Group #27284244 — subgroups

Subgroups — #27284244

Order 13568.

438 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 167 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	#20274867	C₂	1	1	yes	maximal, characteristic
2	#20274568	C₂	1	1	yes	maximal, characteristic
3	#20272598	C₂	1	4	yes	maximal
4	C₅₃×#2859	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88322	—	2	2	no	—
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	C₂×#13133128	—	2	1	no	—
9	#16810378	—	2	1	no	—
10	C₂×#125×C₅₃	—	2	1	no	—
11	C₅₃×#389	—	2	2	no	—
12	C₅₃×#389	C₄	1	4	yes	—
13	C₅₃×#384	C₂²	1	2	yes	—
14	C₅₃×#382	—	2	1	no	—
15	C₄³×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×#88322	—	2	2	no	—
17	C₂²×#88322	—	2	2	no	—
18	C₄⋊#88322	—	2	2	no	—
19	C₄⋊#88322	—	2	4	no	—
20	C₄⋊#88322	—	4	4	no	—
21	C₄⋊#88322	Q₈	1	4	yes	—
22	#13133128	—	4	2	no	—
23	#125×C₅₃	—	4	2	no	—
24	#125×C₅₃	—	4	2	no	—
25	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
26	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
27	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	D₄	1	2	yes	—
32	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	#95×C₅₃	—	2	4	no	—
35	C₂²×C₄×C₅₃	—	2	4	no	—
36	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₄⋊C₄×C₅₃	—	4	4	no	—
42	C₄⋊C₄×C₅₃	—	4	8	no	—
43	C₄⋊C₄×C₅₃	—	4	2	no	—
44	C₄⋊C₄×C₅₃	—	4	2	no	—
45	C₄⋊C₄×C₅₃	—	4	2	no	—
46	C₄⋊C₄×C₅₃	—	4	2	no	—
47	C₄²×C₅₃	—	2	2	no	—
48	C₄²×C₅₃	—	2	2	no	—
49	C₄²×C₅₃	—	2	2	no	—
50	C₄²×C₅₃	—	4	2	no	—
51	C₄²×C₅₃	D₈	1	2	yes	—
52	C₄²×C₅₃	SD₁₆	1	2	yes	—
53	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
54	C₄²×C₅₃	#41	1	2	yes	—
55	C₂×#88322	—	2	4	no	—
56	C₂×#88322	—	4	2	no	—
57	C₂×#88322	—	4	2	no	—
58	C₂×#88322	—	4	2	no	—
59	C₂×#88322	—	4	4	no	—
60	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	2	2	no	—
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	4	4	no	—
68	C₂×C₄×C₅₃	—	4	4	no	—
69	C₂×C₄×C₅₃	—	4	4	no	—
70	C₂×C₄×C₅₃	#102	1	4	yes	—
71	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
72	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
73	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
78	C₂×C₄×C₅₃	#137	1	2	yes	—
79	C₂×C₄×C₅₃	#137	1	2	yes	—
80	#88322	—	4	4	no	—
81	#88322	—	8	2	no	—
82	#9904	—	53	1	no	maximal, Sylow
83	C₂²×C₅₃	#391	1	1	yes	characteristic
84	C₂²×C₅₃	#413	1	2	yes	—
85	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
86	C₂²×C₅₃	#417	1	2	yes	—
87	C₂²×C₅₃	#419	1	1	yes	characteristic
88	C₄×C₅₃	—	2	4	no	—
89	C₄×C₅₃	—	2	2	no	—
90	C₄×C₅₃	—	4	2	no	—
91	C₄×C₅₃	—	8	4	no	—
92	C₄×C₅₃	#474	1	2	yes	—
93	C₄×C₅₃	#475	1	2	yes	—
94	C₄×C₅₃	#490	1	1	yes	characteristic
95	C₄×C₅₃	#491	1	1	yes	characteristic
96	C₄×C₅₃	#501	1	2	yes	—
97	#2859	D₅₃	1	1	yes	characteristic
98	#1842	—	53	1	no	—
99	#1767	—	53	1	no	—
100	#1296	—	53	4	no	—
101	C₂×C₅₃	#1547	1	2	yes	—
102	C₂×C₅₃	#1559	1	2	yes	—
103	C₂×C₅₃	#1988	1	1	yes	characteristic
104	C₂×C₅₃	#1992	1	1	yes	characteristic
105	C₂×C₅₃	#1998	1	1	yes	characteristic
106	C₂×#125	—	2	1	no	—
107	C₂×C₄⋊C₈	—	53	2	no	—
108	C₂×C₄⋊C₈	—	53	2	no	—
109	C₂×C₄⋊C₈	—	106	2	no	—
110	C₂×#97	—	106	1	no	—
111	#389	—	2	2	no	—
112	#389	Dic₅₃	1	4	yes	—
113	#384	C₂×D₅₃	1	2	yes	—
114	#382	—	2	1	no	—
115	C₄³	C₂×D₅₃	1	1	yes	characteristic
116	#336	—	106	1	no	—
117	C₅₃	#9904	1	1	yes	characteristic, complemented, Sylow
118	C₂²×C₈	—	106	2	no	—
119	C₂²×C₈	—	106	2	no	—
120	#125	—	4	2	no	—
121	#125	—	4	2	no	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	2	4	no	—
125	C₂×C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
126	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
127	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
128	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
129	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
130	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
131	C₄⋊C₈	—	53	4	no	—
132	C₄⋊C₈	—	106	2	no	—
133	C₄⋊C₈	—	106	4	no	—
134	C₄⋊C₈	—	212	4	no	—
135	#97	—	212	2	no	—
136	#95	—	2	4	no	—
137	C₂²×C₄	—	2	4	no	—
138	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
139	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
140	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
142	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
143	C₂×C₈	—	106	4	no	—
144	C₂×C₈	—	212	2	no	—
145	C₂×C₈	—	212	4	no	—
146	C₂×C₈	—	212	2	no	—
147	C₂×C₈	—	212	2	no	—
148	C₄⋊C₄	—	4	4	no	—
149	C₄⋊C₄	—	4	8	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄²	—	2	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	4	2	no	—
158	C₄²	#11719922	1	2	yes	—
159	C₄²	#11719923	1	2	yes	—
160	C₄²	Q₈×D₅₃	1	2	yes	—
161	C₄²	#11719949	1	2	yes	—
162	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	#13133138	1	4	yes	—
173	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
174	C₂×C₄	#13133193	1	2	yes	—
175	C₂×C₄	#13133206	1	1	yes	characteristic
176	C₂×C₄	#13133215	1	1	yes	characteristic
177	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
178	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
179	C₂×C₄	#13133242	1	2	yes	—
180	C₂×C₄	#13133247	1	2	yes	—
181	C₂×C₄	#13133248	1	2	yes	—
182	C₈	—	424	2	no	—
183	C₈	—	212	4	no	—
184	C₂²	#16810530	1	1	yes	characteristic
185	C₂²	#16810607	1	2	yes	—
186	C₂²	C₂×#13133138	1	1	yes	characteristic
187	C₂²	#16810624	1	2	yes	—
188	C₂²	#16810632	1	1	yes	characteristic
189	C₄	—	2	4	no	—
190	C₄	—	2	2	no	—
191	C₄	—	4	2	no	—
192	C₄	—	8	4	no	—
193	C₄	#16810900	1	2	yes	—
194	C₄	#16810905	1	2	yes	—
195	C₄	#16810989	1	1	yes	characteristic
196	C₄	#16810993	1	1	yes	characteristic
197	C₄	#16811033	1	2	yes	—
198	C₂	#20273392	1	2	yes	—
199	C₂	#20273445	1	2	yes	—
200	C₂	#20275668	1	1	yes	characteristic
201	C₂	#20275685	1	1	yes	characteristic
202	C₂	#20275709	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274867

C₂

yes

maximal, characteristic

#20274568

C₂

yes

maximal, characteristic

#20272598

C₂

yes

maximal

C₅₃×#2859

C₂

yes

maximal, characteristic

C₂×C₄⋊#88322

—

C₂×C₄⋊#88322

C₂²

yes

—

C₂×C₄⋊#88322

C₂²

yes

—

C₂×#13133128

—

#16810378

—

C₂×#125×C₅₃

—

C₅₃×#389

—

C₅₃×#389

C₄

yes

—

C₅₃×#384

C₂²

yes

—

C₅₃×#382

—

C₄³×C₅₃

C₂²

yes

characteristic

C₂²×#88322

—

C₂²×#88322

—

—

—

—

Q₈

yes

—

#13133128

—

#125×C₅₃

—

#125×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

D₈

yes

—

C₄²×C₅₃

SD₁₆

yes

—

C₄²×C₅₃

C₂×Q₈

yes

—

C₄²×C₅₃

#41

yes

—

—

—

—

—

—

C₂³×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#102

yes

—

C₂×C₄×C₅₃

C₄×Q₈

yes

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

#124

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₂×C₄×C₅₃

#137

yes

—

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

C₂×#102

yes

characteristic

C₂²×C₅₃

#417

yes

—

C₂²×C₅₃

#419

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#474

yes

—

C₄×C₅₃

#475

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

#2859

D₅₃

yes

characteristic

#1842

—

#1767

—

100

#1296

—

101

C₂×C₅₃

#1547

yes

—

102

C₂×C₅₃

#1559

yes

—

103

C₂×C₅₃

#1988

yes

characteristic

104

C₂×C₅₃

#1992

yes

characteristic

105

C₂×C₅₃

#1998

yes

characteristic

106

C₂×#125

—

107

C₂×C₄⋊C₈

—

108

C₂×C₄⋊C₈

—

109

C₂×C₄⋊C₈

—

106

—

110

C₂×#97

—

106

—

111

#389

—

112

#389

Dic₅₃

yes

—

113

#384

C₂×D₅₃

yes

—

114

#382

—

115

C₄³

C₂×D₅₃

yes

characteristic

116

#336

—

106

—

117

C₅₃

#9904

yes

characteristic, complemented, Sylow

118

C₂²×C₈

—

106

—

119

C₂²×C₈

—

106

—

120

#125

—

121

#125

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

C₂×Dic₅₃

yes

—

126

C₂×C₄²

C₂×Dic₅₃

yes

—

127

C₂×C₄²

C₂²⋊D₅₃

yes

characteristic

128

C₂×C₄²

C₂²⋊D₅₃

yes

—

129

C₂×C₄²

C₂²⋊D₅₃

yes

characteristic

130

C₂×C₄²

C₂²×D₅₃

yes

characteristic

131

C₄⋊C₈

—

132

C₄⋊C₈

—

106

—

133

C₄⋊C₈

—

106

—

134

C₄⋊C₈

—

212

—

135

#97

—

212

—

136

#95

—

137

C₂²×C₄

—

138

C₂²×C₄

C₂²⋊Dic₅₃

yes

—

139

C₂²×C₄

C₂²⋊Dic₅₃

yes

—

140

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

141

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

142

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

143

C₂×C₈

—

106

—

144

C₂×C₈

—

212

—

145

C₂×C₈

—

212

—

146

C₂×C₈

—

212

—

147

C₂×C₈

—

212

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄²

—

155

C₄²

—

156

C₄²

—

157

C₄²

—

158

C₄²

#11719922

yes

—

159

C₄²

#11719923

yes

—

160

C₄²

Q₈×D₅₃

yes

—

161

C₄²

#11719949

yes

—

162

C₂³

C₂×C₂²⋊Dic₅₃

yes

characteristic

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

#13133138

yes

—

173

C₂×C₄

Q₈×Dic₅₃

yes

—

174

C₂×C₄

#13133193

yes

—

175

C₂×C₄

#13133206

yes

characteristic

176

C₂×C₄

#13133215

yes

characteristic

177

C₂×C₄

C₂×#11719922

yes

characteristic

178

C₂×C₄

C₂×#11719923

yes

characteristic

179

C₂×C₄

#13133242

yes

—

180

C₂×C₄

#13133247

yes

—

181

C₂×C₄

#13133248

yes

—

182

C₈

—

424

—

183

C₈

—

212

—

184

C₂²

#16810530

yes

characteristic

185

C₂²

#16810607

yes

—

186

C₂²

C₂×#13133138

yes

characteristic

187

C₂²

#16810624

yes

—

188

C₂²

#16810632

yes

characteristic

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

#16810900

yes

—

194

C₄

#16810905

yes

—

195

C₄

#16810989

yes

characteristic

196

C₄

#16810993

yes

characteristic

197

C₄

#16811033

yes

—

198

C₂

#20273392

yes

—

199

C₂

#20273445

yes

—

200

C₂

#20275668

yes

characteristic

201

C₂

#20275685

yes

characteristic

202

C₂

#20275709

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2859	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2859	D₅₃
hypocenter	C₅₃	#9904
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₅₃

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#2859

C₂

socle, prime radical

C₂³×C₅₃

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

N	H ≅ G/N	type
117. C₅₃	#9904	semidirect

H ≅ G/N

type

117. C₅₃

#9904

semidirect

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

subgroup	index
#20274867	2
#20274568	2
#20272598	2
C₅₃×#2859	2
#9904	53

subgroup

index

C₅₃×#2859

Subgroups — #27284244

Conjugacy classes of subgroups

Complements

Maximal subgroups