Group #22610609 — subgroups

Subgroups — #22610609

Order 9344.

256 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 101 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×#14556515	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
3	#17988494	C₂	1	1	yes	maximal, characteristic
4	#17988486	C₂	1	1	yes	maximal, characteristic
5	#17988350	C₂	1	1	yes	maximal, characteristic
6	#17988349	C₂	1	1	yes	maximal, characteristic
7	C₂×#103×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10596831	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₇₃	—	2	1	no	—
10	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
13	#14556515	—	2	2	no	—
14	C₈×Dic₇₃	—	2	2	no	—
15	#14556498	—	2	1	no	—
16	#14556498	—	2	1	no	—
17	#14556498	—	2	1	no	—
18	C₂²×Q₈×C₇₃	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
21	#103×C₇₃	C₄	1	4	yes	complemented
22	C₂²×Dic₇₃	—	2	1	no	—
23	C₂²×Dic₇₃	—	2	1	no	—
24	C₂²×Dic₇₃	D₄	1	1	yes	characteristic
25	C₂²×Dic₇₃	D₄	1	1	yes	characteristic
26	C₂×#10596831	—	2	1	no	—
27	C₂×#10596831	—	2	1	no	—
28	C₂×#10596831	D₄	1	2	yes	—
29	C₄⋊Dic₇₃	—	2	1	no	—
30	C₄⋊Dic₇₃	—	2	1	no	—
31	C₄⋊Dic₇₃	—	2	1	no	—
32	C₄⋊Dic₇₃	—	2	1	no	—
33	C₄⋊Dic₇₃	—	4	2	no	—
34	C₄×Dic₇₃	—	2	1	no	—
35	C₄×Dic₇₃	—	2	1	no	—
36	C₂×Q₈×C₇₃	—	2	1	no	—
37	C₂×Q₈×C₇₃	—	2	1	no	—
38	C₂×Q₈×C₇₃	—	2	1	no	—
39	C₂×Q₈×C₇₃	—	2	1	no	—
40	C₂×Q₈×C₇₃	—	2	1	no	—
41	C₂×Q₈×C₇₃	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₇₃	—	2	1	no	—
43	C₂²×C₄×C₇₃	—	2	1	no	—
44	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
45	C₂×C₈×C₇₃	—	2	1	no	—
46	C₂×C₈×C₇₃	—	2	1	no	—
47	C₂×C₈×C₇₃	C₂×C₄	1	2	yes	—
48	C₄⋊C₄×C₇₃	—	2	1	no	—
49	C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
50	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
51	Q₈×C₇₃	—	2	2	no	—
52	Q₈×C₇₃	—	2	2	no	—
53	Q₈×C₇₃	—	4	1	no	—
54	Q₈×C₇₃	—	4	1	no	—
55	C₂×C₄×C₇₃	—	2	2	no	—
56	C₂×C₄×C₇₃	—	2	2	no	—
57	C₂×C₄×C₇₃	—	4	1	no	—
58	C₂×C₄×C₇₃	—	4	1	no	—
59	C₂×C₄×C₇₃	—	4	1	no	—
60	C₂×C₄×C₇₃	—	4	1	no	—
61	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
64	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
65	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
66	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
67	C₂×Dic₇₃	—	2	1	no	—
68	C₂×Dic₇₃	—	2	1	no	—
69	C₂×Dic₇₃	—	2	1	no	—
70	C₂×Dic₇₃	—	2	1	no	—
71	C₂×Dic₇₃	—	4	1	no	—
72	C₂×Dic₇₃	—	4	1	no	—
73	C₂×Dic₇₃	—	4	1	no	—
74	C₂×Dic₇₃	—	4	1	no	—
75	C₂×Dic₇₃	—	4	1	no	—
76	C₂×Dic₇₃	—	4	1	no	—
77	C₂×Dic₇₃	SD₁₆	1	2	yes	—
78	C₂×Dic₇₃	Q₁₆	1	2	yes	—
79	C₈×C₇₃	—	2	2	no	—
80	#10596831	—	2	2	no	—
81	C₂²×C₇₃	C₄×D₄	1	1	yes	characteristic
82	C₂²×C₇₃	#121	1	1	yes	characteristic
83	C₂²×C₇₃	#124	1	1	yes	characteristic
84	C₂²×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
85	C₂²×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
86	C₂²×C₇₃	#135	1	1	yes	characteristic
87	C₂²×C₇₃	#135	1	1	yes	characteristic
88	C₄×C₇₃	—	4	2	no	—
89	C₄×C₇₃	—	4	2	no	—
90	C₄×C₇₃	#117	1	1	yes	characteristic
91	C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
92	C₄×C₇₃	#123	1	1	yes	characteristic
93	C₄×C₇₃	#126	1	1	yes	characteristic
94	Dic₇₃	—	2	2	no	—
95	Dic₇₃	—	2	2	no	—
96	Dic₇₃	—	8	1	no	—
97	Dic₇₃	—	8	1	no	—
98	C₂×C₇₃	#388	1	1	yes	characteristic
99	C₂×C₇₃	C₄×SD₁₆	1	1	yes	characteristic
100	C₂×C₇₃	C₄×Q₁₆	1	1	yes	characteristic
101	C₂×C₇₃	#465	1	1	yes	characteristic
102	C₂×C₇₃	#467	1	1	yes	characteristic
103	C₂×C₇₃	#487	1	1	yes	characteristic
104	C₂×C₇₃	#488	1	1	yes	characteristic
105	#1933	—	73	1	no	maximal, Sylow
106	C₇₃	#1933	1	1	yes	characteristic, complemented, Sylow
107	C₂×#103	—	73	1	no	—
108	C₂×#103	D₇₃	1	1	yes	characteristic
109	C₂×C₄×C₈	—	73	1	no	—
110	#391	—	73	1	no	—
111	#389	—	73	1	no	—
112	#340	—	73	1	no	—
113	#340	—	73	1	no	—
114	C₂²×Q₈	C₂×D₇₃	1	1	yes	characteristic
115	C₂²×C₈	—	73	1	no	—
116	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	73	1	no	—
118	C₂×C₄⋊C₄	—	73	1	no	—
119	C₂×C₄⋊C₄	—	146	1	no	—
120	C₂×C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
121	C₂×C₄²	—	73	1	no	—
122	#103	—	146	2	no	—
123	#103	Dic₇₃	1	4	yes	complemented
124	C₄×C₈	—	146	2	no	—
125	#95	—	146	1	no	—
126	#95	—	146	1	no	—
127	#95	—	146	1	no	—
128	C₂×Q₈	—	2	1	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	C₂×Dic₇₃	1	2	yes	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	—	73	1	no	—
137	C₂²×C₄	—	73	1	no	—
138	C₂²×C₄	—	146	1	no	—
139	C₂²×C₄	—	146	1	no	—
140	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	73	2	no	—
144	C₂×C₈	—	146	1	no	—
145	C₂×C₈	—	146	1	no	—
146	C₂×C₈	C₂×Dic₇₃	1	2	yes	—
147	C₄⋊C₄	—	2	1	no	—
148	C₄⋊C₄	—	292	2	no	—
149	C₄⋊C₄	—	146	1	no	—
150	C₄⋊C₄	—	146	1	no	—
151	C₄⋊C₄	—	146	1	no	—
152	C₄⋊C₄	—	146	1	no	—
153	C₄⋊C₄	C₂×Dic₇₃	1	2	yes	—
154	C₄²	—	146	1	no	—
155	C₄²	—	146	1	no	—
156	C₂³	D₄×D₇₃	1	1	yes	characteristic
157	Q₈	—	2	2	no	—
158	Q₈	—	2	2	no	—
159	Q₈	—	4	1	no	—
160	Q₈	—	4	1	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	4	1	no	—
165	C₂×C₄	—	4	1	no	—
166	C₂×C₄	—	4	1	no	—
167	C₂×C₄	—	292	1	no	—
168	C₂×C₄	—	292	1	no	—
169	C₂×C₄	—	292	1	no	—
170	C₂×C₄	—	292	1	no	—
171	C₂×C₄	—	292	1	no	—
172	C₂×C₄	—	292	1	no	—
173	C₂×C₄	—	73	2	no	—
174	C₂×C₄	—	73	2	no	—
175	C₂×C₄	—	146	1	no	—
176	C₂×C₄	—	146	1	no	—
177	C₂×C₄	—	146	1	no	—
178	C₂×C₄	—	146	1	no	—
179	C₂×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
180	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
181	C₂×C₄	#11924027	1	1	yes	characteristic
182	C₂×C₄	#11924027	1	1	yes	characteristic
183	C₂×C₄	#11924028	1	1	yes	characteristic
184	C₂×C₄	#11924028	1	1	yes	characteristic
185	C₈	—	2	2	no	—
186	C₈	—	146	2	no	—
187	C₂²	D₄×Dic₇₃	1	1	yes	characteristic
188	C₂²	#14556564	1	1	yes	characteristic
189	C₂²	#14556577	1	1	yes	characteristic
190	C₂²	SD₁₆×D₇₃	1	1	yes	characteristic
191	C₂²	Q₁₆×D₇₃	1	1	yes	characteristic
192	C₂²	#14556612	1	1	yes	characteristic
193	C₂²	#14556614	1	1	yes	characteristic
194	C₄	—	4	2	no	—
195	C₄	—	4	2	no	—
196	C₄	—	584	1	no	—
197	C₄	—	584	1	no	—
198	C₄	—	146	2	no	—
199	C₄	—	146	2	no	—
200	C₄	#14556550	1	1	yes	characteristic
201	C₄	D₄×Dic₇₃	1	1	yes	characteristic
202	C₄	#14556573	1	1	yes	characteristic
203	C₄	#14556584	1	1	yes	characteristic
204	C₂	#17988480	1	1	yes	characteristic
205	C₂	SD₁₆×Dic₇₃	1	1	yes	characteristic
206	C₂	Q₁₆×Dic₇₃	1	1	yes	characteristic
207	C₂	#17988819	1	1	yes	characteristic
208	C₂	#17988827	1	1	yes	characteristic
209	C₂	#17988936	1	1	yes	characteristic
210	C₂	#17988943	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556515

C₂

yes

maximal, characteristic

C₂×C₈×Dic₇₃

C₂

yes

maximal, characteristic

#17988494

C₂

yes

maximal, characteristic

#17988486

C₂

yes

maximal, characteristic

#17988350

C₂

yes

maximal, characteristic

#17988349

C₂

yes

maximal, characteristic

C₂×#103×C₇₃

C₂

yes

maximal, characteristic

C₂²×#10596831

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

characteristic

C₂×C₄×Dic₇₃

C₂²

yes

characteristic

#14556515

—

C₈×Dic₇₃

—

#14556498

—

#14556498

—

#14556498

—

C₂²×Q₈×C₇₃

C₂²

yes

characteristic

C₂²×C₈×C₇₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₃

C₂²

yes

characteristic

#103×C₇₃

C₄

yes

complemented

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

D₄

yes

characteristic

C₂²×Dic₇₃

D₄

yes

—

—

D₄

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

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

C₂×C₄

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂×C₄

yes

—

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

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

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

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

SD₁₆

yes

—

C₂×Dic₇₃

Q₁₆

yes

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

C₄×D₄

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

C₂×SD₁₆

yes

characteristic

C₂²×C₇₃

C₂×Q₁₆

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#117

yes

characteristic

C₄×C₇₃

C₄×D₄

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

#388

yes

characteristic

C₂×C₇₃

C₄×SD₁₆

yes

characteristic

100

C₂×C₇₃

C₄×Q₁₆

yes

characteristic

101

C₂×C₇₃

#465

yes

characteristic

102

C₂×C₇₃

#467

yes

characteristic

103

C₂×C₇₃

#487

yes

characteristic

104

C₂×C₇₃

yes

105

—

106

C₇₃

#1933

yes

characteristic, complemented, Sylow

107

C₂×#103

—

108

C₂×#103

D₇₃

yes

characteristic

109

C₂×C₄×C₈

—

110

#391

—

111

#389

—

112

#340

—

113

#340

—

114

C₂²×Q₈

C₂×D₇₃

yes

characteristic

115

C₂²×C₈

—

116

C₂²×C₈

C₂×D₇₃

yes

characteristic

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

146

—

120

C₂×C₄⋊C₄

C₂×D₇₃

yes

characteristic

121

C₂×C₄²

—

122

#103

—

146

—

123

#103

Dic₇₃

yes

complemented

124

C₄×C₈

—

146

—

125

#95

—

146

—

126

#95

—

146

—

127

#95

—

146

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

C₂×Dic₇₃

yes

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

146

—

139

C₂²×C₄

—

146

—

140

C₂²×C₄

C₂²×D₇₃

yes

characteristic

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

146

—

145

C₂×C₈

—

146

—

146

C₂×C₈

C₂×Dic₇₃

yes

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

292

—

149

C₄⋊C₄

—

146

—

150

C₄⋊C₄

—

146

—

151

C₄⋊C₄

—

146

—

152

C₄⋊C₄

—

146

—

153

C₄⋊C₄

C₂×Dic₇₃

yes

—

154

C₄²

—

146

—

155

C₄²

—

146

—

156

C₂³

D₄×D₇₃

yes

characteristic

157

Q₈

—

158

Q₈

—

159

Q₈

—

160

Q₈

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

292

—

168

C₂×C₄

—

292

—

169

C₂×C₄

—

292

—

170

C₂×C₄

—

292

—

171

C₂×C₄

—

292

—

172

C₂×C₄

—

292

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

146

—

176

C₂×C₄

—

146

—

177

C₂×C₄

—

146

—

178

C₂×C₄

—

146

—

179

C₂×C₄

C₂²×Dic₇₃

yes

characteristic

180

C₂×C₄

D₄×D₇₃

yes

characteristic

181

C₂×C₄

#11924027

yes

characteristic

182

C₂×C₄

#11924027

yes

characteristic

183

C₂×C₄

#11924028

yes

characteristic

184

C₂×C₄

#11924028

yes

characteristic

185

C₈

—

186

C₈

—

146

—

187

C₂²

D₄×Dic₇₃

yes

characteristic

188

C₂²

#14556564

yes

characteristic

189

C₂²

#14556577

yes

characteristic

190

C₂²

SD₁₆×D₇₃

yes

characteristic

191

C₂²

Q₁₆×D₇₃

yes

characteristic

192

C₂²

#14556612

yes

characteristic

193

C₂²

#14556614

yes

characteristic

194

C₄

—

195

C₄

—

196

C₄

—

584

—

197

C₄

—

584

—

198

C₄

—

146

—

199

C₄

—

146

—

200

C₄

#14556550

yes

characteristic

201

C₄

D₄×Dic₇₃

yes

characteristic

202

C₄

#14556573

yes

characteristic

203

C₄

#14556584

yes

characteristic

204

C₂

#17988480

yes

characteristic

205

C₂

SD₁₆×Dic₇₃

yes

characteristic

206

C₂

Q₁₆×Dic₇₃

yes

characteristic

207

C₂

#17988819

yes

characteristic

208

C₂

#17988827

yes

characteristic

209

C₂

#17988936

yes

characteristic

210

C₂

#17988943

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	#1933
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
21. #103×C₇₃	C₄	semidirect
106. C₇₃	#1933	semidirect
123. #103	Dic₇₃	semidirect

H ≅ G/N

type

21. #103×C₇₃

C₄

semidirect

106. C₇₃

#1933

semidirect

123. #103

Dic₇₃

semidirect

Subgroups — #22610609

Conjugacy classes of subgroups

Complements