Group #22610789 — subgroups

Subgroups — #22610789

Order 9344.

312 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 103 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	C₂²×#11924018	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556524	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556516	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
5	#17988494	C₂	1	1	yes	maximal, characteristic
6	#17988492	C₂	1	1	yes	maximal, characteristic
7	C₂×#103×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₄₆	C₂²	1	1	yes	characteristic
9	C₂×#11924018	—	2	2	no	—
10	C₂×#11924018	—	2	2	no	—
11	C₂×#11924018	C₄	1	4	yes	complemented
12	C₂²×#10596831	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
15	#14556524	—	2	2	no	—
16	#14556516	—	2	2	no	—
17	C₈×Dic₇₃	—	2	2	no	—
18	#14556498	—	2	2	no	—
19	#14556498	—	2	2	no	—
20	C₂²×Q₈×C₇₃	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
23	#103×C₇₃	—	2	2	no	—
24	C₂×Dic₁₄₆	—	2	1	no	—
25	C₂×Dic₁₄₆	—	2	1	no	—
26	C₂×Dic₁₄₆	—	2	1	no	—
27	C₂×Dic₁₄₆	—	2	1	no	—
28	C₂×Dic₁₄₆	—	2	1	no	—
29	C₂×Dic₁₄₆	C₂×C₄	1	2	yes	—
30	C₂²×Dic₇₃	—	2	1	no	—
31	C₂²×Dic₇₃	—	2	1	no	—
32	C₂²×Dic₇₃	D₄	1	2	yes	—
33	#11924018	—	2	8	no	—
34	C₂×#10596831	C₂×C₄	1	2	yes	—
35	C₂×#10596831	D₄	1	2	yes	—
36	C₂×#10596831	D₄	1	2	yes	—
37	C₄⋊Dic₇₃	—	2	1	no	—
38	C₄⋊Dic₇₃	—	2	1	no	—
39	C₄×Dic₇₃	—	2	1	no	—
40	C₄×Dic₇₃	—	2	1	no	—
41	C₂×Q₈×C₇₃	—	2	1	no	—
42	C₂×Q₈×C₇₃	—	2	1	no	—
43	C₂×Q₈×C₇₃	—	2	1	no	—
44	C₂×Q₈×C₇₃	—	2	1	no	—
45	C₂×Q₈×C₇₃	—	2	1	no	—
46	C₂×Q₈×C₇₃	C₂×C₄	1	2	yes	—
47	C₂²×C₄×C₇₃	—	2	1	no	—
48	C₂²×C₄×C₇₃	—	2	1	no	—
49	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₇₃	—	2	1	no	—
51	C₂×C₈×C₇₃	—	2	1	no	—
52	C₂×C₈×C₇₃	D₄	1	2	yes	—
53	C₄⋊C₄×C₇₃	—	2	1	no	—
54	C₄⋊C₄×C₇₃	—	2	1	no	—
55	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
56	Q₈×C₇₃	—	2	2	no	—
57	Q₈×C₇₃	—	2	2	no	—
58	Q₈×C₇₃	—	4	1	no	—
59	Q₈×C₇₃	—	4	1	no	—
60	C₂×C₄×C₇₃	—	2	2	no	—
61	C₂×C₄×C₇₃	—	4	1	no	—
62	C₂×C₄×C₇₃	—	4	1	no	—
63	C₂×C₄×C₇₃	—	4	1	no	—
64	C₂×C₄×C₇₃	—	4	2	no	—
65	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
70	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
71	C₂×Dic₇₃	—	2	2	no	—
72	C₂×Dic₇₃	—	2	2	no	—
73	C₂×Dic₇₃	—	2	2	no	—
74	C₂×Dic₇₃	—	4	1	no	—
75	C₂×Dic₇₃	—	4	1	no	—
76	C₂×Dic₇₃	—	4	1	no	—
77	C₂×Dic₇₃	—	4	2	no	—
78	C₂×Dic₇₃	Q₁₆	1	4	yes	complemented
79	Dic₁₄₆	—	2	2	no	—
80	Dic₁₄₆	—	2	2	no	—
81	Dic₁₄₆	—	4	1	no	—
82	Dic₁₄₆	—	4	1	no	—
83	C₈×C₇₃	—	2	2	no	—
84	#10596831	C₂²⋊C₄	1	4	yes	—
85	C₂²×C₇₃	C₄×D₄	1	1	yes	characteristic
86	C₂²×C₇₃	#121	1	1	yes	characteristic
87	C₂²×C₇₃	#127	1	1	yes	characteristic
88	C₂²×C₇₃	C₂×Q₁₆	1	2	yes	—
89	C₂²×C₇₃	#135	1	2	yes	—
90	C₄×C₇₃	—	4	2	no	—
91	C₄×C₇₃	—	8	1	no	—
92	C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
94	C₄×C₇₃	#121	1	1	yes	characteristic
95	C₄×C₇₃	#124	1	1	yes	characteristic
96	Dic₇₃	—	2	4	no	—
97	Dic₇₃	—	4	2	no	—
98	Dic₇₃	—	8	1	no	—
99	C₂×C₇₃	#390	1	1	yes	characteristic
100	C₂×C₇₃	C₄×Q₁₆	1	2	yes	—
101	C₂×C₇₃	#467	1	2	yes	—
102	C₂×C₇₃	#494	1	1	yes	characteristic
103	C₂×C₇₃	#495	1	1	yes	characteristic
104	#1969	—	73	1	no	maximal, Sylow
105	C₇₃	#1969	1	1	yes	characteristic, complemented, Sylow
106	C₂²×Q₁₆	—	73	1	no	—
107	C₂×#107	—	73	1	no	—
108	C₂×#103	—	73	1	no	—
109	C₂×#103	D₇₃	1	1	yes	characteristic
110	C₂×C₄×C₈	—	73	1	no	—
111	#391	—	73	1	no	—
112	#391	—	73	1	no	—
113	C₂²×Q₈	—	73	1	no	—
114	C₂²×Q₈	C₂×D₇₃	1	1	yes	characteristic
115	C₂×Q₁₆	—	73	4	no	—
116	C₂×Q₁₆	—	146	2	no	—
117	C₂×Q₁₆	—	146	2	no	—
118	C₂²×C₈	—	73	1	no	—
119	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
120	C₂×C₄⋊C₄	—	73	1	no	—
121	C₂×C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
122	C₂×C₄²	—	73	1	no	—
123	#107	—	146	2	no	—
124	#103	—	2	2	no	—
125	#103	—	146	2	no	—
126	C₄×C₈	—	146	2	no	—
127	#95	—	146	2	no	—
128	#95	—	146	2	no	—
129	C₂×Q₈	—	2	1	no	—
130	C₂×Q₈	—	2	1	no	—
131	C₂×Q₈	—	2	1	no	—
132	C₂×Q₈	—	2	1	no	—
133	C₂×Q₈	—	2	1	no	—
134	C₂×Q₈	—	73	2	no	—
135	C₂×Q₈	—	146	1	no	—
136	C₂×Q₈	—	146	1	no	—
137	C₂×Q₈	—	146	1	no	—
138	C₂×Q₈	—	146	1	no	—
139	C₂×Q₈	—	146	1	no	—
140	C₂×Q₈	C₄×D₇₃	1	2	yes	—
141	C₂²×C₄	—	2	1	no	—
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	—	73	2	no	—
144	C₂²×C₄	—	146	1	no	—
145	C₂²×C₄	—	146	1	no	—
146	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
147	Q₁₆	—	146	8	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	73	2	no	—
151	C₂×C₈	—	73	2	no	—
152	C₂×C₈	—	73	2	no	—
153	C₂×C₈	C₂²⋊D₇₃	1	2	yes	—
154	C₄⋊C₄	—	2	1	no	—
155	C₄⋊C₄	—	2	1	no	—
156	C₄⋊C₄	—	146	1	no	—
157	C₄⋊C₄	—	146	1	no	—
158	C₄²	—	146	1	no	—
159	C₄²	—	146	1	no	—
160	C₂³	D₄×D₇₃	1	1	yes	characteristic
161	Q₈	—	2	2	no	—
162	Q₈	—	2	2	no	—
163	Q₈	—	4	1	no	—
164	Q₈	—	4	1	no	—
165	Q₈	—	292	1	no	—
166	Q₈	—	292	1	no	—
167	Q₈	—	146	2	no	—
168	Q₈	—	146	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	292	1	no	—
175	C₂×C₄	—	292	1	no	—
176	C₂×C₄	—	292	1	no	—
177	C₂×C₄	—	292	2	no	—
178	C₂×C₄	—	73	4	no	—
179	C₂×C₄	—	146	2	no	—
180	C₂×C₄	—	146	2	no	—
181	C₂×C₄	—	146	2	no	—
182	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
183	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
184	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
185	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
186	C₂×C₄	#11924027	1	1	yes	characteristic
187	C₂×C₄	#11924028	1	1	yes	characteristic
188	C₈	—	2	2	no	—
189	C₈	—	73	4	no	—
190	C₂²	#14556551	1	1	yes	characteristic
191	C₂²	#14556564	1	1	yes	characteristic
192	C₂²	#14556587	1	1	yes	characteristic
193	C₂²	Q₁₆×D₇₃	1	2	yes	—
194	C₂²	#14556612	1	2	yes	—
195	C₄	—	4	2	no	—
196	C₄	—	8	1	no	—
197	C₄	—	292	2	no	—
198	C₄	—	584	1	no	—
199	C₄	—	146	4	no	—
200	C₄	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
201	C₄	C₄×C₂²⋊D₇₃	1	1	yes	characteristic
202	C₄	#14556563	1	1	yes	characteristic
203	C₄	#14556577	1	1	yes	characteristic
204	C₂	#17988487	1	1	yes	characteristic
205	C₂	#17988679	1	2	yes	—
206	C₂	#17988827	1	2	yes	—
207	C₂	#17988965	1	1	yes	characteristic
208	C₂	#17988968	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11924018

C₂

yes

maximal, characteristic

C₂×#14556524

C₂

yes

maximal, characteristic

C₂×#14556516

C₂

yes

maximal, characteristic

C₂×C₈×Dic₇₃

C₂

yes

maximal, characteristic

#17988494

C₂

yes

maximal, characteristic

#17988492

C₂

yes

maximal, characteristic

C₂×#103×C₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₆

C₂²

yes

—

—

C₄

yes

complemented

C₂²×#10596831

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

characteristic

C₂×C₄×Dic₇₃

C₂²

yes

characteristic

#14556524

—

#14556516

—

C₈×Dic₇₃

—

#14556498

—

#14556498

—

C₂²×Q₈×C₇₃

C₂²

yes

characteristic

C₂²×C₈×C₇₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₃

C₂²

yes

characteristic

#103×C₇₃

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

C₂×C₄

yes

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

D₄

yes

—

#11924018

—

C₂×#10596831

C₂×C₄

yes

—

C₂×#10596831

D₄

yes

—

C₂×#10596831

D₄

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

C₂×C₄

yes

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

Q₁₆

yes

complemented

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

#10596831

C₂²⋊C₄

yes

—

C₂²×C₇₃

C₄×D₄

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

C₂×Q₁₆

yes

—

C₂²×C₇₃

#135

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₃

C₄×D₄

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

#390

yes

characteristic

100

C₂×C₇₃

C₄×Q₁₆

yes

—

101

C₂×C₇₃

#467

yes

—

102

C₂×C₇₃

#494

yes

characteristic

103

C₂×C₇₃

yes

104

—

105

C₇₃

#1969

yes

characteristic, complemented, Sylow

106

C₂²×Q₁₆

—

107

C₂×#107

—

108

C₂×#103

—

109

C₂×#103

D₇₃

yes

characteristic

110

C₂×C₄×C₈

—

111

#391

—

112

#391

—

113

C₂²×Q₈

—

114

C₂²×Q₈

C₂×D₇₃

yes

characteristic

115

C₂×Q₁₆

—

116

C₂×Q₁₆

—

146

—

117

C₂×Q₁₆

—

146

—

118

C₂²×C₈

—

119

C₂²×C₈

C₂×D₇₃

yes

characteristic

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

C₂×D₇₃

yes

characteristic

122

C₂×C₄²

—

123

#107

—

146

—

124

#103

—

125

#103

—

146

—

126

C₄×C₈

—

146

—

127

#95

—

146

—

128

#95

—

146

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

146

—

136

C₂×Q₈

—

146

—

137

C₂×Q₈

—

146

—

138

C₂×Q₈

—

146

—

139

C₂×Q₈

—

146

—

140

C₂×Q₈

C₄×D₇₃

yes

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

146

—

145

C₂²×C₄

—

146

—

146

C₂²×C₄

C₂²×D₇₃

yes

characteristic

147

Q₁₆

—

146

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

C₂²⋊D₇₃

yes

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

146

—

157

C₄⋊C₄

—

146

—

158

C₄²

—

146

—

159

C₄²

—

146

—

160

C₂³

D₄×D₇₃

yes

characteristic

161

Q₈

—

162

Q₈

—

163

Q₈

—

164

Q₈

—

165

Q₈

—

292

—

166

Q₈

—

292

—

167

Q₈

—

146

—

168

Q₈

—

146

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

292

—

175

C₂×C₄

—

292

—

176

C₂×C₄

—

292

—

177

C₂×C₄

—

292

—

178

C₂×C₄

—

179

C₂×C₄

—

146

—

180

C₂×C₄

—

146

—

181

C₂×C₄

—

146

—

182

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

183

C₂×C₄

D₄×D₇₃

yes

characteristic

184

C₂×C₄

D₄×D₇₃

yes

characteristic

185

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

186

C₂×C₄

#11924027

yes

characteristic

187

C₂×C₄

#11924028

yes

characteristic

188

C₈

—

189

C₈

—

190

C₂²

#14556551

yes

characteristic

191

C₂²

#14556564

yes

characteristic

192

C₂²

#14556587

yes

characteristic

193

C₂²

Q₁₆×D₇₃

yes

—

194

C₂²

#14556612

yes

—

195

C₄

—

196

C₄

—

197

C₄

—

292

—

198

C₄

—

584

—

199

C₄

—

146

—

200

C₄

C₂²⋊C₄×D₇₃

yes

characteristic

201

C₄

C₄×C₂²⋊D₇₃

yes

characteristic

202

C₄

#14556563

yes

characteristic

203

C₄

#14556577

yes

characteristic

204

C₂

#17988487

yes

characteristic

205

C₂

#17988679

yes

—

206

C₂

#17988827

yes

—

207

C₂

#17988965

yes

characteristic

208

C₂

#17988968

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	C₂×#103×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₄
hypercenter	C₂×#103	D₇₃
hypocenter	C₇₃	#1969
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#103

subgroup

G/H

center

C₂³

D₄×D₇₃

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

C₂×#103×C₇₃

C₂

socle

C₂³×C₇₃

C₂×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#103

N	H ≅ G/N	type
11. C₂×#11924018	C₄	semidirect
78. C₂×Dic₇₃	Q₁₆	semidirect
105. C₇₃	#1969	semidirect

H ≅ G/N

type

11. C₂×#11924018

C₄

semidirect

78. C₂×Dic₇₃

Q₁₆

semidirect

105. C₇₃

#1969

semidirect

Subgroups — #22610789

Conjugacy classes of subgroups

Complements