Group #22589839 — subgroups

Subgroups — #22589839

Order 9088.

224 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 85 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×Dic₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#17985895	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	#17985761	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	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	Dic₇₁⋊C₈	—	2	2	no	—
15	C₇₁⋊#95	—	2	1	no	—
16	C₇₁⋊#95	—	2	1	no	—
17	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
20	#106×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	Q₈	1	2	yes	—
31	C₄⋊Dic₇₁	Q₈	1	2	yes	—
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	C₂²×C₄×C₇₁	—	2	1	no	—
37	C₂²×C₄×C₇₁	—	2	1	no	—
38	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
39	C₂×C₈×C₇₁	—	2	1	no	—
40	C₂×C₈×C₇₁	—	2	1	no	—
41	C₂×C₈×C₇₁	D₄	1	2	yes	—
42	C₄⋊C₄×C₇₁	—	2	1	no	—
43	C₄⋊C₄×C₇₁	—	2	1	no	—
44	C₄⋊C₄×C₇₁	—	2	1	no	—
45	C₄⋊C₄×C₇₁	—	2	1	no	—
46	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₇₁	—	4	1	no	—
48	C₂×C₄×C₇₁	—	4	1	no	—
49	C₂×C₄×C₇₁	—	4	1	no	—
50	C₂×C₄×C₇₁	—	4	1	no	—
51	C₂×C₄×C₇₁	—	4	1	no	—
52	C₂×C₄×C₇₁	—	4	1	no	—
53	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
57	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
58	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
59	C₂×Dic₇₁	—	2	1	no	—
60	C₂×Dic₇₁	—	2	1	no	—
61	C₂×Dic₇₁	—	2	1	no	—
62	C₂×Dic₇₁	—	2	2	no	—
63	C₂×Dic₇₁	—	2	2	no	—
64	C₂×Dic₇₁	—	2	1	no	—
65	C₂×Dic₇₁	—	2	1	no	—
66	C₂×Dic₇₁	—	4	1	no	—
67	C₂×Dic₇₁	—	4	1	no	—
68	C₂×Dic₇₁	—	4	1	no	—
69	C₂×Dic₇₁	—	4	1	no	—
70	C₂×Dic₇₁	SD₁₆	1	2	yes	—
71	C₈×C₇₁	—	2	2	no	—
72	C₇₁⋊C₈	—	4	1	no	—
73	C₂²×C₇₁	#121	1	1	yes	characteristic
74	C₂²×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂²×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂²×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
77	C₂²×C₇₁	#135	1	1	yes	characteristic
78	C₂²×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂²×C₇₁	#137	1	1	yes	characteristic
80	C₄×C₇₁	—	8	1	no	—
81	C₄×C₇₁	—	8	1	no	—
82	C₄×C₇₁	#123	1	1	yes	characteristic
83	C₄×C₇₁	#125	1	1	yes	characteristic
84	C₄×C₇₁	#125	1	1	yes	characteristic
85	C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
86	Dic₇₁	—	2	2	no	—
87	Dic₇₁	—	4	1	no	—
88	Dic₇₁	—	4	2	no	—
89	Dic₇₁	—	4	2	no	—
90	C₂×C₇₁	#398	1	1	yes	characteristic
91	C₂×C₇₁	#465	1	1	yes	characteristic
92	C₂×C₇₁	#468	1	1	yes	characteristic
93	C₂×C₇₁	#475	1	1	yes	characteristic
94	C₂×C₇₁	#476	1	1	yes	characteristic
95	C₂×C₇₁	#478	1	1	yes	characteristic
96	C₂×C₇₁	#479	1	1	yes	characteristic
97	#2088	—	71	1	no	maximal, Sylow
98	C₇₁	#2088	1	1	yes	characteristic, complemented, Sylow
99	C₂×#106	D₇₁	1	1	yes	characteristic
100	C₂×C₄⋊C₈	—	71	1	no	—
101	#389	—	71	1	no	—
102	#389	—	71	1	no	—
103	#340	—	71	1	no	—
104	#340	—	71	1	no	—
105	#340	—	71	1	no	—
106	C₂²×C₈	—	71	1	no	—
107	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
108	C₂×C₄⋊C₄	—	71	1	no	—
109	C₂×C₄⋊C₄	—	71	1	no	—
110	C₂×C₄⋊C₄	—	142	1	no	—
111	C₂×C₄⋊C₄	—	142	1	no	—
112	C₂×C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
113	C₂×C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
114	C₂×C₄²	—	71	1	no	—
115	#106	—	2	2	no	—
116	C₄⋊C₈	—	142	2	no	—
117	#95	—	142	1	no	—
118	#95	—	142	1	no	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	—	71	1	no	—
122	C₂²×C₄	—	71	1	no	—
123	C₂²×C₄	—	142	1	no	—
124	C₂²×C₄	—	142	1	no	—
125	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	142	1	no	—
129	C₂×C₈	—	142	1	no	—
130	C₂×C₈	—	142	1	no	—
131	C₂×C₈	C₂²⋊D₇₁	1	2	yes	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	2	1	no	—
134	C₄⋊C₄	—	2	1	no	—
135	C₄⋊C₄	—	2	1	no	—
136	C₄⋊C₄	—	284	2	no	—
137	C₄⋊C₄	—	284	2	no	—
138	C₄⋊C₄	—	71	2	no	—
139	C₄⋊C₄	—	71	2	no	—
140	C₄⋊C₄	—	142	1	no	—
141	C₄⋊C₄	—	142	1	no	—
142	C₄²	—	142	1	no	—
143	C₄²	—	142	1	no	—
144	C₂³	D₄×D₇₁	1	1	yes	characteristic
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	284	1	no	—
152	C₂×C₄	—	284	1	no	—
153	C₂×C₄	—	284	1	no	—
154	C₂×C₄	—	284	1	no	—
155	C₂×C₄	—	71	2	no	—
156	C₂×C₄	—	142	1	no	—
157	C₂×C₄	—	142	2	no	—
158	C₂×C₄	—	142	2	no	—
159	C₂×C₄	—	142	1	no	—
160	C₂×C₄	—	142	1	no	—
161	C₂×C₄	—	142	1	no	—
162	C₂×C₄	—	142	1	no	—
163	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
164	C₂×C₄	Q₈×D₇₁	1	1	yes	characteristic
165	C₂×C₄	Q₈×D₇₁	1	1	yes	characteristic
166	C₂×C₄	#11765864	1	1	yes	characteristic
167	C₂×C₄	#11765865	1	1	yes	characteristic
168	C₂×C₄	#11765865	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	—	284	1	no	—
171	C₂²	#14556099	1	1	yes	characteristic
172	C₂²	#14556106	1	1	yes	characteristic
173	C₂²	#14556106	1	1	yes	characteristic
174	C₂²	SD₁₆×D₇₁	1	1	yes	characteristic
175	C₂²	#14556149	1	1	yes	characteristic
176	C₂²	#14556155	1	1	yes	characteristic
177	C₂²	#14556157	1	1	yes	characteristic
178	C₄	—	8	1	no	—
179	C₄	—	8	1	no	—
180	C₄	—	284	1	no	—
181	C₄	—	284	2	no	—
182	C₄	—	284	2	no	—
183	C₄	—	142	2	no	—
184	C₄	#14556111	1	1	yes	characteristic
185	C₄	#14556118	1	1	yes	characteristic
186	C₄	#14556118	1	1	yes	characteristic
187	C₄	#14556126	1	1	yes	characteristic
188	C₂	#17985930	1	1	yes	characteristic
189	C₂	#17986225	1	1	yes	characteristic
190	C₂	#17986242	1	1	yes	characteristic
191	C₂	#17986283	1	1	yes	characteristic
192	C₂	#17986288	1	1	yes	characteristic
193	C₂	#17986298	1	1	yes	characteristic
194	C₂	#17986309	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₇₁⋊C₈

C₂

yes

maximal, characteristic

#17985895

C₂

yes

maximal, characteristic

#17985895

C₂

yes

maximal, characteristic

#17985762

C₂

yes

maximal, characteristic

#17985761

C₂

yes

maximal, characteristic

#17985761

C₂

yes

maximal, characteristic

C₂×#106×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

Dic₇₁⋊C₈

—

C₇₁⋊#95

—

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

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

Q₈

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

D₄

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

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×Q₈

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

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

—

C₂×Dic₇₁

SD₁₆

yes

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

#121

yes

characteristic

C₂²×C₇₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₇₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₇₁

C₂×SD₁₆

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

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

—

C₇₁

yes

characteristic, complemented, Sylow

C₂×#106

D₇₁

yes

characteristic

100

C₂×C₄⋊C₈

—

101

#389

—

102

#389

—

103

#340

—

104

#340

—

105

#340

—

106

C₂²×C₈

—

107

C₂²×C₈

C₂×D₇₁

yes

characteristic

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

142

—

111

C₂×C₄⋊C₄

—

142

—

112

C₂×C₄⋊C₄

C₂×D₇₁

yes

characteristic

113

C₂×C₄⋊C₄

C₂×D₇₁

yes

characteristic

114

C₂×C₄²

—

115

#106

—

116

C₄⋊C₈

—

142

—

117

#95

—

142

—

118

#95

—

142

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

142

—

124

C₂²×C₄

—

142

—

125

C₂²×C₄

C₂²×D₇₁

yes

characteristic

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

142

—

129

C₂×C₈

—

142

—

130

C₂×C₈

—

142

—

131

C₂×C₈

C₂²⋊D₇₁

yes

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

284

—

137

C₄⋊C₄

—

284

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

142

—

141

C₄⋊C₄

—

142

—

142

C₄²

—

142

—

143

C₄²

—

142

—

144

C₂³

D₄×D₇₁

yes

characteristic

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

284

—

152

C₂×C₄

—

284

—

153

C₂×C₄

—

284

—

154

C₂×C₄

—

284

—

155

C₂×C₄

—

156

C₂×C₄

—

142

—

157

C₂×C₄

—

142

—

158

C₂×C₄

—

142

—

159

C₂×C₄

—

142

—

160

C₂×C₄

—

142

—

161

C₂×C₄

—

142

—

162

C₂×C₄

—

142

—

163

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

164

C₂×C₄

Q₈×D₇₁

yes

characteristic

165

C₂×C₄

Q₈×D₇₁

yes

characteristic

166

C₂×C₄

#11765864

yes

characteristic

167

C₂×C₄

#11765865

yes

characteristic

168

C₂×C₄

#11765865

yes

characteristic

169

C₈

—

170

C₈

—

284

—

171

C₂²

#14556099

yes

characteristic

172

C₂²

#14556106

yes

characteristic

173

C₂²

#14556106

yes

characteristic

174

C₂²

SD₁₆×D₇₁

yes

characteristic

175

C₂²

#14556149

yes

characteristic

176

C₂²

#14556155

yes

characteristic

177

C₂²

#14556157

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

—

284

—

181

C₄

—

284

—

182

C₄

—

284

—

183

C₄

—

142

—

184

C₄

#14556111

yes

characteristic

185

C₄

#14556118

yes

characteristic

186

C₄

#14556118

yes

characteristic

187

C₄

#14556126

yes

characteristic

188

C₂

#17985930

yes

characteristic

189

C₂

#17986225

yes

characteristic

190

C₂

#17986242

yes

characteristic

191

C₂

#17986283

yes

characteristic

192

C₂

#17986288

yes

characteristic

193

C₂

#17986298

yes

characteristic

194

C₂

#17986309

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	#2088
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
98. C₇₁	#2088	semidirect

H ≅ G/N

type

98. C₇₁

#2088

semidirect

Subgroups — #22589839

Conjugacy classes of subgroups

Complements