Group #22610777 — subgroups

Subgroups — #22610777

Order 9344.

312 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 83 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×#14556578	C₂	1	1	yes	maximal, characteristic
2	#17988610	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14556509	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14556509	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14556508	C₂	1	1	yes	maximal, characteristic
6	M₁₆×Dic₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#100×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₄₆	C₂²	1	1	yes	characteristic
9	C₂×#11924012	C₂²	1	1	yes	characteristic
10	C₂×#11924012	C₂²	1	1	yes	characteristic
11	#14556578	—	2	1	no	—
12	#14556578	—	2	1	no	—
13	#14556578	—	2	2	no	—
14	C₂×C₂²⋊Dic₇₃	C₄	1	4	yes	—
15	C₂×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
16	#14556520	—	2	2	no	—
17	#14556509	—	2	2	no	—
18	#14556509	—	2	2	no	—
19	#14556508	—	2	2	no	—
20	#14556501	—	2	1	no	—
21	C₈×Dic₇₃	—	2	1	no	—
22	C₂²×D₄×C₇₃	C₂²	1	1	yes	characteristic
23	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
24	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
25	#100×C₇₃	—	2	2	no	—
26	C₂×Dic₁₄₆	—	2	1	no	—
27	C₂×Dic₁₄₆	—	2	1	no	—
28	C₂×Dic₁₄₆	—	2	1	no	—
29	C₂×Dic₁₄₆	—	2	1	no	—
30	C₂×Dic₁₄₆	—	2	1	no	—
31	C₂×Dic₁₄₆	—	2	1	no	—
32	C₂²×Dic₇₃	C₂×C₄	1	2	yes	—
33	C₂²×Dic₇₃	C₂×C₄	1	2	yes	—
34	#11924012	—	2	1	no	—
35	#11924012	—	2	1	no	—
36	#11924012	—	2	1	no	—
37	#11924012	D₄	1	2	yes	—
38	C₂×#10596831	—	2	1	no	—
39	C₂×#10596831	—	2	1	no	—
40	C₂²⋊Dic₇₃	—	2	4	no	—
41	C₂²⋊Dic₇₃	—	2	4	no	—
42	C₄×Dic₇₃	D₄	1	1	yes	characteristic
43	C₄×Dic₇₃	D₄	1	1	yes	characteristic
44	C₄×Dic₇₃	D₄	1	2	yes	—
45	C₂⁴×C₇₃	C₂×C₄	1	2	yes	—
46	C₂×D₄×C₇₃	—	2	1	no	—
47	C₂×D₄×C₇₃	—	2	1	no	—
48	C₂×D₄×C₇₃	—	2	1	no	—
49	C₂×D₄×C₇₃	—	2	1	no	—
50	C₂×D₄×C₇₃	—	2	1	no	—
51	C₂×D₄×C₇₃	—	2	1	no	—
52	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
53	M₁₆×C₇₃	—	2	1	no	—
54	M₁₆×C₇₃	—	2	1	no	—
55	M₁₆×C₇₃	—	2	1	no	—
56	M₁₆×C₇₃	D₄	1	2	yes	—
57	C₂×C₈×C₇₃	—	2	1	no	—
58	C₂×C₈×C₇₃	—	2	1	no	—
59	C₂³×C₇₃	—	2	2	no	—
60	C₂³×C₇₃	—	2	2	no	—
61	C₂³×C₇₃	—	2	2	no	—
62	C₂³×C₇₃	—	4	4	no	—
63	C₂³×C₇₃	C₂²×C₄	1	1	yes	characteristic
64	D₄×C₇₃	—	4	1	no	—
65	D₄×C₇₃	—	4	1	no	—
66	D₄×C₇₃	—	4	1	no	—
67	D₄×C₇₃	—	4	1	no	—
68	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
73	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
74	C₂×Dic₇₃	—	2	2	no	—
75	C₂×Dic₇₃	—	2	2	no	—
76	C₂×Dic₇₃	—	2	2	no	—
77	C₂×Dic₇₃	—	4	2	no	—
78	C₂×Dic₇₃	C₂²⋊C₄	1	2	yes	—
79	C₂×Dic₇₃	C₂²⋊C₄	1	2	yes	—
80	Dic₁₄₆	—	4	1	no	—
81	Dic₁₄₆	—	4	1	no	—
82	Dic₁₄₆	—	4	1	no	—
83	Dic₁₄₆	—	4	1	no	—
84	C₈×C₇₃	—	2	2	no	—
85	C₈×C₇₃	—	4	1	no	—
86	#10596831	—	2	2	no	—
87	#10596831	—	4	1	no	—
88	C₂²×C₇₃	—	2	2	no	—
89	C₂²×C₇₃	—	4	2	no	—
90	C₂²×C₇₃	—	4	4	no	—
91	C₂²×C₇₃	—	8	2	no	—
92	C₂²×C₇₃	—	8	2	no	—
93	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂²×C₇₃	C₄×D₄	1	1	yes	characteristic
95	C₂²×C₇₃	C₄×D₄	1	1	yes	characteristic
96	C₄×C₇₃	#121	1	1	yes	characteristic
97	C₄×C₇₃	#121	1	1	yes	characteristic
98	C₄×C₇₃	#124	1	1	yes	characteristic
99	C₄×C₇₃	#127	1	1	yes	characteristic
100	Dic₇₃	—	2	2	no	—
101	Dic₇₃	—	2	2	no	—
102	Dic₇₃	—	4	2	no	—
103	C₂×C₇₃	—	2	1	no	—
104	C₂×C₇₃	—	2	1	no	—
105	C₂×C₇₃	—	8	2	no	—
106	C₂×C₇₃	#390	1	1	yes	characteristic
107	C₂×C₇₃	#413	1	2	yes	—
108	#1967	—	73	1	no	maximal, Sylow
109	C₇₃	#1967	1	1	yes	characteristic, complemented, Sylow
110	C₂×#124	—	73	1	no	—
111	#423	—	73	1	no	—
112	C₂×#101	—	73	1	no	—
113	C₂×#101	—	73	1	no	—
114	C₂×#100	—	73	1	no	—
115	C₂×#100	D₇₃	1	1	yes	characteristic
116	C₄×M₁₆	—	73	1	no	—
117	C₂²×Q₈	—	73	1	no	—
118	C₂²×D₄	C₂×D₇₃	1	1	yes	characteristic
119	C₂×M₁₆	—	73	1	no	—
120	C₂×M₁₆	—	73	1	no	—
121	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
122	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
123	#124	—	146	1	no	—
124	#124	—	146	1	no	—
125	#124	—	146	2	no	—
126	C₂×C₂²⋊C₄	—	73	4	no	—
127	C₂×C₄²	—	73	1	no	—
128	C₄⋊C₈	—	146	2	no	—
129	#101	—	146	2	no	—
130	#101	—	146	2	no	—
131	#100	—	2	2	no	—
132	#100	—	146	2	no	—
133	#97	—	146	1	no	—
134	C₄×C₈	—	146	1	no	—
135	C₂⁴	C₄×D₇₃	1	2	yes	—
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	—	146	1	no	—
141	C₂×Q₈	—	146	1	no	—
142	C₂×D₄	—	2	1	no	—
143	C₂×D₄	—	2	1	no	—
144	C₂×D₄	—	2	1	no	—
145	C₂×D₄	—	2	1	no	—
146	C₂×D₄	—	2	1	no	—
147	C₂×D₄	—	2	1	no	—
148	C₂²×C₄	—	73	2	no	—
149	C₂²×C₄	—	73	2	no	—
150	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
151	M₁₆	—	2	1	no	—
152	M₁₆	—	2	1	no	—
153	M₁₆	—	2	1	no	—
154	M₁₆	—	73	2	no	—
155	M₁₆	—	146	1	no	—
156	M₁₆	—	146	1	no	—
157	M₁₆	—	146	1	no	—
158	M₁₆	C₂²⋊D₇₃	1	2	yes	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	146	1	no	—
162	C₂×C₈	—	146	1	no	—
163	C₂²⋊C₄	—	146	4	no	—
164	C₂²⋊C₄	—	146	4	no	—
165	C₄²	—	73	1	no	—
166	C₄²	—	73	1	no	—
167	C₄²	—	73	2	no	—
168	C₂³	—	2	2	no	—
169	C₂³	—	2	2	no	—
170	C₂³	—	2	2	no	—
171	C₂³	—	4	4	no	—
172	C₂³	C₂×C₄×D₇₃	1	1	yes	characteristic
173	Q₈	—	292	1	no	—
174	Q₈	—	292	1	no	—
175	Q₈	—	292	1	no	—
176	Q₈	—	292	1	no	—
177	D₄	—	4	1	no	—
178	D₄	—	4	1	no	—
179	D₄	—	4	1	no	—
180	D₄	—	4	1	no	—
181	C₂×C₄	—	292	2	no	—
182	C₂×C₄	—	73	2	no	—
183	C₂×C₄	—	73	2	no	—
184	C₂×C₄	—	146	2	no	—
185	C₂×C₄	—	146	2	no	—
186	C₂×C₄	—	146	2	no	—
187	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
188	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
189	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
190	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
191	C₂×C₄	#11924027	1	1	yes	characteristic
192	C₂×C₄	#11924028	1	1	yes	characteristic
193	C₈	—	2	2	no	—
194	C₈	—	4	1	no	—
195	C₈	—	292	1	no	—
196	C₈	—	146	2	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	4	2	no	—
199	C₂²	—	4	4	no	—
200	C₂²	—	8	2	no	—
201	C₂²	—	8	2	no	—
202	C₂²	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
203	C₂²	#14556551	1	1	yes	characteristic
204	C₂²	C₄×C₂²⋊D₇₃	1	1	yes	characteristic
205	C₄	—	292	2	no	—
206	C₄	—	146	2	no	—
207	C₄	—	146	2	no	—
208	C₄	#14556563	1	1	yes	characteristic
209	C₄	#14556564	1	1	yes	characteristic
210	C₄	#14556577	1	1	yes	characteristic
211	C₄	#14556587	1	1	yes	characteristic
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	—	8	2	no	—
215	C₂	#17988487	1	1	yes	characteristic
216	C₂	#17988568	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556578

C₂

yes

maximal, characteristic

#17988610

C₂

yes

maximal, characteristic, complemented

C₂×#14556509

C₂

yes

maximal, characteristic, complemented

C₂×#14556509

C₂

yes

maximal, characteristic, complemented

C₂×#14556508

C₂

yes

maximal, characteristic

M₁₆×Dic₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#100×C₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₆

C₂²

yes

characteristic

C₂×#11924012

C₂²

yes

characteristic

C₂×#11924012

C₂²

yes

—

—

—

C₂×C₂²⋊Dic₇₃

C₄

yes

—

C₂×C₄×Dic₇₃

C₂²

yes

—

—

—

—

—

C₈×Dic₇₃

—

C₂²×D₄×C₇₃

C₂²

yes

characteristic

C₂×M₁₆×C₇₃

C₂²

yes

characteristic

C₂×M₁₆×C₇₃

C₂²

yes

characteristic

#100×C₇₃

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

C₂×C₄

yes

—

C₂²×Dic₇₃

C₂×C₄

yes

—

—

—

—

D₄

yes

—

C₂×#10596831

—

C₂×#10596831

—

C₂²⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

—

C₂⁴×C₇₃

C₂×C₄

yes

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

M₁₆×C₇₃

—

M₁₆×C₇₃

D₄

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

C₂²×C₄

yes

characteristic

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

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

C₂²⋊C₄

yes

—

C₂×Dic₇₃

C₂²⋊C₄

yes

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₃

C₄×D₄

yes

characteristic

C₂²×C₇₃

C₄×D₄

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

100

Dic₇₃

—

101

Dic₇₃

—

102

Dic₇₃

—

103

C₂×C₇₃

—

104

C₂×C₇₃

—

105

C₂×C₇₃

—

106

C₂×C₇₃

#390

yes

characteristic

107

C₂×C₇₃

#413

yes

—

108

#1967

—

maximal, Sylow

109

C₇₃

#1967

yes

characteristic, complemented, Sylow

110

C₂×#124

—

111

#423

—

112

C₂×#101

—

113

C₂×#101

—

114

C₂×#100

—

115

C₂×#100

D₇₃

yes

characteristic

116

C₄×M₁₆

—

117

C₂²×Q₈

—

118

C₂²×D₄

C₂×D₇₃

yes

characteristic

119

C₂×M₁₆

—

120

C₂×M₁₆

—

121

C₂×M₁₆

C₂×D₇₃

yes

characteristic

122

C₂×M₁₆

C₂×D₇₃

yes

characteristic

123

#124

—

146

—

124

#124

—

146

—

125

#124

—

146

—

126

C₂×C₂²⋊C₄

—

127

C₂×C₄²

—

128

C₄⋊C₈

—

146

—

129

#101

—

146

—

130

#101

—

146

—

131

#100

—

132

#100

—

146

—

133

#97

—

146

—

134

C₄×C₈

—

146

—

135

C₂⁴

C₄×D₇₃

yes

—

136

C₂×Q₈

—

146

—

137

C₂×Q₈

—

146

—

138

C₂×Q₈

—

146

—

139

C₂×Q₈

—

146

—

140

C₂×Q₈

—

146

—

141

C₂×Q₈

—

146

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂²×C₄

—

149

C₂²×C₄

—

150

C₂²×C₄

C₂²×D₇₃

yes

characteristic

151

M₁₆

—

152

M₁₆

—

153

M₁₆

—

154

M₁₆

—

155

M₁₆

—

146

—

156

M₁₆

—

146

—

157

M₁₆

—

146

—

158

M₁₆

C₂²⋊D₇₃

yes

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

146

—

162

C₂×C₈

—

146

—

163

C₂²⋊C₄

—

146

—

164

C₂²⋊C₄

—

146

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₂³

—

169

C₂³

—

170

C₂³

—

171

C₂³

—

172

C₂³

C₂×C₄×D₇₃

yes

characteristic

173

Q₈

—

292

—

174

Q₈

—

292

—

175

Q₈

—

292

—

176

Q₈

—

292

—

177

D₄

—

178

D₄

—

179

D₄

—

180

D₄

—

181

C₂×C₄

—

292

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

146

—

185

C₂×C₄

—

146

—

186

C₂×C₄

—

146

—

187

C₂×C₄

D₄×D₇₃

yes

characteristic

188

C₂×C₄

D₄×D₇₃

yes

characteristic

189

C₂×C₄

D₄×D₇₃

yes

characteristic

190

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

191

C₂×C₄

#11924027

yes

characteristic

192

C₂×C₄

#11924028

yes

characteristic

193

C₈

—

194

C₈

—

195

C₈

—

292

—

196

C₈

—

146

—

197

C₂²

—

198

C₂²

—

199

C₂²

—

200

C₂²

—

201

C₂²

—

202

C₂²

C₂²⋊C₄×D₇₃

yes

characteristic

203

C₂²

#14556551

yes

characteristic

204

C₂²

C₄×C₂²⋊D₇₃

yes

characteristic

205

C₄

—

292

—

206

C₄

—

146

—

207

C₄

—

146

—

208

C₄

#14556563

yes

characteristic

209

C₄

#14556564

yes

characteristic

210

C₄

#14556577

yes

characteristic

211

C₄

#14556587

yes

characteristic

212

C₂

—

213

C₂

—

214

C₂

—

215

C₂

#17988487

yes

characteristic

216

C₂

#17988568

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₇₃
commutator	C₂³×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	C₂×#100×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₂²⋊C₄
hypercenter	C₂×#100	D₇₃
hypocenter	C₇₃	#1967
LCS	G ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#100

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₇₃

commutator

C₂³×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

C₂×#100×C₇₃

C₂

socle

C₂²×C₇₃

C₂×C₂²⋊C₄

D₇₃

C₇₃

G ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#100

N	H ≅ G/N	type
2. #17988610	C₂	semidirect
3. C₂×#14556509	C₂	semidirect
4. C₂×#14556509	C₂	semidirect
6. M₁₆×Dic₇₃	C₂	semidirect
109. C₇₃	#1967	semidirect

H ≅ G/N

type

2. #17988610

C₂

semidirect

3. C₂×#14556509

C₂

semidirect

4. C₂×#14556509

C₂

semidirect

6. M₁₆×Dic₇₃

C₂

semidirect

109. C₇₃

#1967

semidirect

Subgroups — #22610777

Conjugacy classes of subgroups

Complements