Group #27278597 — subgroups

Subgroups — #27278597

Order 13568.

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

normal subgroups only

This group has 412 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	#20283782	C₂	1	1	yes	maximal, characteristic
2	#20275544	C₂	1	1	yes	maximal, characteristic
3	#20273504	C₂	1	2	yes	maximal, complemented
4	#20272547	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1543	C₂	1	1	yes	maximal, characteristic
6	#16810701	—	2	2	no	—
7	D₄×#88322	—	2	2	no	—
8	#16810676	C₂²	1	2	yes	—
9	#16810647	C₂²	1	1	yes	characteristic
10	C₂×#13133144	—	2	2	no	—
11	#16810590	—	2	2	no	—
12	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
13	#16810376	—	2	2	no	—
14	#16810374	—	2	2	no	—
15	C₅₃×#532	C₂²	1	1	yes	characteristic
16	C₅₃×#431	C₂²	1	2	yes	—
17	C₅₃×#333	C₄	1	2	yes	complemented
18	C₅₃×#331	C₄	1	1	yes	characteristic, complemented
19	C₅₃×#331	C₄	1	1	yes	characteristic, complemented
20	C₂×#11719917	—	2	2	no	—
21	C₂×#11719917	—	2	1	no	—
22	C₂²×#88322	—	2	2	no	—
23	C₂×C₄×Dic₅₃	—	2	1	no	—
24	C₂×C₄×Dic₅₃	—	2	2	no	—
25	C₄⋊#88322	—	2	2	no	—
26	C₄⋊#88322	D₄	1	2	yes	—
27	C₄⋊#88322	D₄	1	2	yes	—
28	#13133144	—	4	4	no	—
29	C₂²⋊#88322	—	2	4	no	—
30	C₂²⋊#88322	—	2	2	no	—
31	C₄×#88322	—	2	2	no	—
32	C₂×#41×C₅₃	D₄	1	2	yes	—
33	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	#127×C₅₃	C₂×C₄	1	1	yes	characteristic
35	#124×C₅₃	Q₈	1	2	yes	—
36	#121×C₅₃	—	2	2	no	—
37	C₄×D₄×C₅₃	—	2	2	no	—
38	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
40	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
41	C₂²⋊C₈×C₅₃	—	2	2	no	—
42	C₄×C₈×C₅₃	—	2	2	no	—
43	#41×C₅₃	—	4	2	no	—
44	#41×C₅₃	—	4	2	no	—
45	C₂×Q₈×C₅₃	C₄⋊C₄	1	2	yes	—
46	C₂×D₄×C₅₃	—	2	2	no	—
47	C₂×D₄×C₅₃	—	2	2	no	—
48	C₂×D₄×C₅₃	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₅₃	—	2	2	no	—
50	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	#41	1	2	yes	—
52	C₂²×Dic₅₃	—	4	1	no	—
53	C₂×C₈×C₅₃	—	2	2	no	—
54	C₂×C₈×C₅₃	—	2	2	no	—
55	C₄⋊C₄×C₅₃	—	2	2	no	—
56	C₂²⋊C₄×C₅₃	—	2	4	no	—
57	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₅₃	C₂×D₄	1	2	yes	—
59	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
60	C₄×Dic₅₃	—	2	2	no	—
61	C₄×Dic₅₃	—	2	4	no	—
62	C₄×Dic₅₃	—	4	2	no	—
63	C₄×Dic₅₃	—	4	1	no	—
64	#11719917	—	4	2	no	—
65	#11719917	—	8	2	no	—
66	C₂×#88322	—	2	2	no	—
67	C₂×#88322	—	2	2	no	—
68	C₂×#88322	—	4	1	no	—
69	C₂×#88322	—	8	2	no	—
70	C₂³×C₅₃	—	2	2	no	—
71	C₂³×C₅₃	#123	1	1	yes	characteristic
72	Q₈×C₅₃	—	4	2	no	—
73	D₄×C₅₃	—	4	2	no	—
74	D₄×C₅₃	—	4	2	no	—
75	D₄×C₅₃	—	4	2	no	—
76	D₄×C₅₃	—	8	2	no	—
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	2	1	no	—
80	C₂×C₄×C₅₃	—	2	1	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	4	2	no	—
83	C₂×C₄×C₅₃	—	4	2	no	—
84	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
86	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
88	C₂×Dic₅₃	—	4	2	no	—
89	C₂×Dic₅₃	—	4	4	no	—
90	C₈×C₅₃	—	4	2	no	—
91	C₈×C₅₃	—	4	2	no	—
92	#88322	—	4	2	no	—
93	#88322	—	4	2	no	—
94	#88322	—	8	1	no	—
95	#8934	—	53	1	no	maximal, Sylow
96	C₂²×C₅₃	—	2	1	no	—
97	C₂²×C₅₃	—	2	2	no	—
98	C₂²×C₅₃	—	4	2	no	—
99	C₂²×C₅₃	—	4	4	no	—
100	C₂²×C₅₃	#385	1	1	yes	characteristic
101	C₄×C₅₃	—	2	1	no	—
102	C₄×C₅₃	—	2	2	no	—
103	C₄×C₅₃	—	4	2	no	—
104	C₄×C₅₃	#443	1	2	yes	—
105	C₄×C₅₃	#444	1	2	yes	—
106	C₄×C₅₃	#453	1	1	yes	characteristic
107	C₄×C₅₃	#455	1	1	yes	characteristic
108	Dic₅₃	—	4	4	no	—
109	#2979	—	53	1	no	—
110	#1962	—	53	1	no	—
111	#1574	—	53	2	no	—
112	#1543	D₅₃	1	1	yes	characteristic
113	#1276	—	53	2	no	—
114	C₂×C₅₃	—	4	1	no	—
115	C₂×C₅₃	—	8	2	no	—
116	C₂×C₅₃	#1856	1	1	yes	characteristic
117	C₂×C₅₃	#1866	1	2	yes	—
118	#532	C₂×D₅₃	1	1	yes	characteristic
119	#436	—	106	2	no	—
120	C₈×D₄	—	106	2	no	—
121	#431	—	53	2	no	—
122	#431	C₂×D₅₃	1	2	yes	—
123	#423	—	53	1	no	—
124	C₂×#104	—	106	2	no	—
125	#408	—	106	2	no	—
126	C₄³	—	53	1	no	—
127	#335	—	106	2	no	—
128	#334	—	106	2	no	—
129	#333	Dic₅₃	1	2	yes	complemented
130	#331	Dic₅₃	1	1	yes	characteristic, complemented
131	#331	Dic₅₃	1	1	yes	characteristic, complemented
132	C₅₃	#8934	1	1	yes	characteristic, complemented, Sylow
133	C₂×#41	D₂₁₂	1	2	yes	—
134	C₂×M₁₆	—	106	2	no	—
135	C₂×M₁₆	—	106	1	no	—
136	C₂²×C₈	—	106	2	no	—
137	C₄⋊Q₈	C₂×Dic₅₃	1	1	yes	characteristic
138	#127	C₂×Dic₅₃	1	1	yes	characteristic
139	#124	Dic₁₀₆	1	2	yes	—
140	#121	—	2	2	no	—
141	C₄×D₄	—	2	2	no	—
142	C₂×C₄²	—	106	1	no	—
143	C₂×C₄²	—	106	2	no	—
144	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
145	C₄⋊C₈	—	53	2	no	—
146	C₄⋊C₈	—	53	2	no	—
147	C₄⋊C₈	—	106	2	no	—
148	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
149	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
150	#104	—	212	4	no	—
151	C₂²⋊C₈	—	2	2	no	—
152	C₂²⋊C₈	—	106	4	no	—
153	C₂²⋊C₈	—	106	2	no	—
154	C₄×C₈	—	2	2	no	—
155	C₄×C₈	—	106	2	no	—
156	#41	—	4	2	no	—
157	#41	—	4	2	no	—
158	C₂×Q₈	C₄⋊Dic₅₃	1	2	yes	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	—	2	2	no	—
161	C₂×D₄	C₄⋊Dic₅₃	1	2	yes	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	—	212	1	no	—
164	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
165	C₂²×C₄	#11719947	1	2	yes	—
166	M₁₆	—	424	2	no	—
167	M₁₆	—	212	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	106	2	no	—
171	C₂×C₈	—	106	2	no	—
172	C₂×C₈	—	424	2	no	—
173	C₂×C₈	—	212	1	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₂²⋊C₄	—	2	4	no	—
176	C₄²	—	106	2	no	—
177	C₄²	—	106	4	no	—
178	C₄²	—	212	1	no	—
179	C₄²	—	212	2	no	—
180	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
181	C₄²	D₄×D₅₃	1	2	yes	—
182	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
183	C₂³	—	2	2	no	—
184	C₂³	#13133201	1	1	yes	characteristic
185	Q₈	—	4	2	no	—
186	D₄	—	4	2	no	—
187	D₄	—	4	2	no	—
188	D₄	—	4	2	no	—
189	D₄	—	8	2	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	212	4	no	—
198	C₂×C₄	—	212	2	no	—
199	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
200	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
201	C₂×C₄	#13133187	1	1	yes	characteristic
202	C₂×C₄	#13133194	1	2	yes	—
203	C₈	—	4	2	no	—
204	C₈	—	4	2	no	—
205	C₈	—	424	1	no	—
206	C₈	—	212	2	no	—
207	C₈	—	212	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	4	4	no	—
211	C₂²	—	4	2	no	—
212	C₂²	#16810498	1	1	yes	characteristic
213	C₄	—	2	2	no	—
214	C₄	—	2	1	no	—
215	C₄	—	4	2	no	—
216	C₄	—	212	4	no	—
217	C₄	#16810740	1	2	yes	—
218	C₄	#16810745	1	2	yes	—
219	C₄	#16810793	1	1	yes	characteristic
220	C₄	#16810803	1	1	yes	characteristic
221	C₂	—	4	1	no	—
222	C₂	—	8	2	no	—
223	C₂	#20274927	1	1	yes	characteristic
224	C₂	#20274992	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20283782

C₂

yes

maximal, characteristic

#20275544

C₂

yes

maximal, characteristic

#20273504

C₂

yes

maximal, complemented

#20272547

C₂

yes

maximal, complemented

C₅₃×#1543

C₂

yes

maximal, characteristic

#16810701

—

D₄×#88322

—

#16810676

C₂²

yes

—

#16810647

C₂²

yes

characteristic

C₂×#13133144

—

#16810590

—

C₄²×Dic₅₃

C₂²

yes

characteristic

#16810376

—

#16810374

—

C₅₃×#532

C₂²

yes

characteristic

C₅₃×#431

C₂²

yes

—

C₅₃×#333

C₄

yes

complemented

C₅₃×#331

C₄

yes

characteristic, complemented

C₅₃×#331

C₄

yes

characteristic, complemented

C₂×#11719917

—

C₂×#11719917

—

C₂²×#88322

—

C₂×C₄×Dic₅₃

—

C₂×C₄×Dic₅₃

—

C₄⋊#88322

—

C₄⋊#88322

D₄

yes

—

C₄⋊#88322

D₄

yes

—

—

—

—

—

C₂×#41×C₅₃

D₄

yes

—

C₄⋊Q₈×C₅₃

C₂×C₄

yes

characteristic

#127×C₅₃

C₂×C₄

yes

characteristic

#124×C₅₃

Q₈

yes

—

#121×C₅₃

—

C₄×D₄×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

C₂×C₄

yes

—

C₄⋊C₈×C₅₃

C₂×C₄

yes

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

C₄⋊C₄

yes

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

—

—

—

—

—

—

C₂³×C₅₃

—

C₂³×C₅₃

#123

yes

characteristic

Q₈×C₅₃

—

D₄×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

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

#120

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

100

C₂²×C₅₃

#385

yes

characteristic

101

C₄×C₅₃

—

102

C₄×C₅₃

—

103

C₄×C₅₃

—

104

C₄×C₅₃

#443

yes

—

105

C₄×C₅₃

#444

yes

—

106

C₄×C₅₃

#453

yes

characteristic

107

C₄×C₅₃

#455

yes

characteristic

108

Dic₅₃

—

109

#2979

—

110

#1962

—

111

#1574

—

112

#1543

D₅₃

yes

characteristic

113

#1276

—

114

C₂×C₅₃

—

115

C₂×C₅₃

—

116

C₂×C₅₃

#1856

yes

characteristic

117

C₂×C₅₃

#1866

yes

—

118

#532

C₂×D₅₃

yes

characteristic

119

#436

—

106

—

120

C₈×D₄

—

106

—

121

#431

—

122

#431

C₂×D₅₃

yes

—

123

#423

—

124

C₂×#104

—

106

—

125

#408

—

106

—

126

C₄³

—

127

#335

—

106

—

128

#334

—

106

—

129

#333

Dic₅₃

yes

complemented

130

#331

Dic₅₃

yes

characteristic, complemented

131

#331

Dic₅₃

yes

characteristic, complemented

132

C₅₃

#8934

yes

characteristic, complemented, Sylow

133

C₂×#41

D₂₁₂

yes

—

134

C₂×M₁₆

—

106

—

135

C₂×M₁₆

—

106

—

136

C₂²×C₈

—

106

—

137

C₄⋊Q₈

C₂×Dic₅₃

yes

characteristic

138

#127

C₂×Dic₅₃

yes

characteristic

139

#124

Dic₁₀₆

yes

—

140

#121

—

141

C₄×D₄

—

142

C₂×C₄²

—

106

—

143

C₂×C₄²

—

106

—

144

C₂×C₄²

C₂²×D₅₃

yes

characteristic

145

C₄⋊C₈

—

146

C₄⋊C₈

—

147

C₄⋊C₈

—

106

—

148

C₄⋊C₈

C₂×Dic₅₃

yes

—

149

C₄⋊C₈

C₂×Dic₅₃

yes

—

150

#104

—

212

—

151

C₂²⋊C₈

—

152

C₂²⋊C₈

—

106

—

153

C₂²⋊C₈

—

106

—

154

C₄×C₈

—

155

C₄×C₈

—

106

—

156

#41

—

157

#41

—

158

C₂×Q₈

C₄⋊Dic₅₃

yes

—

159

C₂×D₄

—

160

C₂×D₄

—

161

C₂×D₄

C₄⋊Dic₅₃

yes

—

162

C₂²×C₄

—

163

C₂²×C₄

—

212

—

164

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

165

C₂²×C₄

#11719947

yes

—

166

M₁₆

—

424

—

167

M₁₆

—

212

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

106

—

171

C₂×C₈

—

106

—

172

C₂×C₈

—

424

—

173

C₂×C₈

—

212

—

174

C₄⋊C₄

—

175

C₂²⋊C₄

—

176

C₄²

—

106

—

177

C₄²

—

106

—

178

C₄²

—

212

—

179

C₄²

—

212

—

180

C₄²

C₂×Dic₁₀₆

yes

characteristic

181

C₄²

D₄×D₅₃

yes

—

182

C₄²

C₂²×Dic₅₃

yes

characteristic

183

C₂³

—

184

C₂³

#13133201

yes

characteristic

185

Q₈

—

186

D₄

—

187

D₄

—

188

D₄

—

189

D₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

212

—

198

C₂×C₄

—

212

—

199

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

200

C₂×C₄

D₄×Dic₅₃

yes

—

201

C₂×C₄

#13133187

yes

characteristic

202

C₂×C₄

#13133194

yes

—

203

C₈

—

204

C₈

—

205

C₈

—

424

—

206

C₈

—

212

—

207

C₈

—

212

—

208

C₂²

—

209

C₂²

—

210

C₂²

—

211

C₂²

—

212

C₂²

#16810498

yes

characteristic

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

—

212

—

217

C₄

#16810740

yes

—

218

C₄

#16810745

yes

—

219

C₄

#16810793

yes

characteristic

220

C₄

#16810803

yes

characteristic

221

C₂

—

222

C₂

—

223

C₂

#20274927

yes

characteristic

224

C₂

#20274992

yes

—

	subgroup	G/H
center	C₂×C₄	#13133187
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1543	C₂
socle	C₂²×C₅₃	#385
prime radical	C₅₃×#532	C₂²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1543	D₅₃
hypocenter	C₅₃	#8934
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1543

subgroup

G/H

center

C₂×C₄

#13133187

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1543

C₂

socle

C₂²×C₅₃

#385

prime radical

C₅₃×#532

C₂²

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1543

N	H ≅ G/N	type
3. #20273504	C₂	semidirect
4. #20272547	C₂	semidirect
17. C₅₃×#333	C₄	semidirect
18. C₅₃×#331	C₄	semidirect
19. C₅₃×#331	C₄	semidirect
129. #333	Dic₅₃	semidirect
130. #331	Dic₅₃	semidirect
131. #331	Dic₅₃	semidirect
132. C₅₃	#8934	semidirect

H ≅ G/N

type

3. #20273504

C₂

semidirect

4. #20272547

C₂

semidirect

17. C₅₃×#333

C₄

semidirect

18. C₅₃×#331

C₄

semidirect

19. C₅₃×#331

C₄

semidirect

129. #333

Dic₅₃

semidirect

130. #331

Dic₅₃

semidirect

131. #331

Dic₅₃

semidirect

132. C₅₃

#8934

semidirect

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

subgroup	index
#20283782	2
#20275544	2
#20273504	2
#20272547	2
C₅₃×#1543	2
#8934	53

subgroup

index

C₅₃×#1543

Subgroups — #27278597

Conjugacy classes of subgroups

Complements

Maximal subgroups