Group #34004029 — subgroups

Subgroups — #34004029

Order 18688.

398 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 132 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	C₄×#14556659	C₂	1	1	yes	maximal, characteristic
2	#22625728	C₂	1	2	yes	maximal
3	#22625125	C₂	1	1	yes	maximal, characteristic
4	#22624765	C₂	1	2	yes	maximal
5	#335×D₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556659	C₄	1	2	yes	—
7	C₂×#14556645	—	2	2	no	—
8	C₄×#11924036	C₂²	1	1	yes	characteristic
9	C₄×#11924036	C₂²	1	1	yes	characteristic
10	C₄×C₄⋊#64408	C₂²	1	2	yes	—
11	#17989461	C₄	1	2	yes	—
12	#17989452	—	2	2	no	—
13	#17989428	C₄	1	2	yes	complemented
14	#17989428	C₄	1	2	yes	complemented
15	C₄⋊C₈×D₇₃	C₂²	1	2	yes	—
16	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
17	#17988340	C₄	1	1	yes	characteristic, complemented
18	#17988339	C₄	1	2	yes	complemented
19	#335×C₇₃	C₄	1	1	yes	characteristic, complemented
20	C₂×#11924036	—	2	1	no	—
21	C₂×#11924036	D₄	1	2	yes	—
22	C₂×C₄⋊#64408	C₂×C₄	1	4	yes	—
23	C₂×C₄×#64408	—	2	2	no	—
24	#14556659	—	2	4	no	—
25	#14556645	—	4	4	no	—
26	C₄×#10596833	—	2	1	no	—
27	C₄×#10596833	C₂×C₄	1	2	yes	—
28	C₄×#10596833	C₂×C₄	1	2	yes	—
29	C₄×#10596833	Q₈	1	2	yes	—
30	(C₂×#64408)⋊C₄	—	2	2	no	—
31	C₂×C₈×D₇₃	—	2	2	no	—
32	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
34	C₄²×D₇₃	C₂³	1	1	yes	characteristic
35	C₄⋊#10596831	C₂×C₄	1	2	yes	—
36	#14556520	—	2	2	no	—
37	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
38	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
39	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
40	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×#64408	—	2	4	no	—
43	#11924036	—	2	2	no	—
44	#11924036	—	2	2	no	—
45	#11924036	—	4	2	no	—
46	C₂×#10596833	—	2	2	no	—
47	C₂×#10596833	C₄⋊C₄	1	4	yes	—
48	C₄⋊#64408	—	2	4	no	—
49	C₄⋊#64408	—	2	4	no	—
50	C₄×#64408	—	4	2	no	—
51	C₄×#64408	—	4	2	no	—
52	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
55	C₂×#10596831	—	2	2	no	—
56	C₂×#10596831	C₄²	1	1	yes	characteristic
57	C₂×#10596831	C₄²	1	1	yes	characteristic
58	C₈×D₇₃	—	2	2	no	—
59	C₈×D₇₃	—	4	2	no	—
60	C₈×D₇₃	—	4	2	no	—
61	C₈×D₇₃	M₁₆	1	4	yes	—
62	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
63	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
64	C₄×Dic₇₃	#41	1	1	yes	characteristic
65	C₂×C₈×C₇₃	—	2	2	no	—
66	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
67	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
68	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×D₇₃	C₄×D₄	1	1	yes	characteristic
70	C₂×#64408	—	4	4	no	—
71	C₂×#64408	—	4	4	no	—
72	C₂×#64408	—	4	4	no	—
73	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
76	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×Dic₇₃	#117	1	1	yes	characteristic
78	C₂×Dic₇₃	C₄×Q₈	1	1	yes	characteristic
79	C₄×D₇₃	—	2	2	no	—
80	C₄×D₇₃	C₂×M₁₆	1	2	yes	—
81	C₄×D₇₃	#131	1	2	yes	—
82	C₄×D₇₃	AGL(1,ℤ/8)	1	2	yes	—
83	C₄×D₇₃	#137	1	2	yes	—
84	#10596833	—	2	4	no	—
85	#10596833	—	4	2	no	—
86	C₈×C₇₃	—	2	1	no	—
87	C₈×C₇₃	—	4	2	no	—
88	C₈×C₇₃	#97	1	2	yes	complemented
89	#10596831	—	2	1	no	—
90	#10596831	—	4	2	no	—
91	#10596831	#97	1	2	yes	complemented
92	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
93	C₂×D₇₃	#436	1	2	yes	—
94	C₂×D₇₃	#440	1	2	yes	—
95	C₂×D₇₃	#444	1	2	yes	—
96	#64408	—	8	4	no	—
97	C₄×C₇₃	—	2	1	no	—
98	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
99	C₄×C₇₃	#405	1	1	yes	characteristic
100	C₄×C₇₃	#441	1	1	yes	characteristic
101	C₄×C₇₃	#442	1	1	yes	characteristic
102	Dic₇₃	—	2	1	no	—
103	Dic₇₃	#428	1	2	yes	—
104	Dic₇₃	#446	1	2	yes	—
105	#8286	—	73	1	no	maximal, Sylow
106	C₂×C₇₃	#1770	1	1	yes	characteristic
107	C₂×C₇₃	#1773	1	1	yes	characteristic
108	C₂×C₇₃	#1776	1	1	yes	characteristic
109	D₇₃	#1592	1	4	yes	—
110	C₄×#106	—	73	1	no	—
111	#1770	—	73	2	no	—
112	C₂×#335	—	73	1	no	—
113	C₂×#322	—	73	1	no	—
114	#1279	—	73	2	no	—
115	C₇₃	#8286	1	1	yes	characteristic, complemented, Sylow
116	C₂×#106	—	73	2	no	—
117	C₂×C₄⋊C₈	—	73	2	no	—
118	C₂×#97	—	146	2	no	—
119	C₂×C₄×C₈	—	73	1	no	—
120	C₂×C₄×C₈	—	73	1	no	—
121	C₂×C₄×C₈	—	73	1	no	—
122	C₄×C₄⋊C₄	—	73	2	no	—
123	#340	—	73	2	no	—
124	#336	—	146	2	no	—
125	#335	—	73	2	no	—
126	#335	—	73	1	no	—
127	#335	#64408	1	1	yes	characteristic, complemented
128	#322	—	73	2	no	—
129	#322	—	73	2	no	—
130	C₂²×C₈	—	73	1	no	—
131	C₂²×C₈	—	73	2	no	—
132	C₂²×C₈	—	73	1	no	—
133	C₂²×C₈	—	146	2	no	—
134	C₂²×C₈	—	146	1	no	—
135	C₂×C₄⋊C₄	—	73	4	no	—
136	C₂×C₄²	—	73	1	no	—
137	C₂×C₄²	—	146	2	no	—
138	#106	—	146	4	no	—
139	C₄⋊C₈	—	73	2	no	—
140	C₄⋊C₈	—	146	2	no	—
141	C₄⋊C₈	C₂×#64408	1	2	yes	—
142	#97	—	292	4	no	—
143	C₄×C₈	—	73	2	no	—
144	C₄×C₈	—	73	2	no	—
145	C₄×C₈	—	73	1	no	—
146	C₄×C₈	—	73	1	no	—
147	C₄×C₈	—	73	2	no	—
148	C₄×C₈	—	73	1	no	—
149	C₄×C₈	—	146	1	no	—
150	C₄×C₈	C₂×#64408	1	1	yes	characteristic
151	#95	—	146	2	no	—
152	C₂²×C₄	—	73	1	no	—
153	C₂²×C₄	—	73	1	no	—
154	C₂²×C₄	—	73	1	no	—
155	C₂²×C₄	—	146	4	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	292	2	no	—
158	C₂×C₈	—	292	2	no	—
159	C₂×C₈	—	292	2	no	—
160	C₂×C₈	—	73	1	no	—
161	C₂×C₈	—	73	4	no	—
162	C₂×C₈	—	73	4	no	—
163	C₂×C₈	—	73	1	no	—
164	C₂×C₈	—	146	2	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	C₄×#64408	1	1	yes	characteristic
170	C₂×C₈	C₄×#64408	1	1	yes	characteristic
171	C₄⋊C₄	—	146	4	no	—
172	C₄⋊C₄	—	146	4	no	—
173	C₄²	—	292	2	no	—
174	C₄²	—	292	2	no	—
175	C₄²	—	73	1	no	—
176	C₄²	—	73	1	no	—
177	C₄²	—	73	1	no	—
178	C₄²	C₂²×#64408	1	1	yes	characteristic
179	C₂³	—	73	1	no	—
180	C₂×C₄	—	292	4	no	—
181	C₂×C₄	—	292	4	no	—
182	C₂×C₄	—	292	4	no	—
183	C₂×C₄	—	73	1	no	—
184	C₂×C₄	—	73	1	no	—
185	C₂×C₄	—	73	2	no	—
186	C₂×C₄	—	73	2	no	—
187	C₂×C₄	—	73	2	no	—
188	C₂×C₄	—	73	2	no	—
189	C₂×C₄	—	73	1	no	—
190	C₂×C₄	—	146	2	no	—
191	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
192	C₂×C₄	D₄×#64408	1	1	yes	characteristic
193	C₂×C₄	Q₈×#64408	1	1	yes	characteristic
194	C₈	—	2	1	no	—
195	C₈	—	4	2	no	—
196	C₈	—	292	2	no	—
197	C₈	—	292	2	no	—
198	C₈	—	73	2	no	—
199	C₈	—	146	4	no	—
200	C₈	—	146	1	no	—
201	C₈	#14556645	1	2	yes	complemented
202	C₂²	—	73	2	no	—
203	C₂²	—	73	2	no	—
204	C₂²	—	73	2	no	—
205	C₂²	C₄⋊C₄×#64408	1	1	yes	characteristic
206	C₄	—	2	1	no	—
207	C₄	—	584	4	no	—
208	C₄	—	73	2	no	—
209	C₄	—	73	2	no	—
210	C₄	—	146	1	no	—
211	C₄	C₂×#14556645	1	1	yes	characteristic
212	C₄	#17989553	1	1	yes	characteristic
213	C₄	#17989627	1	1	yes	characteristic
214	C₄	#17989628	1	1	yes	characteristic
215	C₂	—	73	4	no	—
216	C₂	#22625728	1	1	yes	characteristic
217	C₂	#22625740	1	1	yes	characteristic
218	C₂	#22625746	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14556659

C₂

yes

maximal, characteristic

#22625728

C₂

yes

maximal

#22625125

C₂

yes

maximal, characteristic

#22624765

C₂

yes

maximal

#335×D₇₃

C₂

yes

maximal, characteristic

C₂×#14556659

C₄

yes

—

C₂×#14556645

—

C₄×#11924036

C₂²

yes

characteristic

C₄×#11924036

C₂²

yes

characteristic

C₄×C₄⋊#64408

C₂²

yes

—

#17989461

C₄

yes

—

#17989452

—

#17989428

C₄

yes

complemented

#17989428

C₄

yes

complemented

C₄⋊C₈×D₇₃

C₂²

yes

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988340

C₄

yes

characteristic, complemented

#17988339

C₄

yes

complemented

#335×C₇₃

C₄

yes

characteristic, complemented

C₂×#11924036

—

C₂×#11924036

D₄

yes

—

C₂×C₄⋊#64408

C₂×C₄

yes

—

C₂×C₄×#64408

—

—

—

—

C₂×C₄

yes

—

C₄×#10596833

C₂×C₄

yes

—

C₄×#10596833

Q₈

yes

—

(C₂×#64408)⋊C₄

—

C₂×C₈×D₇₃

—

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₄²×D₇₃

C₂³

yes

characteristic

C₄⋊#10596831

C₂×C₄

yes

—

#14556520

—

C₄×#10596831

C₂×C₄

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×#64408

—

—

—

—

—

C₄⋊C₄

yes

—

—

—

—

—

C₂×C₄×D₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×D₇₃

C₂×D₄

yes

C₂×C₄×D₇₃

yes

—

C₄²

yes

characteristic

C₂×#10596831

C₄²

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₈×D₇₃

—

C₈×D₇₃

M₁₆

yes

—

C₄×Dic₇₃

C₂²×C₄

yes

characteristic

C₄×Dic₇₃

C₂×Q₈

yes

characteristic

C₄×Dic₇₃

#41

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₂²×D₇₃

C₄×D₄

yes

—

—

—

C₂×C₄×C₇₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₃

C₄×D₄

yes

characteristic

C₂×C₄×C₇₃

C₄×Q₈

yes

characteristic

C₂×Dic₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₇₃

#117

yes

characteristic

C₂×Dic₇₃

C₄×Q₈

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

C₂×M₁₆

yes

—

C₄×D₇₃

#131

yes

—

C₄×D₇₃

AGL(1,ℤ/8)

yes

—

C₄×D₇₃

#137

yes

—

#10596833

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

yes

—

—

yes

C₂²×C₇₃

C₄×C₄⋊C₄

yes

characteristic

C₂×D₇₃

#436

yes

—

C₂×D₇₃

#440

yes

—

C₂×D₇₃

#444

yes

—

#64408

—

C₄×C₇₃

—

C₄×C₇₃

C₂×#97

yes

characteristic

C₄×C₇₃

#405

yes

characteristic

100

C₄×C₇₃

#441

yes

characteristic

101

C₄×C₇₃

#442

yes

characteristic

102

Dic₇₃

—

103

Dic₇₃

#428

yes

—

104

Dic₇₃

#446

yes

—

105

#8286

—

maximal, Sylow

106

C₂×C₇₃

#1770

yes

characteristic

107

C₂×C₇₃

#1773

yes

characteristic

108

C₂×C₇₃

#1776

yes

characteristic

109

D₇₃

#1592

yes

—

110

C₄×#106

—

111

#1770

—

112

C₂×#335

—

113

C₂×#322

—

114

#1279

—

115

C₇₃

#8286

yes

characteristic, complemented, Sylow

116

C₂×#106

—

117

C₂×C₄⋊C₈

—

118

C₂×#97

—

146

—

119

C₂×C₄×C₈

—

120

C₂×C₄×C₈

—

121

C₂×C₄×C₈

—

122

C₄×C₄⋊C₄

—

123

#340

—

124

#336

—

146

—

125

#335

—

126

#335

—

127

#335

#64408

yes

characteristic, complemented

128

#322

—

129

#322

—

130

C₂²×C₈

—

131

C₂²×C₈

—

132

C₂²×C₈

—

133

C₂²×C₈

—

146

—

134

C₂²×C₈

—

146

—

135

C₂×C₄⋊C₄

—

136

C₂×C₄²

—

137

C₂×C₄²

—

146

—

138

#106

—

146

—

139

C₄⋊C₈

—

140

C₄⋊C₈

—

146

—

141

C₄⋊C₈

C₂×#64408

yes

—

142

#97

—

292

—

143

C₄×C₈

—

144

C₄×C₈

—

145

C₄×C₈

—

146

C₄×C₈

—

147

C₄×C₈

—

148

C₄×C₈

—

149

C₄×C₈

—

146

—

150

C₄×C₈

C₂×#64408

yes

characteristic

151

#95

—

146

—

152

C₂²×C₄

—

153

C₂²×C₄

—

154

C₂²×C₄

—

155

C₂²×C₄

—

146

—

156

C₂×C₈

—

157

C₂×C₈

—

292

—

158

C₂×C₈

—

292

—

159

C₂×C₈

—

292

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

146

—

165

C₂×C₈

—

146

—

166

C₂×C₈

—

146

—

167

C₂×C₈

—

146

—

168

C₂×C₈

—

146

—

169

C₂×C₈

C₄×#64408

yes

characteristic

170

C₂×C₈

C₄×#64408

yes

characteristic

171

C₄⋊C₄

—

146

—

172

C₄⋊C₄

—

146

—

173

C₄²

—

292

—

174

C₄²

—

292

—

175

C₄²

—

176

C₄²

—

177

C₄²

—

178

C₄²

C₂²×#64408

yes

characteristic

179

C₂³

—

180

C₂×C₄

—

292

—

181

C₂×C₄

—

292

—

182

C₂×C₄

—

292

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

146

—

191

C₂×C₄

C₂×C₄×#64408

yes

characteristic

192

C₂×C₄

D₄×#64408

yes

characteristic

193

C₂×C₄

Q₈×#64408

yes

characteristic

194

C₈

—

195

C₈

—

196

C₈

—

292

—

197

C₈

—

292

—

198

C₈

—

199

C₈

—

146

—

200

C₈

—

146

—

201

C₈

#14556645

yes

complemented

202

C₂²

—

203

C₂²

—

204

C₂²

—

205

C₂²

C₄⋊C₄×#64408

yes

characteristic

206

C₄

—

207

C₄

—

584

—

208

C₄

—

209

C₄

—

210

C₄

—

146

—

211

C₄

C₂×#14556645

yes

characteristic

212

C₄

#17989553

yes

characteristic

213

C₄

#17989627

yes

characteristic

214

C₄

#17989628

yes

characteristic

215

C₂

—

216

C₂

#22625728

yes

characteristic

217

C₂

#22625740

yes

characteristic

218

C₂

#22625746

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#64408
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₄²	C₂²×#64408
Fitting	#335×C₇₃	C₄
socle	C₂²×C₇₃	C₄×C₄⋊C₄
prime radical	C₂²×D₇₃	C₄×D₄
cosocle	C₄²×D₇₃	C₂³
hypercenter	#335	#64408
hypocenter	C₇₃	#8286
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

subgroup

G/H

center

C₂×C₄

D₄×#64408

commutator

C₂×C₄×C₇₃

C₂×C₄²

Frattini

C₄²

C₂²×#64408

Fitting

#335×C₇₃

C₄

socle

C₂²×C₇₃

C₄×C₄⋊C₄

prime radical

C₂²×D₇₃

C₄×D₄

cosocle

C₄²×D₇₃

C₂³

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

N	H ≅ G/N	type
13. #17989428	C₄	semidirect
14. #17989428	C₄	semidirect
17. #17988340	C₄	semidirect
18. #17988339	C₄	semidirect
19. #335×C₇₃	C₄	semidirect
88. C₈×C₇₃	#97	semidirect
91. #10596831	#97	semidirect
115. C₇₃	#8286	semidirect
127. #335	#64408	semidirect
201. C₈	#14556645	semidirect

H ≅ G/N

type

13. #17989428

C₄

semidirect

14. #17989428

C₄

semidirect

17. #17988340

C₄

semidirect

18. #17988339

C₄

semidirect

19. #335×C₇₃

C₄

semidirect

88. C₈×C₇₃

#97

semidirect

91. #10596831

#97

semidirect

115. C₇₃

#8286

semidirect

127. #335

#64408

semidirect

201. C₈

#14556645

semidirect

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

subgroup	index
C₄×#14556659	2
#22625728	2
#22625125	2
#22624765	2
#335×D₇₃	2
#8286	73

subgroup

index

#335×D₇₃

Subgroups — #34004029

Conjugacy classes of subgroups

Complements

Maximal subgroups