Group #34004017 — subgroups

Subgroups — #34004017

Order 18688.

414 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 140 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	C₈×#11924036	C₂	1	1	yes	maximal, characteristic
4	#22624765	C₂	1	2	yes	maximal
5	#334×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	C₈×#10596833	C₄	1	2	yes	complemented
14	C₈×#10596833	C₄	1	2	yes	complemented
15	C₄⋊C₈×D₇₃	C₂²	1	2	yes	—
16	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
17	#17988338	C₄	1	1	yes	characteristic, complemented
18	#17988337	C₄	1	2	yes	complemented
19	#334×C₇₃	C₄	1	1	yes	characteristic, complemented
20	C₂×#11924036	—	2	1	no	—
21	C₂×#11924036	D₄	1	1	yes	characteristic
22	C₂×#11924036	D₄	1	1	yes	characteristic
23	C₂×C₄⋊#64408	C₂×C₄	1	4	yes	—
24	C₂×C₄×#64408	—	2	2	no	—
25	#14556659	—	2	4	no	—
26	#14556645	—	4	4	no	—
27	C₄×#10596833	—	2	1	no	—
28	C₄×#10596833	C₂×C₄	1	2	yes	—
29	C₄×#10596833	C₂×C₄	1	2	yes	—
30	C₄×#10596833	Q₈	1	2	yes	—
31	(C₂×#64408)⋊C₄	—	2	2	no	—
32	C₂×C₈×D₇₃	—	2	2	no	—
33	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
35	C₄²×D₇₃	C₂³	1	1	yes	characteristic
36	C₄⋊#10596831	C₂×C₄	1	2	yes	—
37	#14556520	—	2	2	no	—
38	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
39	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
40	C₈×Dic₇₃	C₂×C₄	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	4	no	—
44	#11924036	—	2	4	no	—
45	#11924036	—	2	2	no	—
46	#11924036	SD₁₆	1	4	yes	—
47	C₂×#10596833	—	2	2	no	—
48	C₂×#10596833	C₄⋊C₄	1	2	yes	—
49	C₂×#10596833	C₄⋊C₄	1	2	yes	—
50	C₄⋊#64408	—	2	4	no	—
51	C₄⋊#64408	—	2	4	no	—
52	C₄×#64408	—	4	2	no	—
53	C₄×#64408	—	4	2	no	—
54	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
57	C₂×#10596831	—	2	2	no	—
58	C₂×#10596831	C₄²	1	1	yes	characteristic
59	C₂×#10596831	C₄²	1	1	yes	characteristic
60	C₈×D₇₃	—	2	2	no	—
61	C₈×D₇₃	—	4	2	no	—
62	C₈×D₇₃	—	4	2	no	—
63	C₈×D₇₃	M₁₆	1	4	yes	—
64	C₄×Dic₇₃	C₂²×C₄	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	C₄²	1	1	yes	characteristic
69	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
70	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×D₇₃	C₄×D₄	1	1	yes	characteristic
72	C₂×#64408	—	4	4	no	—
73	C₂×#64408	—	4	4	no	—
74	C₂×#64408	—	4	4	no	—
75	C₂×C₄×C₇₃	C₂×C₄²	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	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×Dic₇₃	#117	1	1	yes	characteristic
80	C₂×Dic₇₃	C₄×Q₈	1	1	yes	characteristic
81	C₄×D₇₃	—	2	2	no	—
82	C₄×D₇₃	C₂×M₁₆	1	2	yes	—
83	C₄×D₇₃	#131	1	2	yes	—
84	C₄×D₇₃	C₂×SD₁₆	1	2	yes	—
85	C₄×D₇₃	#135	1	2	yes	—
86	#10596833	—	2	4	no	—
87	#10596833	—	2	2	no	—
88	#10596833	#106	1	4	yes	complemented
89	C₈×C₇₃	—	2	1	no	—
90	C₈×C₇₃	—	4	2	no	—
91	C₈×C₇₃	#97	1	2	yes	complemented
92	#10596831	—	2	1	no	—
93	#10596831	—	4	2	no	—
94	#10596831	#97	1	2	yes	complemented
95	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
96	C₂×D₇₃	#436	1	2	yes	—
97	C₂×D₇₃	C₄×SD₁₆	1	2	yes	—
98	C₂×D₇₃	#443	1	2	yes	—
99	#64408	—	8	4	no	—
100	C₄×C₇₃	—	2	1	no	—
101	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
102	C₄×C₇₃	#405	1	1	yes	characteristic
103	C₄×C₇₃	C₄×SD₁₆	1	2	yes	—
104	Dic₇₃	—	2	1	no	—
105	Dic₇₃	C₂×#106	1	1	yes	characteristic
106	Dic₇₃	#427	1	1	yes	characteristic
107	Dic₇₃	#446	1	2	yes	—
108	#8283	—	73	1	no	maximal, Sylow
109	C₂×C₇₃	#1770	1	1	yes	characteristic
110	C₂×C₇₃	#1772	1	1	yes	characteristic
111	C₂×C₇₃	C₄×#106	1	1	yes	characteristic
112	D₇₃	#1590	1	4	yes	—
113	C₄×#106	—	73	1	no	—
114	#1770	—	73	2	no	—
115	C₂×#334	—	73	1	no	—
116	C₂×C₈²	—	73	1	no	—
117	#1279	—	73	2	no	—
118	C₇₃	#8283	1	1	yes	characteristic, complemented, Sylow
119	C₂×#106	—	73	2	no	—
120	C₂×C₄⋊C₈	—	73	2	no	—
121	C₂×#97	—	146	2	no	—
122	C₂×C₄×C₈	—	73	1	no	—
123	C₂×C₄×C₈	—	73	1	no	—
124	C₂×C₄×C₈	—	73	1	no	—
125	C₄×C₄⋊C₄	—	73	2	no	—
126	#340	—	73	2	no	—
127	#336	—	146	2	no	—
128	#334	—	73	2	no	—
129	#334	—	73	1	no	—
130	#334	#64408	1	1	yes	characteristic, complemented
131	C₈²	—	73	2	no	—
132	C₈²	—	73	2	no	—
133	C₂²×C₈	—	73	1	no	—
134	C₂²×C₈	—	73	1	no	—
135	C₂²×C₈	—	73	1	no	—
136	C₂²×C₈	—	73	1	no	—
137	C₂²×C₈	—	146	2	no	—
138	C₂²×C₈	—	146	1	no	—
139	C₂×C₄⋊C₄	—	73	4	no	—
140	C₂×C₄²	—	73	1	no	—
141	C₂×C₄²	—	146	2	no	—
142	#106	—	146	4	no	—
143	C₄⋊C₈	—	73	2	no	—
144	C₄⋊C₈	—	146	2	no	—
145	C₄⋊C₈	C₂×#64408	1	2	yes	—
146	#97	—	292	4	no	—
147	C₄×C₈	—	73	2	no	—
148	C₄×C₈	—	73	2	no	—
149	C₄×C₈	—	73	1	no	—
150	C₄×C₈	—	73	1	no	—
151	C₄×C₈	—	73	2	no	—
152	C₄×C₈	—	73	1	no	—
153	C₄×C₈	—	146	1	no	—
154	C₄×C₈	C₂×#64408	1	1	yes	characteristic
155	#95	—	146	2	no	—
156	C₂²×C₄	—	73	1	no	—
157	C₂²×C₄	—	73	1	no	—
158	C₂²×C₄	—	73	1	no	—
159	C₂²×C₄	—	146	4	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	292	2	no	—
162	C₂×C₈	—	292	2	no	—
163	C₂×C₈	—	73	2	no	—
164	C₂×C₈	—	73	4	no	—
165	C₂×C₈	—	73	1	no	—
166	C₂×C₈	—	73	4	no	—
167	C₂×C₈	—	73	2	no	—
168	C₂×C₈	—	73	1	no	—
169	C₂×C₈	—	146	2	no	—
170	C₂×C₈	—	146	2	no	—
171	C₂×C₈	—	146	2	no	—
172	C₂×C₈	—	146	2	no	—
173	C₂×C₈	—	146	4	no	—
174	C₂×C₈	C₄×#64408	1	1	yes	characteristic
175	C₂×C₈	C₄×#64408	1	1	yes	characteristic
176	C₄⋊C₄	—	146	4	no	—
177	C₄⋊C₄	—	146	4	no	—
178	C₄²	—	292	2	no	—
179	C₄²	—	292	2	no	—
180	C₄²	—	73	1	no	—
181	C₄²	—	73	1	no	—
182	C₄²	—	73	1	no	—
183	C₄²	C₂²×#64408	1	1	yes	characteristic
184	C₂³	—	73	1	no	—
185	C₂×C₄	—	292	4	no	—
186	C₂×C₄	—	292	4	no	—
187	C₂×C₄	—	292	4	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	2	no	—
193	C₂×C₄	—	73	2	no	—
194	C₂×C₄	—	73	1	no	—
195	C₂×C₄	—	146	2	no	—
196	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
197	C₂×C₄	D₄×#64408	1	1	yes	characteristic
198	C₂×C₄	Q₈×#64408	1	1	yes	characteristic
199	C₈	—	2	1	no	—
200	C₈	—	4	2	no	—
201	C₈	—	292	2	no	—
202	C₈	—	73	4	no	—
203	C₈	—	73	2	no	—
204	C₈	—	146	2	no	—
205	C₈	—	146	4	no	—
206	C₈	—	146	1	no	—
207	C₈	#14556645	1	2	yes	complemented
208	C₂²	—	73	2	no	—
209	C₂²	—	73	2	no	—
210	C₂²	—	73	2	no	—
211	C₂²	C₄⋊C₄×#64408	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	584	4	no	—
214	C₄	—	73	2	no	—
215	C₄	—	73	1	no	—
216	C₄	—	73	1	no	—
217	C₄	—	146	1	no	—
218	C₄	C₂×#14556645	1	1	yes	characteristic
219	C₄	#17989553	1	1	yes	characteristic
220	C₄	SD₁₆×#64408	1	2	yes	—
221	C₂	—	73	4	no	—
222	C₂	#22625728	1	1	yes	characteristic
223	C₂	#22625736	1	1	yes	characteristic
224	C₂	#106×#64408	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14556659

C₂

yes

maximal, characteristic

#22625728

C₂

yes

maximal

C₈×#11924036

C₂

yes

maximal, characteristic

#22624765

C₂

yes

maximal

#334×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

—

C₈×#10596833

C₄

yes

complemented

C₈×#10596833

C₄

yes

complemented

C₄⋊C₈×D₇₃

C₂²

yes

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988338

C₄

yes

characteristic, complemented

#17988337

C₄

yes

complemented

#334×C₇₃

C₄

yes

characteristic, complemented

C₂×#11924036

—

C₂×#11924036

D₄

yes

characteristic

C₂×#11924036

D₄

yes

characteristic

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

—

#11924036

—

#11924036

—

#11924036

SD₁₆

yes

—

C₂×#10596833

—

C₂×#10596833

C₄⋊C₄

yes

—

C₂×#10596833

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

C₂×SD₁₆

yes

—

C₄×D₇₃

#135

yes

—

—

—

yes

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

yes

—

—

yes

C₂²×C₇₃

C₄×C₄⋊C₄

yes

characteristic

C₂×D₇₃

#436

yes

—

C₂×D₇₃

C₄×SD₁₆

yes

—

C₂×D₇₃

#443

yes

—

#64408

—

100

C₄×C₇₃

—

101

C₄×C₇₃

C₂×#97

yes

characteristic

102

C₄×C₇₃

#405

yes

characteristic

103

C₄×C₇₃

C₄×SD₁₆

yes

—

104

Dic₇₃

—

105

Dic₇₃

C₂×#106

yes

characteristic

106

Dic₇₃

#427

yes

characteristic

107

Dic₇₃

#446

yes

—

108

#8283

—

maximal, Sylow

109

C₂×C₇₃

#1770

yes

characteristic

110

C₂×C₇₃

#1772

yes

characteristic

111

C₂×C₇₃

C₄×#106

yes

characteristic

112

D₇₃

#1590

yes

—

113

C₄×#106

—

114

#1770

—

115

C₂×#334

—

116

C₂×C₈²

—

117

#1279

—

118

C₇₃

#8283

yes

characteristic, complemented, Sylow

119

C₂×#106

—

120

C₂×C₄⋊C₈

—

121

C₂×#97

—

146

—

122

C₂×C₄×C₈

—

123

C₂×C₄×C₈

—

124

C₂×C₄×C₈

—

125

C₄×C₄⋊C₄

—

126

#340

—

127

#336

—

146

—

128

#334

—

129

#334

—

130

#334

#64408

yes

characteristic, complemented

131

C₈²

—

132

C₈²

—

133

C₂²×C₈

—

134

C₂²×C₈

—

135

C₂²×C₈

—

136

C₂²×C₈

—

137

C₂²×C₈

—

146

—

138

C₂²×C₈

—

146

—

139

C₂×C₄⋊C₄

—

140

C₂×C₄²

—

141

C₂×C₄²

—

146

—

142

#106

—

146

—

143

C₄⋊C₈

—

144

C₄⋊C₈

—

146

—

145

C₄⋊C₈

C₂×#64408

yes

—

146

#97

—

292

—

147

C₄×C₈

—

148

C₄×C₈

—

149

C₄×C₈

—

150

C₄×C₈

—

151

C₄×C₈

—

152

C₄×C₈

—

153

C₄×C₈

—

146

—

154

C₄×C₈

C₂×#64408

yes

characteristic

155

#95

—

146

—

156

C₂²×C₄

—

157

C₂²×C₄

—

158

C₂²×C₄

—

159

C₂²×C₄

—

146

—

160

C₂×C₈

—

161

C₂×C₈

—

292

—

162

C₂×C₈

—

292

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

146

—

170

C₂×C₈

—

146

—

171

C₂×C₈

—

146

—

172

C₂×C₈

—

146

—

173

C₂×C₈

—

146

—

174

C₂×C₈

C₄×#64408

yes

characteristic

175

C₂×C₈

C₄×#64408

yes

characteristic

176

C₄⋊C₄

—

146

—

177

C₄⋊C₄

—

146

—

178

C₄²

—

292

—

179

C₄²

—

292

—

180

C₄²

—

181

C₄²

—

182

C₄²

—

183

C₄²

C₂²×#64408

yes

characteristic

184

C₂³

—

185

C₂×C₄

—

292

—

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

—

195

C₂×C₄

—

146

—

196

C₂×C₄

C₂×C₄×#64408

yes

characteristic

197

C₂×C₄

D₄×#64408

yes

characteristic

198

C₂×C₄

Q₈×#64408

yes

characteristic

199

C₈

—

200

C₈

—

201

C₈

—

292

—

202

C₈

—

203

C₈

—

204

C₈

—

146

—

205

C₈

—

146

—

206

C₈

—

146

—

207

C₈

#14556645

yes

complemented

208

C₂²

—

209

C₂²

—

210

C₂²

—

211

C₂²

C₄⋊C₄×#64408

yes

characteristic

212

C₄

—

213

C₄

—

584

—

214

C₄

—

215

C₄

—

216

C₄

—

217

C₄

—

146

—

218

C₄

C₂×#14556645

yes

characteristic

219

C₄

#17989553

yes

characteristic

220

C₄

SD₁₆×#64408

yes

—

221

C₂

—

222

C₂

#22625728

yes

characteristic

223

C₂

#22625736

yes

characteristic

224

C₂

#106×#64408

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#64408
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₄²	C₂²×#64408
Fitting	#334×C₇₃	C₄
socle	C₂²×C₇₃	C₄×C₄⋊C₄
prime radical	C₂²×D₇₃	C₄×D₄
cosocle	C₄²×D₇₃	C₂³
hypercenter	#334	#64408
hypocenter	C₇₃	#8283
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

subgroup

G/H

center

C₂×C₄

D₄×#64408

commutator

C₂×C₄×C₇₃

C₂×C₄²

Frattini

C₄²

C₂²×#64408

Fitting

#334×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 ◁ #334

N	H ≅ G/N	type
13. C₈×#10596833	C₄	semidirect
14. C₈×#10596833	C₄	semidirect
17. #17988338	C₄	semidirect
18. #17988337	C₄	semidirect
19. #334×C₇₃	C₄	semidirect
88. #10596833	#106	semidirect
91. C₈×C₇₃	#97	semidirect
94. #10596831	#97	semidirect
118. C₇₃	#8283	semidirect
130. #334	#64408	semidirect
207. C₈	#14556645	semidirect

H ≅ G/N

type

13. C₈×#10596833

C₄

semidirect

14. C₈×#10596833

C₄

semidirect

17. #17988338

C₄

semidirect

18. #17988337

C₄

semidirect

19. #334×C₇₃

C₄

semidirect

88. #10596833

#106

semidirect

91. C₈×C₇₃

#97

semidirect

94. #10596831

#97

semidirect

118. C₇₃

#8283

semidirect

130. #334

#64408

semidirect

207. 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
C₈×#11924036	2
#22624765	2
#334×D₇₃	2
#8283	73

subgroup

index

#334×D₇₃

Subgroups — #34004017

Conjugacy classes of subgroups

Complements

Maximal subgroups