Group #31911729 — subgroups

Subgroups — #31911729

Order 18176.

212 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 67 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	#22590592	C₂	1	1	yes	maximal, characteristic
2	#22590249	C₂	1	1	yes	maximal, characteristic
3	#22587710	C₂	1	1	yes	maximal, characteristic
4	#22586413	C₂	1	1	yes	maximal, characteristic
5	#22586389	C₂	1	1	yes	maximal, characteristic
6	#22586388	C₂	1	1	yes	maximal, characteristic
7	C₇₁×#1670	C₂	1	1	yes	maximal, characteristic
8	C₂×#14556074	—	2	1	no	—
9	C₂×#14556074	—	2	1	no	—
10	#17986379	C₂²	1	1	yes	characteristic
11	#17986284	—	2	1	no	—
12	C₄⋊(C₇₁⋊C₁₆)	—	2	1	no	—
13	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	1	yes	characteristic
14	#17985775	—	2	1	no	—
15	C₄×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
16	#17985752	C₂²	1	1	yes	characteristic
17	#17985746	—	2	1	no	—
18	#494×C₇₁	C₂²	1	1	yes	characteristic
19	#475×C₇₁	—	2	1	no	—
20	C₈×Q₈×C₇₁	C₂²	1	1	yes	characteristic
21	#335×C₇₁	C₂²	1	1	yes	characteristic
22	#332×C₇₁	—	2	1	no	—
23	C₂×Dic₂₈₄	—	2	1	no	—
24	C₂×Dic₂₈₄	—	2	1	no	—
25	#14556128	—	2	1	no	—
26	#14556074	—	4	2	no	—
27	#14556074	—	4	2	no	—
28	C₂×C₇₁⋊C₁₆	—	2	1	no	—
29	C₂×C₇₁⋊C₁₆	D₄	1	1	yes	characteristic
30	C₂×C₇₁⋊C₁₆	D₄	1	1	yes	characteristic
31	#14556059	—	4	1	no	—
32	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
33	#14556052	—	4	1	no	—
34	C₂×Q₁₆×C₇₁	—	2	1	no	—
35	C₂×Q₁₆×C₇₁	—	2	1	no	—
36	C₄⋊Q₈×C₇₁	—	2	1	no	—
37	C₄×Q₈×C₇₁	D₄	1	1	yes	characteristic
38	#106×C₇₁	—	4	1	no	—
39	C₄⋊C₈×C₇₁	—	2	1	no	—
40	C₄⋊C₈×C₇₁	—	2	1	no	—
41	C₄⋊C₈×C₇₁	D₄	1	1	yes	characteristic
42	#103×C₇₁	—	4	1	no	—
43	C₄×C₈×C₇₁	—	2	1	no	—
44	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
45	C₂×Dic₁₄₂	—	4	1	no	—
46	Dic₂₈₄	—	4	1	no	—
47	Dic₂₈₄	—	4	1	no	—
48	Dic₂₈₄	—	8	1	no	—
49	C₂×C₇₁⋊C₈	—	4	1	no	—
50	C₄⋊Dic₇₁	—	4	1	no	—
51	C₄⋊Dic₇₁	—	4	1	no	—
52	C₇₁⋊C₁₆	—	2	1	no	—
53	C₇₁⋊C₁₆	—	4	1	no	—
54	C₇₁⋊C₁₆	SD₁₆	1	2	yes	—
55	C₂×Q₈×C₇₁	—	4	1	no	—
56	C₂×Q₈×C₇₁	D₈	1	1	yes	characteristic
57	Q₁₆×C₇₁	—	4	1	no	—
58	Q₁₆×C₇₁	—	4	1	no	—
59	Q₁₆×C₇₁	—	8	1	no	—
60	C₂×C₈×C₇₁	—	2	1	no	—
61	C₂×C₈×C₇₁	—	4	1	no	—
62	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
64	C₄⋊C₄×C₇₁	—	2	1	no	—
65	C₄⋊C₄×C₇₁	—	4	1	no	—
66	C₄⋊C₄×C₇₁	—	4	1	no	—
67	C₄⋊C₄×C₇₁	D₈	1	1	yes	characteristic
68	C₄²×C₇₁	—	2	1	no	—
69	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
70	Q₈×C₇₁	—	2	1	no	—
71	Q₈×C₇₁	—	2	1	no	—
72	Q₈×C₇₁	—	8	1	no	—
73	Q₈×C₇₁	—	8	1	no	—
74	C₂×C₄×C₇₁	—	2	1	no	—
75	C₂×C₄×C₇₁	—	2	1	no	—
76	C₂×C₄×C₇₁	—	8	1	no	—
77	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
78	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
80	C₂×Dic₇₁	—	8	1	no	—
81	Dic₁₄₂	—	8	1	no	—
82	Dic₁₄₂	—	8	1	no	—
83	C₈×C₇₁	—	2	1	no	—
84	C₈×C₇₁	—	4	1	no	—
85	C₈×C₇₁	—	8	1	no	—
86	C₈×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
87	C₈×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
88	C₇₁⋊C₈	—	8	1	no	—
89	C₂²×C₇₁	#466	1	1	yes	characteristic
90	C₄×C₇₁	—	2	1	no	—
91	C₄×C₇₁	—	4	1	no	—
92	C₄×C₇₁	—	4	1	no	—
93	C₄×C₇₁	—	16	1	no	—
94	C₄×C₇₁	#460	1	1	yes	characteristic
95	C₄×C₇₁	#472	1	1	yes	characteristic
96	C₄×C₇₁	#510	1	1	yes	characteristic
97	C₄×C₇₁	#510	1	1	yes	characteristic
98	Dic₇₁	—	16	1	no	—
99	#12443	—	71	1	no	maximal, Sylow
100	C₂×C₇₁	#1668	1	1	yes	characteristic
101	C₂×C₇₁	#2219	1	1	yes	characteristic
102	C₂×C₇₁	#2222	1	1	yes	characteristic
103	#2247	—	71	1	no	—
104	#2183	—	71	1	no	—
105	#1670	—	71	1	no	—
106	#1670	D₇₁	1	1	yes	characteristic
107	#1372	—	71	1	no	—
108	#1362	—	71	1	no	—
109	#1362	—	71	1	no	—
110	C₇₁	#12443	1	1	yes	characteristic, complemented, Sylow
111	C₂×Q₃₂	—	142	1	no	—
112	C₂×Q₃₂	—	142	1	no	—
113	#494	—	71	1	no	—
114	#494	C₂×D₇₁	1	1	yes	characteristic
115	#475	—	2	1	no	—
116	#475	—	142	1	no	—
117	C₈×Q₈	C₂×D₇₁	1	1	yes	characteristic
118	C₄⋊C₁₆	—	71	1	no	—
119	C₄⋊C₁₆	—	142	1	no	—
120	#346	—	142	1	no	—
121	C₄×C₁₆	—	71	1	no	—
122	#335	—	71	1	no	—
123	#335	C₂×D₇₁	1	1	yes	characteristic
124	#332	—	2	1	no	—
125	#332	—	142	1	no	—
126	C₂×Q₁₆	—	2	1	no	—
127	C₂×Q₁₆	—	2	1	no	—
128	C₂×Q₁₆	—	142	1	no	—
129	C₂×Q₁₆	—	142	1	no	—
130	C₄⋊Q₈	—	2	1	no	—
131	C₄⋊Q₈	—	142	1	no	—
132	C₄×Q₈	C₂²⋊D₇₁	1	1	yes	characteristic
133	Q₃₂	—	284	2	no	—
134	Q₃₂	—	284	2	no	—
135	C₂×C₁₆	—	71	1	no	—
136	C₂×C₁₆	—	71	1	no	—
137	C₂×C₁₆	—	142	1	no	—
138	#106	—	4	1	no	—
139	#106	—	284	1	no	—
140	C₄⋊C₈	—	2	1	no	—
141	C₄⋊C₈	—	2	1	no	—
142	C₄⋊C₈	—	142	1	no	—
143	C₄⋊C₈	C₂²⋊D₇₁	1	1	yes	characteristic
144	#103	—	4	1	no	—
145	#103	—	284	1	no	—
146	C₄×C₈	—	2	1	no	—
147	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
148	C₂×Q₈	—	4	1	no	—
149	C₂×Q₈	—	284	1	no	—
150	C₂×Q₈	#11765850	1	1	yes	characteristic
151	Q₁₆	—	4	1	no	—
152	Q₁₆	—	4	1	no	—
153	Q₁₆	—	8	1	no	—
154	Q₁₆	—	284	1	no	—
155	Q₁₆	—	284	1	no	—
156	Q₁₆	—	568	1	no	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	4	1	no	—
159	C₂×C₈	—	284	1	no	—
160	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
161	C₂×C₈	#11765864	1	1	yes	characteristic
162	C₄⋊C₄	—	2	1	no	—
163	C₄⋊C₄	—	4	1	no	—
164	C₄⋊C₄	—	4	1	no	—
165	C₄⋊C₄	—	284	1	no	—
166	C₄⋊C₄	—	284	1	no	—
167	C₄⋊C₄	#11765850	1	1	yes	characteristic
168	C₄²	—	2	1	no	—
169	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
170	C₁₆	—	284	1	no	—
171	C₁₆	—	71	2	no	—
172	C₁₆	—	142	1	no	—
173	Q₈	—	2	1	no	—
174	Q₈	—	2	1	no	—
175	Q₈	—	8	1	no	—
176	Q₈	—	8	1	no	—
177	Q₈	—	568	1	no	—
178	Q₈	—	568	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	8	1	no	—
182	C₂×C₄	—	568	1	no	—
183	C₂×C₄	#14556100	1	1	yes	characteristic
184	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
185	C₂×C₄	#14556148	1	1	yes	characteristic
186	C₈	—	2	1	no	—
187	C₈	—	4	1	no	—
188	C₈	—	8	1	no	—
189	C₈	—	568	1	no	—
190	C₈	SD₁₆×D₇₁	1	1	yes	characteristic
191	C₈	#14556155	1	1	yes	characteristic
192	C₂²	#17986233	1	1	yes	characteristic
193	C₄	—	2	1	no	—
194	C₄	—	4	1	no	—
195	C₄	—	4	1	no	—
196	C₄	—	16	1	no	—
197	C₄	—	1136	1	no	—
198	C₄	#17986197	1	1	yes	characteristic
199	C₄	#17986267	1	1	yes	characteristic
200	C₄	#17986439	1	1	yes	characteristic
201	C₄	#17986440	1	1	yes	characteristic
202	C₂	#22587698	1	1	yes	characteristic
203	C₂	#22590446	1	1	yes	characteristic
204	C₂	#22590463	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590592

C₂

yes

maximal, characteristic

#22590249

C₂

yes

maximal, characteristic

#22587710

C₂

yes

maximal, characteristic

#22586413

C₂

yes

maximal, characteristic

#22586389

C₂

yes

maximal, characteristic

#22586388

C₂

yes

maximal, characteristic

C₇₁×#1670

C₂

yes

maximal, characteristic

C₂×#14556074

—

C₂×#14556074

—

#17986379

C₂²

yes

characteristic

#17986284

—

C₄⋊(C₇₁⋊C₁₆)

—

C₄⋊(C₇₁⋊C₁₆)

C₂²

yes

characteristic

#17985775

—

C₄×C₇₁⋊C₁₆

C₂²

yes

characteristic

#17985752

C₂²

yes

characteristic

#17985746

—

#494×C₇₁

C₂²

yes

characteristic

#475×C₇₁

—

C₈×Q₈×C₇₁

C₂²

yes

characteristic

#335×C₇₁

C₂²

yes

characteristic

#332×C₇₁

—

C₂×Dic₂₈₄

—

C₂×Dic₂₈₄

—

#14556128

—

#14556074

—

#14556074

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

D₄

yes

characteristic

C₂×C₇₁⋊C₁₆

D₄

yes

characteristic

#14556059

—

C₄⋊(C₇₁⋊C₈)

—

#14556052

—

C₂×Q₁₆×C₇₁

—

C₂×Q₁₆×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₄×Q₈×C₇₁

D₄

yes

characteristic

#106×C₇₁

—

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

D₄

yes

characteristic

#103×C₇₁

—

C₄×C₈×C₇₁

—

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×Dic₁₄₂

—

Dic₂₈₄

—

Dic₂₈₄

—

Dic₂₈₄

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₇₁⋊C₁₆

—

C₇₁⋊C₁₆

—

C₇₁⋊C₁₆

SD₁₆

yes

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

D₈

yes

characteristic

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

#41

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

D₈

yes

characteristic

C₄²×C₇₁

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#121

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

#135

yes

characteristic

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

C₂×SD₁₆

yes

characteristic

C₈×C₇₁

AGL(1,ℤ/8)

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

#466

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

#12443

—

maximal, Sylow

100

C₂×C₇₁

#1668

yes

characteristic

101

C₂×C₇₁

#2219

yes

characteristic

102

C₂×C₇₁

#2222

yes

characteristic

103

#2247

—

104

#2183

—

105

#1670

—

106

#1670

D₇₁

yes

characteristic

107

#1372

—

108

#1362

—

109

#1362

—

110

C₇₁

#12443

yes

characteristic, complemented, Sylow

111

C₂×Q₃₂

—

142

—

112

C₂×Q₃₂

—

142

—

113

#494

—

114

#494

C₂×D₇₁

yes

characteristic

115

#475

—

116

#475

—

142

—

117

C₈×Q₈

C₂×D₇₁

yes

characteristic

118

C₄⋊C₁₆

—

119

C₄⋊C₁₆

—

142

—

120

#346

—

142

—

121

C₄×C₁₆

—

122

#335

—

123

#335

C₂×D₇₁

yes

characteristic

124

#332

—

125

#332

—

142

—

126

C₂×Q₁₆

—

127

C₂×Q₁₆

—

128

C₂×Q₁₆

—

142

—

129

C₂×Q₁₆

—

142

—

130

C₄⋊Q₈

—

131

C₄⋊Q₈

—

142

—

132

C₄×Q₈

C₂²⋊D₇₁

yes

characteristic

133

Q₃₂

—

284

—

134

Q₃₂

—

284

—

135

C₂×C₁₆

—

136

C₂×C₁₆

—

137

C₂×C₁₆

—

142

—

138

#106

—

139

#106

—

284

—

140

C₄⋊C₈

—

141

C₄⋊C₈

—

142

C₄⋊C₈

—

142

—

143

C₄⋊C₈

C₂²⋊D₇₁

yes

characteristic

144

#103

—

145

#103

—

284

—

146

C₄×C₈

—

147

C₄×C₈

C₂²×D₇₁

yes

characteristic

148

C₂×Q₈

—

149

C₂×Q₈

—

284

—

150

C₂×Q₈

#11765850

yes

characteristic

151

Q₁₆

—

152

Q₁₆

—

153

Q₁₆

—

154

Q₁₆

—

284

—

155

Q₁₆

—

284

—

156

Q₁₆

—

568

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

284

—

160

C₂×C₈

D₄×D₇₁

yes

characteristic

161

C₂×C₈

#11765864

yes

characteristic

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

284

—

166

C₄⋊C₄

—

284

—

167

C₄⋊C₄

#11765850

yes

characteristic

168

C₄²

—

169

C₄²

C₂×C₂²⋊D₇₁

yes

characteristic

170

C₁₆

—

284

—

171

C₁₆

—

172

C₁₆

—

142

—

173

Q₈

—

174

Q₈

—

175

Q₈

—

176

Q₈

—

177

Q₈

—

568

—

178

Q₈

—

568

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

568

—

183

C₂×C₄

#14556100

yes

characteristic

184

C₂×C₄

C₂×#11765850

yes

characteristic

185

C₂×C₄

#14556148

yes

characteristic

186

C₈

—

187

C₈

—

188

C₈

—

189

C₈

—

568

—

190

C₈

SD₁₆×D₇₁

yes

characteristic

191

C₈

#14556155

yes

characteristic

192

C₂²

#17986233

yes

characteristic

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

—

1136

—

198

C₄

#17986197

yes

characteristic

199

C₄

#17986267

yes

characteristic

200

C₄

#17986439

yes

characteristic

201

C₄

#17986440

yes

characteristic

202

C₂

#22587698

yes

characteristic

203

C₂

#22590446

yes

characteristic

204

C₂

#22590463

yes

characteristic

	subgroup	G/H
center	C₂²	#17986233
commutator, cosocle	C₄×C₈×C₇₁	C₂³
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1670	C₂
socle, prime radical	C₂²×C₇₁	#466
hypercenter	#1670	D₇₁
hypocenter	C₇₁	#12443
LCS	G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ #1670

subgroup

G/H

center

C₂²

#17986233

commutator, cosocle

C₄×C₈×C₇₁

C₂³

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#1670

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ #1670

N	H ≅ G/N	type
110. C₇₁	#12443	semidirect

H ≅ G/N

type

110. C₇₁

#12443

semidirect

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

subgroup	index
#22590592	2
#22590249	2
#22587710	2
#22586413	2
#22586389	2
#22586388	2
C₇₁×#1670	2
#12443	71

subgroup

index

C₇₁×#1670

Subgroups — #31911729

Conjugacy classes of subgroups

Complements

Maximal subgroups