Group #31913926 — subgroups

Subgroups — #31913926

Order 18176.

374 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 131 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	#22601443	C₂	1	1	yes	maximal, characteristic
2	#22590098	C₂	1	2	yes	maximal, complemented
3	#22588140	C₂	1	1	yes	maximal, characteristic
4	#22586427	C₂	1	2	yes	maximal, complemented
5	C₇₁×#2113	C₂	1	1	yes	maximal, characteristic
6	#17986416	—	2	4	no	—
7	C₂×#14556068	—	2	2	no	—
8	C₂×#14556068	C₂²	1	2	yes	—
9	#17985983	C₄	1	1	yes	characteristic
10	#17985973	C₄	1	1	yes	characteristic
11	#17985954	C₂²	1	2	yes	—
12	Dic₇₁⋊C₁₆	—	2	4	no	—
13	#17985770	C₄	1	2	yes	—
14	C₂×#131×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
17	C₂²⋊C₁₆×C₇₁	—	2	2	no	—
18	C₂×#11765849	C₂×C₄	1	2	yes	—
19	#14556087	C₂×C₄	1	2	yes	—
20	#14556068	—	2	4	no	—
21	#14556068	—	2	2	no	—
22	#14556068	—	2	2	no	—
23	C₂×C₇₁⋊C₁₆	—	2	4	no	—
24	C₂×C₇₁⋊C₁₆	D₄	1	4	yes	—
25	#14556047	—	2	1	no	—
26	#14556045	—	2	1	no	—
27	#14556043	C₈	1	4	yes	—
28	#14556038	—	2	2	no	—
29	C₈×Dic₇₁	—	2	2	no	—
30	C₂×#41×C₇₁	C₂×C₄	1	1	yes	characteristic
31	#131×C₇₁	—	2	2	no	—
32	#131×C₇₁	—	2	2	no	—
33	C₂×M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₂×M₁₆×C₇₁	Q₈	1	2	yes	—
35	C₂²×C₈×C₇₁	D₄	1	2	yes	—
36	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
37	M₃₂×C₇₁	—	2	2	no	—
38	C₂×C₁₆×C₇₁	—	2	2	no	—
39	C₂×C₁₆×C₇₁	—	2	1	no	—
40	C₂×C₁₆×C₇₁	—	2	1	no	—
41	#11765849	—	2	2	no	—
42	#11765849	—	2	2	no	—
43	C₂×C₇₁⋊C₈	—	2	2	no	—
44	C₄⋊Dic₇₁	—	2	2	no	—
45	C₂²⋊Dic₇₁	C₂×C₈	1	4	yes	—
46	C₄×Dic₇₁	—	2	2	no	—
47	C₇₁⋊C₁₆	—	2	4	no	—
48	#41×C₇₁	—	2	2	no	—
49	#41×C₇₁	—	2	2	no	—
50	C₂×Q₈×C₇₁	M₁₆	1	1	yes	characteristic
51	C₂×D₄×C₇₁	C₂×C₈	1	2	yes	—
52	C₂×D₄×C₇₁	M₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
55	M₁₆×C₇₁	—	2	2	no	—
56	M₁₆×C₇₁	—	2	4	no	—
57	C₂×C₈×C₇₁	—	2	2	no	—
58	C₂×C₈×C₇₁	—	2	2	no	—
59	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
60	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₈×C₇₁	#41	1	2	yes	—
64	C₁₆×C₇₁	—	4	1	no	—
65	C₁₆×C₇₁	—	4	1	no	—
66	C₂³×C₇₁	C₄⋊C₈	1	2	yes	—
67	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
68	Q₈×C₇₁	—	2	2	no	—
69	D₄×C₇₁	—	2	4	no	—
70	D₄×C₇₁	—	2	2	no	—
71	C₂×C₄×C₇₁	—	2	2	no	—
72	C₂×C₄×C₇₁	—	2	2	no	—
73	C₂×C₄×C₇₁	C₄⋊C₈	1	2	yes	—
74	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₇₁	C₄×D₄	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	4	no	—
79	C₈×C₇₁	—	2	2	no	—
80	C₈×C₇₁	#120	1	1	yes	characteristic
81	C₈×C₇₁	C₂²⋊Q₈	1	2	yes	—
82	C₈×C₇₁	#123	1	1	yes	characteristic
83	C₇₁⋊C₈	—	4	2	no	—
84	C₂²×C₇₁	—	2	2	no	—
85	C₂²×C₇₁	—	4	1	no	—
86	C₂²×C₇₁	—	4	2	no	—
87	C₂²×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
88	C₂²×C₇₁	C₈×D₄	1	2	yes	—
89	C₄×C₇₁	—	2	2	no	—
90	C₄×C₇₁	#385	1	1	yes	characteristic
91	C₄×C₇₁	#424	1	1	yes	characteristic
92	C₄×C₇₁	#435	1	2	yes	—
93	Dic₇₁	—	4	4	no	—
94	#12915	—	71	1	no	maximal, Sylow
95	C₂×C₇₁	—	2	2	no	—
96	C₂×C₇₁	—	2	1	no	—
97	C₂×C₇₁	—	4	2	no	—
98	C₂×C₇₁	#1850	1	1	yes	characteristic
99	#3407	—	71	1	no	—
100	#2151	—	71	2	no	—
101	#2113	D₇₁	1	1	yes	characteristic
102	#1757	—	71	1	no	—
103	#1378	—	71	2	no	—
104	C₇₁	#12915	1	1	yes	characteristic, complemented, Sylow
105	C₂×#131	C₂×D₇₁	1	1	yes	characteristic
106	#504	—	142	4	no	—
107	C₂×M₃₂	—	71	2	no	—
108	C₂×M₃₂	—	142	2	no	—
109	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
110	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
111	#413	—	71	1	no	—
112	#410	—	71	1	no	—
113	#405	—	71	2	no	—
114	C₄⋊C₁₆	—	142	4	no	—
115	C₂²⋊C₁₆	—	2	2	no	—
116	#344	—	71	2	no	—
117	C₂×#41	C₄×D₇₁	1	1	yes	characteristic
118	#131	—	2	2	no	—
119	#131	—	2	2	no	—
120	C₂×M₁₆	—	71	2	no	—
121	C₂×M₁₆	Dic₁₄₂	1	2	yes	—
122	C₂×M₁₆	C₄×D₇₁	1	1	yes	characteristic
123	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
124	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
125	#117	—	71	2	no	—
126	M₃₂	—	2	2	no	—
127	M₃₂	—	142	4	no	—
128	M₃₂	—	142	2	no	—
129	M₃₂	—	142	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	2	1	no	—
132	C₂×C₁₆	—	2	1	no	—
133	C₂×C₁₆	—	71	4	no	—
134	C₂×C₁₆	—	142	4	no	—
135	#101	—	142	1	no	—
136	#100	—	142	1	no	—
137	#99	—	71	4	no	—
138	#97	—	142	2	no	—
139	C₄×C₈	—	142	2	no	—
140	#41	—	2	2	no	—
141	#41	—	2	2	no	—
142	C₂×Q₈	#11765848	1	1	yes	characteristic
143	C₂×D₄	C₈×D₇₁	1	2	yes	—
144	C₂×D₄	#11765848	1	1	yes	characteristic
145	C₂²×C₄	Dic₇₁⋊C₄	1	2	yes	—
146	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
147	M₁₆	—	2	2	no	—
148	M₁₆	—	2	4	no	—
149	M₁₆	—	142	2	no	—
150	M₁₆	—	142	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	142	2	no	—
154	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
155	C₂×C₈	D₄×D₇₁	1	2	yes	—
156	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
157	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
158	C₂×C₈	#11765864	1	2	yes	—
159	C₄⋊C₄	—	142	2	no	—
160	C₂²⋊C₄	—	71	4	no	—
161	C₄²	—	142	2	no	—
162	C₁₆	—	4	1	no	—
163	C₁₆	—	4	1	no	—
164	C₁₆	—	142	4	no	—
165	C₂³	Dic₇₁⋊C₈	1	2	yes	—
166	C₂³	C₂×C₈×D₇₁	1	1	yes	characteristic
167	Q₈	—	2	2	no	—
168	D₄	—	2	4	no	—
169	D₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	142	4	no	—
173	C₂×C₄	Dic₇₁⋊C₈	1	2	yes	—
174	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
175	C₂×C₄	(C₂²⋊D₇₁)⋊C₄	1	2	yes	—
176	C₂×C₄	C₂×#11765848	1	1	yes	characteristic
177	C₂×C₄	#14556136	1	2	yes	—
178	C₈	—	2	2	no	—
179	C₈	—	284	2	no	—
180	C₈	#14556096	1	1	yes	characteristic
181	C₈	#14556105	1	2	yes	—
182	C₈	#14556109	1	1	yes	characteristic
183	C₂²	—	2	2	no	—
184	C₂²	—	4	1	no	—
185	C₂²	—	4	2	no	—
186	C₂²	C₂×Dic₇₁⋊C₈	1	1	yes	characteristic
187	C₂²	(C₂²⋊D₇₁)⋊C₈	1	2	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	284	4	no	—
190	C₄	#17985871	1	1	yes	characteristic
191	C₄	#17986027	1	1	yes	characteristic
192	C₄	#17986067	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	—	2	1	no	—
195	C₂	—	4	2	no	—
196	C₂	#22588517	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22601443

C₂

yes

maximal, characteristic

#22590098

C₂

yes

maximal, complemented

#22588140

C₂

yes

maximal, characteristic

#22586427

C₂

yes

maximal, complemented

C₇₁×#2113

C₂

yes

maximal, characteristic

#17986416

—

C₂×#14556068

—

C₂×#14556068

C₂²

yes

—

#17985983

C₄

yes

characteristic

#17985973

C₄

yes

characteristic

#17985954

C₂²

yes

—

Dic₇₁⋊C₁₆

—

#17985770

C₄

yes

—

C₂×#131×C₇₁

C₂²

yes

characteristic

C₂×M₃₂×C₇₁

C₂²

yes

characteristic

C₂²×C₁₆×C₇₁

C₂²

yes

characteristic

C₂²⋊C₁₆×C₇₁

—

C₂×#11765849

C₂×C₄

yes

—

#14556087

C₂×C₄

yes

—

#14556068

—

#14556068

—

#14556068

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

D₄

yes

—

#14556047

—

#14556045

—

#14556043

C₈

yes

—

#14556038

—

C₈×Dic₇₁

—

C₂×#41×C₇₁

C₂×C₄

yes

characteristic

#131×C₇₁

—

#131×C₇₁

—

C₂×M₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×M₁₆×C₇₁

Q₈

yes

—

C₂²×C₈×C₇₁

D₄

yes

—

C₂²×C₈×C₇₁

C₂³

yes

characteristic

M₃₂×C₇₁

—

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₂²⋊Dic₇₁

C₂×C₈

yes

—

C₄×Dic₇₁

—

C₇₁⋊C₁₆

—

#41×C₇₁

—

#41×C₇₁

—

C₂×Q₈×C₇₁

M₁₆

yes

characteristic

C₂×D₄×C₇₁

C₂×C₈

yes

—

C₂×D₄×C₇₁

M₁₆

yes

characteristic

C₂²×C₄×C₇₁

C₄⋊C₄

yes

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₁

#41

yes

—

C₁₆×C₇₁

—

C₁₆×C₇₁

—

C₂³×C₇₁

C₄⋊C₈

yes

—

C₂³×C₇₁

C₂²×C₈

yes

characteristic

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄⋊C₈

yes

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

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

#120

yes

characteristic

C₈×C₇₁

C₂²⋊Q₈

yes

—

C₈×C₇₁

#123

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₇₁

C₈×D₄

yes

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

Dic₇₁

—

#12915

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#1850

yes

characteristic

#3407

—

100

#2151

—

101

#2113

D₇₁

yes

characteristic

102

#1757

—

103

#1378

—

104

C₇₁

#12915

yes

characteristic, complemented, Sylow

105

C₂×#131

C₂×D₇₁

yes

characteristic

106

#504

—

142

—

107

C₂×M₃₂

—

108

C₂×M₃₂

—

142

—

109

C₂×M₃₂

C₂×D₇₁

yes

characteristic

110

C₂²×C₁₆

C₂×D₇₁

yes

characteristic

111

#413

—

112

#410

—

113

#405

—

114

C₄⋊C₁₆

—

142

—

115

C₂²⋊C₁₆

—

116

#344

—

117

C₂×#41

C₄×D₇₁

yes

characteristic

118

#131

—

119

#131

—

120

C₂×M₁₆

—

121

C₂×M₁₆

Dic₁₄₂

yes

—

122

C₂×M₁₆

C₄×D₇₁

yes

characteristic

123

C₂²×C₈

C₂²⋊D₇₁

yes

—

124

C₂²×C₈

C₂²×D₇₁

yes

characteristic

125

#117

—

126

M₃₂

—

127

M₃₂

—

142

—

128

M₃₂

—

142

—

129

M₃₂

—

142

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

132

C₂×C₁₆

—

133

C₂×C₁₆

—

134

C₂×C₁₆

—

142

—

135

#101

—

142

—

136

#100

—

142

—

137

#99

—

138

#97

—

142

—

139

C₄×C₈

—

142

—

140

#41

—

141

#41

—

142

C₂×Q₈

#11765848

yes

characteristic

143

C₂×D₄

C₈×D₇₁

yes

—

144

C₂×D₄

#11765848

yes

characteristic

145

C₂²×C₄

Dic₇₁⋊C₄

yes

—

146

C₂²×C₄

C₂×C₄×D₇₁

yes

characteristic

147

M₁₆

—

148

M₁₆

—

149

M₁₆

—

142

—

150

M₁₆

—

142

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

142

—

154

C₂×C₈

Dic₇₁⋊C₄

yes

—

155

C₂×C₈

D₄×D₇₁

yes

—

156

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

157

C₂×C₈

C₂×Dic₁₄₂

yes

characteristic

158

C₂×C₈

#11765864

yes

—

159

C₄⋊C₄

—

142

—

160

C₂²⋊C₄

—

161

C₄²

—

142

—

162

C₁₆

—

163

C₁₆

—

164

C₁₆

—

142

—

165

C₂³

Dic₇₁⋊C₈

yes

—

166

C₂³

C₂×C₈×D₇₁

yes

characteristic

167

Q₈

—

168

D₄

—

169

D₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

142

—

173

C₂×C₄

Dic₇₁⋊C₈

yes

—

174

C₂×C₄

C₂×Dic₇₁⋊C₄

yes

characteristic

175

C₂×C₄

(C₂²⋊D₇₁)⋊C₄

yes

—

176

C₂×C₄

C₂×#11765848

yes

characteristic

177

C₂×C₄

#14556136

yes

—

178

C₈

—

179

C₈

—

284

—

180

C₈

#14556096

yes

characteristic

181

C₈

#14556105

yes

—

182

C₈

#14556109

yes

characteristic

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

C₂×Dic₇₁⋊C₈

yes

characteristic

187

C₂²

(C₂²⋊D₇₁)⋊C₈

yes

—

188

C₄

—

189

C₄

—

284

—

190

C₄

#17985871

yes

characteristic

191

C₄

#17986027

yes

characteristic

192

C₄

#17986067

yes

—

193

C₂

—

194

C₂

—

195

C₂

—

196

C₂

#22588517

yes

characteristic

	subgroup	G/H
center	C₈	#14556096
commutator	C₂³×C₇₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#2113	C₂
socle	C₂×C₇₁	#1850
prime radical	C₂×#41×C₇₁	C₂×C₄
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#2113	D₇₁
hypocenter	C₇₁	#12915
LCS	G ▷ C₂³×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2113

subgroup

G/H

center

C₈

#14556096

commutator

C₂³×C₇₁

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₇₁×#2113

C₂

socle

C₂×C₇₁

#1850

prime radical

C₂×#41×C₇₁

C₂×C₄

cosocle

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂³×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2113

N	H ≅ G/N	type
2. #22590098	C₂	semidirect
4. #22586427	C₂	semidirect
104. C₇₁	#12915	semidirect

H ≅ G/N

type

2. #22590098

C₂

semidirect

4. #22586427

C₂

semidirect

104. C₇₁

#12915

semidirect

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

subgroup	index
#22601443	2
#22590098	2
#22588140	2
#22586427	2
C₇₁×#2113	2
#12915	71

subgroup

index

C₇₁×#2113

Subgroups — #31913926

Conjugacy classes of subgroups

Complements

Maximal subgroups