Group #31913811 — subgroups

Subgroups — #31913811

Order 18176.

342 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 127 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	#22590095	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	#22586422	C₂	1	1	yes	maximal, characteristic
4	#22586421	C₂	1	2	yes	maximal
5	C₇₁×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556068	C₂²	1	2	yes	—
7	C₂×C₄×C₇₁⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₇₁	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₇₁	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
12	C₄²×Dic₇₁	C₄	1	1	yes	characteristic
13	Dic₇₁⋊C₁₆	—	2	4	no	—
14	C₂×M₃₂×C₇₁	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₇₁	—	2	2	no	—
19	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
21	#14556068	—	2	2	no	—
22	#14556068	—	2	2	no	—
23	C₂×C₇₁⋊C₁₆	—	2	2	no	—
24	C₄×C₇₁⋊C₈	—	2	2	no	—
25	C₈×Dic₇₁	—	2	2	no	—
26	C₈×Dic₇₁	—	2	2	no	—
27	C₈×Dic₇₁	C₈	1	2	yes	—
28	C₈×Dic₇₁	C₈	1	2	yes	—
29	C₈×Dic₇₁	D₄	1	2	yes	—
30	C₈×Dic₇₁	Q₈	1	2	yes	—
31	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
34	M₃₂×C₇₁	—	2	2	no	—
35	M₃₂×C₇₁	—	2	2	no	—
36	C₂×C₁₆×C₇₁	—	2	2	no	—
37	C₄×C₈×C₇₁	—	2	2	no	—
38	C₂²×Dic₇₁	—	2	1	no	—
39	C₂²×Dic₇₁	M₁₆	1	2	yes	—
40	C₂×C₇₁⋊C₈	—	2	1	no	—
41	C₂×C₇₁⋊C₈	—	2	1	no	—
42	C₂×C₇₁⋊C₈	—	2	1	no	—
43	C₂×C₇₁⋊C₈	—	2	1	no	—
44	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
45	C₂×C₇₁⋊C₈	C₂×C₈	1	2	yes	—
46	C₄×Dic₇₁	—	2	8	no	—
47	C₄×Dic₇₁	—	2	1	no	—
48	C₄×Dic₇₁	C₄⋊C₄	1	2	yes	—
49	C₄×Dic₇₁	C₂×C₈	1	1	yes	characteristic
50	C₄×Dic₇₁	C₂×C₈	1	1	yes	characteristic
51	C₄×Dic₇₁	M₁₆	1	2	yes	—
52	C₇₁⋊C₁₆	—	4	2	no	—
53	C₂²×C₄×C₇₁	—	2	1	no	—
54	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₇₁	—	2	1	no	—
56	C₂×C₈×C₇₁	—	2	1	no	—
57	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
58	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₇₁	#41	1	2	yes	—
61	C₂×C₈×C₇₁	#41	1	2	yes	—
62	C₄²×C₇₁	—	2	1	no	—
63	C₄²×C₇₁	M₁₆	1	2	yes	—
64	C₁₆×C₇₁	—	4	2	no	—
65	C₂³×C₇₁	C₂×M₁₆	1	1	yes	characteristic
66	C₂×C₄×C₇₁	—	2	4	no	—
67	C₂×C₄×C₇₁	—	2	2	no	—
68	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
70	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
71	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
74	C₂×Dic₇₁	—	2	4	no	—
75	C₂×Dic₇₁	—	2	2	no	—
76	C₂×Dic₇₁	—	2	4	no	—
77	C₂×Dic₇₁	C₄⋊C₈	1	2	yes	—
78	C₂×Dic₇₁	C₄⋊C₈	1	2	yes	—
79	C₈×C₇₁	—	2	2	no	—
80	C₈×C₇₁	#124	1	2	yes	—
81	C₈×C₇₁	#125	1	2	yes	—
82	C₇₁⋊C₈	—	2	2	no	—
83	C₇₁⋊C₈	—	2	2	no	—
84	C₂²×C₇₁	—	2	2	no	—
85	C₂²×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
86	C₂²×C₇₁	#432	1	1	yes	characteristic
87	C₂²×C₇₁	#432	1	1	yes	characteristic
88	C₄×C₇₁	—	2	4	no	—
89	C₄×C₇₁	#382	1	1	yes	characteristic
90	C₄×C₇₁	#423	1	1	yes	characteristic
91	C₄×C₇₁	#431	1	1	yes	characteristic
92	C₄×C₇₁	#431	1	1	yes	characteristic
93	Dic₇₁	—	2	4	no	—
94	Dic₇₁	—	2	2	no	—
95	Dic₇₁	#364	1	4	yes	—
96	#12883	—	71	1	no	maximal, Sylow
97	C₂×C₇₁	—	2	1	no	—
98	C₂×C₇₁	—	2	1	no	—
99	C₂×C₇₁	#1831	1	1	yes	characteristic
100	C₂×C₇₁	C₂×#364	1	2	yes	—
101	#2150	—	71	2	no	—
102	C₄²×C₈	—	71	1	no	—
103	#1376	—	71	2	no	—
104	#1376	—	71	1	no	—
105	#1376	D₇₁	1	1	yes	characteristic
106	C₇₁	#12883	1	1	yes	characteristic, complemented, Sylow
107	C₂×M₃₂	—	71	2	no	—
108	C₂×M₃₂	C₂×D₇₁	1	2	yes	—
109	C₂×C₄×C₈	—	71	1	no	—
110	C₂×C₄×C₈	—	71	1	no	—
111	C₂×C₄×C₈	—	71	1	no	—
112	C₂×C₄×C₈	—	71	1	no	—
113	C₂×C₄×C₈	—	71	1	no	—
114	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
115	C₄³	—	71	1	no	—
116	C₄⋊C₁₆	—	142	4	no	—
117	C₂²×C₈	—	71	1	no	—
118	C₂²×C₈	—	71	1	no	—
119	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
120	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
121	C₂×C₄²	—	71	1	no	—
122	C₂×C₄²	—	71	1	no	—
123	C₂×C₄²	—	142	2	no	—
124	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
125	M₃₂	—	2	2	no	—
126	M₃₂	—	2	2	no	—
127	M₃₂	—	142	2	no	—
128	M₃₂	—	142	2	no	—
129	C₂×C₁₆	—	2	2	no	—
130	C₂×C₁₆	—	142	2	no	—
131	C₄×C₈	—	2	2	no	—
132	C₄×C₈	—	71	2	no	—
133	C₄×C₈	—	71	2	no	—
134	C₄×C₈	—	71	2	no	—
135	C₄×C₈	—	71	2	no	—
136	C₄×C₈	—	142	2	no	—
137	C₄×C₈	—	142	2	no	—
138	C₄×C₈	—	142	2	no	—
139	C₂²×C₄	—	2	1	no	—
140	C₂²×C₄	—	71	2	no	—
141	C₂²×C₄	—	142	1	no	—
142	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	71	2	no	—
146	C₂×C₈	—	71	2	no	—
147	C₂×C₈	—	142	1	no	—
148	C₂×C₈	—	142	1	no	—
149	C₂×C₈	—	142	1	no	—
150	C₂×C₈	—	142	1	no	—
151	C₂×C₈	C₈×D₇₁	1	2	yes	—
152	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
153	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
154	C₂×C₈	#11765864	1	2	yes	—
155	C₂×C₈	#11765865	1	2	yes	—
156	C₄²	—	2	1	no	—
157	C₄²	—	71	2	no	—
158	C₄²	—	71	1	no	—
159	C₄²	—	71	1	no	—
160	C₄²	—	71	2	no	—
161	C₄²	—	142	8	no	—
162	C₄²	—	142	1	no	—
163	C₄²	#11765848	1	2	yes	—
164	C₁₆	—	4	2	no	—
165	C₁₆	—	284	2	no	—
166	C₂³	M₁₆×D₇₁	1	1	yes	characteristic
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	71	2	no	—
170	C₂×C₄	—	71	2	no	—
171	C₂×C₄	—	142	4	no	—
172	C₂×C₄	—	142	4	no	—
173	C₂×C₄	—	142	2	no	—
174	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
175	C₂×C₄	#14556084	1	1	yes	characteristic
176	C₂×C₄	#14556085	1	1	yes	characteristic
177	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
178	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
179	C₂×C₄	C₂×#11765848	1	1	yes	characteristic
180	C₈	—	2	2	no	—
181	C₈	—	142	2	no	—
182	C₈	—	142	2	no	—
183	C₈	#14556114	1	2	yes	—
184	C₈	#14556118	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	C₄⋊C₈×D₇₁	1	1	yes	characteristic
187	C₂²	#17986053	1	1	yes	characteristic
188	C₂²	#17986054	1	1	yes	characteristic
189	C₄	—	2	4	no	—
190	C₄	—	71	4	no	—
191	C₄	—	142	4	no	—
192	C₄	—	142	2	no	—
193	C₄	#17985863	1	1	yes	characteristic
194	C₄	#17986021	1	1	yes	characteristic
195	C₄	#17986049	1	1	yes	characteristic
196	C₄	#17986050	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	#22588455	1	1	yes	characteristic
200	C₂	#364×D₇₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22590095

C₂

yes

maximal

C₄×C₈×Dic₇₁

C₂

yes

maximal, characteristic

#22586422

C₂

yes

maximal, characteristic

#22586421

C₂

yes

maximal

C₇₁×#1376

C₂

yes

maximal, characteristic

C₂×#14556068

C₂²

yes

—

C₂×C₄×C₇₁⋊C₈

C₄

yes

characteristic

C₂×C₈×Dic₇₁

C₄

yes

characteristic

C₂×C₈×Dic₇₁

C₄

yes

characteristic

C₂×C₈×Dic₇₁

C₂²

yes

characteristic

C₂×C₈×Dic₇₁

C₂²

yes

characteristic

C₄²×Dic₇₁

C₄

yes

characteristic

Dic₇₁⋊C₁₆

—

C₂×M₃₂×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂²×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₄×Dic₇₁

—

C₂×C₄×Dic₇₁

C₂×C₄

yes

characteristic

C₂×C₄×Dic₇₁

C₂×C₄

yes

characteristic

#14556068

—

#14556068

—

C₂×C₇₁⋊C₁₆

—

C₄×C₇₁⋊C₈

—

C₈×Dic₇₁

—

C₈×Dic₇₁

—

C₈×Dic₇₁

C₈

yes

—

C₈×Dic₇₁

C₈

yes

—

C₈×Dic₇₁

D₄

yes

—

C₈×Dic₇₁

Q₈

yes

—

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄²×C₇₁

C₂×C₄

yes

characteristic

M₃₂×C₇₁

—

M₃₂×C₇₁

—

C₂×C₁₆×C₇₁

—

C₄×C₈×C₇₁

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

M₁₆

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

—

C₂×C₇₁⋊C₈

C₂×C₈

yes

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

C₄⋊C₄

yes

—

C₄×Dic₇₁

C₂×C₈

yes

characteristic

C₄×Dic₇₁

C₂×C₈

yes

characteristic

C₄×Dic₇₁

M₁₆

yes

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×C₈

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₁

#41

yes

—

C₂×C₈×C₇₁

#41

yes

—

C₄²×C₇₁

—

C₄²×C₇₁

M₁₆

yes

—

C₁₆×C₇₁

—

C₂³×C₇₁

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₁

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×M₁₆

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

C₄⋊C₈

yes

—

C₂×Dic₇₁

C₄⋊C₈

yes

—

C₈×C₇₁

—

C₈×C₇₁

#124

yes

—

C₈×C₇₁

#125

yes

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×C₄⋊C₈

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

Dic₇₁

#364

yes

—

#12883

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#1831

yes

characteristic

100

C₂×C₇₁

C₂×#364

yes

—

101

#2150

—

102

C₄²×C₈

—

103

#1376

—

104

#1376

—

105

#1376

D₇₁

yes

characteristic

106

C₇₁

#12883

yes

characteristic, complemented, Sylow

107

C₂×M₃₂

—

108

C₂×M₃₂

C₂×D₇₁

yes

—

109

C₂×C₄×C₈

—

110

C₂×C₄×C₈

—

111

C₂×C₄×C₈

—

112

C₂×C₄×C₈

—

113

C₂×C₄×C₈

—

114

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

115

C₄³

—

116

C₄⋊C₁₆

—

142

—

117

C₂²×C₈

—

118

C₂²×C₈

—

119

C₂²×C₈

C₄×D₇₁

yes

characteristic

120

C₂²×C₈

C₂²×D₇₁

yes

characteristic

121

C₂×C₄²

—

122

C₂×C₄²

—

123

C₂×C₄²

—

142

—

124

C₂×C₄²

C₄×D₇₁

yes

characteristic

125

M₃₂

—

126

M₃₂

—

127

M₃₂

—

142

—

128

M₃₂

—

142

—

129

C₂×C₁₆

—

130

C₂×C₁₆

—

142

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

—

134

C₄×C₈

—

135

C₄×C₈

—

136

C₄×C₈

—

142

—

137

C₄×C₈

—

142

—

138

C₄×C₈

—

142

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

—

142

C₂²×C₄

C₂×C₄×D₇₁

yes

characteristic

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

142

—

148

C₂×C₈

—

142

—

149

C₂×C₈

—

142

—

150

C₂×C₈

—

142

—

151

C₂×C₈

C₈×D₇₁

yes

—

152

C₂×C₈

D₄×D₇₁

yes

characteristic

153

C₂×C₈

Q₈×D₇₁

yes

characteristic

154

C₂×C₈

#11765864

yes

—

155

C₂×C₈

#11765865

yes

—

156

C₄²

—

157

C₄²

—

158

C₄²

—

159

C₄²

—

160

C₄²

—

161

C₄²

—

142

—

162

C₄²

—

142

—

163

C₄²

#11765848

yes

—

164

C₁₆

—

165

C₁₆

—

284

—

166

C₂³

M₁₆×D₇₁

yes

characteristic

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

142

—

172

C₂×C₄

—

142

—

173

C₂×C₄

—

142

—

174

C₂×C₄

C₄⋊C₄×D₇₁

yes

characteristic

175

C₂×C₄

#14556084

yes

characteristic

176

C₂×C₄

#14556085

yes

characteristic

177

C₂×C₄

C₂×C₈×D₇₁

yes

characteristic

178

C₂×C₄

M₁₆×D₇₁

yes

characteristic

179

C₂×C₄

C₂×#11765848

yes

characteristic

180

C₈

—

181

C₈

—

142

—

182

C₈

—

142

—

183

C₈

#14556114

yes

—

184

C₈

#14556118

yes

—

185

C₂²

—

186

C₂²

C₄⋊C₈×D₇₁

yes

characteristic

187

C₂²

#17986053

yes

characteristic

188

C₂²

#17986054

yes

characteristic

189

C₄

—

190

C₄

—

191

C₄

—

142

—

192

C₄

—

142

—

193

C₄

#17985863

yes

characteristic

194

C₄

#17986021

yes

characteristic

195

C₄

#17986049

yes

characteristic

196

C₄

#17986050

yes

characteristic

197

C₂

—

198

C₂

—

199

C₂

#22588455

yes

characteristic

200

C₂

#364×D₇₁

yes

—

	subgroup	G/H
center	C₂×C₈	D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1376	C₂
socle	C₂²×C₇₁	C₂×C₄⋊C₈
prime radical	C₂³×C₇₁	C₂×M₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1376	D₇₁
hypocenter	C₇₁	#12883
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

subgroup

G/H

center

C₂×C₈

D₄×D₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₇₁×#1376

C₂

socle

C₂²×C₇₁

C₂×C₄⋊C₈

prime radical

C₂³×C₇₁

C₂×M₁₆

cosocle

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

N	H ≅ G/N	type
106. C₇₁	#12883	semidirect

H ≅ G/N

type

106. C₇₁

#12883

semidirect

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

subgroup	index
#22590095	2
C₄×C₈×Dic₇₁	2
#22586422	2
#22586421	2
C₇₁×#1376	2
#12883	71

subgroup

index

C₄×C₈×Dic₇₁

C₇₁×#1376

Subgroups — #31913811

Conjugacy classes of subgroups

Complements

Maximal subgroups