Group #27284236 — subgroups

Subgroups — #27284236

Order 13568.

438 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 167 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	#20274867	C₂	1	1	yes	maximal, characteristic
2	#20274568	C₂	1	1	yes	maximal, characteristic
3	#20272598	C₂	1	4	yes	maximal
4	C₅₃×#2859	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88322	—	2	2	no	—
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	C₂×#13133128	—	2	1	no	—
9	#16810378	—	2	1	no	—
10	C₂×#125×C₅₃	—	2	1	no	—
11	C₅₃×#389	—	2	2	no	—
12	C₅₃×#389	C₄	1	4	yes	—
13	C₅₃×#384	C₂²	1	2	yes	—
14	C₅₃×#382	—	2	1	no	—
15	C₄³×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×#88322	—	2	2	no	—
17	C₂²×#88322	—	2	2	no	—
18	C₄⋊#88322	—	2	2	no	—
19	C₄⋊#88322	—	2	4	no	—
20	C₄⋊#88322	—	4	4	no	—
21	C₄⋊#88322	Q₈	1	4	yes	—
22	#13133128	—	4	2	no	—
23	#125×C₅₃	—	4	4	no	—
24	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
25	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
26	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
27	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	D₄	1	2	yes	—
31	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	#95×C₅₃	—	2	4	no	—
34	C₂²×C₄×C₅₃	—	2	4	no	—
35	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
36	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₄⋊C₄×C₅₃	—	4	8	no	—
41	C₄⋊C₄×C₅₃	—	4	4	no	—
42	C₄⋊C₄×C₅₃	—	4	2	no	—
43	C₄⋊C₄×C₅₃	—	4	4	no	—
44	C₄⋊C₄×C₅₃	—	4	2	no	—
45	C₄²×C₅₃	—	2	2	no	—
46	C₄²×C₅₃	—	2	2	no	—
47	C₄²×C₅₃	—	2	2	no	—
48	C₄²×C₅₃	—	4	2	no	—
49	C₄²×C₅₃	SD₁₆	1	2	yes	—
50	C₄²×C₅₃	Q₁₆	1	2	yes	—
51	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
52	C₄²×C₅₃	#41	1	2	yes	—
53	C₂×#88322	—	2	4	no	—
54	C₂×#88322	—	4	2	no	—
55	C₂×#88322	—	4	2	no	—
56	C₂×#88322	—	4	2	no	—
57	C₂×#88322	—	4	4	no	—
58	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	2	2	no	—
65	C₂×C₄×C₅₃	—	4	4	no	—
66	C₂×C₄×C₅₃	—	4	4	no	—
67	C₂×C₄×C₅₃	—	4	4	no	—
68	C₂×C₄×C₅₃	#103	1	4	yes	—
69	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
70	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
71	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
76	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
77	C₂×C₄×C₅₃	#137	1	2	yes	—
78	#88322	—	4	4	no	—
79	#88322	—	8	2	no	—
80	#9902	—	53	1	no	maximal, Sylow
81	C₂²×C₅₃	#391	1	1	yes	characteristic
82	C₂²×C₅₃	#413	1	2	yes	—
83	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
84	C₂²×C₅₃	#417	1	2	yes	—
85	C₂²×C₅₃	#418	1	1	yes	characteristic
86	C₄×C₅₃	—	2	4	no	—
87	C₄×C₅₃	—	2	2	no	—
88	C₄×C₅₃	—	4	2	no	—
89	C₄×C₅₃	—	8	4	no	—
90	C₄×C₅₃	#476	1	2	yes	—
91	C₄×C₅₃	#477	1	2	yes	—
92	C₄×C₅₃	#490	1	1	yes	characteristic
93	C₄×C₅₃	#491	1	1	yes	characteristic
94	C₄×C₅₃	#501	1	2	yes	—
95	#2859	D₅₃	1	1	yes	characteristic
96	#1842	—	53	1	no	—
97	#1767	—	53	1	no	—
98	#1296	—	53	4	no	—
99	C₂×C₅₃	#1550	1	2	yes	—
100	C₂×C₅₃	#1557	1	2	yes	—
101	C₂×C₅₃	#1987	1	1	yes	characteristic
102	C₂×C₅₃	#1992	1	1	yes	characteristic
103	C₂×C₅₃	#1998	1	1	yes	characteristic
104	C₂×#125	—	2	1	no	—
105	C₂×C₄⋊C₈	—	53	2	no	—
106	C₂×C₄⋊C₈	—	53	2	no	—
107	C₂×C₄⋊C₈	—	106	2	no	—
108	C₂×#97	—	106	1	no	—
109	#389	—	2	2	no	—
110	#389	Dic₅₃	1	4	yes	—
111	#384	C₂×D₅₃	1	2	yes	—
112	#382	—	2	1	no	—
113	C₄³	C₂×D₅₃	1	1	yes	characteristic
114	#336	—	106	1	no	—
115	C₅₃	#9902	1	1	yes	characteristic, complemented, Sylow
116	C₂²×C₈	—	106	2	no	—
117	C₂²×C₈	—	106	2	no	—
118	#125	—	4	4	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	2	4	no	—
122	C₂×C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
123	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
124	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
125	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
126	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
127	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
128	C₄⋊C₈	—	53	4	no	—
129	C₄⋊C₈	—	106	2	no	—
130	C₄⋊C₈	—	106	4	no	—
131	C₄⋊C₈	—	212	4	no	—
132	#97	—	212	2	no	—
133	#95	—	2	4	no	—
134	C₂²×C₄	—	2	4	no	—
135	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
136	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
137	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
140	C₂×C₈	—	106	4	no	—
141	C₂×C₈	—	212	2	no	—
142	C₂×C₈	—	212	4	no	—
143	C₂×C₈	—	212	2	no	—
144	C₂×C₈	—	212	2	no	—
145	C₄⋊C₄	—	4	4	no	—
146	C₄⋊C₄	—	4	8	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	4	4	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	2	2	no	—
153	C₄²	—	4	2	no	—
154	C₄²	#11719924	1	2	yes	—
155	C₄²	#11719925	1	2	yes	—
156	C₄²	Q₈×D₅₃	1	2	yes	—
157	C₄²	#11719949	1	2	yes	—
158	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	#13133141	1	4	yes	—
169	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
170	C₂×C₄	#13133193	1	2	yes	—
171	C₂×C₄	#13133206	1	1	yes	characteristic
172	C₂×C₄	#13133215	1	1	yes	characteristic
173	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
174	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
175	C₂×C₄	#13133242	1	2	yes	—
176	C₂×C₄	#13133243	1	2	yes	—
177	C₂×C₄	#13133247	1	2	yes	—
178	C₈	—	424	2	no	—
179	C₈	—	212	4	no	—
180	C₂²	#16810530	1	1	yes	characteristic
181	C₂²	#16810607	1	2	yes	—
182	C₂²	C₂×#13133141	1	1	yes	characteristic
183	C₂²	#16810624	1	2	yes	—
184	C₂²	#16810628	1	1	yes	characteristic
185	C₄	—	2	4	no	—
186	C₄	—	2	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	8	4	no	—
189	C₄	#16810910	1	2	yes	—
190	C₄	#16810915	1	2	yes	—
191	C₄	#16810989	1	1	yes	characteristic
192	C₄	#16810993	1	1	yes	characteristic
193	C₄	#16811033	1	2	yes	—
194	C₂	#20273405	1	2	yes	—
195	C₂	#20273437	1	2	yes	—
196	C₂	#20275664	1	1	yes	characteristic
197	C₂	#20275685	1	1	yes	characteristic
198	C₂	#20275709	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274867

C₂

yes

maximal, characteristic

#20274568

C₂

yes

maximal, characteristic

#20272598

C₂

yes

maximal

C₅₃×#2859

C₂

yes

maximal, characteristic

C₂×C₄⋊#88322

—

C₂×C₄⋊#88322

C₂²

yes

—

C₂×C₄⋊#88322

C₂²

yes

—

C₂×#13133128

—

#16810378

—

C₂×#125×C₅₃

—

C₅₃×#389

—

C₅₃×#389

C₄

yes

—

C₅₃×#384

C₂²

yes

—

C₅₃×#382

—

C₄³×C₅₃

C₂²

yes

characteristic

C₂²×#88322

—

C₂²×#88322

—

—

—

—

Q₈

yes

—

#13133128

—

#125×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

SD₁₆

yes

—

C₄²×C₅₃

Q₁₆

yes

—

C₄²×C₅₃

C₂×Q₈

yes

—

C₄²×C₅₃

#41

yes

—

—

—

—

—

—

C₂³×C₅₃

C₂×C₂²⋊C₄

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

#103

yes

—

C₂×C₄×C₅₃

C₄×Q₈

yes

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

#124

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅₃

#137

yes

—

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

C₂×#103

yes

characteristic

C₂²×C₅₃

#417

yes

—

C₂²×C₅₃

#418

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#476

yes

—

C₄×C₅₃

#477

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

#2859

D₅₃

yes

—

—

—

C₂×C₅₃

#1550

yes

—

100

C₂×C₅₃

#1557

yes

—

101

C₂×C₅₃

#1987

yes

characteristic

102

C₂×C₅₃

#1992

yes

characteristic

103

C₂×C₅₃

#1998

yes

characteristic

104

C₂×#125

—

105

C₂×C₄⋊C₈

—

106

C₂×C₄⋊C₈

—

107

C₂×C₄⋊C₈

—

106

—

108

C₂×#97

—

106

—

109

#389

—

110

#389

Dic₅₃

yes

—

111

#384

C₂×D₅₃

yes

—

112

#382

—

113

C₄³

C₂×D₅₃

yes

characteristic

114

#336

—

106

—

115

C₅₃

#9902

yes

characteristic, complemented, Sylow

116

C₂²×C₈

—

106

—

117

C₂²×C₈

—

106

—

118

#125

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

C₂×Dic₅₃

yes

—

123

C₂×C₄²

C₂×Dic₅₃

yes

—

124

C₂×C₄²

C₂²⋊D₅₃

yes

characteristic

125

C₂×C₄²

C₂²⋊D₅₃

yes

—

126

C₂×C₄²

C₂²⋊D₅₃

yes

characteristic

127

C₂×C₄²

C₂²×D₅₃

yes

characteristic

128

C₄⋊C₈

—

129

C₄⋊C₈

—

106

—

130

C₄⋊C₈

—

106

—

131

C₄⋊C₈

—

212

—

132

#97

—

212

—

133

#95

—

134

C₂²×C₄

—

135

C₂²×C₄

C₂²⋊Dic₅₃

yes

—

136

C₂²×C₄

C₂²⋊Dic₅₃

yes

—

137

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

138

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

139

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

140

C₂×C₈

—

106

—

141

C₂×C₈

—

212

—

142

C₂×C₈

—

212

—

143

C₂×C₈

—

212

—

144

C₂×C₈

—

212

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄²

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

#11719924

yes

—

155

C₄²

#11719925

yes

—

156

C₄²

Q₈×D₅₃

yes

—

157

C₄²

#11719949

yes

—

158

C₂³

C₂×C₂²⋊Dic₅₃

yes

characteristic

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

#13133141

yes

—

169

C₂×C₄

Q₈×Dic₅₃

yes

—

170

C₂×C₄

#13133193

yes

—

171

C₂×C₄

#13133206

yes

characteristic

172

C₂×C₄

#13133215

yes

characteristic

173

C₂×C₄

C₂×#11719924

yes

characteristic

174

C₂×C₄

C₂×#11719925

yes

characteristic

175

C₂×C₄

#13133242

yes

—

176

C₂×C₄

#13133243

yes

—

177

C₂×C₄

#13133247

yes

—

178

C₈

—

424

—

179

C₈

—

212

—

180

C₂²

#16810530

yes

characteristic

181

C₂²

#16810607

yes

—

182

C₂²

C₂×#13133141

yes

characteristic

183

C₂²

#16810624

yes

—

184

C₂²

#16810628

yes

characteristic

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

#16810910

yes

—

190

C₄

#16810915

yes

—

191

C₄

#16810989

yes

characteristic

192

C₄

#16810993

yes

characteristic

193

C₄

#16811033

yes

—

194

C₂

#20273405

yes

—

195

C₂

#20273437

yes

—

196

C₂

#20275664

yes

characteristic

197

C₂

#20275685

yes

characteristic

198

C₂

#20275709

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2859	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2859	D₅₃
hypocenter	C₅₃	#9902
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₅₃

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#2859

C₂

socle, prime radical

C₂³×C₅₃

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

N	H ≅ G/N	type
115. C₅₃	#9902	semidirect

H ≅ G/N

type

115. C₅₃

#9902

semidirect

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

subgroup	index
#20274867	2
#20274568	2
#20272598	2
C₅₃×#2859	2
#9902	53

subgroup

index

C₅₃×#2859

Subgroups — #27284236

Conjugacy classes of subgroups

Complements

Maximal subgroups