Group #22619460 — subgroups

Subgroups — #22619460

Order 9344.

458 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14556624	C₂	1	1	yes	maximal, characteristic
2	C₂²×#11924018	C₂	1	1	yes	maximal, characteristic, complemented
3	#17989161	C₂	1	1	yes	maximal, characteristic
4	#17989064	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988812	C₂	1	4	yes	maximal, complemented
6	#17988796	C₂	1	4	yes	maximal, complemented
7	#17988615	C₂	1	1	yes	maximal, characteristic, complemented
8	#17988581	C₂	1	1	yes	maximal, characteristic
9	#419×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11924028	C₂²	1	1	yes	characteristic
11	C₂²×Dic₁₄₆	C₂²	1	1	yes	characteristic
12	#14556624	—	2	4	no	—
13	C₂×Dic₂₉₂	C₂²	1	2	yes	—
14	C₂×#11924018	—	2	4	no	—
15	C₂×#11924018	C₂²	1	4	yes	—
16	C₂×#11924017	C₂²	1	2	yes	—
17	C₂²×#10596831	C₂²	1	1	yes	characteristic
18	#14556589	C₂²	1	2	yes	—
19	#14556580	C₂²	1	2	yes	—
20	#14556574	—	2	2	no	—
21	#14556567	—	2	2	no	—
22	C₄×Dic₁₄₆	—	2	2	no	—
23	#14556556	C₂²	1	4	yes	—
24	#14556550	C₂²	1	1	yes	characteristic
25	#14556547	—	2	2	no	—
26	#14556520	C₂²	1	4	yes	—
27	#14556514	C₂²	1	2	yes	—
28	#14556511	C₂²	1	2	yes	—
29	C₂²×Q₈×C₇₃	C₂²	1	1	yes	characteristic
30	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
31	#117×C₇₃	C₂²	1	1	yes	characteristic
32	#103×C₇₃	C₂²	1	4	yes	—
33	#11924028	—	2	2	no	—
34	#11924028	—	2	2	no	—
35	C₂×Dic₁₄₆	—	2	1	no	—
36	C₂×Dic₁₄₆	—	2	2	no	—
37	C₂×Dic₁₄₆	D₄	1	4	yes	—
38	C₂×Dic₁₄₆	C₂³	1	1	yes	characteristic
39	C₂×Dic₁₄₆	C₂³	1	2	yes	—
40	C₂×D₂₉₂	C₂³	1	1	yes	characteristic
41	C₂×C₂²⋊D₇₃	—	2	1	no	—
42	C₂²×Dic₇₃	—	2	1	no	—
43	C₂×C₄×D₇₃	—	2	1	no	—
44	Dic₂₉₂	—	2	4	no	—
45	#11924018	—	2	8	no	—
46	#11924017	—	2	4	no	—
47	C₂×#10596831	—	2	2	no	—
48	C₂×#10596831	C₂³	1	2	yes	—
49	C₄⋊Dic₇₃	D₄	1	2	yes	—
50	#11924007	—	2	4	no	—
51	#11924007	—	2	4	no	—
52	C₂²⋊Dic₇₃	D₄	1	2	yes	—
53	#11924005	—	2	4	no	—
54	C₄×Dic₇₃	—	2	4	no	—
55	C₄×Dic₇₃	C₂³	1	2	yes	—
56	C₂×Q₈×C₇₃	—	2	1	no	—
57	C₂×Q₈×C₇₃	—	2	2	no	—
58	C₂×Q₈×C₇₃	—	2	2	no	—
59	C₂×Q₈×C₇₃	C₂³	1	2	yes	—
60	C₂²×C₄×C₇₃	—	2	1	no	—
61	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
62	M₁₆×C₇₃	—	2	2	no	—
63	C₂×C₈×C₇₃	C₂³	1	2	yes	—
64	C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
65	C₂²⋊C₄×C₇₃	—	2	1	no	—
66	C₄²×C₇₃	—	2	1	no	—
67	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×D₇₃	—	2	1	no	—
69	Q₈×C₇₃	—	2	2	no	—
70	Q₈×C₇₃	—	2	2	no	—
71	Q₈×C₇₃	—	4	2	no	—
72	C₂×C₄×C₇₃	—	2	2	no	—
73	C₂×C₄×C₇₃	—	2	2	no	—
74	C₂×C₄×C₇₃	—	4	2	no	—
75	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₇₃	#41	1	2	yes	—
78	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
79	C₂²⋊D₇₃	—	4	2	no	—
80	C₂×Dic₇₃	—	2	4	no	—
81	C₂×Dic₇₃	—	2	2	no	—
82	C₂×Dic₇₃	—	2	1	no	—
83	C₂×Dic₇₃	C₂×D₄	1	4	yes	—
84	D₂₉₂	—	2	1	no	—
85	D₂₉₂	—	2	1	no	—
86	C₄×D₇₃	—	4	2	no	—
87	Dic₁₄₆	—	2	4	no	—
88	Dic₁₄₆	—	2	2	no	—
89	Dic₁₄₆	—	2	1	no	—
90	Dic₁₄₆	—	2	1	no	—
91	Dic₁₄₆	C₂×D₄	1	4	yes	—
92	C₈×C₇₃	—	2	2	no	—
93	#10596831	—	2	2	no	—
94	C₂²×C₇₃	—	2	2	no	—
95	C₂²×C₇₃	#121	1	2	yes	—
96	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
97	C₂×D₇₃	—	4	1	no	—
98	C₂×D₇₃	—	4	2	no	—
99	C₄×C₇₃	—	4	2	no	—
100	C₄×C₇₃	—	4	2	no	—
101	C₄×C₇₃	#121	1	2	yes	—
102	C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
103	C₄×C₇₃	C₂×#41	1	1	yes	characteristic
104	Dic₇₃	—	2	4	no	—
105	Dic₇₃	—	2	4	no	—
106	Dic₇₃	—	4	1	no	—
107	C₂×C₇₃	—	2	2	no	—
108	C₂×C₇₃	C₂×#121	1	1	yes	characteristic
109	C₂×C₇₃	#578	1	2	yes	—
110	D₇₃	—	8	1	no	—
111	#3044	—	73	1	no	maximal, Sylow
112	C₇₃	#3044	1	1	yes	characteristic, complemented, Sylow
113	C₂×#137	—	73	1	no	—
114	C₂²×Q₁₆	—	73	1	no	—
115	#536	—	73	1	no	—
116	#519	—	73	1	no	—
117	#464	—	73	4	no	—
118	#462	—	73	4	no	—
119	#424	—	73	1	no	—
120	#419	D₇₃	1	1	yes	characteristic, complemented
121	#416	—	73	1	no	—
122	C₂×#41	—	73	1	no	—
123	C₂²×Q₈	—	73	1	no	—
124	C₂²×Q₈	C₂×D₇₃	1	1	yes	characteristic
125	#137	—	146	4	no	—
126	C₂×Q₁₆	—	73	4	no	—
127	C₂×Q₁₆	—	73	2	no	—
128	C₂×Q₁₆	—	146	4	no	—
129	C₂×SD₁₆	—	73	2	no	—
130	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
131	C₂²×C₈	—	73	1	no	—
132	C₄⋊Q₈	—	73	2	no	—
133	#124	—	73	2	no	—
134	#123	—	146	2	no	—
135	C₂²⋊Q₈	—	146	2	no	—
136	C₄×Q₈	—	73	4	no	—
137	C₄×Q₈	—	146	2	no	—
138	#117	—	73	1	no	—
139	#117	—	146	2	no	—
140	#117	C₂×D₇₃	1	1	yes	characteristic
141	C₄⋊C₈	—	73	4	no	—
142	#103	—	73	2	no	—
143	#103	C₂×D₇₃	1	4	yes	—
144	#102	—	73	2	no	—
145	#41	—	146	2	no	—
146	#41	—	146	2	no	—
147	C₂×Q₈	—	2	1	no	—
148	C₂×Q₈	—	2	2	no	—
149	C₂×Q₈	—	2	2	no	—
150	C₂×Q₈	—	73	4	no	—
151	C₂×Q₈	—	73	2	no	—
152	C₂×Q₈	—	73	1	no	—
153	C₂×Q₈	—	146	1	no	—
154	C₂×Q₈	—	146	2	no	—
155	C₂×Q₈	C₂²×D₇₃	1	2	yes	—
156	C₂×D₄	—	73	1	no	—
157	C₂×D₄	—	146	1	no	—
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	146	1	no	—
160	C₂²×C₄	—	146	1	no	—
161	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
162	Q₁₆	—	146	4	no	—
163	Q₁₆	—	146	8	no	—
164	SD₁₆	—	146	4	no	—
165	M₁₆	—	2	2	no	—
166	C₂×C₈	—	73	2	no	—
167	C₂×C₈	—	146	2	no	—
168	C₂×C₈	C₂²×D₇₃	1	2	yes	—
169	C₄⋊C₄	—	73	2	no	—
170	C₄⋊C₄	—	146	4	no	—
171	C₄⋊C₄	—	146	4	no	—
172	C₄⋊C₄	C₂²×D₇₃	1	2	yes	—
173	C₂²⋊C₄	—	2	1	no	—
174	C₂²⋊C₄	—	73	2	no	—
175	C₂²⋊C₄	—	146	4	no	—
176	C₄²	—	2	1	no	—
177	C₄²	—	73	2	no	—
178	C₄²	—	146	4	no	—
179	C₂³	—	146	1	no	—
180	C₂³	D₄×D₇₃	1	1	yes	characteristic
181	Q₈	—	2	2	no	—
182	Q₈	—	2	2	no	—
183	Q₈	—	4	2	no	—
184	Q₈	—	73	4	no	—
185	Q₈	—	146	1	no	—
186	Q₈	—	146	1	no	—
187	Q₈	—	146	4	no	—
188	Q₈	—	146	2	no	—
189	D₄	—	292	2	no	—
190	D₄	—	146	1	no	—
191	D₄	—	146	1	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	292	2	no	—
196	C₂×C₄	—	73	4	no	—
197	C₂×C₄	—	146	1	no	—
198	C₂×C₄	—	146	4	no	—
199	C₂×C₄	—	146	2	no	—
200	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
201	C₂×C₄	D₄×D₇₃	1	2	yes	—
202	C₂×C₄	#11924028	1	2	yes	—
203	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	146	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	292	2	no	—
208	C₂²	—	292	1	no	—
209	C₂²	#14556564	1	2	yes	—
210	C₂²	C₂×D₄×D₇₃	1	1	yes	characteristic
211	C₄	—	4	2	no	—
212	C₄	—	4	2	no	—
213	C₄	—	292	1	no	—
214	C₄	—	146	4	no	—
215	C₄	—	146	4	no	—
216	C₄	#14556564	1	2	yes	—
217	C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
218	C₄	C₂×#11924028	1	1	yes	characteristic
219	C₂	—	2	2	no	—
220	C₂	—	584	1	no	—
221	C₂	C₂×#14556564	1	1	yes	characteristic
222	C₂	#17989403	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556624

C₂

yes

maximal, characteristic

C₂²×#11924018

C₂

yes

maximal, characteristic, complemented

#17989161

C₂

yes

maximal, characteristic

#17989064

C₂

yes

maximal, characteristic, complemented

#17988812

C₂

yes

maximal, complemented

#17988796

C₂

yes

maximal, complemented

#17988615

C₂

yes

maximal, characteristic, complemented

#17988581

C₂

yes

maximal, characteristic

#419×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924028

C₂²

yes

characteristic

C₂²×Dic₁₄₆

C₂²

yes

characteristic

#14556624

—

C₂×Dic₂₉₂

C₂²

yes

—

C₂×#11924018

—

C₂×#11924018

C₂²

yes

—

C₂×#11924017

C₂²

yes

—

C₂²×#10596831

C₂²

yes

characteristic

#14556589

C₂²

yes

—

#14556580

C₂²

yes

—

#14556574

—

#14556567

—

C₄×Dic₁₄₆

—

#14556556

C₂²

yes

—

#14556550

C₂²

yes

characteristic

#14556547

—

#14556520

C₂²

yes

—

#14556514

C₂²

yes

—

#14556511

C₂²

yes

—

C₂²×Q₈×C₇₃

C₂²

yes

characteristic

C₂×M₁₆×C₇₃

C₂²

yes

characteristic

#117×C₇₃

C₂²

yes

characteristic

#103×C₇₃

C₂²

yes

—

#11924028

—

#11924028

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

D₄

yes

—

C₂×Dic₁₄₆

C₂³

yes

characteristic

C₂×Dic₁₄₆

C₂³

yes

—

C₂×D₂₉₂

C₂³

yes

characteristic

C₂×C₂²⋊D₇₃

—

C₂²×Dic₇₃

—

C₂×C₄×D₇₃

—

Dic₂₉₂

—

—

—

—

C₂³

yes

—

C₄⋊Dic₇₃

D₄

yes

—

#11924007

—

#11924007

—

C₂²⋊Dic₇₃

D₄

yes

—

#11924005

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₂³

yes

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

C₂³

yes

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

C₂³

yes

—

C₄⋊C₄×C₇₃

C₂³

yes

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂²×D₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

#41

yes

—

C₂×C₄×C₇₃

C₂⁴

yes

characteristic

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂×D₄

yes

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

C₂×D₄

yes

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

#121

yes

—

C₂²×C₇₃

C₂²×D₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

100

C₄×C₇₃

—

101

C₄×C₇₃

#121

yes

—

102

C₄×C₇₃

C₂²×D₄

yes

characteristic

103

C₄×C₇₃

C₂×#41

yes

characteristic

104

Dic₇₃

—

105

Dic₇₃

—

106

Dic₇₃

—

107

C₂×C₇₃

—

108

C₂×C₇₃

C₂×#121

yes

characteristic

109

C₂×C₇₃

#578

yes

—

110

D₇₃

—

111

#3044

—

maximal, Sylow

112

C₇₃

#3044

yes

characteristic, complemented, Sylow

113

C₂×#137

—

114

C₂²×Q₁₆

—

115

#536

—

116

#519

—

117

#464

—

118

#462

—

119

#424

—

120

#419

D₇₃

yes

characteristic, complemented

121

#416

—

122

C₂×#41

—

123

C₂²×Q₈

—

124

C₂²×Q₈

C₂×D₇₃

yes

characteristic

125

#137

—

146

—

126

C₂×Q₁₆

—

127

C₂×Q₁₆

—

128

C₂×Q₁₆

—

146

—

129

C₂×SD₁₆

—

130

C₂×M₁₆

C₂×D₇₃

yes

characteristic

131

C₂²×C₈

—

132

C₄⋊Q₈

—

133

#124

—

134

#123

—

146

—

135

C₂²⋊Q₈

—

146

—

136

C₄×Q₈

—

137

C₄×Q₈

—

146

—

138

#117

—

139

#117

—

146

—

140

#117

C₂×D₇₃

yes

characteristic

141

C₄⋊C₈

—

142

#103

—

143

#103

C₂×D₇₃

yes

—

144

#102

—

145

#41

—

146

—

146

#41

—

146

—

147

C₂×Q₈

—

148

C₂×Q₈

—

149

C₂×Q₈

—

150

C₂×Q₈

—

151

C₂×Q₈

—

152

C₂×Q₈

—

153

C₂×Q₈

—

146

—

154

C₂×Q₈

—

146

—

155

C₂×Q₈

C₂²×D₇₃

yes

—

156

C₂×D₄

—

157

C₂×D₄

—

146

—

158

C₂²×C₄

—

159

C₂²×C₄

—

146

—

160

C₂²×C₄

—

146

—

161

C₂²×C₄

C₂²×D₇₃

yes

characteristic

162

Q₁₆

—

146

—

163

Q₁₆

—

146

—

164

SD₁₆

—

146

—

165

M₁₆

—

166

C₂×C₈

—

167

C₂×C₈

—

146

—

168

C₂×C₈

C₂²×D₇₃

yes

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

146

—

171

C₄⋊C₄

—

146

—

172

C₄⋊C₄

C₂²×D₇₃

yes

—

173

C₂²⋊C₄

—

174

C₂²⋊C₄

—

175

C₂²⋊C₄

—

146

—

176

C₄²

—

177

C₄²

—

178

C₄²

—

146

—

179

C₂³

—

146

—

180

C₂³

D₄×D₇₃

yes

characteristic

181

Q₈

—

182

Q₈

—

183

Q₈

—

184

Q₈

—

185

Q₈

—

146

—

186

Q₈

—

146

—

187

Q₈

—

146

—

188

Q₈

—

146

—

189

D₄

—

292

—

190

D₄

—

146

—

191

D₄

—

146

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

292

—

196

C₂×C₄

—

197

C₂×C₄

—

146

—

198

C₂×C₄

—

146

—

199

C₂×C₄

—

146

—

200

C₂×C₄

D₄×D₇₃

yes

characteristic

201

C₂×C₄

D₄×D₇₃

yes

—

202

C₂×C₄

#11924028

yes

—

203

C₂×C₄

C₂³×D₇₃

yes

characteristic

204

C₈

—

205

C₈

—

146

—

206

C₂²

—

207

C₂²

—

292

—

208

C₂²

—

292

—

209

C₂²

#14556564

yes

—

210

C₂²

C₂×D₄×D₇₃

yes

characteristic

211

C₄

—

212

C₄

—

213

C₄

—

292

—

214

C₄

—

146

—

215

C₄

—

146

—

216

C₄

#14556564

yes

—

217

C₄

C₂×D₄×D₇₃

yes

characteristic

218

C₄

C₂×#11924028

yes

characteristic

219

C₂

—

220

C₂

—

584

—

221

C₂

C₂×#14556564

yes

characteristic

222

C₂

#17989403

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₃
Fitting	#419×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₄
hypercenter	#419	D₇₃
hypocenter	C₇₃	#3044
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

subgroup

G/H

center

C₂²

C₂×D₄×D₇₃

commutator

C₂×C₄×C₇₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₃

Fitting

#419×C₇₃

C₂

socle

C₂²×C₇₃

C₂²×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

N	H ≅ G/N	type
2. C₂²×#11924018	C₂	semidirect
4. #17989064	C₂	semidirect
5. #17988812	C₂	semidirect
6. #17988796	C₂	semidirect
7. #17988615	C₂	semidirect
9. #419×C₇₃	C₂	semidirect
112. C₇₃	#3044	semidirect
120. #419	D₇₃	semidirect

H ≅ G/N

type

2. C₂²×#11924018

C₂

semidirect

4. #17989064

C₂

semidirect

5. #17988812

C₂

semidirect

6. #17988796

C₂

semidirect

7. #17988615

C₂

semidirect

9. #419×C₇₃

C₂

semidirect

112. C₇₃

#3044

semidirect

120. #419

D₇₃

semidirect

Subgroups — #22619460

Conjugacy classes of subgroups

Complements