Group #34007494 — subgroups

Subgroups — #34007494

Order 18688.

378 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 120 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	#22626029	C₂	1	2	yes	maximal
2	C₂×#17989450	C₂	1	1	yes	maximal, characteristic
3	C₂×#17989429	C₂	1	1	yes	maximal, characteristic
4	#22624767	C₂	1	2	yes	maximal
5	#22609819	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556658	—	2	2	no	—
7	C₂×C₄⋊#10596833	C₂²	1	2	yes	—
8	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
9	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
10	#17989453	—	2	2	no	—
11	#17989450	—	2	2	no	—
12	#17989429	—	2	2	no	—
13	C₂×#14556522	C₄	1	2	yes	—
14	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×Dic₇₃	C₂²	1	2	yes	—
16	#17988350	C₄	1	1	yes	characteristic
17	#340×C₇₃	C₄	1	1	yes	characteristic
18	C₂²×#10596833	—	2	2	no	—
19	C₂²×#10596833	—	2	1	no	—
20	C₂²×#10596833	D₄	1	1	yes	characteristic
21	C₂²×#10596833	D₄	1	1	yes	characteristic
22	#14556658	—	4	4	no	—
23	C₄⋊#10596833	—	2	2	no	—
24	C₄⋊#10596833	—	2	2	no	—
25	C₄×#10596833	—	2	1	no	—
26	C₄×#10596833	—	2	1	no	—
27	C₄×#10596833	—	2	1	no	—
28	C₄×#10596833	Q₈	1	2	yes	—
29	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
31	C₂×C₄×Dic₇₃	—	2	2	no	—
32	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
33	#14556522	—	2	4	no	—
34	C₈×Dic₇₃	D₄	1	2	yes	—
35	C₈×Dic₇₃	Q₈	1	2	yes	—
36	#14556498	—	2	2	no	—
37	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
39	C₂×#10596833	—	2	2	no	—
40	C₂×#10596833	—	2	2	no	—
41	C₂×#10596833	—	2	2	no	—
42	C₂×#10596833	—	2	1	no	—
43	C₂×#10596833	—	2	1	no	—
44	C₂×#10596833	—	2	2	no	—
45	C₂×#10596833	—	2	2	no	—
46	C₂×#10596833	—	4	2	no	—
47	C₂×#10596833	—	4	2	no	—
48	C₂×#10596833	—	4	2	no	—
49	C₂²×Dic₇₃	—	2	2	no	—
50	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₇₃	#41	1	1	yes	characteristic
52	C₂×#10596831	—	2	2	no	—
53	C₂×#10596831	C₄⋊C₄	1	2	yes	—
54	C₄⋊Dic₇₃	—	2	2	no	—
55	C₄⋊Dic₇₃	M₁₆	1	4	yes	—
56	C₄×Dic₇₃	—	4	2	no	—
57	C₄×Dic₇₃	—	4	2	no	—
58	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
59	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
60	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
61	C₄×Dic₇₃	#41	1	1	yes	characteristic
62	C₂²×C₄×C₇₃	—	2	2	no	—
63	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₇₃	—	2	2	no	—
65	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
66	C₄⋊C₄×C₇₃	—	2	2	no	—
67	C₄⋊C₄×C₇₃	—	2	2	no	—
68	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₇₃	—	4	2	no	—
70	C₂×C₄×C₇₃	—	4	2	no	—
71	C₂×C₄×C₇₃	—	4	2	no	—
72	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
75	C₂×C₄×C₇₃	#131	1	2	yes	—
76	C₂×Dic₇₃	—	2	1	no	—
77	C₂×Dic₇₃	—	2	1	no	—
78	C₂×Dic₇₃	—	2	4	no	—
79	C₂×Dic₇₃	—	4	2	no	—
80	C₂×Dic₇₃	—	4	2	no	—
81	C₂×Dic₇₃	#121	1	1	yes	characteristic
82	C₂×Dic₇₃	C₂²⋊Q₈	1	1	yes	characteristic
83	C₂×Dic₇₃	#125	1	1	yes	characteristic
84	C₂×Dic₇₃	C₄⋊Q₈	1	1	yes	characteristic
85	C₂×Dic₇₃	AGL(1,ℤ/8)	1	2	yes	—
86	C₂×Dic₇₃	#137	1	2	yes	—
87	#10596833	—	2	4	no	—
88	#10596833	—	4	1	no	—
89	#10596833	—	4	1	no	—
90	#10596833	—	8	2	no	—
91	C₈×C₇₃	—	2	2	no	—
92	#10596831	—	2	2	no	—
93	C₂²×C₇₃	#389	1	1	yes	characteristic
94	C₂²×C₇₃	#435	1	2	yes	—
95	C₂²×C₇₃	#440	1	2	yes	—
96	C₂²×C₇₃	#443	1	1	yes	characteristic
97	C₂²×C₇₃	#443	1	1	yes	characteristic
98	C₄×C₇₃	—	8	2	no	—
99	C₄×C₇₃	#423	1	1	yes	characteristic
100	C₄×C₇₃	#424	1	1	yes	characteristic
101	C₄×C₇₃	#446	1	2	yes	—
102	Dic₇₃	—	2	1	no	—
103	Dic₇₃	—	2	1	no	—
104	Dic₇₃	—	4	4	no	—
105	Dic₇₃	#469	1	1	yes	characteristic
106	Dic₇₃	#470	1	1	yes	characteristic
107	Dic₇₃	#501	1	2	yes	—
108	#9600	—	73	1	no	maximal, Sylow
109	C₂×C₇₃	#1585	1	2	yes	—
110	C₂×C₇₃	#1588	1	2	yes	—
111	C₂×C₇₃	#1940	1	1	yes	characteristic
112	C₂×C₇₃	#1944	1	1	yes	characteristic
113	C₂×C₇₃	#1945	1	1	yes	characteristic
114	#1939	—	73	2	no	—
115	C₄×#106	—	73	1	no	—
116	C₂×#335	—	73	1	no	—
117	C₂×#322	—	73	1	no	—
118	#1279	—	73	2	no	—
119	C₇₃	#9600	1	1	yes	characteristic, complemented, Sylow
120	C₂×#106	—	73	2	no	—
121	C₂×C₄⋊C₈	—	73	2	no	—
122	C₂×C₄⋊C₈	—	146	2	no	—
123	C₂×C₄×C₈	—	73	1	no	—
124	C₂×C₄×C₈	—	73	1	no	—
125	C₂×C₄×C₈	—	73	1	no	—
126	C₄×C₄⋊C₄	—	73	2	no	—
127	#340	—	73	1	no	—
128	#340	#64408	1	1	yes	characteristic
129	#336	—	146	2	no	—
130	#335	—	146	2	no	—
131	#322	—	146	2	no	—
132	C₂²×C₈	—	73	1	no	—
133	C₂²×C₈	—	73	1	no	—
134	C₂²×C₈	—	73	1	no	—
135	C₂²×C₈	—	146	2	no	—
136	C₂²×C₈	—	146	1	no	—
137	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
138	C₂×C₄⋊C₄	—	73	2	no	—
139	C₂×C₄⋊C₄	C₂×#64408	1	2	yes	—
140	C₂×C₄²	—	73	1	no	—
141	C₂×C₄²	—	146	2	no	—
142	#106	—	146	4	no	—
143	C₄⋊C₈	—	292	4	no	—
144	C₄⋊C₈	—	146	2	no	—
145	C₄⋊C₈	—	146	2	no	—
146	C₄×C₈	—	73	2	no	—
147	C₄×C₈	—	73	2	no	—
148	C₄×C₈	—	73	2	no	—
149	C₄×C₈	—	146	1	no	—
150	C₄×C₈	—	146	1	no	—
151	C₄×C₈	—	146	1	no	—
152	#95	—	146	2	no	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	73	1	no	—
155	C₂²×C₄	—	73	1	no	—
156	C₂²×C₄	—	146	2	no	—
157	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	292	2	no	—
160	C₂×C₈	—	292	2	no	—
161	C₂×C₈	—	292	2	no	—
162	C₂×C₈	—	73	2	no	—
163	C₂×C₈	—	146	1	no	—
164	C₂×C₈	—	146	1	no	—
165	C₂×C₈	—	146	2	no	—
166	C₂×C₈	—	146	2	no	—
167	C₂×C₈	—	146	2	no	—
168	C₂×C₈	—	146	2	no	—
169	C₂×C₈	—	146	2	no	—
170	C₂×C₈	—	146	2	no	—
171	C₂×C₈	C₄⋊#64408	1	2	yes	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	73	4	no	—
175	C₄⋊C₄	—	146	2	no	—
176	C₄²	—	292	2	no	—
177	C₄²	—	292	2	no	—
178	C₄²	—	73	1	no	—
179	C₄²	—	73	1	no	—
180	C₄²	—	73	1	no	—
181	C₄²	—	73	1	no	—
182	C₂³	D₄×#64408	1	1	yes	characteristic
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	292	2	no	—
187	C₂×C₄	—	292	2	no	—
188	C₂×C₄	—	73	1	no	—
189	C₂×C₄	—	73	1	no	—
190	C₂×C₄	—	73	2	no	—
191	C₂×C₄	—	73	2	no	—
192	C₂×C₄	—	73	1	no	—
193	C₂×C₄	—	73	1	no	—
194	C₂×C₄	—	146	4	no	—
195	C₂×C₄	—	146	1	no	—
196	C₂×C₄	—	146	1	no	—
197	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
198	C₂×C₄	Q₈×#64408	1	1	yes	characteristic
199	C₂×C₄	D₇₃⋊M₁₆	1	2	yes	—
200	C₂×C₄	#14556678	1	2	yes	—
201	C₈	—	2	2	no	—
202	C₈	—	292	1	no	—
203	C₈	—	292	1	no	—
204	C₈	—	584	2	no	—
205	C₈	—	146	4	no	—
206	C₈	—	146	2	no	—
207	C₂²	#17989538	1	1	yes	characteristic
208	C₂²	#17989619	1	2	yes	—
209	C₂²	#17989626	1	2	yes	—
210	C₂²	#17989629	1	1	yes	characteristic
211	C₂²	#17989631	1	1	yes	characteristic
212	C₄	—	8	2	no	—
213	C₄	—	292	4	no	—
214	C₄	—	73	1	no	—
215	C₄	—	73	1	no	—
216	C₄	—	73	2	no	—
217	C₄	—	146	1	no	—
218	C₄	—	146	1	no	—
219	C₄	#17989592	1	1	yes	characteristic
220	C₄	#17989593	1	1	yes	characteristic
221	C₄	#17989639	1	2	yes	—
222	C₂	#22625549	1	2	yes	—
223	C₂	#22625561	1	2	yes	—
224	C₂	#22626032	1	1	yes	characteristic
225	C₂	#22626037	1	1	yes	characteristic
226	C₂	#22626039	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626029

C₂

yes

maximal

C₂×#17989450

C₂

yes

maximal, characteristic

C₂×#17989429

C₂

yes

maximal, characteristic

#22624767

C₂

yes

maximal

#22609819

C₂

yes

maximal, characteristic

C₂×#14556658

—

C₂×C₄⋊#10596833

C₂²

yes

—

C₂×C₄×#10596833

C₂²

yes

characteristic

C₂×C₄×#10596833

C₂²

yes

—

—

—

C₄

yes

—

C₂×C₈×Dic₇₃

C₂²

yes

characteristic

C₄⋊C₄×Dic₇₃

C₂²

yes

—

#17988350

C₄

yes

characteristic

#340×C₇₃

C₄

yes

—

—

D₄

yes

characteristic

C₂²×#10596833

D₄

yes

—

—

—

—

—

—

Q₈

yes

—

C₂²×#10596831

C₂×C₄

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂×C₄

yes

—

C₂×C₄×Dic₇₃

—

C₂×C₄×Dic₇₃

C₂³

yes

characteristic

#14556522

—

C₈×Dic₇₃

D₄

yes

—

C₈×Dic₇₃

Q₈

yes

—

#14556498

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₄⋊C₄×C₇₃

C₂×C₄

yes

—

—

—

—

—

—

—

—

—

—

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

C₂×D₄

yes

C₂²×Dic₇₃

yes

—

C₄⋊C₄

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

M₁₆

yes

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₂×D₄

yes

characteristic

C₄×Dic₇₃

C₂×Q₈

yes

characteristic

C₄×Dic₇₃

C₂×Q₈

yes

characteristic

C₄×Dic₇₃

#41

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₂³×C₇₃

C₄×D₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₄×Q₈

yes

characteristic

C₂×C₄×C₇₃

C₂×M₁₆

yes

—

C₂×C₄×C₇₃

#131

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

#121

yes

characteristic

C₂×Dic₇₃

C₂²⋊Q₈

yes

characteristic

C₂×Dic₇₃

#125

yes

characteristic

C₂×Dic₇₃

C₄⋊Q₈

yes

characteristic

C₂×Dic₇₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₇₃

#137

yes

—

—

—

—

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

#389

yes

characteristic

C₂²×C₇₃

#435

yes

—

C₂²×C₇₃

#440

yes

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

#423

yes

characteristic

100

C₄×C₇₃

#424

yes

characteristic

101

C₄×C₇₃

#446

yes

—

102

Dic₇₃

—

103

Dic₇₃

—

104

Dic₇₃

—

105

Dic₇₃

#469

yes

characteristic

106

Dic₇₃

#470

yes

characteristic

107

Dic₇₃

#501

yes

—

108

#9600

—

maximal, Sylow

109

C₂×C₇₃

#1585

yes

—

110

C₂×C₇₃

#1588

yes

—

111

C₂×C₇₃

#1940

yes

characteristic

112

C₂×C₇₃

#1944

yes

characteristic

113

C₂×C₇₃

#1945

yes

characteristic

114

#1939

—

115

C₄×#106

—

116

C₂×#335

—

117

C₂×#322

—

118

#1279

—

119

C₇₃

#9600

yes

characteristic, complemented, Sylow

120

C₂×#106

—

121

C₂×C₄⋊C₈

—

122

C₂×C₄⋊C₈

—

146

—

123

C₂×C₄×C₈

—

124

C₂×C₄×C₈

—

125

C₂×C₄×C₈

—

126

C₄×C₄⋊C₄

—

127

#340

—

128

yes

129

—

146

—

130

#335

—

146

—

131

#322

—

146

—

132

C₂²×C₈

—

133

C₂²×C₈

—

134

C₂²×C₈

—

135

C₂²×C₈

—

146

—

136

C₂²×C₈

—

146

—

137

C₂²×C₈

C₂×#64408

yes

characteristic

138

C₂×C₄⋊C₄

—

139

C₂×C₄⋊C₄

C₂×#64408

yes

—

140

C₂×C₄²

—

141

C₂×C₄²

—

146

—

142

#106

—

146

—

143

C₄⋊C₈

—

292

—

144

C₄⋊C₈

—

146

—

145

C₄⋊C₈

—

146

—

146

C₄×C₈

—

147

C₄×C₈

—

148

C₄×C₈

—

149

C₄×C₈

—

146

—

150

C₄×C₈

—

146

—

151

C₄×C₈

—

146

—

152

#95

—

146

—

153

C₂²×C₄

—

154

C₂²×C₄

—

155

C₂²×C₄

—

156

C₂²×C₄

—

146

—

157

C₂²×C₄

C₂²×#64408

yes

characteristic

158

C₂×C₈

—

159

C₂×C₈

—

292

—

160

C₂×C₈

—

292

—

161

C₂×C₈

—

292

—

162

C₂×C₈

—

163

C₂×C₈

—

146

—

164

C₂×C₈

—

146

—

165

C₂×C₈

—

146

—

166

C₂×C₈

—

146

—

167

C₂×C₈

—

146

—

168

C₂×C₈

—

146

—

169

C₂×C₈

—

146

—

170

C₂×C₈

—

146

—

171

C₂×C₈

C₄⋊#64408

yes

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

—

175

C₄⋊C₄

—

146

—

176

C₄²

—

292

—

177

C₄²

—

292

—

178

C₄²

—

179

C₄²

—

180

C₄²

—

181

C₄²

—

182

C₂³

D₄×#64408

yes

characteristic

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

292

—

187

C₂×C₄

—

292

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

146

—

195

C₂×C₄

—

146

—

196

C₂×C₄

—

146

—

197

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

198

C₂×C₄

Q₈×#64408

yes

characteristic

199

C₂×C₄

D₇₃⋊M₁₆

yes

—

200

C₂×C₄

#14556678

yes

—

201

C₈

—

202

C₈

—

292

—

203

C₈

—

292

—

204

C₈

—

584

—

205

C₈

—

146

—

206

C₈

—

146

—

207

C₂²

#17989538

yes

characteristic

208

C₂²

#17989619

yes

—

209

C₂²

#17989626

yes

—

210

C₂²

#17989629

yes

characteristic

211

C₂²

#17989631

yes

characteristic

212

C₄

—

213

C₄

—

292

—

214

C₄

—

215

C₄

—

216

C₄

—

217

C₄

—

146

—

218

C₄

—

146

—

219

C₄

#17989592

yes

characteristic

220

C₄

#17989593

yes

characteristic

221

C₄

#17989639

yes

—

222

C₂

#22625549

yes

—

223

C₂

#22625561

yes

—

224

C₂

#22626032

yes

characteristic

225

C₂

#22626037

yes

characteristic

226

C₂

#22626039

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×#64408
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#64408
Fitting	#340×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₄×D₄
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	#340	#64408
hypocenter	C₇₃	#9600
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×#64408

commutator

C₂²×C₄×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×#64408

Fitting

#340×C₇₃

C₄

socle, prime radical

C₂³×C₇₃

C₄×D₄

cosocle

C₂×C₄×Dic₇₃

C₂³

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
119. C₇₃	#9600	semidirect

H ≅ G/N

type

119. C₇₃

#9600

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22626029	2
C₂×#17989450	2
C₂×#17989429	2
#22624767	2
#22609819	2
#9600	73

subgroup

index

Subgroups — #34007494

Conjugacy classes of subgroups

Complements

Maximal subgroups