Group #32990752 — subgroups

Subgroups — #32990752

Order 18688.

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

normal subgroups only

This group has 330 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	#22610085	C₂	1	1	yes	maximal, characteristic, complemented
2	#22608580	C₂	1	2	yes	maximal
3	#22607826	C₂	1	2	yes	maximal, complemented
4	#22607760	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#1963	C₂	1	1	yes	maximal, characteristic
6	#17988640	C₂²	1	2	yes	—
7	C₂×C₄⋊#10596831	—	2	1	no	—
8	C₂×C₄⋊#10596831	—	2	1	no	—
9	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
10	#17988333	—	2	2	no	—
11	#17988331	—	2	2	no	—
12	#532×C₇₃	C₂²	1	1	yes	characteristic
13	#423×C₇₃	C₂²	1	1	yes	characteristic
14	C₂×#104×C₇₃	—	2	1	no	—
15	C₂×#104×C₇₃	—	2	1	no	—
16	#376×C₇₃	C₂²	1	1	yes	characteristic
17	C₂²×#10596831	—	2	2	no	—
18	C₄⋊#10596831	—	2	2	no	—
19	C₄⋊#10596831	—	2	2	no	—
20	C₄⋊#10596831	—	2	2	no	—
21	C₄⋊#10596831	—	4	2	no	—
22	C₄⋊#10596831	—	4	2	no	—
23	C₂²⋊#10596831	—	2	2	no	—
24	C₄×#10596831	—	2	2	no	—
25	C₂×#41×C₇₃	—	2	1	no	—
26	C₂×M₁₆×C₇₃	—	2	1	no	—
27	C₄⋊Q₈×C₇₃	D₄	1	1	yes	characteristic
28	#127×C₇₃	D₄	1	1	yes	characteristic
29	#124×C₇₃	—	2	1	no	—
30	#121×C₇₃	—	2	2	no	—
31	C₄×D₄×C₇₃	—	2	2	no	—
32	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₇₃	—	2	2	no	—
36	#104×C₇₃	—	4	2	no	—
37	#104×C₇₃	—	4	2	no	—
38	#95×C₇₃	—	2	2	no	—
39	C₂×#10596831	—	4	2	no	—
40	C₂×#10596831	—	4	2	no	—
41	C₂×#10596831	—	4	2	no	—
42	C₂×#10596831	—	4	2	no	—
43	#41×C₇₃	—	4	1	no	—
44	#41×C₇₃	—	4	1	no	—
45	#41×C₇₃	—	4	1	no	—
46	#41×C₇₃	—	4	1	no	—
47	C₂×Q₈×C₇₃	—	2	1	no	—
48	C₂×D₄×C₇₃	—	2	1	no	—
49	C₂×D₄×C₇₃	—	2	2	no	—
50	C₂×D₄×C₇₃	—	4	1	no	—
51	C₂²×C₄×C₇₃	—	2	1	no	—
52	C₂²×C₄×C₇₃	—	2	1	no	—
53	C₂²×C₄×C₇₃	—	4	1	no	—
54	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₃	#41	1	2	yes	—
56	M₁₆×C₇₃	—	4	1	no	—
57	M₁₆×C₇₃	—	4	1	no	—
58	C₂×C₈×C₇₃	—	4	1	no	—
59	C₄⋊C₄×C₇₃	—	2	2	no	—
60	C₂²⋊C₄×C₇₃	—	4	2	no	—
61	C₄²×C₇₃	—	2	1	no	—
62	C₄²×C₇₃	—	2	1	no	—
63	C₄²×C₇₃	D₈	1	2	yes	—
64	C₄²×C₇₃	SD₁₆	1	2	yes	—
65	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₇₃	#41	1	2	yes	—
67	C₄²×C₇₃	#41	1	1	yes	characteristic
68	C₂³×C₇₃	—	4	1	no	—
69	C₂³×C₇₃	#123	1	1	yes	characteristic
70	Q₈×C₇₃	—	4	1	no	—
71	Q₈×C₇₃	—	4	1	no	—
72	D₄×C₇₃	—	4	1	no	—
73	D₄×C₇₃	—	4	1	no	—
74	D₄×C₇₃	—	8	2	no	—
75	D₄×C₇₃	—	8	1	no	—
76	D₄×C₇₃	—	8	1	no	—
77	C₂×C₄×C₇₃	—	2	2	no	—
78	C₂×C₄×C₇₃	—	2	2	no	—
79	C₂×C₄×C₇₃	—	4	2	no	—
80	C₂×C₄×C₇₃	—	4	2	no	—
81	C₂×C₄×C₇₃	—	4	2	no	—
82	C₂×C₄×C₇₃	—	4	1	no	—
83	C₂×C₄×C₇₃	—	8	1	no	—
84	C₂×C₄×C₇₃	—	8	1	no	—
85	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
86	C₂×C₄×C₇₃	#123	1	2	yes	—
87	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
88	C₂×C₄×C₇₃	#126	1	2	yes	—
89	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
91	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
92	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
93	C₈×C₇₃	—	8	1	no	—
94	#10596831	—	8	2	no	—
95	#10596831	—	8	2	no	—
96	C₂²×C₇₃	—	2	2	no	—
97	C₂²×C₇₃	—	8	1	no	—
98	C₂²×C₇₃	—	8	2	no	—
99	C₂²×C₇₃	#397	1	1	yes	characteristic
100	C₂²×C₇₃	#480	1	1	yes	characteristic
101	C₂²×C₇₃	#481	1	1	yes	characteristic
102	C₄×C₇₃	—	2	2	no	—
103	C₄×C₇₃	—	2	2	no	—
104	C₄×C₇₃	—	4	2	no	—
105	C₄×C₇₃	—	4	2	no	—
106	C₄×C₇₃	—	8	1	no	—
107	C₄×C₇₃	#459	1	1	yes	characteristic
108	C₄×C₇₃	#461	1	1	yes	characteristic
109	C₄×C₇₃	#486	1	1	yes	characteristic
110	C₄×C₇₃	#489	1	1	yes	characteristic
111	#10627	—	73	1	no	maximal, Sylow
112	C₂×C₇₃	—	2	1	no	—
113	C₂×C₇₃	—	2	1	no	—
114	C₂×C₇₃	—	16	1	no	—
115	C₂×C₇₃	#2071	1	1	yes	characteristic
116	C₂×C₇₃	#2080	1	2	yes	—
117	#1963	D₇₃	1	1	yes	characteristic
118	#1843	—	73	1	no	—
119	#1540	—	73	2	no	—
120	#1301	—	73	2	no	—
121	#1274	—	73	1	no	—
122	C₇₃	#10627	1	1	yes	characteristic, complemented, Sylow
123	#532	C₂×D₇₃	1	1	yes	characteristic
124	#431	—	73	2	no	—
125	#423	C₂×D₇₃	1	1	yes	characteristic
126	C₂×C₄⋊C₈	—	73	2	no	—
127	C₂×C₄⋊C₈	—	146	1	no	—
128	C₂×C₄⋊C₈	—	146	1	no	—
129	C₂×#104	—	2	1	no	—
130	C₂×#104	—	2	1	no	—
131	#376	C₂×D₇₃	1	1	yes	characteristic
132	#332	—	146	2	no	—
133	#331	—	146	2	no	—
134	C₂×#41	—	2	1	no	—
135	C₂×M₁₆	—	2	1	no	—
136	C₂²×C₈	—	146	2	no	—
137	C₄⋊Q₈	D₂₉₂	1	1	yes	characteristic
138	#127	D₂₉₂	1	1	yes	characteristic
139	#124	—	2	1	no	—
140	#121	—	2	2	no	—
141	C₄×D₄	—	2	2	no	—
142	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
143	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
144	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	—	292	2	no	—
147	C₄⋊C₈	—	292	2	no	—
148	C₄⋊C₈	—	146	2	no	—
149	C₄⋊C₈	—	146	2	no	—
150	C₄⋊C₈	—	146	2	no	—
151	#104	—	4	2	no	—
152	#104	—	4	2	no	—
153	C₂²⋊C₈	—	146	2	no	—
154	C₄×C₈	—	146	2	no	—
155	#95	—	2	2	no	—
156	#41	—	4	1	no	—
157	#41	—	4	1	no	—
158	#41	—	4	1	no	—
159	#41	—	4	1	no	—
160	C₂×Q₈	—	2	1	no	—
161	C₂×D₄	—	2	1	no	—
162	C₂×D₄	—	2	2	no	—
163	C₂×D₄	—	4	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	4	1	no	—
167	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
168	C₂²×C₄	#11924027	1	2	yes	—
169	M₁₆	—	4	1	no	—
170	M₁₆	—	4	1	no	—
171	C₂×C₈	—	4	1	no	—
172	C₂×C₈	—	292	2	no	—
173	C₂×C₈	—	292	2	no	—
174	C₂×C₈	—	292	2	no	—
175	C₂×C₈	—	292	2	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₂²⋊C₄	—	4	2	no	—
178	C₄²	—	2	1	no	—
179	C₄²	—	2	1	no	—
180	C₄²	#11924013	1	2	yes	—
181	C₄²	#11924016	1	2	yes	—
182	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
183	C₄²	#11924027	1	2	yes	—
184	C₄²	#11924028	1	1	yes	characteristic
185	C₂³	—	4	1	no	—
186	C₂³	#14556572	1	1	yes	characteristic
187	Q₈	—	4	1	no	—
188	Q₈	—	4	1	no	—
189	D₄	—	4	1	no	—
190	D₄	—	4	1	no	—
191	D₄	—	8	2	no	—
192	D₄	—	8	1	no	—
193	D₄	—	8	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	8	1	no	—
201	C₂×C₄	—	8	1	no	—
202	C₂×C₄	#14556565	1	1	yes	characteristic
203	C₂×C₄	#14556575	1	2	yes	—
204	C₂×C₄	#14556578	1	1	yes	characteristic
205	C₂×C₄	#14556584	1	2	yes	—
206	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
207	C₂×C₄	C₂×#11924016	1	1	yes	characteristic
208	C₂×C₄	#14556616	1	1	yes	characteristic
209	C₂×C₄	#14556621	1	1	yes	characteristic
210	C₈	—	8	1	no	—
211	C₈	—	584	2	no	—
212	C₈	—	584	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	8	1	no	—
215	C₂²	—	8	2	no	—
216	C₂²	#17988512	1	1	yes	characteristic
217	C₂²	#17988898	1	1	yes	characteristic
218	C₂²	#17988903	1	1	yes	characteristic
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	4	2	no	—
222	C₄	—	4	2	no	—
223	C₄	—	8	1	no	—
224	C₄	#17988782	1	1	yes	characteristic
225	C₄	#17988792	1	1	yes	characteristic
226	C₄	#17988933	1	1	yes	characteristic
227	C₄	#17988946	1	1	yes	characteristic
228	C₂	—	2	1	no	—
229	C₂	—	2	1	no	—
230	C₂	—	16	1	no	—
231	C₂	#22611334	1	1	yes	characteristic
232	C₂	#22611381	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22610085

C₂

yes

maximal, characteristic, complemented

#22608580

C₂

yes

maximal

#22607826

C₂

yes

maximal, complemented

#22607760

C₂

yes

maximal, characteristic, complemented

C₇₃×#1963

C₂

yes

maximal, characteristic

#17988640

C₂²

yes

—

C₂×C₄⋊#10596831

—

C₂×C₄⋊#10596831

—

C₂×C₄⋊#10596831

C₂²

yes

—

#17988333

—

#17988331

—

#532×C₇₃

C₂²

yes

characteristic

#423×C₇₃

C₂²

yes

characteristic

C₂×#104×C₇₃

—

C₂×#104×C₇₃

—

#376×C₇₃

C₂²

yes

—

—

—

—

—

—

—

—

C₂×#41×C₇₃

—

C₂×M₁₆×C₇₃

—

C₄⋊Q₈×C₇₃

D₄

yes

characteristic

#127×C₇₃

D₄

yes

characteristic

#124×C₇₃

—

#121×C₇₃

—

C₄×D₄×C₇₃

—

C₂×C₄²×C₇₃

D₄

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

characteristic

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

—

#104×C₇₃

—

#104×C₇₃

—

#95×C₇₃

—

—

—

—

—

#41×C₇₃

—

#41×C₇₃

—

#41×C₇₃

—

#41×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

#41

yes

—

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

D₈

yes

—

C₄²×C₇₃

SD₁₆

yes

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

#41

yes

—

C₄²×C₇₃

#41

yes

characteristic

C₂³×C₇₃

—

C₂³×C₇₃

#123

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

D₄×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×C₇₃

#121

yes

characteristic

C₂×C₄×C₇₃

#123

yes

—

C₂×C₄×C₇₃

#124

yes

characteristic

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

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#397

yes

characteristic

100

C₂²×C₇₃

#480

yes

characteristic

101

C₂²×C₇₃

#481

yes

characteristic

102

C₄×C₇₃

—

103

C₄×C₇₃

—

104

C₄×C₇₃

—

105

C₄×C₇₃

—

106

C₄×C₇₃

—

107

C₄×C₇₃

#459

yes

characteristic

108

C₄×C₇₃

#461

yes

characteristic

109

C₄×C₇₃

#486

yes

characteristic

110

C₄×C₇₃

yes

111

—

112

C₂×C₇₃

—

113

C₂×C₇₃

—

114

C₂×C₇₃

—

115

C₂×C₇₃

#2071

yes

characteristic

116

C₂×C₇₃

#2080

yes

—

117

#1963

D₇₃

yes

characteristic

118

#1843

—

119

#1540

—

120

#1301

—

121

#1274

—

122

C₇₃

#10627

yes

characteristic, complemented, Sylow

123

#532

C₂×D₇₃

yes

characteristic

124

#431

—

125

#423

C₂×D₇₃

yes

characteristic

126

C₂×C₄⋊C₈

—

127

C₂×C₄⋊C₈

—

146

—

128

C₂×C₄⋊C₈

—

146

—

129

C₂×#104

—

130

C₂×#104

—

131

#376

C₂×D₇₃

yes

characteristic

132

#332

—

146

—

133

#331

—

146

—

134

C₂×#41

—

135

C₂×M₁₆

—

136

C₂²×C₈

—

146

—

137

C₄⋊Q₈

D₂₉₂

yes

characteristic

138

#127

D₂₉₂

yes

characteristic

139

#124

—

140

#121

—

141

C₄×D₄

—

142

C₂×C₄²

C₂²⋊D₇₃

yes

characteristic

143

C₂×C₄²

C₂²⋊D₇₃

yes

characteristic

144

C₂×C₄²

C₂²×D₇₃

yes

characteristic

145

C₄⋊C₈

—

146

C₄⋊C₈

—

292

—

147

C₄⋊C₈

—

292

—

148

C₄⋊C₈

—

146

—

149

C₄⋊C₈

—

146

—

150

C₄⋊C₈

—

146

—

151

#104

—

152

#104

—

153

C₂²⋊C₈

—

146

—

154

C₄×C₈

—

146

—

155

#95

—

156

#41

—

157

#41

—

158

#41

—

159

#41

—

160

C₂×Q₈

—

161

C₂×D₄

—

162

C₂×D₄

—

163

C₂×D₄

—

164

C₂²×C₄

—

165

C₂²×C₄

—

166

C₂²×C₄

—

167

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

168

C₂²×C₄

#11924027

yes

—

169

M₁₆

—

170

M₁₆

—

171

C₂×C₈

—

172

C₂×C₈

—

292

—

173

C₂×C₈

—

292

—

174

C₂×C₈

—

292

—

175

C₂×C₈

—

292

—

176

C₄⋊C₄

—

177

C₂²⋊C₄

—

178

C₄²

—

179

C₄²

—

180

C₄²

#11924013

yes

—

181

C₄²

#11924016

yes

—

182

C₄²

C₂×D₂₉₂

yes

characteristic

183

C₄²

#11924027

yes

—

184

C₄²

#11924028

yes

characteristic

185

C₂³

—

186

C₂³

#14556572

yes

characteristic

187

Q₈

—

188

Q₈

—

189

D₄

—

190

D₄

—

191

D₄

—

192

D₄

—

193

D₄

—

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

#14556565

yes

characteristic

203

C₂×C₄

#14556575

yes

—

204

C₂×C₄

#14556578

yes

characteristic

205

C₂×C₄

#14556584

yes

—

206

C₂×C₄

C₂×#11924013

yes

characteristic

207

C₂×C₄

C₂×#11924016

yes

characteristic

208

C₂×C₄

#14556616

yes

characteristic

209

C₂×C₄

#14556621

yes

characteristic

210

C₈

—

211

C₈

—

584

—

212

C₈

—

584

—

213

C₂²

—

214

C₂²

—

215

C₂²

—

216

C₂²

#17988512

yes

characteristic

217

C₂²

#17988898

yes

characteristic

218

C₂²

#17988903

yes

characteristic

219

C₄

—

220

C₄

—

221

C₄

—

222

C₄

—

223

C₄

—

224

C₄

#17988782

yes

characteristic

225

C₄

#17988792

yes

characteristic

226

C₄

#17988933

yes

characteristic

227

C₄

#17988946

yes

characteristic

228

C₂

—

229

C₂

—

230

C₂

—

231

C₂

#22611334

yes

characteristic

232

C₂

#22611381

yes

—

	subgroup	G/H
center	C₂×C₄	#14556578
commutator, cosocle	C₂×C₄²×C₇₃	C₂³
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1963	C₂
socle	C₂²×C₇₃	#397
prime radical	#532×C₇₃	C₂²
hypercenter	#1963	D₇₃
hypocenter	C₇₃	#10627
LCS	G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1963

subgroup

G/H

center

C₂×C₄

#14556578

commutator, cosocle

C₂×C₄²×C₇₃

C₂³

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1963

C₂

socle

C₂²×C₇₃

#397

prime radical

#532×C₇₃

C₂²

D₇₃

C₇₃

G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1963

N	H ≅ G/N	type
1. #22610085	C₂	semidirect
3. #22607826	C₂	semidirect
4. #22607760	C₂	semidirect
122. C₇₃	#10627	semidirect

H ≅ G/N

type

1. #22610085

C₂

semidirect

3. #22607826

C₂

semidirect

4. #22607760

C₂

semidirect

122. C₇₃

#10627

semidirect

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

subgroup	index
#22610085	2
#22608580	2
#22607826	2
#22607760	2
C₇₃×#1963	2
#10627	73

subgroup

index

C₇₃×#1963

Subgroups — #32990752

Conjugacy classes of subgroups

Complements

Maximal subgroups