Group #31911605 — subgroups

Subgroups — #31911605

Order 18176.

218 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 69 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	#22590390	C₂	1	1	yes	maximal, characteristic
2	#22587938	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#22586403	C₂	1	1	yes	maximal, characteristic
5	#22586398	C₂	1	1	yes	maximal, characteristic
6	#22586390	C₂	1	1	yes	maximal, characteristic
7	C₇₁×#2227	C₂	1	1	yes	maximal, characteristic
8	C₂×#14556074	—	2	1	no	—
9	#17986395	—	2	1	no	—
10	#17986379	C₂²	1	1	yes	characteristic
11	#17986349	—	2	1	no	—
12	Q₈×C₇₁⋊C₈	—	2	1	no	—
13	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	1	yes	characteristic
14	#17985802	—	2	1	no	—
15	#17985752	C₂²	1	1	yes	characteristic
16	Q₈⋊(C₇₁⋊C₈)	—	2	1	no	—
17	C₈×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
18	#500×C₇₁	C₂²	1	1	yes	characteristic
19	C₄×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
20	#366×C₇₁	—	2	1	no	—
21	C₄⋊C₁₆×C₇₁	C₂²	1	1	yes	characteristic
22	#358×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	C₂×C₇₁⋊C₁₆	—	2	1	no	—
28	#14556059	—	4	1	no	—
29	#14556059	—	4	1	no	—
30	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
31	C₄⋊(C₇₁⋊C₈)	—	4	1	no	—
32	#14556052	—	4	2	no	—
33	C₄×C₇₁⋊C₈	—	4	1	no	—
34	C₄×C₇₁⋊C₈	D₄	1	1	yes	characteristic
35	C₄×C₇₁⋊C₈	D₄	1	1	yes	characteristic
36	C₂×Q₁₆×C₇₁	D₄	1	1	yes	characteristic
37	C₄⋊Q₈×C₇₁	—	2	1	no	—
38	C₄×Q₈×C₇₁	—	2	1	no	—
39	C₂×C₁₆×C₇₁	—	2	1	no	—
40	#107×C₇₁	—	2	1	no	—
41	#107×C₇₁	—	2	1	no	—
42	#107×C₇₁	D₄	1	1	yes	characteristic
43	#103×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	—	2	1	no	—
50	C₂×C₇₁⋊C₈	—	4	1	no	—
51	C₂×C₇₁⋊C₈	D₈	1	1	yes	characteristic
52	C₂×C₇₁⋊C₈	D₈	1	1	yes	characteristic
53	C₄⋊Dic₇₁	—	4	1	no	—
54	C₄⋊Dic₇₁	—	4	1	no	—
55	C₇₁⋊C₁₆	—	4	1	no	—
56	C₂×Q₈×C₇₁	—	2	1	no	—
57	C₂×Q₈×C₇₁	—	4	1	no	—
58	Q₁₆×C₇₁	—	2	1	no	—
59	Q₁₆×C₇₁	SD₁₆	1	2	yes	—
60	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
62	C₄⋊C₄×C₇₁	—	2	1	no	—
63	C₄⋊C₄×C₇₁	—	4	1	no	—
64	C₄⋊C₄×C₇₁	—	4	1	no	—
65	C₄⋊C₄×C₇₁	—	4	1	no	—
66	C₄²×C₇₁	—	4	1	no	—
67	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
68	C₁₆×C₇₁	—	4	1	no	—
69	Q₈×C₇₁	—	2	2	no	—
70	Q₈×C₇₁	—	4	1	no	—
71	Q₈×C₇₁	—	8	2	no	—
72	C₂×C₄×C₇₁	—	4	1	no	—
73	C₂×C₄×C₇₁	—	4	1	no	—
74	C₂×C₄×C₇₁	—	8	1	no	—
75	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
76	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂×Dic₇₁	—	8	1	no	—
79	Dic₁₄₂	—	8	1	no	—
80	Dic₁₄₂	—	8	1	no	—
81	C₈×C₇₁	—	2	1	no	—
82	C₈×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
83	C₈×C₇₁	#135	1	1	yes	characteristic
84	C₇₁⋊C₈	—	2	1	no	—
85	C₇₁⋊C₈	—	4	1	no	—
86	C₇₁⋊C₈	—	8	1	no	—
87	C₇₁⋊C₈	Q₃₂	1	2	yes	complemented
88	C₂²×C₇₁	#459	1	1	yes	characteristic
89	C₄×C₇₁	—	2	1	no	—
90	C₄×C₇₁	—	4	2	no	—
91	C₄×C₇₁	—	8	1	no	—
92	C₄×C₇₁	—	16	1	no	—
93	C₄×C₇₁	#465	1	1	yes	characteristic
94	C₄×C₇₁	#471	1	1	yes	characteristic
95	C₄×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
96	C₄×C₇₁	#510	1	1	yes	characteristic
97	Dic₇₁	—	16	1	no	—
98	#12421	—	71	1	no	maximal, Sylow
99	C₂×C₇₁	#1665	1	1	yes	characteristic
100	C₂×C₇₁	#2209	1	1	yes	characteristic
101	C₂×C₇₁	#2212	1	1	yes	characteristic
102	#2227	D₇₁	1	1	yes	characteristic
103	#2209	—	71	1	no	—
104	#1707	—	71	1	no	—
105	C₈×Q₁₆	—	71	1	no	—
106	#1367	—	71	1	no	—
107	#1365	—	71	1	no	—
108	#1362	—	71	1	no	—
109	C₇₁	#12421	1	1	yes	characteristic, complemented, Sylow
110	C₂×Q₃₂	—	142	1	no	—
111	#500	C₂×D₇₁	1	1	yes	characteristic
112	#498	—	142	1	no	—
113	#494	—	71	1	no	—
114	#487	—	142	1	no	—
115	C₈×Q₈	—	142	1	no	—
116	C₄×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
117	#366	—	2	1	no	—
118	C₄⋊C₁₆	—	71	1	no	—
119	C₄⋊C₁₆	C₂×D₇₁	1	1	yes	characteristic
120	#358	—	2	1	no	—
121	#358	—	142	1	no	—
122	#335	—	71	1	no	—
123	Q₈⋊C₈	—	142	1	no	—
124	C₈²	—	71	1	no	—
125	C₂×Q₁₆	—	142	1	no	—
126	C₂×Q₁₆	—	142	1	no	—
127	C₂×Q₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
128	C₄⋊Q₈	—	2	1	no	—
129	C₄⋊Q₈	—	142	1	no	—
130	C₄×Q₈	—	2	1	no	—
131	Q₃₂	—	284	2	no	—
132	C₂×C₁₆	—	2	1	no	—
133	C₂×C₁₆	—	142	1	no	—
134	#107	—	2	1	no	—
135	#107	—	2	1	no	—
136	#107	C₂²⋊D₇₁	1	1	yes	characteristic
137	#106	—	284	1	no	—
138	#106	—	284	1	no	—
139	C₄⋊C₈	—	284	1	no	—
140	C₄⋊C₈	—	142	1	no	—
141	#103	—	2	1	no	—
142	#103	—	284	2	no	—
143	C₄×C₈	—	284	1	no	—
144	C₄×C₈	—	71	1	no	—
145	C₄×C₈	—	71	1	no	—
146	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
147	C₂×Q₈	—	2	1	no	—
148	C₂×Q₈	—	4	1	no	—
149	C₂×Q₈	—	284	1	no	—
150	Q₁₆	—	2	1	no	—
151	Q₁₆	—	284	1	no	—
152	Q₁₆	—	284	1	no	—
153	Q₁₆	—	568	1	no	—
154	Q₁₆	#11765852	1	2	yes	—
155	C₂×C₈	—	284	1	no	—
156	C₂×C₈	—	71	1	no	—
157	C₂×C₈	—	71	1	no	—
158	C₂×C₈	—	142	1	no	—
159	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
160	C₂×C₈	#11765864	1	1	yes	characteristic
161	C₄⋊C₄	—	2	1	no	—
162	C₄⋊C₄	—	4	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	C₄²	—	4	1	no	—
168	C₄²	D₄×D₇₁	1	1	yes	characteristic
169	C₁₆	—	4	1	no	—
170	C₁₆	—	284	1	no	—
171	Q₈	—	2	2	no	—
172	Q₈	—	4	1	no	—
173	Q₈	—	8	2	no	—
174	Q₈	—	568	1	no	—
175	Q₈	—	568	1	no	—
176	C₂×C₄	—	4	1	no	—
177	C₂×C₄	—	4	1	no	—
178	C₂×C₄	—	8	1	no	—
179	C₂×C₄	—	568	1	no	—
180	C₂×C₄	#14556100	1	1	yes	characteristic
181	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
182	C₂×C₄	#14556152	1	1	yes	characteristic
183	C₈	—	2	1	no	—
184	C₈	—	284	1	no	—
185	C₈	—	568	1	no	—
186	C₈	—	71	2	no	—
187	C₈	—	142	1	no	—
188	C₈	C₂×#11765852	1	1	yes	characteristic
189	C₈	#14556148	1	1	yes	characteristic
190	C₂²	#17986192	1	1	yes	characteristic
191	C₄	—	2	1	no	—
192	C₄	—	4	2	no	—
193	C₄	—	8	1	no	—
194	C₄	—	16	1	no	—
195	C₄	—	1136	1	no	—
196	C₄	#17986230	1	1	yes	characteristic
197	C₄	#17986264	1	1	yes	characteristic
198	C₄	Q₃₂×D₇₁	1	1	yes	characteristic
199	C₄	#17986441	1	1	yes	characteristic
200	C₂	#22587683	1	1	yes	characteristic
201	C₂	#22590388	1	1	yes	characteristic
202	C₂	#22590407	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590390

C₂

yes

maximal, characteristic

#22587938

C₂

yes

maximal, characteristic

Q₁₆×C₇₁⋊C₈

C₂

yes

maximal, characteristic

#22586403

C₂

yes

maximal, characteristic

#22586398

C₂

yes

maximal, characteristic

#22586390

C₂

yes

maximal, characteristic

C₇₁×#2227

C₂

yes

maximal, characteristic

C₂×#14556074

—

#17986395

—

#17986379

C₂²

yes

characteristic

#17986349

—

Q₈×C₇₁⋊C₈

—

C₄⋊(C₇₁⋊C₁₆)

C₂²

yes

characteristic

#17985802

—

#17985752

C₂²

yes

characteristic

Q₈⋊(C₇₁⋊C₈)

—

C₈×C₇₁⋊C₈

C₂²

yes

characteristic

#500×C₇₁

C₂²

yes

characteristic

C₄×Q₁₆×C₇₁

C₂²

yes

characteristic

#366×C₇₁

—

C₄⋊C₁₆×C₇₁

C₂²

yes

characteristic

#358×C₇₁

—

C₂×Dic₂₈₄

—

C₂×Dic₂₈₄

—

#14556128

—

#14556074

—

C₂×C₇₁⋊C₁₆

—

#14556059

—

#14556059

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

#14556052

—

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

D₄

yes

characteristic

C₄×C₇₁⋊C₈

D₄

yes

characteristic

C₂×Q₁₆×C₇₁

D₄

yes

characteristic

C₄⋊Q₈×C₇₁

—

C₄×Q₈×C₇₁

—

C₂×C₁₆×C₇₁

—

#107×C₇₁

—

#107×C₇₁

—

#107×C₇₁

D₄

yes

characteristic

#103×C₇₁

—

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×Dic₁₄₂

—

Dic₂₈₄

—

Dic₂₈₄

—

Dic₂₈₄

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

D₈

yes

characteristic

C₂×C₇₁⋊C₈

D₈

yes

characteristic

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₇₁⋊C₁₆

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

SD₁₆

yes

—

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

—

C₄²×C₇₁

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₁₆×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

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

C₂×SD₁₆

yes

characteristic

C₈×C₇₁

#135

yes

characteristic

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

Q₃₂

yes

complemented

C₂²×C₇₁

#459

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

C₂×Q₃₂

yes

characteristic

C₄×C₇₁

#510

yes

characteristic

Dic₇₁

—

—

C₂×C₇₁

yes

100

C₂×C₇₁

#2209

yes

characteristic

101

C₂×C₇₁

#2212

yes

characteristic

102

#2227

D₇₁

yes

characteristic

103

#2209

—

104

#1707

—

105

C₈×Q₁₆

—

106

#1367

—

107

#1365

—

108

#1362

—

109

C₇₁

#12421

yes

characteristic, complemented, Sylow

110

C₂×Q₃₂

—

142

—

111

#500

C₂×D₇₁

yes

characteristic

112

#498

—

142

—

113

#494

—

114

#487

—

142

—

115

C₈×Q₈

—

142

—

116

C₄×Q₁₆

C₂×D₇₁

yes

characteristic

117

#366

—

118

C₄⋊C₁₆

—

119

C₄⋊C₁₆

C₂×D₇₁

yes

characteristic

120

#358

—

121

#358

—

142

—

122

#335

—

123

Q₈⋊C₈

—

142

—

124

C₈²

—

125

C₂×Q₁₆

—

142

—

126

C₂×Q₁₆

—

142

—

127

C₂×Q₁₆

C₂²⋊D₇₁

yes

characteristic

128

C₄⋊Q₈

—

129

C₄⋊Q₈

—

142

—

130

C₄×Q₈

—

131

Q₃₂

—

284

—

132

C₂×C₁₆

—

133

C₂×C₁₆

—

142

—

134

#107

—

135

#107

—

136

#107

C₂²⋊D₇₁

yes

characteristic

137

#106

—

284

—

138

#106

—

284

—

139

C₄⋊C₈

—

284

—

140

C₄⋊C₈

—

142

—

141

#103

—

142

#103

—

284

—

143

C₄×C₈

—

284

—

144

C₄×C₈

—

145

C₄×C₈

—

146

C₄×C₈

C₂²×D₇₁

yes

characteristic

147

C₂×Q₈

—

148

C₂×Q₈

—

149

C₂×Q₈

—

284

—

150

Q₁₆

—

151

Q₁₆

—

284

—

152

Q₁₆

—

284

—

153

Q₁₆

—

568

—

154

Q₁₆

#11765852

yes

—

155

C₂×C₈

—

284

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

142

—

159

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

160

C₂×C₈

#11765864

yes

characteristic

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

284

—

166

C₄⋊C₄

—

284

—

167

C₄²

—

168

C₄²

D₄×D₇₁

yes

characteristic

169

C₁₆

—

170

C₁₆

—

284

—

171

Q₈

—

172

Q₈

—

173

Q₈

—

174

Q₈

—

568

—

175

Q₈

—

568

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

568

—

180

C₂×C₄

#14556100

yes

characteristic

181

C₂×C₄

D₈×D₇₁

yes

characteristic

182

C₂×C₄

#14556152

yes

characteristic

183

C₈

—

184

C₈

—

284

—

185

C₈

—

568

—

186

C₈

—

187

C₈

—

142

—

188

C₈

C₂×#11765852

yes

characteristic

189

C₈

#14556148

yes

characteristic

190

C₂²

#17986192

yes

characteristic

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

1136

—

196

C₄

#17986230

yes

characteristic

197

C₄

#17986264

yes

characteristic

198

C₄

Q₃₂×D₇₁

yes

characteristic

199

C₄

#17986441

yes

characteristic

200

C₂

#22587683

yes

characteristic

201

C₂

#22590388

yes

characteristic

202

C₂

#22590407

yes

characteristic

	subgroup	G/H
center	C₂²	#17986192
commutator, cosocle	C₄×C₈×C₇₁	C₂³
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#2227	C₂
socle, prime radical	C₂²×C₇₁	#459
hypercenter	#2227	D₇₁
hypocenter	C₇₁	#12421
LCS	G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2227

subgroup

G/H

center

C₂²

#17986192

commutator, cosocle

C₄×C₈×C₇₁

C₂³

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#2227

C₂

C₂²×C₇₁

D₇₁

C₇₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2227

N	H ≅ G/N	type
87. C₇₁⋊C₈	Q₃₂	semidirect
109. C₇₁	#12421	semidirect

H ≅ G/N

type

87. C₇₁⋊C₈

Q₃₂

semidirect

109. C₇₁

#12421

semidirect

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

subgroup	index
#22590390	2
#22587938	2
Q₁₆×C₇₁⋊C₈	2
#22586403	2
#22586398	2
#22586390	2
C₇₁×#2227	2
#12421	71

subgroup

index

Q₁₆×C₇₁⋊C₈

C₇₁×#2227

Subgroups — #31911605

Conjugacy classes of subgroups

Complements

Maximal subgroups