Group #20276108 — subgroups

Subgroups — #20276108

Order 6784.

248 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 87 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×#13133151	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133146	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133141	C₂	1	1	yes	maximal, characteristic
4	#16810530	C₂	1	1	yes	maximal, characteristic
5	#16810519	C₂	1	1	yes	maximal, characteristic
6	#16810385	C₂	1	1	yes	maximal, characteristic
7	C₂×#103×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×#88322	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×#11719919	—	2	1	no	—
12	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
13	#13133151	—	2	2	no	—
14	#13133146	—	2	2	no	—
15	#13133141	—	2	2	no	—
16	#13133124	—	2	1	no	—
17	#13133124	—	2	1	no	—
18	#13133124	—	2	1	no	—
19	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
22	#103×C₅₃	—	2	2	no	—
23	C₂×Q₈×C₅₃	—	2	1	no	—
24	C₂×Q₈×C₅₃	—	2	1	no	—
25	C₂×Q₈×C₅₃	—	2	1	no	—
26	C₂×Q₈×C₅₃	—	2	1	no	—
27	C₂×Q₈×C₅₃	—	2	1	no	—
28	C₂×Q₈×C₅₃	Q₈	1	2	yes	—
29	C₂²×C₄×C₅₃	—	2	1	no	—
30	C₂²×C₄×C₅₃	—	2	1	no	—
31	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
32	C₂²×Dic₅₃	—	2	1	no	—
33	C₂²×Dic₅₃	—	2	1	no	—
34	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
35	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
36	C₂×C₈×C₅₃	—	2	1	no	—
37	C₂×C₈×C₅₃	—	2	1	no	—
38	C₂×C₈×C₅₃	D₄	1	2	yes	—
39	C₄⋊C₄×C₅₃	—	2	1	no	—
40	C₄⋊C₄×C₅₃	—	2	1	no	—
41	C₄⋊Dic₅₃	—	2	1	no	—
42	C₄⋊Dic₅₃	—	2	1	no	—
43	C₄⋊Dic₅₃	—	2	1	no	—
44	C₄⋊Dic₅₃	D₄	1	2	yes	—
45	#11719919	—	4	2	no	—
46	C₄×Dic₅₃	—	2	1	no	—
47	C₄×Dic₅₃	—	2	1	no	—
48	C₂×#88322	—	2	1	no	—
49	C₂×#88322	—	2	1	no	—
50	C₂×#88322	—	2	1	no	—
51	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
52	Q₈×C₅₃	—	2	2	no	—
53	Q₈×C₅₃	—	2	2	no	—
54	Q₈×C₅₃	—	4	1	no	—
55	Q₈×C₅₃	—	4	1	no	—
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	—	4	1	no	—
58	C₂×C₄×C₅₃	—	4	1	no	—
59	C₂×C₄×C₅₃	—	4	1	no	—
60	C₂×C₄×C₅₃	—	4	1	no	—
61	C₂×C₄×C₅₃	—	4	1	no	—
62	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
66	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
67	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
68	C₂×Dic₅₃	—	2	1	no	—
69	C₂×Dic₅₃	—	2	1	no	—
70	C₂×Dic₅₃	—	2	2	no	—
71	C₂×Dic₅₃	—	2	1	no	—
72	C₂×Dic₅₃	—	2	1	no	—
73	C₂×Dic₅₃	—	2	1	no	—
74	C₂×Dic₅₃	—	4	1	no	—
75	C₂×Dic₅₃	—	4	1	no	—
76	C₂×Dic₅₃	—	4	1	no	—
77	C₂×Dic₅₃	—	4	1	no	—
78	C₂×Dic₅₃	—	4	1	no	—
79	C₂×Dic₅₃	Q₁₆	1	2	yes	—
80	C₈×C₅₃	—	2	2	no	—
81	#88322	—	4	1	no	—
82	C₂²×C₅₃	#121	1	1	yes	characteristic
83	C₂²×C₅₃	#121	1	1	yes	characteristic
84	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
85	C₂²×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
86	C₂²×C₅₃	#135	1	1	yes	characteristic
87	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
88	C₂²×C₅₃	#137	1	1	yes	characteristic
89	C₄×C₅₃	—	4	2	no	—
90	C₄×C₅₃	—	8	1	no	—
91	C₄×C₅₃	#121	1	1	yes	characteristic
92	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
93	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
94	C₄×C₅₃	#126	1	1	yes	characteristic
95	Dic₅₃	—	2	2	no	—
96	Dic₅₃	—	4	2	no	—
97	Dic₅₃	—	4	1	no	—
98	Dic₅₃	—	8	1	no	—
99	#2068	—	53	1	no	maximal, Sylow
100	C₂×C₅₃	#396	1	1	yes	characteristic
101	C₂×C₅₃	#462	1	1	yes	characteristic
102	C₂×C₅₃	#463	1	1	yes	characteristic
103	C₂×C₅₃	#467	1	1	yes	characteristic
104	C₂×C₅₃	#468	1	1	yes	characteristic
105	C₂×C₅₃	#477	1	1	yes	characteristic
106	C₂×C₅₃	#479	1	1	yes	characteristic
107	C₂×#107	—	53	1	no	—
108	C₂×C₄⋊C₈	—	53	1	no	—
109	C₂×#103	—	53	1	no	—
110	C₂×#103	D₅₃	1	1	yes	characteristic
111	#391	—	53	1	no	—
112	#389	—	53	1	no	—
113	#340	—	53	1	no	—
114	C₅₃	#2068	1	1	yes	characteristic, complemented, Sylow
115	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
116	C₂²×C₈	—	53	1	no	—
117	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
118	C₂×C₄⋊C₄	—	53	1	no	—
119	C₂×C₄⋊C₄	—	53	1	no	—
120	C₂×C₄⋊C₄	—	106	1	no	—
121	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
122	C₂×C₄²	—	53	1	no	—
123	#107	—	106	2	no	—
124	C₄⋊C₈	—	106	2	no	—
125	#103	—	2	2	no	—
126	#103	—	106	2	no	—
127	#95	—	106	1	no	—
128	#95	—	106	1	no	—
129	#95	—	106	1	no	—
130	C₂×Q₈	—	2	1	no	—
131	C₂×Q₈	—	2	1	no	—
132	C₂×Q₈	—	2	1	no	—
133	C₂×Q₈	—	2	1	no	—
134	C₂×Q₈	—	2	1	no	—
135	C₂×Q₈	Dic₁₀₆	1	2	yes	—
136	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	—	2	1	no	—
138	C₂²×C₄	—	53	1	no	—
139	C₂²×C₄	—	53	1	no	—
140	C₂²×C₄	—	106	1	no	—
141	C₂²×C₄	—	106	1	no	—
142	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	106	1	no	—
146	C₂×C₈	—	106	1	no	—
147	C₂×C₈	—	106	1	no	—
148	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
149	C₄⋊C₄	—	2	1	no	—
150	C₄⋊C₄	—	2	1	no	—
151	C₄⋊C₄	—	53	2	no	—
152	C₄⋊C₄	—	106	1	no	—
153	C₄⋊C₄	—	106	1	no	—
154	C₄⋊C₄	—	106	1	no	—
155	C₄⋊C₄	—	212	2	no	—
156	C₄²	—	106	1	no	—
157	C₄²	—	106	1	no	—
158	C₂³	D₄×D₅₃	1	1	yes	characteristic
159	Q₈	—	2	2	no	—
160	Q₈	—	2	2	no	—
161	Q₈	—	4	1	no	—
162	Q₈	—	4	1	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	4	1	no	—
165	C₂×C₄	—	4	1	no	—
166	C₂×C₄	—	4	1	no	—
167	C₂×C₄	—	4	1	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	53	2	no	—
170	C₂×C₄	—	106	1	no	—
171	C₂×C₄	—	106	1	no	—
172	C₂×C₄	—	106	1	no	—
173	C₂×C₄	—	106	2	no	—
174	C₂×C₄	—	106	1	no	—
175	C₂×C₄	—	106	1	no	—
176	C₂×C₄	—	212	1	no	—
177	C₂×C₄	—	212	1	no	—
178	C₂×C₄	—	212	1	no	—
179	C₂×C₄	—	212	1	no	—
180	C₂×C₄	—	212	1	no	—
181	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
182	C₂×C₄	#11719945	1	1	yes	characteristic
183	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
184	C₂×C₄	#11719947	1	1	yes	characteristic
185	C₂×C₄	#11719947	1	1	yes	characteristic
186	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
187	C₈	—	2	2	no	—
188	C₈	—	212	1	no	—
189	C₂²	#13133188	1	1	yes	characteristic
190	C₂²	#13133190	1	1	yes	characteristic
191	C₂²	#13133194	1	1	yes	characteristic
192	C₂²	Q₁₆×D₅₃	1	1	yes	characteristic
193	C₂²	#13133240	1	1	yes	characteristic
194	C₂²	#13133244	1	1	yes	characteristic
195	C₂²	#13133246	1	1	yes	characteristic
196	C₄	—	4	2	no	—
197	C₄	—	8	1	no	—
198	C₄	—	106	2	no	—
199	C₄	—	424	1	no	—
200	C₄	—	212	2	no	—
201	C₄	—	212	1	no	—
202	C₄	#13133189	1	1	yes	characteristic
203	C₄	#13133194	1	1	yes	characteristic
204	C₄	#13133196	1	1	yes	characteristic
205	C₄	#13133210	1	1	yes	characteristic
206	C₂	#16810546	1	1	yes	characteristic
207	C₂	#16810832	1	1	yes	characteristic
208	C₂	#16810841	1	1	yes	characteristic
209	C₂	#16810861	1	1	yes	characteristic
210	C₂	#16810866	1	1	yes	characteristic
211	C₂	#16810916	1	1	yes	characteristic
212	C₂	#16810932	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133151

C₂

yes

maximal, characteristic

C₂×#13133146

C₂

yes

maximal, characteristic

C₂×#13133141

C₂

yes

maximal, characteristic

#16810530

C₂

yes

maximal, characteristic

#16810519

C₂

yes

maximal, characteristic

#16810385

C₂

yes

maximal, characteristic

C₂×#103×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×#11719919

—

C₂×C₄×Dic₅₃

C₂²

yes

—

—

—

—

—

—

C₂²×Q₈×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

#103×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

D₄

yes

characteristic

C₂²×Dic₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

Q₁₆

yes

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₃

C₂×Q₁₆

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#121

yes

characteristic

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₄×C₅₃

#126

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#2068

—

maximal, Sylow

100

C₂×C₅₃

#396

yes

characteristic

101

C₂×C₅₃

#462

yes

characteristic

102

C₂×C₅₃

#463

yes

characteristic

103

C₂×C₅₃

#467

yes

characteristic

104

C₂×C₅₃

#468

yes

characteristic

105

C₂×C₅₃

#477

yes

characteristic

106

C₂×C₅₃

#479

yes

characteristic

107

C₂×#107

—

108

C₂×C₄⋊C₈

—

109

C₂×#103

—

110

C₂×#103

D₅₃

yes

characteristic

111

#391

—

112

#389

—

113

#340

—

114

C₅₃

#2068

yes

characteristic, complemented, Sylow

115

C₂²×Q₈

C₂×D₅₃

yes

characteristic

116

C₂²×C₈

—

117

C₂²×C₈

C₂×D₅₃

yes

characteristic

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

106

—

121

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

122

C₂×C₄²

—

123

#107

—

106

—

124

C₄⋊C₈

—

106

—

125

#103

—

126

#103

—

106

—

127

#95

—

106

—

128

#95

—

106

—

129

#95

—

106

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

Dic₁₀₆

yes

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

—

106

—

141

C₂²×C₄

—

106

—

142

C₂²×C₄

C₂²×D₅₃

yes

characteristic

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

106

—

146

C₂×C₈

—

106

—

147

C₂×C₈

—

106

—

148

C₂×C₈

C₂²⋊D₅₃

yes

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

106

—

153

C₄⋊C₄

—

106

—

154

C₄⋊C₄

—

106

—

155

C₄⋊C₄

—

212

—

156

C₄²

—

106

—

157

C₄²

—

106

—

158

C₂³

D₄×D₅₃

yes

characteristic

159

Q₈

—

160

Q₈

—

161

Q₈

—

162

Q₈

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

106

—

171

C₂×C₄

—

106

—

172

C₂×C₄

—

106

—

173

C₂×C₄

—

106

—

174

C₂×C₄

—

106

—

175

C₂×C₄

—

106

—

176

C₂×C₄

—

212

—

177

C₂×C₄

—

212

—

178

C₂×C₄

—

212

—

179

C₂×C₄

—

212

—

180

C₂×C₄

—

212

—

181

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

182

C₂×C₄

#11719945

yes

characteristic

183

C₂×C₄

D₄×D₅₃

yes

characteristic

184

C₂×C₄

#11719947

yes

characteristic

185

C₂×C₄

#11719947

yes

characteristic

186

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

187

C₈

—

188

C₈

—

212

—

189

C₂²

#13133188

yes

characteristic

190

C₂²

#13133190

yes

characteristic

191

C₂²

#13133194

yes

characteristic

192

C₂²

Q₁₆×D₅₃

yes

characteristic

193

C₂²

#13133240

yes

characteristic

194

C₂²

#13133244

yes

characteristic

195

C₂²

#13133246

yes

characteristic

196

C₄

—

197

C₄

—

198

C₄

—

106

—

199

C₄

—

424

—

200

C₄

—

212

—

201

C₄

—

212

—

202

C₄

#13133189

yes

characteristic

203

C₄

#13133194

yes

characteristic

204

C₄

#13133196

yes

characteristic

205

C₄

#13133210

yes

characteristic

206

C₂

#16810546

yes

characteristic

207

C₂

#16810832

yes

characteristic

208

C₂

#16810841

yes

characteristic

209

C₂

#16810861

yes

characteristic

210

C₂

#16810866

yes

characteristic

211

C₂

#16810916

yes

characteristic

212

C₂

#16810932

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₅₃
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#103×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×#103	D₅₃
hypocenter	C₅₃	#2068
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#103

subgroup

G/H

center

C₂³

D₄×D₅₃

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×#103×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#103

N	H ≅ G/N	type
114. C₅₃	#2068	semidirect

H ≅ G/N

type

114. C₅₃

#2068

semidirect

Subgroups — #20276108

Conjugacy classes of subgroups

Complements