Group #31830639 — subgroups

Subgroups — C₇₁×#9767 #31830639

Order 18176.

414 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 174 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	C₂×#498×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#1848	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#1549	C₂	1	2	yes	maximal
4	C₇₁×#1548	C₂	1	2	yes	maximal
5	C₇₁×#1450	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₇₁	C₂²	1	2	yes	—
7	#498×C₇₁	—	2	1	no	—
8	#498×C₇₁	—	2	1	no	—
9	#498×C₇₁	C₄	1	4	yes	—
10	#431×C₇₁	C₂²	1	2	yes	—
11	C₂×#108×C₇₁	—	2	2	no	—
12	C₂×#106×C₇₁	—	2	2	no	—
13	#423×C₇₁	C₂²	1	1	yes	characteristic
14	#423×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	#333×C₇₁	—	2	4	no	—
17	#332×C₇₁	—	2	4	no	—
18	#322×C₇₁	—	2	1	no	—
19	C₈²×C₇₁	—	2	1	no	—
20	C₂²×Q₈×C₇₁	—	2	2	no	—
21	C₂×M₁₆×C₇₁	—	2	1	no	—
22	C₂×M₁₆×C₇₁	—	2	1	no	—
23	C₂²×C₈×C₇₁	D₄	1	2	yes	—
24	C₄⋊Q₈×C₇₁	—	2	2	no	—
25	C₄⋊Q₈×C₇₁	—	2	2	no	—
26	C₄⋊Q₈×C₇₁	—	2	2	no	—
27	C₄⋊Q₈×C₇₁	C₂×C₄	1	4	yes	—
28	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
29	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
30	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
31	#108×C₇₁	—	4	4	no	—
32	#106×C₇₁	—	2	4	no	—
33	#106×C₇₁	—	2	4	no	—
34	C₄⋊C₈×C₇₁	—	2	2	no	—
35	C₄⋊C₈×C₇₁	—	2	2	no	—
36	C₄⋊C₈×C₇₁	D₄	1	4	yes	—
37	C₂²⋊C₈×C₇₁	—	2	2	no	—
38	C₄×C₈×C₇₁	—	2	2	no	—
39	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
41	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
42	C₂×Q₈×C₇₁	—	2	4	no	—
43	C₂×Q₈×C₇₁	—	4	2	no	—
44	C₂×Q₈×C₇₁	—	8	4	no	—
45	C₂²×C₄×C₇₁	—	4	2	no	—
46	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₇₁	#41	1	2	yes	—
48	M₁₆×C₇₁	—	4	2	no	—
49	M₁₆×C₇₁	—	4	2	no	—
50	C₂×C₈×C₇₁	—	2	4	no	—
51	C₂×C₈×C₇₁	—	4	1	no	—
52	C₂×C₈×C₇₁	—	4	1	no	—
53	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
55	C₂×C₈×C₇₁	SD₁₆	1	4	yes	—
56	C₂×C₈×C₇₁	Q₁₆	1	4	yes	—
57	C₄⋊C₄×C₇₁	—	2	4	no	—
58	C₄⋊C₄×C₇₁	—	2	4	no	—
59	C₄⋊C₄×C₇₁	—	4	2	no	—
60	C₄⋊C₄×C₇₁	—	4	2	no	—
61	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₇₁	C₂×D₄	1	2	yes	—
64	C₂³×C₇₁	#124	1	1	yes	characteristic
65	Q₈×C₇₁	—	8	4	no	—
66	Q₈×C₇₁	—	8	4	no	—
67	C₂×C₄×C₇₁	—	2	4	no	—
68	C₂×C₄×C₇₁	—	4	4	no	—
69	C₂×C₄×C₇₁	—	8	4	no	—
70	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
72	C₂×C₄×C₇₁	#121	1	2	yes	—
73	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
74	C₂×C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
75	C₂×C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
76	C₈×C₇₁	—	4	4	no	—
77	C₈×C₇₁	—	8	1	no	—
78	C₈×C₇₁	—	8	1	no	—
79	C₈×C₇₁	#103	1	4	yes	—
80	C₈×C₇₁	#103	1	4	yes	—
81	C₂²×C₇₁	—	2	2	no	—
82	C₂²×C₇₁	#390	1	1	yes	characteristic
83	C₂²×C₇₁	#487	1	2	yes	—
84	C₄×C₇₁	—	2	2	no	—
85	C₄×C₇₁	—	8	4	no	—
86	C₄×C₇₁	C₂×#103	1	2	yes	—
87	C₄×C₇₁	#471	1	4	yes	—
88	C₄×C₇₁	#492	1	1	yes	characteristic
89	C₄×C₇₁	#494	1	1	yes	characteristic
90	#9767	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂×C₇₁	—	2	2	no	—
92	C₂×C₇₁	#1956	1	1	yes	characteristic
93	C₂×C₇₁	#1977	1	2	yes	—
94	C₂×#498	C₂×C₇₁	1	1	yes	characteristic
95	#1848	C₂×C₇₁	1	1	yes	characteristic
96	#1549	C₂×C₇₁	1	2	yes	—
97	#1548	C₂×C₇₁	1	2	yes	—
98	#1450	C₂×C₇₁	1	1	yes	characteristic
99	C₇₁	#9767	1	1	yes	characteristic, direct product factor, Sylow
100	C₂×C₄⋊Q₈	C₂²×C₇₁	1	2	yes	—
101	#498	—	2	1	no	—
102	#498	—	2	1	no	—
103	#498	C₄×C₇₁	1	4	yes	—
104	#431	C₂²×C₇₁	1	2	yes	—
105	C₂×#108	—	2	2	no	—
106	C₂×#106	—	2	2	no	—
107	#423	C₂²×C₇₁	1	1	yes	characteristic
108	#423	C₂²×C₇₁	1	1	yes	characteristic
109	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
110	#333	—	2	4	no	—
111	#332	—	2	4	no	—
112	#322	—	2	1	no	—
113	C₈²	—	2	1	no	—
114	C₂²×Q₈	—	2	2	no	—
115	C₂×M₁₆	—	2	1	no	—
116	C₂×M₁₆	—	2	1	no	—
117	C₂²×C₈	D₄×C₇₁	1	2	yes	—
118	C₄⋊Q₈	—	2	2	no	—
119	C₄⋊Q₈	—	2	2	no	—
120	C₄⋊Q₈	—	2	2	no	—
121	C₄⋊Q₈	C₂×C₄×C₇₁	1	4	yes	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
125	#108	—	4	4	no	—
126	#106	—	2	4	no	—
127	#106	—	2	4	no	—
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	D₄×C₇₁	1	4	yes	—
131	C₂²⋊C₈	—	2	2	no	—
132	C₄×C₈	—	2	2	no	—
133	C₄×C₈	C₂×C₄×C₇₁	1	2	yes	—
134	C₄×C₈	D₄×C₇₁	1	1	yes	characteristic
135	C₄×C₈	D₄×C₇₁	1	1	yes	characteristic
136	C₂×Q₈	—	2	4	no	—
137	C₂×Q₈	—	4	2	no	—
138	C₂×Q₈	—	8	4	no	—
139	C₂²×C₄	—	4	2	no	—
140	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
141	C₂²×C₄	#41×C₇₁	1	2	yes	—
142	M₁₆	—	4	2	no	—
143	M₁₆	—	4	2	no	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	—	4	1	no	—
146	C₂×C₈	—	4	1	no	—
147	C₂×C₈	C₂²⋊C₄×C₇₁	1	2	yes	—
148	C₂×C₈	C₂²⋊C₄×C₇₁	1	2	yes	—
149	C₂×C₈	SD₁₆×C₇₁	1	4	yes	—
150	C₂×C₈	Q₁₆×C₇₁	1	4	yes	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄⋊C₄	—	4	2	no	—
155	C₄²	C₂²×C₄×C₇₁	1	1	yes	characteristic
156	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
157	C₄²	C₂×D₄×C₇₁	1	2	yes	—
158	C₂³	#124×C₇₁	1	1	yes	characteristic
159	Q₈	—	8	4	no	—
160	Q₈	—	8	4	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	—	8	4	no	—
164	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
165	C₂×C₄	C₄×D₄×C₇₁	1	2	yes	—
166	C₂×C₄	#121×C₇₁	1	2	yes	—
167	C₂×C₄	#127×C₇₁	1	1	yes	characteristic
168	C₂×C₄	C₂×SD₁₆×C₇₁	1	2	yes	—
169	C₂×C₄	C₂×Q₁₆×C₇₁	1	2	yes	—
170	C₈	—	4	4	no	—
171	C₈	—	8	1	no	—
172	C₈	—	8	1	no	—
173	C₈	#103×C₇₁	1	4	yes	—
174	C₈	#103×C₇₁	1	4	yes	—
175	C₂²	—	2	2	no	—
176	C₂²	#390×C₇₁	1	1	yes	characteristic
177	C₂²	#487×C₇₁	1	2	yes	—
178	C₄	—	2	2	no	—
179	C₄	—	8	4	no	—
180	C₄	C₂×#103×C₇₁	1	2	yes	—
181	C₄	#471×C₇₁	1	4	yes	—
182	C₄	#492×C₇₁	1	1	yes	characteristic
183	C₄	#494×C₇₁	1	1	yes	characteristic
184	C₂	—	2	2	no	—
185	C₂	C₇₁×#1956	1	1	yes	characteristic
186	C₂	C₇₁×#1977	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#498×C₇₁

C₂

yes

maximal, characteristic

C₇₁×#1848

C₂

yes

maximal, characteristic

C₇₁×#1549

C₂

yes

maximal

C₇₁×#1548

C₂

yes

maximal

C₇₁×#1450

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₇₁

C₂²

yes

—

#498×C₇₁

—

#498×C₇₁

—

#498×C₇₁

C₄

yes

—

#431×C₇₁

C₂²

yes

—

C₂×#108×C₇₁

—

C₂×#106×C₇₁

—

#423×C₇₁

C₂²

yes

characteristic

#423×C₇₁

C₂²

yes

characteristic

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#333×C₇₁

—

#332×C₇₁

—

#322×C₇₁

—

C₈²×C₇₁

—

C₂²×Q₈×C₇₁

—

C₂×M₁₆×C₇₁

—

C₂×M₁₆×C₇₁

—

C₂²×C₈×C₇₁

D₄

yes

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

#108×C₇₁

—

#106×C₇₁

—

#106×C₇₁

—

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

D₄

yes

—

C₂²⋊C₈×C₇₁

—

C₄×C₈×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

#41

yes

—

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₂×C₈×C₇₁

SD₁₆

yes

—

C₂×C₈×C₇₁

Q₁₆

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂²×C₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

—

C₂³×C₇₁

#124

yes

characteristic

Q₈×C₇₁

—

Q₈×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

C₂×SD₁₆

yes

—

C₂×C₄×C₇₁

C₂×Q₁₆

yes

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

#103

yes

—

C₈×C₇₁

#103

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

#390

yes

characteristic

C₂²×C₇₁

#487

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×#103

yes

—

C₄×C₇₁

#471

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₇₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₁

—

C₂×C₇₁

#1956

yes

characteristic

C₂×C₇₁

#1977

yes

—

C₂×#498

C₂×C₇₁

yes

characteristic

#1848

C₂×C₇₁

yes

characteristic

#1549

C₂×C₇₁

yes

—

#1548

C₂×C₇₁

yes

—

#1450

C₂×C₇₁

yes

characteristic

C₇₁

#9767

yes

characteristic, direct product factor, Sylow

100

C₂×C₄⋊Q₈

C₂²×C₇₁

yes

—

101

#498

—

102

#498

—

103

#498

C₄×C₇₁

yes

—

104

#431

C₂²×C₇₁

yes

—

105

C₂×#108

—

106

C₂×#106

—

107

#423

C₂²×C₇₁

yes

characteristic

108

#423

C₂²×C₇₁

yes

characteristic

109

C₂×C₄×C₈

C₂²×C₇₁

yes

characteristic

110

#333

—

111

#332

—

112

#322

—

113

C₈²

—

114

C₂²×Q₈

—

115

C₂×M₁₆

—

116

C₂×M₁₆

—

117

C₂²×C₈

D₄×C₇₁

yes

—

118

C₄⋊Q₈

—

119

C₄⋊Q₈

—

120

C₄⋊Q₈

—

121

C₄⋊Q₈

C₂×C₄×C₇₁

yes

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄²

C₂³×C₇₁

yes

characteristic

125

#108

—

126

#106

—

127

#106

—

128

C₄⋊C₈

—

129

C₄⋊C₈

—

130

C₄⋊C₈

D₄×C₇₁

yes

—

131

C₂²⋊C₈

—

132

C₄×C₈

—

133

C₄×C₈

C₂×C₄×C₇₁

yes

—

134

C₄×C₈

D₄×C₇₁

yes

characteristic

135

C₄×C₈

D₄×C₇₁

yes

characteristic

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂²×C₄

—

140

C₂²×C₄

C₂×D₄×C₇₁

yes

characteristic

141

C₂²×C₄

#41×C₇₁

yes

—

142

M₁₆

—

143

M₁₆

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

C₂²⋊C₄×C₇₁

yes

—

148

C₂×C₈

C₂²⋊C₄×C₇₁

yes

—

149

C₂×C₈

SD₁₆×C₇₁

yes

—

150

C₂×C₈

Q₁₆×C₇₁

yes

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄²

C₂²×C₄×C₇₁

yes

characteristic

156

C₄²

C₂×D₄×C₇₁

yes

characteristic

157

C₄²

C₂×D₄×C₇₁

yes

—

158

C₂³

#124×C₇₁

yes

characteristic

159

Q₈

—

160

Q₈

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

C₂×C₂²⋊C₄×C₇₁

yes

characteristic

165

C₂×C₄

C₄×D₄×C₇₁

yes

—

166

C₂×C₄

#121×C₇₁

yes

—

167

C₂×C₄

#127×C₇₁

yes

characteristic

168

C₂×C₄

C₂×SD₁₆×C₇₁

yes

—

169

C₂×C₄

C₂×Q₁₆×C₇₁

yes

—

170

C₈

—

171

C₈

—

172

C₈

—

173

C₈

#103×C₇₁

yes

—

174

C₈

#103×C₇₁

yes

—

175

C₂²

—

176

C₂²

#390×C₇₁

yes

characteristic

177

C₂²

#487×C₇₁

yes

—

178

C₄

—

179

C₄

—

180

C₄

C₂×#103×C₇₁

yes

—

181

C₄

#471×C₇₁

yes

—

182

C₄

#492×C₇₁

yes

characteristic

183

C₄

#494×C₇₁

yes

characteristic

184

C₂

—

185

C₂

C₇₁×#1956

yes

characteristic

186

C₂

C₇₁×#1977

yes

—

	subgroup	G/H
center, socle	C₂²×C₇₁	#390
commutator	C₄²	C₂²×C₄×C₇₁
Frattini, cosocle	C₂×C₄²	C₂³×C₇₁
prime radical	C₂³×C₇₁	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₁

#390

commutator

C₄²

C₂²×C₄×C₇₁

Frattini, cosocle

C₂×C₄²

C₂³×C₇₁

C₂³×C₇₁

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ G

N	H ≅ G/N	type
90. #9767	C₇₁	direct (N × H)
99. C₇₁	#9767	direct (N × H)

H ≅ G/N

type

90. #9767

C₇₁

direct (N × H)

99. C₇₁

#9767

direct (N × H)

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

subgroup	index
C₂×#498×C₇₁	2
C₇₁×#1848	2
C₇₁×#1549	2
C₇₁×#1548	2
C₇₁×#1450	2
#9767	71

subgroup

index

C₂×#498×C₇₁

C₇₁×#1848

C₇₁×#1549

C₇₁×#1548

C₇₁×#1450

#9767

Subgroups — C₇₁×#9767 #31830639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C71×#9767 #31830639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#9767 #31830639