Group #31831532 — subgroups

Subgroups — C₇₁×#10660 #31831532

Order 18176.

372 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 114 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×#1955	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#1849	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₁×#1571	C₂	1	2	yes	maximal, complemented
4	C₇₁×#1486	C₂	1	2	yes	maximal
5	C₇₁×#1275	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₄×C₇₁	C₂²	1	1	yes	characteristic
7	#431×C₇₁	C₂²	1	2	yes	—
8	C₂×#108×C₇₁	—	2	2	no	—
9	#423×C₇₁	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊C₈×C₇₁	—	2	2	no	—
12	C₂×#97×C₇₁	C₂²	1	1	yes	characteristic
13	#335×C₇₁	—	2	4	no	—
14	D₄⋊C₈×C₇₁	—	2	4	no	—
15	C₂²×D₄×C₇₁	Q₈	1	1	yes	characteristic
16	C₂³×C₄×C₇₁	—	2	1	no	—
17	C₂×M₁₆×C₇₁	—	2	2	no	—
18	C₂²×C₈×C₇₁	—	2	1	no	—
19	C₂²×C₈×C₇₁	D₄	1	2	yes	—
20	C₄×D₄×C₇₁	—	2	2	no	—
21	C₄×D₄×C₇₁	—	2	2	no	—
22	C₂×C₄⋊C₄×C₇₁	Q₈	1	1	yes	characteristic
23	C₂×C₂²⋊C₄×C₇₁	—	2	1	no	—
24	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
25	#108×C₇₁	—	4	4	no	—
26	C₄⋊C₈×C₇₁	—	2	2	no	—
27	C₄⋊C₈×C₇₁	—	2	2	no	—
28	C₄⋊C₈×C₇₁	—	2	2	no	—
29	C₄⋊C₈×C₇₁	—	2	2	no	—
30	C₂²⋊C₈×C₇₁	—	2	2	no	—
31	C₂²⋊C₈×C₇₁	—	4	4	no	—
32	#97×C₇₁	—	2	2	no	—
33	C₄×C₈×C₇₁	D₄	1	4	yes	—
34	C₂⁴×C₇₁	—	2	1	no	—
35	C₂×D₄×C₇₁	—	2	2	no	—
36	C₂×D₄×C₇₁	—	4	2	no	—
37	C₂²×C₄×C₇₁	—	2	1	no	—
38	C₂²×C₄×C₇₁	—	4	1	no	—
39	C₂²×C₄×C₇₁	—	4	1	no	—
40	C₂²×C₄×C₇₁	—	4	2	no	—
41	C₂²×C₄×C₇₁	—	4	2	no	—
42	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
45	M₁₆×C₇₁	—	4	4	no	—
46	C₂×C₈×C₇₁	—	2	2	no	—
47	C₂×C₈×C₇₁	—	2	2	no	—
48	C₂×C₈×C₇₁	—	2	4	no	—
49	C₂×C₈×C₇₁	—	4	1	no	—
50	C₂×C₈×C₇₁	—	4	2	no	—
51	C₂×C₈×C₇₁	—	4	2	no	—
52	C₂×C₈×C₇₁	D₈	1	4	yes	—
53	C₄⋊C₄×C₇₁	—	2	2	no	—
54	C₂²⋊C₄×C₇₁	—	4	2	no	—
55	C₄²×C₇₁	C₂×D₄	1	2	yes	—
56	C₄²×C₇₁	#41	1	2	yes	—
57	C₂³×C₇₁	—	2	2	no	—
58	C₂³×C₇₁	—	4	1	no	—
59	C₂³×C₇₁	—	4	2	no	—
60	C₂³×C₇₁	—	4	1	no	—
61	C₂³×C₇₁	—	4	2	no	—
62	C₂³×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
63	D₄×C₇₁	—	2	4	no	—
64	D₄×C₇₁	—	4	2	no	—
65	C₂×C₄×C₇₁	—	4	1	no	—
66	C₂×C₄×C₇₁	—	4	1	no	—
67	C₂×C₄×C₇₁	—	4	1	no	—
68	C₂×C₄×C₇₁	—	4	2	no	—
69	C₂×C₄×C₇₁	—	8	2	no	—
70	C₂×C₄×C₇₁	—	8	1	no	—
71	C₂×C₄×C₇₁	—	8	1	no	—
72	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
73	C₂×C₄×C₇₁	#123	1	2	yes	—
74	C₂×C₄×C₇₁	#123	1	1	yes	characteristic
75	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
76	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
77	C₂×C₄×C₇₁	#137	1	2	yes	—
78	C₈×C₇₁	—	2	4	no	—
79	C₈×C₇₁	—	4	4	no	—
80	C₈×C₇₁	—	8	1	no	—
81	C₈×C₇₁	—	8	2	no	—
82	C₂²×C₇₁	—	2	1	no	—
83	C₂²×C₇₁	—	2	1	no	—
84	C₂²×C₇₁	—	4	4	no	—
85	C₂²×C₇₁	—	4	2	no	—
86	C₂²×C₇₁	—	8	2	no	—
87	C₂²×C₇₁	—	8	1	no	—
88	C₂²×C₇₁	—	8	1	no	—
89	C₂²×C₇₁	#399	1	1	yes	characteristic
90	C₂²×C₇₁	#474	1	2	yes	—
91	C₄×C₇₁	—	2	2	no	—
92	C₄×C₇₁	—	2	2	no	—
93	C₄×C₇₁	—	8	1	no	—
94	C₄×C₇₁	#480	1	2	yes	—
95	C₄×C₇₁	#493	1	1	yes	characteristic
96	C₄×C₇₁	#497	1	1	yes	characteristic
97	#10660	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂×C₇₁	—	2	2	no	—
99	C₂×C₇₁	—	8	2	no	—
100	C₂×C₇₁	#2091	1	1	yes	characteristic
101	C₂×C₇₁	#2093	1	1	yes	characteristic
102	C₂×C₇₁	#2098	1	1	yes	characteristic
103	#1955	C₂×C₇₁	1	1	yes	characteristic
104	#1849	C₂×C₇₁	1	1	yes	characteristic, complemented
105	#1571	C₂×C₇₁	1	2	yes	complemented
106	#1486	C₂×C₇₁	1	2	yes	—
107	#1275	C₂×C₇₁	1	1	yes	characteristic, complemented
108	C₇₁	#10660	1	1	yes	characteristic, direct product factor, Sylow
109	C₂×C₄×D₄	C₂²×C₇₁	1	1	yes	characteristic
110	#431	C₂²×C₇₁	1	2	yes	—
111	C₂×#108	—	2	2	no	—
112	#423	C₂²×C₇₁	1	2	yes	—
113	C₂×C₄⋊C₈	C₂²×C₇₁	1	1	yes	characteristic
114	C₂×C₂²⋊C₈	—	2	2	no	—
115	C₂×#97	C₂²×C₇₁	1	1	yes	characteristic
116	#335	—	2	4	no	—
117	D₄⋊C₈	—	2	4	no	—
118	C₂²×D₄	Q₈×C₇₁	1	1	yes	characteristic
119	C₂³×C₄	—	2	1	no	—
120	C₂×M₁₆	—	2	2	no	—
121	C₂²×C₈	—	2	1	no	—
122	C₂²×C₈	D₄×C₇₁	1	2	yes	—
123	C₄×D₄	—	2	2	no	—
124	C₄×D₄	—	2	2	no	—
125	C₂×C₄⋊C₄	Q₈×C₇₁	1	1	yes	characteristic
126	C₂×C₂²⋊C₄	—	2	1	no	—
127	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
128	#108	—	4	4	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	2	2	no	—
131	C₄⋊C₈	—	2	2	no	—
132	C₄⋊C₈	—	2	2	no	—
133	C₂²⋊C₈	—	2	2	no	—
134	C₂²⋊C₈	—	4	4	no	—
135	#97	—	2	2	no	—
136	C₄×C₈	D₄×C₇₁	1	4	yes	—
137	C₂⁴	—	2	1	no	—
138	C₂×D₄	—	2	2	no	—
139	C₂×D₄	—	4	2	no	—
140	C₂²×C₄	—	2	1	no	—
141	C₂²×C₄	—	4	1	no	—
142	C₂²×C₄	—	4	1	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	—	4	2	no	—
145	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
146	C₂²×C₄	C₂×Q₈×C₇₁	1	1	yes	characteristic
147	C₂²×C₄	#41×C₇₁	1	1	yes	characteristic
148	M₁₆	—	4	4	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	4	1	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	D₈×C₇₁	1	4	yes	—
156	C₄⋊C₄	—	2	2	no	—
157	C₂²⋊C₄	—	4	2	no	—
158	C₄²	C₂×D₄×C₇₁	1	2	yes	—
159	C₄²	#41×C₇₁	1	2	yes	—
160	C₂³	—	2	2	no	—
161	C₂³	—	4	1	no	—
162	C₂³	—	4	2	no	—
163	C₂³	—	4	2	no	—
164	C₂³	—	4	1	no	—
165	C₂³	C₂²⋊Q₈×C₇₁	1	1	yes	characteristic
166	D₄	—	2	4	no	—
167	D₄	—	4	2	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	1	no	—
174	C₂×C₄	—	8	1	no	—
175	C₂×C₄	C₂²⋊Q₈×C₇₁	1	2	yes	—
176	C₂×C₄	#123×C₇₁	1	2	yes	—
177	C₂×C₄	#123×C₇₁	1	1	yes	characteristic
178	C₂×C₄	#127×C₇₁	1	1	yes	characteristic
179	C₂×C₄	C₂×D₈×C₇₁	1	2	yes	—
180	C₂×C₄	#137×C₇₁	1	2	yes	—
181	C₈	—	2	4	no	—
182	C₈	—	4	4	no	—
183	C₈	—	8	1	no	—
184	C₈	—	8	2	no	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	1	no	—
187	C₂²	—	4	4	no	—
188	C₂²	—	4	2	no	—
189	C₂²	—	8	2	no	—
190	C₂²	—	8	1	no	—
191	C₂²	—	8	1	no	—
192	C₂²	#399×C₇₁	1	1	yes	characteristic
193	C₂²	#474×C₇₁	1	2	yes	—
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	—	8	1	no	—
197	C₄	#480×C₇₁	1	2	yes	—
198	C₄	#493×C₇₁	1	1	yes	characteristic
199	C₄	#497×C₇₁	1	1	yes	characteristic
200	C₂	—	2	2	no	—
201	C₂	—	8	2	no	—
202	C₂	C₇₁×#2091	1	1	yes	characteristic
203	C₂	C₇₁×#2093	1	1	yes	characteristic
204	C₂	C₇₁×#2098	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₁×#1955

C₂

yes

maximal, characteristic

C₇₁×#1849

C₂

yes

maximal, characteristic, complemented

C₇₁×#1571

C₂

yes

maximal, complemented

C₇₁×#1486

C₂

yes

maximal

C₇₁×#1275

C₂

yes

maximal, characteristic, complemented

C₂×C₄×D₄×C₇₁

C₂²

yes

characteristic

#431×C₇₁

C₂²

yes

—

C₂×#108×C₇₁

—

#423×C₇₁

C₂²

yes

—

C₂×C₄⋊C₈×C₇₁

C₂²

yes

characteristic

C₂×C₂²⋊C₈×C₇₁

—

C₂×#97×C₇₁

C₂²

yes

characteristic

#335×C₇₁

—

D₄⋊C₈×C₇₁

—

C₂²×D₄×C₇₁

Q₈

yes

characteristic

C₂³×C₄×C₇₁

—

C₂×M₁₆×C₇₁

—

C₂²×C₈×C₇₁

—

C₂²×C₈×C₇₁

D₄

yes

—

C₄×D₄×C₇₁

—

C₄×D₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

Q₈

yes

characteristic

C₂×C₂²⋊C₄×C₇₁

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

#108×C₇₁

—

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

—

C₂²⋊C₈×C₇₁

—

C₂²⋊C₈×C₇₁

—

#97×C₇₁

—

C₄×C₈×C₇₁

D₄

yes

—

C₂⁴×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂²×C₄×C₇₁

#41

yes

characteristic

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

D₈

yes

—

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

C₂×D₄

yes

—

C₄²×C₇₁

#41

yes

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

C₂²⋊Q₈

yes

characteristic

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₁

#123

yes

—

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

C₂×D₈

yes

—

C₂×C₄×C₇₁

#137

yes

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#399

yes

characteristic

C₂²×C₇₁

#474

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#480

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₇₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

100

C₂×C₇₁

#2091

yes

characteristic

101

C₂×C₇₁

#2093

yes

characteristic

102

C₂×C₇₁

#2098

yes

characteristic

103

#1955

C₂×C₇₁

yes

characteristic

104

#1849

C₂×C₇₁

yes

characteristic, complemented

105

#1571

C₂×C₇₁

yes

complemented

106

#1486

C₂×C₇₁

yes

—

107

#1275

C₂×C₇₁

yes

characteristic, complemented

108

C₇₁

#10660

yes

characteristic, direct product factor, Sylow

109

C₂×C₄×D₄

C₂²×C₇₁

yes

characteristic

110

#431

C₂²×C₇₁

yes

—

111

C₂×#108

—

112

#423

C₂²×C₇₁

yes

—

113

C₂×C₄⋊C₈

C₂²×C₇₁

yes

characteristic

114

C₂×C₂²⋊C₈

—

115

C₂×#97

C₂²×C₇₁

yes

characteristic

116

#335

—

117

D₄⋊C₈

—

118

C₂²×D₄

Q₈×C₇₁

yes

characteristic

119

C₂³×C₄

—

120

C₂×M₁₆

—

121

C₂²×C₈

—

122

C₂²×C₈

D₄×C₇₁

yes

—

123

C₄×D₄

—

124

C₄×D₄

—

125

C₂×C₄⋊C₄

Q₈×C₇₁

yes

characteristic

126

C₂×C₂²⋊C₄

—

127

C₂×C₄²

C₂³×C₇₁

yes

characteristic

128

#108

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

131

C₄⋊C₈

—

132

C₄⋊C₈

—

133

C₂²⋊C₈

—

134

C₂²⋊C₈

—

135

#97

—

136

C₄×C₈

D₄×C₇₁

yes

—

137

C₂⁴

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

C₂×D₄×C₇₁

yes

characteristic

146

C₂²×C₄

C₂×Q₈×C₇₁

yes

characteristic

147

C₂²×C₄

#41×C₇₁

yes

characteristic

148

M₁₆

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

D₈×C₇₁

yes

—

156

C₄⋊C₄

—

157

C₂²⋊C₄

—

158

C₄²

C₂×D₄×C₇₁

yes

—

159

C₄²

#41×C₇₁

yes

—

160

C₂³

—

161

C₂³

—

162

C₂³

—

163

C₂³

—

164

C₂³

—

165

C₂³

C₂²⋊Q₈×C₇₁

yes

characteristic

166

D₄

—

167

D₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

C₂²⋊Q₈×C₇₁

yes

—

176

C₂×C₄

#123×C₇₁

yes

—

177

C₂×C₄

#123×C₇₁

yes

characteristic

178

C₂×C₄

#127×C₇₁

yes

characteristic

179

C₂×C₄

C₂×D₈×C₇₁

yes

—

180

C₂×C₄

#137×C₇₁

yes

—

181

C₈

—

182

C₈

—

183

C₈

—

184

C₈

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

192

C₂²

#399×C₇₁

yes

characteristic

193

C₂²

#474×C₇₁

yes

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

#480×C₇₁

yes

—

198

C₄

#493×C₇₁

yes

characteristic

199

C₄

#497×C₇₁

yes

characteristic

200

C₂

—

201

C₂

—

202

C₂

C₇₁×#2091

yes

characteristic

203

C₂

C₇₁×#2093

yes

characteristic

204

C₂

C₇₁×#2098

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₇₁	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇₁
socle	C₂²×C₇₁	#399
prime radical	C₂²×D₄×C₇₁	Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ G

subgroup

G/H

center

C₂×C₄×C₇₁

#127

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₇₁

socle

C₂²×C₇₁

#399

prime radical

C₂²×D₄×C₇₁

Q₈

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ G

N	H ≅ G/N	type
2. C₇₁×#1849	C₂	semidirect
3. C₇₁×#1571	C₂	semidirect
5. C₇₁×#1275	C₂	semidirect
97. #10660	C₇₁	direct (N × H)
104. #1849	C₂×C₇₁	semidirect
105. #1571	C₂×C₇₁	semidirect
107. #1275	C₂×C₇₁	semidirect
108. C₇₁	#10660	direct (N × H)

H ≅ G/N

type

2. C₇₁×#1849

C₂

semidirect

3. C₇₁×#1571

C₂

semidirect

5. C₇₁×#1275

C₂

semidirect

97. #10660

C₇₁

direct (N × H)

104. #1849

C₂×C₇₁

semidirect

105. #1571

C₂×C₇₁

semidirect

107. #1275

C₂×C₇₁

semidirect

108. C₇₁

#10660

direct (N × H)

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

subgroup	index
C₇₁×#1955	2
C₇₁×#1849	2
C₇₁×#1571	2
C₇₁×#1486	2
C₇₁×#1275	2
#10660	71

subgroup

index

C₇₁×#1955

C₇₁×#1849

C₇₁×#1571

C₇₁×#1486

C₇₁×#1275

#10660

Subgroups — C₇₁×#10660 #31831532

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C71×#10660 #31831532

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#10660 #31831532