Group #32981301 — subgroups

Subgroups — #32981301

Order 18688.

404 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 111 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	#22619033	C₂	1	1	yes	maximal, characteristic
2	#22608526	C₂	1	1	yes	maximal, characteristic
3	#22608352	C₂	1	2	yes	maximal
4	#22607768	C₂	1	2	yes	maximal
5	C₇₃×#3160	C₂	1	1	yes	maximal, characteristic
6	#17988718	—	2	2	no	—
7	Q₈×#10596831	—	2	2	no	—
8	#17988648	—	2	2	no	—
9	#17988640	C₂²	1	2	yes	—
10	C₄×#11924012	C₂²	1	2	yes	—
11	#17988329	—	2	2	no	—
12	Q₈⋊#10596831	—	2	2	no	—
13	Q₈⋊#10596831	—	2	2	no	—
14	#533×C₇₃	C₂²	1	1	yes	characteristic
15	#533×C₇₃	C₂²	1	1	yes	characteristic
16	#500×C₇₃	C₄	1	2	yes	—
17	#499×C₇₃	—	2	2	no	—
18	#498×C₇₃	C₄	1	2	yes	—
19	#427×C₇₃	—	2	2	no	—
20	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
21	C₂×#11924012	—	2	2	no	—
22	C₄⋊#10596831	—	2	4	no	—
23	C₄⋊#10596831	D₄	1	2	yes	—
24	C₄⋊#10596831	D₄	1	2	yes	—
25	C₂²⋊#10596831	—	2	2	no	—
26	#14556502	—	2	2	no	—
27	#14556502	—	2	2	no	—
28	C₄×#10596831	—	2	2	no	—
29	C₄×#10596831	—	2	2	no	—
30	C₄×#10596831	—	2	2	no	—
31	C₂²×C₈×C₇₃	—	2	1	no	—
32	C₄⋊Q₈×C₇₃	C₂×C₄	1	2	yes	—
33	C₄⋊Q₈×C₇₃	C₂×C₄	1	2	yes	—
34	#125×C₇₃	—	2	1	no	—
35	#125×C₇₃	D₄	1	2	yes	—
36	C₂²⋊Q₈×C₇₃	—	2	2	no	—
37	C₂²⋊Q₈×C₇₃	—	2	2	no	—
38	C₄×Q₈×C₇₃	—	2	2	no	—
39	C₄×Q₈×C₇₃	—	2	2	no	—
40	#117×C₇₃	—	2	1	no	—
41	#117×C₇₃	D₄	1	2	yes	—
42	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
43	#107×C₇₃	—	2	4	no	—
44	#106×C₇₃	—	2	4	no	—
45	C₄×C₈×C₇₃	—	2	1	no	—
46	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
48	#11924012	—	2	4	no	—
49	#11924012	—	4	2	no	—
50	C₂×#10596831	—	2	2	no	—
51	C₂×#10596831	—	2	2	no	—
52	C₂×#10596831	—	4	2	no	—
53	C₂×Q₈×C₇₃	—	2	2	no	—
54	C₂×Q₈×C₇₃	—	2	2	no	—
55	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₇₃	#41	1	2	yes	—
57	C₂×C₈×C₇₃	—	2	4	no	—
58	C₂×C₈×C₇₃	—	2	1	no	—
59	C₂×C₈×C₇₃	—	2	1	no	—
60	C₄⋊C₄×C₇₃	—	2	2	no	—
61	C₄⋊C₄×C₇₃	—	2	4	no	—
62	C₄⋊C₄×C₇₃	—	2	2	no	—
63	C₄⋊C₄×C₇₃	—	2	2	no	—
64	C₄⋊C₄×C₇₃	—	4	2	no	—
65	C₄⋊C₄×C₇₃	C₂²⋊C₄	1	4	yes	—
66	C₂²⋊C₄×C₇₃	—	2	2	no	—
67	C₂²⋊C₄×C₇₃	—	4	1	no	—
68	C₄²×C₇₃	—	2	2	no	—
69	C₄²×C₇₃	—	4	1	no	—
70	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₇₃	C₂×D₄	1	2	yes	—
72	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
73	C₂³×C₇₃	#123	1	1	yes	characteristic
74	Q₈×C₇₃	—	4	2	no	—
75	Q₈×C₇₃	—	4	2	no	—
76	Q₈×C₇₃	—	4	2	no	—
77	Q₈×C₇₃	—	4	2	no	—
78	C₂×C₄×C₇₃	—	2	2	no	—
79	C₂×C₄×C₇₃	—	2	2	no	—
80	C₂×C₄×C₇₃	—	2	2	no	—
81	C₂×C₄×C₇₃	—	2	4	no	—
82	C₂×C₄×C₇₃	—	4	2	no	—
83	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
86	C₂×C₄×C₇₃	#121	1	2	yes	—
87	C₈×C₇₃	—	2	2	no	—
88	C₈×C₇₃	—	2	2	no	—
89	#10596831	—	4	4	no	—
90	#10596831	—	4	2	no	—
91	#10596831	—	4	2	no	—
92	C₂²×C₇₃	—	2	2	no	—
93	C₂²×C₇₃	—	2	1	no	—
94	C₂²×C₇₃	#386	1	1	yes	characteristic
95	C₄×C₇₃	—	2	2	no	—
96	C₄×C₇₃	—	2	1	no	—
97	C₄×C₇₃	—	4	4	no	—
98	C₄×C₇₃	—	8	2	no	—
99	C₄×C₇₃	#443	1	2	yes	—
100	C₄×C₇₃	#444	1	2	yes	—
101	C₄×C₇₃	#456	1	2	yes	—
102	#9068	—	73	1	no	maximal, Sylow
103	C₂×C₇₃	—	4	1	no	—
104	C₂×C₇₃	#1876	1	2	yes	—
105	C₂×C₇₃	#1886	1	1	yes	characteristic
106	#3160	D₇₃	1	1	yes	characteristic
107	#2986	—	73	1	no	—
108	#1526	—	73	1	no	—
109	#1485	—	73	2	no	—
110	#1278	—	73	2	no	—
111	C₇₃	#9068	1	1	yes	characteristic, complemented, Sylow
112	#533	C₂×D₇₃	1	1	yes	characteristic
113	#533	C₂×D₇₃	1	1	yes	characteristic
114	#500	Dic₇₃	1	2	yes	—
115	#499	—	2	2	no	—
116	#498	Dic₇₃	1	2	yes	—
117	#446	—	146	2	no	—
118	C₈×Q₈	—	146	2	no	—
119	#433	—	146	2	no	—
120	#431	—	73	2	no	—
121	#427	—	2	2	no	—
122	C₄×M₁₆	—	73	2	no	—
123	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
124	#330	—	146	2	no	—
125	Q₈⋊C₈	—	146	2	no	—
126	Q₈⋊C₈	—	146	2	no	—
127	C₂×M₁₆	—	146	2	no	—
128	C₂²×C₈	—	2	1	no	—
129	C₄⋊Q₈	C₂×Dic₇₃	1	2	yes	—
130	C₄⋊Q₈	C₂×Dic₇₃	1	2	yes	—
131	#125	—	2	1	no	—
132	#125	C₂²⋊D₇₃	1	2	yes	—
133	C₂²⋊Q₈	—	2	2	no	—
134	C₂²⋊Q₈	—	2	2	no	—
135	C₄×Q₈	—	2	2	no	—
136	C₄×Q₈	—	2	2	no	—
137	#117	—	2	1	no	—
138	#117	C₂²⋊D₇₃	1	2	yes	—
139	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
140	#107	—	2	4	no	—
141	#106	—	2	4	no	—
142	C₄⋊C₈	—	73	2	no	—
143	C₄⋊C₈	—	73	2	no	—
144	C₄⋊C₈	—	146	4	no	—
145	C₂²⋊C₈	—	146	2	no	—
146	#97	—	146	2	no	—
147	#97	—	146	2	no	—
148	C₄×C₈	—	2	1	no	—
149	C₄×C₈	—	146	2	no	—
150	C₄×C₈	—	146	2	no	—
151	C₄×C₈	—	146	2	no	—
152	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
153	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
154	C₂×Q₈	—	2	2	no	—
155	C₂×Q₈	—	2	2	no	—
156	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
157	C₂²×C₄	#11924027	1	2	yes	—
158	M₁₆	—	292	2	no	—
159	M₁₆	—	146	4	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	292	2	no	—
164	C₂×C₈	—	146	2	no	—
165	C₂×C₈	—	146	2	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	C₂²⋊Dic₇₃	1	4	yes	—
172	C₂²⋊C₄	—	2	2	no	—
173	C₂²⋊C₄	—	4	1	no	—
174	C₄²	—	2	2	no	—
175	C₄²	—	4	1	no	—
176	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
177	C₄²	D₄×D₇₃	1	2	yes	—
178	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
179	C₂³	#14556572	1	1	yes	characteristic
180	Q₈	—	4	2	no	—
181	Q₈	—	4	2	no	—
182	Q₈	—	4	2	no	—
183	Q₈	—	4	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
190	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
191	C₂×C₄	#14556559	1	1	yes	characteristic
192	C₂×C₄	#14556563	1	2	yes	—
193	C₈	—	2	2	no	—
194	C₈	—	2	2	no	—
195	C₈	—	292	4	no	—
196	C₈	—	292	2	no	—
197	C₈	—	292	2	no	—
198	C₂²	—	2	2	no	—
199	C₂²	—	2	1	no	—
200	C₂²	#17988467	1	1	yes	characteristic
201	C₄	—	2	2	no	—
202	C₄	—	2	1	no	—
203	C₄	—	4	4	no	—
204	C₄	—	8	2	no	—
205	C₄	#17988704	1	2	yes	—
206	C₄	#17988709	1	2	yes	—
207	C₄	#17988769	1	2	yes	—
208	C₂	—	4	1	no	—
209	C₂	#22610260	1	2	yes	—
210	C₂	#22610325	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22619033

C₂

yes

maximal, characteristic

#22608526

C₂

yes

maximal, characteristic

#22608352

C₂

yes

maximal

#22607768

C₂

yes

maximal

C₇₃×#3160

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

C₄×#11924012

C₂²

yes

—

#17988329

—

Q₈⋊#10596831

—

Q₈⋊#10596831

—

#533×C₇₃

C₂²

yes

characteristic

#533×C₇₃

C₂²

yes

characteristic

#500×C₇₃

C₄

yes

—

#499×C₇₃

—

#498×C₇₃

C₄

yes

—

#427×C₇₃

—

C₂×C₄×C₈×C₇₃

C₂²

yes

—

—

D₄

yes

—

C₄⋊#10596831

D₄

yes

—

—

—

—

—

—

—

C₂²×C₈×C₇₃

—

C₄⋊Q₈×C₇₃

C₂×C₄

yes

—

C₄⋊Q₈×C₇₃

C₂×C₄

yes

—

#125×C₇₃

—

#125×C₇₃

D₄

yes

—

C₂²⋊Q₈×C₇₃

—

C₂²⋊Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

#117×C₇₃

—

#117×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

#107×C₇₃

—

#106×C₇₃

—

C₄×C₈×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

—

—

—

—

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

#41

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂²⋊C₄

yes

—

C₂²⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₂³×C₇₃

#123

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₄×D₄

yes

—

C₂×C₄×C₇₃

#120

yes

characteristic

C₂×C₄×C₇₃

#121

yes

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#386

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#443

yes

—

100

C₄×C₇₃

#444

yes

—

101

C₄×C₇₃

#456

yes

—

102

#9068

—

maximal, Sylow

103

C₂×C₇₃

—

104

C₂×C₇₃

#1876

yes

—

105

C₂×C₇₃

#1886

yes

characteristic

106

#3160

D₇₃

yes

characteristic

107

#2986

—

108

#1526

—

109

#1485

—

110

#1278

—

111

C₇₃

#9068

yes

characteristic, complemented, Sylow

112

#533

C₂×D₇₃

yes

characteristic

113

#533

C₂×D₇₃

yes

characteristic

114

#500

Dic₇₃

yes

—

115

#499

—

116

#498

Dic₇₃

yes

—

117

#446

—

146

—

118

C₈×Q₈

—

146

—

119

#433

—

146

—

120

#431

—

121

#427

—

122

C₄×M₁₆

—

123

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

124

#330

—

146

—

125

Q₈⋊C₈

—

146

—

126

Q₈⋊C₈

—

146

—

127

C₂×M₁₆

—

146

—

128

C₂²×C₈

—

129

C₄⋊Q₈

C₂×Dic₇₃

yes

—

130

C₄⋊Q₈

C₂×Dic₇₃

yes

—

131

#125

—

132

#125

C₂²⋊D₇₃

yes

—

133

C₂²⋊Q₈

—

134

C₂²⋊Q₈

—

135

C₄×Q₈

—

136

C₄×Q₈

—

137

#117

—

138

#117

C₂²⋊D₇₃

yes

—

139

C₂×C₄²

C₂²×D₇₃

yes

characteristic

140

#107

—

141

#106

—

142

C₄⋊C₈

—

143

C₄⋊C₈

—

144

C₄⋊C₈

—

146

—

145

C₂²⋊C₈

—

146

—

146

#97

—

146

—

147

#97

—

146

—

148

C₄×C₈

—

149

C₄×C₈

—

146

—

150

C₄×C₈

—

146

—

151

C₄×C₈

—

146

—

152

C₄×C₈

C₂×Dic₇₃

yes

characteristic

153

C₄×C₈

C₂×Dic₇₃

yes

characteristic

154

C₂×Q₈

—

155

C₂×Q₈

—

156

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

157

C₂²×C₄

#11924027

yes

—

158

M₁₆

—

292

—

159

M₁₆

—

146

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

292

—

164

C₂×C₈

—

146

—

165

C₂×C₈

—

146

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

C₂²⋊Dic₇₃

yes

—

172

C₂²⋊C₄

—

173

C₂²⋊C₄

—

174

C₄²

—

175

C₄²

—

176

C₄²

C₂²×Dic₇₃

yes

characteristic

177

C₄²

D₄×D₇₃

yes

—

178

C₄²

C₂×C₂²⋊D₇₃

yes

characteristic

179

C₂³

#14556572

yes

characteristic

180

Q₈

—

181

Q₈

—

182

Q₈

—

183

Q₈

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

C₂×C₂²⋊Dic₇₃

yes

characteristic

190

C₂×C₄

D₄×Dic₇₃

yes

—

191

C₂×C₄

#14556559

yes

characteristic

192

C₂×C₄

#14556563

yes

—

193

C₈

—

194

C₈

—

195

C₈

—

292

—

196

C₈

—

292

—

197

C₈

—

292

—

198

C₂²

—

199

C₂²

—

200

C₂²

#17988467

yes

characteristic

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

#17988704

yes

—

206

C₄

#17988709

yes

—

207

C₄

#17988769

yes

—

208

C₂

—

209

C₂

#22610260

yes

—

210

C₂

#22610325

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556559
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#3160	C₂
socle	C₂²×C₇₃	#386
prime radical	C₂³×C₇₃	#123
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#3160	D₇₃
hypocenter	C₇₃	#9068
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

subgroup

G/H

center

C₂×C₄

#14556559

commutator

C₄²×C₇₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#3160

C₂

C₂²×C₇₃

C₂³×C₇₃

C₂×C₄²×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

N	H ≅ G/N	type
111. C₇₃	#9068	semidirect

H ≅ G/N

type

111. C₇₃

#9068

semidirect

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

subgroup	index
#22619033	2
#22608526	2
#22608352	2
#22607768	2
C₇₃×#3160	2
#9068	73

subgroup

index

C₇₃×#3160

Subgroups — #32981301

Conjugacy classes of subgroups

Complements

Maximal subgroups