Group #34007558 — subgroups

Subgroups — #34007558

Order 18688.

310 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 98 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	#22625870	C₂	1	1	yes	maximal, characteristic
2	#22625725	C₂	1	1	yes	maximal, characteristic
3	#22625501	C₂	1	2	yes	maximal
4	#22624828	C₂	1	2	yes	maximal
5	#22608678	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556661	—	2	1	no	—
7	C₂×#14556661	—	2	1	no	—
8	#17989591	C₂²	1	2	yes	—
9	#17989588	C₂²	1	1	yes	characteristic
10	C₂×#14556645	—	2	1	no	—
11	C₄×C₄⋊#64408	C₂²	1	1	yes	characteristic
12	#17989452	—	2	1	no	—
13	#17989451	—	2	2	no	—
14	#17989450	—	2	2	no	—
15	#17988639	C₂²	1	2	yes	—
16	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
17	#17988340	C₄	1	1	yes	characteristic, complemented
18	#17988337	C₄	1	2	yes	complemented
19	#335×C₇₃	C₄	1	1	yes	characteristic, complemented
20	C₂×#11924038	—	2	2	no	—
21	C₂×#11924036	—	2	1	no	—
22	C₂×C₄⋊#64408	—	2	1	no	—
23	C₂×C₄×#64408	—	2	1	no	—
24	#14556661	—	4	2	no	—
25	#14556661	—	4	2	no	—
26	#14556658	—	2	1	no	—
27	C₄⋊#10596833	—	2	2	no	—
28	C₄⋊#10596833	—	2	2	no	—
29	C₄⋊#10596833	D₄	1	2	yes	—
30	#14556645	—	4	2	no	—
31	(C₂×#64408)⋊C₄	Q₈	1	2	yes	—
32	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
33	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
34	C₄²×D₇₃	C₂³	1	1	yes	characteristic
35	C₄⋊#10596831	C₂×C₄	1	2	yes	—
36	#14556503	—	2	2	no	—
37	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
38	C₈×Dic₇₃	—	2	2	no	—
39	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
40	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
41	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×#64408	—	2	2	no	—
44	#11924038	—	4	2	no	—
45	#11924038	—	4	2	no	—
46	#11924036	—	4	1	no	—
47	#11924036	—	4	1	no	—
48	C₂×#10596833	—	2	2	no	—
49	C₂×#10596833	—	4	2	no	—
50	C₄⋊#64408	—	4	1	no	—
51	C₄⋊#64408	—	4	1	no	—
52	C₄×#64408	—	4	1	no	—
53	C₄×#64408	—	4	1	no	—
54	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×D₇₃	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
57	C₂×#10596831	—	2	2	no	—
58	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
59	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
60	C₈×D₇₃	—	2	1	no	—
61	C₈×D₇₃	—	2	1	no	—
62	C₈×D₇₃	D₈	1	2	yes	—
63	C₈×D₇₃	Q₁₆	1	2	yes	—
64	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
65	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
66	C₄×Dic₇₃	#41	1	1	yes	characteristic
67	C₂×C₈×C₇₃	—	2	2	no	—
68	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
70	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×D₇₃	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×#64408	—	4	2	no	—
73	C₂×#64408	—	4	2	no	—
74	C₂×#64408	—	4	2	no	—
75	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
78	C₂×Dic₇₃	#121	1	1	yes	characteristic
79	C₂×Dic₇₃	#125	1	1	yes	characteristic
80	C₂×Dic₇₃	C₄⋊Q₈	1	1	yes	characteristic
81	C₄×D₇₃	—	2	1	no	—
82	C₄×D₇₃	—	2	1	no	—
83	C₄×D₇₃	—	4	1	no	—
84	C₄×D₇₃	C₂×D₈	1	1	yes	characteristic
85	C₄×D₇₃	C₂×Q₁₆	1	1	yes	characteristic
86	C₄×D₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
87	C₄×D₇₃	#137	1	1	yes	characteristic
88	#10596833	—	4	2	no	—
89	#10596833	—	8	2	no	—
90	C₈×C₇₃	—	2	1	no	—
91	C₈×C₇₃	—	4	2	no	—
92	C₈×C₇₃	#107	1	2	yes	—
93	#10596831	—	2	1	no	—
94	#10596831	—	4	2	no	—
95	#10596831	#107	1	2	yes	—
96	C₂²×C₇₃	#389	1	1	yes	characteristic
97	C₂×D₇₃	—	2	1	no	—
98	C₂×D₇₃	—	2	1	no	—
99	C₂×D₇₃	#474	1	1	yes	characteristic
100	C₂×D₇₃	#477	1	1	yes	characteristic
101	#64408	—	8	2	no	—
102	C₄×C₇₃	—	2	1	no	—
103	C₄×C₇₃	C₂×#107	1	1	yes	characteristic
104	C₄×C₇₃	#428	1	1	yes	characteristic
105	C₄×C₇₃	#444	1	1	yes	characteristic
106	C₄×C₇₃	#444	1	1	yes	characteristic
107	Dic₇₃	—	2	1	no	—
108	Dic₇₃	—	2	1	no	—
109	Dic₇₃	#459	1	1	yes	characteristic
110	Dic₇₃	#462	1	1	yes	characteristic
111	#9630	—	73	1	no	maximal, Sylow
112	C₂×C₇₃	#1946	1	1	yes	characteristic
113	C₂×C₇₃	#1947	1	1	yes	characteristic
114	C₂×C₇₃	#1953	1	1	yes	characteristic
115	D₇₃	—	2	1	no	—
116	D₇₃	—	2	1	no	—
117	#1848	—	73	1	no	—
118	#1770	—	73	1	no	—
119	#1571	—	73	2	no	—
120	#1564	—	73	1	no	—
121	#1303	—	73	2	no	—
122	C₇₃	#9630	1	1	yes	characteristic, complemented, Sylow
123	#431	—	73	2	no	—
124	C₂×#108	—	146	1	no	—
125	C₂×#108	—	146	1	no	—
126	#423	—	73	2	no	—
127	C₂×C₄⋊C₈	—	73	1	no	—
128	C₂×#97	—	146	1	no	—
129	C₂×C₄×C₈	—	73	1	no	—
130	C₄×C₄⋊C₄	—	73	1	no	—
131	#336	—	146	1	no	—
132	#335	—	73	1	no	—
133	#335	—	146	2	no	—
134	#335	—	146	2	no	—
135	#335	#64408	1	1	yes	characteristic, complemented
136	#334	—	73	2	no	—
137	C₂×M₁₆	—	146	2	no	—
138	C₂²×C₈	—	73	1	no	—
139	C₂²×C₈	—	73	1	no	—
140	C₂²×C₈	—	146	1	no	—
141	C₂×C₄⋊C₄	—	146	1	no	—
142	C₂×C₄²	—	73	1	no	—
143	C₂×C₄²	—	146	1	no	—
144	#108	—	292	2	no	—
145	#108	—	292	2	no	—
146	C₄⋊C₈	—	73	2	no	—
147	C₄⋊C₈	—	73	2	no	—
148	C₄⋊C₈	—	146	2	no	—
149	C₄⋊C₈	—	146	2	no	—
150	C₄⋊C₈	—	146	1	no	—
151	C₄⋊C₈	C₂×#64408	1	2	yes	—
152	C₂²⋊C₈	—	146	2	no	—
153	#97	—	292	2	no	—
154	C₄×C₈	—	73	1	no	—
155	C₄×C₈	—	73	1	no	—
156	C₄×C₈	—	73	1	no	—
157	C₄×C₈	—	146	2	no	—
158	C₄×C₈	C₂×#64408	1	1	yes	characteristic
159	#95	—	73	2	no	—
160	C₂²×C₄	—	73	1	no	—
161	C₂²×C₄	—	73	1	no	—
162	C₂²×C₄	—	73	1	no	—
163	C₂²×C₄	—	146	2	no	—
164	M₁₆	—	292	2	no	—
165	M₁₆	—	292	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	292	1	no	—
168	C₂×C₈	—	292	1	no	—
169	C₂×C₈	—	292	2	no	—
170	C₂×C₈	—	73	1	no	—
171	C₂×C₈	—	73	2	no	—
172	C₂×C₈	—	73	2	no	—
173	C₂×C₈	—	73	1	no	—
174	C₂×C₈	—	146	1	no	—
175	C₂×C₈	—	146	1	no	—
176	C₂×C₈	—	146	2	no	—
177	C₂×C₈	—	146	2	no	—
178	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
179	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
180	C₄⋊C₄	—	292	1	no	—
181	C₄⋊C₄	—	292	1	no	—
182	C₄²	—	292	1	no	—
183	C₄²	—	292	1	no	—
184	C₄²	—	73	1	no	—
185	C₄²	—	73	1	no	—
186	C₄²	—	73	1	no	—
187	C₄²	C₂²×#64408	1	1	yes	characteristic
188	C₂³	—	73	1	no	—
189	C₂×C₄	—	292	1	no	—
190	C₂×C₄	—	292	2	no	—
191	C₂×C₄	—	292	2	no	—
192	C₂×C₄	—	292	2	no	—
193	C₂×C₄	—	73	1	no	—
194	C₂×C₄	—	73	1	no	—
195	C₂×C₄	—	73	1	no	—
196	C₂×C₄	—	73	1	no	—
197	C₂×C₄	—	73	1	no	—
198	C₂×C₄	—	73	1	no	—
199	C₂×C₄	—	73	1	no	—
200	C₂×C₄	—	146	1	no	—
201	C₂×C₄	—	146	1	no	—
202	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
203	C₂×C₄	D₄×#64408	1	1	yes	characteristic
204	C₂×C₄	Q₈×#64408	1	1	yes	characteristic
205	C₈	—	2	1	no	—
206	C₈	—	4	2	no	—
207	C₈	—	292	2	no	—
208	C₈	—	292	2	no	—
209	C₈	—	584	2	no	—
210	C₈	—	73	2	no	—
211	C₈	—	146	1	no	—
212	C₈	#14556660	1	2	yes	—
213	C₂²	—	73	1	no	—
214	C₂²	—	73	1	no	—
215	C₂²	—	146	1	no	—
216	C₂²	—	146	1	no	—
217	C₂²	#17989538	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	584	2	no	—
220	C₄	—	73	1	no	—
221	C₄	—	73	1	no	—
222	C₄	—	146	1	no	—
223	C₄	—	146	1	no	—
224	C₄	C₂×#14556660	1	1	yes	characteristic
225	C₄	#17989598	1	1	yes	characteristic
226	C₄	#17989633	1	1	yes	characteristic
227	C₄	#17989634	1	1	yes	characteristic
228	C₂	—	146	1	no	—
229	C₂	—	146	1	no	—
230	C₂	#22626040	1	1	yes	characteristic
231	C₂	#22626042	1	1	yes	characteristic
232	C₂	#22626050	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22625870

C₂

yes

maximal, characteristic

#22625725

C₂

yes

maximal, characteristic

#22625501

C₂

yes

maximal

#22624828

C₂

yes

maximal

#22608678

C₂

yes

maximal, characteristic

C₂×#14556661

—

C₂×#14556661

—

#17989591

C₂²

yes

—

#17989588

C₂²

yes

characteristic

C₂×#14556645

—

C₄×C₄⋊#64408

C₂²

yes

—

—

—

C₂²

yes

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988340

C₄

yes

characteristic, complemented

#17988337

C₄

yes

complemented

#335×C₇₃

C₄

yes

characteristic, complemented

C₂×#11924038

—

C₂×#11924036

—

C₂×C₄⋊#64408

—

C₂×C₄×#64408

—

—

—

—

—

—

D₄

yes

—

#14556645

—

(C₂×#64408)⋊C₄

Q₈

yes

—

C₂×C₈×D₇₃

D₄

yes

characteristic

C₂×C₈×D₇₃

D₄

yes

characteristic

C₄²×D₇₃

C₂³

yes

characteristic

C₄⋊#10596831

C₂×C₄

yes

—

#14556503

—

C₄×#10596831

C₂×C₄

yes

characteristic

C₈×Dic₇₃

—

C₈×Dic₇₃

Q₈

yes

characteristic

C₈×Dic₇₃

Q₈

yes

characteristic

C₄⋊C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×#64408

—

—

—

—

—

—

—

—

—

—

—

C₂×C₄×D₇₃

C₂×D₄

yes

characteristic

C₂×C₄×D₇₃

C₂×Q₈

yes

C₂×C₄×D₇₃

yes

—

C₄⋊C₄

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₈×D₇₃

D₈

yes

—

C₈×D₇₃

Q₁₆

yes

—

C₄×Dic₇₃

C₂×D₄

yes

characteristic

C₄×Dic₇₃

C₂×Q₈

yes

characteristic

C₄×Dic₇₃

#41

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₂²×D₇₃

C₂²⋊Q₈

yes

—

—

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₄×D₄

yes

characteristic

C₂×C₄×C₇₃

C₄×Q₈

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

C₄⋊Q₈

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₂×D₈

yes

characteristic

C₄×D₇₃

C₂×Q₁₆

yes

C₄×D₇₃

yes

C₄×D₇₃

yes

—

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

#107

yes

—

—

—

yes

—

C₂²×C₇₃

#389

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

#474

yes

characteristic

100

C₂×D₇₃

#477

yes

characteristic

101

#64408

—

102

C₄×C₇₃

—

103

C₄×C₇₃

C₂×#107

yes

characteristic

104

C₄×C₇₃

#428

yes

characteristic

105

C₄×C₇₃

#444

yes

characteristic

106

C₄×C₇₃

#444

yes

characteristic

107

Dic₇₃

—

108

Dic₇₃

—

109

Dic₇₃

#459

yes

characteristic

110

Dic₇₃

yes

111

—

112

C₂×C₇₃

#1946

yes

characteristic

113

C₂×C₇₃

#1947

yes

characteristic

114

C₂×C₇₃

#1953

yes

characteristic

115

D₇₃

—

116

D₇₃

—

117

#1848

—

118

#1770

—

119

#1571

—

120

#1564

—

121

#1303

—

122

C₇₃

#9630

yes

characteristic, complemented, Sylow

123

#431

—

124

C₂×#108

—

146

—

125

C₂×#108

—

146

—

126

#423

—

127

C₂×C₄⋊C₈

—

128

C₂×#97

—

146

—

129

C₂×C₄×C₈

—

130

C₄×C₄⋊C₄

—

131

#336

—

146

—

132

#335

—

133

#335

—

146

—

134

#335

—

146

—

135

#335

#64408

yes

characteristic, complemented

136

#334

—

137

C₂×M₁₆

—

146

—

138

C₂²×C₈

—

139

C₂²×C₈

—

140

C₂²×C₈

—

146

—

141

C₂×C₄⋊C₄

—

146

—

142

C₂×C₄²

—

143

C₂×C₄²

—

146

—

144

#108

—

292

—

145

#108

—

292

—

146

C₄⋊C₈

—

147

C₄⋊C₈

—

148

C₄⋊C₈

—

146

—

149

C₄⋊C₈

—

146

—

150

C₄⋊C₈

—

146

—

151

C₄⋊C₈

C₂×#64408

yes

—

152

C₂²⋊C₈

—

146

—

153

#97

—

292

—

154

C₄×C₈

—

155

C₄×C₈

—

156

C₄×C₈

—

157

C₄×C₈

—

146

—

158

C₄×C₈

C₂×#64408

yes

characteristic

159

#95

—

160

C₂²×C₄

—

161

C₂²×C₄

—

162

C₂²×C₄

—

163

C₂²×C₄

—

146

—

164

M₁₆

—

292

—

165

M₁₆

—

292

—

166

C₂×C₈

—

167

C₂×C₈

—

292

—

168

C₂×C₈

—

292

—

169

C₂×C₈

—

292

—

170

C₂×C₈

—

171

C₂×C₈

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

146

—

175

C₂×C₈

—

146

—

176

C₂×C₈

—

146

—

177

C₂×C₈

—

146

—

178

C₂×C₈

C₄⋊#64408

yes

characteristic

179

C₂×C₈

C₄⋊#64408

yes

characteristic

180

C₄⋊C₄

—

292

—

181

C₄⋊C₄

—

292

—

182

C₄²

—

292

—

183

C₄²

—

292

—

184

C₄²

—

185

C₄²

—

186

C₄²

—

187

C₄²

C₂²×#64408

yes

characteristic

188

C₂³

—

189

C₂×C₄

—

292

—

190

C₂×C₄

—

292

—

191

C₂×C₄

—

292

—

192

C₂×C₄

—

292

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

146

—

201

C₂×C₄

—

146

—

202

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

203

C₂×C₄

D₄×#64408

yes

characteristic

204

C₂×C₄

Q₈×#64408

yes

characteristic

205

C₈

—

206

C₈

—

207

C₈

—

292

—

208

C₈

—

292

—

209

C₈

—

584

—

210

C₈

—

211

C₈

—

146

—

212

C₈

#14556660

yes

—

213

C₂²

—

214

C₂²

—

215

C₂²

—

146

—

216

C₂²

—

146

—

217

C₂²

#17989538

yes

characteristic

218

C₄

—

219

C₄

—

584

—

220

C₄

—

221

C₄

—

222

C₄

—

146

—

223

C₄

—

146

—

224

C₄

C₂×#14556660

yes

characteristic

225

C₄

#17989598

yes

characteristic

226

C₄

#17989633

yes

characteristic

227

C₄

#17989634

yes

characteristic

228

C₂

—

146

—

229

C₂

—

146

—

230

C₂

#22626040

yes

characteristic

231

C₂

#22626042

yes

characteristic

232

C₂

#22626050

yes

characteristic

	subgroup	G/H
center	C₂²	#17989538
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₄²	C₂²×#64408
Fitting	#335×C₇₃	C₄
socle	C₂²×C₇₃	#389
prime radical	C₂²×D₇₃	C₂²⋊Q₈
cosocle	C₄²×D₇₃	C₂³
hypercenter	#335	#64408
hypocenter	C₇₃	#9630
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

subgroup

G/H

center

C₂²

#17989538

commutator

C₄²×C₇₃

C₂²×C₄

Frattini

C₄²

C₂²×#64408

Fitting

#335×C₇₃

C₄

socle

C₂²×C₇₃

#389

prime radical

C₂²×D₇₃

C₂²⋊Q₈

cosocle

C₄²×D₇₃

C₂³

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #335

N	H ≅ G/N	type
17. #17988340	C₄	semidirect
18. #17988337	C₄	semidirect
19. #335×C₇₃	C₄	semidirect
122. C₇₃	#9630	semidirect
135. #335	#64408	semidirect

H ≅ G/N

type

17. #17988340

C₄

semidirect

18. #17988337

C₄

semidirect

19. #335×C₇₃

C₄

semidirect

122. C₇₃

#9630

semidirect

135. #335

#64408

semidirect

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

subgroup	index
#22625870	2
#22625725	2
#22625501	2
#22624828	2
#22608678	2
#9630	73

subgroup

index

Subgroups — #34007558

Conjugacy classes of subgroups

Complements

Maximal subgroups