Group #33005588 — subgroups

Subgroups — #33005588

Order 18688.

410 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 113 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×#17988405	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988391	C₂	1	2	yes	maximal
3	#22610793	C₂	1	2	yes	maximal
4	#22608023	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₃₂×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924003	C₂²	1	1	yes	characteristic
7	C₂×#14556525	C₂²	1	2	yes	—
8	C₂×#14556515	—	2	2	no	—
9	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
10	#17988494	—	2	2	no	—
11	#17988405	—	2	2	no	—
12	#17988391	—	2	4	no	—
13	C₂²×Q₁₆×C₇₃	C₂²	1	2	yes	—
14	C₂×Q₃₂×C₇₃	—	2	4	no	—
15	C₂×Q₃₂×C₇₃	—	2	2	no	—
16	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
17	C₂²×#10596831	D₄	1	1	yes	characteristic
18	C₂×C₄⋊Dic₇₃	—	2	2	no	—
19	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
20	C₂×#11924003	—	2	2	no	—
21	C₂×#11924003	—	2	1	no	—
22	#14556525	—	2	2	no	—
23	#14556525	—	2	2	no	—
24	#14556515	—	4	4	no	—
25	C₈×Dic₇₃	—	2	1	no	—
26	C₈×Dic₇₃	—	2	1	no	—
27	#14556498	—	4	2	no	—
28	#14556498	—	4	2	no	—
29	C₂²×Q₈×C₇₃	—	2	2	no	—
30	C₂×Q₁₆×C₇₃	—	2	2	no	—
31	C₂×Q₁₆×C₇₃	—	2	2	no	—
32	C₂×Q₁₆×C₇₃	—	2	2	no	—
33	C₂×Q₁₆×C₇₃	—	2	2	no	—
34	C₂×Q₁₆×C₇₃	—	2	2	no	—
35	C₂×Q₁₆×C₇₃	D₄	1	4	yes	—
36	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
37	Q₃₂×C₇₃	—	2	8	no	—
38	C₂×C₁₆×C₇₃	—	2	2	no	—
39	C₂×C₁₆×C₇₃	D₄	1	2	yes	—
40	C₂²×Dic₇₃	—	4	2	no	—
41	C₂²×Dic₇₃	D₈	1	1	yes	characteristic
42	C₂²×Dic₇₃	D₈	1	1	yes	characteristic
43	C₂×#10596831	—	2	1	no	—
44	C₂×#10596831	—	2	2	no	—
45	C₄⋊Dic₇₃	—	4	2	no	—
46	C₄⋊Dic₇₃	—	4	2	no	—
47	C₄×Dic₇₃	—	2	1	no	—
48	C₄×Dic₇₃	—	2	1	no	—
49	#11924003	—	4	1	no	—
50	C₂×Q₈×C₇₃	—	2	4	no	—
51	C₂×Q₈×C₇₃	—	4	2	no	—
52	C₂×Q₈×C₇₃	—	4	2	no	—
53	C₂×Q₈×C₇₃	—	4	2	no	—
54	C₂×Q₈×C₇₃	—	4	2	no	—
55	C₂×Q₈×C₇₃	—	4	2	no	—
56	C₂²×C₄×C₇₃	—	4	2	no	—
57	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
58	Q₁₆×C₇₃	—	2	4	no	—
59	Q₁₆×C₇₃	—	2	4	no	—
60	Q₁₆×C₇₃	—	4	2	no	—
61	Q₁₆×C₇₃	—	4	2	no	—
62	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
65	C₂×C₈×C₇₃	#41	1	2	yes	—
66	C₁₆×C₇₃	—	2	2	no	—
67	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
68	Q₈×C₇₃	—	4	4	no	—
69	Q₈×C₇₃	—	4	4	no	—
70	Q₈×C₇₃	—	8	2	no	—
71	Q₈×C₇₃	—	8	2	no	—
72	C₂×C₄×C₇₃	—	4	4	no	—
73	C₂×C₄×C₇₃	—	8	2	no	—
74	C₂×C₄×C₇₃	—	8	2	no	—
75	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
76	C₂×C₄×C₇₃	#121	1	2	yes	—
77	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
78	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
79	C₂×Dic₇₃	—	2	1	no	—
80	C₂×Dic₇₃	—	2	1	no	—
81	C₂×Dic₇₃	—	2	1	no	—
82	C₂×Dic₇₃	—	2	2	no	—
83	C₂×Dic₇₃	—	8	2	no	—
84	C₂×Dic₇₃	—	8	2	no	—
85	C₂×Dic₇₃	—	8	2	no	—
86	C₂×Dic₇₃	Q₃₂	1	2	yes	complemented
87	C₈×C₇₃	#120	1	1	yes	characteristic
88	C₈×C₇₃	#123	1	2	yes	—
89	C₈×C₇₃	#124	1	1	yes	characteristic
90	#10596831	—	4	1	no	—
91	C₂²×C₇₃	#459	1	2	yes	—
92	C₂²×C₇₃	#466	1	1	yes	characteristic
93	C₂²×C₇₃	C₂×Q₃₂	1	1	yes	characteristic
94	C₂²×C₇₃	#508	1	1	yes	characteristic
95	C₂²×C₇₃	#510	1	2	yes	—
96	C₄×C₇₃	—	8	4	no	—
97	C₄×C₇₃	#394	1	1	yes	characteristic
98	C₄×C₇₃	#463	1	2	yes	—
99	C₄×C₇₃	#469	1	1	yes	characteristic
100	Dic₇₃	—	2	2	no	—
101	Dic₇₃	—	4	1	no	—
102	Dic₇₃	—	16	2	no	—
103	#13450	—	73	1	no	maximal, Sylow
104	C₂×C₇₃	#2054	1	1	yes	characteristic
105	C₂×C₇₃	#2209	1	2	yes	—
106	C₂×C₇₃	#2211	1	2	yes	—
107	C₂×C₇₃	#2216	1	1	yes	characteristic
108	C₂×C₇₃	#2219	1	1	yes	characteristic
109	C₂²×Q₃₂	D₇₃	1	1	yes	characteristic
110	C₂×C₄⋊C₁₆	—	73	1	no	—
111	C₂×#358	—	73	2	no	—
112	#1969	—	73	2	no	—
113	#1380	—	73	1	no	—
114	C₇₃	#13450	1	1	yes	characteristic, complemented, Sylow
115	C₂²×Q₁₆	C₂×D₇₃	1	2	yes	—
116	C₂×Q₃₂	—	2	4	no	—
117	C₂×Q₃₂	—	2	2	no	—
118	C₂²×C₁₆	—	73	1	no	—
119	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
120	C₂×#107	—	73	2	no	—
121	C₂×#103	—	146	2	no	—
122	C₂×C₄×C₈	—	73	1	no	—
123	#391	—	146	2	no	—
124	C₄⋊C₁₆	—	146	2	no	—
125	#358	—	146	4	no	—
126	C₂²×Q₈	—	2	2	no	—
127	C₂×Q₁₆	—	2	2	no	—
128	C₂×Q₁₆	—	2	2	no	—
129	C₂×Q₁₆	—	2	2	no	—
130	C₂×Q₁₆	—	2	2	no	—
131	C₂×Q₁₆	—	2	2	no	—
132	C₂×Q₁₆	C₂²⋊D₇₃	1	4	yes	—
133	C₂²×C₈	—	73	1	no	—
134	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
135	C₂×C₄⋊C₄	—	146	2	no	—
136	C₂×C₄²	—	73	1	no	—
137	Q₃₂	—	2	8	no	—
138	C₂×C₁₆	—	2	2	no	—
139	C₂×C₁₆	—	146	2	no	—
140	C₂×C₁₆	—	146	1	no	—
141	C₂×C₁₆	C₂²⋊D₇₃	1	2	yes	—
142	#107	—	146	2	no	—
143	#107	—	146	2	no	—
144	#103	—	292	4	no	—
145	C₄×C₈	—	146	1	no	—
146	C₄×C₈	—	146	1	no	—
147	#95	—	292	2	no	—
148	#95	—	292	2	no	—
149	C₂×Q₈	—	2	4	no	—
150	C₂×Q₈	—	4	2	no	—
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	—	4	2	no	—
153	C₂×Q₈	—	4	2	no	—
154	C₂×Q₈	—	4	2	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	—	292	2	no	—
157	C₂²×C₄	—	73	1	no	—
158	C₂²×C₄	—	73	1	no	—
159	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
160	Q₁₆	—	2	4	no	—
161	Q₁₆	—	2	4	no	—
162	Q₁₆	—	4	2	no	—
163	Q₁₆	—	4	2	no	—
164	C₂×C₈	—	146	1	no	—
165	C₂×C₈	—	146	2	no	—
166	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
167	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
168	C₂×C₈	#11924027	1	1	yes	characteristic
169	C₂×C₈	#11924027	1	2	yes	—
170	C₄⋊C₄	—	292	2	no	—
171	C₄⋊C₄	—	292	2	no	—
172	C₄²	—	146	1	no	—
173	C₄²	—	146	1	no	—
174	C₁₆	—	2	2	no	—
175	C₁₆	—	292	1	no	—
176	C₂³	D₈×D₇₃	1	1	yes	characteristic
177	Q₈	—	4	4	no	—
178	Q₈	—	4	4	no	—
179	Q₈	—	8	2	no	—
180	Q₈	—	8	2	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	8	2	no	—
183	C₂×C₄	—	8	2	no	—
184	C₂×C₄	—	584	2	no	—
185	C₂×C₄	—	584	2	no	—
186	C₂×C₄	—	584	2	no	—
187	C₂×C₄	—	73	2	no	—
188	C₂×C₄	—	146	2	no	—
189	C₂×C₄	—	146	1	no	—
190	C₂×C₄	—	146	1	no	—
191	C₂×C₄	—	146	1	no	—
192	C₂×C₄	#14556562	1	1	yes	characteristic
193	C₂×C₄	#14556563	1	2	yes	—
194	C₂×C₄	#14556610	1	1	yes	characteristic
195	C₂×C₄	#14556615	1	2	yes	—
196	C₈	—	292	1	no	—
197	C₈	#14556560	1	1	yes	characteristic
198	C₈	#14556572	1	2	yes	—
199	C₈	#14556578	1	1	yes	characteristic
200	C₂²	#17988780	1	2	yes	—
201	C₂²	#17988820	1	1	yes	characteristic
202	C₂²	Q₃₂×D₇₃	1	1	yes	characteristic
203	C₂²	#17989020	1	1	yes	characteristic
204	C₂²	#17989029	1	2	yes	—
205	C₄	—	8	4	no	—
206	C₄	—	292	1	no	—
207	C₄	—	1168	2	no	—
208	C₄	—	146	2	no	—
209	C₄	#17988501	1	1	yes	characteristic
210	C₄	#17988800	1	2	yes	—
211	C₄	#17988837	1	1	yes	characteristic
212	C₂	#22611239	1	1	yes	characteristic
213	C₂	#22611979	1	2	yes	—
214	C₂	#22611995	1	2	yes	—
215	C₂	#22612019	1	1	yes	characteristic
216	C₂	#22612036	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988405

C₂

yes

maximal, characteristic

C₂×#17988391

C₂

yes

maximal

#22610793

C₂

yes

maximal

#22608023

C₂

yes

maximal, characteristic

C₂²×Q₃₂×C₇₃

C₂

yes

maximal, characteristic

C₂²×#11924003

C₂²

yes

characteristic

C₂×#14556525

C₂²

yes

—

C₂×#14556515

—

C₂×C₈×Dic₇₃

C₂²

yes

—

—

—

C₂²×Q₁₆×C₇₃

C₂²

yes

—

C₂×Q₃₂×C₇₃

—

C₂×Q₃₂×C₇₃

—

C₂²×C₁₆×C₇₃

C₂²

yes

characteristic

C₂²×#10596831

D₄

yes

characteristic

C₂×C₄⋊Dic₇₃

—

C₂×C₄×Dic₇₃

D₄

yes

—

—

—

—

—

C₈×Dic₇₃

—

C₈×Dic₇₃

—

#14556498

—

#14556498

—

C₂²×Q₈×C₇₃

—

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

D₄

yes

—

C₂²×C₈×C₇₃

C₂³

yes

characteristic

Q₃₂×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

D₈

yes

characteristic

C₂²×Dic₇₃

D₈

yes

characteristic

C₂×#10596831

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊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

C₂×D₄

yes

characteristic

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

C₂×C₈×C₇₃

yes

C₂×C₈×C₇₃

yes

—

C₁₆×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

#121

yes

characteristic

C₂×C₄×C₇₃

#121

yes

—

C₂×C₄×C₇₃

#135

yes

characteristic

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

Q₃₂

yes

C₈×C₇₃

yes

C₈×C₇₃

yes

—

C₈×C₇₃

#124

yes

characteristic

#10596831

—

C₂²×C₇₃

#459

yes

—

C₂²×C₇₃

#466

yes

characteristic

C₂²×C₇₃

C₂×Q₃₂

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₄×C₇₃

—

C₄×C₇₃

#394

yes

characteristic

C₄×C₇₃

#463

yes

—

C₄×C₇₃

#469

yes

characteristic

100

Dic₇₃

—

101

Dic₇₃

—

102

Dic₇₃

—

103

#13450

—

maximal, Sylow

104

C₂×C₇₃

#2054

yes

characteristic

105

C₂×C₇₃

#2209

yes

—

106

C₂×C₇₃

#2211

yes

—

107

C₂×C₇₃

#2216

yes

characteristic

108

C₂×C₇₃

#2219

yes

characteristic

109

C₂²×Q₃₂

D₇₃

yes

characteristic

110

C₂×C₄⋊C₁₆

—

111

C₂×#358

—

112

#1969

—

113

#1380

—

114

C₇₃

#13450

yes

characteristic, complemented, Sylow

115

C₂²×Q₁₆

C₂×D₇₃

yes

—

116

C₂×Q₃₂

—

117

C₂×Q₃₂

—

118

C₂²×C₁₆

—

119

C₂²×C₁₆

C₂×D₇₃

yes

characteristic

120

C₂×#107

—

121

C₂×#103

—

146

—

122

C₂×C₄×C₈

—

123

#391

—

146

—

124

C₄⋊C₁₆

—

146

—

125

#358

—

146

—

126

C₂²×Q₈

—

127

C₂×Q₁₆

—

128

C₂×Q₁₆

—

129

C₂×Q₁₆

—

130

C₂×Q₁₆

—

131

C₂×Q₁₆

—

132

C₂×Q₁₆

C₂²⋊D₇₃

yes

—

133

C₂²×C₈

—

134

C₂²×C₈

C₂²×D₇₃

yes

characteristic

135

C₂×C₄⋊C₄

—

146

—

136

C₂×C₄²

—

137

Q₃₂

—

138

C₂×C₁₆

—

139

C₂×C₁₆

—

146

—

140

C₂×C₁₆

—

146

—

141

C₂×C₁₆

C₂²⋊D₇₃

yes

—

142

#107

—

146

—

143

#107

—

146

—

144

#103

—

292

—

145

C₄×C₈

—

146

—

146

C₄×C₈

—

146

—

147

#95

—

292

—

148

#95

—

292

—

149

C₂×Q₈

—

150

C₂×Q₈

—

151

C₂×Q₈

—

152

C₂×Q₈

—

153

C₂×Q₈

—

154

C₂×Q₈

—

155

C₂²×C₄

—

156

C₂²×C₄

—

292

—

157

C₂²×C₄

—

158

C₂²×C₄

—

159

C₂²×C₄

D₄×D₇₃

yes

characteristic

160

Q₁₆

—

161

Q₁₆

—

162

Q₁₆

—

163

Q₁₆

—

164

C₂×C₈

—

146

—

165

C₂×C₈

—

146

—

166

C₂×C₈

C₂×C₂²⋊D₇₃

yes

—

167

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

168

C₂×C₈

#11924027

yes

characteristic

169

C₂×C₈

#11924027

yes

—

170

C₄⋊C₄

—

292

—

171

C₄⋊C₄

—

292

—

172

C₄²

—

146

—

173

C₄²

—

146

—

174

C₁₆

—

175

C₁₆

—

292

—

176

C₂³

D₈×D₇₃

yes

characteristic

177

Q₈

—

178

Q₈

—

179

Q₈

—

180

Q₈

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

584

—

185

C₂×C₄

—

584

—

186

C₂×C₄

—

584

—

187

C₂×C₄

—

188

C₂×C₄

—

146

—

189

C₂×C₄

—

146

—

190

C₂×C₄

—

146

—

191

C₂×C₄

—

146

—

192

C₂×C₄

#14556562

yes

characteristic

193

C₂×C₄

#14556563

yes

—

194

C₂×C₄

#14556610

yes

characteristic

195

C₂×C₄

#14556615

yes

—

196

C₈

—

292

—

197

C₈

#14556560

yes

characteristic

198

C₈

#14556572

yes

—

199

C₈

#14556578

yes

characteristic

200

C₂²

#17988780

yes

—

201

C₂²

#17988820

yes

characteristic

202

C₂²

Q₃₂×D₇₃

yes

characteristic

203

C₂²

#17989020

yes

characteristic

204

C₂²

#17989029

yes

—

205

C₄

—

206

C₄

—

292

—

207

C₄

—

1168

—

208

C₄

—

146

—

209

C₄

#17988501

yes

characteristic

210

C₄

#17988800

yes

—

211

C₄

#17988837

yes

characteristic

212

C₂

#22611239

yes

characteristic

213

C₂

#22611979

yes

—

214

C₂

#22611995

yes

—

215

C₂

#22612019

yes

characteristic

216

C₂

#22612036

yes

characteristic

	subgroup	G/H
center	C₂³	D₈×D₇₃
commutator, cosocle	C₂²×C₈×C₇₃	C₂³
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₂²×Q₃₂×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂×D₈
hypercenter	C₂²×Q₃₂	D₇₃
hypocenter	C₇₃	#13450
LCS	G ▷ C₂²×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₃₂

subgroup

G/H

center

C₂³

D₈×D₇₃

commutator, cosocle

C₂²×C₈×C₇₃

C₂³

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₂²×Q₃₂×C₇₃

C₂

socle, prime radical

C₂³×C₇₃

C₂×D₈

hypercenter

C₂²×Q₃₂

D₇₃

hypocenter

C₇₃

#13450

LCS

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

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₃₂

N	H ≅ G/N	type
86. C₂×Dic₇₃	Q₃₂	semidirect
114. C₇₃	#13450	semidirect

H ≅ G/N

type

86. C₂×Dic₇₃

Q₃₂

semidirect

114. C₇₃

#13450

semidirect

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

subgroup	index
C₂×#17988405	2
C₂×#17988391	2
#22610793	2
#22608023	2
C₂²×Q₃₂×C₇₃	2
#13450	73

subgroup

index

C₂²×Q₃₂×C₇₃

Subgroups — #33005588

Conjugacy classes of subgroups

Complements

Maximal subgroups