Group #27286765 — subgroups

Subgroups — #27286765

Order 13568.

566 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 111 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	#20285637	C₂	1	2	yes	maximal
2	#20283093	C₂	1	1	yes	maximal, characteristic, complemented
3	#20273397	C₂	1	2	yes	maximal, complemented
4	#20272548	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1962	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133217	C₂²	1	2	yes	—
7	#16810995	—	2	2	no	—
8	#16810975	—	2	2	no	—
9	#16810828	—	2	4	no	—
10	#16810761	—	2	4	no	—
11	#16810676	C₂²	1	2	yes	—
12	#16810529	—	2	4	no	—
13	#16810489	—	2	2	no	—
14	#16810371	—	2	4	no	—
15	C₅₃×#532	C₂²	1	1	yes	characteristic
16	C₅₃×#423	C₂²	1	1	yes	characteristic
17	C₂×#104×C₅₃	—	2	2	no	—
18	C₄³×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²×Dic₁₀₆	D₄	1	4	yes	—
20	C₂×#11719923	—	2	8	no	—
21	#13133217	—	2	2	no	—
22	#13133217	—	2	2	no	—
23	#13133217	—	2	2	no	—
24	#13133217	—	2	2	no	—
25	C₂×C₄⋊Dic₅₃	—	2	2	no	—
26	C₂×C₄⋊Dic₅₃	—	2	2	no	—
27	C₄⋊#88322	D₄	1	4	yes	—
28	#13133140	—	2	8	no	—
29	C₂²⋊#88322	—	2	2	no	—
30	C₄×#88322	—	2	2	no	—
31	#13133124	—	2	8	no	—
32	C₂×#41×C₅₃	—	2	1	no	—
33	C₂×M₁₆×C₅₃	—	2	1	no	—
34	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
35	#127×C₅₃	D₄	1	1	yes	characteristic
36	#124×C₅₃	—	2	1	no	—
37	#121×C₅₃	—	2	2	no	—
38	C₄×D₄×C₅₃	—	2	2	no	—
39	C₂×C₄²×C₅₃	—	2	2	no	—
40	C₂×C₄²×C₅₃	D₄	1	2	yes	—
41	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₅₃	—	2	2	no	—
43	#104×C₅₃	—	4	4	no	—
44	#41×C₅₃	—	4	2	no	—
45	#41×C₅₃	—	4	2	no	—
46	C₂×Q₈×C₅₃	—	2	1	no	—
47	C₂×D₄×C₅₃	—	2	2	no	—
48	C₂×D₄×C₅₃	—	2	1	no	—
49	C₂×D₄×C₅₃	—	4	1	no	—
50	C₂²×C₄×C₅₃	—	2	2	no	—
51	C₂²×C₄×C₅₃	—	4	1	no	—
52	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×Dic₅₃	—	2	4	no	—
55	C₂×Dic₁₀₆	—	2	4	no	—
56	C₂×Dic₁₀₆	—	2	4	no	—
57	C₂×Dic₁₀₆	—	4	8	no	—
58	M₁₆×C₅₃	—	4	2	no	—
59	C₂×C₈×C₅₃	—	4	1	no	—
60	C₄⋊C₄×C₅₃	—	2	2	no	—
61	C₂²⋊C₄×C₅₃	—	4	2	no	—
62	C₄²×C₅₃	—	2	2	no	—
63	C₄²×C₅₃	—	4	4	no	—
64	C₄²×C₅₃	SD₁₆	1	4	yes	complemented
65	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₅₃	C₂×D₄	1	2	yes	—
67	C₄²×C₅₃	#41	1	1	yes	characteristic
68	#11719923	—	8	8	no	—
69	C₄⋊Dic₅₃	—	4	2	no	—
70	C₄⋊Dic₅₃	—	4	2	no	—
71	C₄⋊Dic₅₃	—	4	2	no	—
72	C₄⋊Dic₅₃	—	4	2	no	—
73	C₂×#88322	—	2	4	no	—
74	C₂³×C₅₃	—	4	1	no	—
75	C₂³×C₅₃	#120	1	1	yes	characteristic
76	Q₈×C₅₃	—	4	2	no	—
77	D₄×C₅₃	—	4	4	no	—
78	D₄×C₅₃	—	4	2	no	—
79	D₄×C₅₃	—	8	2	no	—
80	C₂×C₄×C₅₃	—	2	4	no	—
81	C₂×C₄×C₅₃	—	2	4	no	—
82	C₂×C₄×C₅₃	—	4	4	no	—
83	C₂×C₄×C₅₃	—	4	1	no	—
84	C₂×C₄×C₅₃	—	8	1	no	—
85	C₂×C₄×C₅₃	—	8	1	no	—
86	C₂×C₄×C₅₃	#120	1	2	yes	—
87	C₂×C₄×C₅₃	#121	1	2	yes	—
88	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
91	C₂×C₄×C₅₃	#137	1	2	yes	—
92	C₂×Dic₅₃	—	4	8	no	—
93	C₂×Dic₅₃	—	4	4	no	—
94	Dic₁₀₆	—	4	8	no	—
95	Dic₁₀₆	—	8	4	no	—
96	C₈×C₅₃	—	8	1	no	—
97	#88322	—	4	4	no	—
98	#10287	—	53	1	no	maximal, Sylow
99	C₂²×C₅₃	—	2	2	no	—
100	C₂²×C₅₃	—	8	2	no	—
101	C₂²×C₅₃	—	8	1	no	—
102	C₂²×C₅₃	#392	1	1	yes	characteristic
103	C₂²×C₅₃	#448	1	2	yes	—
104	C₄×C₅₃	—	2	2	no	—
105	C₄×C₅₃	—	4	4	no	—
106	C₄×C₅₃	—	8	1	no	—
107	C₄×C₅₃	#455	1	4	yes	—
108	C₄×C₅₃	#461	1	2	yes	—
109	C₄×C₅₃	#492	1	1	yes	characteristic
110	C₄×C₅₃	#497	1	1	yes	characteristic
111	Dic₅₃	—	8	4	no	—
112	#3206	—	53	2	no	—
113	#2862	—	53	1	no	—
114	#1962	D₅₃	1	1	yes	characteristic
115	#1548	—	53	2	no	—
116	#1276	—	53	1	no	—
117	C₂×C₅₃	—	2	2	no	—
118	C₂×C₅₃	—	16	1	no	—
119	C₂×C₅₃	#2004	1	2	yes	—
120	C₂×C₅₃	#2013	1	1	yes	characteristic
121	#532	C₂×D₅₃	1	1	yes	characteristic
122	C₂×C₄⋊Q₈	—	53	2	no	—
123	#492	—	106	2	no	—
124	#487	—	106	2	no	—
125	#461	—	106	4	no	—
126	#448	—	106	4	no	—
127	#431	—	53	2	no	—
128	#423	C₂×D₅₃	1	1	yes	characteristic
129	C₂×#104	—	2	2	no	—
130	#391	—	106	4	no	—
131	#382	—	106	2	no	—
132	C₄³	C₂×D₅₃	1	1	yes	characteristic
133	#333	—	106	4	no	—
134	C₅₃	#10287	1	1	yes	characteristic, complemented, Sylow
135	C₂×#41	—	2	1	no	—
136	C₂²×Q₈	—	53	4	no	—
137	C₂×SD₁₆	—	106	8	no	—
138	C₂×M₁₆	—	2	1	no	—
139	C₄⋊Q₈	—	106	2	no	—
140	C₄⋊Q₈	—	106	2	no	—
141	C₄⋊Q₈	—	106	2	no	—
142	C₄⋊Q₈	—	106	2	no	—
143	C₄⋊Q₈	D₂₁₂	1	1	yes	characteristic
144	#127	D₂₁₂	1	1	yes	characteristic
145	#124	—	2	1	no	—
146	#121	—	2	2	no	—
147	C₄×D₄	—	2	2	no	—
148	C₂×C₄⋊C₄	—	106	2	no	—
149	C₂×C₄⋊C₄	—	106	2	no	—
150	C₂×C₄²	—	2	2	no	—
151	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
152	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	53	4	no	—
155	#104	—	4	4	no	—
156	#103	—	106	8	no	—
157	C₂²⋊C₈	—	106	2	no	—
158	C₄×C₈	—	106	2	no	—
159	#95	—	106	8	no	—
160	#41	—	4	2	no	—
161	#41	—	4	2	no	—
162	C₂×Q₈	—	2	1	no	—
163	C₂×Q₈	—	106	4	no	—
164	C₂×Q₈	—	106	4	no	—
165	C₂×Q₈	—	212	8	no	—
166	C₂×D₄	—	2	2	no	—
167	C₂×D₄	—	2	1	no	—
168	C₂×D₄	—	4	1	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	4	1	no	—
171	C₂²×C₄	—	106	4	no	—
172	C₂²×C₄	D₄×D₅₃	1	2	yes	—
173	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
174	SD₁₆	—	424	8	no	—
175	M₁₆	—	4	2	no	—
176	C₂×C₈	—	4	1	no	—
177	C₂×C₈	—	106	4	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	212	2	no	—
180	C₄⋊C₄	—	212	2	no	—
181	C₄⋊C₄	—	212	2	no	—
182	C₄⋊C₄	—	212	2	no	—
183	C₂²⋊C₄	—	4	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	4	4	no	—
186	C₄²	#11719923	1	4	yes	complemented
187	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
188	C₄²	#11719945	1	1	yes	characteristic
189	C₄²	D₄×D₅₃	1	2	yes	—
190	C₂³	—	4	1	no	—
191	C₂³	#13133185	1	1	yes	characteristic
192	Q₈	—	4	2	no	—
193	Q₈	—	424	4	no	—
194	Q₈	—	212	8	no	—
195	D₄	—	4	4	no	—
196	D₄	—	4	2	no	—
197	D₄	—	8	2	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	8	1	no	—
203	C₂×C₄	—	8	1	no	—
204	C₂×C₄	—	212	8	no	—
205	C₂×C₄	—	212	4	no	—
206	C₂×C₄	#13133187	1	2	yes	—
207	C₂×C₄	#13133190	1	2	yes	—
208	C₂×C₄	#13133191	1	1	yes	characteristic
209	C₂×C₄	#13133213	1	1	yes	characteristic
210	C₂×C₄	C₂×#11719923	1	2	yes	—
211	C₂×C₄	#13133247	1	2	yes	—
212	C₈	—	8	1	no	—
213	C₈	—	212	4	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	8	2	no	—
216	C₂²	—	8	1	no	—
217	C₂²	#16810532	1	1	yes	characteristic
218	C₂²	#16810761	1	2	yes	—
219	C₄	—	2	2	no	—
220	C₄	—	4	4	no	—
221	C₄	—	8	1	no	—
222	C₄	—	424	4	no	—
223	C₄	#16810802	1	4	yes	—
224	C₄	#16810828	1	2	yes	—
225	C₄	#16810995	1	1	yes	characteristic
226	C₄	#16811018	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	—	16	1	no	—
229	C₂	#20275743	1	2	yes	—
230	C₂	#20275797	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20285637

C₂

yes

maximal

#20283093

C₂

yes

maximal, characteristic, complemented

#20273397

C₂

yes

maximal, complemented

#20272548

C₂

yes

maximal, characteristic, complemented

C₅₃×#1962

C₂

yes

maximal, characteristic

C₂×#13133217

C₂²

yes

—

—

—

—

—

C₂²

yes

—

#16810529

—

#16810489

—

#16810371

—

C₅₃×#532

C₂²

yes

characteristic

C₅₃×#423

C₂²

yes

characteristic

C₂×#104×C₅₃

—

C₄³×C₅₃

C₂²

yes

characteristic

C₂²×Dic₁₀₆

D₄

yes

—

—

—

—

—

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

C₄⋊#88322

D₄

yes

—

—

—

—

—

C₂×#41×C₅₃

—

C₂×M₁₆×C₅₃

—

C₄⋊Q₈×C₅₃

D₄

yes

characteristic

#127×C₅₃

D₄

yes

characteristic

#124×C₅₃

—

#121×C₅₃

—

C₄×D₄×C₅₃

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

#104×C₅₃

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

SD₁₆

yes

complemented

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

#41

yes

characteristic

#11719923

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

#120

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#120

yes

—

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

#10287

—

maximal, Sylow

C₂²×C₅₃

—

100

C₂²×C₅₃

—

101

C₂²×C₅₃

—

102

C₂²×C₅₃

#392

yes

characteristic

103

C₂²×C₅₃

#448

yes

—

104

C₄×C₅₃

—

105

C₄×C₅₃

—

106

C₄×C₅₃

—

107

C₄×C₅₃

#455

yes

—

108

C₄×C₅₃

#461

yes

—

109

C₄×C₅₃

#492

yes

characteristic

110

C₄×C₅₃

#497

yes

characteristic

111

Dic₅₃

—

112

#3206

—

113

#2862

—

114

#1962

D₅₃

yes

characteristic

115

#1548

—

116

#1276

—

117

C₂×C₅₃

—

118

C₂×C₅₃

—

119

C₂×C₅₃

#2004

yes

—

120

C₂×C₅₃

#2013

yes

characteristic

121

#532

C₂×D₅₃

yes

characteristic

122

C₂×C₄⋊Q₈

—

123

#492

—

106

—

124

#487

—

106

—

125

#461

—

106

—

126

#448

—

106

—

127

#431

—

128

#423

C₂×D₅₃

yes

characteristic

129

C₂×#104

—

130

#391

—

106

—

131

#382

—

106

—

132

C₄³

C₂×D₅₃

yes

characteristic

133

#333

—

106

—

134

C₅₃

#10287

yes

characteristic, complemented, Sylow

135

C₂×#41

—

136

C₂²×Q₈

—

137

C₂×SD₁₆

—

106

—

138

C₂×M₁₆

—

139

C₄⋊Q₈

—

106

—

140

C₄⋊Q₈

—

106

—

141

C₄⋊Q₈

—

106

—

142

C₄⋊Q₈

—

106

—

143

C₄⋊Q₈

D₂₁₂

yes

characteristic

144

#127

D₂₁₂

yes

characteristic

145

#124

—

146

#121

—

147

C₄×D₄

—

148

C₂×C₄⋊C₄

—

106

—

149

C₂×C₄⋊C₄

—

106

—

150

C₂×C₄²

—

151

C₂×C₄²

C₂²⋊D₅₃

yes

—

152

C₂×C₄²

C₂²×D₅₃

yes

characteristic

153

C₄⋊C₈

—

154

C₄⋊C₈

—

155

#104

—

156

#103

—

106

—

157

C₂²⋊C₈

—

106

—

158

C₄×C₈

—

106

—

159

#95

—

106

—

160

#41

—

161

#41

—

162

C₂×Q₈

—

163

C₂×Q₈

—

106

—

164

C₂×Q₈

—

106

—

165

C₂×Q₈

—

212

—

166

C₂×D₄

—

167

C₂×D₄

—

168

C₂×D₄

—

169

C₂²×C₄

—

170

C₂²×C₄

—

171

C₂²×C₄

—

106

—

172

C₂²×C₄

D₄×D₅₃

yes

—

173

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

174

SD₁₆

—

424

—

175

M₁₆

—

176

C₂×C₈

—

177

C₂×C₈

—

106

—

178

C₄⋊C₄

—

179

C₄⋊C₄

—

212

—

180

C₄⋊C₄

—

212

—

181

C₄⋊C₄

—

212

—

182

C₄⋊C₄

—

212

—

183

C₂²⋊C₄

—

184

C₄²

—

185

C₄²

—

186

C₄²

#11719923

yes

complemented

187

C₄²

C₂×D₂₁₂

yes

characteristic

188

C₄²

#11719945

yes

characteristic

189

C₄²

D₄×D₅₃

yes

—

190

C₂³

—

191

C₂³

#13133185

yes

characteristic

192

Q₈

—

193

Q₈

—

424

—

194

Q₈

—

212

—

195

D₄

—

196

D₄

—

197

D₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

212

—

205

C₂×C₄

—

212

—

206

C₂×C₄

#13133187

yes

—

207

C₂×C₄

#13133190

yes

—

208

C₂×C₄

#13133191

yes

characteristic

209

C₂×C₄

#13133213

yes

characteristic

210

C₂×C₄

C₂×#11719923

yes

—

211

C₂×C₄

#13133247

yes

—

212

C₈

—

213

C₈

—

212

—

214

C₂²

—

215

C₂²

—

216

C₂²

—

217

C₂²

#16810532

yes

characteristic

218

C₂²

#16810761

yes

—

219

C₄

—

220

C₄

—

221

C₄

—

222

C₄

—

424

—

223

C₄

#16810802

yes

—

224

C₄

#16810828

yes

—

225

C₄

#16810995

yes

characteristic

226

C₄

#16811018

yes

characteristic

227

C₂

—

228

C₂

—

229

C₂

#20275743

yes

—

230

C₂

#20275797

yes

characteristic

	subgroup	G/H
center	C₂²	#16810532
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1962	C₂
socle	C₂²×C₅₃	#392
prime radical	C₅₃×#532	C₂²
hypercenter	#1962	D₅₃
hypocenter	C₅₃	#10287
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1962

subgroup

G/H

center

C₂²

#16810532

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1962

C₂

socle

C₂²×C₅₃

#392

prime radical

C₅₃×#532

C₂²

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1962

N	H ≅ G/N	type
2. #20283093	C₂	semidirect
3. #20273397	C₂	semidirect
4. #20272548	C₂	semidirect
64. C₄²×C₅₃	SD₁₆	semidirect
134. C₅₃	#10287	semidirect
186. C₄²	#11719923	semidirect

H ≅ G/N

type

2. #20283093

C₂

semidirect

3. #20273397

C₂

semidirect

4. #20272548

C₂

semidirect

64. C₄²×C₅₃

SD₁₆

semidirect

134. C₅₃

#10287

semidirect

186. C₄²

#11719923

semidirect

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

subgroup	index
#20285637	2
#20283093	2
#20273397	2
#20272548	2
C₅₃×#1962	2
#10287	53

subgroup

index

C₅₃×#1962

Subgroups — #27286765

Conjugacy classes of subgroups

Complements

Maximal subgroups