Group #32989077 — subgroups

Subgroups — #32989077

Order 18688.

416 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 99 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	#22618309	C₂	1	1	yes	maximal, characteristic
2	#22608495	C₂	1	2	yes	maximal
3	#22607791	C₂	1	2	yes	maximal
4	#22607764	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1994	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556591	—	2	1	no	—
7	#17988640	C₂²	1	2	yes	—
8	#17988493	—	2	4	no	—
9	#17988453	—	2	1	no	—
10	#17988453	C₂²	1	2	yes	—
11	#17988324	—	2	4	no	—
12	#533×C₇₃	C₂²	1	1	yes	characteristic
13	#423×C₇₃	C₂²	1	1	yes	characteristic
14	C₄³×C₇₃	C₂²	1	1	yes	characteristic
15	#339×C₇₃	—	2	1	no	—
16	#339×C₇₃	—	2	1	no	—
17	C₂²×Dic₁₄₆	—	2	2	no	—
18	#14556591	—	4	2	no	—
19	#14556591	—	4	2	no	—
20	C₂×C₄⋊Dic₇₃	—	2	2	no	—
21	C₂×C₄⋊Dic₇₃	—	2	2	no	—
22	C₄⋊#10596831	—	2	2	no	—
23	C₂²⋊#10596831	—	2	2	no	—
24	C₄×#10596831	—	2	2	no	—
25	#14556498	—	2	4	no	—
26	#14556498	—	2	2	no	—
27	#14556498	D₄	1	4	yes	—
28	C₂×M₁₆×C₇₃	—	2	1	no	—
29	C₄⋊Q₈×C₇₃	Q₈	1	2	yes	complemented
30	#125×C₇₃	—	2	1	no	—
31	C₂²⋊Q₈×C₇₃	—	2	2	no	—
32	C₄×Q₈×C₇₃	—	2	2	no	—
33	#117×C₇₃	—	2	1	no	—
34	C₂×C₄²×C₇₃	—	2	2	no	—
35	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₇₃	—	2	2	no	—
39	C₂×Dic₁₄₆	—	4	4	no	—
40	C₂×Dic₁₄₆	—	4	8	no	—
41	C₂²×Dic₇₃	—	2	4	no	—
42	C₂×#10596831	—	4	2	no	—
43	C₄⋊Dic₇₃	—	4	4	no	—
44	C₄⋊Dic₇₃	—	4	2	no	—
45	C₄⋊Dic₇₃	—	4	2	no	—
46	C₂×Q₈×C₇₃	—	2	2	no	—
47	C₂²×C₄×C₇₃	—	2	1	no	—
48	C₂²×C₄×C₇₃	—	2	1	no	—
49	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
51	M₁₆×C₇₃	—	4	1	no	—
52	M₁₆×C₇₃	—	4	1	no	—
53	C₂×C₈×C₇₃	—	4	1	no	—
54	C₄⋊C₄×C₇₃	—	2	2	no	—
55	C₄⋊C₄×C₇₃	—	2	2	no	—
56	C₄⋊C₄×C₇₃	—	4	2	no	—
57	C₂²⋊C₄×C₇₃	—	4	1	no	—
58	C₄²×C₇₃	—	2	1	no	—
59	C₄²×C₇₃	—	2	1	no	—
60	C₄²×C₇₃	—	4	4	no	—
61	C₄²×C₇₃	—	4	1	no	—
62	C₄²×C₇₃	SD₁₆	1	2	yes	—
63	C₄²×C₇₃	Q₁₆	1	2	yes	—
64	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
65	C₄²×C₇₃	#41	1	2	yes	—
66	C₄²×C₇₃	#41	1	1	yes	characteristic
67	C₂³×C₇₃	#120	1	1	yes	characteristic
68	Q₈×C₇₃	—	8	2	no	—
69	C₂×C₄×C₇₃	—	2	4	no	—
70	C₂×C₄×C₇₃	—	2	2	no	—
71	C₂×C₄×C₇₃	—	2	2	no	—
72	C₂×C₄×C₇₃	—	4	2	no	—
73	C₂×C₄×C₇₃	—	4	2	no	—
74	C₂×C₄×C₇₃	—	4	2	no	—
75	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
76	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₇₃	#124	1	2	yes	—
78	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
79	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
80	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
81	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
83	C₂×Dic₇₃	—	4	8	no	—
84	C₂×Dic₇₃	—	4	4	no	—
85	Dic₁₄₆	—	8	8	no	—
86	C₈×C₇₃	—	8	1	no	—
87	#10596831	—	8	2	no	—
88	C₂²×C₇₃	—	2	2	no	—
89	C₂²×C₇₃	#393	1	1	yes	characteristic
90	C₂²×C₇₃	#450	1	1	yes	characteristic
91	C₂²×C₇₃	#451	1	1	yes	characteristic
92	C₄×C₇₃	—	2	4	no	—
93	C₄×C₇₃	—	2	2	no	—
94	C₄×C₇₃	—	2	2	no	—
95	C₄×C₇₃	—	4	2	no	—
96	C₄×C₇₃	—	8	2	no	—
97	C₄×C₇₃	#475	1	1	yes	characteristic
98	C₄×C₇₃	#477	1	1	yes	characteristic
99	C₄×C₇₃	#487	1	1	yes	characteristic
100	C₄×C₇₃	#489	1	1	yes	characteristic
101	Dic₇₃	—	8	4	no	—
102	#10363	—	73	1	no	maximal, Sylow
103	C₂×C₇₃	—	2	1	no	—
104	C₂×C₇₃	—	2	1	no	—
105	C₂×C₇₃	#2024	1	1	yes	characteristic
106	C₂×C₇₃	#2031	1	2	yes	—
107	#2862	—	73	1	no	—
108	#1994	D₇₃	1	1	yes	characteristic
109	#1518	—	73	2	no	—
110	#1286	—	73	2	no	—
111	#1276	—	73	1	no	—
112	C₇₃	#10363	1	1	yes	characteristic, complemented, Sylow
113	#533	C₂×D₇₃	1	1	yes	characteristic
114	C₂×C₄⋊Q₈	—	146	1	no	—
115	#431	—	73	2	no	—
116	#423	C₂×D₇₃	1	1	yes	characteristic
117	#391	—	146	4	no	—
118	#382	—	73	2	no	—
119	#382	—	146	1	no	—
120	C₄³	C₂×D₇₃	1	1	yes	characteristic
121	#339	—	2	1	no	—
122	#339	—	2	1	no	—
123	#328	—	146	4	no	—
124	C₂²×Q₈	—	146	2	no	—
125	C₂×M₁₆	—	2	1	no	—
126	C₄⋊Q₈	—	292	2	no	—
127	C₄⋊Q₈	—	292	2	no	—
128	C₄⋊Q₈	Dic₁₄₆	1	2	yes	complemented
129	#125	—	2	1	no	—
130	C₂²⋊Q₈	—	2	2	no	—
131	C₄×Q₈	—	2	2	no	—
132	#117	—	2	1	no	—
133	C₂×C₄⋊C₄	—	146	2	no	—
134	C₂×C₄⋊C₄	—	146	2	no	—
135	C₂×C₄²	—	2	2	no	—
136	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
137	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
138	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
139	C₄⋊C₈	—	2	2	no	—
140	C₄⋊C₈	—	146	2	no	—
141	C₂²⋊C₈	—	146	2	no	—
142	C₄×C₈	—	146	2	no	—
143	#95	—	73	4	no	—
144	#95	—	146	4	no	—
145	#95	—	146	2	no	—
146	C₂×Q₈	—	2	2	no	—
147	C₂×Q₈	—	292	4	no	—
148	C₂×Q₈	—	292	8	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	2	1	no	—
151	C₂²×C₄	—	146	4	no	—
152	C₂²×C₄	D₄×D₇₃	1	2	yes	—
153	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
154	M₁₆	—	4	1	no	—
155	M₁₆	—	4	1	no	—
156	C₂×C₈	—	4	1	no	—
157	C₂×C₈	—	292	2	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	292	4	no	—
162	C₄⋊C₄	—	292	2	no	—
163	C₄⋊C₄	—	292	2	no	—
164	C₂²⋊C₄	—	4	1	no	—
165	C₄²	—	2	1	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	4	4	no	—
168	C₄²	—	4	1	no	—
169	C₄²	#11924015	1	2	yes	—
170	C₄²	#11924018	1	2	yes	—
171	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
172	C₄²	#11924027	1	2	yes	—
173	C₄²	#11924028	1	1	yes	characteristic
174	C₂³	#14556559	1	1	yes	characteristic
175	Q₈	—	8	2	no	—
176	Q₈	—	584	8	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	292	8	no	—
184	C₂×C₄	—	292	4	no	—
185	C₂×C₄	#14556568	1	2	yes	—
186	C₂×C₄	#14556570	1	1	yes	characteristic
187	C₂×C₄	#14556577	1	2	yes	—
188	C₂×C₄	#14556578	1	1	yes	characteristic
189	C₂×C₄	C₂×#11924015	1	1	yes	characteristic
190	C₂×C₄	C₂×#11924018	1	1	yes	characteristic
191	C₂×C₄	#14556616	1	1	yes	characteristic
192	C₂×C₄	#14556621	1	1	yes	characteristic
193	C₈	—	8	1	no	—
194	C₈	—	584	2	no	—
195	C₂²	—	2	2	no	—
196	C₂²	#17988499	1	1	yes	characteristic
197	C₂²	#17988737	1	1	yes	characteristic
198	C₂²	#17988742	1	1	yes	characteristic
199	C₄	—	2	4	no	—
200	C₄	—	2	2	no	—
201	C₄	—	2	2	no	—
202	C₄	—	4	2	no	—
203	C₄	—	8	2	no	—
204	C₄	—	584	4	no	—
205	C₄	#17988872	1	1	yes	characteristic
206	C₄	#17988882	1	1	yes	characteristic
207	C₄	#17988937	1	1	yes	characteristic
208	C₄	#17988946	1	1	yes	characteristic
209	C₂	—	2	1	no	—
210	C₂	—	2	1	no	—
211	C₂	#22611072	1	1	yes	characteristic
212	C₂	#22611103	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22618309

C₂

yes

maximal, characteristic

#22608495

C₂

yes

maximal

#22607791

C₂

yes

maximal

#22607764

C₂

yes

maximal, characteristic

C₇₃×#1994

C₂

yes

maximal, characteristic

C₂×#14556591

—

#17988640

C₂²

yes

—

#17988493

—

#17988453

—

#17988453

C₂²

yes

—

#17988324

—

#533×C₇₃

C₂²

yes

characteristic

#423×C₇₃

C₂²

yes

characteristic

C₄³×C₇₃

C₂²

yes

characteristic

#339×C₇₃

—

#339×C₇₃

—

C₂²×Dic₁₄₆

—

#14556591

—

#14556591

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

—

—

—

—

—

D₄

yes

—

C₂×M₁₆×C₇₃

—

C₄⋊Q₈×C₇₃

Q₈

yes

complemented

#125×C₇₃

—

C₂²⋊Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

#117×C₇₃

—

C₂×C₄²×C₇₃

—

C₂×C₄²×C₇₃

D₄

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

characteristic

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

SD₁₆

yes

—

C₄²×C₇₃

Q₁₆

yes

—

C₄²×C₇₃

C₂×Q₈

yes

characteristic

C₄²×C₇₃

#41

yes

—

C₄²×C₇₃

yes

C₂³×C₇₃

yes

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

#124

yes

—

C₂×C₄×C₇₃

#124

yes

characteristic

C₂×C₄×C₇₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₁₆

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

100

C₄×C₇₃

#489

yes

characteristic

101

Dic₇₃

—

102

#10363

—

maximal, Sylow

103

C₂×C₇₃

—

104

C₂×C₇₃

—

105

C₂×C₇₃

#2024

yes

characteristic

106

C₂×C₇₃

#2031

yes

—

107

#2862

—

108

#1994

D₇₃

yes

characteristic

109

#1518

—

110

#1286

—

111

#1276

—

112

C₇₃

#10363

yes

characteristic, complemented, Sylow

113

#533

C₂×D₇₃

yes

characteristic

114

C₂×C₄⋊Q₈

—

146

—

115

#431

—

116

#423

C₂×D₇₃

yes

characteristic

117

#391

—

146

—

118

#382

—

119

#382

—

146

—

120

C₄³

C₂×D₇₃

yes

characteristic

121

#339

—

122

#339

—

123

#328

—

146

—

124

C₂²×Q₈

—

146

—

125

C₂×M₁₆

—

126

C₄⋊Q₈

—

292

—

127

C₄⋊Q₈

—

292

—

128

C₄⋊Q₈

Dic₁₄₆

yes

complemented

129

#125

—

130

C₂²⋊Q₈

—

131

C₄×Q₈

—

132

#117

—

133

C₂×C₄⋊C₄

—

146

—

134

C₂×C₄⋊C₄

—

146

—

135

C₂×C₄²

—

136

C₂×C₄²

C₂²⋊D₇₃

yes

characteristic

137

C₂×C₄²

C₂²⋊D₇₃

yes

characteristic

138

C₂×C₄²

C₂²×D₇₃

yes

characteristic

139

C₄⋊C₈

—

140

C₄⋊C₈

—

146

—

141

C₂²⋊C₈

—

146

—

142

C₄×C₈

—

146

—

143

#95

—

144

#95

—

146

—

145

#95

—

146

—

146

C₂×Q₈

—

147

C₂×Q₈

—

292

—

148

C₂×Q₈

—

292

—

149

C₂²×C₄

—

150

C₂²×C₄

—

151

C₂²×C₄

—

146

—

152

C₂²×C₄

D₄×D₇₃

yes

—

153

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

154

M₁₆

—

155

M₁₆

—

156

C₂×C₈

—

157

C₂×C₈

—

292

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

292

—

162

C₄⋊C₄

—

292

—

163

C₄⋊C₄

—

292

—

164

C₂²⋊C₄

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₄²

—

169

C₄²

#11924015

yes

—

170

C₄²

#11924018

yes

—

171

C₄²

C₂×Dic₁₄₆

yes

characteristic

172

C₄²

#11924027

yes

—

173

C₄²

#11924028

yes

characteristic

174

C₂³

#14556559

yes

characteristic

175

Q₈

—

176

Q₈

—

584

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

292

—

184

C₂×C₄

—

292

—

185

C₂×C₄

#14556568

yes

—

186

C₂×C₄

#14556570

yes

characteristic

187

C₂×C₄

#14556577

yes

—

188

C₂×C₄

#14556578

yes

characteristic

189

C₂×C₄

C₂×#11924015

yes

characteristic

190

C₂×C₄

C₂×#11924018

yes

characteristic

191

C₂×C₄

#14556616

yes

characteristic

192

C₂×C₄

#14556621

yes

characteristic

193

C₈

—

194

C₈

—

584

—

195

C₂²

—

196

C₂²

#17988499

yes

characteristic

197

C₂²

#17988737

yes

characteristic

198

C₂²

#17988742

yes

characteristic

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

—

584

—

205

C₄

#17988872

yes

characteristic

206

C₄

#17988882

yes

characteristic

207

C₄

#17988937

yes

characteristic

208

C₄

#17988946

yes

characteristic

209

C₂

—

210

C₂

—

211

C₂

#22611072

yes

characteristic

212

C₂

#22611103

yes

—

	subgroup	G/H
center	C₂²	#17988499
commutator, cosocle	C₂×C₄²×C₇₃	C₂³
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1994	C₂
socle	C₂²×C₇₃	#393
prime radical	C₂³×C₇₃	#120
hypercenter	#1994	D₇₃
hypocenter	C₇₃	#10363
LCS	G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1994

subgroup

G/H

center

C₂²

#17988499

commutator, cosocle

C₂×C₄²×C₇₃

C₂³

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1994

C₂

C₂²×C₇₃

C₂³×C₇₃

D₇₃

C₇₃

G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1994

N	H ≅ G/N	type
29. C₄⋊Q₈×C₇₃	Q₈	semidirect
112. C₇₃	#10363	semidirect
128. C₄⋊Q₈	Dic₁₄₆	semidirect

H ≅ G/N

type

29. C₄⋊Q₈×C₇₃

Q₈

semidirect

112. C₇₃

#10363

semidirect

128. C₄⋊Q₈

Dic₁₄₆

semidirect

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

subgroup	index
#22618309	2
#22608495	2
#22607791	2
#22607764	2
C₇₃×#1994	2
#10363	73

subgroup

index

C₇₃×#1994

Subgroups — #32989077

Conjugacy classes of subgroups

Complements

Maximal subgroups