Group #32919621 — subgroups

Subgroups — C₇₃×#10217 #32919621

Order 18688.

430 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 98 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	C₇₃×#3255	C₂	1	1	yes	maximal, characteristic
2	C₇₃×#3237	C₂	1	2	yes	maximal
3	C₇₃×#1995	C₂	1	1	yes	maximal, characteristic
4	C₇₃×#1506	C₂	1	2	yes	maximal
5	C₇₃×#1307	C₂	1	1	yes	maximal, characteristic
6	#533×C₇₃	C₂²	1	1	yes	characteristic
7	#533×C₇₃	C₂²	1	2	yes	—
8	#501×C₇₃	—	2	2	no	—
9	#491×C₇₃	—	2	2	no	—
10	#485×C₇₃	—	2	4	no	—
11	#477×C₇₃	—	2	2	no	—
12	#475×C₇₃	—	2	2	no	—
13	#475×C₇₃	—	2	2	no	—
14	#470×C₇₃	—	2	2	no	—
15	#462×C₇₃	—	2	2	no	—
16	#432×C₇₃	C₂²	1	2	yes	—
17	#423×C₇₃	C₂²	1	1	yes	characteristic
18	#376×C₇₃	C₂²	1	1	yes	characteristic
19	#339×C₇₃	—	2	2	no	—
20	#328×C₇₃	—	2	4	no	—
21	C₂×Q₁₆×C₇₃	—	2	2	no	—
22	C₂×M₁₆×C₇₃	D₄	1	2	yes	—
23	C₄⋊Q₈×C₇₃	D₄	1	2	yes	—
24	C₄⋊Q₈×C₇₃	D₄	1	2	yes	—
25	C₄⋊Q₈×C₇₃	D₄	1	2	yes	—
26	#125×C₇₃	—	2	2	no	—
27	#125×C₇₃	—	2	1	no	—
28	C₂²⋊Q₈×C₇₃	—	2	2	no	—
29	C₂²⋊Q₈×C₇₃	—	2	2	no	—
30	C₂²⋊Q₈×C₇₃	—	2	2	no	—
31	C₄×Q₈×C₇₃	—	2	2	no	—
32	C₄×Q₈×C₇₃	—	2	2	no	—
33	C₄×Q₈×C₇₃	—	2	2	no	—
34	#117×C₇₃	—	2	2	no	—
35	#117×C₇₃	—	2	1	no	—
36	C₂×C₄²×C₇₃	—	2	1	no	—
37	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
38	#107×C₇₃	—	2	4	no	—
39	#106×C₇₃	—	2	2	no	—
40	#106×C₇₃	—	2	4	no	—
41	C₄⋊C₈×C₇₃	—	2	2	no	—
42	C₄⋊C₈×C₇₃	—	2	2	no	—
43	#103×C₇₃	—	2	4	no	—
44	#103×C₇₃	—	2	4	no	—
45	#103×C₇₃	—	2	4	no	—
46	C₂²⋊C₈×C₇₃	D₄	1	4	yes	—
47	#97×C₇₃	—	2	2	no	—
48	#95×C₇₃	—	2	2	no	—
49	C₂×Q₈×C₇₃	—	2	2	no	—
50	C₂×Q₈×C₇₃	—	2	2	no	—
51	C₂×Q₈×C₇₃	—	2	2	no	—
52	C₂²×C₄×C₇₃	—	4	1	no	—
53	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
55	Q₁₆×C₇₃	—	8	2	no	—
56	M₁₆×C₇₃	—	4	2	no	—
57	C₂×C₈×C₇₃	—	2	4	no	—
58	C₂×C₈×C₇₃	—	2	2	no	—
59	C₄⋊C₄×C₇₃	—	2	2	no	—
60	C₄⋊C₄×C₇₃	—	2	2	no	—
61	C₄⋊C₄×C₇₃	—	2	2	no	—
62	C₄⋊C₄×C₇₃	—	2	2	no	—
63	C₄⋊C₄×C₇₃	—	2	2	no	—
64	C₄⋊C₄×C₇₃	—	2	2	no	—
65	C₄⋊C₄×C₇₃	—	4	2	no	—
66	C₄⋊C₄×C₇₃	—	4	2	no	—
67	C₄⋊C₄×C₇₃	—	4	2	no	—
68	C₂²⋊C₄×C₇₃	—	4	2	no	—
69	C₂²⋊C₄×C₇₃	—	4	1	no	—
70	C₄²×C₇₃	—	2	2	no	—
71	C₄²×C₇₃	—	4	1	no	—
72	C₄²×C₇₃	—	4	1	no	—
73	C₄²×C₇₃	—	4	2	no	—
74	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₇₃	C₂×D₄	1	2	yes	—
76	C₄²×C₇₃	#41	1	1	yes	characteristic
77	C₂³×C₇₃	#127	1	1	yes	characteristic
78	Q₈×C₇₃	—	4	4	no	—
79	Q₈×C₇₃	—	4	2	no	—
80	Q₈×C₇₃	—	4	2	no	—
81	Q₈×C₇₃	—	8	2	no	—
82	C₂×C₄×C₇₃	—	2	1	no	—
83	C₂×C₄×C₇₃	—	2	1	no	—
84	C₂×C₄×C₇₃	—	4	2	no	—
85	C₂×C₄×C₇₃	—	4	2	no	—
86	C₂×C₄×C₇₃	—	4	2	no	—
87	C₂×C₄×C₇₃	—	4	2	no	—
88	C₂×C₄×C₇₃	—	4	2	no	—
89	C₂×C₄×C₇₃	—	8	2	no	—
90	C₂×C₄×C₇₃	#120	1	2	yes	—
91	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
92	C₂×C₄×C₇₃	#121	1	2	yes	—
93	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
94	C₈×C₇₃	—	4	4	no	—
95	C₈×C₇₃	—	4	2	no	—
96	C₂²×C₇₃	—	2	1	no	—
97	C₂²×C₇₃	—	2	2	no	—
98	C₂²×C₇₃	#392	1	1	yes	characteristic
99	C₄×C₇₃	—	2	1	no	—
100	C₄×C₇₃	—	2	2	no	—
101	C₄×C₇₃	—	2	2	no	—
102	C₄×C₇₃	—	8	2	no	—
103	C₄×C₇₃	—	8	2	no	—
104	C₄×C₇₃	—	8	2	no	—
105	C₄×C₇₃	—	8	2	no	—
106	C₄×C₇₃	#455	1	1	yes	characteristic
107	C₄×C₇₃	#456	1	1	yes	characteristic
108	#10217	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂×C₇₃	—	4	1	no	—
110	C₂×C₇₃	#2007	1	1	yes	characteristic
111	C₂×C₇₃	#2009	1	2	yes	—
112	#3255	C₂×C₇₃	1	1	yes	characteristic
113	#3237	C₂×C₇₃	1	2	yes	—
114	#1995	C₂×C₇₃	1	1	yes	characteristic
115	#1506	C₂×C₇₃	1	2	yes	—
116	#1307	C₂×C₇₃	1	1	yes	characteristic
117	C₇₃	#10217	1	1	yes	characteristic, direct product factor, Sylow
118	#533	C₂²×C₇₃	1	1	yes	characteristic
119	#533	C₂²×C₇₃	1	2	yes	—
120	#501	—	2	2	no	—
121	#491	—	2	2	no	—
122	#485	—	2	4	no	—
123	#477	—	2	2	no	—
124	#475	—	2	2	no	—
125	#475	—	2	2	no	—
126	#470	—	2	2	no	—
127	#462	—	2	2	no	—
128	#432	C₂²×C₇₃	1	2	yes	—
129	#423	C₂²×C₇₃	1	1	yes	characteristic
130	#376	C₂²×C₇₃	1	1	yes	characteristic
131	#339	—	2	2	no	—
132	#328	—	2	4	no	—
133	C₂×Q₁₆	—	2	2	no	—
134	C₂×M₁₆	D₄×C₇₃	1	2	yes	—
135	C₄⋊Q₈	D₄×C₇₃	1	2	yes	—
136	C₄⋊Q₈	D₄×C₇₃	1	2	yes	—
137	C₄⋊Q₈	D₄×C₇₃	1	2	yes	—
138	#125	—	2	2	no	—
139	#125	—	2	1	no	—
140	C₂²⋊Q₈	—	2	2	no	—
141	C₂²⋊Q₈	—	2	2	no	—
142	C₂²⋊Q₈	—	2	2	no	—
143	C₄×Q₈	—	2	2	no	—
144	C₄×Q₈	—	2	2	no	—
145	C₄×Q₈	—	2	2	no	—
146	#117	—	2	1	no	—
147	#117	—	2	2	no	—
148	C₂×C₄²	—	2	1	no	—
149	C₂×C₄²	C₂³×C₇₃	1	1	yes	characteristic
150	#107	—	2	4	no	—
151	#106	—	2	2	no	—
152	#106	—	2	4	no	—
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	2	2	no	—
155	#103	—	2	4	no	—
156	#103	—	2	4	no	—
157	#103	—	2	4	no	—
158	C₂²⋊C₈	D₄×C₇₃	1	4	yes	—
159	#97	—	2	2	no	—
160	#95	—	2	2	no	—
161	C₂×Q₈	—	2	2	no	—
162	C₂×Q₈	—	2	2	no	—
163	C₂×Q₈	—	2	2	no	—
164	C₂²×C₄	—	4	1	no	—
165	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
167	Q₁₆	—	8	2	no	—
168	M₁₆	—	4	2	no	—
169	C₂×C₈	—	2	4	no	—
170	C₂×C₈	—	2	2	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₂²⋊C₄	—	4	1	no	—
181	C₂²⋊C₄	—	4	2	no	—
182	C₄²	—	2	2	no	—
183	C₄²	—	4	1	no	—
184	C₄²	—	4	1	no	—
185	C₄²	—	4	2	no	—
186	C₄²	C₂×D₄×C₇₃	1	1	yes	characteristic
187	C₄²	C₂×D₄×C₇₃	1	2	yes	—
188	C₄²	#41×C₇₃	1	1	yes	characteristic
189	C₂³	#127×C₇₃	1	1	yes	characteristic
190	Q₈	—	4	4	no	—
191	Q₈	—	4	2	no	—
192	Q₈	—	4	2	no	—
193	Q₈	—	8	2	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	8	2	no	—
202	C₂×C₄	#120×C₇₃	1	2	yes	—
203	C₂×C₄	#120×C₇₃	1	1	yes	characteristic
204	C₂×C₄	#121×C₇₃	1	2	yes	—
205	C₂×C₄	#121×C₇₃	1	1	yes	characteristic
206	C₈	—	4	4	no	—
207	C₈	—	4	2	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	2	no	—
210	C₂²	#392×C₇₃	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	2	no	—
213	C₄	—	2	2	no	—
214	C₄	—	8	2	no	—
215	C₄	—	8	2	no	—
216	C₄	—	8	2	no	—
217	C₄	—	8	2	no	—
218	C₄	#455×C₇₃	1	1	yes	characteristic
219	C₄	#456×C₇₃	1	1	yes	characteristic
220	C₂	—	4	1	no	—
221	C₂	C₇₃×#2007	1	1	yes	characteristic
222	C₂	C₇₃×#2009	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#3255

C₂

yes

maximal, characteristic

C₇₃×#3237

C₂

yes

maximal

C₇₃×#1995

C₂

yes

maximal, characteristic

C₇₃×#1506

C₂

yes

maximal

C₇₃×#1307

C₂

yes

maximal, characteristic

#533×C₇₃

C₂²

yes

characteristic

#533×C₇₃

C₂²

yes

—

#501×C₇₃

—

#491×C₇₃

—

#485×C₇₃

—

#477×C₇₃

—

#475×C₇₃

—

#475×C₇₃

—

#470×C₇₃

—

#462×C₇₃

—

#432×C₇₃

C₂²

yes

—

#423×C₇₃

C₂²

yes

characteristic

#376×C₇₃

C₂²

yes

characteristic

#339×C₇₃

—

#328×C₇₃

—

C₂×Q₁₆×C₇₃

—

C₂×M₁₆×C₇₃

D₄

yes

—

C₄⋊Q₈×C₇₃

D₄

yes

—

C₄⋊Q₈×C₇₃

D₄

yes

—

C₄⋊Q₈×C₇₃

D₄

yes

—

#125×C₇₃

—

#125×C₇₃

—

C₂²⋊Q₈×C₇₃

—

C₂²⋊Q₈×C₇₃

—

C₂²⋊Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

#117×C₇₃

—

#117×C₇₃

—

C₂×C₄²×C₇₃

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

#107×C₇₃

—

#106×C₇₃

—

#106×C₇₃

—

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

—

#103×C₇₃

—

#103×C₇₃

—

#103×C₇₃

—

C₂²⋊C₈×C₇₃

D₄

yes

—

#97×C₇₃

—

#95×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×D₄

yes

—

Q₁₆×C₇₃

—

M₁₆×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

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

—

C₄²×C₇₃

yes

C₂³×C₇₃

yes

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×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#120

yes

—

C₂×C₄×C₇₃

#120

yes

characteristic

C₂×C₄×C₇₃

#121

yes

—

C₂×C₄×C₇₃

#121

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#392

yes

characteristic

C₄×C₇₃

—

100

C₄×C₇₃

—

101

C₄×C₇₃

—

102

C₄×C₇₃

—

103

C₄×C₇₃

—

104

C₄×C₇₃

—

105

C₄×C₇₃

—

106

C₄×C₇₃

#455

yes

characteristic

107

C₄×C₇₃

#456

yes

characteristic

108

#10217

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

109

C₂×C₇₃

—

110

C₂×C₇₃

#2007

yes

characteristic

111

C₂×C₇₃

#2009

yes

—

112

#3255

C₂×C₇₃

yes

characteristic

113

#3237

C₂×C₇₃

yes

—

114

#1995

C₂×C₇₃

yes

characteristic

115

#1506

C₂×C₇₃

yes

—

116

#1307

C₂×C₇₃

yes

characteristic

117

C₇₃

#10217

yes

characteristic, direct product factor, Sylow

118

#533

C₂²×C₇₃

yes

characteristic

119

#533

C₂²×C₇₃

yes

—

120

#501

—

121

#491

—

122

#485

—

123

#477

—

124

#475

—

125

#475

—

126

#470

—

127

#462

—

128

#432

C₂²×C₇₃

yes

—

129

#423

C₂²×C₇₃

yes

characteristic

130

#376

C₂²×C₇₃

yes

characteristic

131

#339

—

132

#328

—

133

C₂×Q₁₆

—

134

C₂×M₁₆

D₄×C₇₃

yes

—

135

C₄⋊Q₈

D₄×C₇₃

yes

—

136

C₄⋊Q₈

D₄×C₇₃

yes

—

137

C₄⋊Q₈

D₄×C₇₃

yes

—

138

#125

—

139

#125

—

140

C₂²⋊Q₈

—

141

C₂²⋊Q₈

—

142

C₂²⋊Q₈

—

143

C₄×Q₈

—

144

C₄×Q₈

—

145

C₄×Q₈

—

146

#117

—

147

#117

—

148

C₂×C₄²

—

149

C₂×C₄²

C₂³×C₇₃

yes

characteristic

150

#107

—

151

#106

—

152

#106

—

153

C₄⋊C₈

—

154

C₄⋊C₈

—

155

#103

—

156

#103

—

157

#103

—

158

C₂²⋊C₈

D₄×C₇₃

yes

—

159

#97

—

160

#95

—

161

C₂×Q₈

—

162

C₂×Q₈

—

163

C₂×Q₈

—

164

C₂²×C₄

—

165

C₂²×C₄

C₂×D₄×C₇₃

yes

characteristic

166

C₂²×C₄

C₂×D₄×C₇₃

yes

—

167

Q₁₆

—

168

M₁₆

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

—

175

C₄⋊C₄

—

176

C₄⋊C₄

—

177

C₄⋊C₄

—

178

C₄⋊C₄

—

179

C₄⋊C₄

—

180

C₂²⋊C₄

—

181

C₂²⋊C₄

—

182

C₄²

—

183

C₄²

—

184

C₄²

—

185

C₄²

—

186

C₄²

C₂×D₄×C₇₃

yes

characteristic

187

C₄²

C₂×D₄×C₇₃

yes

—

188

C₄²

#41×C₇₃

yes

characteristic

189

C₂³

#127×C₇₃

yes

characteristic

190

Q₈

—

191

Q₈

—

192

Q₈

—

193

Q₈

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

#120×C₇₃

yes

—

203

C₂×C₄

#120×C₇₃

yes

characteristic

204

C₂×C₄

#121×C₇₃

yes

—

205

C₂×C₄

#121×C₇₃

yes

characteristic

206

C₈

—

207

C₈

—

208

C₂²

—

209

C₂²

—

210

C₂²

#392×C₇₃

yes

characteristic

211

C₄

—

212

C₄

—

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

—

217

C₄

—

218

C₄

#455×C₇₃

yes

characteristic

219

C₄

#456×C₇₃

yes

characteristic

220

C₂

—

221

C₂

C₇₃×#2007

yes

characteristic

222

C₂

C₇₃×#2009

yes

—

	subgroup	G/H
center, socle	C₂²×C₇₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇₃
prime radical	C₂³×C₇₃	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₃

#392

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₇₃

prime radical

C₂³×C₇₃

#127

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ G

N	H ≅ G/N	type
108. #10217	C₇₃	direct (N × H)
117. C₇₃	#10217	direct (N × H)

H ≅ G/N

type

108. #10217

C₇₃

direct (N × H)

117. C₇₃

#10217

direct (N × H)

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

subgroup	index
C₇₃×#3255	2
C₇₃×#3237	2
C₇₃×#1995	2
C₇₃×#1506	2
C₇₃×#1307	2
#10217	73

subgroup

index

C₇₃×#3255

C₇₃×#3237

C₇₃×#1995

C₇₃×#1506

C₇₃×#1307

#10217

Subgroups — C₇₃×#10217 #32919621

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#10217 #32919621

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#10217 #32919621