Group #32980494 — subgroups

Subgroups — #32980494

Order 18688.

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

normal subgroups only

This group has 412 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	#22618998	C₂	1	1	yes	maximal, characteristic
2	#22610760	C₂	1	1	yes	maximal, characteristic
3	#22608720	C₂	1	2	yes	maximal, complemented
4	#22607763	C₂	1	2	yes	maximal, complemented
5	C₇₃×#1543	C₂	1	1	yes	maximal, characteristic
6	#17988665	—	2	2	no	—
7	D₄×#10596831	—	2	2	no	—
8	#17988640	C₂²	1	2	yes	—
9	#17988611	C₂²	1	1	yes	characteristic
10	C₂×#14556518	—	2	2	no	—
11	#17988554	—	2	2	no	—
12	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
13	#17988340	—	2	2	no	—
14	#17988338	—	2	2	no	—
15	#532×C₇₃	C₂²	1	1	yes	characteristic
16	#431×C₇₃	C₂²	1	2	yes	—
17	#333×C₇₃	C₄	1	1	yes	characteristic, complemented
18	#333×C₇₃	C₄	1	1	yes	characteristic, complemented
19	#331×C₇₃	C₄	1	2	yes	complemented
20	C₂×#11924012	—	2	2	no	—
21	C₂×#11924012	—	2	1	no	—
22	C₂²×#10596831	—	2	2	no	—
23	C₂×C₄×Dic₇₃	—	2	1	no	—
24	C₂×C₄×Dic₇₃	—	2	2	no	—
25	C₄⋊#10596831	—	2	2	no	—
26	C₄⋊#10596831	D₄	1	2	yes	—
27	C₄⋊#10596831	D₄	1	2	yes	—
28	#14556518	—	4	4	no	—
29	C₂²⋊#10596831	—	2	4	no	—
30	C₂²⋊#10596831	—	2	2	no	—
31	C₄×#10596831	—	2	2	no	—
32	C₂×#41×C₇₃	D₄	1	2	yes	—
33	C₄⋊Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
34	#127×C₇₃	C₂×C₄	1	1	yes	characteristic
35	#124×C₇₃	Q₈	1	2	yes	—
36	#121×C₇₃	—	2	2	no	—
37	C₄×D₄×C₇₃	—	2	2	no	—
38	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
40	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
41	C₂²⋊C₈×C₇₃	—	2	2	no	—
42	C₄×C₈×C₇₃	—	2	2	no	—
43	C₂²×Dic₇₃	—	4	1	no	—
44	#11924012	—	4	2	no	—
45	#11924012	—	8	2	no	—
46	C₂×#10596831	—	2	2	no	—
47	C₂×#10596831	—	2	2	no	—
48	C₂×#10596831	—	4	1	no	—
49	C₂×#10596831	—	8	2	no	—
50	C₄×Dic₇₃	—	2	2	no	—
51	C₄×Dic₇₃	—	2	4	no	—
52	C₄×Dic₇₃	—	4	2	no	—
53	C₄×Dic₇₃	—	4	1	no	—
54	#41×C₇₃	—	4	2	no	—
55	#41×C₇₃	—	4	2	no	—
56	C₂×Q₈×C₇₃	C₄⋊C₄	1	2	yes	—
57	C₂×D₄×C₇₃	—	2	2	no	—
58	C₂×D₄×C₇₃	—	2	2	no	—
59	C₂×D₄×C₇₃	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₇₃	—	2	2	no	—
61	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₇₃	#41	1	2	yes	—
63	C₂×C₈×C₇₃	—	2	2	no	—
64	C₂×C₈×C₇₃	—	2	2	no	—
65	C₄⋊C₄×C₇₃	—	2	2	no	—
66	C₂²⋊C₄×C₇₃	—	2	4	no	—
67	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₇₃	C₂×D₄	1	2	yes	—
69	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
70	C₂³×C₇₃	—	2	2	no	—
71	C₂³×C₇₃	#123	1	1	yes	characteristic
72	Q₈×C₇₃	—	4	2	no	—
73	D₄×C₇₃	—	4	2	no	—
74	D₄×C₇₃	—	4	2	no	—
75	D₄×C₇₃	—	4	2	no	—
76	D₄×C₇₃	—	8	2	no	—
77	C₂×C₄×C₇₃	—	2	2	no	—
78	C₂×C₄×C₇₃	—	2	2	no	—
79	C₂×C₄×C₇₃	—	2	1	no	—
80	C₂×C₄×C₇₃	—	2	1	no	—
81	C₂×C₄×C₇₃	—	2	2	no	—
82	C₂×C₄×C₇₃	—	4	2	no	—
83	C₂×C₄×C₇₃	—	4	2	no	—
84	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
86	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
87	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
88	C₂×Dic₇₃	—	4	2	no	—
89	C₂×Dic₇₃	—	4	4	no	—
90	C₈×C₇₃	—	4	2	no	—
91	C₈×C₇₃	—	4	2	no	—
92	#10596831	—	4	2	no	—
93	#10596831	—	4	2	no	—
94	#10596831	—	8	1	no	—
95	C₂²×C₇₃	—	2	1	no	—
96	C₂²×C₇₃	—	2	2	no	—
97	C₂²×C₇₃	—	4	2	no	—
98	C₂²×C₇₃	—	4	4	no	—
99	C₂²×C₇₃	#385	1	1	yes	characteristic
100	C₄×C₇₃	—	2	1	no	—
101	C₄×C₇₃	—	2	2	no	—
102	C₄×C₇₃	—	4	2	no	—
103	C₄×C₇₃	#443	1	2	yes	—
104	C₄×C₇₃	#444	1	2	yes	—
105	C₄×C₇₃	#453	1	1	yes	characteristic
106	C₄×C₇₃	#455	1	1	yes	characteristic
107	Dic₇₃	—	4	4	no	—
108	#8935	—	73	1	no	maximal, Sylow
109	C₂×C₇₃	—	4	1	no	—
110	C₂×C₇₃	—	8	2	no	—
111	C₂×C₇₃	#1856	1	1	yes	characteristic
112	C₂×C₇₃	#1866	1	2	yes	—
113	#2979	—	73	1	no	—
114	#1962	—	73	1	no	—
115	#1574	—	73	2	no	—
116	#1543	D₇₃	1	1	yes	characteristic
117	#1276	—	73	2	no	—
118	C₇₃	#8935	1	1	yes	characteristic, complemented, Sylow
119	#532	C₂×D₇₃	1	1	yes	characteristic
120	#436	—	146	2	no	—
121	C₈×D₄	—	146	2	no	—
122	#431	—	73	2	no	—
123	#431	C₂×D₇₃	1	2	yes	—
124	#423	—	73	1	no	—
125	C₂×#104	—	146	2	no	—
126	#408	—	146	2	no	—
127	C₄³	—	73	1	no	—
128	#335	—	146	2	no	—
129	#334	—	146	2	no	—
130	#333	Dic₇₃	1	1	yes	characteristic, complemented
131	#333	Dic₇₃	1	1	yes	characteristic, complemented
132	#331	Dic₇₃	1	2	yes	complemented
133	C₂×#41	D₂₉₂	1	2	yes	—
134	C₂×M₁₆	—	146	1	no	—
135	C₂×M₁₆	—	146	2	no	—
136	C₂²×C₈	—	146	2	no	—
137	C₄⋊Q₈	C₂×Dic₇₃	1	1	yes	characteristic
138	#127	C₂×Dic₇₃	1	1	yes	characteristic
139	#124	Dic₁₄₆	1	2	yes	—
140	#121	—	2	2	no	—
141	C₄×D₄	—	2	2	no	—
142	C₂×C₄²	—	146	1	no	—
143	C₂×C₄²	—	146	2	no	—
144	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
145	C₄⋊C₈	—	73	2	no	—
146	C₄⋊C₈	—	73	2	no	—
147	C₄⋊C₈	—	146	2	no	—
148	C₄⋊C₈	C₂×Dic₇₃	1	2	yes	—
149	C₄⋊C₈	C₂×Dic₇₃	1	2	yes	—
150	#104	—	292	4	no	—
151	C₂²⋊C₈	—	2	2	no	—
152	C₂²⋊C₈	—	146	4	no	—
153	C₂²⋊C₈	—	146	2	no	—
154	C₄×C₈	—	2	2	no	—
155	C₄×C₈	—	146	2	no	—
156	#41	—	4	2	no	—
157	#41	—	4	2	no	—
158	C₂×Q₈	C₄⋊Dic₇₃	1	2	yes	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	—	2	2	no	—
161	C₂×D₄	C₄⋊Dic₇₃	1	2	yes	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	—	292	1	no	—
164	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
165	C₂²×C₄	#11924027	1	2	yes	—
166	M₁₆	—	292	2	no	—
167	M₁₆	—	584	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	292	1	no	—
171	C₂×C₈	—	584	2	no	—
172	C₂×C₈	—	146	2	no	—
173	C₂×C₈	—	146	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₂²⋊C₄	—	2	4	no	—
176	C₄²	—	292	2	no	—
177	C₄²	—	292	1	no	—
178	C₄²	—	146	2	no	—
179	C₄²	—	146	4	no	—
180	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
181	C₄²	D₄×D₇₃	1	2	yes	—
182	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
183	C₂³	—	2	2	no	—
184	C₂³	#14556575	1	1	yes	characteristic
185	Q₈	—	4	2	no	—
186	D₄	—	4	2	no	—
187	D₄	—	4	2	no	—
188	D₄	—	4	2	no	—
189	D₄	—	8	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	292	4	no	—
198	C₂×C₄	—	292	2	no	—
199	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
200	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
201	C₂×C₄	#14556561	1	1	yes	characteristic
202	C₂×C₄	#14556568	1	2	yes	—
203	C₈	—	4	2	no	—
204	C₈	—	4	2	no	—
205	C₈	—	292	2	no	—
206	C₈	—	292	2	no	—
207	C₈	—	584	1	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	4	4	no	—
211	C₂²	—	4	2	no	—
212	C₂²	#17988462	1	1	yes	characteristic
213	C₄	—	2	2	no	—
214	C₄	—	2	1	no	—
215	C₄	—	4	2	no	—
216	C₄	—	292	4	no	—
217	C₄	#17988704	1	2	yes	—
218	C₄	#17988709	1	2	yes	—
219	C₄	#17988757	1	1	yes	characteristic
220	C₄	#17988767	1	1	yes	characteristic
221	C₂	—	4	1	no	—
222	C₂	—	8	2	no	—
223	C₂	#22610143	1	1	yes	characteristic
224	C₂	#22610208	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22618998

C₂

yes

maximal, characteristic

#22610760

C₂

yes

maximal, characteristic

#22608720

C₂

yes

maximal, complemented

#22607763

C₂

yes

maximal, complemented

C₇₃×#1543

C₂

yes

maximal, characteristic

#17988665

—

D₄×#10596831

—

#17988640

C₂²

yes

—

#17988611

C₂²

yes

characteristic

C₂×#14556518

—

#17988554

—

C₄²×Dic₇₃

C₂²

yes

characteristic

#17988340

—

#17988338

—

#532×C₇₃

C₂²

yes

characteristic

#431×C₇₃

C₂²

yes

—

#333×C₇₃

C₄

yes

characteristic, complemented

#333×C₇₃

C₄

yes

characteristic, complemented

#331×C₇₃

C₄

yes

—

—

—

C₂×C₄×Dic₇₃

—

C₂×C₄×Dic₇₃

—

C₄⋊#10596831

—

C₄⋊#10596831

D₄

yes

—

C₄⋊#10596831

D₄

yes

—

—

—

—

—

C₂×#41×C₇₃

D₄

yes

—

C₄⋊Q₈×C₇₃

C₂×C₄

yes

characteristic

#127×C₇₃

C₂×C₄

yes

characteristic

#124×C₇₃

Q₈

yes

—

#121×C₇₃

—

C₄×D₄×C₇₃

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

C₂×C₄

yes

—

C₄⋊C₈×C₇₃

C₂×C₄

yes

—

C₂²⋊C₈×C₇₃

—

C₄×C₈×C₇₃

—

C₂²×Dic₇₃

—

—

—

—

—

—

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

#41×C₇₃

—

#41×C₇₃

—

C₂×Q₈×C₇₃

C₄⋊C₄

yes

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

C₄⋊C₄

yes

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

#41

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

—

C₄²×C₇₃

C₂×Q₈

yes

characteristic

C₂³×C₇₃

—

C₂³×C₇₃

#123

yes

characteristic

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×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⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₄×D₄

yes

—

C₂×C₄×C₇₃

#120

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#385

yes

characteristic

100

C₄×C₇₃

—

101

C₄×C₇₃

—

102

C₄×C₇₃

—

103

C₄×C₇₃

#443

yes

—

104

C₄×C₇₃

#444

yes

—

105

C₄×C₇₃

#453

yes

characteristic

106

C₄×C₇₃

#455

yes

characteristic

107

Dic₇₃

—

108

#8935

—

maximal, Sylow

109

C₂×C₇₃

—

110

C₂×C₇₃

—

111

C₂×C₇₃

#1856

yes

characteristic

112

C₂×C₇₃

#1866

yes

—

113

#2979

—

114

#1962

—

115

#1574

—

116

#1543

D₇₃

yes

characteristic

117

#1276

—

118

C₇₃

#8935

yes

characteristic, complemented, Sylow

119

#532

C₂×D₇₃

yes

characteristic

120

#436

—

146

—

121

C₈×D₄

—

146

—

122

#431

—

123

#431

C₂×D₇₃

yes

—

124

#423

—

125

C₂×#104

—

146

—

126

#408

—

146

—

127

C₄³

—

128

#335

—

146

—

129

#334

—

146

—

130

#333

Dic₇₃

yes

characteristic, complemented

131

#333

Dic₇₃

yes

characteristic, complemented

132

#331

Dic₇₃

yes

complemented

133

C₂×#41

D₂₉₂

yes

—

134

C₂×M₁₆

—

146

—

135

C₂×M₁₆

—

146

—

136

C₂²×C₈

—

146

—

137

C₄⋊Q₈

C₂×Dic₇₃

yes

characteristic

138

#127

C₂×Dic₇₃

yes

characteristic

139

#124

Dic₁₄₆

yes

—

140

#121

—

141

C₄×D₄

—

142

C₂×C₄²

—

146

—

143

C₂×C₄²

—

146

—

144

C₂×C₄²

C₂²×D₇₃

yes

characteristic

145

C₄⋊C₈

—

146

C₄⋊C₈

—

147

C₄⋊C₈

—

146

—

148

C₄⋊C₈

C₂×Dic₇₃

yes

—

149

C₄⋊C₈

C₂×Dic₇₃

yes

—

150

#104

—

292

—

151

C₂²⋊C₈

—

152

C₂²⋊C₈

—

146

—

153

C₂²⋊C₈

—

146

—

154

C₄×C₈

—

155

C₄×C₈

—

146

—

156

#41

—

157

#41

—

158

C₂×Q₈

C₄⋊Dic₇₃

yes

—

159

C₂×D₄

—

160

C₂×D₄

—

161

C₂×D₄

C₄⋊Dic₇₃

yes

—

162

C₂²×C₄

—

163

C₂²×C₄

—

292

—

164

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

165

C₂²×C₄

#11924027

yes

—

166

M₁₆

—

292

—

167

M₁₆

—

584

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

292

—

171

C₂×C₈

—

584

—

172

C₂×C₈

—

146

—

173

C₂×C₈

—

146

—

174

C₄⋊C₄

—

175

C₂²⋊C₄

—

176

C₄²

—

292

—

177

C₄²

—

292

—

178

C₄²

—

146

—

179

C₄²

—

146

—

180

C₄²

C₂²×Dic₇₃

yes

characteristic

181

C₄²

D₄×D₇₃

yes

—

182

C₄²

C₂×Dic₁₄₆

yes

characteristic

183

C₂³

—

184

C₂³

#14556575

yes

characteristic

185

Q₈

—

186

D₄

—

187

D₄

—

188

D₄

—

189

D₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

292

—

198

C₂×C₄

—

292

—

199

C₂×C₄

C₂×C₄⋊Dic₇₃

yes

characteristic

200

C₂×C₄

D₄×Dic₇₃

yes

—

201

C₂×C₄

#14556561

yes

characteristic

202

C₂×C₄

#14556568

yes

—

203

C₈

—

204

C₈

—

205

C₈

—

292

—

206

C₈

—

292

—

207

C₈

—

584

—

208

C₂²

—

209

C₂²

—

210

C₂²

—

211

C₂²

—

212

C₂²

#17988462

yes

characteristic

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

—

292

—

217

C₄

#17988704

yes

—

218

C₄

#17988709

yes

—

219

C₄

#17988757

yes

characteristic

220

C₄

#17988767

yes

characteristic

221

C₂

—

222

C₂

—

223

C₂

#22610143

yes

characteristic

224

C₂

#22610208

yes

—

	subgroup	G/H
center	C₂×C₄	#14556561
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1543	C₂
socle	C₂²×C₇₃	#385
prime radical	#532×C₇₃	C₂²
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1543	D₇₃
hypocenter	C₇₃	#8935
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1543

subgroup

G/H

center

C₂×C₄

#14556561

commutator

C₄²×C₇₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1543

C₂

socle

C₂²×C₇₃

#385

prime radical

#532×C₇₃

C₂²

cosocle

C₂×C₄²×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1543

N	H ≅ G/N	type
3. #22608720	C₂	semidirect
4. #22607763	C₂	semidirect
17. #333×C₇₃	C₄	semidirect
18. #333×C₇₃	C₄	semidirect
19. #331×C₇₃	C₄	semidirect
118. C₇₃	#8935	semidirect
130. #333	Dic₇₃	semidirect
131. #333	Dic₇₃	semidirect
132. #331	Dic₇₃	semidirect

H ≅ G/N

type

3. #22608720

C₂

semidirect

4. #22607763

C₂

semidirect

17. #333×C₇₃

C₄

semidirect

18. #333×C₇₃

C₄

semidirect

19. #331×C₇₃

C₄

semidirect

118. C₇₃

#8935

semidirect

130. #333

Dic₇₃

semidirect

131. #333

Dic₇₃

semidirect

132. #331

Dic₇₃

semidirect

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

subgroup	index
#22618998	2
#22610760	2
#22608720	2
#22607763	2
C₇₃×#1543	2
#8935	73

subgroup

index

C₇₃×#1543

Subgroups — #32980494

Conjugacy classes of subgroups

Complements

Maximal subgroups