Group #31830735 — subgroups

Subgroups — C₇₁×#9863 #31830735

Order 18176.

412 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 142 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	C₇₁×#3131	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#1837	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₁×#1493	C₂	1	2	yes	maximal, complemented
4	C₇₁×#1492	C₂	1	2	yes	maximal
5	C₂×#322×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#533×C₇₁	C₂²	1	1	yes	characteristic
7	#532×C₇₁	C₂²	1	1	yes	characteristic
8	#488×C₇₁	—	2	2	no	—
9	#487×C₇₁	C₄	1	2	yes	—
10	#486×C₇₁	C₄	1	2	yes	—
11	#423×C₇₁	C₂²	1	2	yes	—
12	#416×C₇₁	—	2	2	no	—
13	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	#341×C₇₁	—	2	2	no	—
16	Q₈⋊C₈×C₇₁	—	2	4	no	—
17	D₄⋊C₈×C₇₁	—	2	4	no	—
18	#322×C₇₁	—	2	2	no	—
19	C₂×#41×C₇₁	—	2	1	no	—
20	C₂×M₁₆×C₇₁	—	2	2	no	—
21	C₂²×C₈×C₇₁	—	2	2	no	—
22	C₂²×C₈×C₇₁	D₄	1	2	yes	—
23	C₄⋊Q₈×C₇₁	C₂×C₄	1	2	yes	—
24	C₄⋊Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
25	#127×C₇₁	C₂×C₄	1	1	yes	characteristic
26	#125×C₇₁	—	2	1	no	—
27	#124×C₇₁	—	2	1	no	—
28	C₂²⋊Q₈×C₇₁	—	2	2	no	—
29	#121×C₇₁	—	2	2	no	—
30	C₄×Q₈×C₇₁	—	2	2	no	—
31	C₄×D₄×C₇₁	—	2	2	no	—
32	#117×C₇₁	—	2	1	no	—
33	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₇₁	—	2	2	no	—
35	C₄⋊C₈×C₇₁	—	2	2	no	—
36	#103×C₇₁	—	2	4	no	—
37	#102×C₇₁	—	2	4	no	—
38	C₄×C₈×C₇₁	—	2	2	no	—
39	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₇₁	D₄	1	4	yes	—
41	C₄×C₈×C₇₁	D₄	1	2	yes	—
42	#41×C₇₁	—	4	4	no	—
43	C₂×Q₈×C₇₁	—	2	1	no	—
44	C₂×Q₈×C₇₁	—	2	2	no	—
45	C₂×D₄×C₇₁	—	2	1	no	—
46	C₂×D₄×C₇₁	—	2	2	no	—
47	C₂×D₄×C₇₁	—	4	1	no	—
48	C₂²×C₄×C₇₁	—	4	1	no	—
49	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₇₁	#41	1	2	yes	—
51	M₁₆×C₇₁	—	4	4	no	—
52	C₂×C₈×C₇₁	—	2	4	no	—
53	C₂×C₈×C₇₁	—	2	4	no	—
54	C₂×C₈×C₇₁	—	4	4	no	—
55	C₂×C₈×C₇₁	—	4	2	no	—
56	C₂×C₈×C₇₁	C₂²⋊C₄	1	4	yes	—
57	C₄⋊C₄×C₇₁	—	2	2	no	—
58	C₄⋊C₄×C₇₁	—	2	2	no	—
59	C₄⋊C₄×C₇₁	—	2	2	no	—
60	C₄⋊C₄×C₇₁	—	4	2	no	—
61	C₂²⋊C₄×C₇₁	—	4	1	no	—
62	C₂²⋊C₄×C₇₁	—	4	2	no	—
63	C₄²×C₇₁	—	4	1	no	—
64	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₇₁	C₂×D₄	1	2	yes	—
67	C₂³×C₇₁	—	4	1	no	—
68	C₂³×C₇₁	#124	1	1	yes	characteristic
69	Q₈×C₇₁	—	4	2	no	—
70	Q₈×C₇₁	—	4	4	no	—
71	D₄×C₇₁	—	4	2	no	—
72	D₄×C₇₁	—	4	4	no	—
73	D₄×C₇₁	—	8	2	no	—
74	C₂×C₄×C₇₁	—	2	4	no	—
75	C₂×C₄×C₇₁	—	4	2	no	—
76	C₂×C₄×C₇₁	—	4	1	no	—
77	C₂×C₄×C₇₁	—	8	2	no	—
78	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
80	C₂×C₄×C₇₁	#121	1	2	yes	—
81	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
82	C₂×C₄×C₇₁	#135	1	4	yes	—
83	C₈×C₇₁	—	2	4	no	—
84	C₈×C₇₁	—	4	4	no	—
85	C₈×C₇₁	—	8	2	no	—
86	C₂²×C₇₁	—	2	2	no	—
87	C₂²×C₇₁	—	8	1	no	—
88	C₂²×C₇₁	—	8	2	no	—
89	C₂²×C₇₁	#390	1	1	yes	characteristic
90	C₂²×C₇₁	#488	1	2	yes	—
91	C₄×C₇₁	—	2	2	no	—
92	C₄×C₇₁	—	8	1	no	—
93	C₄×C₇₁	—	8	2	no	—
94	C₄×C₇₁	#416	1	2	yes	—
95	C₄×C₇₁	#444	1	4	yes	—
96	C₄×C₇₁	#495	1	2	yes	—
97	#9863	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂×C₇₁	—	2	2	no	—
99	C₂×C₇₁	—	16	1	no	—
100	C₂×C₇₁	#1958	1	1	yes	characteristic
101	C₂×C₇₁	#1975	1	2	yes	—
102	#3131	C₂×C₇₁	1	1	yes	characteristic
103	#1837	C₂×C₇₁	1	1	yes	characteristic, complemented
104	#1493	C₂×C₇₁	1	2	yes	complemented
105	#1492	C₂×C₇₁	1	2	yes	—
106	C₂×#322	C₂×C₇₁	1	1	yes	characteristic, complemented
107	C₇₁	#9863	1	1	yes	characteristic, direct product factor, Sylow
108	#533	C₂²×C₇₁	1	1	yes	characteristic
109	#532	C₂²×C₇₁	1	1	yes	characteristic
110	#488	—	2	2	no	—
111	#487	C₄×C₇₁	1	2	yes	—
112	#486	C₄×C₇₁	1	2	yes	—
113	#423	C₂²×C₇₁	1	2	yes	—
114	#416	—	2	2	no	—
115	C₂×C₄×C₈	C₂²×C₇₁	1	2	yes	—
116	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
117	#341	—	2	2	no	—
118	Q₈⋊C₈	—	2	4	no	—
119	D₄⋊C₈	—	2	4	no	—
120	#322	—	2	2	no	—
121	C₂×#41	—	2	1	no	—
122	C₂×M₁₆	—	2	2	no	—
123	C₂²×C₈	—	2	2	no	—
124	C₂²×C₈	D₄×C₇₁	1	2	yes	—
125	C₄⋊Q₈	C₂×C₄×C₇₁	1	1	yes	characteristic
126	C₄⋊Q₈	C₂×C₄×C₇₁	1	2	yes	—
127	#127	C₂×C₄×C₇₁	1	1	yes	characteristic
128	#125	—	2	1	no	—
129	#124	—	2	1	no	—
130	C₂²⋊Q₈	—	2	2	no	—
131	#121	—	2	2	no	—
132	C₄×Q₈	—	2	2	no	—
133	C₄×D₄	—	2	2	no	—
134	#117	—	2	1	no	—
135	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	—	2	2	no	—
138	#103	—	2	4	no	—
139	#102	—	2	4	no	—
140	C₄×C₈	—	2	2	no	—
141	C₄×C₈	C₂×C₄×C₇₁	1	2	yes	—
142	C₄×C₈	D₄×C₇₁	1	4	yes	—
143	C₄×C₈	D₄×C₇₁	1	2	yes	—
144	#41	—	4	4	no	—
145	C₂×Q₈	—	2	1	no	—
146	C₂×Q₈	—	2	2	no	—
147	C₂×D₄	—	2	1	no	—
148	C₂×D₄	—	2	2	no	—
149	C₂×D₄	—	4	1	no	—
150	C₂²×C₄	—	4	1	no	—
151	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
152	C₂²×C₄	#41×C₇₁	1	2	yes	—
153	M₁₆	—	4	4	no	—
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	—	4	4	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	C₂²⋊C₄×C₇₁	1	4	yes	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₂²⋊C₄	—	4	1	no	—
164	C₂²⋊C₄	—	4	2	no	—
165	C₄²	—	4	1	no	—
166	C₄²	C₂²×C₄×C₇₁	1	1	yes	characteristic
167	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
168	C₄²	C₂×D₄×C₇₁	1	2	yes	—
169	C₂³	—	4	1	no	—
170	C₂³	#124×C₇₁	1	1	yes	characteristic
171	Q₈	—	4	2	no	—
172	Q₈	—	4	4	no	—
173	D₄	—	4	2	no	—
174	D₄	—	4	4	no	—
175	D₄	—	8	2	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	1	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
181	C₂×C₄	C₄×D₄×C₇₁	1	2	yes	—
182	C₂×C₄	#121×C₇₁	1	2	yes	—
183	C₂×C₄	#127×C₇₁	1	1	yes	characteristic
184	C₂×C₄	#135×C₇₁	1	4	yes	—
185	C₈	—	2	4	no	—
186	C₈	—	4	4	no	—
187	C₈	—	8	2	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	8	1	no	—
190	C₂²	—	8	2	no	—
191	C₂²	#390×C₇₁	1	1	yes	characteristic
192	C₂²	#488×C₇₁	1	2	yes	—
193	C₄	—	2	2	no	—
194	C₄	—	8	1	no	—
195	C₄	—	8	2	no	—
196	C₄	#416×C₇₁	1	2	yes	—
197	C₄	#444×C₇₁	1	4	yes	—
198	C₄	#495×C₇₁	1	2	yes	—
199	C₂	—	2	2	no	—
200	C₂	—	16	1	no	—
201	C₂	C₇₁×#1958	1	1	yes	characteristic
202	C₂	C₇₁×#1975	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₁×#3131

C₂

yes

maximal, characteristic

C₇₁×#1837

C₂

yes

maximal, characteristic, complemented

C₇₁×#1493

C₂

yes

maximal, complemented

C₇₁×#1492

C₂

yes

maximal

C₂×#322×C₇₁

C₂

yes

maximal, characteristic, complemented

#533×C₇₁

C₂²

yes

characteristic

#532×C₇₁

C₂²

yes

characteristic

#488×C₇₁

—

#487×C₇₁

C₄

yes

—

#486×C₇₁

C₄

yes

—

#423×C₇₁

C₂²

yes

—

#416×C₇₁

—

C₂×C₄×C₈×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#341×C₇₁

—

Q₈⋊C₈×C₇₁

—

D₄⋊C₈×C₇₁

—

#322×C₇₁

—

C₂×#41×C₇₁

—

C₂×M₁₆×C₇₁

—

C₂²×C₈×C₇₁

—

C₂²×C₈×C₇₁

D₄

yes

—

C₄⋊Q₈×C₇₁

C₂×C₄

yes

—

C₄⋊Q₈×C₇₁

C₂×C₄

yes

characteristic

#127×C₇₁

C₂×C₄

yes

characteristic

#125×C₇₁

—

#124×C₇₁

—

C₂²⋊Q₈×C₇₁

—

#121×C₇₁

—

C₄×Q₈×C₇₁

—

C₄×D₄×C₇₁

—

#117×C₇₁

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

—

#103×C₇₁

—

#102×C₇₁

—

C₄×C₈×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

D₄

yes

—

C₄×C₈×C₇₁

D₄

yes

—

#41×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

#41

yes

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

C₂²×C₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

—

C₂³×C₇₁

—

C₂³×C₇₁

#124

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄×D₄

yes

—

C₂×C₄×C₇₁

#121

yes

—

C₂×C₄×C₇₁

#127

yes

characteristic

C₂×C₄×C₇₁

#135

yes

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#390

yes

characteristic

C₂²×C₇₁

#488

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#416

yes

—

C₄×C₇₁

#444

yes

—

C₄×C₇₁

#495

yes

—

#9863

C₇₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

100

C₂×C₇₁

#1958

yes

characteristic

101

C₂×C₇₁

#1975

yes

—

102

#3131

C₂×C₇₁

yes

characteristic

103

#1837

C₂×C₇₁

yes

characteristic, complemented

104

#1493

C₂×C₇₁

yes

complemented

105

#1492

C₂×C₇₁

yes

—

106

C₂×#322

C₂×C₇₁

yes

characteristic, complemented

107

C₇₁

#9863

yes

characteristic, direct product factor, Sylow

108

#533

C₂²×C₇₁

yes

characteristic

109

#532

C₂²×C₇₁

yes

characteristic

110

#488

—

111

#487

C₄×C₇₁

yes

—

112

#486

C₄×C₇₁

yes

—

113

#423

C₂²×C₇₁

yes

—

114

#416

—

115

C₂×C₄×C₈

C₂²×C₇₁

yes

—

116

C₂×C₄×C₈

C₂²×C₇₁

yes

characteristic

117

#341

—

118

Q₈⋊C₈

—

119

D₄⋊C₈

—

120

#322

—

121

C₂×#41

—

122

C₂×M₁₆

—

123

C₂²×C₈

—

124

C₂²×C₈

D₄×C₇₁

yes

—

125

C₄⋊Q₈

C₂×C₄×C₇₁

yes

characteristic

126

C₄⋊Q₈

C₂×C₄×C₇₁

yes

—

127

#127

C₂×C₄×C₇₁

yes

characteristic

128

#125

—

129

#124

—

130

C₂²⋊Q₈

—

131

#121

—

132

C₄×Q₈

—

133

C₄×D₄

—

134

#117

—

135

C₂×C₄²

C₂³×C₇₁

yes

characteristic

136

C₄⋊C₈

—

137

C₄⋊C₈

—

138

#103

—

139

#102

—

140

C₄×C₈

—

141

C₄×C₈

C₂×C₄×C₇₁

yes

—

142

C₄×C₈

D₄×C₇₁

yes

—

143

C₄×C₈

D₄×C₇₁

yes

—

144

#41

—

145

C₂×Q₈

—

146

C₂×Q₈

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

—

150

C₂²×C₄

—

151

C₂²×C₄

C₂×D₄×C₇₁

yes

characteristic

152

C₂²×C₄

#41×C₇₁

yes

—

153

M₁₆

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

C₂²⋊C₄×C₇₁

yes

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₂²⋊C₄

—

164

C₂²⋊C₄

—

165

C₄²

—

166

C₄²

C₂²×C₄×C₇₁

yes

characteristic

167

C₄²

C₂×D₄×C₇₁

yes

characteristic

168

C₄²

C₂×D₄×C₇₁

yes

—

169

C₂³

—

170

C₂³

#124×C₇₁

yes

characteristic

171

Q₈

—

172

Q₈

—

173

D₄

—

174

D₄

—

175

D₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

C₂×C₂²⋊C₄×C₇₁

yes

characteristic

181

C₂×C₄

C₄×D₄×C₇₁

yes

—

182

C₂×C₄

#121×C₇₁

yes

—

183

C₂×C₄

#127×C₇₁

yes

characteristic

184

C₂×C₄

#135×C₇₁

yes

—

185

C₈

—

186

C₈

—

187

C₈

—

188

C₂²

—

189

C₂²

—

190

C₂²

—

191

C₂²

#390×C₇₁

yes

characteristic

192

C₂²

#488×C₇₁

yes

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

#416×C₇₁

yes

—

197

C₄

#444×C₇₁

yes

—

198

C₄

#495×C₇₁

yes

—

199

C₂

—

200

C₂

—

201

C₂

C₇₁×#1958

yes

characteristic

202

C₂

C₇₁×#1975

yes

—

	subgroup	G/H
center	C₂×C₄×C₇₁	#127
commutator	C₄²	C₂²×C₄×C₇₁
Frattini, cosocle	C₂×C₄²	C₂³×C₇₁
socle	C₂²×C₇₁	#390
prime radical	#532×C₇₁	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ G

subgroup

G/H

center

C₂×C₄×C₇₁

#127

commutator

C₄²

C₂²×C₄×C₇₁

Frattini, cosocle

C₂×C₄²

C₂³×C₇₁

socle

C₂²×C₇₁

#390

prime radical

#532×C₇₁

C₂²

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ G

N	H ≅ G/N	type
2. C₇₁×#1837	C₂	semidirect
3. C₇₁×#1493	C₂	semidirect
5. C₂×#322×C₇₁	C₂	semidirect
97. #9863	C₇₁	direct (N × H)
103. #1837	C₂×C₇₁	semidirect
104. #1493	C₂×C₇₁	semidirect
106. C₂×#322	C₂×C₇₁	semidirect
107. C₇₁	#9863	direct (N × H)

H ≅ G/N

type

2. C₇₁×#1837

C₂

semidirect

3. C₇₁×#1493

C₂

semidirect

5. C₂×#322×C₇₁

C₂

semidirect

97. #9863

C₇₁

direct (N × H)

103. #1837

C₂×C₇₁

semidirect

104. #1493

C₂×C₇₁

semidirect

106. C₂×#322

C₂×C₇₁

semidirect

107. C₇₁

#9863

direct (N × H)

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

subgroup	index
C₇₁×#3131	2
C₇₁×#1837	2
C₇₁×#1493	2
C₇₁×#1492	2
C₂×#322×C₇₁	2
#9863	71

subgroup

index

C₇₁×#3131

C₇₁×#1837

C₇₁×#1493

C₇₁×#1492

C₂×#322×C₇₁

#9863

Subgroups — C₇₁×#9863 #31830735

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C71×#9863 #31830735

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#9863 #31830735