Group #31854699 — subgroups

Subgroups — C₇₁×#33827 #31854699

Order 18176.

464 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 194 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×#3177	C₂	1	2	yes	maximal
2	C₇₁×#2229	C₂	1	4	yes	maximal, complemented
3	C₇₁×#2219	C₂	1	2	yes	maximal
4	C₇₁×#2216	C₂	1	2	yes	maximal
5	C₇₁×#2209	C₂	1	4	yes	maximal, complemented
6	C₇₁×#2151	C₂	1	1	yes	maximal, characteristic
7	#541×C₇₁	—	2	2	no	—
8	C₂×Q₃₂×C₇₁	C₂²	1	2	yes	—
9	C₂×Q₃₂×C₇₁	C₂²	1	2	yes	—
10	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
12	#499×C₇₁	C₂²	1	2	yes	—
13	#494×C₇₁	C₂²	1	2	yes	—
14	#491×C₇₁	—	2	2	no	—
15	#470×C₇₁	—	2	2	no	—
16	#467×C₇₁	C₂²	1	4	yes	—
17	#441×C₇₁	—	2	2	no	—
18	C₄×Q₁₆×C₇₁	C₂²	1	4	yes	—
19	#405×C₇₁	C₂²	1	1	yes	characteristic
20	#367×C₇₁	C₂²	1	2	yes	—
21	#366×C₇₁	C₂²	1	2	yes	—
22	C₄⋊C₁₆×C₇₁	C₂²	1	4	yes	—
23	#358×C₇₁	C₂²	1	4	yes	—
24	#358×C₇₁	C₂²	1	4	yes	—
25	C₂×Q₁₆×C₇₁	—	2	2	no	—
26	C₂×Q₁₆×C₇₁	C₂³	1	4	yes	—
27	C₂×M₁₆×C₇₁	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
29	C₄⋊Q₈×C₇₁	—	2	2	no	—
30	#126×C₇₁	—	4	4	no	—
31	#125×C₇₁	—	2	2	no	—
32	#125×C₇₁	—	4	4	no	—
33	C₂²⋊Q₈×C₇₁	—	2	4	no	—
34	C₄×Q₈×C₇₁	—	2	4	no	—
35	#117×C₇₁	D₄	1	1	yes	characteristic
36	Q₃₂×C₇₁	—	4	2	no	—
37	Q₃₂×C₇₁	—	4	2	no	—
38	M₃₂×C₇₁	—	4	1	no	—
39	C₂×C₁₆×C₇₁	—	4	1	no	—
40	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
41	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
42	#107×C₇₁	C₂³	1	4	yes	—
43	#106×C₇₁	—	2	2	no	—
44	#103×C₇₁	—	2	4	no	—
45	#103×C₇₁	—	2	4	no	—
46	#97×C₇₁	D₄	1	2	yes	—
47	C₄×C₈×C₇₁	C₂³	1	2	yes	—
48	C₂×Q₈×C₇₁	—	2	4	no	—
49	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
50	Q₁₆×C₇₁	—	2	4	no	—
51	Q₁₆×C₇₁	—	2	4	no	—
52	Q₁₆×C₇₁	—	4	4	no	—
53	M₁₆×C₇₁	—	4	1	no	—
54	C₂×C₈×C₇₁	—	2	1	no	—
55	C₂×C₈×C₇₁	—	2	1	no	—
56	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
57	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
59	C₄⋊C₄×C₇₁	—	2	4	no	—
60	C₄⋊C₄×C₇₁	—	4	4	no	—
61	C₄⋊C₄×C₇₁	—	4	4	no	—
62	C₄⋊C₄×C₇₁	—	4	4	no	—
63	C₄⋊C₄×C₇₁	—	4	4	no	—
64	C₄⋊C₄×C₇₁	D₈	1	2	yes	—
65	C₂²⋊C₄×C₇₁	—	4	4	no	—
66	C₂²⋊C₄×C₇₁	D₈	1	2	yes	—
67	C₄²×C₇₁	—	4	4	no	—
68	C₄²×C₇₁	C₂×D₄	1	2	yes	—
69	C₁₆×C₇₁	—	2	2	no	—
70	C₁₆×C₇₁	—	2	2	no	—
71	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
72	Q₈×C₇₁	—	4	4	no	—
73	Q₈×C₇₁	—	4	4	no	—
74	C₂×C₄×C₇₁	—	2	1	no	—
75	C₂×C₄×C₇₁	—	2	1	no	—
76	C₂×C₄×C₇₁	—	4	4	no	—
77	C₂×C₄×C₇₁	—	4	4	no	—
78	C₂×C₄×C₇₁	C₂×D₈	1	4	yes	—
79	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
81	C₈×C₇₁	—	2	1	no	—
82	C₈×C₇₁	—	2	2	no	—
83	C₈×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
84	C₈×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
85	C₂²×C₇₁	—	2	2	no	—
86	C₂²×C₇₁	—	2	1	no	—
87	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
88	C₄×C₇₁	—	2	4	no	—
89	C₄×C₇₁	—	2	1	no	—
90	C₄×C₇₁	—	8	4	no	—
91	C₄×C₇₁	—	8	4	no	—
92	C₄×C₇₁	#537	1	1	yes	characteristic
93	C₄×C₇₁	#575	1	1	yes	characteristic
94	#33827	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂×C₇₁	—	4	1	no	—
96	C₂×C₇₁	#3245	1	1	yes	characteristic
97	C₂×C₇₁	#3417	1	2	yes	—
98	#3177	C₂×C₇₁	1	2	yes	—
99	#2229	C₂×C₇₁	1	4	yes	complemented
100	#2219	C₂×C₇₁	1	2	yes	—
101	#2216	C₂×C₇₁	1	2	yes	—
102	#2209	C₂×C₇₁	1	4	yes	complemented
103	#2151	C₂×C₇₁	1	1	yes	characteristic
104	C₇₁	#33827	1	1	yes	characteristic, direct product factor, Sylow
105	#541	—	2	2	no	—
106	C₂×Q₃₂	C₂²×C₇₁	1	2	yes	—
107	C₂×Q₃₂	C₂²×C₇₁	1	2	yes	—
108	C₂×M₃₂	C₂²×C₇₁	1	1	yes	characteristic
109	C₂²×C₁₆	C₂²×C₇₁	1	1	yes	characteristic
110	#499	C₂²×C₇₁	1	2	yes	—
111	#494	C₂²×C₇₁	1	2	yes	—
112	#491	—	2	2	no	—
113	#470	—	2	2	no	—
114	#467	C₂²×C₇₁	1	4	yes	—
115	#441	—	2	2	no	—
116	C₄×Q₁₆	C₂²×C₇₁	1	4	yes	—
117	#405	C₂²×C₇₁	1	1	yes	characteristic
118	#367	C₂²×C₇₁	1	2	yes	—
119	#366	C₂²×C₇₁	1	2	yes	—
120	C₄⋊C₁₆	C₂²×C₇₁	1	4	yes	—
121	#358	C₂²×C₇₁	1	4	yes	—
122	#358	C₂²×C₇₁	1	4	yes	—
123	C₂×Q₁₆	—	2	2	no	—
124	C₂×Q₁₆	C₂³×C₇₁	1	4	yes	—
125	C₂×M₁₆	D₄×C₇₁	1	1	yes	characteristic
126	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
127	C₄⋊Q₈	—	2	2	no	—
128	#126	—	4	4	no	—
129	#125	—	2	2	no	—
130	#125	—	4	4	no	—
131	C₂²⋊Q₈	—	2	4	no	—
132	C₄×Q₈	—	2	4	no	—
133	#117	D₄×C₇₁	1	1	yes	characteristic
134	Q₃₂	—	4	2	no	—
135	Q₃₂	—	4	2	no	—
136	M₃₂	—	4	1	no	—
137	C₂×C₁₆	—	4	1	no	—
138	C₂×C₁₆	C₂³×C₇₁	1	2	yes	—
139	C₂×C₁₆	C₂³×C₇₁	1	2	yes	—
140	#107	C₂³×C₇₁	1	4	yes	—
141	#106	—	2	2	no	—
142	#103	—	2	4	no	—
143	#103	—	2	4	no	—
144	#97	D₄×C₇₁	1	2	yes	—
145	C₄×C₈	C₂³×C₇₁	1	2	yes	—
146	C₂×Q₈	—	2	4	no	—
147	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
148	Q₁₆	—	2	4	no	—
149	Q₁₆	—	2	4	no	—
150	Q₁₆	—	4	4	no	—
151	M₁₆	—	4	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
155	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
156	C₂×C₈	C₂⁴×C₇₁	1	1	yes	characteristic
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	4	4	no	—
159	C₄⋊C₄	—	4	4	no	—
160	C₄⋊C₄	—	4	4	no	—
161	C₄⋊C₄	—	4	4	no	—
162	C₄⋊C₄	D₈×C₇₁	1	2	yes	—
163	C₂²⋊C₄	—	4	4	no	—
164	C₂²⋊C₄	D₈×C₇₁	1	2	yes	—
165	C₄²	—	4	4	no	—
166	C₄²	C₂×D₄×C₇₁	1	2	yes	—
167	C₁₆	—	2	2	no	—
168	C₁₆	—	2	2	no	—
169	C₂³	C₂×D₈×C₇₁	1	1	yes	characteristic
170	Q₈	—	4	4	no	—
171	Q₈	—	4	4	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	4	4	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	C₂×D₈×C₇₁	1	4	yes	—
177	C₂×C₄	C₂×D₈×C₇₁	1	1	yes	characteristic
178	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
179	C₈	—	2	1	no	—
180	C₈	—	2	2	no	—
181	C₈	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
182	C₈	2₋¹⁺⁴×C₇₁	1	1	yes	characteristic
183	C₂²	—	2	2	no	—
184	C₂²	—	2	1	no	—
185	C₂²	C₂²×D₈×C₇₁	1	1	yes	characteristic
186	C₄	—	2	4	no	—
187	C₄	—	2	1	no	—
188	C₄	—	8	4	no	—
189	C₄	—	8	4	no	—
190	C₄	#537×C₇₁	1	1	yes	characteristic
191	C₄	#575×C₇₁	1	1	yes	characteristic
192	C₂	—	4	1	no	—
193	C₂	C₇₁×#3245	1	1	yes	characteristic
194	C₂	C₇₁×#3417	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₁×#3177

C₂

yes

maximal

C₇₁×#2229

C₂

yes

maximal, complemented

C₇₁×#2219

C₂

yes

maximal

C₇₁×#2216

C₂

yes

maximal

C₇₁×#2209

C₂

yes

maximal, complemented

C₇₁×#2151

C₂

yes

maximal, characteristic

#541×C₇₁

—

C₂×Q₃₂×C₇₁

C₂²

yes

—

C₂×Q₃₂×C₇₁

C₂²

yes

—

C₂×M₃₂×C₇₁

C₂²

yes

characteristic

C₂²×C₁₆×C₇₁

C₂²

yes

characteristic

#499×C₇₁

C₂²

yes

—

#494×C₇₁

C₂²

yes

—

#491×C₇₁

—

#470×C₇₁

—

#467×C₇₁

C₂²

yes

—

#441×C₇₁

—

C₄×Q₁₆×C₇₁

C₂²

yes

—

#405×C₇₁

C₂²

yes

characteristic

#367×C₇₁

C₂²

yes

—

#366×C₇₁

C₂²

yes

—

C₄⋊C₁₆×C₇₁

C₂²

yes

—

#358×C₇₁

C₂²

yes

—

#358×C₇₁

C₂²

yes

—

C₂×Q₁₆×C₇₁

—

C₂×Q₁₆×C₇₁

C₂³

yes

—

C₂×M₁₆×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₄⋊Q₈×C₇₁

—

#126×C₇₁

—

#125×C₇₁

—

#125×C₇₁

—

C₂²⋊Q₈×C₇₁

—

C₄×Q₈×C₇₁

—

#117×C₇₁

D₄

yes

characteristic

Q₃₂×C₇₁

—

Q₃₂×C₇₁

—

M₃₂×C₇₁

—

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

C₂³

yes

—

C₂×C₁₆×C₇₁

C₂³

yes

—

#107×C₇₁

C₂³

yes

—

#106×C₇₁

—

#103×C₇₁

—

#103×C₇₁

—

#97×C₇₁

D₄

yes

—

C₄×C₈×C₇₁

C₂³

yes

—

C₂×Q₈×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×D₄

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂⁴

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

D₈

yes

—

C₂²⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

D₈

yes

—

C₄²×C₇₁

—

C₄²×C₇₁

C₂×D₄

yes

—

C₁₆×C₇₁

—

C₁₆×C₇₁

—

C₂³×C₇₁

C₂×D₈

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₈

yes

—

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

C₂²×D₄

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

H(5,𝔽₂)

yes

characteristic

C₈×C₇₁

2₋¹⁺⁴

yes

characteristic

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂²×D₈

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₇₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₁

—

C₂×C₇₁

#3245

yes

characteristic

C₂×C₇₁

#3417

yes

—

#3177

C₂×C₇₁

yes

—

#2229

C₂×C₇₁

yes

complemented

100

#2219

C₂×C₇₁

yes

—

101

#2216

C₂×C₇₁

yes

—

102

#2209

C₂×C₇₁

yes

complemented

103

#2151

C₂×C₇₁

yes

characteristic

104

C₇₁

#33827

yes

characteristic, direct product factor, Sylow

105

#541

—

106

C₂×Q₃₂

C₂²×C₇₁

yes

—

107

C₂×Q₃₂

C₂²×C₇₁

yes

—

108

C₂×M₃₂

C₂²×C₇₁

yes

characteristic

109

C₂²×C₁₆

C₂²×C₇₁

yes

characteristic

110

#499

C₂²×C₇₁

yes

—

111

#494

C₂²×C₇₁

yes

—

112

#491

—

113

#470

—

114

#467

C₂²×C₇₁

yes

—

115

#441

—

116

C₄×Q₁₆

C₂²×C₇₁

yes

—

117

#405

C₂²×C₇₁

yes

characteristic

118

#367

C₂²×C₇₁

yes

—

119

#366

C₂²×C₇₁

yes

—

120

C₄⋊C₁₆

C₂²×C₇₁

yes

—

121

#358

C₂²×C₇₁

yes

—

122

#358

C₂²×C₇₁

yes

—

123

C₂×Q₁₆

—

124

C₂×Q₁₆

C₂³×C₇₁

yes

—

125

C₂×M₁₆

D₄×C₇₁

yes

characteristic

126

C₂²×C₈

C₂³×C₇₁

yes

characteristic

127

C₄⋊Q₈

—

128

#126

—

129

#125

—

130

#125

—

131

C₂²⋊Q₈

—

132

C₄×Q₈

—

133

#117

D₄×C₇₁

yes

characteristic

134

Q₃₂

—

135

Q₃₂

—

136

M₃₂

—

137

C₂×C₁₆

—

138

C₂×C₁₆

C₂³×C₇₁

yes

—

139

C₂×C₁₆

C₂³×C₇₁

yes

—

140

#107

C₂³×C₇₁

yes

—

141

#106

—

142

#103

—

143

#103

—

144

#97

D₄×C₇₁

yes

—

145

C₄×C₈

C₂³×C₇₁

yes

—

146

C₂×Q₈

—

147

C₂²×C₄

C₂×D₄×C₇₁

yes

characteristic

148

Q₁₆

—

149

Q₁₆

—

150

Q₁₆

—

151

M₁₆

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

C₂×D₄×C₇₁

yes

—

155

C₂×C₈

C₂×D₄×C₇₁

yes

characteristic

156

C₂×C₈

C₂⁴×C₇₁

yes

characteristic

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

D₈×C₇₁

yes

—

163

C₂²⋊C₄

—

164

C₂²⋊C₄

D₈×C₇₁

yes

—

165

C₄²

—

166

C₄²

C₂×D₄×C₇₁

yes

—

167

C₁₆

—

168

C₁₆

—

169

C₂³

C₂×D₈×C₇₁

yes

characteristic

170

Q₈

—

171

Q₈

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

C₂×D₈×C₇₁

yes

—

177

C₂×C₄

C₂×D₈×C₇₁

yes

characteristic

178

C₂×C₄

C₂²×D₄×C₇₁

yes

characteristic

179

C₈

—

180

C₈

—

181

C₈

H(5,𝔽₂)×C₇₁

yes

characteristic

182

C₈

2₋¹⁺⁴×C₇₁

yes

characteristic

183

C₂²

—

184

C₂²

—

185

C₂²

C₂²×D₈×C₇₁

yes

characteristic

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

#537×C₇₁

yes

characteristic

191

C₄

#575×C₇₁

yes

characteristic

192

C₂

—

193

C₂

C₇₁×#3245

yes

characteristic

194

C₂

C₇₁×#3417

yes

—

	subgroup	G/H
center, socle	C₂²×C₇₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₇₁
prime radical	C₂³×C₇₁	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ #117×C₇₁ ◁ #405×C₇₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₁

C₂²×D₈

commutator, Frattini, cosocle

C₂×C₈

C₂⁴×C₇₁

prime radical

C₂³×C₇₁

C₂×D₈

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₇₁ ◁ #117×C₇₁ ◁ #405×C₇₁ ◁ G

N	H ≅ G/N	type
2. C₇₁×#2229	C₂	semidirect
5. C₇₁×#2209	C₂	semidirect
94. #33827	C₇₁	direct (N × H)
99. #2229	C₂×C₇₁	semidirect
102. #2209	C₂×C₇₁	semidirect
104. C₇₁	#33827	direct (N × H)

H ≅ G/N

type

2. C₇₁×#2229

C₂

semidirect

5. C₇₁×#2209

C₂

semidirect

94. #33827

C₇₁

direct (N × H)

99. #2229

C₂×C₇₁

semidirect

102. #2209

C₂×C₇₁

semidirect

104. C₇₁

#33827

direct (N × H)

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

subgroup	index
C₇₁×#3177	2
C₇₁×#2229	2
C₇₁×#2219	2
C₇₁×#2216	2
C₇₁×#2209	2
C₇₁×#2151	2
#33827	71

subgroup

index

C₇₁×#3177

C₇₁×#2229

C₇₁×#2219

C₇₁×#2216

C₇₁×#2209

C₇₁×#2151

#33827

Subgroups — C₇₁×#33827 #31854699

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C71×#33827 #31854699

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#33827 #31854699