Group #31908995 — subgroups

Subgroups — #31908995

Order 18176.

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

normal subgroups only

This group has 270 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	#22590094	C₂	1	2	yes	maximal
2	C₈²×D₇₁	C₂	1	1	yes	maximal, characteristic
3	#22586404	C₂	1	1	yes	maximal, characteristic, complemented
4	#22586403	C₂	1	2	yes	maximal, complemented
5	C₇₁×#1367	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14556067	—	2	2	no	—
7	C₄×C₈×D₇₁	C₄	1	1	yes	characteristic
8	C₄×C₈×D₇₁	C₄	1	1	yes	characteristic
9	C₄×C₈×D₇₁	C₂²	1	1	yes	characteristic
10	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	2	yes	—
11	Dic₇₁⋊C₁₆	—	2	2	no	—
12	C₈×C₇₁⋊C₈	C₄	1	1	yes	characteristic
13	C₈×C₇₁⋊C₈	C₄	1	1	yes	characteristic
14	C₈×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₇₁	C₂²	1	2	yes	—
16	C₈²×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₇₁	—	2	1	no	—
18	C₂×C₈×D₇₁	C₈	1	1	yes	characteristic
19	C₂×C₈×D₇₁	C₈	1	1	yes	characteristic
20	C₂×C₈×D₇₁	D₄	1	1	yes	characteristic
21	C₂×C₈×D₇₁	Q₈	1	1	yes	characteristic
22	C₄²×D₇₁	C₂×C₄	1	1	yes	characteristic
23	#14556067	—	4	4	no	—
24	C₂×C₇₁⋊C₁₆	—	2	2	no	—
25	C₄×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₇₁	—	2	1	no	—
29	C₈×Dic₇₁	C₈	1	1	yes	characteristic
30	C₈×Dic₇₁	C₈	1	1	yes	characteristic
31	C₈×Dic₇₁	D₄	1	1	yes	characteristic
32	C₈×Dic₇₁	Q₈	1	1	yes	characteristic
33	C₂×C₁₆×C₇₁	—	2	2	no	—
34	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
37	C₂×C₄×D₇₁	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×D₇₁	C₂×C₈	1	1	yes	characteristic
39	C₂×C₄×D₇₁	M₁₆	1	1	yes	characteristic
40	C₂×C₇₁⋊C₈	—	2	1	no	—
41	C₂×C₇₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₇₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₇₁⋊C₈	C₂×C₈	1	1	yes	characteristic
44	C₂×C₇₁⋊C₈	C₂×C₈	1	1	yes	characteristic
45	C₈×D₇₁	—	2	4	no	—
46	C₈×D₇₁	—	2	2	no	—
47	C₈×D₇₁	—	2	2	no	—
48	C₈×D₇₁	—	2	2	no	—
49	C₈×D₇₁	SD₁₆	1	4	yes	—
50	C₄×Dic₇₁	C₄⋊C₄	1	1	yes	characteristic
51	C₄×Dic₇₁	C₂×C₈	1	1	yes	characteristic
52	C₄×Dic₇₁	M₁₆	1	1	yes	characteristic
53	C₇₁⋊C₁₆	—	4	2	no	—
54	C₂×C₈×C₇₁	—	2	1	no	—
55	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
56	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
57	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
60	C₁₆×C₇₁	—	4	2	no	—
61	C₂²×D₇₁	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
65	C₂×Dic₇₁	C₄⋊C₈	1	1	yes	characteristic
66	C₂×Dic₇₁	C₄⋊C₈	1	1	yes	characteristic
67	C₂×Dic₇₁	C₄⋊C₈	1	1	yes	characteristic
68	C₄×D₇₁	—	2	1	no	—
69	C₄×D₇₁	—	2	1	no	—
70	C₄×D₇₁	—	2	2	no	—
71	C₄×D₇₁	#106	1	2	yes	—
72	C₄×D₇₁	#108	1	2	yes	—
73	C₈×C₇₁	—	2	2	no	—
74	C₈×C₇₁	—	2	1	no	—
75	C₈×C₇₁	—	2	1	no	—
76	C₈×C₇₁	M₃₂	1	2	yes	complemented
77	C₈×C₇₁	C₂×SD₁₆	1	2	yes	—
78	C₇₁⋊C₈	—	2	2	no	—
79	C₇₁⋊C₈	—	2	1	no	—
80	C₇₁⋊C₈	—	2	1	no	—
81	C₇₁⋊C₈	#106	1	2	yes	—
82	C₇₁⋊C₈	M₃₂	1	2	yes	complemented
83	C₂²×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
84	C₂×D₇₁	—	2	1	no	—
85	C₂×D₇₁	—	2	1	no	—
86	C₂×D₇₁	#334	1	2	yes	—
87	C₄×C₇₁	—	2	1	no	—
88	C₄×C₇₁	C₂×#106	1	1	yes	characteristic
89	C₄×C₇₁	C₂×#108	1	1	yes	characteristic
90	C₄×C₇₁	C₂×M₃₂	1	1	yes	characteristic
91	C₄×C₇₁	C₂×M₃₂	1	1	yes	characteristic
92	Dic₇₁	—	2	1	no	—
93	Dic₇₁	#334	1	2	yes	—
94	Dic₇₁	#364	1	2	yes	—
95	#11996	—	71	1	no	maximal, Sylow
96	C₂×C₇₁	C₂×#334	1	1	yes	characteristic
97	C₂×C₇₁	#2150	1	1	yes	characteristic
98	C₂×C₇₁	C₂×#364	1	1	yes	characteristic
99	D₇₁	—	2	2	no	—
100	#2150	—	71	2	no	—
101	C₂×C₈²	—	71	1	no	—
102	#1367	—	71	2	no	—
103	#1367	—	71	1	no	—
104	#1367	D₇₁	1	1	yes	characteristic, complemented
105	C₇₁	#11996	1	1	yes	characteristic, complemented, Sylow
106	C₂×M₃₂	—	142	2	no	—
107	C₂×C₄×C₈	—	71	1	no	—
108	C₂×C₄×C₈	—	71	1	no	—
109	C₂×C₄×C₈	—	71	1	no	—
110	C₄⋊C₁₆	—	71	2	no	—
111	C₄⋊C₁₆	—	142	2	no	—
112	C₄⋊C₁₆	C₂×D₇₁	1	2	yes	—
113	C₈²	—	71	1	no	—
114	C₈²	—	71	1	no	—
115	C₈²	—	71	1	no	—
116	C₈²	C₂×D₇₁	1	1	yes	characteristic
117	C₂²×C₈	—	71	1	no	—
118	C₂²×C₈	—	71	1	no	—
119	C₂²×C₈	—	71	1	no	—
120	C₂²×C₈	—	71	1	no	—
121	C₂²×C₈	—	142	1	no	—
122	C₂×C₄²	—	71	1	no	—
123	M₃₂	—	284	4	no	—
124	C₂×C₁₆	—	2	2	no	—
125	C₂×C₁₆	—	142	2	no	—
126	C₄×C₈	—	71	1	no	—
127	C₄×C₈	—	71	1	no	—
128	C₄×C₈	—	71	1	no	—
129	C₄×C₈	—	71	1	no	—
130	C₄×C₈	—	71	1	no	—
131	C₄×C₈	—	71	1	no	—
132	C₄×C₈	—	71	1	no	—
133	C₄×C₈	—	142	1	no	—
134	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
135	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
136	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
137	C₂²×C₄	—	71	1	no	—
138	C₂²×C₄	—	71	1	no	—
139	C₂²×C₄	—	71	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	71	1	no	—
142	C₂×C₈	—	71	1	no	—
143	C₂×C₈	—	71	4	no	—
144	C₂×C₈	—	71	1	no	—
145	C₂×C₈	—	71	1	no	—
146	C₂×C₈	—	142	2	no	—
147	C₂×C₈	—	142	2	no	—
148	C₂×C₈	—	142	4	no	—
149	C₂×C₈	—	142	1	no	—
150	C₂×C₈	—	142	2	no	—
151	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
152	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
153	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
154	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
155	C₄²	—	71	1	no	—
156	C₄²	—	71	1	no	—
157	C₄²	—	71	1	no	—
158	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
159	C₁₆	—	4	2	no	—
160	C₁₆	—	284	2	no	—
161	C₂³	—	71	1	no	—
162	C₂×C₄	—	71	1	no	—
163	C₂×C₄	—	71	1	no	—
164	C₂×C₄	—	71	2	no	—
165	C₂×C₄	—	71	2	no	—
166	C₂×C₄	—	71	1	no	—
167	C₂×C₄	—	142	2	no	—
168	C₂×C₄	—	142	1	no	—
169	C₂×C₄	—	142	1	no	—
170	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
171	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
172	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
173	C₈	—	2	2	no	—
174	C₈	—	2	1	no	—
175	C₈	—	2	1	no	—
176	C₈	—	71	2	no	—
177	C₈	—	71	2	no	—
178	C₈	—	142	2	no	—
179	C₈	—	142	1	no	—
180	C₈	—	142	1	no	—
181	C₈	#14556067	1	2	yes	complemented
182	C₈	SD₁₆×D₇₁	1	2	yes	—
183	C₂²	—	71	2	no	—
184	C₂²	—	142	1	no	—
185	C₂²	—	142	1	no	—
186	C₂²	C₄⋊C₈×D₇₁	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	71	2	no	—
189	C₄	—	71	2	no	—
190	C₄	—	142	1	no	—
191	C₄	#106×D₇₁	1	1	yes	characteristic
192	C₄	#108×D₇₁	1	1	yes	characteristic
193	C₄	M₃₂×D₇₁	1	1	yes	characteristic
194	C₄	C₂×#14556067	1	1	yes	characteristic
195	C₂	—	142	2	no	—
196	C₂	#334×D₇₁	1	1	yes	characteristic
197	C₂	#22590094	1	1	yes	characteristic
198	C₂	#364×D₇₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590094

C₂

yes

maximal

C₈²×D₇₁

C₂

yes

maximal, characteristic

#22586404

C₂

yes

maximal, characteristic, complemented

#22586403

C₂

yes

maximal, complemented

C₇₁×#1367

C₂

yes

maximal, characteristic, complemented

C₂×#14556067

—

C₄×C₈×D₇₁

C₄

yes

characteristic

C₄×C₈×D₇₁

C₄

yes

characteristic

C₄×C₈×D₇₁

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₁₆)

C₂²

yes

—

Dic₇₁⋊C₁₆

—

C₈×C₇₁⋊C₈

C₄

yes

characteristic

C₈×C₇₁⋊C₈

C₄

yes

characteristic

C₈×C₇₁⋊C₈

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₁

C₂²

yes

—

C₈²×C₇₁

C₂²

yes

characteristic

C₂×C₈×D₇₁

—

C₂×C₈×D₇₁

C₈

yes

characteristic

C₂×C₈×D₇₁

C₈

yes

characteristic

C₂×C₈×D₇₁

D₄

yes

characteristic

C₂×C₈×D₇₁

Q₈

yes

characteristic

C₄²×D₇₁

C₂×C₄

yes

characteristic

#14556067

—

C₂×C₇₁⋊C₁₆

—

C₄×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₄×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₄×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₈×Dic₇₁

—

C₈×Dic₇₁

C₈

yes

characteristic

C₈×Dic₇₁

C₈

yes

characteristic

C₈×Dic₇₁

D₄

yes

characteristic

C₈×Dic₇₁

Q₈

yes

characteristic

C₂×C₁₆×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄×D₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×D₇₁

C₂×C₈

yes

characteristic

C₂×C₄×D₇₁

M₁₆

yes

characteristic

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₇₁⋊C₈

C₂×C₈

yes

characteristic

C₂×C₇₁⋊C₈

C₂×C₈

yes

characteristic

C₈×D₇₁

—

C₈×D₇₁

—

C₈×D₇₁

—

C₈×D₇₁

—

C₈×D₇₁

SD₁₆

yes

—

C₄×Dic₇₁

C₄⋊C₄

yes

characteristic

C₄×Dic₇₁

C₂×C₈

yes

characteristic

C₄×Dic₇₁

M₁₆

yes

characteristic

C₇₁⋊C₁₆

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×C₈

yes

characteristic

C₂×C₈×C₇₁

C₂×C₈

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

C₄²×C₇₁

C₂²×C₄

yes

characteristic

C₁₆×C₇₁

—

C₂²×D₇₁

C₄⋊C₈

yes

characteristic

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₁

C₂×M₁₆

yes

characteristic

C₂×Dic₇₁

C₄⋊C₈

yes

characteristic

C₂×Dic₇₁

C₄⋊C₈

yes

characteristic

C₂×Dic₇₁

C₄⋊C₈

yes

characteristic

C₄×D₇₁

—

C₄×D₇₁

—

C₄×D₇₁

—

C₄×D₇₁

#106

yes

—

C₄×D₇₁

#108

yes

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

M₃₂

yes

complemented

C₈×C₇₁

C₂×SD₁₆

yes

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

#106

yes

—

C₇₁⋊C₈

M₃₂

yes

complemented

C₂²×C₇₁

C₂×C₄⋊C₈

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

#334

yes

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

C₂×M₃₂

yes

characteristic

C₄×C₇₁

C₂×M₃₂

yes

characteristic

Dic₇₁

—

Dic₇₁

#334

yes

—

Dic₇₁

#364

yes

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

—

100

#2150

—

101

C₂×C₈²

—

102

#1367

—

103

#1367

—

104

#1367

D₇₁

yes

characteristic, complemented

105

C₇₁

#11996

yes

characteristic, complemented, Sylow

106

C₂×M₃₂

—

142

—

107

C₂×C₄×C₈

—

108

C₂×C₄×C₈

—

109

C₂×C₄×C₈

—

110

C₄⋊C₁₆

—

111

C₄⋊C₁₆

—

142

—

112

C₄⋊C₁₆

C₂×D₇₁

yes

—

113

C₈²

—

114

C₈²

—

115

C₈²

—

116

C₈²

C₂×D₇₁

yes

characteristic

117

C₂²×C₈

—

118

C₂²×C₈

—

119

C₂²×C₈

—

120

C₂²×C₈

—

121

C₂²×C₈

—

142

—

122

C₂×C₄²

—

123

M₃₂

—

284

—

124

C₂×C₁₆

—

125

C₂×C₁₆

—

142

—

126

C₄×C₈

—

127

C₄×C₈

—

128

C₄×C₈

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

—

142

—

134

C₄×C₈

C₄×D₇₁

yes

characteristic

135

C₄×C₈

C₄×D₇₁

yes

characteristic

136

C₄×C₈

C₂²×D₇₁

yes

characteristic

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

142

—

147

C₂×C₈

—

142

—

148

C₂×C₈

—

142

—

149

C₂×C₈

—

142

—

150

C₂×C₈

—

142

—

151

C₂×C₈

C₈×D₇₁

yes

characteristic

152

C₂×C₈

C₈×D₇₁

yes

characteristic

153

C₂×C₈

D₄×D₇₁

yes

characteristic

154

C₂×C₈

Q₈×D₇₁

yes

characteristic

155

C₄²

—

156

C₄²

—

157

C₄²

—

158

C₄²

C₂×C₄×D₇₁

yes

characteristic

159

C₁₆

—

160

C₁₆

—

284

—

161

C₂³

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

142

—

168

C₂×C₄

—

142

—

169

C₂×C₄

—

142

—

170

C₂×C₄

C₄⋊C₄×D₇₁

yes

characteristic

171

C₂×C₄

C₂×C₈×D₇₁

yes

characteristic

172

C₂×C₄

M₁₆×D₇₁

yes

characteristic

173

C₈

—

174

C₈

—

175

C₈

—

176

C₈

—

177

C₈

—

178

C₈

—

142

—

179

C₈

—

142

—

180

C₈

—

142

—

181

C₈

#14556067

yes

complemented

182

C₈

SD₁₆×D₇₁

yes

—

183

C₂²

—

184

C₂²

—

142

—

185

C₂²

—

142

—

186

C₂²

C₄⋊C₈×D₇₁

yes

characteristic

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

142

—

191

C₄

#106×D₇₁

yes

characteristic

192

C₄

#108×D₇₁

yes

characteristic

193

C₄

M₃₂×D₇₁

yes

characteristic

194

C₄

C₂×#14556067

yes

characteristic

195

C₂

—

142

—

196

C₂

#334×D₇₁

yes

characteristic

197

C₂

#22590094

yes

characteristic

198

C₂

#364×D₇₁

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1367	C₂
socle	C₂²×C₇₁	C₂×C₄⋊C₈
prime radical	C₂²×D₇₁	C₄⋊C₈
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	#1367	D₇₁
hypocenter	C₇₁	#11996
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1367

subgroup

G/H

center

C₂×C₈

D₄×D₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#1367

C₂

socle

C₂²×C₇₁

C₂×C₄⋊C₈

prime radical

C₂²×D₇₁

C₄⋊C₈

cosocle

C₄×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1367

N	H ≅ G/N	type
3. #22586404	C₂	semidirect
4. #22586403	C₂	semidirect
5. C₇₁×#1367	C₂	semidirect
76. C₈×C₇₁	M₃₂	semidirect
82. C₇₁⋊C₈	M₃₂	semidirect
104. #1367	D₇₁	semidirect
105. C₇₁	#11996	semidirect
181. C₈	#14556067	semidirect

H ≅ G/N

type

3. #22586404

C₂

semidirect

4. #22586403

C₂

semidirect

5. C₇₁×#1367

C₂

semidirect

76. C₈×C₇₁

M₃₂

semidirect

82. C₇₁⋊C₈

M₃₂

semidirect

104. #1367

D₇₁

semidirect

105. C₇₁

#11996

semidirect

181. C₈

#14556067

semidirect

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

subgroup	index
#22590094	2
C₈²×D₇₁	2
#22586404	2
#22586403	2
C₇₁×#1367	2
#11996	71

subgroup

index

C₈²×D₇₁

C₇₁×#1367

Subgroups — #31908995

Conjugacy classes of subgroups

Complements

Maximal subgroups