Group #22621393 — subgroups

Subgroups — #22621393

Order 9344.

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

normal subgroups only

This group has 346 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	#17989210	C₂	1	2	yes	maximal, complemented
2	#17988880	C₂	1	2	yes	maximal, complemented
3	#17988866	C₂	1	2	yes	maximal, complemented
4	#17988835	C₂	1	2	yes	maximal, complemented
5	#17988823	C₂	1	1	yes	maximal, characteristic
6	#17988811	C₂	1	2	yes	maximal, complemented
7	#17988805	C₂	1	2	yes	maximal, complemented
8	#17988613	C₂	1	1	yes	maximal, characteristic, complemented
9	#467×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	#14556590	C₂²	1	2	yes	—
14	#14556586	—	2	2	no	—
15	#14556580	C₂²	1	2	yes	—
16	#14556577	—	2	2	no	—
17	#14556573	—	2	2	no	—
18	#14556571	—	2	2	no	—
19	#14556565	C₂²	1	2	yes	—
20	Q₈×Dic₇₃	C₂²	1	2	yes	—
21	#14556553	C₂²	1	2	yes	—
22	#14556550	C₂²	1	1	yes	characteristic
23	#14556525	C₂²	1	1	yes	characteristic
24	#14556524	C₂²	1	2	yes	—
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	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
32	C₂²⋊Q₈×C₇₃	C₂²	1	2	yes	—
33	#107×C₇₃	C₂²	1	1	yes	characteristic
34	#103×C₇₃	C₂²	1	2	yes	—
35	C₂×Dic₁₄₆	—	2	2	no	—
36	C₂×D₂₉₂	C₂³	1	2	yes	—
37	C₂×C₂²⋊D₇₃	—	2	2	no	—
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	D₄	1	1	yes	characteristic
45	C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
46	C₄⋊Dic₇₃	C₂³	1	2	yes	—
47	#11924007	—	2	2	no	—
48	C₂²⋊Dic₇₃	D₄	1	2	yes	—
49	#11924005	—	2	2	no	—
50	#11924005	—	2	2	no	—
51	#11924005	—	2	2	no	—
52	#11924005	—	2	2	no	—
53	C₄×Dic₇₃	—	2	2	no	—
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	C₂×C₈×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	—	8	2	no	—
75	C₂×Dic₇₃	—	2	2	no	—
76	C₂×Dic₇₃	C₂×D₄	1	2	yes	—
77	C₂×Dic₇₃	C₂×D₄	1	2	yes	—
78	D₂₉₂	—	2	2	no	—
79	D₂₉₂	—	2	2	no	—
80	C₄×D₇₃	—	8	2	no	—
81	Dic₁₄₆	—	8	2	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	1	no	—
87	C₂²×C₇₃	—	2	1	no	—
88	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
89	C₂×D₇₃	—	4	2	no	—
90	C₂×D₇₃	—	4	2	no	—
91	C₂×D₇₃	—	4	2	no	—
92	C₄×C₇₃	—	2	1	no	—
93	C₄×C₇₃	—	4	2	no	—
94	C₄×C₇₃	—	4	2	no	—
95	C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
96	C₄×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
97	Dic₇₃	—	2	2	no	—
98	Dic₇₃	—	2	2	no	—
99	Dic₇₃	—	4	2	no	—
100	C₂×C₇₃	—	4	1	no	—
101	C₂×C₇₃	#537	1	1	yes	characteristic
102	C₂×C₇₃	#576	1	1	yes	characteristic
103	C₂×C₇₃	#578	1	1	yes	characteristic
104	D₇₃	—	8	2	no	—
105	#3262	—	73	1	no	maximal, Sylow
106	C₇₃	#3262	1	1	yes	characteristic, complemented, Sylow
107	#542	—	73	2	no	—
108	#477	—	73	2	no	—
109	#474	—	73	2	no	—
110	#468	—	73	2	no	—
111	#467	D₇₃	1	1	yes	characteristic, complemented
112	#466	—	73	1	no	—
113	#464	—	73	2	no	—
114	#463	—	73	2	no	—
115	#424	—	73	1	no	—
116	C₂×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
117	C₂×SD₁₆	—	73	2	no	—
118	C₂×D₈	—	73	1	no	—
119	C₂×M₁₆	—	73	1	no	—
120	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
121	C₄⋊Q₈	—	73	2	no	—
122	#126	—	146	2	no	—
123	#124	—	73	2	no	—
124	#124	—	146	2	no	—
125	#123	—	146	2	no	—
126	C₂²⋊Q₈	—	146	2	no	—
127	C₂²⋊Q₈	C₂×D₇₃	1	2	yes	—
128	#121	—	73	2	no	—
129	C₄×Q₈	—	73	2	no	—
130	C₄×D₄	—	73	2	no	—
131	#117	—	73	1	no	—
132	#107	—	73	1	no	—
133	#107	—	73	2	no	—
134	#107	C₂×D₇₃	1	1	yes	characteristic
135	C₄⋊C₈	—	73	2	no	—
136	C₄⋊C₈	—	73	2	no	—
137	#103	—	73	2	no	—
138	#103	C₂×D₇₃	1	2	yes	—
139	#102	—	73	2	no	—
140	#102	—	73	2	no	—
141	C₂×Q₈	—	146	2	no	—
142	C₂×Q₈	C₂²×D₇₃	1	2	yes	—
143	C₂×D₄	—	73	2	no	—
144	C₂×D₄	—	146	2	no	—
145	C₂²×C₄	—	146	2	no	—
146	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
147	Q₁₆	—	4	1	no	—
148	SD₁₆	—	292	2	no	—
149	D₈	—	292	1	no	—
150	M₁₆	—	292	1	no	—
151	C₂×C₈	—	4	1	no	—
152	C₂×C₈	—	73	2	no	—
153	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
154	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	73	1	no	—
157	C₄⋊C₄	—	73	1	no	—
158	C₄⋊C₄	—	73	2	no	—
159	C₄⋊C₄	—	146	2	no	—
160	C₄⋊C₄	—	146	2	no	—
161	C₄⋊C₄	C₂²×D₇₃	1	2	yes	—
162	C₂²⋊C₄	—	2	2	no	—
163	C₂²⋊C₄	—	73	2	no	—
164	C₂²⋊C₄	—	146	2	no	—
165	C₂²⋊C₄	—	146	2	no	—
166	C₂²⋊C₄	—	146	2	no	—
167	C₂²⋊C₄	—	146	2	no	—
168	C₄²	—	73	2	no	—
169	C₄²	—	146	2	no	—
170	C₂³	—	146	2	no	—
171	C₂³	D₄×D₇₃	1	1	yes	characteristic
172	Q₈	—	2	2	no	—
173	Q₈	—	2	2	no	—
174	Q₈	—	584	2	no	—
175	D₄	—	584	2	no	—
176	D₄	—	146	2	no	—
177	D₄	—	146	2	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	584	2	no	—
183	C₂×C₄	—	73	2	no	—
184	C₂×C₄	—	73	2	no	—
185	C₂×C₄	—	146	2	no	—
186	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
187	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	—	146	2	no	—
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	292	2	no	—
195	C₂²	—	292	2	no	—
196	C₂²	—	292	2	no	—
197	C₂²	C₂×D₄×D₇₃	1	1	yes	characteristic
198	C₄	—	2	1	no	—
199	C₄	—	4	2	no	—
200	C₄	—	4	2	no	—
201	C₄	—	292	2	no	—
202	C₄	—	146	2	no	—
203	C₄	—	146	2	no	—
204	C₄	#14556636	1	1	yes	characteristic
205	C₄	#14556638	1	1	yes	characteristic
206	C₂	—	4	1	no	—
207	C₂	—	584	2	no	—
208	C₂	#17989164	1	1	yes	characteristic
209	C₂	#17989389	1	1	yes	characteristic
210	C₂	#17989400	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989210

C₂

yes

maximal, complemented

#17988880

C₂

yes

maximal, complemented

#17988866

C₂

yes

maximal, complemented

#17988835

C₂

yes

maximal, complemented

#17988823

C₂

yes

maximal, characteristic

#17988811

C₂

yes

maximal, complemented

#17988805

C₂

yes

maximal, complemented

#17988613

C₂

yes

maximal, characteristic, complemented

#467×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924015

C₂²

yes

—

C₂×D₅₈₄

C₂²

yes

characteristic

C₂×#11924012

C₂²

yes

characteristic

#14556590

C₂²

yes

—

#14556586

—

#14556580

C₂²

yes

—

—

—

—

C₂²

yes

—

Q₈×Dic₇₃

C₂²

yes

—

#14556553

C₂²

yes

—

#14556550

C₂²

yes

characteristic

#14556525

C₂²

yes

characteristic

#14556524

C₂²

yes

—

#14556520

C₂²

yes

—

#14556520

C₂²

yes

—

#14556515

C₂²

yes

—

#14556513

C₂²

yes

—

#14556511

C₂²

yes

—

C₂×Q₁₆×C₇₃

C₂²

yes

characteristic

C₂²×C₈×C₇₃

C₂²

yes

characteristic

C₂²⋊Q₈×C₇₃

C₂²

yes

—

#107×C₇₃

C₂²

yes

characteristic

#103×C₇₃

C₂²

yes

—

C₂×Dic₁₄₆

—

C₂×D₂₉₂

C₂³

yes

—

C₂×C₂²⋊D₇₃

—

C₂×C₄×D₇₃

—

#11924015

—

D₅₈₄

—

#11924012

—

C₂×#10596831

C₂³

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

D₄

yes

characteristic

C₄⋊Dic₇₃

D₄

yes

characteristic

C₄⋊Dic₇₃

C₂³

yes

—

#11924007

—

C₂²⋊Dic₇₃

D₄

yes

—

—

—

—

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₂³

yes

—

C₂×Q₈×C₇₃

C₂³

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

Q₁₆×C₇₃

—

C₂×C₈×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×D₄

yes

—

C₂×Dic₇₃

C₂×D₄

yes

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

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

—

100

C₂×C₇₃

—

101

C₂×C₇₃

#537

yes

characteristic

102

C₂×C₇₃

#576

yes

characteristic

103

C₂×C₇₃

#578

yes

characteristic

104

D₇₃

—

105

#3262

—

maximal, Sylow

106

C₇₃

#3262

yes

characteristic, complemented, Sylow

107

#542

—

108

#477

—

109

#474

—

110

#468

—

111

#467

D₇₃

yes

characteristic, complemented

112

#466

—

113

#464

—

114

#463

—

115

#424

—

116

C₂×Q₁₆

C₂×D₇₃

yes

characteristic

117

C₂×SD₁₆

—

118

C₂×D₈

—

119

C₂×M₁₆

—

120

C₂²×C₈

C₂×D₇₃

yes

characteristic

121

C₄⋊Q₈

—

122

#126

—

146

—

123

#124

—

124

#124

—

146

—

125

#123

—

146

—

126

C₂²⋊Q₈

—

146

—

127

C₂²⋊Q₈

C₂×D₇₃

yes

—

128

#121

—

129

C₄×Q₈

—

130

C₄×D₄

—

131

#117

—

132

#107

—

133

#107

—

134

#107

C₂×D₇₃

yes

characteristic

135

C₄⋊C₈

—

136

C₄⋊C₈

—

137

#103

—

138

#103

C₂×D₇₃

yes

—

139

#102

—

140

#102

—

141

C₂×Q₈

—

146

—

142

C₂×Q₈

C₂²×D₇₃

yes

—

143

C₂×D₄

—

144

C₂×D₄

—

146

—

145

C₂²×C₄

—

146

—

146

C₂²×C₄

C₂²×D₇₃

yes

characteristic

147

Q₁₆

—

148

SD₁₆

—

292

—

149

D₈

—

292

—

150

M₁₆

—

292

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

C₂²×D₇₃

yes

characteristic

154

C₂×C₈

C₂²×D₇₃

yes

characteristic

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

146

—

160

C₄⋊C₄

—

146

—

161

C₄⋊C₄

C₂²×D₇₃

yes

—

162

C₂²⋊C₄

—

163

C₂²⋊C₄

—

164

C₂²⋊C₄

—

146

—

165

C₂²⋊C₄

—

146

—

166

C₂²⋊C₄

—

146

—

167

C₂²⋊C₄

—

146

—

168

C₄²

—

169

C₄²

—

146

—

170

C₂³

—

146

—

171

C₂³

D₄×D₇₃

yes

characteristic

172

Q₈

—

173

Q₈

—

174

Q₈

—

584

—

175

D₄

—

584

—

176

D₄

—

146

—

177

D₄

—

146

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

584

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

146

—

186

C₂×C₄

D₄×D₇₃

yes

characteristic

187

C₂×C₄

C₂³×D₇₃

yes

characteristic

188

C₈

—

189

C₈

—

190

C₈

—

146

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

194

C₂²

—

292

—

195

C₂²

—

292

—

196

C₂²

—

292

—

197

C₂²

C₂×D₄×D₇₃

yes

characteristic

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

—

292

—

202

C₄

—

146

—

203

C₄

—

146

—

204

C₄

#14556636

yes

characteristic

205

C₄

#14556638

yes

characteristic

206

C₂

—

207

C₂

—

584

—

208

C₂

#17989164

yes

characteristic

209

C₂

#17989389

yes

characteristic

210

C₂

#17989400

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₃
Fitting	#467×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₄
hypercenter	#467	D₇₃
hypocenter	C₇₃	#3262
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #467

subgroup

G/H

center

C₂²

C₂×D₄×D₇₃

commutator

C₂×C₄×C₇₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₃

Fitting

#467×C₇₃

C₂

socle

C₂²×C₇₃

C₂²×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #467

N	H ≅ G/N	type
1. #17989210	C₂	semidirect
2. #17988880	C₂	semidirect
3. #17988866	C₂	semidirect
4. #17988835	C₂	semidirect
6. #17988811	C₂	semidirect
7. #17988805	C₂	semidirect
8. #17988613	C₂	semidirect
9. #467×C₇₃	C₂	semidirect
106. C₇₃	#3262	semidirect
111. #467	D₇₃	semidirect

H ≅ G/N

type

1. #17989210

C₂

semidirect

2. #17988880

C₂

semidirect

3. #17988866

C₂

semidirect

4. #17988835

C₂

semidirect

6. #17988811

C₂

semidirect

7. #17988805

C₂

semidirect

8. #17988613

C₂

semidirect

9. #467×C₇₃

C₂

semidirect

106. C₇₃

#3262

semidirect

111. #467

D₇₃

semidirect

Subgroups — #22621393

Conjugacy classes of subgroups

Complements