Group #22589480 — subgroups

Subgroups — #22589480

Order 9088.

290 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 87 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×#14556127	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556053	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556051	C₂	1	1	yes	maximal, characteristic
4	#17985904	C₂	1	1	yes	maximal, characteristic
5	#17985761	C₂	1	1	yes	maximal, characteristic
6	#17985753	C₂	1	1	yes	maximal, characteristic
7	#340×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₄₂	—	2	1	no	—
9	C₂²×Dic₁₄₂	C₂²	1	1	yes	characteristic
10	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
11	#14556127	—	2	2	no	—
12	#14556127	—	2	2	no	—
13	C₂×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
14	C₂×Dic₇₁⋊C₄	—	2	1	no	—
15	C₂×C₄×Dic₇₁	C₂²	1	1	yes	characteristic
16	#14556053	—	2	2	no	—
17	#14556051	—	2	2	no	—
18	C₇₁⋊#95	—	2	1	no	—
19	C₇₁⋊#95	—	2	1	no	—
20	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₄₂	—	2	1	no	—
24	C₂×Dic₁₄₂	—	2	1	no	—
25	C₂×Dic₁₄₂	—	2	1	no	—
26	C₂×Dic₁₄₂	—	2	1	no	—
27	C₂×Dic₁₄₂	—	2	4	no	—
28	C₂×Dic₁₄₂	—	2	1	no	—
29	C₂×Dic₁₄₂	—	4	2	no	—
30	C₂×Dic₁₄₂	—	4	2	no	—
31	C₂×Dic₁₄₂	Q₈	1	2	yes	complemented
32	C₂²×Dic₇₁	—	2	1	no	—
33	C₂²×Dic₇₁	—	2	1	no	—
34	C₂²×Dic₇₁	D₄	1	1	yes	characteristic
35	C₂²×Dic₇₁	D₄	1	1	yes	characteristic
36	C₂×C₇₁⋊C₈	—	2	1	no	—
37	C₂×C₇₁⋊C₈	—	2	1	no	—
38	C₂×C₇₁⋊C₈	—	2	1	no	—
39	C₄⋊Dic₇₁	—	2	1	no	—
40	C₄⋊Dic₇₁	D₄	1	2	yes	—
41	Dic₇₁⋊C₄	—	2	4	no	—
42	C₄×Dic₇₁	—	2	1	no	—
43	C₄×Dic₇₁	—	2	1	no	—
44	C₂²×C₄×C₇₁	—	2	1	no	—
45	C₂²×C₄×C₇₁	—	2	1	no	—
46	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₇₁	—	2	1	no	—
48	C₂×C₈×C₇₁	—	2	1	no	—
49	C₂×C₈×C₇₁	—	2	1	no	—
50	C₄⋊C₄×C₇₁	—	2	1	no	—
51	C₄⋊C₄×C₇₁	—	2	1	no	—
52	C₄⋊C₄×C₇₁	—	2	1	no	—
53	C₄⋊C₄×C₇₁	D₄	1	2	yes	—
54	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₇₁	C₂×Q₈	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	—	2	2	no	—
72	C₂×Dic₇₁	—	2	2	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	Dic₁₄₂	—	2	2	no	—
80	Dic₁₄₂	—	2	2	no	—
81	Dic₁₄₂	—	4	8	no	—
82	Dic₁₄₂	—	4	1	no	—
83	Dic₁₄₂	—	4	1	no	—
84	C₈×C₇₁	—	4	1	no	—
85	C₇₁⋊C₈	—	4	1	no	—
86	C₂²×C₇₁	#120	1	1	yes	characteristic
87	C₂²×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
88	C₂²×C₇₁	#124	1	1	yes	characteristic
89	C₂²×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
90	C₂²×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
91	C₂²×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
92	C₂²×C₇₁	#137	1	1	yes	characteristic
93	C₄×C₇₁	—	4	2	no	—
94	C₄×C₇₁	—	8	1	no	—
95	C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
96	C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
97	C₄×C₇₁	#124	1	1	yes	characteristic
98	C₄×C₇₁	#124	1	1	yes	characteristic
99	Dic₇₁	—	2	2	no	—
100	Dic₇₁	—	2	2	no	—
101	Dic₇₁	—	4	2	no	—
102	Dic₇₁	—	4	2	no	—
103	C₂×C₇₁	#393	1	1	yes	characteristic
104	C₂×C₇₁	#450	1	1	yes	characteristic
105	C₂×C₇₁	#451	1	1	yes	characteristic
106	C₂×C₇₁	#475	1	1	yes	characteristic
107	C₂×C₇₁	#477	1	1	yes	characteristic
108	C₂×C₇₁	#487	1	1	yes	characteristic
109	C₂×C₇₁	#489	1	1	yes	characteristic
110	#2024	—	71	1	no	maximal, Sylow
111	C₇₁	#2024	1	1	yes	characteristic, complemented, Sylow
112	C₂×C₄⋊Q₈	—	71	1	no	—
113	C₂×#103	—	71	1	no	—
114	C₂×#103	—	71	1	no	—
115	#391	—	71	1	no	—
116	#340	—	71	1	no	—
117	#340	D₇₁	1	1	yes	characteristic
118	#336	—	71	1	no	—
119	C₂²×Q₈	—	71	1	no	—
120	C₂²×Q₈	—	142	1	no	—
121	C₂²×C₈	—	71	1	no	—
122	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
123	C₄⋊Q₈	—	142	2	no	—
124	C₄⋊Q₈	—	142	2	no	—
125	C₂×C₄⋊C₄	—	71	1	no	—
126	C₂×C₄⋊C₄	—	142	1	no	—
127	C₂×C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
128	C₂×C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
129	C₂×C₄²	—	71	1	no	—
130	#103	—	142	2	no	—
131	#103	—	142	2	no	—
132	#95	—	142	1	no	—
133	#95	—	142	1	no	—
134	C₂×Q₈	—	284	2	no	—
135	C₂×Q₈	—	284	2	no	—
136	C₂×Q₈	—	71	2	no	—
137	C₂×Q₈	—	142	4	no	—
138	C₂×Q₈	—	142	1	no	—
139	C₂×Q₈	—	142	1	no	—
140	C₂×Q₈	—	142	1	no	—
141	C₂×Q₈	—	142	1	no	—
142	C₂×Q₈	—	142	1	no	—
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	—	71	1	no	—
146	C₂²×C₄	—	71	1	no	—
147	C₂²×C₄	—	142	1	no	—
148	C₂²×C₄	—	142	1	no	—
149	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	142	1	no	—
154	C₂×C₈	—	142	1	no	—
155	C₂×C₈	—	142	1	no	—
156	C₄⋊C₄	—	2	1	no	—
157	C₄⋊C₄	—	2	1	no	—
158	C₄⋊C₄	—	2	1	no	—
159	C₄⋊C₄	—	71	2	no	—
160	C₄⋊C₄	—	142	4	no	—
161	C₄⋊C₄	—	142	1	no	—
162	C₄⋊C₄	D₂₈₄	1	2	yes	—
163	C₄²	—	142	1	no	—
164	C₄²	—	142	1	no	—
165	C₂³	D₄×D₇₁	1	1	yes	characteristic
166	Q₈	—	284	8	no	—
167	Q₈	—	284	1	no	—
168	Q₈	—	284	1	no	—
169	Q₈	—	142	2	no	—
170	Q₈	—	142	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	—	4	1	no	—
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	—	4	1	no	—
177	C₂×C₄	—	284	1	no	—
178	C₂×C₄	—	284	1	no	—
179	C₂×C₄	—	284	1	no	—
180	C₂×C₄	—	284	1	no	—
181	C₂×C₄	—	71	2	no	—
182	C₂×C₄	—	71	2	no	—
183	C₂×C₄	—	142	2	no	—
184	C₂×C₄	—	142	2	no	—
185	C₂×C₄	—	142	1	no	—
186	C₂×C₄	—	142	1	no	—
187	C₂×C₄	—	142	1	no	—
188	C₂×C₄	—	142	1	no	—
189	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
190	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
191	C₂×C₄	Q₈×D₇₁	1	1	yes	characteristic
192	C₂×C₄	#11765864	1	1	yes	characteristic
193	C₂×C₄	#11765865	1	1	yes	characteristic
194	C₂×C₄	#11765865	1	1	yes	characteristic
195	C₈	—	4	1	no	—
196	C₈	—	284	1	no	—
197	C₂²	#14556098	1	1	yes	characteristic
198	C₂²	#14556106	1	1	yes	characteristic
199	C₂²	#14556114	1	1	yes	characteristic
200	C₂²	SD₁₆×D₇₁	1	1	yes	characteristic
201	C₂²	Q₁₆×D₇₁	1	1	yes	characteristic
202	C₂²	#14556152	1	1	yes	characteristic
203	C₂²	#14556157	1	1	yes	characteristic
204	C₄	—	4	2	no	—
205	C₄	—	8	1	no	—
206	C₄	—	284	2	no	—
207	C₄	—	284	2	no	—
208	C₄	—	142	2	no	—
209	C₄	—	142	2	no	—
210	C₄	#14556106	1	1	yes	characteristic
211	C₄	#14556108	1	1	yes	characteristic
212	C₄	#14556114	1	1	yes	characteristic
213	C₄	#14556116	1	1	yes	characteristic
214	C₂	#17985912	1	1	yes	characteristic
215	C₂	#17986151	1	1	yes	characteristic
216	C₂	#17986156	1	1	yes	characteristic
217	C₂	#17986283	1	1	yes	characteristic
218	C₂	#17986293	1	1	yes	characteristic
219	C₂	#17986348	1	1	yes	characteristic
220	C₂	#17986357	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556127

C₂

yes

maximal, characteristic

C₂×#14556053

C₂

yes

maximal, characteristic

C₂×#14556051

C₂

yes

maximal, characteristic

#17985904

C₂

yes

maximal, characteristic

#17985761

C₂

yes

maximal, characteristic

#17985753

C₂

yes

maximal, characteristic

#340×C₇₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₂

—

C₂²×Dic₁₄₂

C₂²

yes

characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

#14556127

—

#14556127

—

C₂×C₄⋊Dic₇₁

C₂²

yes

characteristic

C₂×Dic₇₁⋊C₄

—

C₂×C₄×Dic₇₁

C₂²

yes

characteristic

#14556053

—

#14556051

—

C₇₁⋊#95

—

C₇₁⋊#95

—

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

Q₈

yes

complemented

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

D₄

yes

—

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

—

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

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

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

#120

yes

characteristic

C₂²×C₇₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₇₁

#124

yes

characteristic

C₂²×C₇₁

C₂×SD₁₆

yes

characteristic

C₂²×C₇₁

C₂×Q₁₆

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂²⋊Q₈

yes

characteristic

C₄×C₇₁

C₂²⋊Q₈

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

100

Dic₇₁

—

101

Dic₇₁

—

102

Dic₇₁

—

103

C₂×C₇₁

#393

yes

characteristic

104

C₂×C₇₁

#450

yes

characteristic

105

C₂×C₇₁

#451

yes

characteristic

106

C₂×C₇₁

#475

yes

characteristic

107

C₂×C₇₁

#477

yes

characteristic

108

C₂×C₇₁

#487

yes

characteristic

109

C₂×C₇₁

yes

110

—

111

C₇₁

#2024

yes

characteristic, complemented, Sylow

112

C₂×C₄⋊Q₈

—

113

C₂×#103

—

114

C₂×#103

—

115

#391

—

116

#340

—

117

#340

D₇₁

yes

characteristic

118

#336

—

119

C₂²×Q₈

—

120

C₂²×Q₈

—

142

—

121

C₂²×C₈

—

122

C₂²×C₈

C₂×D₇₁

yes

characteristic

123

C₄⋊Q₈

—

142

—

124

C₄⋊Q₈

—

142

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

142

—

127

C₂×C₄⋊C₄

C₂×D₇₁

yes

characteristic

128

C₂×C₄⋊C₄

C₂×D₇₁

yes

characteristic

129

C₂×C₄²

—

130

#103

—

142

—

131

#103

—

142

—

132

#95

—

142

—

133

#95

—

142

—

134

C₂×Q₈

—

284

—

135

C₂×Q₈

—

284

—

136

C₂×Q₈

—

137

C₂×Q₈

—

142

—

138

C₂×Q₈

—

142

—

139

C₂×Q₈

—

142

—

140

C₂×Q₈

—

142

—

141

C₂×Q₈

—

142

—

142

C₂×Q₈

—

142

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

C₂²×C₄

—

147

C₂²×C₄

—

142

—

148

C₂²×C₄

—

142

—

149

C₂²×C₄

C₂²×D₇₁

yes

characteristic

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

142

—

154

C₂×C₈

—

142

—

155

C₂×C₈

—

142

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

142

—

161

C₄⋊C₄

—

142

—

162

C₄⋊C₄

D₂₈₄

yes

—

163

C₄²

—

142

—

164

C₄²

—

142

—

165

C₂³

D₄×D₇₁

yes

characteristic

166

Q₈

—

284

—

167

Q₈

—

284

—

168

Q₈

—

284

—

169

Q₈

—

142

—

170

Q₈

—

142

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

284

—

178

C₂×C₄

—

284

—

179

C₂×C₄

—

284

—

180

C₂×C₄

—

284

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

142

—

184

C₂×C₄

—

142

—

185

C₂×C₄

—

142

—

186

C₂×C₄

—

142

—

187

C₂×C₄

—

142

—

188

C₂×C₄

—

142

—

189

C₂×C₄

D₄×D₇₁

yes

characteristic

190

C₂×C₄

C₂×D₂₈₄

yes

characteristic

191

C₂×C₄

Q₈×D₇₁

yes

characteristic

192

C₂×C₄

#11765864

yes

characteristic

193

C₂×C₄

#11765865

yes

characteristic

194

C₂×C₄

#11765865

yes

characteristic

195

C₈

—

196

C₈

—

284

—

197

C₂²

#14556098

yes

characteristic

198

C₂²

#14556106

yes

characteristic

199

C₂²

#14556114

yes

characteristic

200

C₂²

SD₁₆×D₇₁

yes

characteristic

201

C₂²

Q₁₆×D₇₁

yes

characteristic

202

C₂²

#14556152

yes

characteristic

203

C₂²

#14556157

yes

characteristic

204

C₄

—

205

C₄

—

206

C₄

—

284

—

207

C₄

—

284

—

208

C₄

—

142

—

209

C₄

—

142

—

210

C₄

#14556106

yes

characteristic

211

C₄

#14556108

yes

characteristic

212

C₄

#14556114

yes

characteristic

213

C₄

#14556116

yes

characteristic

214

C₂

#17985912

yes

characteristic

215

C₂

#17986151

yes

characteristic

216

C₂

#17986156

yes

characteristic

217

C₂

#17986283

yes

characteristic

218

C₂

#17986293

yes

characteristic

219

C₂

#17986348

yes

characteristic

220

C₂

#17986357

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₇₁
commutator	C₂²×C₄×C₇₁	C₂³
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	#340×C₇₁	C₂
socle	C₂³×C₇₁	C₂×D₄
hypercenter	#340	D₇₁
hypocenter	C₇₁	#2024
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₇₁

commutator

C₂²×C₄×C₇₁

C₂³

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

#340×C₇₁

C₂

socle

C₂³×C₇₁

C₂×D₄

D₇₁

C₇₁

G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
31. C₂×Dic₁₄₂	Q₈	semidirect
111. C₇₁	#2024	semidirect

H ≅ G/N

type

31. C₂×Dic₁₄₂

Q₈

semidirect

111. C₇₁

#2024

semidirect

Subgroups — #22589480

Conjugacy classes of subgroups

Complements