Group #27277735 — subgroups

Subgroups — #27277735

Order 13568.

310 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 123 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	#20274403	C₂	1	1	yes	maximal, characteristic
2	#20273461	C₂	1	2	yes	maximal
3	#20272560	C₂	1	1	yes	maximal, characteristic
4	#20272543	C₂	1	2	yes	maximal
5	C₅₃×#1281	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	2	yes	—
7	C₂×#13133127	C₄	1	1	yes	characteristic
8	C₂×#13133127	C₄	1	1	yes	characteristic
9	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
12	#16810469	C₂²	1	1	yes	characteristic
13	#16810375	—	2	2	no	—
14	#16810373	—	2	2	no	—
15	C₅₃×#431	C₂²	1	2	yes	—
16	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
18	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
21	C₄⋊#88322	—	2	2	no	—
22	C₂²⋊#88322	—	2	2	no	—
23	#13133127	—	2	2	no	—
24	#13133127	—	2	2	no	—
25	C₄×#88322	—	2	2	no	—
26	C₄×#88322	D₄	1	1	yes	characteristic
27	C₄×#88322	D₄	1	1	yes	characteristic
28	C₄×#88322	Q₈	1	1	yes	characteristic
29	C₄×#88322	Q₈	1	1	yes	characteristic
30	C₈×Dic₅₃	—	2	2	no	—
31	C₈×Dic₅₃	C₈	1	4	yes	complemented
32	#13133124	—	2	2	no	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₅₃	—	2	2	no	—
37	C₂²⋊C₈×C₅₃	—	2	2	no	—
38	#97×C₅₃	—	2	2	no	—
39	C₄×C₈×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	#41	1	2	yes	—
42	C₂²×Dic₅₃	—	2	1	no	—
43	C₂²×Dic₅₃	—	2	1	no	—
44	C₂×C₈×C₅₃	—	2	1	no	—
45	C₂×C₈×C₅₃	—	2	1	no	—
46	C₂×C₈×C₅₃	—	2	1	no	—
47	C₂×C₈×C₅₃	—	2	1	no	—
48	C₂×C₈×C₅₃	—	4	2	no	—
49	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
50	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
51	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
53	C₄²×C₅₃	#41	1	2	yes	—
54	C₄×Dic₅₃	—	2	1	no	—
55	C₄×Dic₅₃	—	2	1	no	—
56	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
57	C₄×Dic₅₃	M₁₆	1	2	yes	—
58	C₂×#88322	—	2	1	no	—
59	C₂×#88322	—	2	1	no	—
60	C₂×#88322	—	4	2	no	—
61	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
62	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
63	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
64	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
65	C₂×#88322	C₂×C₈	1	2	yes	—
66	C₂×#88322	M₁₆	1	2	yes	—
67	C₂³×C₅₃	#117	1	1	yes	characteristic
68	C₂×C₄×C₅₃	—	4	2	no	—
69	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
71	C₂×C₄×C₅₃	#124	1	2	yes	—
72	C₂×C₄×C₅₃	#125	1	2	yes	—
73	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
77	C₂×Dic₅₃	—	2	2	no	—
78	C₂×Dic₅₃	—	2	2	no	—
79	C₂×Dic₅₃	—	4	2	no	—
80	C₂×Dic₅₃	—	4	2	no	—
81	C₈×C₅₃	—	2	2	no	—
82	C₈×C₅₃	—	2	2	no	—
83	C₈×C₅₃	—	8	2	no	—
84	#88322	—	2	1	no	—
85	#88322	—	2	1	no	—
86	#88322	—	8	2	no	—
87	#88322	C₄⋊C₈	1	2	yes	—
88	#88322	C₄⋊C₈	1	2	yes	—
89	#8780	—	53	1	no	maximal, Sylow
90	C₂²×C₅₃	—	2	2	no	—
91	C₂²×C₅₃	#382	1	1	yes	characteristic
92	C₂²×C₅₃	#431	1	1	yes	characteristic
93	C₂²×C₅₃	#432	1	1	yes	characteristic
94	C₄×C₅₃	—	2	2	no	—
95	C₄×C₅₃	—	2	2	no	—
96	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
97	C₄×C₅₃	#423	1	1	yes	characteristic
98	C₄×C₅₃	#431	1	1	yes	characteristic
99	C₄×C₅₃	#432	1	1	yes	characteristic
100	Dic₅₃	—	4	2	no	—
101	Dic₅₃	—	4	2	no	—
102	#1726	—	53	1	no	—
103	#1564	—	53	2	no	—
104	#1281	—	53	1	no	—
105	#1281	D₅₃	1	1	yes	characteristic
106	#1274	—	53	2	no	—
107	C₂×C₅₃	—	2	1	no	—
108	C₂×C₅₃	—	2	1	no	—
109	C₂×C₅₃	#1831	1	1	yes	characteristic
110	C₂×C₅₃	#1833	1	2	yes	—
111	#431	—	53	2	no	—
112	#431	C₂×D₅₃	1	2	yes	—
113	C₂×#97	—	53	1	no	—
114	C₂×#97	—	53	1	no	—
115	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
116	C₂×C₄×C₈	—	53	1	no	—
117	C₂×C₄×C₈	—	53	1	no	—
118	C₂×C₄×C₈	—	53	1	no	—
119	#376	—	53	1	no	—
120	#335	—	106	2	no	—
121	#334	—	106	2	no	—
122	C₅₃	#8780	1	1	yes	characteristic, complemented, Sylow
123	C₂²×C₈	—	53	1	no	—
124	C₂²×C₈	—	53	1	no	—
125	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
126	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
127	C₂×C₄²	—	53	1	no	—
128	C₂×C₄²	—	53	1	no	—
129	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
130	C₄⋊C₈	—	2	2	no	—
131	C₄⋊C₈	—	106	2	no	—
132	C₂²⋊C₈	—	2	2	no	—
133	C₂²⋊C₈	—	106	2	no	—
134	#97	—	2	2	no	—
135	#97	—	106	2	no	—
136	#97	—	106	2	no	—
137	C₄×C₈	—	2	2	no	—
138	C₄×C₈	—	53	4	no	—
139	C₄×C₈	—	53	1	no	—
140	C₄×C₈	—	53	1	no	—
141	C₄×C₈	—	53	1	no	—
142	C₄×C₈	—	53	1	no	—
143	C₄×C₈	—	106	2	no	—
144	C₄×C₈	—	106	2	no	—
145	#95	—	106	2	no	—
146	C₂²×C₄	—	106	1	no	—
147	C₂²×C₄	—	106	1	no	—
148	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
149	C₂²×C₄	#11719945	1	2	yes	—
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	—	53	1	no	—
156	C₂×C₈	—	53	1	no	—
157	C₂×C₈	—	53	2	no	—
158	C₂×C₈	—	53	2	no	—
159	C₂×C₈	—	53	1	no	—
160	C₂×C₈	—	53	1	no	—
161	C₂×C₈	—	106	1	no	—
162	C₂×C₈	—	106	1	no	—
163	C₂×C₈	—	212	2	no	—
164	C₂×C₈	C₈×D₅₃	1	2	yes	—
165	C₂×C₈	#11719912	1	2	yes	—
166	C₄²	—	53	2	no	—
167	C₄²	—	53	2	no	—
168	C₄²	—	106	1	no	—
169	C₄²	—	106	1	no	—
170	C₄²	D₄×D₅₃	1	1	yes	characteristic
171	C₄²	#11719947	1	2	yes	—
172	C₄²	Q₈×D₅₃	1	1	yes	characteristic
173	C₂³	#13133174	1	1	yes	characteristic
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	106	2	no	—
176	C₂×C₄	—	106	2	no	—
177	C₂×C₄	—	212	2	no	—
178	C₂×C₄	—	212	2	no	—
179	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
180	C₂×C₄	#13133173	1	1	yes	characteristic
181	C₂×C₄	#13133203	1	2	yes	—
182	C₂×C₄	#13133207	1	2	yes	—
183	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
184	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
185	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
186	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
187	C₈	—	2	2	no	—
188	C₈	—	2	2	no	—
189	C₈	—	8	2	no	—
190	C₈	—	53	2	no	—
191	C₈	—	53	2	no	—
192	C₈	—	106	1	no	—
193	C₈	—	106	1	no	—
194	C₈	—	424	2	no	—
195	C₂²	—	2	2	no	—
196	C₂²	#16810487	1	1	yes	characteristic
197	C₂²	#16810674	1	1	yes	characteristic
198	C₂²	#16810678	1	1	yes	characteristic
199	C₄	—	2	2	no	—
200	C₄	—	2	2	no	—
201	C₄	—	212	2	no	—
202	C₄	—	212	2	no	—
203	C₄	C₄⋊C₈×D₅₃	1	1	yes	characteristic
204	C₄	#16810645	1	1	yes	characteristic
205	C₄	#16810673	1	1	yes	characteristic
206	C₄	#16810677	1	1	yes	characteristic
207	C₂	—	2	1	no	—
208	C₂	—	2	1	no	—
209	C₂	#20274830	1	1	yes	characteristic
210	C₂	#20274836	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20274403

C₂

yes

maximal, characteristic

#20273461

C₂

yes

maximal

#20272560

C₂

yes

maximal, characteristic

#20272543

C₂

yes

maximal

C₅₃×#1281

C₂

yes

maximal, characteristic

#16810676

C₂²

yes

—

C₂×#13133127

C₄

yes

characteristic

C₂×#13133127

C₄

yes

characteristic

C₂×C₄×#88322

C₂²

yes

characteristic

C₂×C₈×Dic₅₃

C₄

yes

characteristic

C₂×C₈×Dic₅₃

C₄

yes

characteristic

#16810469

C₂²

yes

characteristic

#16810375

—

#16810373

—

C₅₃×#431

C₂²

yes

—

C₂×#97×C₅₃

C₂²

yes

characteristic

C₂²×#88322

C₂×C₄

yes

characteristic

C₂²×#88322

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

—

—

—

—

D₄

yes

characteristic

C₄×#88322

D₄

yes

characteristic

C₄×#88322

Q₈

yes

characteristic

C₄×#88322

Q₈

yes

characteristic

C₈×Dic₅₃

—

C₈×Dic₅₃

C₈

yes

complemented

#13133124

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

#97×C₅₃

—

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

M₁₆

yes

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₄²×C₅₃

#41

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×C₈

yes

—

C₄×Dic₅₃

M₁₆

yes

—

—

—

—

C₄⋊C₄

yes

characteristic

C₂×#88322

C₄⋊C₄

yes

characteristic

C₂×#88322

C₄⋊C₄

yes

characteristic

C₂×#88322

C₄⋊C₄

yes

characteristic

C₂×#88322

C₂×C₈

yes

—

C₂×#88322

M₁₆

yes

—

C₂³×C₅₃

#117

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

#125

yes

—

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

C₄⋊C₈

yes

—

#88322

C₄⋊C₈

yes

—

#8780

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₈

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

100

Dic₅₃

—

101

Dic₅₃

—

102

#1726

—

103

#1564

—

104

#1281

—

105

#1281

D₅₃

yes

characteristic

106

#1274

—

107

C₂×C₅₃

—

108

C₂×C₅₃

—

109

C₂×C₅₃

#1831

yes

characteristic

110

C₂×C₅₃

#1833

yes

—

111

#431

—

112

#431

C₂×D₅₃

yes

—

113

C₂×#97

—

114

C₂×#97

—

115

C₂×#97

C₂×D₅₃

yes

characteristic

116

C₂×C₄×C₈

—

117

C₂×C₄×C₈

—

118

C₂×C₄×C₈

—

119

#376

—

120

#335

—

106

—

121

#334

—

106

—

122

C₅₃

#8780

yes

characteristic, complemented, Sylow

123

C₂²×C₈

—

124

C₂²×C₈

—

125

C₂²×C₈

C₄×D₅₃

yes

characteristic

126

C₂²×C₈

C₄×D₅₃

yes

characteristic

127

C₂×C₄²

—

128

C₂×C₄²

—

129

C₂×C₄²

C₂²×D₅₃

yes

characteristic

130

C₄⋊C₈

—

131

C₄⋊C₈

—

106

—

132

C₂²⋊C₈

—

133

C₂²⋊C₈

—

106

—

134

#97

—

135

#97

—

106

—

136

#97

—

106

—

137

C₄×C₈

—

138

C₄×C₈

—

139

C₄×C₈

—

140

C₄×C₈

—

141

C₄×C₈

—

142

C₄×C₈

—

143

C₄×C₈

—

106

—

144

C₄×C₈

—

106

—

145

#95

—

106

—

146

C₂²×C₄

—

106

—

147

C₂²×C₄

—

106

—

148

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

149

C₂²×C₄

#11719945

yes

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

106

—

162

C₂×C₈

—

106

—

163

C₂×C₈

—

212

—

164

C₂×C₈

C₈×D₅₃

yes

—

165

C₂×C₈

#11719912

yes

—

166

C₄²

—

167

C₄²

—

168

C₄²

—

106

—

169

C₄²

—

106

—

170

C₄²

D₄×D₅₃

yes

characteristic

171

C₄²

#11719947

yes

—

172

C₄²

Q₈×D₅₃

yes

characteristic

173

C₂³

#13133174

yes

characteristic

174

C₂×C₄

—

175

C₂×C₄

—

106

—

176

C₂×C₄

—

106

—

177

C₂×C₄

—

212

—

178

C₂×C₄

—

212

—

179

C₂×C₄

C₄⋊C₄×D₅₃

yes

characteristic

180

C₂×C₄

#13133173

yes

characteristic

181

C₂×C₄

#13133203

yes

—

182

C₂×C₄

#13133207

yes

—

183

C₂×C₄

C₂×C₈×D₅₃

yes

characteristic

184

C₂×C₄

M₁₆×D₅₃

yes

characteristic

185

C₂×C₄

M₁₆×D₅₃

yes

characteristic

186

C₂×C₄

C₂×#11719912

yes

characteristic

187

C₈

—

188

C₈

—

189

C₈

—

190

C₈

—

191

C₈

—

192

C₈

—

106

—

193

C₈

—

106

—

194

C₈

—

424

—

195

C₂²

—

196

C₂²

#16810487

yes

characteristic

197

C₂²

#16810674

yes

characteristic

198

C₂²

#16810678

yes

characteristic

199

C₄

—

200

C₄

—

201

C₄

—

212

—

202

C₄

—

212

—

203

C₄

C₄⋊C₈×D₅₃

yes

characteristic

204

C₄

#16810645

yes

characteristic

205

C₄

#16810673

yes

characteristic

206

C₄

#16810677

yes

characteristic

207

C₂

—

208

C₂

—

209

C₂

#20274830

yes

characteristic

210

C₂

#20274836

yes

—

	subgroup	G/H
center	C₂×C₄	#13133173
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1281	C₂
socle	C₂²×C₅₃	#382
prime radical	C₂³×C₅₃	#117
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1281	D₅₃
hypocenter	C₅₃	#8780
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1281

subgroup

G/H

center

C₂×C₄

#13133173

commutator

C₂×C₄×C₅₃

C₂²×C₈

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1281

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1281

N	H ≅ G/N	type
31. C₈×Dic₅₃	C₈	semidirect
122. C₅₃	#8780	semidirect

H ≅ G/N

type

31. C₈×Dic₅₃

C₈

semidirect

122. C₅₃

#8780

semidirect

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

subgroup	index
#20274403	2
#20273461	2
#20272560	2
#20272543	2
C₅₃×#1281	2
#8780	53

subgroup

index

C₅₃×#1281

Subgroups — #27277735

Conjugacy classes of subgroups

Complements

Maximal subgroups