Group #31912575 — subgroups

Subgroups — #31912575

Order 18176.

382 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 151 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×#17986416	C₂	1	1	yes	maximal, characteristic
2	#22589944	C₂	1	2	yes	maximal, complemented
3	#22588140	C₂	1	1	yes	maximal, characteristic
4	#22586430	C₂	1	2	yes	maximal, complemented
5	C₇₁×#2115	C₂	1	1	yes	maximal, characteristic
6	#17986416	—	2	4	no	—
7	C₂×#14556068	C₄	1	2	yes	complemented
8	C₂×#14556068	C₂²	1	1	yes	characteristic
9	C₂²×C₇₁⋊C₁₆	C₄	1	2	yes	complemented
10	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
11	#17985983	C₄	1	1	yes	characteristic
12	#17985973	C₄	1	1	yes	characteristic
13	#17985954	C₂²	1	2	yes	—
14	#17985776	—	2	4	no	—
15	#17985770	C₄	1	2	yes	—
16	C₂×#131×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×M₃₂×C₇₁	C₂²	1	2	yes	—
18	#349×C₇₁	C₄	1	4	yes	complemented
19	C₂×#11765849	C₂×C₄	1	2	yes	—
20	#14556087	C₂×C₄	1	2	yes	—
21	#14556068	—	2	4	no	—
22	#14556068	D₄	1	4	yes	—
23	C₂×C₇₁⋊C₁₆	—	2	4	no	—
24	C₂×C₇₁⋊C₁₆	—	2	2	no	—
25	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
28	#14556047	—	2	1	no	—
29	#14556045	—	2	1	no	—
30	#14556043	—	2	2	no	—
31	#14556038	—	2	2	no	—
32	C₈×Dic₇₁	—	2	2	no	—
33	C₂×#41×C₇₁	C₂×C₄	1	1	yes	characteristic
34	#131×C₇₁	—	2	2	no	—
35	#131×C₇₁	—	2	2	no	—
36	C₂×M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
37	C₂×M₁₆×C₇₁	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
40	M₃₂×C₇₁	—	2	2	no	—
41	M₃₂×C₇₁	C₂×C₄	1	4	yes	—
42	C₂×C₁₆×C₇₁	—	2	2	no	—
43	#11765849	—	2	2	no	—
44	#11765849	—	2	2	no	—
45	C₂×C₇₁⋊C₈	—	2	2	no	—
46	C₄⋊Dic₇₁	—	2	2	no	—
47	C₂²⋊Dic₇₁	—	2	2	no	—
48	C₄×Dic₇₁	—	2	2	no	—
49	C₇₁⋊C₁₆	—	2	1	no	—
50	C₇₁⋊C₁₆	—	2	1	no	—
51	C₇₁⋊C₁₆	C₂²⋊C₄	1	4	yes	—
52	#41×C₇₁	—	2	2	no	—
53	#41×C₇₁	—	2	2	no	—
54	C₂×Q₈×C₇₁	M₁₆	1	1	yes	characteristic
55	C₂×D₄×C₇₁	M₁₆	1	2	yes	—
56	C₂×D₄×C₇₁	M₁₆	1	1	yes	characteristic
57	C₂²×C₄×C₇₁	C₄²	1	2	yes	—
58	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
59	M₁₆×C₇₁	—	2	2	no	—
60	M₁₆×C₇₁	—	2	4	no	—
61	C₂×C₈×C₇₁	—	2	2	no	—
62	C₂×C₈×C₇₁	—	2	2	no	—
63	C₂×C₈×C₇₁	C₄²	1	2	yes	—
64	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₇₁	#41	1	2	yes	—
68	C₁₆×C₇₁	—	2	4	no	—
69	C₂³×C₇₁	#97	1	2	yes	—
70	C₂³×C₇₁	C₂×M₁₆	1	1	yes	characteristic
71	Q₈×C₇₁	—	2	2	no	—
72	D₄×C₇₁	—	2	4	no	—
73	D₄×C₇₁	—	2	2	no	—
74	C₂×C₄×C₇₁	—	2	2	no	—
75	C₂×C₄×C₇₁	—	2	2	no	—
76	C₂×C₄×C₇₁	#97	1	2	yes	—
77	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
79	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₇₁	#131	1	2	yes	—
81	C₂×Dic₇₁	—	4	2	no	—
82	C₈×C₇₁	—	2	2	no	—
83	C₈×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₈×C₇₁	#117	1	1	yes	characteristic
85	C₈×C₇₁	C₄×D₄	1	2	yes	—
86	C₇₁⋊C₈	—	4	2	no	—
87	C₂²×C₇₁	—	2	2	no	—
88	C₂²×C₇₁	—	4	1	no	—
89	C₂²×C₇₁	—	4	2	no	—
90	C₂²×C₇₁	C₂×#97	1	1	yes	characteristic
91	C₂²×C₇₁	#435	1	2	yes	—
92	C₄×C₇₁	—	2	2	no	—
93	C₄×C₇₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
94	C₄×C₇₁	#405	1	1	yes	characteristic
95	C₄×C₇₁	#435	1	2	yes	—
96	Dic₇₁	—	8	2	no	—
97	#12617	—	71	1	no	maximal, Sylow
98	C₂×C₇₁	—	2	2	no	—
99	C₂×C₇₁	—	2	1	no	—
100	C₂×C₇₁	—	4	2	no	—
101	C₂×C₇₁	#1752	1	1	yes	characteristic
102	C₂×#504	—	71	1	no	—
103	#2115	D₇₁	1	1	yes	characteristic
104	#2110	—	71	2	no	—
105	#1757	—	71	1	no	—
106	#1379	—	71	2	no	—
107	C₇₁	#12617	1	1	yes	characteristic, complemented, Sylow
108	C₂×#131	C₂×D₇₁	1	1	yes	characteristic
109	#504	—	142	4	no	—
110	C₂×M₃₂	—	71	1	no	—
111	C₂×M₃₂	—	71	2	no	—
112	C₂×M₃₂	C₂×D₇₁	1	2	yes	—
113	C₂²×C₁₆	—	71	2	no	—
114	C₂²×C₁₆	—	71	1	no	—
115	#413	—	71	1	no	—
116	#410	—	71	1	no	—
117	#405	—	71	2	no	—
118	#349	Dic₇₁	1	4	yes	complemented
119	#347	—	142	4	no	—
120	#344	—	71	2	no	—
121	C₂×#41	C₄×D₇₁	1	1	yes	characteristic
122	#131	—	2	2	no	—
123	#131	—	2	2	no	—
124	C₂×M₁₆	—	71	2	no	—
125	C₂×M₁₆	C₄×D₇₁	1	1	yes	characteristic
126	C₂×M₁₆	C₂×Dic₇₁	1	2	yes	—
127	C₂²×C₈	C₄×D₇₁	1	2	yes	—
128	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
129	#117	—	71	2	no	—
130	M₃₂	—	2	2	no	—
131	M₃₂	—	71	4	no	—
132	M₃₂	—	142	4	no	—
133	M₃₂	C₂×Dic₇₁	1	4	yes	—
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	71	2	no	—
136	C₂×C₁₆	—	71	1	no	—
137	C₂×C₁₆	—	71	1	no	—
138	C₂×C₁₆	—	142	4	no	—
139	C₂×C₁₆	—	142	2	no	—
140	#101	—	142	1	no	—
141	#100	—	142	1	no	—
142	#99	—	142	2	no	—
143	#97	—	142	2	no	—
144	C₄×C₈	—	142	2	no	—
145	#41	—	2	2	no	—
146	#41	—	2	2	no	—
147	C₂×Q₈	#11765848	1	1	yes	characteristic
148	C₂×D₄	#11765848	1	2	yes	—
149	C₂×D₄	#11765848	1	1	yes	characteristic
150	C₂²×C₄	C₄×Dic₇₁	1	2	yes	—
151	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
152	M₁₆	—	2	2	no	—
153	M₁₆	—	2	4	no	—
154	M₁₆	—	142	2	no	—
155	M₁₆	—	142	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	142	2	no	—
159	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
160	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
161	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
162	C₂×C₈	D₄×D₇₁	1	2	yes	—
163	C₂×C₈	#11765864	1	2	yes	—
164	C₄⋊C₄	—	142	2	no	—
165	C₂²⋊C₄	—	142	2	no	—
166	C₄²	—	142	2	no	—
167	C₁₆	—	2	4	no	—
168	C₁₆	—	71	4	no	—
169	C₁₆	—	142	1	no	—
170	C₁₆	—	142	1	no	—
171	C₂³	#14556038	1	2	yes	—
172	C₂³	C₂×#11765848	1	1	yes	characteristic
173	Q₈	—	2	2	no	—
174	D₄	—	2	4	no	—
175	D₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	284	2	no	—
179	C₂×C₄	#14556038	1	2	yes	—
180	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
181	C₂×C₄	(C₂²⋊D₇₁)⋊C₄	1	2	yes	—
182	C₂×C₄	C₂×#11765848	1	1	yes	characteristic
183	C₂×C₄	#14556136	1	2	yes	—
184	C₈	—	2	2	no	—
185	C₈	—	284	2	no	—
186	C₈	C₂²⋊C₄×D₇₁	1	1	yes	characteristic
187	C₈	#14556084	1	1	yes	characteristic
188	C₈	D₄×Dic₇₁	1	2	yes	—
189	C₂²	—	2	2	no	—
190	C₂²	—	4	1	no	—
191	C₂²	—	4	2	no	—
192	C₂²	C₂×#14556038	1	1	yes	characteristic
193	C₂²	#17986067	1	2	yes	—
194	C₄	—	2	2	no	—
195	C₄	—	568	2	no	—
196	C₄	C₂²⋊C₄×Dic₇₁	1	1	yes	characteristic
197	C₄	#17985957	1	1	yes	characteristic
198	C₄	#17986067	1	2	yes	—
199	C₂	—	2	2	no	—
200	C₂	—	2	1	no	—
201	C₂	—	4	2	no	—
202	C₂	#22588119	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17986416

C₂

yes

maximal, characteristic

#22589944

C₂

yes

maximal, complemented

#22588140

C₂

yes

maximal, characteristic

#22586430

C₂

yes

maximal, complemented

C₇₁×#2115

C₂

yes

maximal, characteristic

#17986416

—

C₂×#14556068

C₄

yes

complemented

C₂×#14556068

C₂²

yes

characteristic

C₂²×C₇₁⋊C₁₆

C₄

yes

complemented

C₂²×C₇₁⋊C₁₆

C₂²

yes

characteristic

#17985983

C₄

yes

characteristic

#17985973

C₄

yes

characteristic

#17985954

C₂²

yes

—

#17985776

—

#17985770

C₄

yes

—

C₂×#131×C₇₁

C₂²

yes

characteristic

C₂×M₃₂×C₇₁

C₂²

yes

—

#349×C₇₁

C₄

yes

complemented

C₂×#11765849

C₂×C₄

yes

—

#14556087

C₂×C₄

yes

—

#14556068

—

#14556068

D₄

yes

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

—

—

—

—

—

C₈×Dic₇₁

—

C₂×#41×C₇₁

C₂×C₄

yes

characteristic

#131×C₇₁

—

#131×C₇₁

—

C₂×M₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×M₁₆×C₇₁

C₂×C₄

yes

—

C₂²×C₈×C₇₁

C₂×C₄

yes

—

C₂²×C₈×C₇₁

C₂³

yes

characteristic

M₃₂×C₇₁

—

M₃₂×C₇₁

C₂×C₄

yes

—

C₂×C₁₆×C₇₁

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₇₁⋊C₁₆

—

C₇₁⋊C₁₆

—

C₇₁⋊C₁₆

C₂²⋊C₄

yes

—

#41×C₇₁

—

#41×C₇₁

—

C₂×Q₈×C₇₁

M₁₆

yes

characteristic

C₂×D₄×C₇₁

M₁₆

yes

—

C₂×D₄×C₇₁

M₁₆

yes

characteristic

C₂²×C₄×C₇₁

C₄²

yes

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄²

yes

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

—

C₂×C₈×C₇₁

#41

yes

—

C₁₆×C₇₁

—

C₂³×C₇₁

#97

yes

—

C₂³×C₇₁

C₂×M₁₆

yes

characteristic

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#97

yes

—

C₂×C₄×C₇₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₁

C₄×D₄

yes

—

C₂×C₄×C₇₁

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₁

#131

yes

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₈×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₇₁

#117

yes

characteristic

C₈×C₇₁

C₄×D₄

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×#97

yes

characteristic

C₂²×C₇₁

#435

yes

—

C₄×C₇₁

—

C₄×C₇₁

C₄×C₂²⋊C₄

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

Dic₇₁

—

#12617

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

100

C₂×C₇₁

—

101

C₂×C₇₁

#1752

yes

characteristic

102

C₂×#504

—

103

#2115

D₇₁

yes

characteristic

104

#2110

—

105

#1757

—

106

#1379

—

107

C₇₁

#12617

yes

characteristic, complemented, Sylow

108

C₂×#131

C₂×D₇₁

yes

characteristic

109

#504

—

142

—

110

C₂×M₃₂

—

111

C₂×M₃₂

—

112

C₂×M₃₂

C₂×D₇₁

yes

—

113

C₂²×C₁₆

—

114

C₂²×C₁₆

—

115

#413

—

116

#410

—

117

#405

—

118

#349

Dic₇₁

yes

complemented

119

#347

—

142

—

120

#344

—

121

C₂×#41

C₄×D₇₁

yes

characteristic

122

#131

—

123

#131

—

124

C₂×M₁₆

—

125

C₂×M₁₆

C₄×D₇₁

yes

characteristic

126

C₂×M₁₆

C₂×Dic₇₁

yes

—

127

C₂²×C₈

C₄×D₇₁

yes

—

128

C₂²×C₈

C₂²×D₇₁

yes

characteristic

129

#117

—

130

M₃₂

—

131

M₃₂

—

132

M₃₂

—

142

—

133

M₃₂

C₂×Dic₇₁

yes

—

134

C₂×C₁₆

—

135

C₂×C₁₆

—

136

C₂×C₁₆

—

137

C₂×C₁₆

—

138

C₂×C₁₆

—

142

—

139

C₂×C₁₆

—

142

—

140

#101

—

142

—

141

#100

—

142

—

142

#99

—

142

—

143

#97

—

142

—

144

C₄×C₈

—

142

—

145

#41

—

146

#41

—

147

C₂×Q₈

#11765848

yes

characteristic

148

C₂×D₄

#11765848

yes

—

149

C₂×D₄

#11765848

yes

characteristic

150

C₂²×C₄

C₄×Dic₇₁

yes

—

151

C₂²×C₄

C₂×C₄×D₇₁

yes

characteristic

152

M₁₆

—

153

M₁₆

—

154

M₁₆

—

142

—

155

M₁₆

—

142

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

142

—

159

C₂×C₈

C₄×Dic₇₁

yes

—

160

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

161

C₂×C₈

C₂²×Dic₇₁

yes

characteristic

162

C₂×C₈

D₄×D₇₁

yes

—

163

C₂×C₈

#11765864

yes

—

164

C₄⋊C₄

—

142

—

165

C₂²⋊C₄

—

142

—

166

C₄²

—

142

—

167

C₁₆

—

168

C₁₆

—

169

C₁₆

—

142

—

170

C₁₆

—

142

—

171

C₂³

#14556038

yes

—

172

C₂³

C₂×#11765848

yes

characteristic

173

Q₈

—

174

D₄

—

175

D₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

284

—

179

C₂×C₄

#14556038

yes

—

180

C₂×C₄

C₂×C₄×Dic₇₁

yes

characteristic

181

C₂×C₄

(C₂²⋊D₇₁)⋊C₄

yes

—

182

C₂×C₄

C₂×#11765848

yes

characteristic

183

C₂×C₄

#14556136

yes

—

184

C₈

—

185

C₈

—

284

—

186

C₈

C₂²⋊C₄×D₇₁

yes

characteristic

187

C₈

#14556084

yes

characteristic

188

C₈

D₄×Dic₇₁

yes

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

192

C₂²

C₂×#14556038

yes

characteristic

193

C₂²

#17986067

yes

—

194

C₄

—

195

C₄

—

568

—

196

C₄

C₂²⋊C₄×Dic₇₁

yes

characteristic

197

C₄

#17985957

yes

characteristic

198

C₄

#17986067

yes

—

199

C₂

—

200

C₂

—

201

C₂

—

202

C₂

#22588119

yes

characteristic

	subgroup	G/H
center	C₈	#14556084
commutator	C₂×C₄×C₇₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#2115	C₂
socle	C₂×C₇₁	#1752
prime radical	C₂×#41×C₇₁	C₂×C₄
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#2115	D₇₁
hypocenter	C₇₁	#12617
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2115

subgroup

G/H

center

C₈

#14556084

commutator

C₂×C₄×C₇₁

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₇₁×#2115

C₂

socle

C₂×C₇₁

#1752

prime radical

C₂×#41×C₇₁

C₂×C₄

cosocle

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2115

N	H ≅ G/N	type
2. #22589944	C₂	semidirect
4. #22586430	C₂	semidirect
7. C₂×#14556068	C₄	semidirect
9. C₂²×C₇₁⋊C₁₆	C₄	semidirect
18. #349×C₇₁	C₄	semidirect
107. C₇₁	#12617	semidirect
118. #349	Dic₇₁	semidirect

H ≅ G/N

type

2. #22589944

C₂

semidirect

4. #22586430

C₂

semidirect

7. C₂×#14556068

C₄

semidirect

9. C₂²×C₇₁⋊C₁₆

C₄

semidirect

18. #349×C₇₁

C₄

semidirect

107. C₇₁

#12617

semidirect

118. #349

Dic₇₁

semidirect

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

subgroup	index
C₂×#17986416	2
#22589944	2
#22588140	2
#22586430	2
C₇₁×#2115	2
#12617	71

subgroup

index

C₇₁×#2115

Subgroups — #31912575

Conjugacy classes of subgroups

Complements

Maximal subgroups