Group #33005789 — subgroups

Subgroups — #33005789

Order 18688.

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

normal subgroups only

This group has 326 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	C₂×#17988391	C₂	1	2	yes	maximal
2	#22610095	C₂	1	1	yes	maximal, characteristic
3	#22608022	C₂	1	2	yes	maximal
4	#22608009	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1969	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924003	C₂²	1	2	yes	—
7	C₂×#14556525	C₂²	1	2	yes	—
8	C₂×#14556523	—	2	1	no	—
9	#17988453	—	2	1	no	—
10	#17988391	—	2	4	no	—
11	C₂²×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
12	C₂×#107×C₇₃	C₂²	1	1	yes	characteristic
13	C₂×#103×C₇₃	—	2	1	no	—
14	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
15	#391×C₇₃	—	2	1	no	—
16	C₂×C₄⋊Dic₇₃	—	2	2	no	—
17	C₂×#11924003	—	2	2	no	—
18	C₂×#11924003	—	2	2	no	—
19	C₂×#11924003	—	2	2	no	—
20	#14556525	—	2	2	no	—
21	#14556525	—	2	2	no	—
22	#14556523	—	4	2	no	—
23	#14556498	—	4	2	no	—
24	C₂²×Q₈×C₇₃	—	2	1	no	—
25	C₂×Q₁₆×C₇₃	—	2	2	no	—
26	C₂×Q₁₆×C₇₃	—	2	2	no	—
27	C₂×Q₁₆×C₇₃	—	2	1	no	—
28	C₂×Q₁₆×C₇₃	D₄	1	2	yes	—
29	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
32	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
33	#107×C₇₃	—	2	2	no	—
34	#103×C₇₃	—	4	2	no	—
35	C₄×C₈×C₇₃	—	2	2	no	—
36	#95×C₇₃	—	4	2	no	—
37	C₂²×Dic₇₃	—	4	2	no	—
38	C₄⋊Dic₇₃	—	4	2	no	—
39	C₄⋊Dic₇₃	—	4	2	no	—
40	#11924003	—	4	2	no	—
41	C₂×Q₈×C₇₃	—	2	2	no	—
42	C₂×Q₈×C₇₃	—	4	1	no	—
43	C₂×Q₈×C₇₃	—	4	1	no	—
44	C₂×Q₈×C₇₃	—	4	1	no	—
45	C₂×Q₈×C₇₃	—	4	1	no	—
46	C₂×Q₈×C₇₃	—	4	1	no	—
47	C₂²×C₄×C₇₃	—	4	1	no	—
48	C₂²×C₄×C₇₃	—	4	1	no	—
49	C₂²×C₄×C₇₃	D₈	1	2	yes	—
50	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
51	Q₁₆×C₇₃	—	2	4	no	—
52	Q₁₆×C₇₃	—	4	2	no	—
53	C₂×C₈×C₇₃	—	2	1	no	—
54	C₂×C₈×C₇₃	—	2	1	no	—
55	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
56	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₇₃	#41	1	2	yes	—
58	C₂×C₈×C₇₃	#41	1	2	yes	—
59	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
60	C₄⋊C₄×C₇₃	—	4	1	no	—
61	C₄⋊C₄×C₇₃	—	4	1	no	—
62	C₄²×C₇₃	—	2	1	no	—
63	C₄²×C₇₃	—	2	1	no	—
64	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
65	Q₈×C₇₃	—	4	2	no	—
66	Q₈×C₇₃	—	4	2	no	—
67	Q₈×C₇₃	—	8	1	no	—
68	Q₈×C₇₃	—	8	1	no	—
69	C₂×C₄×C₇₃	—	2	2	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	—	8	1	no	—
74	C₂×C₄×C₇₃	—	8	1	no	—
75	C₂×C₄×C₇₃	—	8	1	no	—
76	C₂×C₄×C₇₃	—	8	2	no	—
77	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
78	C₂×C₄×C₇₃	#123	1	1	yes	characteristic
79	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
80	C₂×C₄×C₇₃	C₂×SD₁₆	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	—	8	2	no	—
84	C₂×Dic₇₃	—	8	2	no	—
85	C₂×Dic₇₃	—	8	2	no	—
86	C₈×C₇₃	—	2	2	no	—
87	C₈×C₇₃	#123	1	2	yes	—
88	C₈×C₇₃	#126	1	2	yes	—
89	C₂²×C₇₃	#459	1	1	yes	characteristic
90	C₂²×C₇₃	#480	1	1	yes	characteristic
91	C₂²×C₇₃	#486	1	1	yes	characteristic
92	C₂²×C₇₃	#508	1	2	yes	—
93	C₂²×C₇₃	#510	1	2	yes	—
94	C₄×C₇₃	—	4	2	no	—
95	C₄×C₇₃	—	8	2	no	—
96	C₄×C₇₃	—	16	1	no	—
97	C₄×C₇₃	#397	1	1	yes	characteristic
98	C₄×C₇₃	#461	1	1	yes	characteristic
99	C₄×C₇₃	#481	1	1	yes	characteristic
100	C₄×C₇₃	#489	1	1	yes	characteristic
101	Dic₇₃	—	16	2	no	—
102	#13483	—	73	1	no	maximal, Sylow
103	C₂×C₇₃	#2071	1	1	yes	characteristic
104	C₂×C₇₃	#2211	1	2	yes	—
105	C₂×C₇₃	#2237	1	2	yes	—
106	C₂×C₇₃	#2242	1	1	yes	characteristic
107	C₂×C₇₃	#2245	1	1	yes	characteristic
108	C₂×#358	—	73	2	no	—
109	#1969	D₇₃	1	1	yes	characteristic
110	#1846	—	73	1	no	—
111	#1380	—	73	2	no	—
112	#1374	—	73	1	no	—
113	C₇₃	#13483	1	1	yes	characteristic, complemented, Sylow
114	C₂²×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
115	C₂²×C₁₆	—	73	2	no	—
116	C₂×#107	—	73	2	no	—
117	C₂×#107	C₂×D₇₃	1	1	yes	characteristic
118	C₂×#106	—	146	1	no	—
119	C₂×#103	—	2	1	no	—
120	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
121	#391	—	2	1	no	—
122	#382	—	146	1	no	—
123	#358	—	146	4	no	—
124	C₂²×Q₈	—	2	1	no	—
125	C₂×Q₁₆	—	2	2	no	—
126	C₂×Q₁₆	—	2	2	no	—
127	C₂×Q₁₆	—	2	1	no	—
128	C₂×Q₁₆	D₂₉₂	1	2	yes	—
129	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
130	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
131	C₂×C₄⋊C₄	—	2	1	no	—
132	C₂×C₄⋊C₄	—	146	2	no	—
133	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
134	C₂×C₁₆	—	146	2	no	—
135	C₂×C₁₆	—	146	2	no	—
136	C₂×C₁₆	—	146	2	no	—
137	#107	—	2	2	no	—
138	#107	—	146	2	no	—
139	#107	—	146	2	no	—
140	#106	—	292	2	no	—
141	#103	—	4	2	no	—
142	C₄×C₈	—	2	2	no	—
143	#95	—	4	2	no	—
144	#95	—	292	2	no	—
145	C₂×Q₈	—	2	2	no	—
146	C₂×Q₈	—	4	1	no	—
147	C₂×Q₈	—	4	1	no	—
148	C₂×Q₈	—	4	1	no	—
149	C₂×Q₈	—	4	1	no	—
150	C₂×Q₈	—	4	1	no	—
151	C₂²×C₄	—	4	1	no	—
152	C₂²×C₄	—	4	1	no	—
153	C₂²×C₄	—	292	2	no	—
154	C₂²×C₄	#11924013	1	2	yes	—
155	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
156	Q₁₆	—	2	4	no	—
157	Q₁₆	—	4	2	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	#11924016	1	2	yes	—
161	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
162	C₂×C₈	#11924027	1	2	yes	—
163	C₂×C₈	#11924027	1	2	yes	—
164	C₂×C₈	#11924028	1	1	yes	characteristic
165	C₄⋊C₄	—	4	1	no	—
166	C₄⋊C₄	—	4	1	no	—
167	C₄⋊C₄	—	292	2	no	—
168	C₄⋊C₄	—	292	2	no	—
169	C₄²	—	2	1	no	—
170	C₄²	—	2	1	no	—
171	C₁₆	—	292	2	no	—
172	C₂³	C₂×#11924013	1	1	yes	characteristic
173	Q₈	—	4	2	no	—
174	Q₈	—	4	2	no	—
175	Q₈	—	8	1	no	—
176	Q₈	—	8	1	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	8	1	no	—
182	C₂×C₄	—	8	1	no	—
183	C₂×C₄	—	8	1	no	—
184	C₂×C₄	—	8	2	no	—
185	C₂×C₄	—	584	2	no	—
186	C₂×C₄	—	584	2	no	—
187	C₂×C₄	—	584	2	no	—
188	C₂×C₄	#14556565	1	1	yes	characteristic
189	C₂×C₄	#14556572	1	1	yes	characteristic
190	C₂×C₄	#14556578	1	1	yes	characteristic
191	C₂×C₄	C₂×#11924016	1	1	yes	characteristic
192	C₂×C₄	#14556616	1	1	yes	characteristic
193	C₂×C₄	#14556621	1	1	yes	characteristic
194	C₈	—	2	2	no	—
195	C₈	#14556575	1	2	yes	—
196	C₈	#14556584	1	2	yes	—
197	C₂²	#17988782	1	1	yes	characteristic
198	C₂²	#17988898	1	1	yes	characteristic
199	C₂²	#17988933	1	1	yes	characteristic
200	C₂²	#17989017	1	2	yes	—
201	C₂²	#17989028	1	2	yes	—
202	C₄	—	4	2	no	—
203	C₄	—	8	2	no	—
204	C₄	—	16	1	no	—
205	C₄	—	1168	2	no	—
206	C₄	#17988512	1	1	yes	characteristic
207	C₄	#17988792	1	1	yes	characteristic
208	C₄	#17988903	1	1	yes	characteristic
209	C₄	#17988946	1	1	yes	characteristic
210	C₂	#22611334	1	1	yes	characteristic
211	C₂	#22611993	1	2	yes	—
212	C₂	#22612136	1	2	yes	—
213	C₂	#22612162	1	1	yes	characteristic
214	C₂	#22612176	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988391

C₂

yes

maximal

#22610095

C₂

yes

maximal, characteristic

#22608022

C₂

yes

maximal

#22608009

C₂

yes

maximal, characteristic

C₇₃×#1969

C₂

yes

maximal, characteristic

C₂²×#11924003

C₂²

yes

—

C₂×#14556525

C₂²

yes

—

C₂×#14556523

—

#17988453

—

#17988391

—

C₂²×Q₁₆×C₇₃

C₂²

yes

characteristic

C₂×#107×C₇₃

C₂²

yes

characteristic

C₂×#103×C₇₃

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#391×C₇₃

—

C₂×C₄⋊Dic₇₃

—

—

—

—

—

—

—

—

C₂²×Q₈×C₇₃

—

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

D₄

yes

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

D₄

yes

characteristic

#107×C₇₃

—

#103×C₇₃

—

C₄×C₈×C₇₃

—

#95×C₇₃

—

C₂²×Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924003

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

D₈

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

SD₁₆

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

#41

yes

—

C₂×C₈×C₇₃

#41

yes

—

C₂×C₈×C₇₃

#41

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

C₂×D₈

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×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

C₂×SD₁₆

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

—

C₈×C₇₃

#123

yes

—

C₈×C₇₃

#126

yes

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₂²×C₇₃

#510

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

100

C₄×C₇₃

#489

yes

characteristic

101

Dic₇₃

—

102

#13483

—

maximal, Sylow

103

C₂×C₇₃

#2071

yes

characteristic

104

C₂×C₇₃

#2211

yes

—

105

C₂×C₇₃

#2237

yes

—

106

C₂×C₇₃

#2242

yes

characteristic

107

C₂×C₇₃

#2245

yes

characteristic

108

C₂×#358

—

109

#1969

D₇₃

yes

characteristic

110

#1846

—

111

#1380

—

112

#1374

—

113

C₇₃

#13483

yes

characteristic, complemented, Sylow

114

C₂²×Q₁₆

C₂×D₇₃

yes

characteristic

115

C₂²×C₁₆

—

116

C₂×#107

—

117

C₂×#107

C₂×D₇₃

yes

characteristic

118

C₂×#106

—

146

—

119

C₂×#103

—

120

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

121

#391

—

122

#382

—

146

—

123

#358

—

146

—

124

C₂²×Q₈

—

125

C₂×Q₁₆

—

126

C₂×Q₁₆

—

127

C₂×Q₁₆

—

128

C₂×Q₁₆

D₂₉₂

yes

—

129

C₂²×C₈

C₂²⋊D₇₃

yes

characteristic

130

C₂²×C₈

C₂²×D₇₃

yes

characteristic

131

C₂×C₄⋊C₄

—

132

C₂×C₄⋊C₄

—

146

—

133

C₂×C₄²

C₂²⋊D₇₃

yes

characteristic

134

C₂×C₁₆

—

146

—

135

C₂×C₁₆

—

146

—

136

C₂×C₁₆

—

146

—

137

#107

—

138

#107

—

146

—

139

#107

—

146

—

140

#106

—

292

—

141

#103

—

142

C₄×C₈

—

143

#95

—

144

#95

—

292

—

145

C₂×Q₈

—

146

C₂×Q₈

—

147

C₂×Q₈

—

148

C₂×Q₈

—

149

C₂×Q₈

—

150

C₂×Q₈

—

151

C₂²×C₄

—

152

C₂²×C₄

—

153

C₂²×C₄

—

292

—

154

C₂²×C₄

#11924013

yes

—

155

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

156

Q₁₆

—

157

Q₁₆

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

#11924016

yes

—

161

C₂×C₈

C₂×D₂₉₂

yes

characteristic

162

C₂×C₈

#11924027

yes

—

163

C₂×C₈

#11924027

yes

—

164

C₂×C₈

#11924028

yes

characteristic

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

292

—

168

C₄⋊C₄

—

292

—

169

C₄²

—

170

C₄²

—

171

C₁₆

—

292

—

172

C₂³

C₂×#11924013

yes

characteristic

173

Q₈

—

174

Q₈

—

175

Q₈

—

176

Q₈

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

584

—

186

C₂×C₄

—

584

—

187

C₂×C₄

—

584

—

188

C₂×C₄

#14556565

yes

characteristic

189

C₂×C₄

#14556572

yes

characteristic

190

C₂×C₄

#14556578

yes

characteristic

191

C₂×C₄

C₂×#11924016

yes

characteristic

192

C₂×C₄

#14556616

yes

characteristic

193

C₂×C₄

#14556621

yes

characteristic

194

C₈

—

195

C₈

#14556575

yes

—

196

C₈

#14556584

yes

—

197

C₂²

#17988782

yes

characteristic

198

C₂²

#17988898

yes

characteristic

199

C₂²

#17988933

yes

characteristic

200

C₂²

#17989017

yes

—

201

C₂²

#17989028

yes

—

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

—

1168

—

206

C₄

#17988512

yes

characteristic

207

C₄

#17988792

yes

characteristic

208

C₄

#17988903

yes

characteristic

209

C₄

#17988946

yes

characteristic

210

C₂

#22611334

yes

characteristic

211

C₂

#22611993

yes

—

212

C₂

#22612136

yes

—

213

C₂

#22612162

yes

characteristic

214

C₂

#22612176

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#11924013
commutator, cosocle	C₂²×C₈×C₇₃	C₂³
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1969	C₂
socle, prime radical	C₂³×C₇₃	C₂×D₈
hypercenter	#1969	D₇₃
hypocenter	C₇₃	#13483
LCS	G ▷ C₂²×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1969

subgroup

G/H

center

C₂³

C₂×#11924013

commutator, cosocle

C₂²×C₈×C₇₃

C₂³

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₇₃×#1969

C₂

socle, prime radical

C₂³×C₇₃

C₂×D₈

D₇₃

C₇₃

G ▷ C₂²×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1969

N	H ≅ G/N	type
113. C₇₃	#13483	semidirect

H ≅ G/N

type

113. C₇₃

#13483

semidirect

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

subgroup	index
C₂×#17988391	2
#22610095	2
#22608022	2
#22608009	2
C₇₃×#1969	2
#13483	73

subgroup

index

C₇₃×#1969

Subgroups — #33005789

Conjugacy classes of subgroups

Complements

Maximal subgroups