Group #22589751 — subgroups

Subgroups — #22589751

Order 9088.

238 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 83 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×#14556052	C₂	1	1	yes	maximal, characteristic
2	#17985906	C₂	1	1	yes	maximal, characteristic
3	#17985868	C₂	1	1	yes	maximal, characteristic
4	#17985762	C₂	1	1	yes	maximal, characteristic
5	#17985761	C₂	1	1	yes	maximal, characteristic
6	#17985753	C₂	1	1	yes	maximal, characteristic
7	C₂×#103×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
11	C₂×Dic₇₁⋊C₄	—	2	1	no	—
12	C₂×Dic₇₁⋊C₄	—	2	1	no	—
13	C₂×C₄×Dic₇₁	C₂²	1	1	yes	characteristic
14	#14556052	—	2	2	no	—
15	C₇₁⋊#95	—	2	1	no	—
16	C₇₁⋊#95	—	2	1	no	—
17	C₂²×Q₈×C₇₁	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
20	#103×C₇₁	—	2	2	no	—
21	C₂²×Dic₇₁	—	2	1	no	—
22	C₂²×Dic₇₁	—	2	1	no	—
23	C₂²×Dic₇₁	D₄	1	1	yes	characteristic
24	C₂²×Dic₇₁	D₄	1	1	yes	characteristic
25	C₂×C₇₁⋊C₈	—	2	1	no	—
26	C₂×C₇₁⋊C₈	—	2	1	no	—
27	C₂×C₇₁⋊C₈	—	2	1	no	—
28	C₄⋊Dic₇₁	—	2	1	no	—
29	C₄⋊Dic₇₁	—	2	1	no	—
30	C₄⋊Dic₇₁	—	2	1	no	—
31	C₄⋊Dic₇₁	—	2	1	no	—
32	Dic₇₁⋊C₄	—	4	2	no	—
33	Dic₇₁⋊C₄	—	4	2	no	—
34	C₄×Dic₇₁	—	2	1	no	—
35	C₄×Dic₇₁	—	2	1	no	—
36	C₂×Q₈×C₇₁	—	2	1	no	—
37	C₂×Q₈×C₇₁	—	2	1	no	—
38	C₂×Q₈×C₇₁	—	2	1	no	—
39	C₂×Q₈×C₇₁	—	2	1	no	—
40	C₂×Q₈×C₇₁	—	2	1	no	—
41	C₂×Q₈×C₇₁	D₄	1	2	yes	—
42	C₂²×C₄×C₇₁	—	2	1	no	—
43	C₂²×C₄×C₇₁	—	2	1	no	—
44	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
45	C₂×C₈×C₇₁	—	2	1	no	—
46	C₂×C₈×C₇₁	—	2	1	no	—
47	C₂×C₈×C₇₁	—	2	1	no	—
48	C₄⋊C₄×C₇₁	—	2	1	no	—
49	C₄⋊C₄×C₇₁	—	2	1	no	—
50	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
51	Q₈×C₇₁	—	2	2	no	—
52	Q₈×C₇₁	—	2	2	no	—
53	Q₈×C₇₁	—	4	1	no	—
54	Q₈×C₇₁	—	4	1	no	—
55	C₂×C₄×C₇₁	—	2	2	no	—
56	C₂×C₄×C₇₁	—	4	1	no	—
57	C₂×C₄×C₇₁	—	4	1	no	—
58	C₂×C₄×C₇₁	—	4	1	no	—
59	C₂×C₄×C₇₁	—	4	1	no	—
60	C₂×C₄×C₇₁	—	4	1	no	—
61	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
63	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
64	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
65	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
66	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
67	C₂×Dic₇₁	—	2	1	no	—
68	C₂×Dic₇₁	—	2	1	no	—
69	C₂×Dic₇₁	—	2	1	no	—
70	C₂×Dic₇₁	—	2	1	no	—
71	C₂×Dic₇₁	—	4	1	no	—
72	C₂×Dic₇₁	—	4	1	no	—
73	C₂×Dic₇₁	—	4	1	no	—
74	C₂×Dic₇₁	—	4	1	no	—
75	C₂×Dic₇₁	—	4	1	no	—
76	C₂×Dic₇₁	—	4	1	no	—
77	C₂×Dic₇₁	SD₁₆	1	2	yes	—
78	C₂×Dic₇₁	Q₁₆	1	2	yes	—
79	C₈×C₇₁	—	4	1	no	—
80	C₇₁⋊C₈	—	4	1	no	—
81	C₂²×C₇₁	#121	1	1	yes	characteristic
82	C₂²×C₇₁	#123	1	1	yes	characteristic
83	C₂²×C₇₁	#124	1	1	yes	characteristic
84	C₂²×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
85	C₂²×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
86	C₂²×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
87	C₂²×C₇₁	#137	1	1	yes	characteristic
88	C₄×C₇₁	—	4	2	no	—
89	C₄×C₇₁	—	8	1	no	—
90	C₄×C₇₁	#123	1	1	yes	characteristic
91	C₄×C₇₁	#123	1	1	yes	characteristic
92	C₄×C₇₁	#126	1	1	yes	characteristic
93	C₄×C₇₁	#126	1	1	yes	characteristic
94	Dic₇₁	—	2	2	no	—
95	Dic₇₁	—	2	2	no	—
96	Dic₇₁	—	8	1	no	—
97	Dic₇₁	—	8	1	no	—
98	C₂×C₇₁	#397	1	1	yes	characteristic
99	C₂×C₇₁	#460	1	1	yes	characteristic
100	C₂×C₇₁	#462	1	1	yes	characteristic
101	C₂×C₇₁	#483	1	1	yes	characteristic
102	C₂×C₇₁	#484	1	1	yes	characteristic
103	C₂×C₇₁	#487	1	1	yes	characteristic
104	C₂×C₇₁	#489	1	1	yes	characteristic
105	#2072	—	71	1	no	maximal, Sylow
106	C₇₁	#2072	1	1	yes	characteristic, complemented, Sylow
107	C₂×#103	—	71	1	no	—
108	C₂×#103	D₇₁	1	1	yes	characteristic
109	#391	—	71	1	no	—
110	#384	—	71	1	no	—
111	#340	—	71	1	no	—
112	#340	—	71	1	no	—
113	#336	—	71	1	no	—
114	C₂²×Q₈	C₂×D₇₁	1	1	yes	characteristic
115	C₂²×C₈	—	71	1	no	—
116	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	71	1	no	—
118	C₂×C₄⋊C₄	—	71	1	no	—
119	C₂×C₄⋊C₄	—	142	1	no	—
120	C₂×C₄⋊C₄	—	142	1	no	—
121	C₂×C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
122	C₂×C₄²	—	71	1	no	—
123	#103	—	2	2	no	—
124	#103	—	142	2	no	—
125	#95	—	142	1	no	—
126	#95	—	142	1	no	—
127	C₂×Q₈	—	2	1	no	—
128	C₂×Q₈	—	2	1	no	—
129	C₂×Q₈	—	2	1	no	—
130	C₂×Q₈	—	2	1	no	—
131	C₂×Q₈	—	2	1	no	—
132	C₂×Q₈	C₂²⋊D₇₁	1	2	yes	—
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	71	1	no	—
136	C₂²×C₄	—	71	1	no	—
137	C₂²×C₄	—	142	1	no	—
138	C₂²×C₄	—	142	1	no	—
139	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	142	1	no	—
144	C₂×C₈	—	142	1	no	—
145	C₂×C₈	—	142	1	no	—
146	C₄⋊C₄	—	2	1	no	—
147	C₄⋊C₄	—	2	1	no	—
148	C₄⋊C₄	—	284	2	no	—
149	C₄⋊C₄	—	284	2	no	—
150	C₄⋊C₄	—	142	1	no	—
151	C₄⋊C₄	—	142	1	no	—
152	C₄⋊C₄	—	142	1	no	—
153	C₄⋊C₄	—	142	1	no	—
154	C₄²	—	142	1	no	—
155	C₄²	—	142	1	no	—
156	C₂³	D₄×D₇₁	1	1	yes	characteristic
157	Q₈	—	2	2	no	—
158	Q₈	—	2	2	no	—
159	Q₈	—	4	1	no	—
160	Q₈	—	4	1	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	4	1	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	4	1	no	—
165	C₂×C₄	—	4	1	no	—
166	C₂×C₄	—	4	1	no	—
167	C₂×C₄	—	284	1	no	—
168	C₂×C₄	—	284	1	no	—
169	C₂×C₄	—	284	1	no	—
170	C₂×C₄	—	284	1	no	—
171	C₂×C₄	—	284	1	no	—
172	C₂×C₄	—	284	1	no	—
173	C₂×C₄	—	71	2	no	—
174	C₂×C₄	—	71	2	no	—
175	C₂×C₄	—	142	1	no	—
176	C₂×C₄	—	142	1	no	—
177	C₂×C₄	—	142	1	no	—
178	C₂×C₄	—	142	1	no	—
179	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
180	C₂×C₄	#11765864	1	1	yes	characteristic
181	C₂×C₄	#11765864	1	1	yes	characteristic
182	C₂×C₄	#11765864	1	1	yes	characteristic
183	C₂×C₄	#11765865	1	1	yes	characteristic
184	C₂×C₄	#11765865	1	1	yes	characteristic
185	C₈	—	4	1	no	—
186	C₈	—	284	1	no	—
187	C₂²	#14556100	1	1	yes	characteristic
188	C₂²	#14556110	1	1	yes	characteristic
189	C₂²	#14556114	1	1	yes	characteristic
190	C₂²	SD₁₆×D₇₁	1	1	yes	characteristic
191	C₂²	Q₁₆×D₇₁	1	1	yes	characteristic
192	C₂²	#14556155	1	1	yes	characteristic
193	C₂²	#14556160	1	1	yes	characteristic
194	C₄	—	4	2	no	—
195	C₄	—	8	1	no	—
196	C₄	—	568	1	no	—
197	C₄	—	568	1	no	—
198	C₄	—	142	2	no	—
199	C₄	—	142	2	no	—
200	C₄	#14556109	1	1	yes	characteristic
201	C₄	#14556111	1	1	yes	characteristic
202	C₄	#14556121	1	1	yes	characteristic
203	C₄	#14556121	1	1	yes	characteristic
204	C₂	#17985927	1	1	yes	characteristic
205	C₂	#17986197	1	1	yes	characteristic
206	C₂	#17986207	1	1	yes	characteristic
207	C₂	#17986328	1	1	yes	characteristic
208	C₂	#17986333	1	1	yes	characteristic
209	C₂	#17986348	1	1	yes	characteristic
210	C₂	#17986360	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556052

C₂

yes

maximal, characteristic

#17985906

C₂

yes

maximal, characteristic

#17985868

C₂

yes

maximal, characteristic

#17985762

C₂

yes

maximal, characteristic

#17985761

C₂

yes

maximal, characteristic

#17985753

C₂

yes

maximal, characteristic

C₂×#103×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₁

C₂²

yes

characteristic

C₂×Dic₇₁⋊C₄

—

C₂×Dic₇₁⋊C₄

—

C₂×C₄×Dic₇₁

C₂²

yes

characteristic

#14556052

—

C₇₁⋊#95

—

C₇₁⋊#95

—

C₂²×Q₈×C₇₁

C₂²

yes

characteristic

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₁

C₂²

yes

characteristic

#103×C₇₁

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

D₄

yes

characteristic

C₂²×Dic₇₁

D₄

yes

characteristic

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

D₄

yes

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₂³×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

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

SD₁₆

yes

—

C₂×Dic₇₁

Q₁₆

yes

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

C₂×SD₁₆

yes

characteristic

C₂²×C₇₁

C₂×Q₁₆

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

100

C₂×C₇₁

#462

yes

characteristic

101

C₂×C₇₁

#483

yes

characteristic

102

C₂×C₇₁

#484

yes

characteristic

103

C₂×C₇₁

#487

yes

characteristic

104

C₂×C₇₁

yes

105

—

106

C₇₁

#2072

yes

characteristic, complemented, Sylow

107

C₂×#103

—

108

C₂×#103

D₇₁

yes

characteristic

109

#391

—

110

#384

—

111

#340

—

112

#340

—

113

#336

—

114

C₂²×Q₈

C₂×D₇₁

yes

characteristic

115

C₂²×C₈

—

116

C₂²×C₈

C₂×D₇₁

yes

characteristic

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

142

—

120

C₂×C₄⋊C₄

—

142

—

121

C₂×C₄⋊C₄

C₂×D₇₁

yes

characteristic

122

C₂×C₄²

—

123

#103

—

124

#103

—

142

—

125

#95

—

142

—

126

#95

—

142

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

C₂²⋊D₇₁

yes

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

142

—

138

C₂²×C₄

—

142

—

139

C₂²×C₄

C₂²×D₇₁

yes

characteristic

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

142

—

144

C₂×C₈

—

142

—

145

C₂×C₈

—

142

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

284

—

149

C₄⋊C₄

—

284

—

150

C₄⋊C₄

—

142

—

151

C₄⋊C₄

—

142

—

152

C₄⋊C₄

—

142

—

153

C₄⋊C₄

—

142

—

154

C₄²

—

142

—

155

C₄²

—

142

—

156

C₂³

D₄×D₇₁

yes

characteristic

157

Q₈

—

158

Q₈

—

159

Q₈

—

160

Q₈

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

284

—

168

C₂×C₄

—

284

—

169

C₂×C₄

—

284

—

170

C₂×C₄

—

284

—

171

C₂×C₄

—

284

—

172

C₂×C₄

—

284

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

142

—

176

C₂×C₄

—

142

—

177

C₂×C₄

—

142

—

178

C₂×C₄

—

142

—

179

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

180

C₂×C₄

#11765864

yes

characteristic

181

C₂×C₄

#11765864

yes

characteristic

182

C₂×C₄

#11765864

yes

characteristic

183

C₂×C₄

#11765865

yes

characteristic

184

C₂×C₄

#11765865

yes

characteristic

185

C₈

—

186

C₈

—

284

—

187

C₂²

#14556100

yes

characteristic

188

C₂²

#14556110

yes

characteristic

189

C₂²

#14556114

yes

characteristic

190

C₂²

SD₁₆×D₇₁

yes

characteristic

191

C₂²

Q₁₆×D₇₁

yes

characteristic

192

C₂²

#14556155

yes

characteristic

193

C₂²

#14556160

yes

characteristic

194

C₄

—

195

C₄

—

196

C₄

—

568

—

197

C₄

—

568

—

198

C₄

—

142

—

199

C₄

—

142

—

200

C₄

#14556109

yes

characteristic

201

C₄

#14556111

yes

characteristic

202

C₄

#14556121

yes

characteristic

203

C₄

#14556121

yes

characteristic

204

C₂

#17985927

yes

characteristic

205

C₂

#17986197

yes

characteristic

206

C₂

#17986207

yes

characteristic

207

C₂

#17986328

yes

characteristic

208

C₂

#17986333

yes

characteristic

209

C₂

#17986348

yes

characteristic

210

C₂

#17986360

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₇₁
commutator	C₂²×C₄×C₇₁	C₂³
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	C₂×#103×C₇₁	C₂
socle	C₂³×C₇₁	C₂×D₄
hypercenter	C₂×#103	D₇₁
hypocenter	C₇₁	#2072
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#103

subgroup

G/H

center

C₂³

D₄×D₇₁

commutator

C₂²×C₄×C₇₁

C₂³

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

C₂×#103×C₇₁

C₂

socle

C₂³×C₇₁

C₂×D₄

D₇₁

C₇₁

G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#103

N	H ≅ G/N	type
106. C₇₁	#2072	semidirect

H ≅ G/N

type

106. C₇₁

#2072

semidirect

Subgroups — #22589751

Conjugacy classes of subgroups

Complements