no	—
174	C₂²⋊C₄	—	142	1	no	—
175	C₂²⋊C₄	—	142	1	no	—
176	C₄²	—	2	1	no	—
177	C₄²	—	71	2	no	—
178	C₄²	—	142	1	no	—
179	C₁₆	—	4	1	no	—
180	C₁₆	—	4	1	no	—
181	C₁₆	—	284	1	no	—
182	C₁₆	—	284	1	no	—
183	C₂³	#14556134	1	1	yes	characteristic
184	C₂×C₄	—	4	1	no	—
185	C₂×C₄	—	284	1	no	—
186	C₂×C₄	—	142	2	no	—
187	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
188	C₂×C₄	#14556084	1	1	yes	characteristic
189	C₂×C₄	#14556085	1	1	yes	characteristic
190	C₂×C₄	#14556134	1	1	yes	characteristic
191	C₂×C₄	#14556136	1	1	yes	characteristic
192	C₂×C₄	#14556136	1	1	yes	characteristic
193	C₈	—	4	1	no	—
194	C₈	—	284	1	no	—
195	C₈	—	142	2	no	—
196	C₈	#14556114	1	1	yes	characteristic
197	C₈	#14556114	1	1	yes	characteristic
198	C₈	#14556118	1	1	yes	characteristic
199	C₈	#14556118	1	1	yes	characteristic
200	C₂²	—	4	1	no	—
201	C₂²	#17986024	1	1	yes	characteristic
202	C₂²	#17986057	1	1	yes	characteristic
203	C₂²	#17986058	1	1	yes	characteristic
204	C₄	—	8	1	no	—
205	C₄	—	284	2	no	—
206	C₄	—	568	1	no	—
207	C₄	#17985863	1	1	yes	characteristic
208	C₄	#17986024	1	1	yes	characteristic
209	C₄	#17986057	1	1	yes	characteristic
210	C₄	#17986061	1	1	yes	characteristic
211	C₂	—	2	1	no	—
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	#22588458	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590105

C₂

yes

maximal, characteristic

#22590100

C₂

yes

maximal, characteristic

#22588095

C₂

yes

maximal, characteristic

#22586430

C₂

yes

maximal, characteristic

#22586429

C₂

yes

maximal, characteristic

#22586427

C₂

yes

maximal, characteristic

C₇₁×#1378

C₂

yes

maximal, characteristic

C₂×#14556068

C₂²

yes

characteristic

C₂²×C₇₁⋊C₁₆

C₂²

yes

characteristic

#17985954

C₂²

yes

characteristic

#17985954

C₂²

yes

characteristic

#17985819

—

Dic₇₁⋊C₁₆

—

C₇₁⋊#338

C₄

yes

characteristic

#17985756

C₄

yes

characteristic

#17985755

C₄

yes

characteristic

#17985755

C₄

yes

characteristic

C₂×M₃₂×C₇₁

C₂²

yes

characteristic

C₂×M₃₂×C₇₁

C₂²

yes

characteristic

#405×C₇₁

C₂²

yes

characteristic

C₂×#11765849

C₂×C₄

yes

characteristic

C₂×#11765849

C₂×C₄

yes

characteristic

#14556087

C₂×C₄

yes

characteristic

#14556087

C₂×C₄

yes

characteristic

#14556068

—

#14556068

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

—

#14556038

—

#14556038

Q₈

yes

—

C₈×Dic₇₁

—

C₈×Dic₇₁

D₄

yes

—

C₂×M₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

#117×C₇₁

C₂×C₄

yes

characteristic

M₃₂×C₇₁

—

M₃₂×C₇₁

—

M₃₂×C₇₁

—

M₃₂×C₇₁

—

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

—

#97×C₇₁

—

C₄×C₈×C₇₁

—

—

—

—

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

C₄⋊C₄

yes

—

C₇₁⋊C₁₆

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

C₂×C₈×C₇₁

yes

C₂×C₈×C₇₁

yes

C₂×C₈×C₇₁

yes

C₂×C₈×C₇₁

yes

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₁₆×C₇₁

—

C₁₆×C₇₁

—

C₂³×C₇₁

#131

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₈×C₇₁

yes

C₈×C₇₁

yes

C₈×C₇₁

yes

C₈×C₇₁

yes

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

100

C₄×C₇₁

#433

yes

characteristic

101

Dic₇₁

—

102

Dic₇₁

—

103

#12885

—

maximal, Sylow

104

C₂×C₇₁

—

105

C₂×C₇₁

—

106

C₂×C₇₁

—

107

C₂×C₇₁

#1832

yes

characteristic

108

#2153

—

109

#2151

—

110

#1747

—

111

#1379

—

112

#1379

—

113

#1378

—

114

#1378

D₇₁

yes

characteristic

115

C₇₁

#12885

yes

characteristic, complemented, Sylow

116

C₂×M₃₂

—

117

C₂×M₃₂

C₂×D₇₁

yes

characteristic

118

C₂×M₃₂

C₂×D₇₁

yes

characteristic

119

C₂²×C₁₆

—

120

#405

—

121

#405

—

122

#405

C₂×D₇₁

yes

characteristic

123

#364

—

142

—

124

C₄⋊C₁₆

—

142

—

125

#338

—

126

H(3,ℤ/4)

—

127

H(3,ℤ/4)

—

128

H(3,ℤ/4)

—

129

C₂×M₁₆

—

130

C₂×M₁₆

—

131

C₂×M₁₆

C₄×D₇₁

yes

characteristic

132

C₂²×C₈

C₂²×D₇₁

yes

characteristic

133

#117

—

134

#117

—

135

#117

C₄×D₇₁

yes

characteristic

136

M₃₂

—

137

M₃₂

—

138

M₃₂

—

139

M₃₂

—

140

M₃₂

—

142

—

141

M₃₂

—

142

—

142

C₂×C₁₆

—

143

C₂×C₁₆

—

144

C₂×C₁₆

—

142

—

145

C₂×C₁₆

—

142

—

146

C₂×C₁₆

—

142

—

147

C₂×C₁₆

—

142

—

148

#97

—

149

#97

—

150

#97

—

142

—

151

C₄×C₈

—

152

C₄×C₈

—

153

C₄×C₈

—

142

—

154

C₂²×C₄

C₂×C₄×D₇₁

yes

characteristic

155

M₁₆

—

156

M₁₆

—

157

M₁₆

—

142

—

158

M₁₆

—

142

—

159

M₁₆

—

142

—

160

M₁₆

—

142

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

142

—

164

C₂×C₈

D₄×D₇₁

yes

characteristic

165

C₂×C₈

Q₈×D₇₁

yes

characteristic

166

C₂×C₈

#11765864

yes

characteristic

167

C₂×C₈

#11765864

yes

characteristic

168

C₂×C₈

#11765865

yes

characteristic

169

C₂×C₈

#11765865

yes

characteristic

170

C₄⋊C₄

—

171

C₄⋊C₄

—

142

—

172

C₄⋊C₄

—

142

—

173

C₂²⋊C₄

—

174

C₂²⋊C₄

—

142

—

175

C₂²⋊C₄

—

142

—

176

C₄²

—

177

C₄²

—

178

C₄²

—

142

—

179

C₁₆

—

180

C₁₆

—

181

C₁₆

—

284

—

182

C₁₆

—

284

—

183

C₂³

#14556134

yes

characteristic

184

C₂×C₄

—

185

C₂×C₄

—

284

—

186

C₂×C₄

—

142

—

187

C₂×C₄

C₄⋊C₄×D₇₁

yes

characteristic

188

C₂×C₄

#14556084

yes

characteristic

189

C₂×C₄

#14556085

yes

characteristic

190

C₂×C₄

#14556134

yes

characteristic

191

C₂×C₄

#14556136

yes

characteristic

192

C₂×C₄

#14556136

yes

characteristic

193

C₈

—

194

C₈

—

284

—

195

C₈

—

142

—

196

C₈

#14556114

yes

characteristic

197

C₈

#14556114

yes

characteristic

198

C₈

#14556118

yes

characteristic

199

C₈

#14556118

yes

characteristic

200

C₂²

—

201

C₂²

#17986024

yes

characteristic

202

C₂²

#17986057

yes

characteristic

203

C₂²

#17986058

yes

characteristic

204

C₄

—

205

C₄

—

284

—

206

C₄

—

568

—

207

C₄

#17985863

yes

characteristic

208

C₄

#17986024

yes

characteristic

209

C₄

#17986057

yes

characteristic

210

C₄

#17986061

yes

characteristic

211

C₂

—

212

C₂

—

213

C₂

—

214

C₂

#22588458

yes

characteristic

	subgroup	G/H
center	C₈	#14556114
commutator	C₂²×C₄×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1378	C₂
socle	C₂×C₇₁	#1832
prime radical	C₂³×C₇₁	#131
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1378	D₇₁
hypocenter	C₇₁	#12885
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1378

subgroup

G/H

center

C₈

#14556114

commutator

C₂²×C₄×C₇₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₇₁×#1378

C₂

C₂×C₇₁

C₂³×C₇₁

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1378

N	H ≅ G/N	type
115. C₇₁	#12885	semidirect

H ≅ G/N

type

115. C₇₁

#12885

semidirect

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

subgroup	index
#22590105	2
#22590100	2
#22588095	2
#22586430	2
#22586429	2
#22586427	2
C₇₁×#1378	2
#12885	71

subgroup

index

C₇₁×#1378

Subgroups — #31913820

Conjugacy classes of subgroups

Complements

Maximal subgroups