Group #27286753 — subgroups

Subgroups — #27286753

Order 13568.

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

normal subgroups only

This group has 542 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	#20285840	C₂	1	2	yes	maximal
2	#20283093	C₂	1	1	yes	maximal, characteristic, complemented
3	#20273401	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	#16811019	—	2	2	no	—
8	#16810969	—	2	2	no	—
9	#16810939	—	2	4	no	—
10	#16810913	—	2	4	no	—
11	#16810676	C₂²	1	2	yes	—
12	#16810529	—	2	4	no	—
13	#16810489	—	2	2	no	—
14	#16810368	—	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	—	2	2	no	—
20	#13133217	—	2	2	no	—
21	#13133217	—	2	2	no	—
22	#13133217	—	2	2	no	—
23	#13133217	—	2	2	no	—
24	C₂×C₄⋊Dic₅₃	—	2	2	no	—
25	C₂×C₄⋊Dic₅₃	D₄	1	4	yes	—
26	#13133148	—	2	8	no	—
27	C₄⋊#88322	D₄	1	4	yes	—
28	#13133138	—	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	—	4	8	no	—
56	C₂×Dic₁₀₆	—	4	2	no	—
57	C₂×Dic₁₀₆	—	4	2	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	—
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	C₄⋊Dic₅₃	—	2	4	no	—
69	C₄⋊Dic₅₃	—	2	4	no	—
70	C₄⋊Dic₅₃	—	4	2	no	—
71	C₄⋊Dic₅₃	—	4	2	no	—
72	C₂×#88322	—	2	4	no	—
73	C₂³×C₅₃	—	4	1	no	—
74	C₂³×C₅₃	#120	1	1	yes	characteristic
75	Q₈×C₅₃	—	4	2	no	—
76	D₄×C₅₃	—	4	4	no	—
77	D₄×C₅₃	—	4	2	no	—
78	D₄×C₅₃	—	8	2	no	—
79	C₂×C₄×C₅₃	—	2	4	no	—
80	C₂×C₄×C₅₃	—	2	4	no	—
81	C₂×C₄×C₅₃	—	4	4	no	—
82	C₂×C₄×C₅₃	—	4	1	no	—
83	C₂×C₄×C₅₃	—	8	1	no	—
84	C₂×C₄×C₅₃	—	8	1	no	—
85	C₂×C₄×C₅₃	#120	1	2	yes	—
86	C₂×C₄×C₅₃	#121	1	2	yes	—
87	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
90	C₂×C₄×C₅₃	#137	1	2	yes	—
91	C₂×Dic₅₃	—	4	8	no	—
92	C₂×Dic₅₃	—	4	4	no	—
93	Dic₁₀₆	—	8	4	no	—
94	Dic₁₀₆	—	8	4	no	—
95	C₈×C₅₃	—	8	1	no	—
96	#88322	—	4	4	no	—
97	#10285	—	53	1	no	maximal, Sylow
98	C₂²×C₅₃	—	2	2	no	—
99	C₂²×C₅₃	—	8	2	no	—
100	C₂²×C₅₃	—	8	1	no	—
101	C₂²×C₅₃	#392	1	1	yes	characteristic
102	C₂²×C₅₃	#448	1	2	yes	—
103	C₄×C₅₃	—	2	2	no	—
104	C₄×C₅₃	—	4	4	no	—
105	C₄×C₅₃	—	8	1	no	—
106	C₄×C₅₃	#454	1	4	yes	—
107	C₄×C₅₃	#461	1	2	yes	—
108	C₄×C₅₃	#492	1	1	yes	characteristic
109	C₄×C₅₃	#497	1	1	yes	characteristic
110	Dic₅₃	—	8	4	no	—
111	#3227	—	53	2	no	—
112	#2862	—	53	1	no	—
113	#1962	D₅₃	1	1	yes	characteristic
114	#1549	—	53	2	no	—
115	#1276	—	53	1	no	—
116	C₂×C₅₃	—	2	2	no	—
117	C₂×C₅₃	—	16	1	no	—
118	C₂×C₅₃	#2003	1	2	yes	—
119	C₂×C₅₃	#2013	1	1	yes	characteristic
120	#532	C₂×D₅₃	1	1	yes	characteristic
121	C₂×C₄⋊Q₈	—	53	2	no	—
122	#498	—	106	2	no	—
123	#486	—	106	2	no	—
124	#481	—	106	4	no	—
125	#476	—	106	4	no	—
126	#431	—	53	2	no	—
127	#423	C₂×D₅₃	1	1	yes	characteristic
128	C₂×#104	—	2	2	no	—
129	#391	—	106	4	no	—
130	#382	—	106	2	no	—
131	C₄³	C₂×D₅₃	1	1	yes	characteristic
132	#332	—	106	4	no	—
133	C₅₃	#10285	1	1	yes	characteristic, complemented, Sylow
134	C₂×#41	—	2	1	no	—
135	C₂²×Q₈	—	106	2	no	—
136	C₂×M₁₆	—	2	1	no	—
137	C₄⋊Q₈	—	106	2	no	—
138	C₄⋊Q₈	—	106	2	no	—
139	C₄⋊Q₈	—	106	2	no	—
140	C₄⋊Q₈	—	106	2	no	—
141	C₄⋊Q₈	D₂₁₂	1	1	yes	characteristic
142	#127	D₂₁₂	1	1	yes	characteristic
143	#124	—	2	1	no	—
144	#121	—	2	2	no	—
145	C₄×D₄	—	2	2	no	—
146	C₂×C₄⋊C₄	—	53	4	no	—
147	C₂×C₄⋊C₄	—	106	2	no	—
148	C₂×C₄²	—	2	2	no	—
149	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
150	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
151	#106	—	106	8	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	53	4	no	—
154	#104	—	4	4	no	—
155	#102	—	106	8	no	—
156	C₂²⋊C₈	—	106	2	no	—
157	C₄×C₈	—	106	2	no	—
158	#95	—	106	8	no	—
159	#41	—	4	2	no	—
160	#41	—	4	2	no	—
161	C₂×Q₈	—	2	1	no	—
162	C₂×Q₈	—	212	8	no	—
163	C₂×Q₈	—	212	2	no	—
164	C₂×Q₈	—	212	2	no	—
165	C₂×D₄	—	2	2	no	—
166	C₂×D₄	—	2	1	no	—
167	C₂×D₄	—	4	1	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	4	1	no	—
170	C₂²×C₄	—	106	4	no	—
171	C₂²×C₄	D₄×D₅₃	1	2	yes	—
172	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
173	M₁₆	—	4	2	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	—	106	4	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	106	4	no	—
178	C₄⋊C₄	—	106	4	no	—
179	C₄⋊C₄	—	212	2	no	—
180	C₄⋊C₄	—	212	2	no	—
181	C₂²⋊C₄	—	4	2	no	—
182	C₄²	—	2	2	no	—
183	C₄²	—	4	4	no	—
184	C₄²	#11719923	1	4	yes	—
185	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
186	C₄²	#11719945	1	1	yes	characteristic
187	C₄²	D₄×D₅₃	1	2	yes	—
188	C₂³	—	4	1	no	—
189	C₂³	#13133185	1	1	yes	characteristic
190	Q₈	—	4	2	no	—
191	Q₈	—	424	4	no	—
192	Q₈	—	424	4	no	—
193	D₄	—	4	4	no	—
194	D₄	—	4	2	no	—
195	D₄	—	8	2	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	8	1	no	—
201	C₂×C₄	—	8	1	no	—
202	C₂×C₄	—	212	8	no	—
203	C₂×C₄	—	212	4	no	—
204	C₂×C₄	#13133187	1	2	yes	—
205	C₂×C₄	#13133190	1	2	yes	—
206	C₂×C₄	#13133191	1	1	yes	characteristic
207	C₂×C₄	#13133213	1	1	yes	characteristic
208	C₂×C₄	C₂×#11719923	1	2	yes	—
209	C₂×C₄	#13133247	1	2	yes	—
210	C₈	—	8	1	no	—
211	C₈	—	212	4	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	8	2	no	—
214	C₂²	—	8	1	no	—
215	C₂²	#16810532	1	1	yes	characteristic
216	C₂²	#16810761	1	2	yes	—
217	C₄	—	2	2	no	—
218	C₄	—	4	4	no	—
219	C₄	—	8	1	no	—
220	C₄	—	424	4	no	—
221	C₄	#16810797	1	4	yes	—
222	C₄	#16810828	1	2	yes	—
223	C₄	#16810995	1	1	yes	characteristic
224	C₄	#16811018	1	1	yes	characteristic
225	C₂	—	2	2	no	—
226	C₂	—	16	1	no	—
227	C₂	#20275736	1	2	yes	—
228	C₂	#20275797	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20285840

C₂

yes

maximal

#20283093

C₂

yes

maximal, characteristic, complemented

#20273401

C₂

yes

maximal, complemented

#20272548

C₂

yes

maximal, characteristic, complemented

C₅₃×#1962

C₂

yes

maximal, characteristic

C₂×#13133217

C₂²

yes

—

—

—

—

—

C₂²

yes

—

#16810529

—

#16810489

—

#16810368

—

C₅₃×#532

C₂²

yes

characteristic

C₅₃×#423

C₂²

yes

characteristic

C₂×#104×C₅₃

—

C₄³×C₅₃

C₂²

yes

characteristic

C₂²×Dic₁₀₆

—

—

—

—

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

D₄

yes

—

#13133148

—

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

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

#41

yes

characteristic

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

—

#10285

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

100

C₂²×C₅₃

—

101

C₂²×C₅₃

#392

yes

characteristic

102

C₂²×C₅₃

#448

yes

—

103

C₄×C₅₃

—

104

C₄×C₅₃

—

105

C₄×C₅₃

—

106

C₄×C₅₃

#454

yes

—

107

C₄×C₅₃

#461

yes

—

108

C₄×C₅₃

#492

yes

characteristic

109

C₄×C₅₃

#497

yes

characteristic

110

Dic₅₃

—

111

#3227

—

112

#2862

—

113

#1962

D₅₃

yes

characteristic

114

#1549

—

115

#1276

—

116

C₂×C₅₃

—

117

C₂×C₅₃

—

118

C₂×C₅₃

#2003

yes

—

119

C₂×C₅₃

#2013

yes

characteristic

120

#532

C₂×D₅₃

yes

characteristic

121

C₂×C₄⋊Q₈

—

122

#498

—

106

—

123

#486

—

106

—

124

#481

—

106

—

125

#476

—

106

—

126

#431

—

127

#423

C₂×D₅₃

yes

characteristic

128

C₂×#104

—

129

#391

—

106

—

130

#382

—

106

—

131

C₄³

C₂×D₅₃

yes

characteristic

132

#332

—

106

—

133

C₅₃

#10285

yes

characteristic, complemented, Sylow

134

C₂×#41

—

135

C₂²×Q₈

—

106

—

136

C₂×M₁₆

—

137

C₄⋊Q₈

—

106

—

138

C₄⋊Q₈

—

106

—

139

C₄⋊Q₈

—

106

—

140

C₄⋊Q₈

—

106

—

141

C₄⋊Q₈

D₂₁₂

yes

characteristic

142

#127

D₂₁₂

yes

characteristic

143

#124

—

144

#121

—

145

C₄×D₄

—

146

C₂×C₄⋊C₄

—

147

C₂×C₄⋊C₄

—

106

—

148

C₂×C₄²

—

149

C₂×C₄²

C₂²⋊D₅₃

yes

—

150

C₂×C₄²

C₂²×D₅₃

yes

characteristic

151

#106

—

106

—

152

C₄⋊C₈

—

153

C₄⋊C₈

—

154

#104

—

155

#102

—

106

—

156

C₂²⋊C₈

—

106

—

157

C₄×C₈

—

106

—

158

#95

—

106

—

159

#41

—

160

#41

—

161

C₂×Q₈

—

162

C₂×Q₈

—

212

—

163

C₂×Q₈

—

212

—

164

C₂×Q₈

—

212

—

165

C₂×D₄

—

166

C₂×D₄

—

167

C₂×D₄

—

168

C₂²×C₄

—

169

C₂²×C₄

—

170

C₂²×C₄

—

106

—

171

C₂²×C₄

D₄×D₅₃

yes

—

172

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

173

M₁₆

—

174

C₂×C₈

—

175

C₂×C₈

—

106

—

176

C₄⋊C₄

—

177

C₄⋊C₄

—

106

—

178

C₄⋊C₄

—

106

—

179

C₄⋊C₄

—

212

—

180

C₄⋊C₄

—

212

—

181

C₂²⋊C₄

—

182

C₄²

—

183

C₄²

—

184

C₄²

#11719923

yes

—

185

C₄²

C₂×D₂₁₂

yes

characteristic

186

C₄²

#11719945

yes

characteristic

187

C₄²

D₄×D₅₃

yes

—

188

C₂³

—

189

C₂³

#13133185

yes

characteristic

190

Q₈

—

191

Q₈

—

424

—

192

Q₈

—

424

—

193

D₄

—

194

D₄

—

195

D₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

212

—

203

C₂×C₄

—

212

—

204

C₂×C₄

#13133187

yes

—

205

C₂×C₄

#13133190

yes

—

206

C₂×C₄

#13133191

yes

characteristic

207

C₂×C₄

#13133213

yes

characteristic

208

C₂×C₄

C₂×#11719923

yes

—

209

C₂×C₄

#13133247

yes

—

210

C₈

—

211

C₈

—

212

—

212

C₂²

—

213

C₂²

—

214

C₂²

—

215

C₂²

#16810532

yes

characteristic

216

C₂²

#16810761

yes

—

217

C₄

—

218

C₄

—

219

C₄

—

220

C₄

—

424

—

221

C₄

#16810797

yes

—

222

C₄

#16810828

yes

—

223

C₄

#16810995

yes

characteristic

224

C₄

#16811018

yes

characteristic

225

C₂

—

226

C₂

—

227

C₂

#20275736

yes

—

228

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	#10285
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. #20273401	C₂	semidirect
4. #20272548	C₂	semidirect
133. C₅₃	#10285	semidirect

H ≅ G/N

type

2. #20283093

C₂

semidirect

3. #20273401

C₂

semidirect

4. #20272548

C₂

semidirect

133. C₅₃

#10285

semidirect

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

subgroup	index
#20285840	2
#20283093	2
#20273401	2
#20272548	2
C₅₃×#1962	2
#10285	53

subgroup

index

C₅₃×#1962

Subgroups — #27286753

Conjugacy classes of subgroups

Complements

Maximal subgroups