Group #27284120 — subgroups

Subgroups — #27284120

Order 13568.

434 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 159 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	#20274832	C₂	1	1	yes	maximal, characteristic
2	#20274461	C₂	1	1	yes	maximal, characteristic
3	#20272556	C₂	1	4	yes	maximal
4	C₄×#125×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88322	C₂²	1	2	yes	—
6	C₂×C₄×#88322	C₂²	1	2	yes	—
7	#16810378	—	2	2	no	—
8	#16810378	—	2	2	no	—
9	C₂×#125×C₅₃	C₄	1	1	yes	characteristic
10	C₅₃×#389	C₄	1	2	yes	—
11	C₅₃×#387	—	2	2	no	—
12	C₅₃×#382	C₄	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
14	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
15	C₄³×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×#88322	—	2	2	no	—
17	C₂²×#88322	—	2	2	no	—
18	C₄⋊#88322	—	2	4	no	—
19	C₄×#88322	—	2	2	no	—
20	C₄×#88322	Q₈	1	4	yes	—
21	#125×C₅₃	—	2	4	no	—
22	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
23	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
24	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
25	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₅₃	—	2	2	no	—
27	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	D₄	1	2	yes	—
30	C₂×C₄²×C₅₃	D₄	1	2	yes	—
31	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	#95×C₅₃	—	2	2	no	—
33	#95×C₅₃	—	2	2	no	—
34	#95×C₅₃	—	2	2	no	—
35	C₂²×C₄×C₅₃	—	2	2	no	—
36	C₂²×C₄×C₅₃	—	2	2	no	—
37	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₄⋊C₄×C₅₃	—	2	8	no	—
43	C₄⋊C₄×C₅₃	—	2	2	no	—
44	C₄⋊C₄×C₅₃	—	2	4	no	—
45	C₄⋊C₄×C₅₃	—	4	4	no	—
46	C₄⋊C₄×C₅₃	M₁₆	1	4	yes	—
47	C₄²×C₅₃	—	2	2	no	—
48	C₄²×C₅₃	—	2	2	no	—
49	C₄²×C₅₃	—	2	4	no	—
50	C₄²×C₅₃	—	4	2	no	—
51	C₄²×C₅₃	—	4	4	no	—
52	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
53	C₄²×C₅₃	#41	1	2	yes	—
54	C₂×#88322	—	2	4	no	—
55	C₂×#88322	—	4	4	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	4	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	—	2	2	no	—
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	4	2	no	—
68	C₂×C₄×C₅₃	—	4	2	no	—
69	C₂×C₄×C₅₃	—	4	2	no	—
70	C₂×C₄×C₅₃	—	4	2	no	—
71	C₂×C₄×C₅₃	—	4	2	no	—
72	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
73	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
74	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
77	C₂×C₄×C₅₃	#131	1	2	yes	—
78	C₂×C₄×C₅₃	#135	1	4	yes	—
79	#88322	—	4	4	no	—
80	#88322	—	8	2	no	—
81	#9875	—	53	1	no	maximal, Sylow
82	C₂²×C₅₃	#391	1	1	yes	characteristic
83	C₂²×C₅₃	#407	1	1	yes	characteristic
84	C₂²×C₅₃	#408	1	1	yes	characteristic
85	C₂²×C₅₃	#416	1	2	yes	—
86	C₂²×C₅₃	#421	1	2	yes	—
87	C₄×C₅₃	—	2	2	no	—
88	C₄×C₅₃	—	4	2	no	—
89	C₄×C₅₃	—	4	2	no	—
90	C₄×C₅₃	—	4	4	no	—
91	C₄×C₅₃	—	8	2	no	—
92	C₄×C₅₃	#446	1	4	yes	—
93	C₄×C₅₃	#488	1	2	yes	—
94	C₄×C₅₃	#499	1	2	yes	—
95	C₄×#125	D₅₃	1	1	yes	characteristic
96	#1831	—	53	1	no	—
97	#1744	—	53	1	no	—
98	#1279	—	53	4	no	—
99	C₂×C₅₃	#1482	1	2	yes	—
100	C₂×C₅₃	#1488	1	2	yes	—
101	C₂×C₅₃	#1985	1	1	yes	characteristic
102	C₂×C₅₃	#1991	1	1	yes	characteristic
103	C₂×C₅₃	#1995	1	1	yes	characteristic
104	C₂×#125	Dic₅₃	1	1	yes	characteristic
105	C₂×C₄⋊C₈	—	53	2	no	—
106	C₂×C₄×C₈	—	53	2	no	—
107	#389	Dic₅₃	1	2	yes	—
108	#387	—	2	2	no	—
109	#382	Dic₅₃	1	1	yes	characteristic
110	C₄×C₄⋊C₄	—	2	2	no	—
111	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
112	C₄³	C₂×D₅₃	1	1	yes	characteristic
113	#336	—	106	2	no	—
114	#336	—	106	2	no	—
115	C₅₃	#9875	1	1	yes	characteristic, complemented, Sylow
116	C₂²×C₈	—	106	2	no	—
117	C₂²×C₈	—	106	2	no	—
118	#125	—	2	4	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
122	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
123	C₂×C₄²	—	2	2	no	—
124	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
125	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
126	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
127	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
128	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
129	C₄⋊C₈	—	106	4	no	—
130	C₄×C₈	—	53	4	no	—
131	C₄×C₈	—	106	2	no	—
132	#95	—	2	2	no	—
133	#95	—	2	2	no	—
134	#95	—	2	2	no	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
138	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
139	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
142	C₂×C₈	—	106	4	no	—
143	C₂×C₈	—	212	2	no	—
144	C₂×C₈	—	212	4	no	—
145	C₂×C₈	—	212	4	no	—
146	C₄⋊C₄	—	2	8	no	—
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	4	4	no	—
150	C₄⋊C₄	#11719917	1	4	yes	—
151	C₄²	—	2	2	no	—
152	C₄²	—	2	2	no	—
153	C₄²	—	2	4	no	—
154	C₄²	—	4	2	no	—
155	C₄²	—	4	4	no	—
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	4	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
173	C₂×C₄	#13133193	1	2	yes	—
174	C₂×C₄	#13133206	1	1	yes	characteristic
175	C₂×C₄	#13133215	1	1	yes	characteristic
176	C₂×C₄	C₂×#11719917	1	2	yes	—
177	C₂×C₄	#13133224	1	2	yes	—
178	C₂×C₄	#13133237	1	4	yes	—
179	C₈	—	424	2	no	—
180	C₈	—	212	4	no	—
181	C₂²	#16810530	1	1	yes	characteristic
182	C₂²	#16810587	1	1	yes	characteristic
183	C₂²	#16810590	1	1	yes	characteristic
184	C₂²	#16810619	1	2	yes	—
185	C₂²	#16810640	1	2	yes	—
186	C₄	—	2	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	4	2	no	—
189	C₄	—	4	4	no	—
190	C₄	—	8	2	no	—
191	C₄	#16810754	1	4	yes	—
192	C₄	#16810980	1	2	yes	—
193	C₄	#16811022	1	2	yes	—
194	C₂	#20273123	1	2	yes	—
195	C₂	#20273149	1	2	yes	—
196	C₂	#20275656	1	1	yes	characteristic
197	C₂	#20275681	1	1	yes	characteristic
198	C₂	#20275697	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274832

C₂

yes

maximal, characteristic

#20274461

C₂

yes

maximal, characteristic

#20272556

C₂

yes

maximal

C₄×#125×C₅₃

C₂

yes

maximal, characteristic

C₂×C₄⋊#88322

C₂²

yes

—

C₂×C₄×#88322

C₂²

yes

—

#16810378

—

#16810378

—

C₂×#125×C₅₃

C₄

yes

characteristic

C₅₃×#389

C₄

yes

—

C₅₃×#387

—

C₅₃×#382

C₄

yes

characteristic

C₄×C₄⋊C₄×C₅₃

—

C₄×C₄⋊C₄×C₅₃

C₂²

yes

—

C₄³×C₅₃

C₂²

yes

characteristic

C₂²×#88322

—

C₂²×#88322

—

C₄⋊#88322

—

C₄×#88322

—

C₄×#88322

Q₈

yes

—

#125×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#95×C₅₃

—

#95×C₅₃

—

#95×C₅₃

—

C₂²×C₄×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

M₁₆

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

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

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

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×M₁₆

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₂×C₄×C₅₃

#135

yes

—

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#421

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#446

yes

—

C₄×C₅₃

#488

yes

—

C₄×C₅₃

#499

yes

—

C₄×#125

D₅₃

yes

—

—

—

C₂×C₅₃

#1482

yes

—

100

C₂×C₅₃

#1488

yes

—

101

C₂×C₅₃

#1985

yes

characteristic

102

C₂×C₅₃

#1991

yes

characteristic

103

C₂×C₅₃

#1995

yes

characteristic

104

C₂×#125

Dic₅₃

yes

characteristic

105

C₂×C₄⋊C₈

—

106

C₂×C₄×C₈

—

107

#389

Dic₅₃

yes

—

108

#387

—

109

#382

Dic₅₃

yes

characteristic

110

C₄×C₄⋊C₄

—

111

C₄×C₄⋊C₄

C₂×D₅₃

yes

—

112

C₄³

C₂×D₅₃

yes

characteristic

113

#336

—

106

—

114

#336

—

106

—

115

C₅₃

#9875

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

C₂×Dic₅₃

yes

—

122

C₂×C₄⋊C₄

C₂×Dic₅₃

yes

—

123

C₂×C₄²

—

124

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

125

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

126

C₂×C₄²

C₂²⋊D₅₃

yes

—

127

C₂×C₄²

C₂²⋊D₅₃

yes

—

128

C₂×C₄²

C₂²×D₅₃

yes

characteristic

129

C₄⋊C₈

—

106

—

130

C₄×C₈

—

131

C₄×C₈

—

106

—

132

#95

—

133

#95

—

134

#95

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

C₂²⋊Dic₅₃

yes

—

138

C₂²×C₄

C₂²⋊Dic₅₃

yes

—

139

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

140

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

141

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

142

C₂×C₈

—

106

—

143

C₂×C₈

—

212

—

144

C₂×C₈

—

212

—

145

C₂×C₈

—

212

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

#11719917

yes

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

155

C₄²

—

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

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

Q₈×Dic₅₃

yes

—

173

C₂×C₄

#13133193

yes

—

174

C₂×C₄

#13133206

yes

characteristic

175

C₂×C₄

#13133215

yes

characteristic

176

C₂×C₄

C₂×#11719917

yes

—

177

C₂×C₄

#13133224

yes

—

178

C₂×C₄

#13133237

yes

—

179

C₈

—

424

—

180

C₈

—

212

—

181

C₂²

#16810530

yes

characteristic

182

C₂²

#16810587

yes

characteristic

183

C₂²

#16810590

yes

characteristic

184

C₂²

#16810619

yes

—

185

C₂²

#16810640

yes

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

#16810754

yes

—

192

C₄

#16810980

yes

—

193

C₄

#16811022

yes

—

194

C₂

#20273123

yes

—

195

C₂

#20273149

yes

—

196

C₂

#20275656

yes

characteristic

197

C₂

#20275681

yes

characteristic

198

C₂

#20275697

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×#125×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×#125	D₅₃
hypocenter	C₅₃	#9875
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₅₃

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₄×#125×C₅₃

C₂

socle, prime radical

C₂³×C₅₃

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

N	H ≅ G/N	type
115. C₅₃	#9875	semidirect

H ≅ G/N

type

115. C₅₃

#9875

semidirect

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

subgroup	index
#20274832	2
#20274461	2
#20272556	2
C₄×#125×C₅₃	2
#9875	53

subgroup

index

C₄×#125×C₅₃

Subgroups — #27284120

Conjugacy classes of subgroups

Complements

Maximal subgroups