Group #27288543 — subgroups

Subgroups — #27288543

Order 13568.

406 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 129 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	#20275478	C₂	1	2	yes	maximal
2	#20274874	C₂	1	1	yes	maximal, characteristic
3	#20272598	C₂	1	1	yes	maximal, characteristic
4	#20272556	C₂	1	2	yes	maximal
5	C₅₃×#2571	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133150	—	2	2	no	—
7	C₂×#13133148	—	2	2	no	—
8	C₂×C₄⋊#88322	C₂²	1	2	yes	—
9	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
10	#16810493	C₂²	1	1	yes	characteristic
11	#16810385	—	2	2	no	—
12	C₅₃×#400	—	2	2	no	—
13	C₅₃×#389	—	2	2	no	—
14	C₅₃×#384	C₂²	1	1	yes	characteristic
15	C₅₃×#382	—	2	2	no	—
16	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
17	C₂²×#88322	—	2	1	no	—
18	C₂²×#88322	—	2	2	no	—
19	C₂×C₄⋊Dic₅₃	—	2	2	no	—
20	C₂×C₄⋊Dic₅₃	—	2	1	no	—
21	#13133150	—	4	4	no	—
22	#13133148	—	4	4	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	Q₈	1	4	yes	—
25	C₄×#88322	—	2	1	no	—
26	C₄×#88322	Q₈	1	2	yes	—
27	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
30	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
32	C₂×C₄²×C₅₃	—	2	2	no	—
33	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	#95×C₅₃	—	2	2	no	—
35	#95×C₅₃	—	2	4	no	—
36	#95×C₅₃	—	2	2	no	—
37	C₂²×C₄×C₅₃	—	2	2	no	—
38	C₂²×C₄×C₅₃	—	2	2	no	—
39	C₂²×C₄×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
42	C₂²×Dic₅₃	—	4	1	no	—
43	C₄⋊C₄×C₅₃	—	2	2	no	—
44	C₄⋊C₄×C₅₃	—	2	2	no	—
45	C₄⋊C₄×C₅₃	—	2	2	no	—
46	C₄⋊C₄×C₅₃	—	2	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	—	4	2	no	—
52	C₄²×C₅₃	—	4	2	no	—
53	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
54	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
55	C₄²×C₅₃	#41	1	1	yes	characteristic
56	C₄⋊Dic₅₃	—	4	2	no	—
57	C₄⋊Dic₅₃	—	4	2	no	—
58	C₄⋊Dic₅₃	—	4	2	no	—
59	C₂×#88322	—	2	2	no	—
60	C₂×#88322	—	2	2	no	—
61	C₂×#88322	—	2	2	no	—
62	C₂×#88322	—	2	4	no	—
63	C₂×#88322	—	4	2	no	—
64	C₂×#88322	—	4	2	no	—
65	C₂³×C₅₃	#120	1	1	yes	characteristic
66	C₂×C₄×C₅₃	—	2	1	no	—
67	C₂×C₄×C₅₃	—	2	1	no	—
68	C₂×C₄×C₅₃	—	4	2	no	—
69	C₂×C₄×C₅₃	—	4	4	no	—
70	C₂×C₄×C₅₃	—	4	2	no	—
71	C₂×C₄×C₅₃	—	4	2	no	—
72	C₂×C₄×C₅₃	—	4	2	no	—
73	C₂×C₄×C₅₃	—	4	2	no	—
74	C₂×C₄×C₅₃	—	4	2	no	—
75	C₂×C₄×C₅₃	—	4	2	no	—
76	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
78	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
80	C₂×C₄×C₅₃	#135	1	2	yes	—
81	C₂×C₄×C₅₃	#135	1	2	yes	—
82	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
83	C₂×C₄×C₅₃	#137	1	2	yes	—
84	C₂×Dic₅₃	—	8	2	no	—
85	C₂×Dic₅₃	—	8	1	no	—
86	#88322	—	2	4	no	—
87	#88322	—	4	4	no	—
88	#10576	—	53	1	no	maximal, Sylow
89	C₂²×C₅₃	#395	1	1	yes	characteristic
90	C₂²×C₅₃	#449	1	2	yes	—
91	C₂²×C₅₃	#452	1	2	yes	—
92	C₂²×C₅₃	#453	1	1	yes	characteristic
93	C₂²×C₅₃	#456	1	1	yes	characteristic
94	C₄×C₅₃	—	2	1	no	—
95	C₄×C₅₃	—	2	1	no	—
96	C₄×C₅₃	—	8	2	no	—
97	C₄×C₅₃	—	8	2	no	—
98	C₄×C₅₃	—	8	2	no	—
99	C₄×C₅₃	#478	1	2	yes	—
100	C₄×C₅₃	#479	1	2	yes	—
101	C₄×C₅₃	#499	1	2	yes	—
102	C₄×C₅₃	#501	1	2	yes	—
103	Dic₅₃	—	16	1	no	—
104	#2571	D₅₃	1	1	yes	characteristic
105	#1948	—	53	2	no	—
106	#1845	—	53	1	no	—
107	#1296	—	53	1	no	—
108	#1279	—	53	2	no	—
109	C₂×C₅₃	#1620	1	1	yes	characteristic
110	C₂×C₅₃	#1634	1	1	yes	characteristic
111	C₂×C₅₃	#1650	1	2	yes	—
112	C₂×C₅₃	#2059	1	2	yes	—
113	C₂×C₅₃	#2060	1	1	yes	characteristic
114	C₂×#107	—	106	2	no	—
115	C₂×#106	—	106	2	no	—
116	C₂×C₄⋊C₈	—	53	2	no	—
117	C₂×C₄×C₈	—	53	1	no	—
118	#400	—	2	2	no	—
119	#389	—	2	2	no	—
120	#384	—	53	1	no	—
121	#384	C₂×D₅₃	1	1	yes	characteristic
122	#382	—	2	2	no	—
123	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
124	#340	—	106	2	no	—
125	C₅₃	#10576	1	1	yes	characteristic, complemented, Sylow
126	C₂²×C₈	—	106	1	no	—
127	C₂²×C₈	—	106	2	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	106	2	no	—
131	C₂×C₄⋊C₄	—	106	1	no	—
132	C₂×C₄⋊C₄	C₂²⋊D₅₃	1	2	yes	—
133	C₂×C₄⋊C₄	C₂²⋊D₅₃	1	2	yes	—
134	C₂×C₄⋊C₄	C₂²⋊D₅₃	1	2	yes	—
135	C₂×C₄²	—	2	2	no	—
136	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
137	#107	—	212	4	no	—
138	#106	—	212	4	no	—
139	C₄⋊C₈	—	53	4	no	—
140	C₄⋊C₈	—	106	2	no	—
141	C₄×C₈	—	53	2	no	—
142	C₄×C₈	—	106	1	no	—
143	#95	—	2	2	no	—
144	#95	—	2	4	no	—
145	#95	—	2	2	no	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	212	1	no	—
150	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
152	C₂×C₈	—	106	2	no	—
153	C₂×C₈	—	106	2	no	—
154	C₂×C₈	—	106	2	no	—
155	C₂×C₈	—	106	4	no	—
156	C₂×C₈	—	212	2	no	—
157	C₂×C₈	—	212	2	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	8	2	no	—
166	C₄⋊C₄	—	212	2	no	—
167	C₄⋊C₄	—	212	2	no	—
168	C₄⋊C₄	—	212	2	no	—
169	C₄²	—	4	2	no	—
170	C₄²	—	4	2	no	—
171	C₄²	Q₈×D₅₃	1	1	yes	characteristic
172	C₄²	Q₈×D₅₃	1	2	yes	—
173	C₄²	#11719949	1	1	yes	characteristic
174	C₂³	#13133186	1	1	yes	characteristic
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	—	424	2	no	—
186	C₂×C₄	—	424	1	no	—
187	C₂×C₄	#13133193	1	1	yes	characteristic
188	C₂×C₄	#13133193	1	2	yes	—
189	C₂×C₄	#13133215	1	1	yes	characteristic
190	C₂×C₄	#13133215	1	2	yes	—
191	C₂×C₄	#13133237	1	2	yes	—
192	C₂×C₄	#13133237	1	2	yes	—
193	C₂×C₄	#13133242	1	2	yes	—
194	C₂×C₄	#13133247	1	2	yes	—
195	C₈	—	106	4	no	—
196	C₈	—	212	4	no	—
197	C₂²	#16810542	1	1	yes	characteristic
198	C₂²	#16810768	1	2	yes	—
199	C₂²	#16810785	1	2	yes	—
200	C₂²	#16810791	1	1	yes	characteristic
201	C₂²	#16810806	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	8	2	no	—
205	C₄	—	8	2	no	—
206	C₄	—	8	2	no	—
207	C₄	—	848	1	no	—
208	C₄	#16810920	1	2	yes	—
209	C₄	#16810927	1	2	yes	—
210	C₄	#16811022	1	2	yes	—
211	C₄	#16811033	1	2	yes	—
212	C₂	#20273774	1	1	yes	characteristic
213	C₂	#20273860	1	1	yes	characteristic
214	C₂	#20273971	1	2	yes	—
215	C₂	#20276050	1	2	yes	—
216	C₂	#20276057	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20275478

C₂

yes

maximal

#20274874

C₂

yes

maximal, characteristic

#20272598

C₂

yes

maximal, characteristic

#20272556

C₂

yes

maximal

C₅₃×#2571

C₂

yes

maximal, characteristic

C₂×#13133150

—

C₂×#13133148

—

C₂×C₄⋊#88322

C₂²

yes

—

C₂×C₄×#88322

C₂²

yes

characteristic

#16810493

C₂²

yes

characteristic

#16810385

—

C₅₃×#400

—

C₅₃×#389

—

C₅₃×#384

C₂²

yes

characteristic

C₅₃×#382

—

C₄×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂²×#88322

—

C₂²×#88322

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

—

—

—

Q₈

yes

—

C₄×#88322

—

C₄×#88322

Q₈

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#95×C₅₃

—

#95×C₅₃

—

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

—

C₄²×C₅₃

#41

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

—

—

—

—

—

—

C₂³×C₅₃

#120

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

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

—

C₂×C₄×C₅₃

#135

yes

—

C₂×C₄×C₅₃

#135

yes

—

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#452

yes

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#478

yes

—

100

C₄×C₅₃

#479

yes

—

101

C₄×C₅₃

#499

yes

—

102

C₄×C₅₃

#501

yes

—

103

Dic₅₃

—

104

#2571

D₅₃

yes

characteristic

105

#1948

—

106

#1845

—

107

#1296

—

108

#1279

—

109

C₂×C₅₃

#1620

yes

characteristic

110

C₂×C₅₃

#1634

yes

characteristic

111

C₂×C₅₃

#1650

yes

—

112

C₂×C₅₃

#2059

yes

—

113

C₂×C₅₃

#2060

yes

characteristic

114

C₂×#107

—

106

—

115

C₂×#106

—

106

—

116

C₂×C₄⋊C₈

—

117

C₂×C₄×C₈

—

118

#400

—

119

#389

—

120

#384

—

121

#384

C₂×D₅₃

yes

characteristic

122

#382

—

123

C₄×C₄⋊C₄

C₂×D₅₃

yes

—

124

#340

—

106

—

125

C₅₃

#10576

yes

characteristic, complemented, Sylow

126

C₂²×C₈

—

106

—

127

C₂²×C₈

—

106

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

106

—

131

C₂×C₄⋊C₄

—

106

—

132

C₂×C₄⋊C₄

C₂²⋊D₅₃

yes

—

133

C₂×C₄⋊C₄

C₂²⋊D₅₃

yes

—

134

C₂×C₄⋊C₄

C₂²⋊D₅₃

yes

—

135

C₂×C₄²

—

136

C₂×C₄²

C₂²×D₅₃

yes

characteristic

137

#107

—

212

—

138

#106

—

212

—

139

C₄⋊C₈

—

140

C₄⋊C₈

—

106

—

141

C₄×C₈

—

142

C₄×C₈

—

106

—

143

#95

—

144

#95

—

145

#95

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

—

212

—

150

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

151

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

—

152

C₂×C₈

—

106

—

153

C₂×C₈

—

106

—

154

C₂×C₈

—

106

—

155

C₂×C₈

—

106

—

156

C₂×C₈

—

212

—

157

C₂×C₈

—

212

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

212

—

167

C₄⋊C₄

—

212

—

168

C₄⋊C₄

—

212

—

169

C₄²

—

170

C₄²

—

171

C₄²

Q₈×D₅₃

yes

characteristic

172

C₄²

Q₈×D₅₃

yes

—

173

C₄²

#11719949

yes

characteristic

174

C₂³

#13133186

yes

characteristic

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

424

—

186

C₂×C₄

—

424

—

187

C₂×C₄

#13133193

yes

characteristic

188

C₂×C₄

#13133193

yes

—

189

C₂×C₄

#13133215

yes

characteristic

190

C₂×C₄

#13133215

yes

—

191

C₂×C₄

#13133237

yes

—

192

C₂×C₄

#13133237

yes

—

193

C₂×C₄

#13133242

yes

—

194

C₂×C₄

#13133247

yes

—

195

C₈

—

106

—

196

C₈

—

212

—

197

C₂²

#16810542

yes

characteristic

198

C₂²

#16810768

yes

—

199

C₂²

#16810785

yes

—

200

C₂²

#16810791

yes

characteristic

201

C₂²

#16810806

yes

characteristic

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

—

206

C₄

—

207

C₄

—

848

—

208

C₄

#16810920

yes

—

209

C₄

#16810927

yes

—

210

C₄

#16811022

yes

—

211

C₄

#16811033

yes

—

212

C₂

#20273774

yes

characteristic

213

C₂

#20273860

yes

characteristic

214

C₂

#20273971

yes

—

215

C₂

#20276050

yes

—

216

C₂

#20276057

yes

characteristic

	subgroup	G/H
center	C₂³	#13133186
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2571	C₂
socle, prime radical	C₂³×C₅₃	#120
hypercenter	#2571	D₅₃
hypocenter	C₅₃	#10576
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2571

subgroup

G/H

center

C₂³

#13133186

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#2571

C₂

C₂³×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2571

N	H ≅ G/N	type
125. C₅₃	#10576	semidirect

H ≅ G/N

type

125. C₅₃

#10576

semidirect

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

subgroup	index
#20275478	2
#20274874	2
#20272598	2
#20272556	2
C₅₃×#2571	2
#10576	53

subgroup

index

C₅₃×#2571

Subgroups — #27288543

Conjugacy classes of subgroups

Complements

Maximal subgroups