Group #22589680 — subgroups

Subgroups — #22589680

Order 9088.

250 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 89 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×#14556119	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556061	C₂	1	1	yes	maximal, characteristic
3	#17985895	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×#106×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
9	#14556119	—	2	2	no	—
10	#14556119	—	2	2	no	—
11	C₂×C₄⋊Dic₇₁	—	2	1	no	—
12	C₂×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
14	C₂×Dic₇₁⋊C₄	—	2	1	no	—
15	C₂×Dic₇₁⋊C₄	—	2	1	no	—
16	C₂×C₄×Dic₇₁	C₂²	1	1	yes	characteristic
17	#14556061	—	2	2	no	—
18	C₇₁⋊#95	—	2	1	no	—
19	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
22	#106×C₇₁	—	2	2	no	—
23	C₂²×Dic₇₁	—	2	1	no	—
24	C₂²×Dic₇₁	—	2	1	no	—
25	C₂²×Dic₇₁	D₄	1	1	yes	characteristic
26	C₂²×Dic₇₁	D₄	1	1	yes	characteristic
27	C₂×C₇₁⋊C₈	—	2	1	no	—
28	C₂×C₇₁⋊C₈	—	2	1	no	—
29	C₂×C₇₁⋊C₈	Q₈	1	2	yes	—
30	C₄⋊Dic₇₁	—	2	4	no	—
31	C₄⋊Dic₇₁	—	2	1	no	—
32	C₄⋊Dic₇₁	—	2	1	no	—
33	C₄⋊Dic₇₁	D₄	1	2	yes	—
34	C₄⋊Dic₇₁	Q₈	1	2	yes	—
35	Dic₇₁⋊C₄	—	2	4	no	—
36	Dic₇₁⋊C₄	—	4	2	no	—
37	C₄×Dic₇₁	—	2	1	no	—
38	C₄×Dic₇₁	—	2	1	no	—
39	C₂²×C₄×C₇₁	—	2	1	no	—
40	C₂²×C₄×C₇₁	—	2	1	no	—
41	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
42	C₂×C₈×C₇₁	—	2	1	no	—
43	C₂×C₈×C₇₁	—	2	1	no	—
44	C₂×C₈×C₇₁	Q₈	1	2	yes	—
45	C₄⋊C₄×C₇₁	—	2	1	no	—
46	C₄⋊C₄×C₇₁	—	2	1	no	—
47	C₄⋊C₄×C₇₁	—	2	1	no	—
48	C₄⋊C₄×C₇₁	D₄	1	2	yes	—
49	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₇₁	—	2	2	no	—
51	C₂×C₄×C₇₁	—	4	1	no	—
52	C₂×C₄×C₇₁	—	4	1	no	—
53	C₂×C₄×C₇₁	—	4	1	no	—
54	C₂×C₄×C₇₁	—	4	1	no	—
55	C₂×C₄×C₇₁	—	4	1	no	—
56	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
62	C₂×Dic₇₁	—	2	1	no	—
63	C₂×Dic₇₁	—	2	1	no	—
64	C₂×Dic₇₁	—	2	2	no	—
65	C₂×Dic₇₁	—	2	2	no	—
66	C₂×Dic₇₁	—	2	1	no	—
67	C₂×Dic₇₁	—	2	1	no	—
68	C₂×Dic₇₁	—	2	1	no	—
69	C₂×Dic₇₁	—	2	1	no	—
70	C₂×Dic₇₁	—	4	1	no	—
71	C₂×Dic₇₁	—	4	1	no	—
72	C₂×Dic₇₁	—	4	1	no	—
73	C₂×Dic₇₁	—	4	1	no	—
74	C₈×C₇₁	—	2	2	no	—
75	C₇₁⋊C₈	—	2	2	no	—
76	C₂²×C₇₁	#120	1	1	yes	characteristic
77	C₂²×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
78	C₂²×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
79	C₂²×C₇₁	#135	1	1	yes	characteristic
80	C₂²×C₇₁	#135	1	1	yes	characteristic
81	C₂²×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₂²×C₇₁	#137	1	1	yes	characteristic
83	C₄×C₇₁	—	4	2	no	—
84	C₄×C₇₁	—	8	1	no	—
85	C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
86	C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
87	C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
88	C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
89	Dic₇₁	—	4	1	no	—
90	Dic₇₁	—	4	2	no	—
91	Dic₇₁	—	4	1	no	—
92	Dic₇₁	—	4	2	no	—
93	C₂×C₇₁	#395	1	1	yes	characteristic
94	C₂×C₇₁	#449	1	1	yes	characteristic
95	C₂×C₇₁	#452	1	1	yes	characteristic
96	C₂×C₇₁	#478	1	1	yes	characteristic
97	C₂×C₇₁	#479	1	1	yes	characteristic
98	C₂×C₇₁	#499	1	1	yes	characteristic
99	C₂×C₇₁	#501	1	1	yes	characteristic
100	#2059	—	71	1	no	maximal, Sylow
101	C₇₁	#2059	1	1	yes	characteristic, complemented, Sylow
102	C₂×#125	—	71	1	no	—
103	C₂×#107	—	71	1	no	—
104	C₂×#106	D₇₁	1	1	yes	characteristic
105	#389	—	71	1	no	—
106	#340	—	71	1	no	—
107	#340	—	71	1	no	—
108	#336	—	71	1	no	—
109	C₂²×C₈	—	71	1	no	—
110	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
111	#125	—	142	2	no	—
112	#125	—	142	2	no	—
113	C₂×C₄⋊C₄	—	71	1	no	—
114	C₂×C₄⋊C₄	—	71	1	no	—
115	C₂×C₄⋊C₄	—	142	1	no	—
116	C₂×C₄⋊C₄	—	142	1	no	—
117	C₂×C₄⋊C₄	—	142	1	no	—
118	C₂×C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
120	C₂×C₄²	—	71	1	no	—
121	#107	—	142	2	no	—
122	#106	—	2	2	no	—
123	#95	—	142	1	no	—
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	—	71	1	no	—
127	C₂²×C₄	—	71	1	no	—
128	C₂²×C₄	—	142	1	no	—
129	C₂²×C₄	—	142	1	no	—
130	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	71	2	no	—
134	C₂×C₈	—	142	1	no	—
135	C₂×C₈	—	142	1	no	—
136	C₂×C₈	Dic₁₄₂	1	2	yes	—
137	C₄⋊C₄	—	2	1	no	—
138	C₄⋊C₄	—	2	1	no	—
139	C₄⋊C₄	—	2	1	no	—
140	C₄⋊C₄	—	284	2	no	—
141	C₄⋊C₄	—	71	2	no	—
142	C₄⋊C₄	—	71	2	no	—
143	C₄⋊C₄	—	142	4	no	—
144	C₄⋊C₄	—	142	4	no	—
145	C₄⋊C₄	—	142	1	no	—
146	C₄⋊C₄	—	142	1	no	—
147	C₄⋊C₄	C₂²⋊D₇₁	1	2	yes	—
148	C₄²	—	142	1	no	—
149	C₄²	—	142	1	no	—
150	C₂³	D₄×D₇₁	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	284	1	no	—
158	C₂×C₄	—	284	1	no	—
159	C₂×C₄	—	284	1	no	—
160	C₂×C₄	—	284	1	no	—
161	C₂×C₄	—	142	1	no	—
162	C₂×C₄	—	142	1	no	—
163	C₂×C₄	—	142	2	no	—
164	C₂×C₄	—	142	2	no	—
165	C₂×C₄	—	142	1	no	—
166	C₂×C₄	—	142	1	no	—
167	C₂×C₄	—	142	1	no	—
168	C₂×C₄	—	142	1	no	—
169	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
170	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
171	C₂×C₄	Q₈×D₇₁	1	1	yes	characteristic
172	C₂×C₄	Q₈×D₇₁	1	1	yes	characteristic
173	C₂×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
174	C₂×C₄	#11765865	1	1	yes	characteristic
175	C₈	—	2	2	no	—
176	C₈	—	142	2	no	—
177	C₂²	#14556096	1	1	yes	characteristic
178	C₂²	#14556106	1	1	yes	characteristic
179	C₂²	#14556127	1	1	yes	characteristic
180	C₂²	#14556149	1	1	yes	characteristic
181	C₂²	#14556149	1	1	yes	characteristic
182	C₂²	#14556155	1	1	yes	characteristic
183	C₂²	#14556157	1	1	yes	characteristic
184	C₄	—	4	2	no	—
185	C₄	—	8	1	no	—
186	C₄	—	284	2	no	—
187	C₄	—	284	1	no	—
188	C₄	—	284	1	no	—
189	C₄	—	284	2	no	—
190	C₄	#14556106	1	1	yes	characteristic
191	C₄	#14556107	1	1	yes	characteristic
192	C₄	#14556126	1	1	yes	characteristic
193	C₄	#14556127	1	1	yes	characteristic
194	C₂	#17985919	1	1	yes	characteristic
195	C₂	#17986143	1	1	yes	characteristic
196	C₂	#17986158	1	1	yes	characteristic
197	C₂	#17986298	1	1	yes	characteristic
198	C₂	#17986309	1	1	yes	characteristic
199	C₂	#17986399	1	1	yes	characteristic
200	C₂	#17986406	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556119

C₂

yes

maximal, characteristic

C₂×#14556061

C₂

yes

maximal, characteristic

#17985895

C₂

yes

maximal, characteristic

#17985762

C₂

yes

maximal, characteristic

#17985761

C₂

yes

maximal, characteristic

#17985753

C₂

yes

maximal, characteristic

C₂×#106×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

#14556119

—

#14556119

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₁

C₂²

yes

characteristic

C₂×Dic₇₁⋊C₄

—

C₂×Dic₇₁⋊C₄

—

C₂×C₄×Dic₇₁

C₂²

yes

characteristic

#14556061

—

C₇₁⋊#95

—

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₁

C₂²

yes

characteristic

#106×C₇₁

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

D₄

yes

characteristic

C₂²×Dic₇₁

D₄

yes

characteristic

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

Q₈

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

D₄

yes

—

C₄⋊Dic₇₁

Q₈

yes

—

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

Q₈

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

D₄

yes

—

C₂³×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₁

#41

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

#120

yes

characteristic

C₂²×C₇₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₇₁

C₄⋊Q₈

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂²⋊Q₈

yes

characteristic

C₄×C₇₁

C₂²⋊Q₈

yes

characteristic

C₄×C₇₁

C₄⋊Q₈

yes

characteristic

C₄×C₇₁

C₄⋊Q₈

yes

characteristic

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

100

—

101

C₇₁

#2059

yes

characteristic, complemented, Sylow

102

C₂×#125

—

103

C₂×#107

—

104

C₂×#106

D₇₁

yes

characteristic

105

#389

—

106

#340

—

107

#340

—

108

#336

—

109

C₂²×C₈

—

110

C₂²×C₈

C₂×D₇₁

yes

characteristic

111

#125

—

142

—

112

#125

—

142

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

142

—

116

C₂×C₄⋊C₄

—

142

—

117

C₂×C₄⋊C₄

—

142

—

118

C₂×C₄⋊C₄

C₂×D₇₁

yes

characteristic

119

C₂×C₄⋊C₄

C₂×D₇₁

yes

characteristic

120

C₂×C₄²

—

121

#107

—

142

—

122

#106

—

123

#95

—

142

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

142

—

129

C₂²×C₄

—

142

—

130

C₂²×C₄

C₂²×D₇₁

yes

characteristic

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

142

—

135

C₂×C₈

—

142

—

136

C₂×C₈

Dic₁₄₂

yes

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

284

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

142

—

144

C₄⋊C₄

—

142

—

145

C₄⋊C₄

—

142

—

146

C₄⋊C₄

—

142

—

147

C₄⋊C₄

C₂²⋊D₇₁

yes

—

148

C₄²

—

142

—

149

C₄²

—

142

—

150

C₂³

D₄×D₇₁

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

284

—

158

C₂×C₄

—

284

—

159

C₂×C₄

—

284

—

160

C₂×C₄

—

284

—

161

C₂×C₄

—

142

—

162

C₂×C₄

—

142

—

163

C₂×C₄

—

142

—

164

C₂×C₄

—

142

—

165

C₂×C₄

—

142

—

166

C₂×C₄

—

142

—

167

C₂×C₄

—

142

—

168

C₂×C₄

—

142

—

169

C₂×C₄

D₄×D₇₁

yes

characteristic

170

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

171

C₂×C₄

Q₈×D₇₁

yes

characteristic

172

C₂×C₄

Q₈×D₇₁

yes

characteristic

173

C₂×C₄

C₂×Dic₁₄₂

yes

characteristic

174

C₂×C₄

#11765865

yes

characteristic

175

C₈

—

176

C₈

—

142

—

177

C₂²

#14556096

yes

characteristic

178

C₂²

#14556106

yes

characteristic

179

C₂²

#14556127

yes

characteristic

180

C₂²

#14556149

yes

characteristic

181

C₂²

#14556149

yes

characteristic

182

C₂²

#14556155

yes

characteristic

183

C₂²

#14556157

yes

characteristic

184

C₄

—

185

C₄

—

186

C₄

—

284

—

187

C₄

—

284

—

188

C₄

—

284

—

189

C₄

—

284

—

190

C₄

#14556106

yes

characteristic

191

C₄

#14556107

yes

characteristic

192

C₄

#14556126

yes

characteristic

193

C₄

#14556127

yes

characteristic

194

C₂

#17985919

yes

characteristic

195

C₂

#17986143

yes

characteristic

196

C₂

#17986158

yes

characteristic

197

C₂

#17986298

yes

characteristic

198

C₂

#17986309

yes

characteristic

199

C₂

#17986399

yes

characteristic

200

C₂

#17986406

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₇₁
commutator	C₂²×C₄×C₇₁	C₂³
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	C₂×#106×C₇₁	C₂
socle	C₂³×C₇₁	C₂×D₄
hypercenter	C₂×#106	D₇₁
hypocenter	C₇₁	#2059
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₇₁

commutator

C₂²×C₄×C₇₁

C₂³

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

C₂×#106×C₇₁

C₂

socle

C₂³×C₇₁

C₂×D₄

D₇₁

C₇₁

G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
101. C₇₁	#2059	semidirect

H ≅ G/N

type

101. C₇₁

#2059

semidirect

Subgroups — #22589680

Conjugacy classes of subgroups

Complements