Group #33005791 — subgroups

Subgroups — #33005791

Order 18688.

326 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 109 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×#17988392	C₂	1	2	yes	maximal
2	#22610792	C₂	1	1	yes	maximal, characteristic
3	#22610095	C₂	1	1	yes	maximal, characteristic
4	#22608023	C₂	1	2	yes	maximal
5	C₇₃×#1374	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₉₂	C₂²	1	1	yes	characteristic
7	C₂×#14556525	C₂²	1	1	yes	characteristic
8	C₂×#14556525	C₂²	1	2	yes	—
9	C₂×#14556523	—	2	1	no	—
10	C₂×#14556516	—	2	1	no	—
11	#17988493	—	2	1	no	—
12	#17988453	—	2	1	no	—
13	#17988392	—	2	4	no	—
14	C₂²×C₁₆×C₇₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
16	C₂²×Dic₁₄₆	—	2	1	no	—
17	C₂×Dic₂₉₂	—	2	2	no	—
18	C₂×Dic₂₉₂	—	2	2	no	—
19	C₂×Dic₂₉₂	—	2	1	no	—
20	C₂×Dic₂₉₂	D₄	1	2	yes	—
21	C₂×C₄⋊Dic₇₃	—	2	2	no	—
22	C₂×C₄⋊Dic₇₃	—	2	1	no	—
23	#14556525	—	2	2	no	—
24	#14556525	—	2	2	no	—
25	#14556525	—	2	2	no	—
26	#14556523	—	4	2	no	—
27	#14556516	—	4	2	no	—
28	#14556498	—	4	2	no	—
29	#14556498	—	4	2	no	—
30	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
33	C₂×C₁₆×C₇₃	—	2	2	no	—
34	C₂×C₁₆×C₇₃	—	2	2	no	—
35	C₂×C₁₆×C₇₃	—	2	2	no	—
36	C₄×C₈×C₇₃	—	2	2	no	—
37	C₂×Dic₁₄₆	—	2	2	no	—
38	C₂×Dic₁₄₆	—	4	1	no	—
39	C₂×Dic₁₄₆	—	4	1	no	—
40	C₂×Dic₁₄₆	—	4	1	no	—
41	C₂×Dic₁₄₆	—	4	1	no	—
42	C₂×Dic₁₄₆	—	4	1	no	—
43	C₂²×Dic₇₃	—	4	1	no	—
44	C₂²×Dic₇₃	—	4	2	no	—
45	C₂²×Dic₇₃	—	4	1	no	—
46	Dic₂₉₂	—	2	4	no	—
47	Dic₂₉₂	—	4	2	no	—
48	C₄⋊Dic₇₃	—	4	2	no	—
49	C₄⋊Dic₇₃	—	4	1	no	—
50	C₄⋊Dic₇₃	—	4	1	no	—
51	C₄⋊Dic₇₃	—	4	2	no	—
52	C₂²×C₄×C₇₃	D₈	1	2	yes	—
53	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₇₃	—	2	1	no	—
55	C₂×C₈×C₇₃	—	2	1	no	—
56	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
57	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₇₃	#41	1	2	yes	—
59	C₂×C₈×C₇₃	#41	1	2	yes	—
60	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
61	C₄²×C₇₃	—	2	1	no	—
62	C₄²×C₇₃	—	2	1	no	—
63	C₁₆×C₇₃	—	4	2	no	—
64	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₇₃	—	2	2	no	—
66	C₂×C₄×C₇₃	—	2	2	no	—
67	C₂×C₄×C₇₃	—	2	2	no	—
68	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
69	C₂×C₄×C₇₃	#123	1	1	yes	characteristic
70	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
74	C₂×Dic₇₃	—	4	2	no	—
75	C₂×Dic₇₃	—	8	1	no	—
76	C₂×Dic₇₃	—	8	1	no	—
77	C₂×Dic₇₃	—	8	2	no	—
78	C₂×Dic₇₃	—	8	1	no	—
79	C₂×Dic₇₃	—	8	2	no	—
80	C₂×Dic₇₃	—	8	2	no	—
81	C₂×Dic₇₃	—	8	2	no	—
82	Dic₁₄₆	—	4	2	no	—
83	Dic₁₄₆	—	4	2	no	—
84	Dic₁₄₆	—	8	1	no	—
85	Dic₁₄₆	—	8	1	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	#397	1	1	yes	characteristic
96	C₄×C₇₃	#461	1	1	yes	characteristic
97	C₄×C₇₃	#481	1	1	yes	characteristic
98	C₄×C₇₃	#489	1	1	yes	characteristic
99	Dic₇₃	—	8	2	no	—
100	Dic₇₃	—	16	2	no	—
101	Dic₇₃	—	16	1	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	—	73	1	no	—
110	#1846	—	73	1	no	—
111	#1380	—	73	2	no	—
112	#1374	D₇₃	1	1	yes	characteristic
113	C₇₃	#13483	1	1	yes	characteristic, complemented, Sylow
114	C₂²×Q₁₆	—	73	1	no	—
115	C₂²×C₁₆	C₂×D₇₃	1	2	yes	—
116	C₂×#107	—	73	2	no	—
117	C₂×#107	—	73	1	no	—
118	C₂×#106	—	146	1	no	—
119	C₂×#103	—	146	1	no	—
120	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
121	#391	—	146	1	no	—
122	#382	—	146	1	no	—
123	#358	—	146	4	no	—
124	C₂²×Q₈	—	146	1	no	—
125	C₂×Q₁₆	—	73	2	no	—
126	C₂×Q₁₆	—	146	2	no	—
127	C₂×Q₁₆	—	146	2	no	—
128	C₂×Q₁₆	—	146	1	no	—
129	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
130	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
131	C₂×C₄⋊C₄	—	146	2	no	—
132	C₂×C₄⋊C₄	—	146	1	no	—
133	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	—	2	2	no	—
137	#107	—	146	2	no	—
138	#107	—	146	2	no	—
139	#107	—	146	2	no	—
140	#106	—	292	2	no	—
141	#103	—	292	2	no	—
142	C₄×C₈	—	2	2	no	—
143	#95	—	292	2	no	—
144	#95	—	292	2	no	—
145	C₂×Q₈	—	292	1	no	—
146	C₂×Q₈	—	292	1	no	—
147	C₂×Q₈	—	292	1	no	—
148	C₂×Q₈	—	292	1	no	—
149	C₂×Q₈	—	292	1	no	—
150	C₂×Q₈	—	146	2	no	—
151	C₂²×C₄	—	292	1	no	—
152	C₂²×C₄	—	292	1	no	—
153	C₂²×C₄	—	292	2	no	—
154	C₂²×C₄	D₅₈₄	1	2	yes	—
155	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
156	Q₁₆	—	292	2	no	—
157	Q₁₆	—	146	4	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	#11924017	1	2	yes	—
161	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
162	C₂×C₈	#11924027	1	2	yes	—
163	C₂×C₈	#11924028	1	2	yes	—
164	C₂×C₈	#11924029	1	1	yes	characteristic
165	C₄⋊C₄	—	292	2	no	—
166	C₄⋊C₄	—	292	1	no	—
167	C₄⋊C₄	—	292	1	no	—
168	C₄⋊C₄	—	292	2	no	—
169	C₄²	—	2	1	no	—
170	C₄²	—	2	1	no	—
171	C₁₆	—	4	2	no	—
172	C₂³	C₂×D₅₈₄	1	1	yes	characteristic
173	Q₈	—	292	2	no	—
174	Q₈	—	292	2	no	—
175	Q₈	—	584	1	no	—
176	Q₈	—	584	1	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	292	2	no	—
181	C₂×C₄	—	584	1	no	—
182	C₂×C₄	—	584	1	no	—
183	C₂×C₄	—	584	1	no	—
184	C₂×C₄	—	584	2	no	—
185	C₂×C₄	—	584	2	no	—
186	C₂×C₄	—	584	2	no	—
187	C₂×C₄	—	584	2	no	—
188	C₂×C₄	#14556566	1	1	yes	characteristic
189	C₂×C₄	#14556575	1	1	yes	characteristic
190	C₂×C₄	#14556579	1	1	yes	characteristic
191	C₂×C₄	C₂×#11924017	1	1	yes	characteristic
192	C₂×C₄	#14556619	1	1	yes	characteristic
193	C₂×C₄	#14556624	1	1	yes	characteristic
194	C₈	—	2	2	no	—
195	C₈	#14556573	1	2	yes	—
196	C₈	#14556586	1	2	yes	—
197	C₂²	#17988783	1	1	yes	characteristic
198	C₂²	#17988901	1	1	yes	characteristic
199	C₂²	#17988934	1	1	yes	characteristic
200	C₂²	#17989019	1	2	yes	—
201	C₂²	#17989030	1	2	yes	—
202	C₄	—	4	2	no	—
203	C₄	—	584	2	no	—
204	C₄	—	1168	2	no	—
205	C₄	—	1168	1	no	—
206	C₄	#17988514	1	1	yes	characteristic
207	C₄	#17988793	1	1	yes	characteristic
208	C₄	#17988906	1	1	yes	characteristic
209	C₄	#17988947	1	1	yes	characteristic
210	C₂	#22611336	1	1	yes	characteristic
211	C₂	#22611994	1	2	yes	—
212	C₂	#22612140	1	2	yes	—
213	C₂	#22612163	1	1	yes	characteristic
214	C₂	#22612177	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988392

C₂

yes

maximal

#22610792

C₂

yes

maximal, characteristic

#22610095

C₂

yes

maximal, characteristic

#22608023

C₂

yes

maximal

C₇₃×#1374

C₂

yes

maximal, characteristic

C₂²×Dic₂₉₂

C₂²

yes

characteristic

C₂×#14556525

C₂²

yes

characteristic

C₂×#14556525

C₂²

yes

—

—

—

—

—

—

C₂²×C₁₆×C₇₃

C₂²

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂²×Dic₁₄₆

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

D₄

yes

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

—

—

—

—

—

—

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

—

C₄×C₈×C₇₃

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

Dic₂₉₂

—

Dic₂₉₂

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂²×C₄×C₇₃

D₈

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

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

C₄²×C₇₃

—

C₄²×C₇₃

—

C₁₆×C₇₃

—

C₂³×C₇₃

C₂×D₈

yes

characteristic

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

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

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

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

100

Dic₇₃

—

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

—

110

#1846

—

111

#1380

—

112

#1374

D₇₃

yes

characteristic

113

C₇₃

#13483

yes

characteristic, complemented, Sylow

114

C₂²×Q₁₆

—

115

C₂²×C₁₆

C₂×D₇₃

yes

—

116

C₂×#107

—

117

C₂×#107

—

118

C₂×#106

—

146

—

119

C₂×#103

—

146

—

120

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

121

#391

—

146

—

122

#382

—

146

—

123

#358

—

146

—

124

C₂²×Q₈

—

146

—

125

C₂×Q₁₆

—

126

C₂×Q₁₆

—

146

—

127

C₂×Q₁₆

—

146

—

128

C₂×Q₁₆

—

146

—

129

C₂²×C₈

D₂₉₂

yes

characteristic

130

C₂²×C₈

C₂²×D₇₃

yes

characteristic

131

C₂×C₄⋊C₄

—

146

—

132

C₂×C₄⋊C₄

—

146

—

133

C₂×C₄²

D₂₉₂

yes

characteristic

134

C₂×C₁₆

—

135

C₂×C₁₆

—

136

C₂×C₁₆

—

137

#107

—

146

—

138

#107

—

146

—

139

#107

—

146

—

140

#106

—

292

—

141

#103

—

292

—

142

C₄×C₈

—

143

#95

—

292

—

144

#95

—

292

—

145

C₂×Q₈

—

292

—

146

C₂×Q₈

—

292

—

147

C₂×Q₈

—

292

—

148

C₂×Q₈

—

292

—

149

C₂×Q₈

—

292

—

150

C₂×Q₈

—

146

—

151

C₂²×C₄

—

292

—

152

C₂²×C₄

—

292

—

153

C₂²×C₄

—

292

—

154

C₂²×C₄

D₅₈₄

yes

—

155

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

156

Q₁₆

—

292

—

157

Q₁₆

—

146

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

#11924017

yes

—

161

C₂×C₈

D₄×D₇₃

yes

characteristic

162

C₂×C₈

#11924027

yes

—

163

C₂×C₈

#11924028

yes

—

164

C₂×C₈

#11924029

yes

characteristic

165

C₄⋊C₄

—

292

—

166

C₄⋊C₄

—

292

—

167

C₄⋊C₄

—

292

—

168

C₄⋊C₄

—

292

—

169

C₄²

—

170

C₄²

—

171

C₁₆

—

172

C₂³

C₂×D₅₈₄

yes

characteristic

173

Q₈

—

292

—

174

Q₈

—

292

—

175

Q₈

—

584

—

176

Q₈

—

584

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

292

—

181

C₂×C₄

—

584

—

182

C₂×C₄

—

584

—

183

C₂×C₄

—

584

—

184

C₂×C₄

—

584

—

185

C₂×C₄

—

584

—

186

C₂×C₄

—

584

—

187

C₂×C₄

—

584

—

188

C₂×C₄

#14556566

yes

characteristic

189

C₂×C₄

#14556575

yes

characteristic

190

C₂×C₄

#14556579

yes

characteristic

191

C₂×C₄

C₂×#11924017

yes

characteristic

192

C₂×C₄

#14556619

yes

characteristic

193

C₂×C₄

#14556624

yes

characteristic

194

C₈

—

195

C₈

#14556573

yes

—

196

C₈

#14556586

yes

—

197

C₂²

#17988783

yes

characteristic

198

C₂²

#17988901

yes

characteristic

199

C₂²

#17988934

yes

characteristic

200

C₂²

#17989019

yes

—

201

C₂²

#17989030

yes

—

202

C₄

—

203

C₄

—

584

—

204

C₄

—

1168

—

205

C₄

—

1168

—

206

C₄

#17988514

yes

characteristic

207

C₄

#17988793

yes

characteristic

208

C₄

#17988906

yes

characteristic

209

C₄

#17988947

yes

characteristic

210

C₂

#22611336

yes

characteristic

211

C₂

#22611994

yes

—

212

C₂

#22612140

yes

—

213

C₂

#22612163

yes

characteristic

214

C₂

#22612177

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₈₄
commutator, cosocle	C₂²×C₈×C₇₃	C₂³
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1374	C₂
socle, prime radical	C₂³×C₇₃	C₂×D₈
hypercenter	#1374	D₇₃
hypocenter	C₇₃	#13483
LCS	G ▷ C₂²×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1374

subgroup

G/H

center

C₂³

C₂×D₅₈₄

commutator, cosocle

C₂²×C₈×C₇₃

C₂³

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₇₃×#1374

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 ◁ #1374

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×#17988392	2
#22610792	2
#22610095	2
#22608023	2
C₇₃×#1374	2
#13483	73

subgroup

index

C₇₃×#1374

Subgroups — #33005791

Conjugacy classes of subgroups

Complements

Maximal subgroups