Group #33005592 — subgroups

Subgroups — #33005592

Order 18688.

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

normal subgroups only

This group has 410 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	C₂²×Dic₅₈₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988405	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988390	C₂	1	2	yes	maximal
4	#22610790	C₂	1	2	yes	maximal
5	C₇₃×#1380	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₉₂	C₂²	1	2	yes	—
7	C₂×Dic₅₈₄	—	2	4	no	—
8	C₂×Dic₅₈₄	—	2	2	no	—
9	C₂²×#11924003	C₂²	1	1	yes	characteristic
10	C₂×#14556514	—	2	2	no	—
11	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
12	#17988492	—	2	2	no	—
13	#17988405	—	2	2	no	—
14	#17988390	—	2	4	no	—
15	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
16	C₂×#107×C₇₃	C₂²	1	2	yes	—
17	C₂²×Dic₁₄₆	—	2	2	no	—
18	C₂×Dic₂₉₂	—	2	2	no	—
19	C₂×Dic₂₉₂	—	2	2	no	—
20	C₂×Dic₂₉₂	—	2	2	no	—
21	C₂×Dic₂₉₂	—	2	2	no	—
22	C₂×Dic₂₉₂	—	2	2	no	—
23	C₂×Dic₂₉₂	D₄	1	4	yes	—
24	C₂²×#10596831	D₄	1	1	yes	characteristic
25	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
26	Dic₅₈₄	—	2	8	no	—
27	C₂×#11924003	—	2	2	no	—
28	C₂×#11924003	—	2	1	no	—
29	#14556514	—	4	4	no	—
30	C₈×Dic₇₃	—	2	1	no	—
31	C₈×Dic₇₃	—	2	1	no	—
32	#14556498	—	4	2	no	—
33	#14556498	—	4	2	no	—
34	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
36	C₂×C₁₆×C₇₃	—	2	2	no	—
37	C₂×C₁₆×C₇₃	D₄	1	2	yes	—
38	#107×C₇₃	—	2	2	no	—
39	#107×C₇₃	—	2	2	no	—
40	C₂×Dic₁₄₆	—	2	4	no	—
41	C₂×Dic₁₄₆	—	4	2	no	—
42	C₂×Dic₁₄₆	—	4	2	no	—
43	C₂×Dic₁₄₆	—	4	2	no	—
44	C₂×Dic₁₄₆	—	4	2	no	—
45	C₂×Dic₁₄₆	—	4	2	no	—
46	C₂²×Dic₇₃	—	4	2	no	—
47	C₂²×Dic₇₃	D₈	1	1	yes	characteristic
48	C₂²×Dic₇₃	D₈	1	1	yes	characteristic
49	Dic₂₉₂	—	2	4	no	—
50	Dic₂₉₂	—	2	4	no	—
51	Dic₂₉₂	—	4	2	no	—
52	Dic₂₉₂	—	4	2	no	—
53	C₂×#10596831	—	2	1	no	—
54	C₂×#10596831	—	2	2	no	—
55	C₄×Dic₇₃	—	2	1	no	—
56	C₄×Dic₇₃	—	2	1	no	—
57	#11924003	—	4	1	no	—
58	C₂²×C₄×C₇₃	—	4	2	no	—
59	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₇₃	#41	1	2	yes	—
63	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
64	C₄⋊C₄×C₇₃	—	4	2	no	—
65	C₄⋊C₄×C₇₃	—	4	2	no	—
66	C₁₆×C₇₃	—	2	2	no	—
67	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₇₃	—	8	2	no	—
69	C₂×C₄×C₇₃	—	8	2	no	—
70	C₂×C₄×C₇₃	—	8	2	no	—
71	C₂×C₄×C₇₃	#121	1	2	yes	—
72	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
73	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
74	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
75	C₂×Dic₇₃	—	2	1	no	—
76	C₂×Dic₇₃	—	2	1	no	—
77	C₂×Dic₇₃	—	2	1	no	—
78	C₂×Dic₇₃	—	2	2	no	—
79	C₂×Dic₇₃	—	4	4	no	—
80	C₂×Dic₇₃	—	8	2	no	—
81	C₂×Dic₇₃	—	8	2	no	—
82	C₂×Dic₇₃	Q₃₂	1	2	yes	complemented
83	Dic₁₄₆	—	4	4	no	—
84	Dic₁₄₆	—	4	4	no	—
85	Dic₁₄₆	—	8	2	no	—
86	Dic₁₄₆	—	8	2	no	—
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	—	16	2	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	—	8	4	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	—	73	1	no	—
110	C₂×C₄⋊C₁₆	—	73	1	no	—
111	C₂×#358	—	73	2	no	—
112	#1969	—	73	2	no	—
113	#1380	D₇₃	1	1	yes	characteristic
114	C₇₃	#13450	1	1	yes	characteristic, complemented, Sylow
115	C₂²×Q₁₆	—	73	2	no	—
116	C₂×Q₃₂	—	146	4	no	—
117	C₂×Q₃₂	—	146	2	no	—
118	C₂²×C₁₆	—	73	1	no	—
119	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
120	C₂×#107	C₂×D₇₃	1	2	yes	—
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	—	146	2	no	—
127	C₂×Q₁₆	—	73	4	no	—
128	C₂×Q₁₆	—	146	2	no	—
129	C₂×Q₁₆	—	146	2	no	—
130	C₂×Q₁₆	—	146	2	no	—
131	C₂×Q₁₆	—	146	2	no	—
132	C₂×Q₁₆	—	146	2	no	—
133	C₂²×C₈	—	73	1	no	—
134	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄²	—	73	1	no	—
137	Q₃₂	—	146	8	no	—
138	C₂×C₁₆	—	2	2	no	—
139	C₂×C₁₆	—	146	2	no	—
140	C₂×C₁₆	—	146	1	no	—
141	C₂×C₁₆	D₂₉₂	1	2	yes	—
142	#107	—	2	2	no	—
143	#107	—	2	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	—	292	2	no	—
150	C₂×Q₈	—	292	2	no	—
151	C₂×Q₈	—	292	2	no	—
152	C₂×Q₈	—	292	2	no	—
153	C₂×Q₈	—	292	2	no	—
154	C₂×Q₈	—	146	4	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	—	292	2	no	—
161	Q₁₆	—	292	2	no	—
162	Q₁₆	—	146	4	no	—
163	Q₁₆	—	146	4	no	—
164	C₂×C₈	—	146	1	no	—
165	C₂×C₈	—	146	2	no	—
166	C₂×C₈	D₄×D₇₃	1	2	yes	—
167	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
168	C₂×C₈	#11924028	1	2	yes	—
169	C₂×C₈	#11924029	1	1	yes	characteristic
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	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	—	292	4	no	—
178	Q₈	—	292	4	no	—
179	Q₈	—	584	2	no	—
180	Q₈	—	584	2	no	—
181	C₂×C₄	—	8	2	no	—
182	C₂×C₄	—	8	2	no	—
183	C₂×C₄	—	8	2	no	—
184	C₂×C₄	—	292	4	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	#14556564	1	2	yes	—
193	C₂×C₄	#14556566	1	1	yes	characteristic
194	C₂×C₄	#14556614	1	1	yes	characteristic
195	C₂×C₄	#14556618	1	2	yes	—
196	C₈	—	292	1	no	—
197	C₈	#14556561	1	1	yes	characteristic
198	C₈	#14556576	1	2	yes	—
199	C₈	#14556579	1	1	yes	characteristic
200	C₂²	#17988781	1	2	yes	—
201	C₂²	#17988824	1	1	yes	characteristic
202	C₂²	Q₃₂×D₇₃	1	1	yes	characteristic
203	C₂²	#17989016	1	1	yes	characteristic
204	C₂²	#17989026	1	2	yes	—
205	C₄	—	16	2	no	—
206	C₄	—	292	1	no	—
207	C₄	—	584	4	no	—
208	C₄	—	146	2	no	—
209	C₄	#17988505	1	1	yes	characteristic
210	C₄	#17988805	1	2	yes	—
211	C₄	#17988841	1	1	yes	characteristic
212	C₂	#22611243	1	1	yes	characteristic
213	C₂	#22611980	1	2	yes	—
214	C₂	#22611992	1	2	yes	—
215	C₂	#22612023	1	1	yes	characteristic
216	C₂	#22612040	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₅₈₄

C₂

yes

maximal, characteristic

C₂×#17988405

C₂

yes

maximal, characteristic

C₂×#17988390

C₂

yes

maximal

#22610790

C₂

yes

maximal

C₇₃×#1380

C₂

yes

maximal, characteristic

C₂²×Dic₂₉₂

C₂²

yes

—

C₂×Dic₅₈₄

—

C₂×Dic₅₈₄

—

C₂²×#11924003

C₂²

yes

characteristic

C₂×#14556514

—

C₂×C₈×Dic₇₃

C₂²

yes

—

—

—

C₂²×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×#107×C₇₃

C₂²

yes

—

C₂²×Dic₁₄₆

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

D₄

yes

—

C₂²×#10596831

D₄

yes

characteristic

C₂×C₄×Dic₇₃

D₄

yes

characteristic

Dic₅₈₄

—

C₂×#11924003

—

C₂×#11924003

—

#14556514

—

C₈×Dic₇₃

—

C₈×Dic₇₃

—

#14556498

—

#14556498

—

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

—

#107×C₇₃

—

#107×C₇₃

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

D₈

yes

characteristic

C₂²×Dic₇₃

D₈

yes

characteristic

Dic₂₉₂

—

Dic₂₉₂

—

Dic₂₉₂

—

Dic₂₉₂

—

C₂×#10596831

—

C₂×#10596831

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

#11924003

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

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

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#121

yes

—

C₂×C₄×C₇₃

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

Q₃₂

yes

complemented

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

#120

yes

characteristic

C₈×C₇₃

#123

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

—

110

C₂×C₄⋊C₁₆

—

111

C₂×#358

—

112

#1969

—

113

#1380

D₇₃

yes

characteristic

114

C₇₃

#13450

yes

characteristic, complemented, Sylow

115

C₂²×Q₁₆

—

116

C₂×Q₃₂

—

146

—

117

C₂×Q₃₂

—

146

—

118

C₂²×C₁₆

—

119

C₂²×C₁₆

C₂×D₇₃

yes

characteristic

120

C₂×#107

C₂×D₇₃

yes

—

121

C₂×#103

—

146

—

122

C₂×C₄×C₈

—

123

#391

—

146

—

124

C₄⋊C₁₆

—

146

—

125

#358

—

146

—

126

C₂²×Q₈

—

146

—

127

C₂×Q₁₆

—

128

C₂×Q₁₆

—

146

—

129

C₂×Q₁₆

—

146

—

130

C₂×Q₁₆

—

146

—

131

C₂×Q₁₆

—

146

—

132

C₂×Q₁₆

—

146

—

133

C₂²×C₈

—

134

C₂²×C₈

C₂²×D₇₃

yes

characteristic

135

C₂×C₄⋊C₄

—

136

C₂×C₄²

—

137

Q₃₂

—

146

—

138

C₂×C₁₆

—

139

C₂×C₁₆

—

146

—

140

C₂×C₁₆

—

146

—

141

C₂×C₁₆

D₂₉₂

yes

—

142

#107

—

143

#107

—

144

#103

—

292

—

145

C₄×C₈

—

146

—

146

C₄×C₈

—

146

—

147

#95

—

292

—

148

#95

—

292

—

149

C₂×Q₈

—

292

—

150

C₂×Q₈

—

292

—

151

C₂×Q₈

—

292

—

152

C₂×Q₈

—

292

—

153

C₂×Q₈

—

292

—

154

C₂×Q₈

—

146

—

155

C₂²×C₄

—

156

C₂²×C₄

—

292

—

157

C₂²×C₄

—

158

C₂²×C₄

—

159

C₂²×C₄

D₄×D₇₃

yes

characteristic

160

Q₁₆

—

292

—

161

Q₁₆

—

292

—

162

Q₁₆

—

146

—

163

Q₁₆

—

146

—

164

C₂×C₈

—

146

—

165

C₂×C₈

—

146

—

166

C₂×C₈

D₄×D₇₃

yes

—

167

C₂×C₈

C₂×D₂₉₂

yes

characteristic

168

C₂×C₈

#11924028

yes

—

169

C₂×C₈

#11924029

yes

characteristic

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄²

—

146

—

173

C₄²

—

146

—

174

C₁₆

—

175

C₁₆

—

292

—

176

C₂³

D₈×D₇₃

yes

characteristic

177

Q₈

—

292

—

178

Q₈

—

292

—

179

Q₈

—

584

—

180

Q₈

—

584

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

292

—

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

#14556564

yes

—

193

C₂×C₄

#14556566

yes

characteristic

194

C₂×C₄

#14556614

yes

characteristic

195

C₂×C₄

#14556618

yes

—

196

C₈

—

292

—

197

C₈

#14556561

yes

characteristic

198

C₈

#14556576

yes

—

199

C₈

#14556579

yes

characteristic

200

C₂²

#17988781

yes

—

201

C₂²

#17988824

yes

characteristic

202

C₂²

Q₃₂×D₇₃

yes

characteristic

203

C₂²

#17989016

yes

characteristic

204

C₂²

#17989026

yes

—

205

C₄

—

206

C₄

—

292

—

207

C₄

—

584

—

208

C₄

—

146

—

209

C₄

#17988505

yes

characteristic

210

C₄

#17988805

yes

—

211

C₄

#17988841

yes

characteristic

212

C₂

#22611243

yes

characteristic

213

C₂

#22611980

yes

—

214

C₂

#22611992

yes

—

215

C₂

#22612023

yes

characteristic

216

C₂

#22612040

yes

characteristic

	subgroup	G/H
center	C₂³	D₈×D₇₃
commutator, cosocle	C₂²×C₈×C₇₃	C₂³
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1380	C₂
socle, prime radical	C₂³×C₇₃	C₂×D₈
hypercenter	#1380	D₇₃
hypocenter	C₇₃	#13450
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, prime radical

C₂³×C₇₃

C₂×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
82. C₂×Dic₇₃	Q₃₂	semidirect
114. C₇₃	#13450	semidirect

H ≅ G/N

type

82. 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
C₂²×Dic₅₈₄	2
C₂×#17988405	2
C₂×#17988390	2
#22610790	2
C₇₃×#1380	2
#13450	73

subgroup

index

C₂²×Dic₅₈₄

C₇₃×#1380

Subgroups — #33005592

Conjugacy classes of subgroups

Complements

Maximal subgroups