Group #34007555 — subgroups

Subgroups — #34007555

Order 18688.

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

normal subgroups only

This group has 322 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	#22625872	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊#64408	C₂	1	1	yes	maximal, characteristic
3	#22625499	C₂	1	2	yes	maximal
4	#22624828	C₂	1	2	yes	maximal
5	#22608678	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556662	—	2	1	no	—
7	C₂×#14556662	—	2	1	no	—
8	#17989591	C₂²	1	2	yes	—
9	#17989588	C₂²	1	1	yes	characteristic
10	C₂×C₈×#64408	—	2	1	no	—
11	C₄×C₄⋊#64408	C₂²	1	1	yes	characteristic
12	#17989452	—	2	1	no	—
13	#17989449	—	2	2	no	—
14	#17989448	—	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	#14556662	—	4	2	no	—
25	#14556662	—	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	1	yes	characteristic
30	C₄⋊#10596833	D₄	1	1	yes	characteristic
31	C₈×#64408	—	4	2	no	—
32	(C₂×#64408)⋊C₄	Q₈	1	2	yes	—
33	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
34	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
35	C₄²×D₇₃	C₂³	1	1	yes	characteristic
36	C₄⋊#10596831	C₂×C₄	1	2	yes	—
37	#14556503	—	2	2	no	—
38	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
39	C₈×Dic₇₃	—	2	2	no	—
40	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
41	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
42	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×#64408	—	2	2	no	—
45	#11924038	—	4	2	no	—
46	#11924038	—	4	2	no	—
47	#11924036	—	4	1	no	—
48	#11924036	—	4	1	no	—
49	C₂×#10596833	—	2	1	no	—
50	C₂×#10596833	—	2	1	no	—
51	C₂×#10596833	—	4	2	no	—
52	C₄⋊#64408	—	2	2	no	—
53	C₄⋊#64408	—	2	2	no	—
54	C₄×#64408	—	4	1	no	—
55	C₄×#64408	—	4	1	no	—
56	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×D₇₃	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
59	C₂×#10596831	—	2	2	no	—
60	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
61	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
62	C₈×D₇₃	—	2	1	no	—
63	C₈×D₇₃	—	2	1	no	—
64	C₈×D₇₃	SD₁₆	1	2	yes	—
65	C₈×D₇₃	SD₁₆	1	2	yes	—
66	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
67	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
68	C₄×Dic₇₃	#41	1	1	yes	characteristic
69	C₂×C₈×C₇₃	—	2	2	no	—
70	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
72	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×D₇₃	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂×#64408	—	2	4	no	—
75	C₂×#64408	—	4	2	no	—
76	C₂×#64408	—	4	2	no	—
77	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
80	C₂×Dic₇₃	#121	1	1	yes	characteristic
81	C₂×Dic₇₃	#125	1	1	yes	characteristic
82	C₂×Dic₇₃	C₄⋊Q₈	1	1	yes	characteristic
83	C₄×D₇₃	—	2	1	no	—
84	C₄×D₇₃	—	2	1	no	—
85	C₄×D₇₃	—	4	1	no	—
86	C₄×D₇₃	C₂×SD₁₆	1	1	yes	characteristic
87	C₄×D₇₃	C₂×SD₁₆	1	1	yes	characteristic
88	C₄×D₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
89	C₄×D₇₃	#137	1	1	yes	characteristic
90	#10596833	—	4	1	no	—
91	#10596833	—	4	1	no	—
92	#10596833	—	8	2	no	—
93	C₈×C₇₃	—	2	1	no	—
94	C₈×C₇₃	—	4	2	no	—
95	C₈×C₇₃	#106	1	2	yes	—
96	#10596831	—	2	1	no	—
97	#10596831	—	4	2	no	—
98	#10596831	#106	1	2	yes	—
99	C₂²×C₇₃	#389	1	1	yes	characteristic
100	C₂×D₇₃	—	2	1	no	—
101	C₂×D₇₃	—	2	1	no	—
102	C₂×D₇₃	#475	1	1	yes	characteristic
103	C₂×D₇₃	#476	1	1	yes	characteristic
104	#64408	—	4	4	no	—
105	C₄×C₇₃	—	2	1	no	—
106	C₄×C₇₃	C₂×#106	1	1	yes	characteristic
107	C₄×C₇₃	#428	1	1	yes	characteristic
108	C₄×C₇₃	#443	1	1	yes	characteristic
109	C₄×C₇₃	#443	1	1	yes	characteristic
110	Dic₇₃	—	2	1	no	—
111	Dic₇₃	—	2	1	no	—
112	Dic₇₃	#460	1	1	yes	characteristic
113	Dic₇₃	#461	1	1	yes	characteristic
114	#9628	—	73	1	no	maximal, Sylow
115	C₂×C₇₃	#1945	1	1	yes	characteristic
116	C₂×C₇₃	#1948	1	1	yes	characteristic
117	C₂×C₇₃	#1953	1	1	yes	characteristic
118	D₇₃	—	2	1	no	—
119	D₇₃	—	2	1	no	—
120	#1848	—	73	1	no	—
121	C₈×C₄⋊C₄	—	73	1	no	—
122	#1570	—	73	2	no	—
123	#1564	—	73	1	no	—
124	#1303	—	73	2	no	—
125	C₇₃	#9628	1	1	yes	characteristic, complemented, Sylow
126	#431	—	73	2	no	—
127	C₂×#108	—	146	1	no	—
128	C₂×#108	—	146	1	no	—
129	#423	—	73	2	no	—
130	C₂×C₄⋊C₈	—	73	1	no	—
131	C₂×C₄×C₈	—	73	1	no	—
132	C₂×C₄×C₈	—	146	1	no	—
133	C₄×C₄⋊C₄	—	73	1	no	—
134	#336	—	146	1	no	—
135	#335	—	73	1	no	—
136	#335	#64408	1	1	yes	characteristic, complemented
137	#334	—	73	2	no	—
138	#334	—	146	2	no	—
139	#334	—	146	2	no	—
140	C₂×M₁₆	—	146	2	no	—
141	C₂²×C₈	—	73	1	no	—
142	C₂²×C₈	—	73	1	no	—
143	C₂²×C₈	—	146	1	no	—
144	C₂×C₄⋊C₄	—	146	1	no	—
145	C₂×C₄²	—	73	1	no	—
146	C₂×C₄²	—	146	1	no	—
147	#108	—	292	2	no	—
148	#108	—	292	2	no	—
149	C₄⋊C₈	—	73	1	no	—
150	C₄⋊C₈	—	73	1	no	—
151	C₄⋊C₈	—	73	2	no	—
152	C₄⋊C₈	—	146	2	no	—
153	C₄⋊C₈	—	146	2	no	—
154	C₄⋊C₈	—	146	1	no	—
155	C₄⋊C₈	C₂×#64408	1	2	yes	—
156	C₂²⋊C₈	—	146	2	no	—
157	C₄×C₈	—	292	2	no	—
158	C₄×C₈	—	73	1	no	—
159	C₄×C₈	—	73	1	no	—
160	C₄×C₈	—	73	1	no	—
161	C₄×C₈	—	146	2	no	—
162	C₄×C₈	C₂×#64408	1	1	yes	characteristic
163	#95	—	73	2	no	—
164	C₂²×C₄	—	73	1	no	—
165	C₂²×C₄	—	73	1	no	—
166	C₂²×C₄	—	73	1	no	—
167	C₂²×C₄	—	146	2	no	—
168	M₁₆	—	292	2	no	—
169	M₁₆	—	292	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	292	1	no	—
172	C₂×C₈	—	292	1	no	—
173	C₂×C₈	—	292	2	no	—
174	C₂×C₈	—	73	1	no	—
175	C₂×C₈	—	73	2	no	—
176	C₂×C₈	—	73	2	no	—
177	C₂×C₈	—	73	1	no	—
178	C₂×C₈	—	146	1	no	—
179	C₂×C₈	—	146	1	no	—
180	C₂×C₈	—	146	1	no	—
181	C₂×C₈	—	146	1	no	—
182	C₂×C₈	—	146	2	no	—
183	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
184	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
185	C₄⋊C₄	—	146	2	no	—
186	C₄⋊C₄	—	146	2	no	—
187	C₄²	—	292	1	no	—
188	C₄²	—	292	1	no	—
189	C₄²	—	73	1	no	—
190	C₄²	—	73	1	no	—
191	C₄²	—	73	1	no	—
192	C₄²	C₂²×#64408	1	1	yes	characteristic
193	C₂³	—	73	1	no	—
194	C₂×C₄	—	292	1	no	—
195	C₂×C₄	—	292	2	no	—
196	C₂×C₄	—	292	2	no	—
197	C₂×C₄	—	73	1	no	—
198	C₂×C₄	—	73	1	no	—
199	C₂×C₄	—	73	1	no	—
200	C₂×C₄	—	73	1	no	—
201	C₂×C₄	—	73	1	no	—
202	C₂×C₄	—	73	1	no	—
203	C₂×C₄	—	73	1	no	—
204	C₂×C₄	—	146	4	no	—
205	C₂×C₄	—	146	1	no	—
206	C₂×C₄	—	146	1	no	—
207	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
208	C₂×C₄	D₄×#64408	1	1	yes	characteristic
209	C₂×C₄	Q₈×#64408	1	1	yes	characteristic
210	C₈	—	2	1	no	—
211	C₈	—	4	2	no	—
212	C₈	—	292	1	no	—
213	C₈	—	292	1	no	—
214	C₈	—	292	2	no	—
215	C₈	—	584	2	no	—
216	C₈	—	73	2	no	—
217	C₈	—	146	1	no	—
218	C₈	#14556659	1	2	yes	—
219	C₂²	—	73	1	no	—
220	C₂²	—	73	1	no	—
221	C₂²	—	146	1	no	—
222	C₂²	—	146	1	no	—
223	C₂²	#17989538	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	292	4	no	—
226	C₄	—	73	1	no	—
227	C₄	—	73	1	no	—
228	C₄	—	146	1	no	—
229	C₄	—	146	1	no	—
230	C₄	C₂×#14556659	1	1	yes	characteristic
231	C₄	#17989598	1	1	yes	characteristic
232	C₄	#17989629	1	1	yes	characteristic
233	C₄	#17989631	1	1	yes	characteristic
234	C₂	—	146	1	no	—
235	C₂	—	146	1	no	—
236	C₂	#22626039	1	1	yes	characteristic
237	C₂	#22626043	1	1	yes	characteristic
238	C₂	#22626051	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22625872

C₂

yes

maximal, characteristic

C₈×C₄⋊#64408

C₂

yes

maximal, characteristic

#22625499

C₂

yes

maximal

#22624828

C₂

yes

maximal

#22608678

C₂

yes

maximal, characteristic

C₂×#14556662

—

C₂×#14556662

—

#17989591

C₂²

yes

—

#17989588

C₂²

yes

characteristic

C₂×C₈×#64408

—

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

characteristic

C₄⋊#10596833

D₄

yes

characteristic

C₈×#64408

—

(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

SD₁₆

yes

—

C₈×D₇₃

SD₁₆

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×SD₁₆

yes

characteristic

C₄×D₇₃

C₂×SD₁₆

yes

C₄×D₇₃

yes

C₄×D₇₃

yes

—

—

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

#106

yes

—

—

—

yes

—

C₂²×C₇₃

#389

yes

characteristic

100

C₂×D₇₃

—

101

C₂×D₇₃

—

102

C₂×D₇₃

#475

yes

characteristic

103

C₂×D₇₃

#476

yes

characteristic

104

#64408

—

105

C₄×C₇₃

—

106

C₄×C₇₃

C₂×#106

yes

characteristic

107

C₄×C₇₃

#428

yes

characteristic

108

C₄×C₇₃

#443

yes

characteristic

109

C₄×C₇₃

#443

yes

characteristic

110

Dic₇₃

—

111

Dic₇₃

—

112

Dic₇₃

#460

yes

characteristic

113

Dic₇₃

yes

114

—

115

C₂×C₇₃

#1945

yes

characteristic

116

C₂×C₇₃

#1948

yes

characteristic

117

C₂×C₇₃

#1953

yes

characteristic

118

D₇₃

—

119

D₇₃

—

120

#1848

—

121

C₈×C₄⋊C₄

—

122

#1570

—

123

#1564

—

124

#1303

—

125

C₇₃

#9628

yes

characteristic, complemented, Sylow

126

#431

—

127

C₂×#108

—

146

—

128

C₂×#108

—

146

—

129

#423

—

130

C₂×C₄⋊C₈

—

131

C₂×C₄×C₈

—

132

C₂×C₄×C₈

—

146

—

133

C₄×C₄⋊C₄

—

134

#336

—

146

—

135

#335

—

136

#335

#64408

yes

characteristic, complemented

137

#334

—

138

#334

—

146

—

139

#334

—

146

—

140

C₂×M₁₆

—

146

—

141

C₂²×C₈

—

142

C₂²×C₈

—

143

C₂²×C₈

—

146

—

144

C₂×C₄⋊C₄

—

146

—

145

C₂×C₄²

—

146

C₂×C₄²

—

146

—

147

#108

—

292

—

148

#108

—

292

—

149

C₄⋊C₈

—

150

C₄⋊C₈

—

151

C₄⋊C₈

—

152

C₄⋊C₈

—

146

—

153

C₄⋊C₈

—

146

—

154

C₄⋊C₈

—

146

—

155

C₄⋊C₈

C₂×#64408

yes

—

156

C₂²⋊C₈

—

146

—

157

C₄×C₈

—

292

—

158

C₄×C₈

—

159

C₄×C₈

—

160

C₄×C₈

—

161

C₄×C₈

—

146

—

162

C₄×C₈

C₂×#64408

yes

characteristic

163

#95

—

164

C₂²×C₄

—

165

C₂²×C₄

—

166

C₂²×C₄

—

167

C₂²×C₄

—

146

—

168

M₁₆

—

292

—

169

M₁₆

—

292

—

170

C₂×C₈

—

171

C₂×C₈

—

292

—

172

C₂×C₈

—

292

—

173

C₂×C₈

—

292

—

174

C₂×C₈

—

175

C₂×C₈

—

176

C₂×C₈

—

177

C₂×C₈

—

178

C₂×C₈

—

146

—

179

C₂×C₈

—

146

—

180

C₂×C₈

—

146

—

181

C₂×C₈

—

146

—

182

C₂×C₈

—

146

—

183

C₂×C₈

C₄⋊#64408

yes

characteristic

184

C₂×C₈

C₄⋊#64408

yes

characteristic

185

C₄⋊C₄

—

146

—

186

C₄⋊C₄

—

146

—

187

C₄²

—

292

—

188

C₄²

—

292

—

189

C₄²

—

190

C₄²

—

191

C₄²

—

192

C₄²

C₂²×#64408

yes

characteristic

193

C₂³

—

194

C₂×C₄

—

292

—

195

C₂×C₄

—

292

—

196

C₂×C₄

—

292

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

146

—

205

C₂×C₄

—

146

—

206

C₂×C₄

—

146

—

207

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

208

C₂×C₄

D₄×#64408

yes

characteristic

209

C₂×C₄

Q₈×#64408

yes

characteristic

210

C₈

—

211

C₈

—

212

C₈

—

292

—

213

C₈

—

292

—

214

C₈

—

292

—

215

C₈

—

584

—

216

C₈

—

217

C₈

—

146

—

218

C₈

#14556659

yes

—

219

C₂²

—

220

C₂²

—

221

C₂²

—

146

—

222

C₂²

—

146

—

223

C₂²

#17989538

yes

characteristic

224

C₄

—

225

C₄

—

292

—

226

C₄

—

227

C₄

—

228

C₄

—

146

—

229

C₄

—

146

—

230

C₄

C₂×#14556659

yes

characteristic

231

C₄

#17989598

yes

characteristic

232

C₄

#17989629

yes

characteristic

233

C₄

#17989631

yes

characteristic

234

C₂

—

146

—

235

C₂

—

146

—

236

C₂

#22626039

yes

characteristic

237

C₂

#22626043

yes

characteristic

238

C₂

#22626051

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	#9628
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
125. C₇₃	#9628	semidirect
136. #335	#64408	semidirect

H ≅ G/N

type

17. #17988340

C₄

semidirect

18. #17988337

C₄

semidirect

19. #335×C₇₃

C₄

semidirect

125. C₇₃

#9628

semidirect

136. #335

#64408

semidirect

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

subgroup	index
#22625872	2
C₈×C₄⋊#64408	2
#22625499	2
#22624828	2
#22608678	2
#9628	73

subgroup

index

C₈×C₄⋊#64408

Subgroups — #34007555

Conjugacy classes of subgroups

Complements

Maximal subgroups