Group #27300451 — subgroups

Subgroups — #27300451

Order 13568.

342 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 127 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	#20276470	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#20272797	C₂	1	1	yes	maximal, characteristic
4	#20272796	C₂	1	2	yes	maximal
5	C₅₃×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	2	yes	—
7	C₂×C₄×#88322	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
12	C₄²×Dic₅₃	C₄	1	1	yes	characteristic
13	#16810441	—	2	4	no	—
14	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
17	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₅₃	—	2	2	no	—
19	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
21	#13133157	—	2	2	no	—
22	#13133157	—	2	2	no	—
23	C₂×#11719908	—	2	2	no	—
24	C₄×#88322	—	2	2	no	—
25	C₈×Dic₅₃	—	2	2	no	—
26	C₈×Dic₅₃	—	2	2	no	—
27	C₈×Dic₅₃	C₈	1	2	yes	—
28	C₈×Dic₅₃	C₈	1	2	yes	—
29	C₈×Dic₅₃	D₄	1	2	yes	—
30	C₈×Dic₅₃	Q₈	1	2	yes	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	M₃₂×C₅₃	—	2	2	no	—
35	M₃₂×C₅₃	—	2	2	no	—
36	C₂×C₁₆×C₅₃	—	2	2	no	—
37	C₄×C₈×C₅₃	—	2	2	no	—
38	C₂²×C₄×C₅₃	—	2	1	no	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×Dic₅₃	—	2	1	no	—
41	C₂²×Dic₅₃	M₁₆	1	2	yes	—
42	C₂×C₈×C₅₃	—	2	1	no	—
43	C₂×C₈×C₅₃	—	2	1	no	—
44	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
45	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₅₃	#41	1	2	yes	—
48	C₂×C₈×C₅₃	#41	1	2	yes	—
49	C₄²×C₅₃	—	2	1	no	—
50	C₄²×C₅₃	M₁₆	1	2	yes	—
51	C₄×Dic₅₃	—	2	8	no	—
52	C₄×Dic₅₃	—	2	1	no	—
53	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
54	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
55	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
56	C₄×Dic₅₃	M₁₆	1	2	yes	—
57	C₂×#88322	—	2	1	no	—
58	C₂×#88322	—	2	1	no	—
59	C₂×#88322	—	2	1	no	—
60	C₂×#88322	—	2	1	no	—
61	C₂×#88322	C₄⋊C₄	1	2	yes	—
62	C₂×#88322	C₂×C₈	1	2	yes	—
63	C₁₆×C₅₃	—	4	2	no	—
64	#11719908	—	4	2	no	—
65	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
66	C₂×C₄×C₅₃	—	2	4	no	—
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
70	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
74	C₂×Dic₅₃	—	2	4	no	—
75	C₂×Dic₅₃	—	2	2	no	—
76	C₂×Dic₅₃	—	2	4	no	—
77	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
78	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
79	C₈×C₅₃	—	2	2	no	—
80	C₈×C₅₃	#124	1	2	yes	—
81	C₈×C₅₃	#125	1	2	yes	—
82	#88322	—	2	2	no	—
83	#88322	—	2	2	no	—
84	#12883	—	53	1	no	maximal, Sylow
85	C₂²×C₅₃	—	2	2	no	—
86	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
87	C₂²×C₅₃	#432	1	1	yes	characteristic
88	C₂²×C₅₃	#432	1	1	yes	characteristic
89	C₄×C₅₃	—	2	4	no	—
90	C₄×C₅₃	#382	1	1	yes	characteristic
91	C₄×C₅₃	#423	1	1	yes	characteristic
92	C₄×C₅₃	#431	1	1	yes	characteristic
93	C₄×C₅₃	#431	1	1	yes	characteristic
94	Dic₅₃	—	2	4	no	—
95	Dic₅₃	—	2	2	no	—
96	Dic₅₃	#364	1	4	yes	—
97	#2150	—	53	2	no	—
98	C₄²×C₈	—	53	1	no	—
99	#1376	—	53	2	no	—
100	#1376	—	53	1	no	—
101	#1376	D₅₃	1	1	yes	characteristic
102	C₂×C₅₃	—	2	1	no	—
103	C₂×C₅₃	—	2	1	no	—
104	C₂×C₅₃	#1831	1	1	yes	characteristic
105	C₂×C₅₃	C₂×#364	1	2	yes	—
106	C₂×M₃₂	—	53	2	no	—
107	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
108	C₂×C₄×C₈	—	53	1	no	—
109	C₂×C₄×C₈	—	53	1	no	—
110	C₂×C₄×C₈	—	53	1	no	—
111	C₂×C₄×C₈	—	53	1	no	—
112	C₂×C₄×C₈	—	53	1	no	—
113	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
114	C₄³	—	53	1	no	—
115	C₄⋊C₁₆	—	106	4	no	—
116	C₅₃	#12883	1	1	yes	characteristic, complemented, Sylow
117	C₂²×C₈	—	53	1	no	—
118	C₂²×C₈	—	53	1	no	—
119	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
120	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
121	C₂×C₄²	—	53	1	no	—
122	C₂×C₄²	—	53	1	no	—
123	C₂×C₄²	—	106	2	no	—
124	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
125	M₃₂	—	2	2	no	—
126	M₃₂	—	2	2	no	—
127	M₃₂	—	106	2	no	—
128	M₃₂	—	106	2	no	—
129	C₂×C₁₆	—	2	2	no	—
130	C₂×C₁₆	—	106	2	no	—
131	C₄×C₈	—	2	2	no	—
132	C₄×C₈	—	53	2	no	—
133	C₄×C₈	—	53	2	no	—
134	C₄×C₈	—	53	2	no	—
135	C₄×C₈	—	53	2	no	—
136	C₄×C₈	—	106	2	no	—
137	C₄×C₈	—	106	2	no	—
138	C₄×C₈	—	106	2	no	—
139	C₂²×C₄	—	2	1	no	—
140	C₂²×C₄	—	53	2	no	—
141	C₂²×C₄	—	106	1	no	—
142	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	53	2	no	—
146	C₂×C₈	—	53	2	no	—
147	C₂×C₈	—	106	1	no	—
148	C₂×C₈	—	106	1	no	—
149	C₂×C₈	—	106	1	no	—
150	C₂×C₈	—	106	1	no	—
151	C₂×C₈	C₈×D₅₃	1	2	yes	—
152	C₂×C₈	#11719945	1	2	yes	—
153	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
154	C₂×C₈	#11719947	1	2	yes	—
155	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
156	C₄²	—	2	1	no	—
157	C₄²	—	53	2	no	—
158	C₄²	—	53	1	no	—
159	C₄²	—	53	1	no	—
160	C₄²	—	53	2	no	—
161	C₄²	—	106	8	no	—
162	C₄²	—	106	1	no	—
163	C₄²	#11719912	1	2	yes	—
164	C₁₆	—	4	2	no	—
165	C₁₆	—	212	2	no	—
166	C₂³	M₁₆×D₅₃	1	1	yes	characteristic
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	53	2	no	—
170	C₂×C₄	—	53	2	no	—
171	C₂×C₄	—	106	4	no	—
172	C₂×C₄	—	106	4	no	—
173	C₂×C₄	—	106	2	no	—
174	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
175	C₂×C₄	#13133173	1	1	yes	characteristic
176	C₂×C₄	#13133174	1	1	yes	characteristic
177	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
178	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
179	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
180	C₈	—	2	2	no	—
181	C₈	—	106	2	no	—
182	C₈	—	106	2	no	—
183	C₈	#13133203	1	2	yes	—
184	C₈	#13133207	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	C₄⋊C₈×D₅₃	1	1	yes	characteristic
187	C₂²	#16810677	1	1	yes	characteristic
188	C₂²	#16810678	1	1	yes	characteristic
189	C₄	—	2	4	no	—
190	C₄	—	53	4	no	—
191	C₄	—	106	4	no	—
192	C₄	—	106	2	no	—
193	C₄	#16810487	1	1	yes	characteristic
194	C₄	#16810645	1	1	yes	characteristic
195	C₄	#16810673	1	1	yes	characteristic
196	C₄	#16810674	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	#20274830	1	1	yes	characteristic
200	C₂	#364×D₅₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20276470

C₂

yes

maximal

C₄×C₈×Dic₅₃

C₂

yes

maximal, characteristic

#20272797

C₂

yes

maximal, characteristic

#20272796

C₂

yes

maximal

C₅₃×#1376

C₂

yes

maximal, characteristic

C₂×#13133157

C₂²

yes

—

C₂×C₄×#88322

C₄

yes

characteristic

C₂×C₈×Dic₅₃

C₄

yes

characteristic

C₂×C₈×Dic₅₃

C₄

yes

characteristic

C₂×C₈×Dic₅₃

C₂²

yes

characteristic

C₂×C₈×Dic₅₃

C₂²

yes

characteristic

C₄²×Dic₅₃

C₄

yes

characteristic

#16810441

—

C₂×M₃₂×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂²×#88322

C₂×C₄

yes

characteristic

C₂²×#88322

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅₃

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

—

—

—

C₈×Dic₅₃

—

C₈×Dic₅₃

—

C₈×Dic₅₃

C₈

yes

—

C₈×Dic₅₃

C₈

yes

—

C₈×Dic₅₃

D₄

yes

—

C₈×Dic₅₃

Q₈

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

M₃₂×C₅₃

—

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

M₁₆

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₂×C₈×C₅₃

#41

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

M₁₆

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₄⋊C₄

yes

—

C₄×Dic₅₃

C₂×C₈

yes

characteristic

C₄×Dic₅₃

C₂×C₈

yes

characteristic

C₄×Dic₅₃

M₁₆

yes

—

—

—

—

—

C₄⋊C₄

yes

—

C₂×#88322

C₂×C₈

yes

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₄⋊C₈

yes

—

C₂×Dic₅₃

C₄⋊C₈

yes

—

C₈×C₅₃

—

C₈×C₅₃

#124

yes

—

C₈×C₅₃

#125

yes

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₈

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#364

yes

—

#2150

—

C₄²×C₈

—

#1376

—

100

#1376

—

101

#1376

D₅₃

yes

characteristic

102

C₂×C₅₃

—

103

C₂×C₅₃

—

104

C₂×C₅₃

#1831

yes

characteristic

105

C₂×C₅₃

C₂×#364

yes

—

106

C₂×M₃₂

—

107

C₂×M₃₂

C₂×D₅₃

yes

—

108

C₂×C₄×C₈

—

109

C₂×C₄×C₈

—

110

C₂×C₄×C₈

—

111

C₂×C₄×C₈

—

112

C₂×C₄×C₈

—

113

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

114

C₄³

—

115

C₄⋊C₁₆

—

106

—

116

C₅₃

#12883

yes

characteristic, complemented, Sylow

117

C₂²×C₈

—

118

C₂²×C₈

—

119

C₂²×C₈

C₄×D₅₃

yes

characteristic

120

C₂²×C₈

C₂²×D₅₃

yes

characteristic

121

C₂×C₄²

—

122

C₂×C₄²

—

123

C₂×C₄²

—

106

—

124

C₂×C₄²

C₄×D₅₃

yes

characteristic

125

M₃₂

—

126

M₃₂

—

127

M₃₂

—

106

—

128

M₃₂

—

106

—

129

C₂×C₁₆

—

130

C₂×C₁₆

—

106

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

—

134

C₄×C₈

—

135

C₄×C₈

—

136

C₄×C₈

—

106

—

137

C₄×C₈

—

106

—

138

C₄×C₈

—

106

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

—

106

—

142

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

106

—

148

C₂×C₈

—

106

—

149

C₂×C₈

—

106

—

150

C₂×C₈

—

106

—

151

C₂×C₈

C₈×D₅₃

yes

—

152

C₂×C₈

#11719945

yes

—

153

C₂×C₈

D₄×D₅₃

yes

characteristic

154

C₂×C₈

#11719947

yes

—

155

C₂×C₈

Q₈×D₅₃

yes

characteristic

156

C₄²

—

157

C₄²

—

158

C₄²

—

159

C₄²

—

160

C₄²

—

161

C₄²

—

106

—

162

C₄²

—

106

—

163

C₄²

#11719912

yes

—

164

C₁₆

—

165

C₁₆

—

212

—

166

C₂³

M₁₆×D₅₃

yes

characteristic

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

106

—

172

C₂×C₄

—

106

—

173

C₂×C₄

—

106

—

174

C₂×C₄

C₄⋊C₄×D₅₃

yes

characteristic

175

C₂×C₄

#13133173

yes

characteristic

176

C₂×C₄

#13133174

yes

characteristic

177

C₂×C₄

C₂×C₈×D₅₃

yes

characteristic

178

C₂×C₄

M₁₆×D₅₃

yes

characteristic

179

C₂×C₄

C₂×#11719912

yes

characteristic

180

C₈

—

181

C₈

—

106

—

182

C₈

—

106

—

183

C₈

#13133203

yes

—

184

C₈

#13133207

yes

—

185

C₂²

—

186

C₂²

C₄⋊C₈×D₅₃

yes

characteristic

187

C₂²

#16810677

yes

characteristic

188

C₂²

#16810678

yes

characteristic

189

C₄

—

190

C₄

—

191

C₄

—

106

—

192

C₄

—

106

—

193

C₄

#16810487

yes

characteristic

194

C₄

#16810645

yes

characteristic

195

C₄

#16810673

yes

characteristic

196

C₄

#16810674

yes

characteristic

197

C₂

—

198

C₂

—

199

C₂

#20274830

yes

characteristic

200

C₂

#364×D₅₃

yes

—

	subgroup	G/H
center	C₂×C₈	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1376	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂³×C₅₃	C₂×M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1376	D₅₃
hypocenter	C₅₃	#12883
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

subgroup

G/H

center

C₂×C₈

D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#1376

C₂

socle

C₂²×C₅₃

C₂×C₄⋊C₈

prime radical

C₂³×C₅₃

C₂×M₁₆

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

N	H ≅ G/N	type
116. C₅₃	#12883	semidirect

H ≅ G/N

type

116. C₅₃

#12883

semidirect

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

subgroup	index
#20276470	2
C₄×C₈×Dic₅₃	2
#20272797	2
#20272796	2
C₅₃×#1376	2
#12883	53

subgroup

index

C₄×C₈×Dic₅₃

C₅₃×#1376

Subgroups — #27300451

Conjugacy classes of subgroups

Complements

Maximal subgroups