Group #27288837 — subgroups

Subgroups — #27288837

Order 13568.

350 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 119 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	#20275651	C₂	1	1	yes	maximal, characteristic
2	#20274717	C₂	1	2	yes	maximal
3	#20272598	C₂	1	2	yes	maximal
4	#20272550	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1834	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
9	C₂×#13133140	—	2	2	no	—
10	#16810385	—	2	2	no	—
11	#16810377	—	2	2	no	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#384	C₂²	1	2	yes	—
14	C₅₃×#340	—	2	2	no	—
15	C₂²×Dic₁₀₆	D₄	1	1	yes	characteristic
16	C₂²×#88322	—	2	2	no	—
17	C₂²×#88322	—	2	1	no	—
18	C₄×Dic₁₀₆	—	2	2	no	—
19	C₄×Dic₁₀₆	—	2	2	no	—
20	C₂×C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
21	C₂×#11719919	—	2	1	no	—
22	C₂×C₄×Dic₅₃	—	2	1	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	2	2	no	—
25	C₄⋊#88322	—	2	2	no	—
26	#13133140	—	4	4	no	—
27	C₂²×C₈×C₅₃	D₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
31	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₅₃	—	2	2	no	—
33	C₂²×C₄×C₅₃	—	4	2	no	—
34	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
37	C₂²×Dic₅₃	—	2	1	no	—
38	C₂²×Dic₅₃	—	2	1	no	—
39	C₂×Dic₁₀₆	—	2	2	no	—
40	C₂×Dic₁₀₆	—	2	2	no	—
41	C₂×Dic₁₀₆	—	2	1	no	—
42	C₂×Dic₁₀₆	D₈	1	2	yes	—
43	C₂×C₈×C₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	—	2	2	no	—
45	C₂×C₈×C₅₃	—	2	2	no	—
46	C₄⋊C₄×C₅₃	—	4	2	no	—
47	C₄⋊C₄×C₅₃	—	4	2	no	—
48	C₄⋊C₄×C₅₃	—	4	2	no	—
49	C₄⋊C₄×C₅₃	—	4	2	no	—
50	C₄⋊C₄×C₅₃	—	8	2	no	—
51	C₄²×C₅₃	#41	1	2	yes	—
52	C₄²×C₅₃	#41	1	2	yes	—
53	C₄⋊Dic₅₃	—	2	2	no	—
54	#11719919	—	4	2	no	—
55	C₄×Dic₅₃	—	4	1	no	—
56	C₄×Dic₅₃	—	4	1	no	—
57	C₂×#88322	—	4	2	no	—
58	C₂×#88322	—	4	1	no	—
59	C₂×#88322	—	4	2	no	—
60	C₂×#88322	—	4	2	no	—
61	C₂×#88322	—	4	2	no	—
62	C₂³×C₅₃	#121	1	1	yes	characteristic
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	8	2	no	—
65	C₂×C₄×C₅₃	—	8	2	no	—
66	C₂×C₄×C₅₃	—	8	2	no	—
67	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
68	C₂×C₄×C₅₃	#123	1	2	yes	—
69	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
70	C₂×C₄×C₅₃	#126	1	2	yes	—
71	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
73	C₂×C₄×C₅₃	#135	1	2	yes	—
74	C₂×C₄×C₅₃	#137	1	2	yes	—
75	C₂×C₄×C₅₃	#137	1	2	yes	—
76	C₂×Dic₅₃	—	2	2	no	—
77	C₂×Dic₅₃	—	4	1	no	—
78	C₂×Dic₅₃	—	4	1	no	—
79	C₂×Dic₅₃	—	4	1	no	—
80	C₂×Dic₅₃	—	4	2	no	—
81	Dic₁₀₆	—	2	4	no	—
82	Dic₁₀₆	—	4	2	no	—
83	C₈×C₅₃	—	4	2	no	—
84	#88322	—	8	2	no	—
85	#88322	—	8	1	no	—
86	#10623	—	53	1	no	maximal, Sylow
87	C₂²×C₅₃	#397	1	1	yes	characteristic
88	C₂²×C₅₃	#464	1	2	yes	—
89	C₂²×C₅₃	#466	1	1	yes	characteristic
90	C₂²×C₅₃	#470	1	1	yes	characteristic
91	C₂²×C₅₃	#471	1	2	yes	—
92	C₄×C₅₃	—	2	2	no	—
93	C₄×C₅₃	—	16	2	no	—
94	C₄×C₅₃	#480	1	2	yes	—
95	C₄×C₅₃	#485	1	2	yes	—
96	C₄×C₅₃	#485	1	2	yes	—
97	C₄×C₅₃	#488	1	1	yes	characteristic
98	C₄×C₅₃	#491	1	1	yes	characteristic
99	Dic₅₃	—	4	2	no	—
100	Dic₅₃	—	8	1	no	—
101	#1984	—	53	1	no	—
102	#1834	D₅₃	1	1	yes	characteristic
103	#1803	—	53	2	no	—
104	#1296	—	53	2	no	—
105	#1277	—	53	1	no	—
106	C₂×C₅₃	#1682	1	2	yes	—
107	C₂×C₅₃	#1704	1	2	yes	—
108	C₂×C₅₃	#2074	1	1	yes	characteristic
109	C₂×C₅₃	#2075	1	1	yes	characteristic
110	C₂×C₅₃	#2079	1	1	yes	characteristic
111	C₂×C₄×Q₈	—	53	1	no	—
112	C₂×C₄⋊C₈	—	53	2	no	—
113	C₂×C₄⋊C₈	—	53	1	no	—
114	C₂×#103	—	106	2	no	—
115	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
116	#384	C₂×D₅₃	1	2	yes	—
117	#340	—	2	2	no	—
118	#340	—	106	2	no	—
119	#336	—	106	2	no	—
120	C₅₃	#10623	1	1	yes	characteristic, complemented, Sylow
121	C₂²×Q₈	—	53	1	no	—
122	C₂²×C₈	—	106	2	no	—
123	C₂²×C₈	—	106	1	no	—
124	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
125	C₄×Q₈	—	106	2	no	—
126	C₄×Q₈	—	106	2	no	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	53	1	no	—
131	C₂×C₄⋊C₄	—	106	1	no	—
132	C₂×C₄²	—	106	1	no	—
133	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
134	C₄⋊C₈	—	106	2	no	—
135	C₄⋊C₈	—	106	2	no	—
136	C₄⋊C₈	—	106	2	no	—
137	#103	—	212	4	no	—
138	C₄×C₈	—	2	2	no	—
139	C₂×Q₈	—	53	2	no	—
140	C₂×Q₈	—	106	1	no	—
141	C₂×Q₈	—	106	2	no	—
142	C₂×Q₈	—	106	2	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	—	106	1	no	—
145	C₂²×C₄	—	106	1	no	—
146	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
147	C₂²×C₄	#11719947	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	212	2	no	—
153	C₂×C₈	—	212	1	no	—
154	C₂×C₈	—	212	2	no	—
155	C₂×C₈	—	212	2	no	—
156	C₂×C₈	—	212	2	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	8	2	no	—
162	C₄⋊C₄	—	106	2	no	—
163	C₄⋊C₄	—	212	2	no	—
164	C₄²	—	212	1	no	—
165	C₄²	—	212	1	no	—
166	C₄²	#11719945	1	2	yes	—
167	C₄²	#11719949	1	2	yes	—
168	C₂³	#13133188	1	1	yes	characteristic
169	Q₈	—	106	4	no	—
170	Q₈	—	212	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	8	2	no	—
175	C₂×C₄	—	106	2	no	—
176	C₂×C₄	—	212	1	no	—
177	C₂×C₄	—	212	2	no	—
178	C₂×C₄	—	212	1	no	—
179	C₂×C₄	—	212	1	no	—
180	C₂×C₄	#13133200	1	1	yes	characteristic
181	C₂×C₄	#13133202	1	2	yes	—
182	C₂×C₄	#13133206	1	1	yes	characteristic
183	C₂×C₄	#13133212	1	2	yes	—
184	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
185	C₂×C₄	#13133236	1	1	yes	characteristic
186	C₂×C₄	#13133237	1	2	yes	—
187	C₂×C₄	#13133247	1	2	yes	—
188	C₂×C₄	#13133249	1	2	yes	—
189	C₈	—	4	2	no	—
190	C₈	—	424	2	no	—
191	C₈	—	424	1	no	—
192	C₂²	#16810549	1	1	yes	characteristic
193	C₂²	#16810842	1	2	yes	—
194	C₂²	#16810856	1	1	yes	characteristic
195	C₂²	#16810878	1	1	yes	characteristic
196	C₂²	#16810883	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	16	2	no	—
199	C₄	—	424	1	no	—
200	C₄	—	212	2	no	—
201	C₄	#16810938	1	2	yes	—
202	C₄	#16810964	1	2	yes	—
203	C₄	#16810966	1	2	yes	—
204	C₄	#16810980	1	1	yes	characteristic
205	C₄	#16810993	1	1	yes	characteristic
206	C₂	#20274156	1	2	yes	—
207	C₂	#20274294	1	2	yes	—
208	C₂	#20276134	1	1	yes	characteristic
209	C₂	#20276139	1	1	yes	characteristic
210	C₂	#20276161	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20275651

C₂

yes

maximal, characteristic

#20274717

C₂

yes

maximal

#20272598

C₂

yes

maximal

#20272550

C₂

yes

maximal, characteristic

C₅₃×#1834

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₀₆

C₂²

yes

characteristic

C₂×C₄⋊#88322

C₂²

yes

—

C₂×C₄⋊#88322

C₂²

yes

—

—

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#384

C₂²

yes

—

C₅₃×#340

—

C₂²×Dic₁₀₆

D₄

yes

characteristic

C₂²×#88322

—

C₂²×#88322

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

—

C₂×C₄⋊Dic₅₃

D₄

yes

characteristic

C₂×#11719919

—

C₂×C₄×Dic₅₃

—

—

—

—

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

D₈

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

#41

yes

—

C₄²×C₅₃

#41

yes

—

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

—

—

—

—

—

C₂³×C₅₃

#121

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#123

yes

characteristic

C₂×C₄×C₅₃

#123

yes

—

C₂×C₄×C₅₃

#124

yes

characteristic

C₂×C₄×C₅₃

#126

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#480

yes

—

C₄×C₅₃

#485

yes

—

C₄×C₅₃

#485

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

100

Dic₅₃

—

101

#1984

—

102

#1834

D₅₃

yes

characteristic

103

#1803

—

104

#1296

—

105

#1277

—

106

C₂×C₅₃

#1682

yes

—

107

C₂×C₅₃

#1704

yes

—

108

C₂×C₅₃

#2074

yes

characteristic

109

C₂×C₅₃

#2075

yes

characteristic

110

C₂×C₅₃

#2079

yes

characteristic

111

C₂×C₄×Q₈

—

112

C₂×C₄⋊C₈

—

113

C₂×C₄⋊C₈

—

114

C₂×#103

—

106

—

115

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

116

#384

C₂×D₅₃

yes

—

117

#340

—

118

#340

—

106

—

119

#336

—

106

—

120

C₅₃

#10623

yes

characteristic, complemented, Sylow

121

C₂²×Q₈

—

122

C₂²×C₈

—

106

—

123

C₂²×C₈

—

106

—

124

C₂²×C₈

C₂²⋊D₅₃

yes

—

125

C₄×Q₈

—

106

—

126

C₄×Q₈

—

106

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

131

C₂×C₄⋊C₄

—

106

—

132

C₂×C₄²

—

106

—

133

C₂×C₄²

C₂²×D₅₃

yes

characteristic

134

C₄⋊C₈

—

106

—

135

C₄⋊C₈

—

106

—

136

C₄⋊C₈

—

106

—

137

#103

—

212

—

138

C₄×C₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

106

—

141

C₂×Q₈

—

106

—

142

C₂×Q₈

—

106

—

143

C₂²×C₄

—

144

C₂²×C₄

—

106

—

145

C₂²×C₄

—

106

—

146

C₂²×C₄

D₄×D₅₃

yes

characteristic

147

C₂²×C₄

#11719947

yes

characteristic

148

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

212

—

153

C₂×C₈

—

212

—

154

C₂×C₈

—

212

—

155

C₂×C₈

—

212

—

156

C₂×C₈

—

212

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

106

—

163

C₄⋊C₄

—

212

—

164

C₄²

—

212

—

165

C₄²

—

212

—

166

C₄²

#11719945

yes

—

167

C₄²

#11719949

yes

—

168

C₂³

#13133188

yes

characteristic

169

Q₈

—

106

—

170

Q₈

—

212

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

106

—

176

C₂×C₄

—

212

—

177

C₂×C₄

—

212

—

178

C₂×C₄

—

212

—

179

C₂×C₄

—

212

—

180

C₂×C₄

#13133200

yes

characteristic

181

C₂×C₄

#13133202

yes

—

182

C₂×C₄

#13133206

yes

characteristic

183

C₂×C₄

#13133212

yes

—

184

C₂×C₄

D₈×D₅₃

yes

characteristic

185

C₂×C₄

#13133236

yes

characteristic

186

C₂×C₄

#13133237

yes

—

187

C₂×C₄

#13133247

yes

—

188

C₂×C₄

#13133249

yes

—

189

C₈

—

190

C₈

—

424

—

191

C₈

—

424

—

192

C₂²

#16810549

yes

characteristic

193

C₂²

#16810842

yes

—

194

C₂²

#16810856

yes

characteristic

195

C₂²

#16810878

yes

characteristic

196

C₂²

#16810883

yes

—

197

C₄

—

198

C₄

—

199

C₄

—

424

—

200

C₄

—

212

—

201

C₄

#16810938

yes

—

202

C₄

#16810964

yes

—

203

C₄

#16810966

yes

—

204

C₄

#16810980

yes

characteristic

205

C₄

#16810993

yes

characteristic

206

C₂

#20274156

yes

—

207

C₂

#20274294

yes

—

208

C₂

#20276134

yes

characteristic

209

C₂

#20276139

yes

characteristic

210

C₂

#20276161

yes

characteristic

	subgroup	G/H
center	C₂³	#13133188
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1834	C₂
socle, prime radical	C₂³×C₅₃	#121
hypercenter	#1834	D₅₃
hypocenter	C₅₃	#10623
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

subgroup

G/H

center

C₂³

#13133188

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1834

C₂

C₂³×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

N	H ≅ G/N	type
120. C₅₃	#10623	semidirect

H ≅ G/N

type

120. C₅₃

#10623

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20275651	2
#20274717	2
#20272598	2
#20272550	2
C₅₃×#1834	2
#10623	53

subgroup

index

C₅₃×#1834

Subgroups — #27288837

Conjugacy classes of subgroups

Complements

Maximal subgroups