Group #33005847 — subgroups

Subgroups — #33005847

Order 18688.

342 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 105 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×#17988413	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17988366	C₂	1	1	yes	maximal, characteristic
3	#22609957	C₂	1	2	yes	maximal
4	#22608022	C₂	1	2	yes	maximal, complemented
5	C₇₃×#1380	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
7	C₂²×#11924003	C₂²	1	1	yes	characteristic
8	C₂×#14556525	C₂²	1	2	yes	—
9	#17988444	—	2	2	no	—
10	#17988413	—	2	2	no	—
11	#17988366	—	2	2	no	—
12	#17988349	—	2	2	no	—
13	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
14	C₂×#107×C₇₃	C₂²	1	2	yes	—
15	C₂²×C₄×D₇₃	D₄	1	1	yes	characteristic
16	C₂²×#10596831	D₄	1	1	yes	characteristic
17	C₂×C₈×D₇₃	—	2	1	no	—
18	C₂×C₈×D₇₃	—	2	2	no	—
19	C₂×C₈×D₇₃	—	2	2	no	—
20	C₂×C₈×D₇₃	—	2	1	no	—
21	C₂×C₄⋊Dic₇₃	—	2	2	no	—
22	C₂×#11924005	—	4	2	no	—
23	C₂×#11924003	—	2	1	no	—
24	C₂×#11924003	—	2	2	no	—
25	#14556525	—	2	2	no	—
26	#14556525	—	2	2	no	—
27	#14556498	—	4	2	no	—
28	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
30	C₂×C₁₆×C₇₃	—	2	2	no	—
31	C₂×C₁₆×C₇₃	D₄	1	2	yes	—
32	#107×C₇₃	—	2	2	no	—
33	#107×C₇₃	—	2	2	no	—
34	C₂³×D₇₃	D₈	1	1	yes	characteristic
35	C₂²×Dic₇₃	—	4	2	no	—
36	C₂²×Dic₇₃	D₈	1	1	yes	characteristic
37	C₂×C₄×D₇₃	—	2	1	no	—
38	C₂×C₄×D₇₃	—	2	2	no	—
39	C₂×C₄×D₇₃	—	2	1	no	—
40	C₂×C₄×D₇₃	—	2	2	no	—
41	C₂×#10596831	—	2	1	no	—
42	C₂×#10596831	—	2	2	no	—
43	C₈×D₇₃	—	4	1	no	—
44	C₈×D₇₃	—	4	1	no	—
45	C₈×D₇₃	—	4	2	no	—
46	C₄⋊Dic₇₃	—	4	2	no	—
47	C₄⋊Dic₇₃	—	4	2	no	—
48	#11924005	—	8	4	no	—
49	#11924003	—	4	1	no	—
50	C₂²×C₄×C₇₃	—	4	2	no	—
51	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₇₃	#41	1	2	yes	—
54	C₂×C₈×C₇₃	#41	1	2	yes	—
55	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
56	C₄⋊C₄×C₇₃	—	4	2	no	—
57	C₄⋊C₄×C₇₃	—	4	2	no	—
58	C₁₆×C₇₃	—	2	2	no	—
59	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
60	C₂²×D₇₃	—	2	1	no	—
61	C₂²×D₇₃	—	2	2	no	—
62	C₂²×D₇₃	—	2	1	no	—
63	C₂²×D₇₃	—	2	2	no	—
64	C₂²×D₇₃	Q₃₂	1	2	yes	—
65	C₂×C₄×C₇₃	—	8	2	no	—
66	C₂×C₄×C₇₃	—	8	2	no	—
67	C₂×C₄×C₇₃	—	8	2	no	—
68	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
69	C₂×C₄×C₇₃	#123	1	2	yes	—
70	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
71	C₂×C₄×C₇₃	#137	1	2	yes	—
72	C₂×Dic₇₃	—	2	1	no	—
73	C₂×Dic₇₃	—	2	2	no	—
74	C₂×Dic₇₃	—	8	2	no	—
75	C₂×Dic₇₃	—	8	2	no	—
76	C₂×Dic₇₃	—	8	2	no	—
77	C₄×D₇₃	—	4	1	no	—
78	C₄×D₇₃	—	4	1	no	—
79	C₄×D₇₃	—	4	2	no	—
80	C₈×C₇₃	#123	1	1	yes	characteristic
81	C₈×C₇₃	#124	1	1	yes	characteristic
82	C₈×C₇₃	#126	1	2	yes	—
83	#10596831	—	4	1	no	—
84	C₂²×C₇₃	#466	1	1	yes	characteristic
85	C₂²×C₇₃	#480	1	2	yes	—
86	C₂²×C₇₃	C₂×Q₃₂	1	1	yes	characteristic
87	C₂²×C₇₃	#508	1	1	yes	characteristic
88	C₂²×C₇₃	#509	1	2	yes	—
89	C₂×D₇₃	—	2	2	no	—
90	C₂×D₇₃	—	2	4	no	—
91	C₂×D₇₃	—	4	1	no	—
92	C₂×D₇₃	—	4	1	no	—
93	C₂×D₇₃	—	4	2	no	—
94	C₄×C₇₃	—	16	2	no	—
95	C₄×C₇₃	#397	1	1	yes	characteristic
96	C₄×C₇₃	#470	1	1	yes	characteristic
97	C₄×C₇₃	#485	1	2	yes	—
98	Dic₇₃	—	4	1	no	—
99	Dic₇₃	—	16	2	no	—
100	#13493	—	73	1	no	maximal, Sylow
101	C₂×C₇₃	#2075	1	1	yes	characteristic
102	C₂×C₇₃	#2216	1	1	yes	characteristic
103	C₂×C₇₃	#2218	1	1	yes	characteristic
104	C₂×C₇₃	#2234	1	2	yes	—
105	C₂×C₇₃	#2236	1	2	yes	—
106	D₇₃	—	4	2	no	—
107	C₂×#366	—	73	1	no	—
108	C₂×C₂²⋊C₁₆	—	73	1	no	—
109	#1808	—	73	2	no	—
110	#1380	—	73	2	no	—
111	#1380	D₇₃	1	1	yes	characteristic, complemented
112	C₇₃	#13493	1	1	yes	characteristic, complemented, Sylow
113	C₂³×C₈	—	73	1	no	—
114	C₂²×C₁₆	—	73	1	no	—
115	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
116	C₂×#107	—	73	2	no	—
117	C₂×#107	C₂×D₇₃	1	2	yes	—
118	#380	—	146	2	no	—
119	#366	—	146	2	no	—
120	C₂²⋊C₁₆	—	146	2	no	—
121	#340	—	146	2	no	—
122	C₂³×C₄	—	73	1	no	—
123	C₂²×C₈	—	73	1	no	—
124	C₂²×C₈	—	146	1	no	—
125	C₂²×C₈	—	146	2	no	—
126	C₂²×C₈	—	146	2	no	—
127	C₂²×C₈	—	146	1	no	—
128	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	146	2	no	—
131	C₂×C₂²⋊C₄	—	292	2	no	—
132	C₂×C₁₆	—	2	2	no	—
133	C₂×C₁₆	—	146	1	no	—
134	C₂×C₁₆	—	146	2	no	—
135	C₂×C₁₆	D₂₉₂	1	2	yes	—
136	#107	—	2	2	no	—
137	#107	—	2	2	no	—
138	#107	—	146	2	no	—
139	#107	—	146	2	no	—
140	#95	—	292	2	no	—
141	C₂⁴	—	73	1	no	—
142	C₂²×C₄	—	4	2	no	—
143	C₂²×C₄	—	292	2	no	—
144	C₂²×C₄	—	73	1	no	—
145	C₂²×C₄	—	146	1	no	—
146	C₂²×C₄	—	146	2	no	—
147	C₂²×C₄	—	146	1	no	—
148	C₂²×C₄	—	146	2	no	—
149	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
150	C₂×C₈	—	292	1	no	—
151	C₂×C₈	—	292	1	no	—
152	C₂×C₈	—	292	2	no	—
153	C₂×C₈	—	146	1	no	—
154	C₂×C₈	—	146	2	no	—
155	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
156	C₂×C₈	#11924027	1	2	yes	—
157	C₂×C₈	#11924028	1	2	yes	—
158	C₂×C₈	#11924029	1	1	yes	characteristic
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	292	2	no	—
162	C₄⋊C₄	—	292	2	no	—
163	C₂²⋊C₄	—	584	4	no	—
164	C₁₆	—	2	2	no	—
165	C₁₆	—	292	1	no	—
166	C₂³	—	73	2	no	—
167	C₂³	—	146	2	no	—
168	C₂³	—	146	1	no	—
169	C₂³	—	146	2	no	—
170	C₂³	—	146	1	no	—
171	C₂³	D₈×D₇₃	1	1	yes	characteristic
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	8	2	no	—
175	C₂×C₄	—	292	2	no	—
176	C₂×C₄	—	292	1	no	—
177	C₂×C₄	—	292	1	no	—
178	C₂×C₄	—	584	2	no	—
179	C₂×C₄	—	584	2	no	—
180	C₂×C₄	—	584	2	no	—
181	C₂×C₄	—	146	2	no	—
182	C₂×C₄	—	146	1	no	—
183	C₂×C₄	#14556566	1	1	yes	characteristic
184	C₂×C₄	#14556573	1	2	yes	—
185	C₂×C₄	#14556614	1	1	yes	characteristic
186	C₂×C₄	#14556620	1	2	yes	—
187	C₈	—	292	1	no	—
188	C₈	#14556575	1	1	yes	characteristic
189	C₈	#14556579	1	1	yes	characteristic
190	C₈	#14556586	1	2	yes	—
191	C₂²	—	292	1	no	—
192	C₂²	—	292	2	no	—
193	C₂²	—	292	1	no	—
194	C₂²	—	146	2	no	—
195	C₂²	—	146	4	no	—
196	C₂²	#17988824	1	1	yes	characteristic
197	C₂²	#17988899	1	2	yes	—
198	C₂²	Q₃₂×D₇₃	1	1	yes	characteristic
199	C₂²	#17989016	1	1	yes	characteristic
200	C₂²	#17989024	1	2	yes	—
201	C₄	—	16	2	no	—
202	C₄	—	292	1	no	—
203	C₄	—	1168	2	no	—
204	C₄	#17988514	1	1	yes	characteristic
205	C₄	#17988846	1	1	yes	characteristic
206	C₄	#17988927	1	2	yes	—
207	C₂	—	292	2	no	—
208	C₂	#22611356	1	1	yes	characteristic
209	C₂	#22612023	1	1	yes	characteristic
210	C₂	#22612035	1	1	yes	characteristic
211	C₂	#22612120	1	2	yes	—
212	C₂	#22612132	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988413

C₂

yes

maximal, characteristic, complemented

C₂×#17988366

C₂

yes

maximal, characteristic

#22609957

C₂

yes

maximal

#22608022

C₂

yes

maximal, complemented

C₇₃×#1380

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×D₇₃

C₂²

yes

characteristic

C₂²×#11924003

C₂²

yes

characteristic

C₂×#14556525

C₂²

yes

—

—

—

—

—

C₂²×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×#107×C₇₃

C₂²

yes

—

C₂²×C₄×D₇₃

D₄

yes

characteristic

C₂²×#10596831

D₄

yes

characteristic

C₂×C₈×D₇₃

—

C₂×C₈×D₇₃

—

C₂×C₈×D₇₃

—

C₂×C₈×D₇₃

—

C₂×C₄⋊Dic₇₃

—

—

—

—

—

—

—

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

—

#107×C₇₃

—

#107×C₇₃

—

C₂³×D₇₃

D₈

yes

characteristic

C₂²×Dic₇₃

—

C₂²×Dic₇₃

D₈

yes

characteristic

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×#10596831

—

C₂×#10596831

—

C₈×D₇₃

—

C₈×D₇₃

—

C₈×D₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924005

—

#11924003

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

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

—

C₁₆×C₇₃

—

C₂³×C₇₃

C₂×D₈

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

Q₃₂

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#121

yes

characteristic

C₂×C₄×C₇₃

#123

yes

—

C₂×C₄×C₇₃

#135

yes

characteristic

C₂×C₄×C₇₃

#137

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

C₈×C₇₃

yes

—

#10596831

—

C₂²×C₇₃

#466

yes

characteristic

C₂²×C₇₃

#480

yes

—

C₂²×C₇₃

C₂×Q₃₂

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

Dic₇₃

—

100

#13493

—

maximal, Sylow

101

C₂×C₇₃

#2075

yes

characteristic

102

C₂×C₇₃

#2216

yes

characteristic

103

C₂×C₇₃

#2218

yes

characteristic

104

C₂×C₇₃

#2234

yes

—

105

C₂×C₇₃

#2236

yes

—

106

D₇₃

—

107

C₂×#366

—

108

C₂×C₂²⋊C₁₆

—

109

#1808

—

110

#1380

—

111

#1380

D₇₃

yes

characteristic, complemented

112

C₇₃

#13493

yes

characteristic, complemented, Sylow

113

C₂³×C₈

—

114

C₂²×C₁₆

—

115

C₂²×C₁₆

C₂×D₇₃

yes

characteristic

116

C₂×#107

—

117

C₂×#107

C₂×D₇₃

yes

—

118

#380

—

146

—

119

#366

—

146

—

120

C₂²⋊C₁₆

—

146

—

121

#340

—

146

—

122

C₂³×C₄

—

123

C₂²×C₈

—

124

C₂²×C₈

—

146

—

125

C₂²×C₈

—

146

—

126

C₂²×C₈

—

146

—

127

C₂²×C₈

—

146

—

128

C₂²×C₈

C₂²×D₇₃

yes

characteristic

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

146

—

131

C₂×C₂²⋊C₄

—

292

—

132

C₂×C₁₆

—

133

C₂×C₁₆

—

146

—

134

C₂×C₁₆

—

146

—

135

C₂×C₁₆

D₂₉₂

yes

—

136

#107

—

137

#107

—

138

#107

—

146

—

139

#107

—

146

—

140

#95

—

292

—

141

C₂⁴

—

142

C₂²×C₄

—

143

C₂²×C₄

—

292

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

—

146

C₂²×C₄

—

146

—

147

C₂²×C₄

—

146

—

148

C₂²×C₄

—

146

—

149

C₂²×C₄

D₄×D₇₃

yes

characteristic

150

C₂×C₈

—

292

—

151

C₂×C₈

—

292

—

152

C₂×C₈

—

292

—

153

C₂×C₈

—

146

—

154

C₂×C₈

—

146

—

155

C₂×C₈

C₂×D₂₉₂

yes

characteristic

156

C₂×C₈

#11924027

yes

—

157

C₂×C₈

#11924028

yes

—

158

C₂×C₈

#11924029

yes

characteristic

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

292

—

162

C₄⋊C₄

—

292

—

163

C₂²⋊C₄

—

584

—

164

C₁₆

—

165

C₁₆

—

292

—

166

C₂³

—

167

C₂³

—

146

—

168

C₂³

—

146

—

169

C₂³

—

146

—

170

C₂³

—

146

—

171

C₂³

D₈×D₇₃

yes

characteristic

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

292

—

176

C₂×C₄

—

292

—

177

C₂×C₄

—

292

—

178

C₂×C₄

—

584

—

179

C₂×C₄

—

584

—

180

C₂×C₄

—

584

—

181

C₂×C₄

—

146

—

182

C₂×C₄

—

146

—

183

C₂×C₄

#14556566

yes

characteristic

184

C₂×C₄

#14556573

yes

—

185

C₂×C₄

#14556614

yes

characteristic

186

C₂×C₄

#14556620

yes

—

187

C₈

—

292

—

188

C₈

#14556575

yes

characteristic

189

C₈

#14556579

yes

characteristic

190

C₈

#14556586

yes

—

191

C₂²

—

292

—

192

C₂²

—

292

—

193

C₂²

—

292

—

194

C₂²

—

146

—

195

C₂²

—

146

—

196

C₂²

#17988824

yes

characteristic

197

C₂²

#17988899

yes

—

198

C₂²

Q₃₂×D₇₃

yes

characteristic

199

C₂²

#17989016

yes

characteristic

200

C₂²

#17989024

yes

—

201

C₄

—

202

C₄

—

292

—

203

C₄

—

1168

—

204

C₄

#17988514

yes

characteristic

205

C₄

#17988846

yes

characteristic

206

C₄

#17988927

yes

—

207

C₂

—

292

—

208

C₂

#22611356

yes

characteristic

209

C₂

#22612023

yes

characteristic

210

C₂

#22612035

yes

characteristic

211

C₂

#22612120

yes

—

212

C₂

#22612132

yes

—

	subgroup	G/H
center	C₂³	D₈×D₇₃
commutator, cosocle	C₂²×C₈×C₇₃	C₂³
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1380	C₂
socle	C₂³×C₇₃	C₂×D₈
prime radical	C₂³×D₇₃	D₈
hypercenter	#1380	D₇₃
hypocenter	C₇₃	#13493
LCS	G ▷ C₂²×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

subgroup

G/H

center

C₂³

D₈×D₇₃

commutator, cosocle

C₂²×C₈×C₇₃

C₂³

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₇₃×#1380

C₂

socle

C₂³×C₇₃

C₂×D₈

prime radical

C₂³×D₇₃

D₈

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

N	H ≅ G/N	type
1. C₂×#17988413	C₂	semidirect
4. #22608022	C₂	semidirect
5. C₇₃×#1380	C₂	semidirect
111. #1380	D₇₃	semidirect
112. C₇₃	#13493	semidirect

H ≅ G/N

type

1. C₂×#17988413

C₂

semidirect

4. #22608022

C₂

semidirect

5. C₇₃×#1380

C₂

semidirect

111. #1380

D₇₃

semidirect

112. C₇₃

#13493

semidirect

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

subgroup	index
C₂×#17988413	2
C₂×#17988366	2
#22609957	2
#22608022	2
C₇₃×#1380	2
#13493	73

subgroup

index

C₇₃×#1380

Subgroups — #33005847

Conjugacy classes of subgroups

Complements

Maximal subgroups