Group #22620647 — subgroups

Subgroups — #22620647

Order 9344.

402 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 125 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	#17989142	C₂	1	1	yes	maximal, characteristic, complemented
2	#17989135	C₂	1	1	yes	maximal, characteristic
3	#17988992	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988937	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988933	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988830	C₂	1	4	yes	maximal, complemented
7	#17988688	C₂	1	4	yes	maximal, complemented
8	C₄×#11924011	C₂	1	1	yes	maximal, characteristic, complemented
9	#492×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11924027	—	2	1	no	—
11	C₂×#11924011	C₂²	1	2	yes	—
12	#14556591	C₂²	1	1	yes	characteristic
13	#14556591	C₂²	1	1	yes	characteristic
14	#14556582	—	2	1	no	—
15	#14556578	—	2	1	no	—
16	#14556571	—	2	1	no	—
17	#14556567	C₂²	1	2	yes	—
18	#14556562	—	2	1	no	—
19	#14556562	C₂²	1	2	yes	—
20	Q₈×Dic₇₃	C₂²	1	2	yes	—
21	#14556556	—	2	1	no	—
22	D₄×Dic₇₃	—	2	1	no	—
23	D₄×Dic₇₃	C₂²	1	2	yes	—
24	#14556549	—	2	1	no	—
25	C₄²×D₇₃	C₂²	1	1	yes	characteristic
26	#14556525	C₂²	1	4	yes	—
27	#14556515	C₂²	1	4	yes	—
28	#14556512	C₂²	1	4	yes	—
29	#14556501	C₂²	1	2	yes	—
30	C₄×#10596831	C₂²	1	1	yes	characteristic
31	C₂×SD₁₆×C₇₃	C₂²	1	4	yes	—
32	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
33	#127×C₇₃	C₂²	1	1	yes	characteristic
34	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
35	#11924027	—	4	4	no	—
36	C₂×Dic₁₄₆	—	2	1	no	—
37	C₂×Dic₁₄₆	—	2	1	no	—
38	C₂×C₂²⋊D₇₃	—	2	2	no	—
39	C₂²×Dic₇₃	—	2	2	no	—
40	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
41	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
42	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
43	#11924011	—	2	4	no	—
44	C₂×#10596831	C₂³	1	2	yes	—
45	C₄⋊Dic₇₃	—	2	1	no	—
46	C₄⋊Dic₇₃	—	2	1	no	—
47	C₄⋊Dic₇₃	—	2	2	no	—
48	C₄⋊Dic₇₃	C₂³	1	2	yes	—
49	C₄⋊Dic₇₃	C₂³	1	2	yes	—
50	#11924007	—	2	2	no	—
51	C₂²⋊Dic₇₃	—	2	2	no	—
52	C₂²⋊Dic₇₃	—	2	2	no	—
53	#11924005	—	2	2	no	—
54	C₄×Dic₇₃	—	2	2	no	—
55	C₄×Dic₇₃	D₄	1	1	yes	characteristic
56	C₄×Dic₇₃	D₄	1	1	yes	characteristic
57	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
58	C₂×Q₈×C₇₃	C₂³	1	2	yes	—
59	C₂×D₄×C₇₃	—	2	1	no	—
60	C₂×D₄×C₇₃	—	2	1	no	—
61	C₂×D₄×C₇₃	C₂³	1	2	yes	—
62	SD₁₆×C₇₃	—	2	8	no	—
63	C₂×C₈×C₇₃	C₂³	1	2	yes	—
64	C₄⋊C₄×C₇₃	—	2	1	no	—
65	C₄⋊C₄×C₇₃	—	2	1	no	—
66	C₄²×C₇₃	C₂³	1	1	yes	characteristic
67	C₂³×C₇₃	—	2	2	no	—
68	C₂²×D₇₃	C₂×D₄	1	1	yes	characteristic
69	Q₈×C₇₃	—	2	4	no	—
70	D₄×C₇₃	—	2	4	no	—
71	D₄×C₇₃	—	4	2	no	—
72	D₄×C₇₃	—	4	2	no	—
73	C₂×C₄×C₇₃	—	2	2	no	—
74	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
77	C₂²⋊D₇₃	—	4	4	no	—
78	C₂×Dic₇₃	—	2	2	no	—
79	C₂×Dic₇₃	—	2	2	no	—
80	C₂×Dic₇₃	—	4	4	no	—
81	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
82	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
83	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
84	C₄×D₇₃	—	2	2	no	—
85	C₄×D₇₃	—	2	2	no	—
86	C₄×D₇₃	—	2	2	no	—
87	Dic₁₄₆	—	4	2	no	—
88	Dic₁₄₆	—	4	2	no	—
89	C₈×C₇₃	#41	1	4	yes	complemented
90	#10596831	—	2	2	no	—
91	C₂²×C₇₃	—	4	2	no	—
92	C₂²×C₇₃	—	4	2	no	—
93	C₂²×C₇₃	—	4	2	no	—
94	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
95	C₂×D₇₃	—	2	1	no	—
96	C₂×D₇₃	—	2	1	no	—
97	C₂×D₇₃	—	2	1	no	—
98	C₄×C₇₃	—	4	2	no	—
99	C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
100	C₄×C₇₃	#137	1	2	yes	—
101	C₄×C₇₃	C₂×#41	1	2	yes	—
102	Dic₇₃	—	2	1	no	—
103	Dic₇₃	—	2	1	no	—
104	Dic₇₃	—	2	1	no	—
105	Dic₇₃	—	4	2	no	—
106	Dic₇₃	—	4	2	no	—
107	C₂×C₇₃	—	8	2	no	—
108	C₂×C₇₃	#532	1	1	yes	characteristic
109	C₂×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₂×C₇₃	C₂×#137	1	1	yes	characteristic
111	D₇₃	—	4	1	no	—
112	#3182	—	73	1	no	maximal, Sylow
113	C₇₃	#3182	1	1	yes	characteristic, complemented, Sylow
114	#533	—	73	1	no	—
115	#532	—	73	1	no	—
116	#500	—	73	1	no	—
117	#492	D₇₃	1	1	yes	characteristic, complemented
118	#487	—	73	1	no	—
119	#486	—	73	1	no	—
120	#468	—	73	4	no	—
121	#440	—	73	4	no	—
122	C₄×M₁₆	—	73	1	no	—
123	C₂×#41	—	146	1	no	—
124	C₂×SD₁₆	C₂×D₇₃	1	4	yes	—
125	C₂×M₁₆	—	73	2	no	—
126	C₄⋊Q₈	—	73	1	no	—
127	C₄⋊Q₈	—	73	1	no	—
128	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
129	#127	C₂×D₇₃	1	1	yes	characteristic
130	#125	—	146	1	no	—
131	#124	—	146	1	no	—
132	C₂²⋊Q₈	—	73	2	no	—
133	C₂²⋊Q₈	—	146	1	no	—
134	#121	—	73	2	no	—
135	#121	—	146	1	no	—
136	C₄×Q₈	—	73	2	no	—
137	C₄×Q₈	—	146	1	no	—
138	C₄×D₄	—	73	2	no	—
139	C₄×D₄	—	146	1	no	—
140	#117	—	146	1	no	—
141	C₂×C₄²	—	73	1	no	—
142	#107	—	73	4	no	—
143	#103	—	73	4	no	—
144	#102	—	73	4	no	—
145	#97	—	73	2	no	—
146	C₄×C₈	—	73	1	no	—
147	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
148	#41	—	292	4	no	—
149	C₂×Q₈	—	146	1	no	—
150	C₂×Q₈	—	146	1	no	—
151	C₂×Q₈	C₂²×D₇₃	1	2	yes	—
152	C₂×D₄	—	2	1	no	—
153	C₂×D₄	—	2	1	no	—
154	C₂×D₄	—	146	2	no	—
155	C₂×D₄	C₂²×D₇₃	1	2	yes	—
156	C₂²×C₄	—	73	1	no	—
157	C₂²×C₄	—	73	1	no	—
158	C₂²×C₄	—	73	1	no	—
159	C₂²×C₄	—	146	2	no	—
160	SD₁₆	—	2	8	no	—
161	M₁₆	—	146	4	no	—
162	C₂×C₈	—	73	2	no	—
163	C₂×C₈	C₂²×D₇₃	1	2	yes	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	1	no	—
166	C₄⋊C₄	—	73	2	no	—
167	C₄⋊C₄	—	73	2	no	—
168	C₄⋊C₄	—	146	2	no	—
169	C₄⋊C₄	—	146	2	no	—
170	C₄⋊C₄	—	146	1	no	—
171	C₄⋊C₄	—	146	1	no	—
172	C₂²⋊C₄	—	146	2	no	—
173	C₂²⋊C₄	—	146	2	no	—
174	C₂²⋊C₄	—	146	2	no	—
175	C₄²	—	73	1	no	—
176	C₄²	—	73	1	no	—
177	C₄²	—	73	1	no	—
178	C₄²	—	146	2	no	—
179	C₄²	C₂²×D₇₃	1	1	yes	characteristic
180	C₂³	—	2	2	no	—
181	C₂³	—	73	1	no	—
182	Q₈	—	2	4	no	—
183	Q₈	—	292	2	no	—
184	Q₈	—	292	2	no	—
185	D₄	—	2	4	no	—
186	D₄	—	4	2	no	—
187	D₄	—	4	2	no	—
188	D₄	—	292	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	292	4	no	—
191	C₂×C₄	—	73	1	no	—
192	C₂×C₄	—	73	1	no	—
193	C₂×C₄	—	73	1	no	—
194	C₂×C₄	—	146	2	no	—
195	C₂×C₄	—	146	2	no	—
196	C₂×C₄	—	146	2	no	—
197	C₂×C₄	—	146	2	no	—
198	C₂×C₄	—	146	2	no	—
199	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
200	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
201	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
202	C₈	—	146	2	no	—
203	C₈	#11924027	1	4	yes	complemented
204	C₂²	—	4	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	—	4	2	no	—
207	C₂²	—	146	1	no	—
208	C₂²	—	146	1	no	—
209	C₂²	—	146	1	no	—
210	C₂²	C₂×D₄×D₇₃	1	1	yes	characteristic
211	C₄	—	4	2	no	—
212	C₄	—	292	2	no	—
213	C₄	—	292	2	no	—
214	C₄	—	146	1	no	—
215	C₄	—	146	1	no	—
216	C₄	—	146	1	no	—
217	C₄	#14556618	1	2	yes	—
218	C₄	#14556620	1	2	yes	—
219	C₄	C₂×#11924027	1	2	yes	—
220	C₂	—	8	2	no	—
221	C₂	—	292	1	no	—
222	C₂	#17989135	1	1	yes	characteristic
223	C₂	C₂×#14556618	1	1	yes	characteristic
224	C₂	C₂×#14556620	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989142

C₂

yes

maximal, characteristic, complemented

#17989135

C₂

yes

maximal, characteristic

#17988992

C₂

yes

maximal, characteristic, complemented

#17988937

C₂

yes

maximal, characteristic, complemented

#17988933

C₂

yes

maximal, characteristic, complemented

#17988830

C₂

yes

maximal, complemented

#17988688

C₂

yes

maximal, complemented

C₄×#11924011

C₂

yes

maximal, characteristic, complemented

#492×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924027

—

C₂×#11924011

C₂²

yes

—

#14556591

C₂²

yes

characteristic

#14556591

C₂²

yes

—

—

—

C₂²

yes

—

#14556562

—

#14556562

C₂²

yes

—

Q₈×Dic₇₃

C₂²

yes

—

#14556556

—

D₄×Dic₇₃

—

D₄×Dic₇₃

C₂²

yes

—

#14556549

—

C₄²×D₇₃

C₂²

yes

characteristic

#14556525

C₂²

yes

—

#14556515

C₂²

yes

—

#14556512

C₂²

yes

—

#14556501

C₂²

yes

—

C₄×#10596831

C₂²

yes

characteristic

C₂×SD₁₆×C₇₃

C₂²

yes

—

C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

#127×C₇₃

C₂²

yes

characteristic

C₄×C₈×C₇₃

C₂²

yes

characteristic

#11924027

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×C₂²⋊D₇₃

—

C₂²×Dic₇₃

—

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×C₄×D₇₃

C₂³

yes

characteristic

#11924011

—

C₂×#10596831

C₂³

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂³

yes

—

C₄⋊Dic₇₃

C₂³

yes

—

#11924007

—

C₂²⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

#11924005

—

C₄×Dic₇₃

—

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

C₂³

yes

characteristic

C₂×Q₈×C₇₃

C₂³

yes

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

C₂³

yes

—

SD₁₆×C₇₃

—

C₂×C₈×C₇₃

C₂³

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

—

C₂²×D₇₃

C₂×D₄

yes

characteristic

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂⁴

yes

characteristic

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

#41

yes

complemented

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂²×D₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

AGL(1,ℤ/8)

yes

—

100

C₄×C₇₃

#137

yes

—

101

C₄×C₇₃

C₂×#41

yes

—

102

Dic₇₃

—

103

Dic₇₃

—

104

Dic₇₃

—

105

Dic₇₃

—

106

Dic₇₃

—

107

C₂×C₇₃

—

108

C₂×C₇₃

#532

yes

characteristic

109

C₂×C₇₃

C₂×AGL(1,ℤ/8)

yes

characteristic

110

C₂×C₇₃

C₂×#137

yes

characteristic

111

D₇₃

—

112

#3182

—

maximal, Sylow

113

C₇₃

#3182

yes

characteristic, complemented, Sylow

114

#533

—

115

#532

—

116

#500

—

117

#492

D₇₃

yes

characteristic, complemented

118

#487

—

119

#486

—

120

#468

—

121

#440

—

122

C₄×M₁₆

—

123

C₂×#41

—

146

—

124

C₂×SD₁₆

C₂×D₇₃

yes

—

125

C₂×M₁₆

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

C₂×D₇₃

yes

characteristic

129

#127

C₂×D₇₃

yes

characteristic

130

#125

—

146

—

131

#124

—

146

—

132

C₂²⋊Q₈

—

133

C₂²⋊Q₈

—

146

—

134

#121

—

135

#121

—

146

—

136

C₄×Q₈

—

137

C₄×Q₈

—

146

—

138

C₄×D₄

—

139

C₄×D₄

—

146

—

140

#117

—

146

—

141

C₂×C₄²

—

142

#107

—

143

#103

—

144

#102

—

145

#97

—

146

C₄×C₈

—

147

C₄×C₈

C₂×D₇₃

yes

characteristic

148

#41

—

292

—

149

C₂×Q₈

—

146

—

150

C₂×Q₈

—

146

—

151

C₂×Q₈

C₂²×D₇₃

yes

—

152

C₂×D₄

—

153

C₂×D₄

—

154

C₂×D₄

—

146

—

155

C₂×D₄

C₂²×D₇₃

yes

—

156

C₂²×C₄

—

157

C₂²×C₄

—

158

C₂²×C₄

—

159

C₂²×C₄

—

146

—

160

SD₁₆

—

161

M₁₆

—

146

—

162

C₂×C₈

—

163

C₂×C₈

C₂²×D₇₃

yes

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

146

—

169

C₄⋊C₄

—

146

—

170

C₄⋊C₄

—

146

—

171

C₄⋊C₄

—

146

—

172

C₂²⋊C₄

—

146

—

173

C₂²⋊C₄

—

146

—

174

C₂²⋊C₄

—

146

—

175

C₄²

—

176

C₄²

—

177

C₄²

—

178

C₄²

—

146

—

179

C₄²

C₂²×D₇₃

yes

characteristic

180

C₂³

—

181

C₂³

—

182

Q₈

—

183

Q₈

—

292

—

184

Q₈

—

292

—

185

D₄

—

186

D₄

—

187

D₄

—

188

D₄

—

292

—

189

C₂×C₄

—

190

C₂×C₄

—

292

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

146

—

195

C₂×C₄

—

146

—

196

C₂×C₄

—

146

—

197

C₂×C₄

—

146

—

198

C₂×C₄

—

146

—

199

C₂×C₄

D₄×D₇₃

yes

characteristic

200

C₂×C₄

D₄×D₇₃

yes

characteristic

201

C₂×C₄

C₂³×D₇₃

yes

characteristic

202

C₈

—

146

—

203

C₈

#11924027

yes

complemented

204

C₂²

—

205

C₂²

—

206

C₂²

—

207

C₂²

—

146

—

208

C₂²

—

146

—

209

C₂²

—

146

—

210

C₂²

C₂×D₄×D₇₃

yes

characteristic

211

C₄

—

212

C₄

—

292

—

213

C₄

—

292

—

214

C₄

—

146

—

215

C₄

—

146

—

216

C₄

—

146

—

217

C₄

#14556618

yes

—

218

C₄

#14556620

yes

—

219

C₄

C₂×#11924027

yes

—

220

C₂

—

221

C₂

—

292

—

222

C₂

#17989135

yes

characteristic

223

C₂

C₂×#14556618

yes

characteristic

224

C₂

C₂×#14556620

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₃
Fitting	#492×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₄
hypercenter	#492	D₇₃
hypocenter	C₇₃	#3182
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

C₂×D₄×D₇₃

commutator

C₂×C₄×C₇₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₃

Fitting

#492×C₇₃

C₂

socle

C₂²×C₇₃

C₂²×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #17989142	C₂	semidirect
3. #17988992	C₂	semidirect
4. #17988937	C₂	semidirect
5. #17988933	C₂	semidirect
6. #17988830	C₂	semidirect
7. #17988688	C₂	semidirect
8. C₄×#11924011	C₂	semidirect
9. #492×C₇₃	C₂	semidirect
89. C₈×C₇₃	#41	semidirect
113. C₇₃	#3182	semidirect
117. #492	D₇₃	semidirect
203. C₈	#11924027	semidirect

H ≅ G/N

type

1. #17989142

C₂

semidirect

3. #17988992

C₂

semidirect

4. #17988937

C₂

semidirect

5. #17988933

C₂

semidirect

6. #17988830

C₂

semidirect

7. #17988688

C₂

semidirect

8. C₄×#11924011

C₂

semidirect

9. #492×C₇₃

C₂

semidirect

89. C₈×C₇₃

#41

semidirect

113. C₇₃

#3182

semidirect

117. #492

D₇₃

semidirect

203. C₈

#11924027

semidirect

Subgroups — #22620647

Conjugacy classes of subgroups

Complements