Group #27289164 — subgroups

Subgroups — #27289164

Order 13568.

494 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 147 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	#20275522	C₂	1	2	yes	maximal
2	#20274872	C₂	1	1	yes	maximal, characteristic, complemented
3	#20274842	C₂	1	2	yes	maximal, complemented
4	#20272542	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1957	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133149	—	2	2	no	—
7	C₂×#13133138	—	2	4	no	—
8	C₂×C₄×#88322	C₂²	1	2	yes	—
9	#16810493	C₂²	1	2	yes	—
10	#16810385	—	2	4	no	—
11	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×#102×C₅₃	—	2	2	no	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#384	C₂²	1	1	yes	characteristic
15	C₂²×#88322	D₄	1	4	yes	—
16	C₂×C₄⋊Dic₅₃	—	2	2	no	—
17	C₂×C₄⋊Dic₅₃	—	2	2	no	—
18	C₂×C₄⋊Dic₅₃	—	2	2	no	—
19	#13133149	—	4	4	no	—
20	#13133138	—	4	8	no	—
21	C₄×#88322	—	2	2	no	—
22	C₄×#88322	—	2	2	no	—
23	C₂²×D₄×C₅₃	—	2	2	no	—
24	C₂²×D₄×C₅₃	—	2	1	no	—
25	C₂²×C₈×C₅₃	D₄	1	2	yes	—
26	#127×C₅₃	—	2	1	no	—
27	#127×C₅₃	—	2	2	no	—
28	#127×C₅₃	Q₈	1	2	yes	—
29	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
31	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	#102×C₅₃	—	4	4	no	—
33	C₄×C₈×C₅₃	—	2	2	no	—
34	C₂⁴×C₅₃	—	4	1	no	—
35	C₂×D₄×C₅₃	—	2	4	no	—
36	C₂×D₄×C₅₃	—	2	2	no	—
37	C₂×D₄×C₅₃	—	4	1	no	—
38	C₂×D₄×C₅₃	—	4	2	no	—
39	C₂×D₄×C₅₃	—	4	2	no	—
40	C₂×D₄×C₅₃	—	4	4	no	—
41	C₂×D₄×C₅₃	—	4	2	no	—
42	C₂×D₄×C₅₃	—	4	4	no	—
43	C₂²×C₄×C₅₃	—	4	1	no	—
44	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
46	C₂²×Dic₅₃	—	4	2	no	—
47	C₂×C₈×C₅₃	—	2	2	no	—
48	C₂×C₈×C₅₃	—	2	2	no	—
49	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
50	C₄⋊C₄×C₅₃	—	4	2	no	—
51	C₄⋊C₄×C₅₃	—	4	4	no	—
52	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
53	C₄²×C₅₃	#41	1	1	yes	characteristic
54	C₄²×C₅₃	#41	1	2	yes	—
55	C₄⋊Dic₅₃	—	4	4	no	—
56	C₄⋊Dic₅₃	—	4	4	no	—
57	C₄⋊Dic₅₃	—	4	4	no	—
58	C₂×#88322	—	2	4	no	—
59	C₂×#88322	—	2	4	no	—
60	C₂×#88322	—	2	4	no	—
61	C₂³×C₅₃	—	4	2	no	—
62	C₂³×C₅₃	—	8	1	no	—
63	C₂³×C₅₃	—	8	2	no	—
64	C₂³×C₅₃	—	8	2	no	—
65	C₂³×C₅₃	—	8	1	no	—
66	C₂³×C₅₃	#127	1	1	yes	characteristic
67	D₄×C₅₃	—	4	4	no	—
68	D₄×C₅₃	—	4	8	no	—
69	D₄×C₅₃	—	8	2	no	—
70	D₄×C₅₃	—	8	4	no	—
71	C₂×C₄×C₅₃	—	8	1	no	—
72	C₂×C₄×C₅₃	—	8	2	no	—
73	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
74	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#123	1	2	yes	—
77	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
78	C₂×C₄×C₅₃	#135	1	4	yes	—
79	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
80	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
81	C₂×Dic₅₃	—	8	2	no	—
82	C₂×Dic₅₃	—	8	2	no	—
83	C₂×Dic₅₃	—	8	2	no	—
84	C₈×C₅₃	—	2	4	no	—
85	#88322	—	4	4	no	—
86	#10677	—	53	1	no	maximal, Sylow
87	C₂²×C₅₃	—	8	4	no	—
88	C₂²×C₅₃	—	8	2	no	—
89	C₂²×C₅₃	—	16	1	no	—
90	C₂²×C₅₃	—	16	1	no	—
91	C₂²×C₅₃	—	16	2	no	—
92	C₂²×C₅₃	#399	1	1	yes	characteristic
93	C₂²×C₅₃	#492	1	1	yes	characteristic
94	C₂²×C₅₃	#495	1	2	yes	—
95	C₂²×C₅₃	#496	1	2	yes	—
96	C₂²×C₅₃	#496	1	1	yes	characteristic
97	C₄×C₅₃	—	16	1	no	—
98	C₄×C₅₃	#476	1	2	yes	—
99	C₄×C₅₃	#478	1	4	yes	—
100	C₄×C₅₃	#481	1	2	yes	—
101	C₄×C₅₃	#482	1	4	yes	—
102	Dic₅₃	—	16	2	no	—
103	#1957	—	53	2	no	—
104	#1957	D₅₃	1	1	yes	characteristic
105	#1844	—	53	1	no	—
106	#1834	—	53	2	no	—
107	#1273	—	53	1	no	—
108	C₂×C₅₃	—	16	2	no	—
109	C₂×C₅₃	#1708	1	1	yes	characteristic
110	C₂×C₅₃	#1715	1	1	yes	characteristic
111	C₂×C₅₃	#1717	1	2	yes	—
112	C₂×C₅₃	#2092	1	2	yes	—
113	C₂×C₅₃	#2094	1	1	yes	characteristic
114	C₂×#127	C₂×D₅₃	1	1	yes	characteristic
115	C₂×#106	—	106	2	no	—
116	C₂×#102	—	2	2	no	—
117	C₂×#102	—	106	4	no	—
118	C₂×C₄×C₈	—	53	2	no	—
119	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
120	#384	—	53	2	no	—
121	#384	C₂×D₅₃	1	1	yes	characteristic
122	#340	—	106	4	no	—
123	C₅₃	#10677	1	1	yes	characteristic, complemented, Sylow
124	C₂²×D₄	—	2	2	no	—
125	C₂²×D₄	—	2	1	no	—
126	C₂²×C₈	—	53	4	no	—
127	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
128	#127	—	2	1	no	—
129	#127	—	2	2	no	—
130	#127	Dic₁₀₆	1	2	yes	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	—	2	1	no	—
133	C₂×C₄⋊C₄	—	106	2	no	—
134	C₂×C₄⋊C₄	—	106	2	no	—
135	C₂×C₄⋊C₄	—	106	2	no	—
136	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
137	#106	—	212	4	no	—
138	#102	—	4	4	no	—
139	#102	—	212	8	no	—
140	C₄×C₈	—	2	2	no	—
141	C₄×C₈	—	106	2	no	—
142	C₄×C₈	—	106	2	no	—
143	C₂⁴	—	4	1	no	—
144	C₂×D₄	—	2	4	no	—
145	C₂×D₄	—	2	2	no	—
146	C₂×D₄	—	4	1	no	—
147	C₂×D₄	—	4	2	no	—
148	C₂×D₄	—	4	2	no	—
149	C₂×D₄	—	4	4	no	—
150	C₂×D₄	—	4	4	no	—
151	C₂×D₄	—	4	2	no	—
152	C₂²×C₄	—	4	1	no	—
153	C₂²×C₄	—	212	2	no	—
154	C₂²×C₄	D₄×D₅₃	1	2	yes	—
155	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	106	4	no	—
159	C₂×C₈	—	106	4	no	—
160	C₂×C₈	—	106	4	no	—
161	C₂×C₈	#11719923	1	4	yes	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	4	no	—
164	C₄⋊C₄	—	212	4	no	—
165	C₄⋊C₄	—	212	4	no	—
166	C₄⋊C₄	—	212	4	no	—
167	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
168	C₄²	#11719945	1	1	yes	characteristic
169	C₄²	#11719947	1	2	yes	—
170	C₂³	—	4	2	no	—
171	C₂³	—	8	2	no	—
172	C₂³	—	8	1	no	—
173	C₂³	—	8	1	no	—
174	C₂³	—	8	2	no	—
175	C₂³	#13133213	1	1	yes	characteristic
176	D₄	—	4	4	no	—
177	D₄	—	4	8	no	—
178	D₄	—	8	2	no	—
179	D₄	—	8	4	no	—
180	C₂×C₄	—	8	2	no	—
181	C₂×C₄	—	8	1	no	—
182	C₂×C₄	—	424	2	no	—
183	C₂×C₄	—	424	2	no	—
184	C₂×C₄	—	424	2	no	—
185	C₂×C₄	#13133194	1	2	yes	—
186	C₂×C₄	#13133196	1	1	yes	characteristic
187	C₂×C₄	#13133198	1	1	yes	characteristic
188	C₂×C₄	#13133199	1	2	yes	—
189	C₂×C₄	C₂×#11719923	1	2	yes	—
190	C₂×C₄	#13133238	1	4	yes	—
191	C₂×C₄	#13133241	1	4	yes	—
192	C₂×C₄	#13133242	1	2	yes	—
193	C₈	—	2	4	no	—
194	C₈	—	212	4	no	—
195	C₂²	—	8	4	no	—
196	C₂²	—	8	2	no	—
197	C₂²	—	16	1	no	—
198	C₂²	—	16	1	no	—
199	C₂²	—	16	2	no	—
200	C₂²	#16810559	1	1	yes	characteristic
201	C₂²	#16810995	1	1	yes	characteristic
202	C₂²	#16811004	1	2	yes	—
203	C₂²	#16811011	1	2	yes	—
204	C₂²	#16811013	1	1	yes	characteristic
205	C₄	—	16	1	no	—
206	C₄	—	848	2	no	—
207	C₄	#16810913	1	2	yes	—
208	C₄	#16810925	1	4	yes	—
209	C₄	#16810939	1	2	yes	—
210	C₄	#16810946	1	4	yes	—
211	C₂	—	16	2	no	—
212	C₂	#20274314	1	1	yes	characteristic
213	C₂	#20274341	1	1	yes	characteristic
214	C₂	#20274356	1	2	yes	—
215	C₂	#20276238	1	2	yes	—
216	C₂	#20276248	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20275522

C₂

yes

maximal

#20274872

C₂

yes

maximal, characteristic, complemented

#20274842

C₂

yes

maximal, complemented

#20272542

C₂

yes

maximal, characteristic, complemented

C₅₃×#1957

C₂

yes

maximal, characteristic

C₂×#13133149

—

C₂×#13133138

—

C₂×C₄×#88322

C₂²

yes

—

#16810493

C₂²

yes

—

#16810385

—

C₂×#127×C₅₃

C₂²

yes

characteristic

C₂×#102×C₅₃

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#384

C₂²

yes

characteristic

C₂²×#88322

D₄

yes

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

—

—

—

—

C₂²×D₄×C₅₃

—

C₂²×D₄×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

#127×C₅₃

—

#127×C₅₃

—

#127×C₅₃

Q₈

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#102×C₅₃

—

C₄×C₈×C₅₃

—

C₂⁴×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×Q₈

yes

C₄²×C₅₃

yes

C₄²×C₅₃

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

#127

yes

characteristic

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₂×C₄×C₅₃

#135

yes

—

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

#88322

—

#10677

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#496

yes

—

C₂²×C₅₃

#496

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#476

yes

—

C₄×C₅₃

#478

yes

—

100

C₄×C₅₃

#481

yes

—

101

C₄×C₅₃

#482

yes

—

102

Dic₅₃

—

103

#1957

—

104

#1957

D₅₃

yes

characteristic

105

#1844

—

106

#1834

—

107

#1273

—

108

C₂×C₅₃

—

109

C₂×C₅₃

#1708

yes

characteristic

110

C₂×C₅₃

#1715

yes

characteristic

111

C₂×C₅₃

#1717

yes

—

112

C₂×C₅₃

#2092

yes

—

113

C₂×C₅₃

#2094

yes

characteristic

114

C₂×#127

C₂×D₅₃

yes

characteristic

115

C₂×#106

—

106

—

116

C₂×#102

—

117

C₂×#102

—

106

—

118

C₂×C₄×C₈

—

119

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

120

#384

—

121

#384

C₂×D₅₃

yes

characteristic

122

#340

—

106

—

123

C₅₃

#10677

yes

characteristic, complemented, Sylow

124

C₂²×D₄

—

125

C₂²×D₄

—

126

C₂²×C₈

—

127

C₂²×C₈

C₂²⋊D₅₃

yes

—

128

#127

—

129

#127

—

130

#127

Dic₁₀₆

yes

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

106

—

134

C₂×C₄⋊C₄

—

106

—

135

C₂×C₄⋊C₄

—

106

—

136

C₂×C₄²

C₂²×D₅₃

yes

characteristic

137

#106

—

212

—

138

#102

—

139

#102

—

212

—

140

C₄×C₈

—

141

C₄×C₈

—

106

—

142

C₄×C₈

—

106

—

143

C₂⁴

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

—

150

C₂×D₄

—

151

C₂×D₄

—

152

C₂²×C₄

—

153

C₂²×C₄

—

212

—

154

C₂²×C₄

D₄×D₅₃

yes

—

155

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

106

—

159

C₂×C₈

—

106

—

160

C₂×C₈

—

106

—

161

C₂×C₈

#11719923

yes

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

212

—

165

C₄⋊C₄

—

212

—

166

C₄⋊C₄

—

212

—

167

C₄²

C₂×Dic₁₀₆

yes

characteristic

168

C₄²

#11719945

yes

characteristic

169

C₄²

#11719947

yes

—

170

C₂³

—

171

C₂³

—

172

C₂³

—

173

C₂³

—

174

C₂³

—

175

C₂³

#13133213

yes

characteristic

176

D₄

—

177

D₄

—

178

D₄

—

179

D₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

424

—

183

C₂×C₄

—

424

—

184

C₂×C₄

—

424

—

185

C₂×C₄

#13133194

yes

—

186

C₂×C₄

#13133196

yes

characteristic

187

C₂×C₄

#13133198

yes

characteristic

188

C₂×C₄

#13133199

yes

—

189

C₂×C₄

C₂×#11719923

yes

—

190

C₂×C₄

#13133238

yes

—

191

C₂×C₄

#13133241

yes

—

192

C₂×C₄

#13133242

yes

—

193

C₈

—

194

C₈

—

212

—

195

C₂²

—

196

C₂²

—

197

C₂²

—

198

C₂²

—

199

C₂²

—

200

C₂²

#16810559

yes

characteristic

201

C₂²

#16810995

yes

characteristic

202

C₂²

#16811004

yes

—

203

C₂²

#16811011

yes

—

204

C₂²

#16811013

yes

characteristic

205

C₄

—

206

C₄

—

848

—

207

C₄

#16810913

yes

—

208

C₄

#16810925

yes

—

209

C₄

#16810939

yes

—

210

C₄

#16810946

yes

—

211

C₂

—

212

C₂

#20274314

yes

characteristic

213

C₂

#20274341

yes

characteristic

214

C₂

#20274356

yes

—

215

C₂

#20276238

yes

—

216

C₂

#20276248

yes

characteristic

	subgroup	G/H
center	C₂³	#13133213
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1957	C₂
socle	C₂³×C₅₃	#127
prime radical	C₂×#127×C₅₃	C₂²
hypercenter	#1957	D₅₃
hypocenter	C₅₃	#10677
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1957

subgroup

G/H

center

C₂³

#13133213

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1957

C₂

socle

C₂³×C₅₃

#127

prime radical

C₂×#127×C₅₃

C₂²

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1957

N	H ≅ G/N	type
2. #20274872	C₂	semidirect
3. #20274842	C₂	semidirect
4. #20272542	C₂	semidirect
123. C₅₃	#10677	semidirect

H ≅ G/N

type

2. #20274872

C₂

semidirect

3. #20274842

C₂

semidirect

4. #20272542

C₂

semidirect

123. C₅₃

#10677

semidirect

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

subgroup	index
#20275522	2
#20274872	2
#20274842	2
#20272542	2
C₅₃×#1957	2
#10677	53

subgroup

index

C₅₃×#1957

Subgroups — #27289164

Conjugacy classes of subgroups

Complements

Maximal subgroups