Group #22621125 — subgroups

Subgroups — #22621125

Order 9344.

366 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 121 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	#17989144	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988993	C₂	1	2	yes	maximal, complemented
3	#17988951	C₂	1	2	yes	maximal, complemented
4	#17988908	C₂	1	4	yes	maximal, complemented
5	#17988877	C₂	1	2	yes	maximal, complemented
6	#17988867	C₂	1	2	yes	maximal, complemented
7	#17988644	C₂	1	1	yes	maximal, characteristic, complemented
8	#532×C₇₃	C₂	1	1	yes	maximal, characteristic
9	#14556591	C₂²	1	1	yes	characteristic
10	#14556591	C₂²	1	1	yes	characteristic
11	#14556583	C₂²	1	2	yes	—
12	#14556570	—	2	1	no	—
13	#14556570	—	2	1	no	—
14	C₄×Dic₁₄₆	—	2	1	no	—
15	C₄×Dic₁₄₆	—	2	1	no	—
16	#14556550	C₂²	1	2	yes	—
17	#14556524	C₂²	1	4	yes	—
18	#14556522	C₂²	1	4	yes	—
19	C₄⋊#10596831	C₂²	1	2	yes	—
20	#14556512	C₂²	1	4	yes	—
21	#14556512	C₂²	1	4	yes	—
22	C₂²⋊#10596831	C₂²	1	2	yes	—
23	#14556502	C₂²	1	2	yes	—
24	C₂×#41×C₇₃	—	2	1	no	—
25	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
26	#127×C₇₃	C₂²	1	1	yes	characteristic
27	#124×C₇₃	—	2	1	no	—
28	#121×C₇₃	—	2	1	no	—
29	#121×C₇₃	C₂²	1	2	yes	—
30	C₄×D₄×C₇₃	—	2	1	no	—
31	C₄×D₄×C₇₃	C₂²	1	2	yes	—
32	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
33	C₂×Dic₁₄₆	—	2	1	no	—
34	C₂×Dic₁₄₆	—	2	1	no	—
35	C₂×#10596831	C₂³	1	4	yes	—
36	C₄⋊Dic₇₃	—	2	1	no	—
37	C₄⋊Dic₇₃	—	2	1	no	—
38	C₄⋊Dic₇₃	C₂³	1	2	yes	—
39	C₄⋊Dic₇₃	C₂³	1	2	yes	—
40	#11924007	—	2	2	no	—
41	#11924007	—	2	2	no	—
42	C₂²⋊Dic₇₃	—	2	2	no	—
43	C₄×Dic₇₃	—	2	2	no	—
44	#41×C₇₃	—	4	4	no	—
45	C₂×Q₈×C₇₃	—	2	1	no	—
46	C₂×D₄×C₇₃	—	2	2	no	—
47	C₂×D₄×C₇₃	—	2	1	no	—
48	C₂×D₄×C₇₃	—	2	1	no	—
49	C₂×D₄×C₇₃	C₂³	1	2	yes	—
50	C₂²×C₄×C₇₃	—	2	2	no	—
51	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₇₃	—	2	1	no	—
55	C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
56	C₂²⋊C₄×C₇₃	—	2	2	no	—
57	C₂²⋊C₄×C₇₃	—	2	2	no	—
58	C₄²×C₇₃	D₄	1	1	yes	characteristic
59	C₄²×C₇₃	D₄	1	1	yes	characteristic
60	C₄²×C₇₃	C₂³	1	1	yes	characteristic
61	C₄²×C₇₃	C₂³	1	1	yes	characteristic
62	C₂³×C₇₃	—	2	2	no	—
63	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
64	Q₈×C₇₃	—	4	2	no	—
65	D₄×C₇₃	—	2	4	no	—
66	D₄×C₇₃	—	4	2	no	—
67	D₄×C₇₃	—	4	2	no	—
68	D₄×C₇₃	—	4	4	no	—
69	C₂×C₄×C₇₃	—	2	2	no	—
70	C₂×C₄×C₇₃	—	2	2	no	—
71	C₂×C₄×C₇₃	—	2	2	no	—
72	C₂×C₄×C₇₃	—	4	1	no	—
73	C₂×C₄×C₇₃	—	4	4	no	—
74	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
80	C₂×Dic₇₃	—	2	2	no	—
81	C₂×Dic₇₃	—	2	2	no	—
82	Dic₁₄₆	—	4	2	no	—
83	Dic₁₄₆	—	4	2	no	—
84	#10596831	—	2	4	no	—
85	C₂²×C₇₃	—	2	1	no	—
86	C₂²×C₇₃	—	2	1	no	—
87	C₂²×C₇₃	—	2	1	no	—
88	C₂²×C₇₃	—	4	2	no	—
89	C₂²×C₇₃	—	4	2	no	—
90	C₂²×C₇₃	—	4	2	no	—
91	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
92	C₄×C₇₃	—	2	1	no	—
93	C₄×C₇₃	—	2	1	no	—
94	C₄×C₇₃	—	2	1	no	—
95	C₄×C₇₃	—	2	1	no	—
96	C₄×C₇₃	—	4	2	no	—
97	C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
98	C₄×C₇₃	2₋¹⁺⁴	1	2	yes	—
99	Dic₇₃	—	4	2	no	—
100	Dic₇₃	—	4	2	no	—
101	C₂×C₇₃	—	4	1	no	—
102	C₂×C₇₃	—	8	2	no	—
103	C₂×C₇₃	#536	1	1	yes	characteristic
104	C₂×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₂×C₇₃	#575	1	1	yes	characteristic
106	#3235	—	73	1	no	maximal, Sylow
107	C₇₃	#3235	1	1	yes	characteristic, complemented, Sylow
108	#533	—	73	1	no	—
109	#532	D₇₃	1	1	yes	characteristic
110	#501	—	73	2	no	—
111	#490	—	73	2	no	—
112	#482	—	73	4	no	—
113	#476	—	73	2	no	—
114	#474	—	73	2	no	—
115	#432	—	73	1	no	—
116	C₂×#41	—	2	1	no	—
117	C₄⋊Q₈	—	73	1	no	—
118	C₄⋊Q₈	—	73	1	no	—
119	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
120	#127	C₂×D₇₃	1	1	yes	characteristic
121	#125	—	73	2	no	—
122	#124	—	2	1	no	—
123	C₂²⋊Q₈	—	146	1	no	—
124	C₂²⋊Q₈	—	146	1	no	—
125	#121	—	2	1	no	—
126	#121	C₂×D₇₃	1	2	yes	—
127	C₄×Q₈	—	146	1	no	—
128	C₄×Q₈	—	146	1	no	—
129	C₄×D₄	—	2	1	no	—
130	C₄×D₄	C₂×D₇₃	1	2	yes	—
131	#117	—	73	2	no	—
132	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
133	#107	—	73	4	no	—
134	#106	—	73	4	no	—
135	C₄⋊C₈	—	73	2	no	—
136	#102	—	73	4	no	—
137	#102	—	73	4	no	—
138	C₂²⋊C₈	—	73	2	no	—
139	#97	—	73	2	no	—
140	#41	—	4	4	no	—
141	C₂×Q₈	—	2	1	no	—
142	C₂×Q₈	—	146	1	no	—
143	C₂×Q₈	—	146	1	no	—
144	C₂×D₄	—	2	2	no	—
145	C₂×D₄	—	2	1	no	—
146	C₂×D₄	—	2	1	no	—
147	C₂×D₄	C₂²×D₇₃	1	2	yes	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
150	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
151	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
152	C₂×C₈	—	73	4	no	—
153	C₄⋊C₄	—	2	1	no	—
154	C₄⋊C₄	—	73	2	no	—
155	C₄⋊C₄	—	73	2	no	—
156	C₄⋊C₄	—	146	2	no	—
157	C₄⋊C₄	—	146	2	no	—
158	C₄⋊C₄	—	146	1	no	—
159	C₄⋊C₄	—	146	1	no	—
160	C₄⋊C₄	C₂²×D₇₃	1	2	yes	—
161	C₂²⋊C₄	—	2	2	no	—
162	C₂²⋊C₄	—	2	2	no	—
163	C₂²⋊C₄	—	146	2	no	—
164	C₄²	—	146	2	no	—
165	C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
166	C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
167	C₄²	C₂²×D₇₃	1	1	yes	characteristic
168	C₄²	C₂²×D₇₃	1	1	yes	characteristic
169	C₂³	—	2	2	no	—
170	C₂³	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
171	Q₈	—	4	2	no	—
172	Q₈	—	292	2	no	—
173	Q₈	—	292	2	no	—
174	D₄	—	2	4	no	—
175	D₄	—	4	2	no	—
176	D₄	—	4	2	no	—
177	D₄	—	4	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	4	1	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	146	2	no	—
184	C₂×C₄	—	146	2	no	—
185	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
186	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
187	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
188	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
189	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
190	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
191	C₈	—	146	4	no	—
192	C₂²	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	4	2	no	—
196	C₂²	—	4	2	no	—
197	C₂²	—	4	2	no	—
198	C₂²	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
199	C₄	—	2	1	no	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	4	2	no	—
204	C₄	—	292	2	no	—
205	C₄	—	292	2	no	—
206	C₄	#14556617	1	2	yes	—
207	C₄	#14556638	1	2	yes	—
208	C₂	—	4	1	no	—
209	C₂	—	8	2	no	—
210	C₂	#17989158	1	1	yes	characteristic
211	C₂	C₂×#14556617	1	1	yes	characteristic
212	C₂	#17989385	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989144

C₂

yes

maximal, characteristic, complemented

#17988993

C₂

yes

maximal, complemented

#17988951

C₂

yes

maximal, complemented

#17988908

C₂

yes

maximal, complemented

#17988877

C₂

yes

maximal, complemented

#17988867

C₂

yes

maximal, complemented

#17988644

C₂

yes

maximal, characteristic, complemented

#532×C₇₃

C₂

yes

maximal, characteristic

#14556591

C₂²

yes

characteristic

#14556591

C₂²

yes

characteristic

#14556583

C₂²

yes

—

#14556570

—

#14556570

—

C₄×Dic₁₄₆

—

C₄×Dic₁₄₆

—

#14556550

C₂²

yes

—

#14556524

C₂²

yes

—

#14556522

C₂²

yes

—

C₄⋊#10596831

C₂²

yes

—

#14556512

C₂²

yes

—

#14556512

C₂²

yes

—

C₂²⋊#10596831

C₂²

yes

—

#14556502

C₂²

yes

—

C₂×#41×C₇₃

—

C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

#127×C₇₃

C₂²

yes

characteristic

#124×C₇₃

—

#121×C₇₃

—

#121×C₇₃

C₂²

yes

—

C₄×D₄×C₇₃

—

C₄×D₄×C₇₃

C₂²

yes

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×#10596831

C₂³

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂³

yes

—

C₄⋊Dic₇₃

C₂³

yes

—

#11924007

—

#11924007

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

#41×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

C₂³

yes

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂³

yes

—

C₂²⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

C₂³

yes

characteristic

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂⁴

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂²×D₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₄×C₇₃

2₋¹⁺⁴

yes

—

Dic₇₃

—

100

Dic₇₃

—

101

C₂×C₇₃

—

102

C₂×C₇₃

—

103

C₂×C₇₃

#536

yes

characteristic

104

C₂×C₇₃

C₂×AGL(1,ℤ/8)

yes

characteristic

105

C₂×C₇₃

yes

106

—

107

C₇₃

#3235

yes

characteristic, complemented, Sylow

108

#533

—

109

#532

D₇₃

yes

characteristic

110

#501

—

111

#490

—

112

#482

—

113

#476

—

114

#474

—

115

#432

—

116

C₂×#41

—

117

C₄⋊Q₈

—

118

C₄⋊Q₈

—

119

C₄⋊Q₈

C₂×D₇₃

yes

characteristic

120

#127

C₂×D₇₃

yes

characteristic

121

#125

—

122

#124

—

123

C₂²⋊Q₈

—

146

—

124

C₂²⋊Q₈

—

146

—

125

#121

—

126

#121

C₂×D₇₃

yes

—

127

C₄×Q₈

—

146

—

128

C₄×Q₈

—

146

—

129

C₄×D₄

—

130

C₄×D₄

C₂×D₇₃

yes

—

131

#117

—

132

C₂×C₄²

C₂×D₇₃

yes

characteristic

133

#107

—

134

#106

—

135

C₄⋊C₈

—

136

#102

—

137

#102

—

138

C₂²⋊C₈

—

139

#97

—

140

#41

—

141

C₂×Q₈

—

142

C₂×Q₈

—

146

—

143

C₂×Q₈

—

146

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

C₂²×D₇₃

yes

—

148

C₂²×C₄

—

149

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

150

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

151

C₂²×C₄

C₂²×D₇₃

yes

characteristic

152

C₂×C₈

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

146

—

157

C₄⋊C₄

—

146

—

158

C₄⋊C₄

—

146

—

159

C₄⋊C₄

—

146

—

160

C₄⋊C₄

C₂²×D₇₃

yes

—

161

C₂²⋊C₄

—

162

C₂²⋊C₄

—

163

C₂²⋊C₄

—

146

—

164

C₄²

—

146

—

165

C₄²

C₂²⋊D₇₃

yes

characteristic

166

C₄²

C₂²⋊D₇₃

yes

characteristic

167

C₄²

C₂²×D₇₃

yes

characteristic

168

C₄²

C₂²×D₇₃

yes

characteristic

169

C₂³

—

170

C₂³

C₂×C₂²⋊D₇₃

yes

characteristic

171

Q₈

—

172

Q₈

—

292

—

173

Q₈

—

292

—

174

D₄

—

175

D₄

—

176

D₄

—

177

D₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

146

—

184

C₂×C₄

—

146

—

185

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

186

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

187

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

188

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

189

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

190

C₂×C₄

C₂³×D₇₃

yes

characteristic

191

C₈

—

146

—

192

C₂²

—

193

C₂²

—

194

C₂²

—

195

C₂²

—

196

C₂²

—

197

C₂²

—

198

C₂²

C₂²×C₂²⋊D₇₃

yes

characteristic

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

—

292

—

205

C₄

—

292

—

206

C₄

#14556617

yes

—

207

C₄

#14556638

yes

—

208

C₂

—

209

C₂

—

210

C₂

#17989158

yes

characteristic

211

C₂

C₂×#14556617

yes

characteristic

212

C₂

#17989385

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₃
Fitting	#532×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₄
hypercenter	#532	D₇₃
hypocenter	C₇₃	#3235
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₇₃

commutator

C₂×C₄×C₇₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₃

Fitting

#532×C₇₃

C₂

socle

C₂²×C₇₃

C₂²×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #17989144	C₂	semidirect
2. #17988993	C₂	semidirect
3. #17988951	C₂	semidirect
4. #17988908	C₂	semidirect
5. #17988877	C₂	semidirect
6. #17988867	C₂	semidirect
7. #17988644	C₂	semidirect
107. C₇₃	#3235	semidirect

H ≅ G/N

type

1. #17989144

C₂

semidirect

2. #17988993

C₂

semidirect

3. #17988951

C₂

semidirect

4. #17988908

C₂

semidirect

5. #17988877

C₂

semidirect

6. #17988867

C₂

semidirect

7. #17988644

C₂

semidirect

107. C₇₃

#3235

semidirect

Subgroups — #22621125

Conjugacy classes of subgroups

Complements