Group #31831235 — subgroups

Subgroups — C₇₁×#10363 #31831235

Order 18176.

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

normal subgroups only

This group has 416 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×#2862	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#1994	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#1518	C₂	1	2	yes	maximal
4	C₇₁×#1286	C₂	1	2	yes	maximal
5	C₇₁×#1276	C₂	1	1	yes	maximal, characteristic
6	#533×C₇₁	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Q₈×C₇₁	—	2	1	no	—
8	#431×C₇₁	C₂²	1	2	yes	—
9	#423×C₇₁	C₂²	1	1	yes	characteristic
10	#391×C₇₁	—	2	4	no	—
11	#382×C₇₁	—	2	1	no	—
12	#382×C₇₁	C₂²	1	2	yes	—
13	C₄³×C₇₁	C₂²	1	1	yes	characteristic
14	#339×C₇₁	—	2	1	no	—
15	#339×C₇₁	—	2	1	no	—
16	#328×C₇₁	—	2	4	no	—
17	C₂²×Q₈×C₇₁	—	2	2	no	—
18	C₂×M₁₆×C₇₁	—	2	1	no	—
19	C₄⋊Q₈×C₇₁	—	4	2	no	—
20	C₄⋊Q₈×C₇₁	—	4	2	no	—
21	C₄⋊Q₈×C₇₁	Q₈	1	2	yes	complemented
22	#125×C₇₁	—	2	1	no	—
23	C₂²⋊Q₈×C₇₁	—	2	2	no	—
24	C₄×Q₈×C₇₁	—	2	2	no	—
25	#117×C₇₁	—	2	1	no	—
26	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
27	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
28	C₂×C₄²×C₇₁	—	2	2	no	—
29	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₇₁	—	2	2	no	—
33	C₄⋊C₈×C₇₁	—	2	2	no	—
34	C₂²⋊C₈×C₇₁	—	2	2	no	—
35	C₄×C₈×C₇₁	—	2	2	no	—
36	#95×C₇₁	—	2	4	no	—
37	#95×C₇₁	—	2	2	no	—
38	#95×C₇₁	D₄	1	4	yes	—
39	C₂×Q₈×C₇₁	—	2	2	no	—
40	C₂×Q₈×C₇₁	—	4	4	no	—
41	C₂×Q₈×C₇₁	—	4	8	no	—
42	C₂²×C₄×C₇₁	—	2	1	no	—
43	C₂²×C₄×C₇₁	—	2	1	no	—
44	C₂²×C₄×C₇₁	—	2	4	no	—
45	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
46	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
47	M₁₆×C₇₁	—	4	1	no	—
48	M₁₆×C₇₁	—	4	1	no	—
49	C₂×C₈×C₇₁	—	4	2	no	—
50	C₂×C₈×C₇₁	—	4	1	no	—
51	C₄⋊C₄×C₇₁	—	2	2	no	—
52	C₄⋊C₄×C₇₁	—	2	2	no	—
53	C₄⋊C₄×C₇₁	—	4	4	no	—
54	C₄⋊C₄×C₇₁	—	4	2	no	—
55	C₄⋊C₄×C₇₁	—	4	2	no	—
56	C₄⋊C₄×C₇₁	—	4	2	no	—
57	C₂²⋊C₄×C₇₁	—	4	1	no	—
58	C₄²×C₇₁	—	2	1	no	—
59	C₄²×C₇₁	—	2	1	no	—
60	C₄²×C₇₁	—	4	4	no	—
61	C₄²×C₇₁	—	4	1	no	—
62	C₄²×C₇₁	SD₁₆	1	2	yes	—
63	C₄²×C₇₁	Q₁₆	1	2	yes	—
64	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
65	C₄²×C₇₁	#41	1	1	yes	characteristic
66	C₄²×C₇₁	#41	1	2	yes	—
67	C₂³×C₇₁	#120	1	1	yes	characteristic
68	Q₈×C₇₁	—	8	2	no	—
69	Q₈×C₇₁	—	8	8	no	—
70	C₂×C₄×C₇₁	—	2	4	no	—
71	C₂×C₄×C₇₁	—	2	2	no	—
72	C₂×C₄×C₇₁	—	2	2	no	—
73	C₂×C₄×C₇₁	—	4	2	no	—
74	C₂×C₄×C₇₁	—	4	2	no	—
75	C₂×C₄×C₇₁	—	4	8	no	—
76	C₂×C₄×C₇₁	—	4	2	no	—
77	C₂×C₄×C₇₁	—	4	4	no	—
78	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
79	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₇₁	#124	1	2	yes	—
81	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
82	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
83	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
84	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
85	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
86	C₈×C₇₁	—	8	2	no	—
87	C₈×C₇₁	—	8	1	no	—
88	C₂²×C₇₁	—	2	2	no	—
89	C₂²×C₇₁	#393	1	1	yes	characteristic
90	C₂²×C₇₁	#450	1	1	yes	characteristic
91	C₂²×C₇₁	#451	1	1	yes	characteristic
92	C₄×C₇₁	—	2	4	no	—
93	C₄×C₇₁	—	2	2	no	—
94	C₄×C₇₁	—	2	2	no	—
95	C₄×C₇₁	—	4	2	no	—
96	C₄×C₇₁	—	8	4	no	—
97	C₄×C₇₁	—	8	2	no	—
98	C₄×C₇₁	#475	1	1	yes	characteristic
99	C₄×C₇₁	#477	1	1	yes	characteristic
100	C₄×C₇₁	#487	1	1	yes	characteristic
101	C₄×C₇₁	#489	1	1	yes	characteristic
102	#10363	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂×C₇₁	—	2	1	no	—
104	C₂×C₇₁	—	2	1	no	—
105	C₂×C₇₁	#2024	1	1	yes	characteristic
106	C₂×C₇₁	#2031	1	2	yes	—
107	#2862	C₂×C₇₁	1	1	yes	characteristic
108	#1994	C₂×C₇₁	1	1	yes	characteristic
109	#1518	C₂×C₇₁	1	2	yes	—
110	#1286	C₂×C₇₁	1	2	yes	—
111	#1276	C₂×C₇₁	1	1	yes	characteristic
112	C₇₁	#10363	1	1	yes	characteristic, direct product factor, Sylow
113	#533	C₂²×C₇₁	1	1	yes	characteristic
114	C₂×C₄⋊Q₈	—	2	1	no	—
115	#431	C₂²×C₇₁	1	2	yes	—
116	#423	C₂²×C₇₁	1	1	yes	characteristic
117	#391	—	2	4	no	—
118	#382	—	2	1	no	—
119	#382	C₂²×C₇₁	1	2	yes	—
120	C₄³	C₂²×C₇₁	1	1	yes	characteristic
121	#339	—	2	1	no	—
122	#339	—	2	1	no	—
123	#328	—	2	4	no	—
124	C₂²×Q₈	—	2	2	no	—
125	C₂×M₁₆	—	2	1	no	—
126	C₄⋊Q₈	—	4	2	no	—
127	C₄⋊Q₈	—	4	2	no	—
128	C₄⋊Q₈	Q₈×C₇₁	1	2	yes	complemented
129	#125	—	2	1	no	—
130	C₂²⋊Q₈	—	2	2	no	—
131	C₄×Q₈	—	2	2	no	—
132	#117	—	2	1	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄²	—	2	2	no	—
136	C₂×C₄²	D₄×C₇₁	1	1	yes	characteristic
137	C₂×C₄²	D₄×C₇₁	1	1	yes	characteristic
138	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
139	C₄⋊C₈	—	2	2	no	—
140	C₄⋊C₈	—	2	2	no	—
141	C₂²⋊C₈	—	2	2	no	—
142	C₄×C₈	—	2	2	no	—
143	#95	—	2	4	no	—
144	#95	—	2	2	no	—
145	#95	D₄×C₇₁	1	4	yes	—
146	C₂×Q₈	—	2	2	no	—
147	C₂×Q₈	—	4	4	no	—
148	C₂×Q₈	—	4	8	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	2	1	no	—
151	C₂²×C₄	—	2	4	no	—
152	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
153	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
154	M₁₆	—	4	1	no	—
155	M₁₆	—	4	1	no	—
156	C₂×C₈	—	4	2	no	—
157	C₂×C₈	—	4	1	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	4	4	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₂²⋊C₄	—	4	1	no	—
165	C₄²	—	2	1	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	4	4	no	—
168	C₄²	—	4	1	no	—
169	C₄²	SD₁₆×C₇₁	1	2	yes	—
170	C₄²	Q₁₆×C₇₁	1	2	yes	—
171	C₄²	C₂×Q₈×C₇₁	1	1	yes	characteristic
172	C₄²	#41×C₇₁	1	1	yes	characteristic
173	C₄²	#41×C₇₁	1	2	yes	—
174	C₂³	#120×C₇₁	1	1	yes	characteristic
175	Q₈	—	8	2	no	—
176	Q₈	—	8	8	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	8	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	C₂²⋊Q₈×C₇₁	1	2	yes	—
186	C₂×C₄	C₂²⋊Q₈×C₇₁	1	1	yes	characteristic
187	C₂×C₄	#124×C₇₁	1	2	yes	—
188	C₂×C₄	#124×C₇₁	1	1	yes	characteristic
189	C₂×C₄	C₂×SD₁₆×C₇₁	1	1	yes	characteristic
190	C₂×C₄	C₂×Q₁₆×C₇₁	1	1	yes	characteristic
191	C₂×C₄	AGL(1,ℤ/8)×C₇₁	1	1	yes	characteristic
192	C₂×C₄	#137×C₇₁	1	1	yes	characteristic
193	C₈	—	8	2	no	—
194	C₈	—	8	1	no	—
195	C₂²	—	2	2	no	—
196	C₂²	#393×C₇₁	1	1	yes	characteristic
197	C₂²	#450×C₇₁	1	1	yes	characteristic
198	C₂²	#451×C₇₁	1	1	yes	characteristic
199	C₄	—	2	4	no	—
200	C₄	—	2	2	no	—
201	C₄	—	2	2	no	—
202	C₄	—	4	2	no	—
203	C₄	—	8	4	no	—
204	C₄	—	8	2	no	—
205	C₄	#475×C₇₁	1	1	yes	characteristic
206	C₄	#477×C₇₁	1	1	yes	characteristic
207	C₄	#487×C₇₁	1	1	yes	characteristic
208	C₄	#489×C₇₁	1	1	yes	characteristic
209	C₂	—	2	1	no	—
210	C₂	—	2	1	no	—
211	C₂	C₇₁×#2024	1	1	yes	characteristic
212	C₂	C₇₁×#2031	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₁×#2862

C₂

yes

maximal, characteristic

C₇₁×#1994

C₂

yes

maximal, characteristic

C₇₁×#1518

C₂

yes

maximal

C₇₁×#1286

C₂

yes

maximal

C₇₁×#1276

C₂

yes

maximal, characteristic

#533×C₇₁

C₂²

yes

characteristic

C₂×C₄⋊Q₈×C₇₁

—

#431×C₇₁

C₂²

yes

—

#423×C₇₁

C₂²

yes

characteristic

#391×C₇₁

—

#382×C₇₁

—

#382×C₇₁

C₂²

yes

—

C₄³×C₇₁

C₂²

yes

characteristic

#339×C₇₁

—

#339×C₇₁

—

#328×C₇₁

—

C₂²×Q₈×C₇₁

—

C₂×M₁₆×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

Q₈

yes

complemented

#125×C₇₁

—

C₂²⋊Q₈×C₇₁

—

C₄×Q₈×C₇₁

—

#117×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄²×C₇₁

—

C₂×C₄²×C₇₁

D₄

yes

characteristic

C₂×C₄²×C₇₁

D₄

yes

characteristic

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

—

C₂²⋊C₈×C₇₁

—

C₄×C₈×C₇₁

—

#95×C₇₁

—

#95×C₇₁

—

#95×C₇₁

D₄

yes

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

SD₁₆

yes

—

C₄²×C₇₁

Q₁₆

yes

—

C₄²×C₇₁

C₂×Q₈

yes

C₄²×C₇₁

yes

C₄²×C₇₁

yes

—

C₂³×C₇₁

#120

yes

characteristic

Q₈×C₇₁

—

Q₈×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

—

C₂×C₄×C₇₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₁

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₁

#124

yes

—

C₂×C₄×C₇₁

#124

yes

characteristic

C₂×C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₁₆

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₈×C₇₁

—

C₈×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

100

C₄×C₇₁

#487

yes

characteristic

101

C₄×C₇₁

#489

yes

characteristic

102

#10363

C₇₁

yes

maximal, characteristic, direct product factor, Sylow

103

C₂×C₇₁

—

104

C₂×C₇₁

—

105

C₂×C₇₁

#2024

yes

characteristic

106

C₂×C₇₁

#2031

yes

—

107

#2862

C₂×C₇₁

yes

characteristic

108

#1994

C₂×C₇₁

yes

characteristic

109

#1518

C₂×C₇₁

yes

—

110

#1286

C₂×C₇₁

yes

—

111

#1276

C₂×C₇₁

yes

characteristic

112

C₇₁

#10363

yes

characteristic, direct product factor, Sylow

113

#533

C₂²×C₇₁

yes

characteristic

114

C₂×C₄⋊Q₈

—

115

#431

C₂²×C₇₁

yes

—

116

#423

C₂²×C₇₁

yes

characteristic

117

#391

—

118

#382

—

119

#382

C₂²×C₇₁

yes

—

120

C₄³

C₂²×C₇₁

yes

characteristic

121

#339

—

122

#339

—

123

#328

—

124

C₂²×Q₈

—

125

C₂×M₁₆

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

Q₈×C₇₁

yes

complemented

129

#125

—

130

C₂²⋊Q₈

—

131

C₄×Q₈

—

132

#117

—

133

C₂×C₄⋊C₄

—

134

C₂×C₄⋊C₄

—

135

C₂×C₄²

—

136

C₂×C₄²

D₄×C₇₁

yes

characteristic

137

C₂×C₄²

D₄×C₇₁

yes

characteristic

138

C₂×C₄²

C₂³×C₇₁

yes

characteristic

139

C₄⋊C₈

—

140

C₄⋊C₈

—

141

C₂²⋊C₈

—

142

C₄×C₈

—

143

#95

—

144

#95

—

145

#95

D₄×C₇₁

yes

—

146

C₂×Q₈

—

147

C₂×Q₈

—

148

C₂×Q₈

—

149

C₂²×C₄

—

150

C₂²×C₄

—

151

C₂²×C₄

—

152

C₂²×C₄

C₂×D₄×C₇₁

yes

—

153

C₂²×C₄

C₂×D₄×C₇₁

yes

characteristic

154

M₁₆

—

155

M₁₆

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₂²⋊C₄

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₄²

—

169

C₄²

SD₁₆×C₇₁

yes

—

170

C₄²

Q₁₆×C₇₁

yes

—

171

C₄²

C₂×Q₈×C₇₁

yes

characteristic

172

C₄²

#41×C₇₁

yes

characteristic

173

C₄²

#41×C₇₁

yes

—

174

C₂³

#120×C₇₁

yes

characteristic

175

Q₈

—

176

Q₈

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

C₂²⋊Q₈×C₇₁

yes

—

186

C₂×C₄

C₂²⋊Q₈×C₇₁

yes

characteristic

187

C₂×C₄

#124×C₇₁

yes

—

188

C₂×C₄

#124×C₇₁

yes

characteristic

189

C₂×C₄

C₂×SD₁₆×C₇₁

yes

characteristic

190

C₂×C₄

C₂×Q₁₆×C₇₁

yes

characteristic

191

C₂×C₄

AGL(1,ℤ/8)×C₇₁

yes

characteristic

192

C₂×C₄

#137×C₇₁

yes

characteristic

193

C₈

—

194

C₈

—

195

C₂²

—

196

C₂²

#393×C₇₁

yes

characteristic

197

C₂²

#450×C₇₁

yes

characteristic

198

C₂²

#451×C₇₁

yes

characteristic

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

#475×C₇₁

yes

characteristic

206

C₄

#477×C₇₁

yes

characteristic

207

C₄

#487×C₇₁

yes

characteristic

208

C₄

#489×C₇₁

yes

characteristic

209

C₂

—

210

C₂

—

211

C₂

C₇₁×#2024

yes

characteristic

212

C₂

C₇₁×#2031

yes

—

	subgroup	G/H
center, socle	C₂²×C₇₁	#393
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇₁
prime radical	C₂³×C₇₁	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₁

#393

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₇₁

prime radical

C₂³×C₇₁

#120

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ G

N	H ≅ G/N	type
21. C₄⋊Q₈×C₇₁	Q₈	semidirect
102. #10363	C₇₁	direct (N × H)
112. C₇₁	#10363	direct (N × H)
128. C₄⋊Q₈	Q₈×C₇₁	semidirect

H ≅ G/N

type

21. C₄⋊Q₈×C₇₁

Q₈

semidirect

102. #10363

C₇₁

direct (N × H)

112. C₇₁

#10363

direct (N × H)

128. C₄⋊Q₈

Q₈×C₇₁

semidirect

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

subgroup	index
C₇₁×#2862	2
C₇₁×#1994	2
C₇₁×#1518	2
C₇₁×#1286	2
C₇₁×#1276	2
#10363	71

subgroup

index

C₇₁×#2862

C₇₁×#1994

C₇₁×#1518

C₇₁×#1286

C₇₁×#1276

#10363

Subgroups — C₇₁×#10363 #31831235

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C71×#10363 #31831235

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#10363 #31831235