Group #22621333 — subgroups

Subgroups — #22621333

Order 9344.

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

normal subgroups only

This group has 354 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	#17989123	C₂	1	2	yes	maximal, complemented
2	#17988892	C₂	1	2	yes	maximal, complemented
3	#17988886	C₂	1	2	yes	maximal, complemented
4	#17988840	C₂	1	1	yes	maximal, characteristic
5	#17988835	C₂	1	2	yes	maximal, complemented
6	#17988811	C₂	1	2	yes	maximal, complemented
7	#17988805	C₂	1	2	yes	maximal, complemented
8	#17988610	C₂	1	1	yes	maximal, characteristic, complemented
9	#470×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11924015	C₂²	1	2	yes	—
11	C₂×D₅₈₄	C₂²	1	1	yes	characteristic
12	C₂×#11924012	C₂²	1	1	yes	characteristic
13	#14556586	—	2	2	no	—
14	#14556582	C₂²	1	2	yes	—
15	#14556580	C₂²	1	2	yes	—
16	#14556575	—	2	2	no	—
17	#14556565	C₂²	1	2	yes	—
18	Q₈×Dic₇₃	C₂²	1	2	yes	—
19	#14556553	C₂²	1	2	yes	—
20	#14556551	—	2	2	no	—
21	#14556549	—	2	2	no	—
22	C₂×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
23	#14556524	C₂²	1	2	yes	—
24	#14556523	C₂²	1	1	yes	characteristic
25	#14556520	C₂²	1	2	yes	—
26	#14556520	C₂²	1	2	yes	—
27	#14556515	C₂²	1	2	yes	—
28	#14556513	C₂²	1	2	yes	—
29	#14556511	C₂²	1	2	yes	—
30	C₂×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
31	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
32	C₂²⋊Q₈×C₇₃	C₂²	1	2	yes	—
33	#106×C₇₃	C₂²	1	1	yes	characteristic
34	#103×C₇₃	C₂²	1	2	yes	—
35	C₂×D₂₉₂	C₂³	1	2	yes	—
36	C₂×C₂²⋊D₇₃	—	2	2	no	—
37	C₂²×Dic₇₃	D₄	1	2	yes	—
38	C₂×C₄×D₇₃	—	2	2	no	—
39	#11924015	—	4	2	no	—
40	D₅₈₄	—	4	1	no	—
41	#11924012	—	4	1	no	—
42	C₂×#10596831	C₂³	1	2	yes	—
43	C₄⋊Dic₇₃	—	2	2	no	—
44	C₄⋊Dic₇₃	—	2	2	no	—
45	C₄⋊Dic₇₃	C₂³	1	2	yes	—
46	#11924007	—	2	2	no	—
47	#11924005	—	2	2	no	—
48	#11924005	—	2	2	no	—
49	#11924005	—	2	2	no	—
50	#11924005	—	2	2	no	—
51	C₄×Dic₇₃	—	2	2	no	—
52	C₄×Dic₇₃	D₄	1	1	yes	characteristic
53	C₄×Dic₇₃	D₄	1	1	yes	characteristic
54	C₄×Dic₇₃	C₂³	1	2	yes	—
55	C₂×Q₈×C₇₃	C₂³	1	2	yes	—
56	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
57	Q₁₆×C₇₃	—	4	1	no	—
58	M₁₆×C₇₃	—	4	1	no	—
59	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₇₃	—	2	2	no	—
62	C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
63	C₂²⋊C₄×C₇₃	—	2	2	no	—
64	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×D₇₃	—	2	2	no	—
66	Q₈×C₇₃	—	2	2	no	—
67	Q₈×C₇₃	—	2	2	no	—
68	C₂×C₄×C₇₃	—	2	1	no	—
69	C₂×C₄×C₇₃	—	2	1	no	—
70	C₂×C₄×C₇₃	—	2	2	no	—
71	C₂×C₄×C₇₃	—	2	2	no	—
72	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
74	C₂²⋊D₇₃	—	4	4	no	—
75	C₂×Dic₇₃	—	2	2	no	—
76	C₂×Dic₇₃	—	4	2	no	—
77	C₂×Dic₇₃	C₂×D₄	1	2	yes	—
78	C₂×Dic₇₃	C₂×D₄	1	2	yes	—
79	D₂₉₂	—	2	2	no	—
80	D₂₉₂	—	2	2	no	—
81	C₄×D₇₃	—	4	4	no	—
82	C₈×C₇₃	—	2	1	no	—
83	C₈×C₇₃	—	2	1	no	—
84	#10596831	—	2	2	no	—
85	C₂²×C₇₃	—	2	1	no	—
86	C₂²×C₇₃	—	2	2	no	—
87	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
88	C₂×D₇₃	—	4	2	no	—
89	C₂×D₇₃	—	4	2	no	—
90	C₂×D₇₃	—	4	2	no	—
91	C₄×C₇₃	—	2	1	no	—
92	C₄×C₇₃	—	4	2	no	—
93	C₄×C₇₃	—	4	2	no	—
94	C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
95	C₄×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
96	Dic₇₃	—	2	2	no	—
97	Dic₇₃	—	2	2	no	—
98	Dic₇₃	—	4	2	no	—
99	C₂×C₇₃	—	4	1	no	—
100	C₂×C₇₃	#537	1	1	yes	characteristic
101	C₂×C₇₃	#575	1	2	yes	—
102	D₇₃	—	8	2	no	—
103	#3256	—	73	1	no	maximal, Sylow
104	C₇₃	#3256	1	1	yes	characteristic, complemented, Sylow
105	#529	—	73	2	no	—
106	#479	—	73	2	no	—
107	#478	—	73	2	no	—
108	#470	D₇₃	1	1	yes	characteristic, complemented
109	#469	—	73	1	no	—
110	#468	—	73	2	no	—
111	#464	—	73	2	no	—
112	#463	—	73	2	no	—
113	#423	—	73	1	no	—
114	C₂×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
115	C₂×SD₁₆	—	73	2	no	—
116	C₂×D₈	—	73	1	no	—
117	C₂×M₁₆	—	73	1	no	—
118	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
119	#126	—	146	2	no	—
120	#125	—	73	2	no	—
121	#124	—	73	2	no	—
122	#123	—	146	2	no	—
123	C₂²⋊Q₈	C₂×D₇₃	1	2	yes	—
124	#121	—	73	2	no	—
125	C₄×Q₈	—	73	2	no	—
126	C₄×D₄	—	73	2	no	—
127	C₄×D₄	—	146	2	no	—
128	#117	—	146	2	no	—
129	C₂×C₄²	—	73	1	no	—
130	#107	—	73	2	no	—
131	#106	—	73	1	no	—
132	#106	C₂×D₇₃	1	1	yes	characteristic
133	C₄⋊C₈	—	73	2	no	—
134	C₄⋊C₈	—	73	2	no	—
135	#103	—	73	2	no	—
136	#103	C₂×D₇₃	1	2	yes	—
137	#102	—	73	2	no	—
138	#102	—	73	2	no	—
139	C₂×Q₈	C₂²×D₇₃	1	2	yes	—
140	C₂×D₄	—	73	2	no	—
141	C₂×D₄	—	146	2	no	—
142	C₂²×C₄	—	73	2	no	—
143	C₂²×C₄	—	146	2	no	—
144	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
145	Q₁₆	—	4	1	no	—
146	SD₁₆	—	292	2	no	—
147	D₈	—	292	1	no	—
148	M₁₆	—	4	1	no	—
149	M₁₆	—	292	1	no	—
150	C₂×C₈	—	73	2	no	—
151	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
152	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	73	2	no	—
155	C₄⋊C₄	—	146	2	no	—
156	C₄⋊C₄	—	146	2	no	—
157	C₄⋊C₄	—	146	2	no	—
158	C₄⋊C₄	C₂²×D₇₃	1	2	yes	—
159	C₂²⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	146	2	no	—
161	C₂²⋊C₄	—	146	2	no	—
162	C₂²⋊C₄	—	146	2	no	—
163	C₂²⋊C₄	—	146	2	no	—
164	C₄²	—	73	1	no	—
165	C₄²	—	73	1	no	—
166	C₄²	—	73	2	no	—
167	C₄²	—	146	2	no	—
168	C₂³	—	146	2	no	—
169	C₂³	D₄×D₇₃	1	1	yes	characteristic
170	Q₈	—	2	2	no	—
171	Q₈	—	2	2	no	—
172	D₄	—	292	4	no	—
173	D₄	—	146	2	no	—
174	D₄	—	146	2	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	292	4	no	—
180	C₂×C₄	—	292	2	no	—
181	C₂×C₄	—	73	2	no	—
182	C₂×C₄	—	73	2	no	—
183	C₂×C₄	—	146	2	no	—
184	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
185	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
186	C₈	—	2	1	no	—
187	C₈	—	2	1	no	—
188	C₈	—	146	2	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	292	2	no	—
192	C₂²	—	292	2	no	—
193	C₂²	—	292	2	no	—
194	C₂²	C₂×D₄×D₇₃	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	4	2	no	—
197	C₄	—	4	2	no	—
198	C₄	—	292	2	no	—
199	C₄	—	146	2	no	—
200	C₄	—	146	2	no	—
201	C₄	#14556636	1	1	yes	characteristic
202	C₄	#14556638	1	1	yes	characteristic
203	C₂	—	4	1	no	—
204	C₂	—	584	2	no	—
205	C₂	#17989164	1	1	yes	characteristic
206	C₂	#17989386	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17989123

C₂

yes

maximal, complemented

#17988892

C₂

yes

maximal, complemented

#17988886

C₂

yes

maximal, complemented

#17988840

C₂

yes

maximal, characteristic

#17988835

C₂

yes

maximal, complemented

#17988811

C₂

yes

maximal, complemented

#17988805

C₂

yes

maximal, complemented

#17988610

C₂

yes

maximal, characteristic, complemented

#470×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924015

C₂²

yes

—

C₂×D₅₈₄

C₂²

yes

characteristic

C₂×#11924012

C₂²

yes

characteristic

#14556586

—

#14556582

C₂²

yes

—

#14556580

C₂²

yes

—

#14556575

—

#14556565

C₂²

yes

—

Q₈×Dic₇₃

C₂²

yes

—

#14556553

C₂²

yes

—

#14556551

—

#14556549

—

C₂×C₄×Dic₇₃

C₂²

yes

characteristic

#14556524

C₂²

yes

—

#14556523

C₂²

yes

characteristic

#14556520

C₂²

yes

—

#14556520

C₂²

yes

—

#14556515

C₂²

yes

—

#14556513

C₂²

yes

—

#14556511

C₂²

yes

—

C₂×Q₁₆×C₇₃

C₂²

yes

characteristic

C₂×M₁₆×C₇₃

C₂²

yes

characteristic

C₂²⋊Q₈×C₇₃

C₂²

yes

—

#106×C₇₃

C₂²

yes

characteristic

#103×C₇₃

C₂²

yes

—

C₂×D₂₉₂

C₂³

yes

—

C₂×C₂²⋊D₇₃

—

C₂²×Dic₇₃

D₄

yes

—

C₂×C₄×D₇₃

—

#11924015

—

D₅₈₄

—

#11924012

—

C₂×#10596831

C₂³

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂³

yes

—

—

—

—

—

—

C₄×Dic₇₃

—

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

C₂³

yes

—

C₂×Q₈×C₇₃

C₂³

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

Q₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂³

yes

—

C₂²⋊C₄×C₇₃

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂²×D₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂⁴

yes

characteristic

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂×D₄

yes

—

C₂×Dic₇₃

C₂×D₄

yes

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂²×D₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

H(5,𝔽₂)

yes

characteristic

C₄×C₇₃

2₋¹⁺⁴

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

100

C₂×C₇₃

#537

yes

characteristic

101

C₂×C₇₃

#575

yes

—

102

D₇₃

—

103

#3256

—

maximal, Sylow

104

C₇₃

#3256

yes

characteristic, complemented, Sylow

105

#529

—

106

#479

—

107

#478

—

108

#470

D₇₃

yes

characteristic, complemented

109

#469

—

110

#468

—

111

#464

—

112

#463

—

113

#423

—

114

C₂×Q₁₆

C₂×D₇₃

yes

characteristic

115

C₂×SD₁₆

—

116

C₂×D₈

—

117

C₂×M₁₆

—

118

C₂×M₁₆

C₂×D₇₃

yes

characteristic

119

#126

—

146

—

120

#125

—

121

#124

—

122

#123

—

146

—

123

C₂²⋊Q₈

C₂×D₇₃

yes

—

124

#121

—

125

C₄×Q₈

—

126

C₄×D₄

—

127

C₄×D₄

—

146

—

128

#117

—

146

—

129

C₂×C₄²

—

130

#107

—

131

#106

—

132

#106

C₂×D₇₃

yes

characteristic

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

#103

—

136

#103

C₂×D₇₃

yes

—

137

#102

—

138

#102

—

139

C₂×Q₈

C₂²×D₇₃

yes

—

140

C₂×D₄

—

141

C₂×D₄

—

146

—

142

C₂²×C₄

—

143

C₂²×C₄

—

146

—

144

C₂²×C₄

C₂²×D₇₃

yes

characteristic

145

Q₁₆

—

146

SD₁₆

—

292

—

147

D₈

—

292

—

148

M₁₆

—

149

M₁₆

—

292

—

150

C₂×C₈

—

151

C₂×C₈

C₂²×D₇₃

yes

characteristic

152

C₂×C₈

C₂²×D₇₃

yes

characteristic

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

146

—

156

C₄⋊C₄

—

146

—

157

C₄⋊C₄

—

146

—

158

C₄⋊C₄

C₂²×D₇₃

yes

—

159

C₂²⋊C₄

—

160

C₂²⋊C₄

—

146

—

161

C₂²⋊C₄

—

146

—

162

C₂²⋊C₄

—

146

—

163

C₂²⋊C₄

—

146

—

164

C₄²

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

146

—

168

C₂³

—

146

—

169

C₂³

D₄×D₇₃

yes

characteristic

170

Q₈

—

171

Q₈

—

172

D₄

—

292

—

173

D₄

—

146

—

174

D₄

—

146

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

292

—

180

C₂×C₄

—

292

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

146

—

184

C₂×C₄

D₄×D₇₃

yes

characteristic

185

C₂×C₄

C₂³×D₇₃

yes

characteristic

186

C₈

—

187

C₈

—

188

C₈

—

146

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

292

—

192

C₂²

—

292

—

193

C₂²

—

292

—

194

C₂²

C₂×D₄×D₇₃

yes

characteristic

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

292

—

199

C₄

—

146

—

200

C₄

—

146

—

201

C₄

#14556636

yes

characteristic

202

C₄

#14556638

yes

characteristic

203

C₂

—

204

C₂

—

584

—

205

C₂

#17989164

yes

characteristic

206

C₂

#17989386

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₃
Fitting	#470×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₄
hypercenter	#470	D₇₃
hypocenter	C₇₃	#3256
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #470

subgroup

G/H

center

C₂²

C₂×D₄×D₇₃

commutator

C₂×C₄×C₇₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₃

Fitting

#470×C₇₃

C₂

socle

C₂²×C₇₃

C₂²×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #470

N	H ≅ G/N	type
1. #17989123	C₂	semidirect
2. #17988892	C₂	semidirect
3. #17988886	C₂	semidirect
5. #17988835	C₂	semidirect
6. #17988811	C₂	semidirect
7. #17988805	C₂	semidirect
8. #17988610	C₂	semidirect
9. #470×C₇₃	C₂	semidirect
104. C₇₃	#3256	semidirect
108. #470	D₇₃	semidirect

H ≅ G/N

type

1. #17989123

C₂

semidirect

2. #17988892

C₂

semidirect

3. #17988886

C₂

semidirect

5. #17988835

C₂

semidirect

6. #17988811

C₂

semidirect

7. #17988805

C₂

semidirect

8. #17988610

C₂

semidirect

9. #470×C₇₃

C₂

semidirect

104. C₇₃

#3256

semidirect

108. #470

D₇₃

semidirect

Subgroups — #22621333

Conjugacy classes of subgroups

Complements