Group #31902504 — subgroups

Subgroups — #31902504

Order 18176.

442 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 155 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	#22589302	C₂	1	2	yes	maximal
2	#22588500	C₂	1	1	yes	maximal, characteristic
3	#22588499	C₂	1	2	yes	maximal
4	C₇₁⋊#1273	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1956	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556062	—	2	2	no	—
7	C₂×#14556061	—	2	4	no	—
8	C₂×C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
9	#17985869	C₂²	1	2	yes	—
10	#17985761	—	2	4	no	—
11	C₂×C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
12	C₂×#103×C₇₁	—	2	2	no	—
13	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	#384×C₇₁	C₂²	1	1	yes	characteristic
15	C₂²×C₇₁⋊C₈	D₄	1	4	yes	—
16	C₂×C₄⋊Dic₇₁	—	2	2	no	—
17	C₂×C₄⋊Dic₇₁	—	2	2	no	—
18	C₂×C₄⋊Dic₇₁	—	2	2	no	—
19	#14556062	—	4	4	no	—
20	#14556061	—	4	8	no	—
21	C₄×C₇₁⋊C₈	—	2	2	no	—
22	C₄×C₇₁⋊C₈	—	2	2	no	—
23	C₂²×Q₈×C₇₁	—	2	1	no	—
24	C₂²×C₈×C₇₁	D₄	1	2	yes	—
25	C₄⋊Q₈×C₇₁	—	2	2	no	—
26	C₄⋊Q₈×C₇₁	—	2	1	no	—
27	C₄⋊Q₈×C₇₁	Q₈	1	2	yes	—
28	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
29	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
30	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
31	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
32	#103×C₇₁	—	4	4	no	—
33	C₄×C₈×C₇₁	—	2	2	no	—
34	C₂²×Dic₇₁	—	4	2	no	—
35	C₂×C₇₁⋊C₈	—	2	4	no	—
36	C₂×C₇₁⋊C₈	—	2	4	no	—
37	C₂×C₇₁⋊C₈	Q₁₆	1	8	yes	—
38	C₄⋊Dic₇₁	—	4	4	no	—
39	C₄⋊Dic₇₁	—	4	4	no	—
40	C₄⋊Dic₇₁	—	4	4	no	—
41	C₂×Q₈×C₇₁	—	2	2	no	—
42	C₂×Q₈×C₇₁	—	4	1	no	—
43	C₂×Q₈×C₇₁	—	4	2	no	—
44	C₂×Q₈×C₇₁	—	4	2	no	—
45	C₂²×C₄×C₇₁	—	4	1	no	—
46	C₂²×C₄×C₇₁	—	4	1	no	—
47	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₇₁	—	2	2	no	—
50	C₂×C₈×C₇₁	—	2	2	no	—
51	C₂×C₈×C₇₁	D₈	1	4	yes	—
52	C₄⋊C₄×C₇₁	—	2	4	no	—
53	C₄⋊C₄×C₇₁	—	4	2	no	—
54	C₄⋊C₄×C₇₁	—	4	2	no	—
55	C₄⋊C₄×C₇₁	—	4	4	no	—
56	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₇₁	#41	1	2	yes	—
58	C₄²×C₇₁	#41	1	1	yes	characteristic
59	C₂³×C₇₁	#127	1	1	yes	characteristic
60	Q₈×C₇₁	—	4	4	no	—
61	Q₈×C₇₁	—	8	2	no	—
62	C₂×C₄×C₇₁	—	4	2	no	—
63	C₂×C₄×C₇₁	—	8	1	no	—
64	C₂×C₄×C₇₁	—	8	1	no	—
65	C₂×C₄×C₇₁	—	8	1	no	—
66	C₂×C₄×C₇₁	—	8	2	no	—
67	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
68	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₇₁	#123	1	1	yes	characteristic
70	C₂×C₄×C₇₁	#123	1	2	yes	—
71	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
72	C₂×C₄×C₇₁	C₂×Q₁₆	1	4	yes	—
73	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	4	yes	—
74	C₂×C₄×C₇₁	#137	1	2	yes	—
75	C₂×Dic₇₁	—	8	2	no	—
76	C₂×Dic₇₁	—	8	2	no	—
77	C₂×Dic₇₁	—	8	2	no	—
78	C₈×C₇₁	—	2	4	no	—
79	C₇₁⋊C₈	—	2	8	no	—
80	C₂²×C₇₁	#399	1	1	yes	characteristic
81	C₂²×C₇₁	#493	1	1	yes	characteristic
82	C₂²×C₇₁	#494	1	2	yes	—
83	C₂²×C₇₁	#496	1	2	yes	—
84	C₂²×C₇₁	#497	1	1	yes	characteristic
85	C₄×C₇₁	—	8	2	no	—
86	C₄×C₇₁	—	16	1	no	—
87	C₄×C₇₁	#474	1	2	yes	—
88	C₄×C₇₁	#477	1	4	yes	—
89	C₄×C₇₁	#480	1	2	yes	—
90	C₄×C₇₁	#484	1	4	yes	—
91	Dic₇₁	—	16	2	no	—
92	#10674	—	71	1	no	maximal, Sylow
93	C₂×C₇₁	#1713	1	1	yes	characteristic
94	C₂×C₇₁	#1721	1	1	yes	characteristic
95	C₂×C₇₁	#1723	1	2	yes	—
96	C₂×C₇₁	#2093	1	1	yes	characteristic
97	C₂×C₇₁	#2095	1	2	yes	—
98	#1990	—	71	2	no	—
99	#1956	D₇₁	1	1	yes	characteristic
100	#1845	—	71	1	no	—
101	#1845	—	71	2	no	—
102	#1273	—	71	1	no	—
103	C₇₁	#10674	1	1	yes	characteristic, complemented, Sylow
104	C₂×C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
105	C₂×#107	—	142	4	no	—
106	C₂×#107	—	142	2	no	—
107	C₂×#103	—	2	2	no	—
108	C₂×C₄×C₈	—	71	2	no	—
109	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
110	#384	—	71	2	no	—
111	#384	C₂×D₇₁	1	1	yes	characteristic
112	#340	—	142	4	no	—
113	C₂²×Q₈	—	2	1	no	—
114	C₂²×C₈	—	71	4	no	—
115	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
116	C₄⋊Q₈	—	2	2	no	—
117	C₄⋊Q₈	—	2	1	no	—
118	C₄⋊Q₈	Dic₁₄₂	1	2	yes	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	2	1	no	—
122	C₂×C₄⋊C₄	—	142	2	no	—
123	C₂×C₄⋊C₄	—	142	2	no	—
124	C₂×C₄⋊C₄	—	142	2	no	—
125	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
126	#107	—	284	8	no	—
127	#107	—	284	4	no	—
128	#103	—	4	4	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	142	2	no	—
131	C₄×C₈	—	142	2	no	—
132	C₂×Q₈	—	2	2	no	—
133	C₂×Q₈	—	4	1	no	—
134	C₂×Q₈	—	4	2	no	—
135	C₂×Q₈	—	4	2	no	—
136	C₂²×C₄	—	4	1	no	—
137	C₂²×C₄	—	4	1	no	—
138	C₂²×C₄	—	284	2	no	—
139	C₂²×C₄	D₄×D₇₁	1	2	yes	—
140	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	71	8	no	—
144	C₂×C₈	—	142	4	no	—
145	C₂×C₈	—	142	4	no	—
146	C₂×C₈	#11765850	1	4	yes	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	284	4	no	—
152	C₄⋊C₄	—	284	4	no	—
153	C₄⋊C₄	—	284	4	no	—
154	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
155	C₄²	#11765864	1	2	yes	—
156	C₄²	#11765865	1	1	yes	characteristic
157	C₂³	#14556124	1	1	yes	characteristic
158	Q₈	—	4	4	no	—
159	Q₈	—	8	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	8	1	no	—
163	C₂×C₄	—	8	1	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	—	568	2	no	—
166	C₂×C₄	—	568	2	no	—
167	C₂×C₄	—	568	2	no	—
168	C₂×C₄	#14556105	1	2	yes	—
169	C₂×C₄	#14556107	1	1	yes	characteristic
170	C₂×C₄	#14556109	1	1	yes	characteristic
171	C₂×C₄	#14556110	1	2	yes	—
172	C₂×C₄	C₂×#11765850	1	2	yes	—
173	C₂×C₄	Q₁₆×D₇₁	1	4	yes	—
174	C₂×C₄	#14556155	1	4	yes	—
175	C₂×C₄	#14556158	1	2	yes	—
176	C₈	—	2	4	no	—
177	C₈	—	142	8	no	—
178	C₂²	#17985935	1	1	yes	characteristic
179	C₂²	#17986375	1	1	yes	characteristic
180	C₂²	#17986377	1	2	yes	—
181	C₂²	#17986385	1	2	yes	—
182	C₂²	#17986394	1	1	yes	characteristic
183	C₄	—	8	2	no	—
184	C₄	—	16	1	no	—
185	C₄	—	1136	2	no	—
186	C₄	#17986279	1	2	yes	—
187	C₄	#17986292	1	4	yes	—
188	C₄	#17986310	1	2	yes	—
189	C₄	#17986333	1	4	yes	—
190	C₂	#22587956	1	1	yes	characteristic
191	C₂	#22588004	1	1	yes	characteristic
192	C₂	#22588020	1	2	yes	—
193	C₂	#22589868	1	1	yes	characteristic
194	C₂	#22589876	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22589302

C₂

yes

maximal

#22588500

C₂

yes

maximal, characteristic

#22588499

C₂

yes

maximal

C₇₁⋊#1273

C₂

yes

maximal, characteristic

C₇₁×#1956

C₂

yes

maximal, characteristic

C₂×#14556062

—

C₂×#14556061

—

C₂×C₄×C₇₁⋊C₈

C₂²

yes

—

#17985869

C₂²

yes

—

#17985761

—

C₂×C₄⋊Q₈×C₇₁

C₂²

yes

characteristic

C₂×#103×C₇₁

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#384×C₇₁

C₂²

yes

characteristic

C₂²×C₇₁⋊C₈

D₄

yes

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

—

#14556062

—

#14556061

—

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

—

C₂²×Q₈×C₇₁

—

C₂²×C₈×C₇₁

D₄

yes

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

Q₈

yes

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

#103×C₇₁

—

C₄×C₈×C₇₁

—

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

Q₁₆

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

D₈

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂×Q₈

yes

characteristic

C₄²×C₇₁

#41

yes

—

C₄²×C₇₁

yes

C₂³×C₇₁

yes

Q₈×C₇₁

—

Q₈×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

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

—

C₂×C₄×C₇₁

C₂×D₈

yes

—

C₂×C₄×C₇₁

C₂×Q₁₆

yes

—

C₂×C₄×C₇₁

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇₁

#137

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

—

C₂²×C₇₁

#496

yes

—

C₂²×C₇₁

#497

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#474

yes

—

C₄×C₇₁

#477

yes

—

C₄×C₇₁

#480

yes

—

C₄×C₇₁

#484

yes

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₂×C₇₁

#2093

yes

characteristic

C₂×C₇₁

#2095

yes

—

#1990

—

#1956

D₇₁

yes

characteristic

100

#1845

—

101

#1845

—

102

#1273

—

103

C₇₁

#10674

yes

characteristic, complemented, Sylow

104

C₂×C₄⋊Q₈

C₂×D₇₁

yes

characteristic

105

C₂×#107

—

142

—

106

C₂×#107

—

142

—

107

C₂×#103

—

108

C₂×C₄×C₈

—

109

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

110

#384

—

111

#384

C₂×D₇₁

yes

characteristic

112

#340

—

142

—

113

C₂²×Q₈

—

114

C₂²×C₈

—

115

C₂²×C₈

C₂²⋊D₇₁

yes

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

—

118

C₄⋊Q₈

Dic₁₄₂

yes

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

142

—

123

C₂×C₄⋊C₄

—

142

—

124

C₂×C₄⋊C₄

—

142

—

125

C₂×C₄²

C₂²×D₇₁

yes

characteristic

126

#107

—

284

—

127

#107

—

284

—

128

#103

—

129

C₄×C₈

—

130

C₄×C₈

—

142

—

131

C₄×C₈

—

142

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

284

—

139

C₂²×C₄

D₄×D₇₁

yes

—

140

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

142

—

145

C₂×C₈

—

142

—

146

C₂×C₈

#11765850

yes

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

284

—

152

C₄⋊C₄

—

284

—

153

C₄⋊C₄

—

284

—

154

C₄²

C₂×Dic₁₄₂

yes

characteristic

155

C₄²

#11765864

yes

—

156

C₄²

#11765865

yes

characteristic

157

C₂³

#14556124

yes

characteristic

158

Q₈

—

159

Q₈

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

568

—

166

C₂×C₄

—

568

—

167

C₂×C₄

—

568

—

168

C₂×C₄

#14556105

yes

—

169

C₂×C₄

#14556107

yes

characteristic

170

C₂×C₄

#14556109

yes

characteristic

171

C₂×C₄

#14556110

yes

—

172

C₂×C₄

C₂×#11765850

yes

—

173

C₂×C₄

Q₁₆×D₇₁

yes

—

174

C₂×C₄

#14556155

yes

—

175

C₂×C₄

#14556158

yes

—

176

C₈

—

177

C₈

—

142

—

178

C₂²

#17985935

yes

characteristic

179

C₂²

#17986375

yes

characteristic

180

C₂²

#17986377

yes

—

181

C₂²

#17986385

yes

—

182

C₂²

#17986394

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

1136

—

186

C₄

#17986279

yes

—

187

C₄

#17986292

yes

—

188

C₄

#17986310

yes

—

189

C₄

#17986333

yes

—

190

C₂

#22587956

yes

characteristic

191

C₂

#22588004

yes

characteristic

192

C₂

#22588020

yes

—

193

C₂

#22589868

yes

characteristic

194

C₂

#22589876

yes

—

	subgroup	G/H
center	C₂³	#14556124
commutator, cosocle	C₂×C₄²×C₇₁	C₂³
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1956	C₂
socle, prime radical	C₂³×C₇₁	#127
hypercenter	#1956	D₇₁
hypocenter	C₇₁	#10674
LCS	G ▷ C₂×C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

subgroup

G/H

center

C₂³

#14556124

commutator, cosocle

C₂×C₄²×C₇₁

C₂³

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#1956

C₂

C₂³×C₇₁

D₇₁

C₇₁

G ▷ C₂×C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

N	H ≅ G/N	type
103. C₇₁	#10674	semidirect

H ≅ G/N

type

103. C₇₁

#10674

semidirect

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

subgroup	index
#22589302	2
#22588500	2
#22588499	2
C₇₁⋊#1273	2
C₇₁×#1956	2
#10674	71

subgroup

index

#22589302

#22588500

#22588499

C₇₁⋊#1273

C₇₁×#1956

#10674

Subgroups — #31902504

Conjugacy classes of subgroups

Complements

Maximal subgroups