Group #32990772 — subgroups

Subgroups — #32990772

Order 18688.

362 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 95 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	#22610106	C₂	1	1	yes	maximal, characteristic, complemented
2	#22608599	C₂	1	2	yes	maximal
3	#22607789	C₂	1	2	yes	maximal, complemented
4	#22607776	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#1961	C₂	1	1	yes	maximal, characteristic
6	#17988640	C₂²	1	2	yes	—
7	C₂×#14556527	—	2	1	no	—
8	C₂×#14556527	—	2	1	no	—
9	#17988611	C₂²	1	2	yes	—
10	#17988331	—	2	4	no	—
11	C₂×#127×C₇₃	C₂²	1	1	yes	characteristic
12	C₂×#102×C₇₃	—	2	1	no	—
13	C₂×#102×C₇₃	—	2	1	no	—
14	C₂×#97×C₇₃	C₂²	1	1	yes	characteristic
15	#382×C₇₃	C₂²	1	1	yes	characteristic
16	C₂×#11924012	—	2	2	no	—
17	#14556527	—	4	2	no	—
18	#14556527	—	4	2	no	—
19	C₄⋊#10596831	—	2	2	no	—
20	C₄⋊#10596831	—	2	2	no	—
21	C₄⋊#10596831	—	2	2	no	—
22	C₂²⋊#10596831	—	2	2	no	—
23	C₄×#10596831	—	2	2	no	—
24	C₂²×D₄×C₇₃	—	2	1	no	—
25	C₂²×D₄×C₇₃	—	2	2	no	—
26	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
28	#127×C₇₃	—	2	2	no	—
29	#127×C₇₃	—	2	1	no	—
30	#127×C₇₃	D₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
32	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
33	#102×C₇₃	—	4	2	no	—
34	#102×C₇₃	—	4	2	no	—
35	#97×C₇₃	—	2	2	no	—
36	#95×C₇₃	—	2	2	no	—
37	#11924012	—	4	2	no	—
38	#11924012	—	4	2	no	—
39	C₂×#10596831	—	4	2	no	—
40	C₂×#10596831	—	4	2	no	—
41	C₂⁴×C₇₃	—	4	1	no	—
42	C₂×D₄×C₇₃	—	2	2	no	—
43	C₂×D₄×C₇₃	—	2	4	no	—
44	C₂×D₄×C₇₃	—	4	1	no	—
45	C₂×D₄×C₇₃	—	4	2	no	—
46	C₂×D₄×C₇₃	—	4	1	no	—
47	C₂×D₄×C₇₃	—	4	1	no	—
48	C₂×D₄×C₇₃	—	4	1	no	—
49	C₂×D₄×C₇₃	—	4	1	no	—
50	C₂×D₄×C₇₃	—	8	4	no	—
51	C₂²×C₄×C₇₃	—	4	1	no	—
52	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₇₃	#41	1	2	yes	—
54	C₂×C₈×C₇₃	—	2	1	no	—
55	C₂×C₈×C₇₃	—	2	1	no	—
56	C₂×C₈×C₇₃	—	2	1	no	—
57	C₂×C₈×C₇₃	—	2	1	no	—
58	C₂×C₈×C₇₃	D₈	1	2	yes	—
59	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
60	C₄⋊C₄×C₇₃	—	4	1	no	—
61	C₄⋊C₄×C₇₃	—	4	1	no	—
62	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₇₃	#41	1	2	yes	—
64	C₄²×C₇₃	#41	1	1	yes	characteristic
65	C₂³×C₇₃	—	4	2	no	—
66	C₂³×C₇₃	—	8	4	no	—
67	C₂³×C₇₃	—	8	1	no	—
68	C₂³×C₇₃	—	8	1	no	—
69	C₂³×C₇₃	#124	1	1	yes	characteristic
70	D₄×C₇₃	—	4	2	no	—
71	D₄×C₇₃	—	4	2	no	—
72	D₄×C₇₃	—	8	4	no	—
73	D₄×C₇₃	—	8	4	no	—
74	D₄×C₇₃	—	8	1	no	—
75	D₄×C₇₃	—	8	1	no	—
76	C₂×C₄×C₇₃	—	4	2	no	—
77	C₂×C₄×C₇₃	—	8	2	no	—
78	C₂×C₄×C₇₃	—	8	1	no	—
79	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
80	C₂×C₄×C₇₃	#123	1	1	yes	characteristic
81	C₂×C₄×C₇₃	#123	1	2	yes	—
82	C₂×C₄×C₇₃	#126	1	2	yes	—
83	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
85	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
86	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
87	C₈×C₇₃	—	2	2	no	—
88	C₈×C₇₃	—	2	2	no	—
89	#10596831	—	8	2	no	—
90	#10596831	—	8	2	no	—
91	C₂²×C₇₃	—	2	2	no	—
92	C₂²×C₇₃	—	8	2	no	—
93	C₂²×C₇₃	—	8	4	no	—
94	C₂²×C₇₃	—	16	2	no	—
95	C₂²×C₇₃	—	16	2	no	—
96	C₂²×C₇₃	#397	1	1	yes	characteristic
97	C₂²×C₇₃	#486	1	1	yes	characteristic
98	C₂²×C₇₃	#489	1	1	yes	characteristic
99	C₄×C₇₃	—	2	2	no	—
100	C₄×C₇₃	—	2	2	no	—
101	C₄×C₇₃	—	16	1	no	—
102	C₄×C₇₃	#459	1	1	yes	characteristic
103	C₄×C₇₃	#461	1	1	yes	characteristic
104	C₄×C₇₃	#480	1	1	yes	characteristic
105	C₄×C₇₃	#481	1	1	yes	characteristic
106	#10631	—	73	1	no	maximal, Sylow
107	C₂×C₇₃	—	2	1	no	—
108	C₂×C₇₃	—	2	1	no	—
109	C₂×C₇₃	—	16	2	no	—
110	C₂×C₇₃	#2071	1	1	yes	characteristic
111	C₂×C₇₃	#2081	1	2	yes	—
112	#1961	D₇₃	1	1	yes	characteristic
113	#1849	—	73	1	no	—
114	#1545	—	73	2	no	—
115	#1286	—	73	2	no	—
116	#1281	—	73	1	no	—
117	C₇₃	#10631	1	1	yes	characteristic, complemented, Sylow
118	C₂×#127	C₂×D₇₃	1	1	yes	characteristic
119	#431	—	73	2	no	—
120	C₂×#108	—	146	1	no	—
121	C₂×#108	—	146	1	no	—
122	#423	—	73	2	no	—
123	C₂×#102	—	2	1	no	—
124	C₂×#102	—	2	1	no	—
125	C₂×#97	C₂×D₇₃	1	1	yes	characteristic
126	#382	C₂×D₇₃	1	1	yes	characteristic
127	#331	—	146	4	no	—
128	C₂²×D₄	—	2	1	no	—
129	C₂²×D₄	—	2	2	no	—
130	C₂×M₁₆	—	146	2	no	—
131	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
132	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
133	#127	—	2	2	no	—
134	#127	—	2	1	no	—
135	#127	D₂₉₂	1	2	yes	—
136	C₂×C₄⋊C₄	—	2	1	no	—
137	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
138	#108	—	292	2	no	—
139	#108	—	292	2	no	—
140	C₄⋊C₈	—	146	2	no	—
141	C₄⋊C₈	—	146	2	no	—
142	C₄⋊C₈	—	146	2	no	—
143	#102	—	4	2	no	—
144	#102	—	4	2	no	—
145	C₂²⋊C₈	—	146	2	no	—
146	#97	—	2	2	no	—
147	C₄×C₈	—	146	2	no	—
148	#95	—	2	2	no	—
149	C₂⁴	—	4	1	no	—
150	C₂×D₄	—	2	2	no	—
151	C₂×D₄	—	2	4	no	—
152	C₂×D₄	—	4	1	no	—
153	C₂×D₄	—	4	2	no	—
154	C₂×D₄	—	4	1	no	—
155	C₂×D₄	—	4	1	no	—
156	C₂×D₄	—	4	1	no	—
157	C₂×D₄	—	4	1	no	—
158	C₂×D₄	—	8	4	no	—
159	C₂²×C₄	—	4	1	no	—
160	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
161	C₂²×C₄	#11924027	1	2	yes	—
162	M₁₆	—	292	2	no	—
163	M₁₆	—	292	2	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	292	2	no	—
169	C₂×C₈	—	292	2	no	—
170	C₂×C₈	#11924013	1	2	yes	—
171	C₂×C₈	#11924016	1	2	yes	—
172	C₄⋊C₄	—	4	1	no	—
173	C₄⋊C₄	—	4	1	no	—
174	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
175	C₄²	#11924027	1	2	yes	—
176	C₄²	#11924028	1	1	yes	characteristic
177	C₂³	—	4	2	no	—
178	C₂³	—	8	4	no	—
179	C₂³	—	8	1	no	—
180	C₂³	—	8	1	no	—
181	C₂³	#14556578	1	1	yes	characteristic
182	D₄	—	4	2	no	—
183	D₄	—	4	2	no	—
184	D₄	—	8	4	no	—
185	D₄	—	8	4	no	—
186	D₄	—	8	1	no	—
187	D₄	—	8	1	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	8	2	no	—
190	C₂×C₄	—	8	1	no	—
191	C₂×C₄	#14556565	1	1	yes	characteristic
192	C₂×C₄	#14556572	1	1	yes	characteristic
193	C₂×C₄	#14556575	1	2	yes	—
194	C₂×C₄	#14556584	1	2	yes	—
195	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
196	C₂×C₄	C₂×#11924016	1	1	yes	characteristic
197	C₂×C₄	#14556616	1	1	yes	characteristic
198	C₂×C₄	#14556621	1	1	yes	characteristic
199	C₈	—	2	2	no	—
200	C₈	—	2	2	no	—
201	C₈	—	584	2	no	—
202	C₈	—	584	2	no	—
203	C₂²	—	2	2	no	—
204	C₂²	—	8	2	no	—
205	C₂²	—	8	4	no	—
206	C₂²	—	16	2	no	—
207	C₂²	—	16	2	no	—
208	C₂²	#17988512	1	1	yes	characteristic
209	C₂²	#17988933	1	1	yes	characteristic
210	C₂²	#17988946	1	1	yes	characteristic
211	C₄	—	2	2	no	—
212	C₄	—	2	2	no	—
213	C₄	—	16	1	no	—
214	C₄	#17988782	1	1	yes	characteristic
215	C₄	#17988792	1	1	yes	characteristic
216	C₄	#17988898	1	1	yes	characteristic
217	C₄	#17988903	1	1	yes	characteristic
218	C₂	—	2	1	no	—
219	C₂	—	2	1	no	—
220	C₂	—	16	2	no	—
221	C₂	#22611334	1	1	yes	characteristic
222	C₂	#22611388	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22610106

C₂

yes

maximal, characteristic, complemented

#22608599

C₂

yes

maximal

#22607789

C₂

yes

maximal, complemented

#22607776

C₂

yes

maximal, characteristic, complemented

C₇₃×#1961

C₂

yes

maximal, characteristic

#17988640

C₂²

yes

—

C₂×#14556527

—

C₂×#14556527

—

#17988611

C₂²

yes

—

#17988331

—

C₂×#127×C₇₃

C₂²

yes

characteristic

C₂×#102×C₇₃

—

C₂×#102×C₇₃

—

C₂×#97×C₇₃

C₂²

yes

characteristic

#382×C₇₃

C₂²

yes

—

—

—

—

—

—

—

—

C₂²×D₄×C₇₃

—

C₂²×D₄×C₇₃

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

D₄

yes

characteristic

#127×C₇₃

—

#127×C₇₃

—

#127×C₇₃

D₄

yes

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

#102×C₇₃

—

#102×C₇₃

—

#97×C₇₃

—

#95×C₇₃

—

—

—

—

—

C₂⁴×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

#41

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

SD₁₆

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

#41

yes

—

C₄²×C₇₃

#41

yes

characteristic

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

#124

yes

characteristic

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

—

C₂×C₄×C₇₃

#126

yes

—

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×SD₁₆

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

100

C₄×C₇₃

—

101

C₄×C₇₃

—

102

C₄×C₇₃

#459

yes

characteristic

103

C₄×C₇₃

#461

yes

characteristic

104

C₄×C₇₃

#480

yes

characteristic

105

C₄×C₇₃

yes

106

—

107

C₂×C₇₃

—

108

C₂×C₇₃

—

109

C₂×C₇₃

—

110

C₂×C₇₃

#2071

yes

characteristic

111

C₂×C₇₃

#2081

yes

—

112

#1961

D₇₃

yes

characteristic

113

#1849

—

114

#1545

—

115

#1286

—

116

#1281

—

117

C₇₃

#10631

yes

characteristic, complemented, Sylow

118

C₂×#127

C₂×D₇₃

yes

characteristic

119

#431

—

120

C₂×#108

—

146

—

121

C₂×#108

—

146

—

122

#423

—

123

C₂×#102

—

124

C₂×#102

—

125

C₂×#97

C₂×D₇₃

yes

characteristic

126

#382

C₂×D₇₃

yes

characteristic

127

#331

—

146

—

128

C₂²×D₄

—

129

C₂²×D₄

—

130

C₂×M₁₆

—

146

—

131

C₂²×C₈

C₂²⋊D₇₃

yes

characteristic

132

C₂²×C₈

C₂²⋊D₇₃

yes

characteristic

133

#127

—

134

#127

—

135

#127

D₂₉₂

yes

—

136

C₂×C₄⋊C₄

—

137

C₂×C₄²

C₂²×D₇₃

yes

characteristic

138

#108

—

292

—

139

#108

—

292

—

140

C₄⋊C₈

—

146

—

141

C₄⋊C₈

—

146

—

142

C₄⋊C₈

—

146

—

143

#102

—

144

#102

—

145

C₂²⋊C₈

—

146

—

146

#97

—

147

C₄×C₈

—

146

—

148

#95

—

149

C₂⁴

—

150

C₂×D₄

—

151

C₂×D₄

—

152

C₂×D₄

—

153

C₂×D₄

—

154

C₂×D₄

—

155

C₂×D₄

—

156

C₂×D₄

—

157

C₂×D₄

—

158

C₂×D₄

—

159

C₂²×C₄

—

160

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

161

C₂²×C₄

#11924027

yes

—

162

M₁₆

—

292

—

163

M₁₆

—

292

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

292

—

169

C₂×C₈

—

292

—

170

C₂×C₈

#11924013

yes

—

171

C₂×C₈

#11924016

yes

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄²

C₂×D₂₉₂

yes

characteristic

175

C₄²

#11924027

yes

—

176

C₄²

#11924028

yes

characteristic

177

C₂³

—

178

C₂³

—

179

C₂³

—

180

C₂³

—

181

C₂³

#14556578

yes

characteristic

182

D₄

—

183

D₄

—

184

D₄

—

185

D₄

—

186

D₄

—

187

D₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

#14556565

yes

characteristic

192

C₂×C₄

#14556572

yes

characteristic

193

C₂×C₄

#14556575

yes

—

194

C₂×C₄

#14556584

yes

—

195

C₂×C₄

C₂×#11924013

yes

characteristic

196

C₂×C₄

C₂×#11924016

yes

characteristic

197

C₂×C₄

#14556616

yes

characteristic

198

C₂×C₄

#14556621

yes

characteristic

199

C₈

—

200

C₈

—

201

C₈

—

584

—

202

C₈

—

584

—

203

C₂²

—

204

C₂²

—

205

C₂²

—

206

C₂²

—

207

C₂²

—

208

C₂²

#17988512

yes

characteristic

209

C₂²

#17988933

yes

characteristic

210

C₂²

#17988946

yes

characteristic

211

C₄

—

212

C₄

—

213

C₄

—

214

C₄

#17988782

yes

characteristic

215

C₄

#17988792

yes

characteristic

216

C₄

#17988898

yes

characteristic

217

C₄

#17988903

yes

characteristic

218

C₂

—

219

C₂

—

220

C₂

—

221

C₂

#22611334

yes

characteristic

222

C₂

#22611388

yes

—

	subgroup	G/H
center	C₂²	#17988512
commutator, cosocle	C₂×C₄²×C₇₃	C₂³
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1961	C₂
socle	C₂²×C₇₃	#397
prime radical	C₂×#127×C₇₃	C₂²
hypercenter	#1961	D₇₃
hypocenter	C₇₃	#10631
LCS	G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1961

subgroup

G/H

center

C₂²

#17988512

commutator, cosocle

C₂×C₄²×C₇₃

C₂³

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1961

C₂

socle

C₂²×C₇₃

#397

prime radical

C₂×#127×C₇₃

C₂²

D₇₃

C₇₃

G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1961

N	H ≅ G/N	type
1. #22610106	C₂	semidirect
3. #22607789	C₂	semidirect
4. #22607776	C₂	semidirect
117. C₇₃	#10631	semidirect

H ≅ G/N

type

1. #22610106

C₂

semidirect

3. #22607789

C₂

semidirect

4. #22607776

C₂

semidirect

117. C₇₃

#10631

semidirect

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

subgroup	index
#22610106	2
#22608599	2
#22607789	2
#22607776	2
C₇₃×#1961	2
#10631	73

subgroup

index

C₇₃×#1961

Subgroups — #32990772

Conjugacy classes of subgroups

Complements

Maximal subgroups