Group #22625703 — subgroups

Subgroups — #22625703

Order 9344.

324 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 100 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×#14556654	C₂	1	2	yes	maximal
2	#17989546	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊#64408	C₂	1	1	yes	maximal, characteristic
4	#17989460	C₂	1	2	yes	maximal
5	C₄×Q₈×D₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11924036	C₂²	1	2	yes	—
7	C₂×C₄⋊#64408	C₂²	1	2	yes	—
8	C₂×C₄×#64408	—	2	1	no	—
9	#14556654	C₄	1	8	yes	complemented
10	#14556646	—	2	1	no	—
11	#14556646	—	2	1	no	—
12	(C₂×#64408)⋊C₄	—	2	1	no	—
13	C₂×Q₈×D₇₃	C₂²	1	1	yes	characteristic
14	Q₈×Dic₇₃	C₄	1	1	yes	characteristic, complemented
15	C₄×Dic₁₄₆	—	2	1	no	—
16	C₄×Dic₁₄₆	C₄	1	1	yes	characteristic, complemented
17	#14556556	—	2	1	no	—
18	#14556556	C₄	1	1	yes	characteristic, complemented
19	C₄⋊C₄×D₇₃	—	2	1	no	—
20	C₄⋊C₄×D₇₃	C₂²	1	1	yes	characteristic
21	C₄²×D₇₃	—	2	1	no	—
22	C₄²×D₇₃	C₂²	1	1	yes	characteristic
23	C₄×Q₈×C₇₃	C₄	1	1	yes	characteristic, complemented
24	C₂²×#64408	—	2	2	no	—
25	#11924036	—	2	2	no	—
26	#11924036	C₂×C₄	1	4	yes	—
27	C₂×#10596833	—	2	2	no	—
28	C₄⋊#64408	—	2	2	no	—
29	C₄⋊#64408	C₂×C₄	1	4	yes	—
30	C₄×#64408	—	4	2	no	—
31	C₂×Dic₁₄₆	—	2	1	no	—
32	C₂×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
33	Q₈×D₇₃	—	2	2	no	—
34	Q₈×D₇₃	C₂×C₄	1	4	yes	—
35	C₂×C₄×D₇₃	—	2	1	no	—
36	C₂×C₄×D₇₃	—	2	1	no	—
37	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
38	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
39	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
40	C₄⋊Dic₇₃	—	2	1	no	—
41	C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
42	#11924007	—	2	1	no	—
43	#11924007	—	2	1	no	—
44	#11924007	—	2	1	no	—
45	C₄×Dic₇₃	—	2	1	no	—
46	C₄×Dic₇₃	—	2	1	no	—
47	C₄×Dic₇₃	—	2	1	no	—
48	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
49	C₄×Dic₇₃	D₄	1	1	yes	characteristic
50	C₄×Dic₇₃	D₄	1	1	yes	characteristic
51	C₂×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
52	C₄⋊C₄×C₇₃	—	2	1	no	—
53	C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
54	C₄²×C₇₃	—	2	1	no	—
55	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
56	C₂²×D₇₃	C₂×D₄	1	1	yes	characteristic
57	C₂×#64408	—	2	4	no	—
58	C₂×#64408	—	4	2	no	—
59	C₂×#64408	—	4	2	no	—
60	Q₈×C₇₃	—	2	1	no	—
61	Q₈×C₇₃	C₄²	1	2	yes	complemented
62	C₂×C₄×C₇₃	—	2	1	no	—
63	C₂×C₄×C₇₃	—	2	1	no	—
64	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×Dic₇₃	—	2	1	no	—
68	C₂×Dic₇₃	—	2	1	no	—
69	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
72	C₄×D₇₃	—	2	4	no	—
73	C₄×D₇₃	—	2	1	no	—
74	C₄×D₇₃	—	2	1	no	—
75	C₄×D₇₃	—	2	2	no	—
76	C₄×D₇₃	—	4	1	no	—
77	C₄×D₇₃	—	4	1	no	—
78	C₄×D₇₃	C₂²×C₄	1	2	yes	—
79	C₄×D₇₃	#41	1	2	yes	—
80	Dic₁₄₆	—	2	1	no	—
81	Dic₁₄₆	—	2	4	no	—
82	Dic₁₄₆	C₄²	1	2	yes	complemented
83	#10596833	—	2	4	no	—
84	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×D₇₃	C₄×D₄	1	2	yes	—
86	C₂×D₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
87	C₂×D₇₃	#137	1	2	yes	—
88	C₂×D₇₃	#137	1	1	yes	characteristic
89	#64408	—	4	4	no	—
90	C₄×C₇₃	—	2	2	no	—
91	C₄×C₇₃	—	2	1	no	—
92	C₄×C₇₃	—	2	1	no	—
93	C₄×C₇₃	—	4	1	no	—
94	C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
95	C₄×C₇₃	#117	1	1	yes	characteristic
96	Dic₇₃	—	2	2	no	—
97	Dic₇₃	—	2	1	no	—
98	Dic₇₃	—	2	1	no	—
99	Dic₇₃	—	4	1	no	—
100	Dic₇₃	C₄×D₄	1	2	yes	—
101	C₂×C₇₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₇₃	#417	1	1	yes	characteristic
103	C₂×C₇₃	#419	1	1	yes	characteristic
104	D₇₃	#440	1	2	yes	—
105	D₇₃	#441	1	2	yes	—
106	#1763	—	73	1	no	maximal, Sylow
107	C₇₃	#1763	1	1	yes	characteristic, complemented, Sylow
108	C₂×C₄×Q₈	—	73	1	no	—
109	C₂×#103	—	73	2	no	—
110	C₂×#97	—	73	1	no	—
111	C₄×C₄⋊C₄	—	73	1	no	—
112	#340	—	73	2	no	—
113	C₂²×Q₈	—	73	1	no	—
114	C₂²×C₈	—	73	2	no	—
115	C₄×Q₈	—	73	1	no	—
116	C₄×Q₈	—	73	1	no	—
117	C₄×Q₈	—	73	1	no	—
118	C₄×Q₈	—	146	1	no	—
119	C₄×Q₈	—	146	1	no	—
120	C₄×Q₈	#64408	1	1	yes	characteristic, complemented
121	C₂×C₄⋊C₄	—	73	2	no	—
122	C₂×C₄⋊C₄	—	73	1	no	—
123	C₂×C₄⋊C₄	—	146	1	no	—
124	C₂×C₄²	—	73	1	no	—
125	C₂×C₄²	—	146	1	no	—
126	C₂×C₄²	—	146	1	no	—
127	#103	—	73	8	no	—
128	#97	—	146	1	no	—
129	#97	—	146	1	no	—
130	#95	—	146	1	no	—
131	C₂×Q₈	—	73	4	no	—
132	C₂×Q₈	—	73	1	no	—
133	C₂×Q₈	—	146	1	no	—
134	C₂×Q₈	—	146	2	no	—
135	C₂×Q₈	C₂×#64408	1	1	yes	characteristic
136	C₂²×C₄	—	73	1	no	—
137	C₂²×C₄	—	73	1	no	—
138	C₂²×C₄	—	73	1	no	—
139	C₂²×C₄	—	146	1	no	—
140	C₂²×C₄	—	146	1	no	—
141	C₂²×C₄	—	146	2	no	—
142	C₂×C₈	—	73	4	no	—
143	C₂×C₈	—	146	2	no	—
144	C₂×C₈	—	146	2	no	—
145	C₄⋊C₄	—	2	1	no	—
146	C₄⋊C₄	—	73	4	no	—
147	C₄⋊C₄	—	73	1	no	—
148	C₄⋊C₄	—	146	1	no	—
149	C₄⋊C₄	—	146	1	no	—
150	C₄⋊C₄	—	146	1	no	—
151	C₄⋊C₄	—	146	2	no	—
152	C₄⋊C₄	—	146	1	no	—
153	C₄⋊C₄	C₂×#64408	1	1	yes	characteristic
154	C₄²	—	2	1	no	—
155	C₄²	—	292	2	no	—
156	C₄²	—	73	1	no	—
157	C₄²	—	73	1	no	—
158	C₄²	—	73	1	no	—
159	C₄²	—	146	1	no	—
160	C₄²	—	146	1	no	—
161	C₄²	—	146	1	no	—
162	C₄²	C₂×#64408	1	1	yes	characteristic
163	C₂³	—	73	1	no	—
164	Q₈	—	2	1	no	—
165	Q₈	—	73	2	no	—
166	Q₈	—	146	4	no	—
167	Q₈	—	146	1	no	—
168	Q₈	C₄×#64408	1	2	yes	complemented
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	292	1	no	—
172	C₂×C₄	—	292	1	no	—
173	C₂×C₄	—	292	2	no	—
174	C₂×C₄	—	292	2	no	—
175	C₂×C₄	—	73	1	no	—
176	C₂×C₄	—	73	2	no	—
177	C₂×C₄	—	73	2	no	—
178	C₂×C₄	—	73	1	no	—
179	C₂×C₄	—	73	1	no	—
180	C₂×C₄	—	146	1	no	—
181	C₂×C₄	—	146	1	no	—
182	C₂×C₄	—	146	2	no	—
183	C₂×C₄	—	146	4	no	—
184	C₂×C₄	—	146	4	no	—
185	C₂×C₄	—	146	1	no	—
186	C₂×C₄	—	146	1	no	—
187	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
188	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
189	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
190	C₈	—	146	4	no	—
191	C₂²	—	73	2	no	—
192	C₂²	—	73	1	no	—
193	C₂²	—	73	1	no	—
194	C₂²	—	73	2	no	—
195	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
196	C₄	—	2	2	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	1	no	—
199	C₄	—	4	1	no	—
200	C₄	—	292	4	no	—
201	C₄	—	292	1	no	—
202	C₄	—	73	2	no	—
203	C₄	—	146	2	no	—
204	C₄	—	146	1	no	—
205	C₄	—	146	1	no	—
206	C₄	C₂×C₄×#64408	1	1	yes	characteristic
207	C₄	#14556670	1	1	yes	characteristic
208	C₂	—	73	2	no	—
209	C₂	—	73	2	no	—
210	C₂	C₄×C₂²⋊#64408	1	1	yes	characteristic
211	C₂	#17989578	1	1	yes	characteristic
212	C₂	#17989580	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556654

C₂

yes

maximal

#17989546

C₂

yes

maximal, characteristic

C₄×C₄⋊#64408

C₂

yes

maximal, characteristic

#17989460

C₂

yes

maximal

C₄×Q₈×D₇₃

C₂

yes

maximal, characteristic

C₂×#11924036

C₂²

yes

—

C₂×C₄⋊#64408

C₂²

yes

—

C₂×C₄×#64408

—

#14556654

C₄

yes

—

—

—

C₂×Q₈×D₇₃

C₂²

yes

characteristic

Q₈×Dic₇₃

C₄

yes

characteristic, complemented

C₄×Dic₁₄₆

—

C₄×Dic₁₄₆

C₄

yes

characteristic, complemented

#14556556

—

#14556556

C₄

yes

characteristic, complemented

C₄⋊C₄×D₇₃

—

C₄⋊C₄×D₇₃

C₂²

yes

characteristic

C₄²×D₇₃

—

C₄²×D₇₃

C₂²

yes

characteristic

C₄×Q₈×C₇₃

C₄

yes

characteristic, complemented

C₂²×#64408

—

#11924036

—

#11924036

C₂×C₄

yes

—

C₂×#10596833

—

C₄⋊#64408

—

C₄⋊#64408

C₂×C₄

yes

—

C₄×#64408

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

C₂×C₄

yes

characteristic

Q₈×D₇₃

—

Q₈×D₇₃

C₂×C₄

yes

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×C₄×D₇₃

C₂³

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

—

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₂×Q₈×C₇₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂×C₄

yes

characteristic

C₄²×C₇₃

—

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂²×D₇₃

C₂×D₄

yes

—

—

—

Q₈×C₇₃

—

Q₈×C₇₃

C₄²

yes

complemented

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₂²×C₄

yes

—

C₄×D₇₃

#41

yes

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

C₄²

yes

complemented

#10596833

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₇₃

C₄×D₄

yes

—

C₂×D₇₃

AGL(1,ℤ/8)

yes

characteristic

C₂×D₇₃

#137

yes

—

C₂×D₇₃

#137

yes

characteristic

#64408

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×C₄²

yes

characteristic

C₄×C₇₃

#117

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

100

Dic₇₃

C₄×D₄

yes

—

101

C₂×C₇₃

C₄×C₂²⋊C₄

yes

characteristic

102

C₂×C₇₃

#417

yes

characteristic

103

C₂×C₇₃

#419

yes

characteristic

104

D₇₃

#440

yes

—

105

D₇₃

#441

yes

—

106

#1763

—

maximal, Sylow

107

C₇₃

#1763

yes

characteristic, complemented, Sylow

108

C₂×C₄×Q₈

—

109

C₂×#103

—

110

C₂×#97

—

111

C₄×C₄⋊C₄

—

112

#340

—

113

C₂²×Q₈

—

114

C₂²×C₈

—

115

C₄×Q₈

—

116

C₄×Q₈

—

117

C₄×Q₈

—

118

C₄×Q₈

—

146

—

119

C₄×Q₈

—

146

—

120

C₄×Q₈

#64408

yes

characteristic, complemented

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

146

—

124

C₂×C₄²

—

125

C₂×C₄²

—

146

—

126

C₂×C₄²

—

146

—

127

#103

—

128

#97

—

146

—

129

#97

—

146

—

130

#95

—

146

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

146

—

134

C₂×Q₈

—

146

—

135

C₂×Q₈

C₂×#64408

yes

characteristic

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

146

—

140

C₂²×C₄

—

146

—

141

C₂²×C₄

—

146

—

142

C₂×C₈

—

143

C₂×C₈

—

146

—

144

C₂×C₈

—

146

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

146

—

149

C₄⋊C₄

—

146

—

150

C₄⋊C₄

—

146

—

151

C₄⋊C₄

—

146

—

152

C₄⋊C₄

—

146

—

153

C₄⋊C₄

C₂×#64408

yes

characteristic

154

C₄²

—

155

C₄²

—

292

—

156

C₄²

—

157

C₄²

—

158

C₄²

—

159

C₄²

—

146

—

160

C₄²

—

146

—

161

C₄²

—

146

—

162

C₄²

C₂×#64408

yes

characteristic

163

C₂³

—

164

Q₈

—

165

Q₈

—

166

Q₈

—

146

—

167

Q₈

—

146

—

168

Q₈

C₄×#64408

yes

complemented

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

292

—

172

C₂×C₄

—

292

—

173

C₂×C₄

—

292

—

174

C₂×C₄

—

292

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

146

—

181

C₂×C₄

—

146

—

182

C₂×C₄

—

146

—

183

C₂×C₄

—

146

—

184

C₂×C₄

—

146

—

185

C₂×C₄

—

146

—

186

C₂×C₄

—

146

—

187

C₂×C₄

C₂²⋊#64408

yes

characteristic

188

C₂×C₄

C₂²⋊#64408

yes

characteristic

189

C₂×C₄

C₂²×#64408

yes

characteristic

190

C₈

—

146

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

194

C₂²

—

195

C₂²

C₂×C₂²⋊#64408

yes

characteristic

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

292

—

201

C₄

—

292

—

202

C₄

—

203

C₄

—

146

—

204

C₄

—

146

—

205

C₄

—

146

—

206

C₄

C₂×C₄×#64408

yes

characteristic

207

C₄

#14556670

yes

characteristic

208

C₂

—

209

C₂

—

210

C₂

C₄×C₂²⋊#64408

yes

characteristic

211

C₂

#17989578

yes

characteristic

212

C₂

#17989580

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#64408
commutator	C₄×C₇₃	C₂×C₄²
Frattini	C₂×C₄	C₂²×#64408
Fitting	C₄×Q₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₂²⋊C₄
hypercenter	C₄×Q₈	#64408
hypocenter	C₇₃	#1763
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#64408

commutator

C₄×C₇₃

C₂×C₄²

Frattini

C₂×C₄

C₂²×#64408

Fitting

C₄×Q₈×C₇₃

C₄

socle

C₂²×C₇₃

C₂×C₂²⋊C₄

C₄×Q₈

C₇₃

G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

N	H ≅ G/N	type
9. #14556654	C₄	semidirect
14. Q₈×Dic₇₃	C₄	semidirect
16. C₄×Dic₁₄₆	C₄	semidirect
18. #14556556	C₄	semidirect
23. C₄×Q₈×C₇₃	C₄	semidirect
61. Q₈×C₇₃	C₄²	semidirect
82. Dic₁₄₆	C₄²	semidirect
107. C₇₃	#1763	semidirect
120. C₄×Q₈	#64408	semidirect
168. Q₈	C₄×#64408	semidirect

H ≅ G/N

type

9. #14556654

C₄

semidirect

14. Q₈×Dic₇₃

C₄

semidirect

16. C₄×Dic₁₄₆

C₄

semidirect

18. #14556556

C₄

semidirect

23. C₄×Q₈×C₇₃

C₄

semidirect

61. Q₈×C₇₃

C₄²

semidirect

82. Dic₁₄₆

C₄²

semidirect

107. C₇₃

#1763

semidirect

120. C₄×Q₈

#64408

semidirect

168. Q₈

C₄×#64408

semidirect

Subgroups — #22625703

Conjugacy classes of subgroups

Complements