Group #34007485 — subgroups

Subgroups — #34007485

Order 18688.

378 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 124 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×#17989448	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	#17989448	—	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	2	yes	—
21	#14556658	—	4	4	no	—
22	C₄⋊#10596833	—	2	2	no	—
23	C₄⋊#10596833	—	2	2	no	—
24	C₄×#10596833	—	2	1	no	—
25	C₄×#10596833	—	2	1	no	—
26	C₄×#10596833	—	2	1	no	—
27	C₄×#10596833	Q₈	1	2	yes	—
28	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₇₃	—	2	2	no	—
31	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
32	#14556522	—	2	4	no	—
33	C₈×Dic₇₃	D₄	1	2	yes	—
34	C₈×Dic₇₃	Q₈	1	2	yes	—
35	#14556498	—	2	2	no	—
36	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
38	C₂×#10596833	—	2	2	no	—
39	C₂×#10596833	—	2	2	no	—
40	C₂×#10596833	—	2	2	no	—
41	C₂×#10596833	—	4	1	no	—
42	C₂×#10596833	—	4	1	no	—
43	C₂×#10596833	—	4	2	no	—
44	C₂×#10596833	—	4	2	no	—
45	C₂×#10596833	—	4	2	no	—
46	C₂×#10596833	SD₁₆	1	4	yes	—
47	C₂²×Dic₇₃	—	2	2	no	—
48	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₇₃	#41	1	1	yes	characteristic
50	C₂×#10596831	—	2	2	no	—
51	C₂×#10596831	C₄⋊C₄	1	2	yes	—
52	C₄⋊Dic₇₃	—	2	2	no	—
53	C₄⋊Dic₇₃	M₁₆	1	4	yes	—
54	C₄×Dic₇₃	—	4	2	no	—
55	C₄×Dic₇₃	—	4	2	no	—
56	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
57	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
58	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
59	C₄×Dic₇₃	#41	1	1	yes	characteristic
60	C₂²×C₄×C₇₃	—	2	2	no	—
61	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₇₃	—	2	2	no	—
63	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
64	C₄⋊C₄×C₇₃	—	2	2	no	—
65	C₄⋊C₄×C₇₃	—	2	2	no	—
66	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₇₃	—	4	2	no	—
68	C₂×C₄×C₇₃	—	4	2	no	—
69	C₂×C₄×C₇₃	—	4	2	no	—
70	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
73	C₂×C₄×C₇₃	#131	1	2	yes	—
74	C₂×Dic₇₃	—	2	1	no	—
75	C₂×Dic₇₃	—	2	1	no	—
76	C₂×Dic₇₃	—	2	4	no	—
77	C₂×Dic₇₃	—	4	2	no	—
78	C₂×Dic₇₃	—	4	2	no	—
79	C₂×Dic₇₃	#121	1	1	yes	characteristic
80	C₂×Dic₇₃	C₂²⋊Q₈	1	1	yes	characteristic
81	C₂×Dic₇₃	#125	1	1	yes	characteristic
82	C₂×Dic₇₃	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×Dic₇₃	C₂×SD₁₆	1	2	yes	—
84	C₂×Dic₇₃	#135	1	2	yes	—
85	#10596833	—	2	4	no	—
86	#10596833	—	4	2	no	—
87	#10596833	—	8	2	no	—
88	C₈×C₇₃	—	2	2	no	—
89	#10596831	—	2	2	no	—
90	C₂²×C₇₃	#389	1	1	yes	characteristic
91	C₂²×C₇₃	#435	1	2	yes	—
92	C₂²×C₇₃	C₄×SD₁₆	1	2	yes	—
93	C₂²×C₇₃	#444	1	1	yes	characteristic
94	C₂²×C₇₃	#444	1	1	yes	characteristic
95	C₄×C₇₃	—	8	2	no	—
96	C₄×C₇₃	#423	1	1	yes	characteristic
97	C₄×C₇₃	#424	1	1	yes	characteristic
98	C₄×C₇₃	#446	1	2	yes	—
99	Dic₇₃	—	2	1	no	—
100	Dic₇₃	—	2	1	no	—
101	Dic₇₃	—	4	4	no	—
102	Dic₇₃	#465	1	2	yes	—
103	Dic₇₃	#498	1	1	yes	characteristic
104	Dic₇₃	#499	1	1	yes	characteristic
105	#9597	—	73	1	no	maximal, Sylow
106	C₂×C₇₃	#1582	1	2	yes	—
107	C₂×C₇₃	#1583	1	2	yes	—
108	C₂×C₇₃	#1940	1	1	yes	characteristic
109	C₂×C₇₃	#1941	1	1	yes	characteristic
110	C₂×C₇₃	#1946	1	1	yes	characteristic
111	#1939	—	73	2	no	—
112	C₄×#106	—	73	1	no	—
113	C₂×#334	—	73	1	no	—
114	C₂×#322	—	73	1	no	—
115	#1279	—	73	2	no	—
116	C₇₃	#9597	1	1	yes	characteristic, complemented, Sylow
117	C₂×#106	—	73	2	no	—
118	C₂×C₄⋊C₈	—	73	2	no	—
119	C₂×C₄⋊C₈	—	146	2	no	—
120	C₂×C₄×C₈	—	73	1	no	—
121	C₂×C₄×C₈	—	73	1	no	—
122	C₂×C₄×C₈	—	73	1	no	—
123	C₄×C₄⋊C₄	—	73	2	no	—
124	#340	—	73	1	no	—
125	#340	#64408	1	1	yes	characteristic
126	#336	—	146	2	no	—
127	#334	—	146	2	no	—
128	#322	—	146	2	no	—
129	C₂²×C₈	—	73	2	no	—
130	C₂²×C₈	—	73	1	no	—
131	C₂²×C₈	—	146	2	no	—
132	C₂²×C₈	—	146	1	no	—
133	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
134	C₂×C₄⋊C₄	—	73	2	no	—
135	C₂×C₄⋊C₄	C₂×#64408	1	2	yes	—
136	C₂×C₄²	—	73	1	no	—
137	C₂×C₄²	—	146	2	no	—
138	#106	—	146	4	no	—
139	C₄⋊C₈	—	292	4	no	—
140	C₄⋊C₈	—	146	2	no	—
141	C₄⋊C₈	—	146	2	no	—
142	C₄×C₈	—	73	2	no	—
143	C₄×C₈	—	73	2	no	—
144	C₄×C₈	—	73	2	no	—
145	C₄×C₈	—	146	1	no	—
146	C₄×C₈	—	146	1	no	—
147	C₄×C₈	—	146	1	no	—
148	#95	—	146	2	no	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	—	73	1	no	—
151	C₂²×C₄	—	73	1	no	—
152	C₂²×C₄	—	146	2	no	—
153	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	292	1	no	—
156	C₂×C₈	—	292	1	no	—
157	C₂×C₈	—	292	2	no	—
158	C₂×C₈	—	292	2	no	—
159	C₂×C₈	—	292	2	no	—
160	C₂×C₈	—	73	4	no	—
161	C₂×C₈	—	73	2	no	—
162	C₂×C₈	—	146	2	no	—
163	C₂×C₈	—	146	2	no	—
164	C₂×C₈	—	146	2	no	—
165	C₂×C₈	—	146	2	no	—
166	C₂×C₈	C₄⋊#64408	1	2	yes	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	73	4	no	—
170	C₄⋊C₄	—	146	2	no	—
171	C₄²	—	292	2	no	—
172	C₄²	—	292	2	no	—
173	C₄²	—	73	1	no	—
174	C₄²	—	73	1	no	—
175	C₄²	—	73	1	no	—
176	C₄²	—	73	1	no	—
177	C₂³	D₄×#64408	1	1	yes	characteristic
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	292	2	no	—
182	C₂×C₄	—	292	2	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	1	no	—
188	C₂×C₄	—	73	1	no	—
189	C₂×C₄	—	146	4	no	—
190	C₂×C₄	—	146	1	no	—
191	C₂×C₄	—	146	1	no	—
192	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
193	C₂×C₄	Q₈×#64408	1	1	yes	characteristic
194	C₂×C₄	D₇₃⋊M₁₆	1	2	yes	—
195	C₂×C₄	#14556678	1	2	yes	—
196	C₈	—	2	2	no	—
197	C₈	—	292	2	no	—
198	C₈	—	584	2	no	—
199	C₈	—	146	4	no	—
200	C₈	—	146	2	no	—
201	C₂²	#17989538	1	1	yes	characteristic
202	C₂²	#17989619	1	2	yes	—
203	C₂²	SD₁₆×#64408	1	2	yes	—
204	C₂²	#17989633	1	1	yes	characteristic
205	C₂²	#17989634	1	1	yes	characteristic
206	C₄	—	8	2	no	—
207	C₄	—	292	4	no	—
208	C₄	—	73	2	no	—
209	C₄	—	73	1	no	—
210	C₄	—	73	1	no	—
211	C₄	—	146	1	no	—
212	C₄	—	146	1	no	—
213	C₄	#17989592	1	1	yes	characteristic
214	C₄	#17989593	1	1	yes	characteristic
215	C₄	#17989639	1	2	yes	—
216	C₂	#22625537	1	2	yes	—
217	C₂	#22625541	1	2	yes	—
218	C₂	#22626032	1	1	yes	characteristic
219	C₂	#22626034	1	1	yes	characteristic
220	C₂	#22626040	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626029

C₂

yes

maximal

C₂×#17989448

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

—

—

—

—

—

—

—

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

—

—

—

—

—

—

—

—

—

SD₁₆

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

C₂×SD₁₆

yes

—

C₂×Dic₇₃

#135

yes

—

#10596833

—

#10596833

—

#10596833

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

#389

yes

characteristic

C₂²×C₇₃

#435

yes

—

C₂²×C₇₃

C₄×SD₁₆

yes

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

100

Dic₇₃

—

101

Dic₇₃

—

102

Dic₇₃

#465

yes

—

103

Dic₇₃

#498

yes

characteristic

104

Dic₇₃

yes

105

—

106

C₂×C₇₃

#1582

yes

—

107

C₂×C₇₃

#1583

yes

—

108

C₂×C₇₃

#1940

yes

characteristic

109

C₂×C₇₃

#1941

yes

characteristic

110

C₂×C₇₃

#1946

yes

characteristic

111

#1939

—

112

C₄×#106

—

113

C₂×#334

—

114

C₂×#322

—

115

#1279

—

116

C₇₃

#9597

yes

characteristic, complemented, Sylow

117

C₂×#106

—

118

C₂×C₄⋊C₈

—

119

C₂×C₄⋊C₈

—

146

—

120

C₂×C₄×C₈

—

121

C₂×C₄×C₈

—

122

C₂×C₄×C₈

—

123

C₄×C₄⋊C₄

—

124

#340

—

125

yes

126

—

146

—

127

#334

—

146

—

128

#322

—

146

—

129

C₂²×C₈

—

130

C₂²×C₈

—

131

C₂²×C₈

—

146

—

132

C₂²×C₈

—

146

—

133

C₂²×C₈

C₂×#64408

yes

characteristic

134

C₂×C₄⋊C₄

—

135

C₂×C₄⋊C₄

C₂×#64408

yes

—

136

C₂×C₄²

—

137

C₂×C₄²

—

146

—

138

#106

—

146

—

139

C₄⋊C₈

—

292

—

140

C₄⋊C₈

—

146

—

141

C₄⋊C₈

—

146

—

142

C₄×C₈

—

143

C₄×C₈

—

144

C₄×C₈

—

145

C₄×C₈

—

146

—

146

C₄×C₈

—

146

—

147

C₄×C₈

—

146

—

148

#95

—

146

—

149

C₂²×C₄

—

150

C₂²×C₄

—

151

C₂²×C₄

—

152

C₂²×C₄

—

146

—

153

C₂²×C₄

C₂²×#64408

yes

characteristic

154

C₂×C₈

—

155

C₂×C₈

—

292

—

156

C₂×C₈

—

292

—

157

C₂×C₈

—

292

—

158

C₂×C₈

—

292

—

159

C₂×C₈

—

292

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

146

—

163

C₂×C₈

—

146

—

164

C₂×C₈

—

146

—

165

C₂×C₈

—

146

—

166

C₂×C₈

C₄⋊#64408

yes

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

146

—

171

C₄²

—

292

—

172

C₄²

—

292

—

173

C₄²

—

174

C₄²

—

175

C₄²

—

176

C₄²

—

177

C₂³

D₄×#64408

yes

characteristic

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

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

—

146

—

190

C₂×C₄

—

146

—

191

C₂×C₄

—

146

—

192

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

193

C₂×C₄

Q₈×#64408

yes

characteristic

194

C₂×C₄

D₇₃⋊M₁₆

yes

—

195

C₂×C₄

#14556678

yes

—

196

C₈

—

197

C₈

—

292

—

198

C₈

—

584

—

199

C₈

—

146

—

200

C₈

—

146

—

201

C₂²

#17989538

yes

characteristic

202

C₂²

#17989619

yes

—

203

C₂²

SD₁₆×#64408

yes

—

204

C₂²

#17989633

yes

characteristic

205

C₂²

#17989634

yes

characteristic

206

C₄

—

207

C₄

—

292

—

208

C₄

—

209

C₄

—

210

C₄

—

211

C₄

—

146

—

212

C₄

—

146

—

213

C₄

#17989592

yes

characteristic

214

C₄

#17989593

yes

characteristic

215

C₄

#17989639

yes

—

216

C₂

#22625537

yes

—

217

C₂

#22625541

yes

—

218

C₂

#22626032

yes

characteristic

219

C₂

#22626034

yes

characteristic

220

C₂

#22626040

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	#9597
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
116. C₇₃	#9597	semidirect

H ≅ G/N

type

116. C₇₃

#9597

semidirect

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

subgroup	index
#22626029	2
C₂×#17989448	2
C₂×#17989429	2
#22624767	2
#22609819	2
#9597	73

subgroup

index

Subgroups — #34007485

Conjugacy classes of subgroups

Complements

Maximal subgroups