Group #32990734 — subgroups

Subgroups — #32990734

Order 18688.

350 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 119 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	#22610867	C₂	1	1	yes	maximal, characteristic
2	#22609933	C₂	1	2	yes	maximal
3	#22607814	C₂	1	2	yes	maximal
4	#22607766	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1834	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₄₆	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
8	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
9	C₂×#14556514	—	2	2	no	—
10	#17988349	—	2	2	no	—
11	#17988341	—	2	2	no	—
12	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
13	#384×C₇₃	C₂²	1	2	yes	—
14	#340×C₇₃	—	2	2	no	—
15	C₂²×Dic₁₄₆	D₄	1	1	yes	characteristic
16	C₂²×#10596831	—	2	2	no	—
17	C₂²×#10596831	—	2	1	no	—
18	C₄×Dic₁₄₆	—	2	2	no	—
19	C₄×Dic₁₄₆	—	2	2	no	—
20	C₂×C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
21	C₂×#11924007	—	2	1	no	—
22	C₂×C₄×Dic₇₃	—	2	1	no	—
23	C₄⋊#10596831	—	2	2	no	—
24	C₄⋊#10596831	—	2	2	no	—
25	C₄⋊#10596831	—	2	2	no	—
26	#14556514	—	4	4	no	—
27	C₂²×C₈×C₇₃	D₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
31	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₇₃	—	2	2	no	—
33	C₂×Dic₁₄₆	—	2	2	no	—
34	C₂×Dic₁₄₆	—	2	2	no	—
35	C₂×Dic₁₄₆	—	2	1	no	—
36	C₂×Dic₁₄₆	Q₁₆	1	2	yes	—
37	C₂²×Dic₇₃	—	2	1	no	—
38	C₂²×Dic₇₃	—	2	1	no	—
39	C₂×#10596831	—	4	2	no	—
40	C₂×#10596831	—	4	1	no	—
41	C₂×#10596831	—	4	2	no	—
42	C₂×#10596831	—	4	2	no	—
43	C₂×#10596831	—	4	2	no	—
44	C₄⋊Dic₇₃	—	2	2	no	—
45	#11924007	—	4	2	no	—
46	C₄×Dic₇₃	—	4	1	no	—
47	C₄×Dic₇₃	—	4	1	no	—
48	C₂²×C₄×C₇₃	—	4	2	no	—
49	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
52	C₂×C₈×C₇₃	—	2	2	no	—
53	C₂×C₈×C₇₃	—	2	2	no	—
54	C₂×C₈×C₇₃	—	2	2	no	—
55	C₄⋊C₄×C₇₃	—	4	2	no	—
56	C₄⋊C₄×C₇₃	—	4	2	no	—
57	C₄⋊C₄×C₇₃	—	4	2	no	—
58	C₄⋊C₄×C₇₃	—	4	2	no	—
59	C₄⋊C₄×C₇₃	—	8	2	no	—
60	C₄²×C₇₃	#41	1	2	yes	—
61	C₄²×C₇₃	#41	1	2	yes	—
62	C₂³×C₇₃	#121	1	1	yes	characteristic
63	C₂×C₄×C₇₃	—	2	2	no	—
64	C₂×C₄×C₇₃	—	8	2	no	—
65	C₂×C₄×C₇₃	—	8	2	no	—
66	C₂×C₄×C₇₃	—	8	2	no	—
67	C₂×C₄×C₇₃	#123	1	1	yes	characteristic
68	C₂×C₄×C₇₃	#123	1	2	yes	—
69	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
70	C₂×C₄×C₇₃	#126	1	2	yes	—
71	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄×C₇₃	#135	1	2	yes	—
73	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
74	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
75	C₂×C₄×C₇₃	#137	1	2	yes	—
76	C₂×Dic₇₃	—	2	2	no	—
77	C₂×Dic₇₃	—	4	1	no	—
78	C₂×Dic₇₃	—	4	1	no	—
79	C₂×Dic₇₃	—	4	1	no	—
80	C₂×Dic₇₃	—	4	2	no	—
81	Dic₁₄₆	—	2	4	no	—
82	Dic₁₄₆	—	4	2	no	—
83	C₈×C₇₃	—	4	2	no	—
84	#10596831	—	8	2	no	—
85	#10596831	—	8	1	no	—
86	C₂²×C₇₃	#397	1	1	yes	characteristic
87	C₂²×C₇₃	#464	1	2	yes	—
88	C₂²×C₇₃	#467	1	1	yes	characteristic
89	C₂²×C₇₃	#469	1	1	yes	characteristic
90	C₂²×C₇₃	#471	1	2	yes	—
91	C₄×C₇₃	—	2	2	no	—
92	C₄×C₇₃	—	16	2	no	—
93	C₄×C₇₃	#482	1	2	yes	—
94	C₄×C₇₃	#484	1	2	yes	—
95	C₄×C₇₃	#485	1	2	yes	—
96	C₄×C₇₃	#488	1	1	yes	characteristic
97	C₄×C₇₃	#491	1	1	yes	characteristic
98	Dic₇₃	—	4	2	no	—
99	Dic₇₃	—	8	1	no	—
100	#10624	—	73	1	no	maximal, Sylow
101	C₂×C₇₃	#1684	1	2	yes	—
102	C₂×C₇₃	#1702	1	2	yes	—
103	C₂×C₇₃	#2074	1	1	yes	characteristic
104	C₂×C₇₃	#2076	1	1	yes	characteristic
105	C₂×C₇₃	#2079	1	1	yes	characteristic
106	#1984	—	73	1	no	—
107	#1834	D₇₃	1	1	yes	characteristic
108	#1803	—	73	2	no	—
109	#1296	—	73	2	no	—
110	#1277	—	73	1	no	—
111	C₇₃	#10624	1	1	yes	characteristic, complemented, Sylow
112	C₂×C₄×Q₈	—	73	1	no	—
113	C₂×C₄⋊C₈	—	73	2	no	—
114	C₂×C₄⋊C₈	—	73	1	no	—
115	C₂×#103	—	146	2	no	—
116	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
117	#384	C₂×D₇₃	1	2	yes	—
118	#340	—	2	2	no	—
119	#340	—	146	2	no	—
120	#336	—	146	2	no	—
121	C₂²×Q₈	—	73	1	no	—
122	C₂²×C₈	—	146	2	no	—
123	C₂²×C₈	—	146	1	no	—
124	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
125	C₄×Q₈	—	146	2	no	—
126	C₄×Q₈	—	146	2	no	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	73	1	no	—
131	C₂×C₄⋊C₄	—	146	1	no	—
132	C₂×C₄²	—	146	1	no	—
133	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
134	C₄⋊C₈	—	146	2	no	—
135	C₄⋊C₈	—	146	2	no	—
136	C₄⋊C₈	—	146	2	no	—
137	#103	—	292	4	no	—
138	C₄×C₈	—	2	2	no	—
139	C₂×Q₈	—	73	2	no	—
140	C₂×Q₈	—	146	2	no	—
141	C₂×Q₈	—	146	2	no	—
142	C₂×Q₈	—	146	1	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	—	146	1	no	—
145	C₂²×C₄	—	146	1	no	—
146	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
147	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
148	C₂²×C₄	#11924027	1	1	yes	characteristic
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	292	2	no	—
153	C₂×C₈	—	292	1	no	—
154	C₂×C₈	—	292	2	no	—
155	C₂×C₈	—	292	2	no	—
156	C₂×C₈	—	292	2	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	8	2	no	—
162	C₄⋊C₄	—	292	2	no	—
163	C₄⋊C₄	—	146	2	no	—
164	C₄²	—	292	1	no	—
165	C₄²	—	292	1	no	—
166	C₄²	#11924028	1	2	yes	—
167	C₄²	#11924029	1	2	yes	—
168	C₂³	#14556562	1	1	yes	characteristic
169	Q₈	—	292	2	no	—
170	Q₈	—	146	4	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	8	2	no	—
175	C₂×C₄	—	292	1	no	—
176	C₂×C₄	—	292	1	no	—
177	C₂×C₄	—	292	1	no	—
178	C₂×C₄	—	292	2	no	—
179	C₂×C₄	—	146	2	no	—
180	C₂×C₄	#14556574	1	1	yes	characteristic
181	C₂×C₄	#14556576	1	2	yes	—
182	C₂×C₄	#14556580	1	1	yes	characteristic
183	C₂×C₄	#14556586	1	2	yes	—
184	C₂×C₄	Q₁₆×D₇₃	1	1	yes	characteristic
185	C₂×C₄	#14556611	1	2	yes	—
186	C₂×C₄	#14556614	1	1	yes	characteristic
187	C₂×C₄	#14556615	1	2	yes	—
188	C₂×C₄	#14556621	1	2	yes	—
189	C₈	—	4	2	no	—
190	C₈	—	584	2	no	—
191	C₈	—	584	1	no	—
192	C₂²	#17988513	1	1	yes	characteristic
193	C₂²	#17988806	1	2	yes	—
194	C₂²	#17988825	1	1	yes	characteristic
195	C₂²	#17988837	1	1	yes	characteristic
196	C₂²	#17988847	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	16	2	no	—
199	C₄	—	292	2	no	—
200	C₄	—	584	1	no	—
201	C₄	#17988914	1	2	yes	—
202	C₄	#17988924	1	2	yes	—
203	C₄	#17988928	1	2	yes	—
204	C₄	#17988944	1	1	yes	characteristic
205	C₄	#17988957	1	1	yes	characteristic
206	C₂	#22609386	1	2	yes	—
207	C₂	#22609496	1	2	yes	—
208	C₂	#22611350	1	1	yes	characteristic
209	C₂	#22611360	1	1	yes	characteristic
210	C₂	#22611377	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22610867

C₂

yes

maximal, characteristic

#22609933

C₂

yes

maximal

#22607814

C₂

yes

maximal

#22607766

C₂

yes

maximal, characteristic

C₇₃×#1834

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₄₆

C₂²

yes

characteristic

C₂×C₄⋊#10596831

C₂²

yes

—

C₂×C₄⋊#10596831

C₂²

yes

—

—

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#384×C₇₃

C₂²

yes

—

#340×C₇₃

—

C₂²×Dic₁₄₆

D₄

yes

characteristic

C₂²×#10596831

—

C₂²×#10596831

—

C₄×Dic₁₄₆

—

C₄×Dic₁₄₆

—

C₂×C₄⋊Dic₇₃

D₄

yes

characteristic

C₂×#11924007

—

C₂×C₄×Dic₇₃

—

—

—

—

—

C₂²×C₈×C₇₃

D₄

yes

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄×C₈×C₇₃

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

Q₁₆

yes

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

—

—

—

—

—

C₄⋊Dic₇₃

—

#11924007

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

#41

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

#41

yes

—

C₄²×C₇₃

#41

yes

—

C₂³×C₇₃

#121

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#123

yes

characteristic

C₂×C₄×C₇₃

#123

yes

—

C₂×C₄×C₇₃

#124

yes

characteristic

C₂×C₄×C₇₃

#126

yes

—

C₂×C₄×C₇₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₃

#135

yes

—

C₂×C₄×C₇₃

#135

yes

characteristic

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇₃

#137

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

#397

yes

characteristic

C₂²×C₇₃

#464

yes

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#482

yes

—

C₄×C₇₃

#484

yes

—

C₄×C₇₃

#485

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

100

#10624

—

maximal, Sylow

101

C₂×C₇₃

#1684

yes

—

102

C₂×C₇₃

#1702

yes

—

103

C₂×C₇₃

#2074

yes

characteristic

104

C₂×C₇₃

#2076

yes

characteristic

105

C₂×C₇₃

#2079

yes

characteristic

106

#1984

—

107

#1834

D₇₃

yes

characteristic

108

#1803

—

109

#1296

—

110

#1277

—

111

C₇₃

#10624

yes

characteristic, complemented, Sylow

112

C₂×C₄×Q₈

—

113

C₂×C₄⋊C₈

—

114

C₂×C₄⋊C₈

—

115

C₂×#103

—

146

—

116

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

117

#384

C₂×D₇₃

yes

—

118

#340

—

119

#340

—

146

—

120

#336

—

146

—

121

C₂²×Q₈

—

122

C₂²×C₈

—

146

—

123

C₂²×C₈

—

146

—

124

C₂²×C₈

C₂²⋊D₇₃

yes

—

125

C₄×Q₈

—

146

—

126

C₄×Q₈

—

146

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

131

C₂×C₄⋊C₄

—

146

—

132

C₂×C₄²

—

146

—

133

C₂×C₄²

C₂²×D₇₃

yes

characteristic

134

C₄⋊C₈

—

146

—

135

C₄⋊C₈

—

146

—

136

C₄⋊C₈

—

146

—

137

#103

—

292

—

138

C₄×C₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

146

—

141

C₂×Q₈

—

146

—

142

C₂×Q₈

—

146

—

143

C₂²×C₄

—

144

C₂²×C₄

—

146

—

145

C₂²×C₄

—

146

—

146

C₂²×C₄

D₄×D₇₃

yes

characteristic

147

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

148

C₂²×C₄

#11924027

yes

characteristic

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

292

—

153

C₂×C₈

—

292

—

154

C₂×C₈

—

292

—

155

C₂×C₈

—

292

—

156

C₂×C₈

—

292

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

292

—

163

C₄⋊C₄

—

146

—

164

C₄²

—

292

—

165

C₄²

—

292

—

166

C₄²

#11924028

yes

—

167

C₄²

#11924029

yes

—

168

C₂³

#14556562

yes

characteristic

169

Q₈

—

292

—

170

Q₈

—

146

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

292

—

176

C₂×C₄

—

292

—

177

C₂×C₄

—

292

—

178

C₂×C₄

—

292

—

179

C₂×C₄

—

146

—

180

C₂×C₄

#14556574

yes

characteristic

181

C₂×C₄

#14556576

yes

—

182

C₂×C₄

#14556580

yes

characteristic

183

C₂×C₄

#14556586

yes

—

184

C₂×C₄

Q₁₆×D₇₃

yes

characteristic

185

C₂×C₄

#14556611

yes

—

186

C₂×C₄

#14556614

yes

characteristic

187

C₂×C₄

#14556615

yes

—

188

C₂×C₄

#14556621

yes

—

189

C₈

—

190

C₈

—

584

—

191

C₈

—

584

—

192

C₂²

#17988513

yes

characteristic

193

C₂²

#17988806

yes

—

194

C₂²

#17988825

yes

characteristic

195

C₂²

#17988837

yes

characteristic

196

C₂²

#17988847

yes

—

197

C₄

—

198

C₄

—

199

C₄

—

292

—

200

C₄

—

584

—

201

C₄

#17988914

yes

—

202

C₄

#17988924

yes

—

203

C₄

#17988928

yes

—

204

C₄

#17988944

yes

characteristic

205

C₄

#17988957

yes

characteristic

206

C₂

#22609386

yes

—

207

C₂

#22609496

yes

—

208

C₂

#22611350

yes

characteristic

209

C₂

#22611360

yes

characteristic

210

C₂

#22611377

yes

characteristic

	subgroup	G/H
center	C₂³	#14556562
commutator, cosocle	C₂×C₄²×C₇₃	C₂³
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1834	C₂
socle, prime radical	C₂³×C₇₃	#121
hypercenter	#1834	D₇₃
hypocenter	C₇₃	#10624
LCS	G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

subgroup

G/H

center

C₂³

#14556562

commutator, cosocle

C₂×C₄²×C₇₃

C₂³

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1834

C₂

C₂³×C₇₃

D₇₃

C₇₃

G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

N	H ≅ G/N	type
111. C₇₃	#10624	semidirect

H ≅ G/N

type

111. C₇₃

#10624

semidirect

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

subgroup	index
#22610867	2
#22609933	2
#22607814	2
#22607766	2
C₇₃×#1834	2
#10624	73

subgroup

index

C₇₃×#1834

Subgroups — #32990734

Conjugacy classes of subgroups

Complements

Maximal subgroups