Group #20285431 — subgroups

Subgroups — #20285431

Order 6784.

402 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 125 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	#16811178	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811171	C₂	1	1	yes	maximal, characteristic
3	#16811028	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810973	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810969	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810866	C₂	1	4	yes	maximal, complemented
7	#16810724	C₂	1	4	yes	maximal, complemented
8	C₄×#11719912	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11719947	—	2	1	no	—
11	C₂×#11719912	C₂²	1	2	yes	—
12	#13133217	C₂²	1	1	yes	characteristic
13	#13133217	C₂²	1	1	yes	characteristic
14	#13133208	—	2	1	no	—
15	#13133204	—	2	1	no	—
16	#13133197	—	2	1	no	—
17	#13133193	C₂²	1	2	yes	—
18	#13133188	—	2	1	no	—
19	#13133188	C₂²	1	2	yes	—
20	Q₈×Dic₅₃	C₂²	1	2	yes	—
21	#13133182	—	2	1	no	—
22	D₄×Dic₅₃	—	2	1	no	—
23	D₄×Dic₅₃	C₂²	1	2	yes	—
24	#13133175	—	2	1	no	—
25	C₄²×D₅₃	C₂²	1	1	yes	characteristic
26	#13133151	C₂²	1	4	yes	—
27	#13133141	C₂²	1	4	yes	—
28	#13133138	C₂²	1	4	yes	—
29	#13133127	C₂²	1	2	yes	—
30	C₄×#88322	C₂²	1	1	yes	characteristic
31	C₂×SD₁₆×C₅₃	C₂²	1	4	yes	—
32	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
33	#127×C₅₃	C₂²	1	1	yes	characteristic
34	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
36	C₂×D₄×C₅₃	—	2	1	no	—
37	C₂×D₄×C₅₃	—	2	1	no	—
38	C₂×D₄×C₅₃	C₂³	1	2	yes	—
39	C₂×C₂²⋊D₅₃	—	2	2	no	—
40	C₂²×Dic₅₃	—	2	2	no	—
41	#11719947	—	4	4	no	—
42	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
43	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
44	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
45	C₂×Dic₁₀₆	—	2	1	no	—
46	C₂×Dic₁₀₆	—	2	1	no	—
47	SD₁₆×C₅₃	—	2	8	no	—
48	C₂×C₈×C₅₃	C₂³	1	2	yes	—
49	C₄⋊C₄×C₅₃	—	2	1	no	—
50	C₄⋊C₄×C₅₃	—	2	1	no	—
51	C₄²×C₅₃	C₂³	1	1	yes	characteristic
52	C₂²⋊Dic₅₃	—	2	2	no	—
53	C₂²⋊Dic₅₃	—	2	2	no	—
54	#11719921	—	2	2	no	—
55	C₄⋊Dic₅₃	—	2	1	no	—
56	C₄⋊Dic₅₃	—	2	1	no	—
57	C₄⋊Dic₅₃	—	2	2	no	—
58	C₄⋊Dic₅₃	C₂³	1	2	yes	—
59	C₄⋊Dic₅₃	C₂³	1	2	yes	—
60	#11719919	—	2	2	no	—
61	C₄×Dic₅₃	—	2	2	no	—
62	C₄×Dic₅₃	D₄	1	1	yes	characteristic
63	C₄×Dic₅₃	D₄	1	1	yes	characteristic
64	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
65	C₂×#88322	C₂³	1	2	yes	—
66	#11719912	—	2	4	no	—
67	C₂³×C₅₃	—	2	2	no	—
68	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
69	Q₈×C₅₃	—	2	4	no	—
70	D₄×C₅₃	—	2	4	no	—
71	D₄×C₅₃	—	4	2	no	—
72	D₄×C₅₃	—	4	2	no	—
73	C₂×C₄×C₅₃	—	2	2	no	—
74	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
77	C₂²⋊D₅₃	—	4	4	no	—
78	C₂×Dic₅₃	—	2	2	no	—
79	C₂×Dic₅₃	—	2	2	no	—
80	C₂×Dic₅₃	—	4	4	no	—
81	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
84	C₄×D₅₃	—	2	2	no	—
85	C₄×D₅₃	—	2	2	no	—
86	C₄×D₅₃	—	2	2	no	—
87	Dic₁₀₆	—	4	2	no	—
88	Dic₁₀₆	—	4	2	no	—
89	C₈×C₅₃	#41	1	4	yes	complemented
90	#88322	—	2	2	no	—
91	C₂²×C₅₃	—	4	2	no	—
92	C₂²×C₅₃	—	4	2	no	—
93	C₂²×C₅₃	—	4	2	no	—
94	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
95	C₂×D₅₃	—	2	1	no	—
96	C₂×D₅₃	—	2	1	no	—
97	C₂×D₅₃	—	2	1	no	—
98	C₄×C₅₃	—	4	2	no	—
99	C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
100	C₄×C₅₃	#137	1	2	yes	—
101	C₄×C₅₃	C₂×#41	1	2	yes	—
102	Dic₅₃	—	2	1	no	—
103	Dic₅₃	—	2	1	no	—
104	Dic₅₃	—	2	1	no	—
105	Dic₅₃	—	4	2	no	—
106	Dic₅₃	—	4	2	no	—
107	#3182	—	53	1	no	maximal, Sylow
108	C₂×C₅₃	—	8	2	no	—
109	C₂×C₅₃	#532	1	1	yes	characteristic
110	C₂×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₂×C₅₃	C₂×#137	1	1	yes	characteristic
112	D₅₃	—	4	1	no	—
113	#533	—	53	1	no	—
114	#532	—	53	1	no	—
115	#500	—	53	1	no	—
116	#492	D₅₃	1	1	yes	characteristic, complemented
117	#487	—	53	1	no	—
118	#486	—	53	1	no	—
119	#468	—	53	4	no	—
120	#440	—	53	4	no	—
121	C₄×M₁₆	—	53	1	no	—
122	C₅₃	#3182	1	1	yes	characteristic, complemented, Sylow
123	C₂×#41	—	106	1	no	—
124	C₂×SD₁₆	C₂×D₅₃	1	4	yes	—
125	C₂×M₁₆	—	53	2	no	—
126	C₄⋊Q₈	—	53	1	no	—
127	C₄⋊Q₈	—	53	1	no	—
128	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
129	#127	C₂×D₅₃	1	1	yes	characteristic
130	#125	—	106	1	no	—
131	#124	—	106	1	no	—
132	C₂²⋊Q₈	—	53	2	no	—
133	C₂²⋊Q₈	—	106	1	no	—
134	#121	—	53	2	no	—
135	#121	—	106	1	no	—
136	C₄×Q₈	—	53	2	no	—
137	C₄×Q₈	—	106	1	no	—
138	C₄×D₄	—	53	2	no	—
139	C₄×D₄	—	106	1	no	—
140	#117	—	106	1	no	—
141	C₂×C₄²	—	53	1	no	—
142	#107	—	53	4	no	—
143	#103	—	53	4	no	—
144	#102	—	53	4	no	—
145	#97	—	53	2	no	—
146	C₄×C₈	—	53	1	no	—
147	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
148	#41	—	212	4	no	—
149	C₂×Q₈	—	106	1	no	—
150	C₂×Q₈	—	106	1	no	—
151	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
152	C₂×D₄	—	2	1	no	—
153	C₂×D₄	—	2	1	no	—
154	C₂×D₄	—	106	2	no	—
155	C₂×D₄	C₂²×D₅₃	1	2	yes	—
156	C₂²×C₄	—	53	1	no	—
157	C₂²×C₄	—	53	1	no	—
158	C₂²×C₄	—	53	1	no	—
159	C₂²×C₄	—	106	2	no	—
160	SD₁₆	—	2	8	no	—
161	M₁₆	—	106	4	no	—
162	C₂×C₈	—	53	2	no	—
163	C₂×C₈	C₂²×D₅₃	1	2	yes	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	1	no	—
166	C₄⋊C₄	—	53	2	no	—
167	C₄⋊C₄	—	53	2	no	—
168	C₄⋊C₄	—	106	2	no	—
169	C₄⋊C₄	—	106	2	no	—
170	C₄⋊C₄	—	106	1	no	—
171	C₄⋊C₄	—	106	1	no	—
172	C₂²⋊C₄	—	106	2	no	—
173	C₂²⋊C₄	—	106	2	no	—
174	C₂²⋊C₄	—	106	2	no	—
175	C₄²	—	53	1	no	—
176	C₄²	—	53	1	no	—
177	C₄²	—	53	1	no	—
178	C₄²	—	106	2	no	—
179	C₄²	C₂²×D₅₃	1	1	yes	characteristic
180	C₂³	—	2	2	no	—
181	C₂³	—	53	1	no	—
182	Q₈	—	2	4	no	—
183	Q₈	—	212	2	no	—
184	Q₈	—	212	2	no	—
185	D₄	—	2	4	no	—
186	D₄	—	4	2	no	—
187	D₄	—	4	2	no	—
188	D₄	—	212	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	53	1	no	—
191	C₂×C₄	—	53	1	no	—
192	C₂×C₄	—	53	1	no	—
193	C₂×C₄	—	106	2	no	—
194	C₂×C₄	—	106	2	no	—
195	C₂×C₄	—	106	2	no	—
196	C₂×C₄	—	106	2	no	—
197	C₂×C₄	—	106	2	no	—
198	C₂×C₄	—	212	4	no	—
199	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
200	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
201	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
202	C₈	—	106	2	no	—
203	C₈	#11719947	1	4	yes	complemented
204	C₂²	—	4	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	—	4	2	no	—
207	C₂²	—	106	1	no	—
208	C₂²	—	106	1	no	—
209	C₂²	—	106	1	no	—
210	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
211	C₄	—	4	2	no	—
212	C₄	—	106	1	no	—
213	C₄	—	106	1	no	—
214	C₄	—	106	1	no	—
215	C₄	—	212	2	no	—
216	C₄	—	212	2	no	—
217	C₄	#13133244	1	2	yes	—
218	C₄	#13133246	1	2	yes	—
219	C₄	C₂×#11719947	1	2	yes	—
220	C₂	—	8	2	no	—
221	C₂	—	212	1	no	—
222	C₂	#16811171	1	1	yes	characteristic
223	C₂	C₂×#13133244	1	1	yes	characteristic
224	C₂	C₂×#13133246	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811178

C₂

yes

maximal, characteristic, complemented

#16811171

C₂

yes

maximal, characteristic

#16811028

C₂

yes

maximal, characteristic, complemented

#16810973

C₂

yes

maximal, characteristic, complemented

#16810969

C₂

yes

maximal, characteristic, complemented

#16810866

C₂

yes

maximal, complemented

#16810724

C₂

yes

maximal, complemented

C₄×#11719912

C₂

yes

maximal, characteristic, complemented

C₅₃×#492

C₂

yes

maximal, characteristic, complemented

C₂×#11719947

—

C₂×#11719912

C₂²

yes

—

#13133217

C₂²

yes

characteristic

#13133217

C₂²

yes

—

—

—

C₂²

yes

—

#13133188

—

#13133188

C₂²

yes

—

Q₈×Dic₅₃

C₂²

yes

—

#13133182

—

D₄×Dic₅₃

—

D₄×Dic₅₃

C₂²

yes

—

#13133175

—

C₄²×D₅₃

C₂²

yes

characteristic

#13133151

C₂²

yes

—

#13133141

C₂²

yes

—

#13133138

C₂²

yes

—

#13133127

C₂²

yes

—

C₄×#88322

C₂²

yes

characteristic

C₂×SD₁₆×C₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

#127×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

C₂³

yes

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂³

yes

—

C₂×C₂²⋊D₅₃

—

C₂²×Dic₅₃

—

#11719947

—

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

SD₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

—

C₂²⋊Dic₅₃

—

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

C₂³

yes

characteristic

C₂×#88322

C₂³

yes

—

#11719912

—

C₂³×C₅₃

—

C₂²×D₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

#41

yes

complemented

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

AGL(1,ℤ/8)

yes

—

100

C₄×C₅₃

#137

yes

—

101

C₄×C₅₃

C₂×#41

yes

—

102

Dic₅₃

—

103

Dic₅₃

—

104

Dic₅₃

—

105

Dic₅₃

—

106

Dic₅₃

—

107

#3182

—

maximal, Sylow

108

C₂×C₅₃

—

109

C₂×C₅₃

#532

yes

characteristic

110

C₂×C₅₃

C₂×AGL(1,ℤ/8)

yes

characteristic

111

C₂×C₅₃

C₂×#137

yes

characteristic

112

D₅₃

—

113

#533

—

114

#532

—

115

#500

—

116

#492

D₅₃

yes

characteristic, complemented

117

#487

—

118

#486

—

119

#468

—

120

#440

—

121

C₄×M₁₆

—

122

C₅₃

#3182

yes

characteristic, complemented, Sylow

123

C₂×#41

—

106

—

124

C₂×SD₁₆

C₂×D₅₃

yes

—

125

C₂×M₁₆

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

129

#127

C₂×D₅₃

yes

characteristic

130

#125

—

106

—

131

#124

—

106

—

132

C₂²⋊Q₈

—

133

C₂²⋊Q₈

—

106

—

134

#121

—

135

#121

—

106

—

136

C₄×Q₈

—

137

C₄×Q₈

—

106

—

138

C₄×D₄

—

139

C₄×D₄

—

106

—

140

#117

—

106

—

141

C₂×C₄²

—

142

#107

—

143

#103

—

144

#102

—

145

#97

—

146

C₄×C₈

—

147

C₄×C₈

C₂×D₅₃

yes

characteristic

148

#41

—

212

—

149

C₂×Q₈

—

106

—

150

C₂×Q₈

—

106

—

151

C₂×Q₈

C₂²×D₅₃

yes

—

152

C₂×D₄

—

153

C₂×D₄

—

154

C₂×D₄

—

106

—

155

C₂×D₄

C₂²×D₅₃

yes

—

156

C₂²×C₄

—

157

C₂²×C₄

—

158

C₂²×C₄

—

159

C₂²×C₄

—

106

—

160

SD₁₆

—

161

M₁₆

—

106

—

162

C₂×C₈

—

163

C₂×C₈

C₂²×D₅₃

yes

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

106

—

169

C₄⋊C₄

—

106

—

170

C₄⋊C₄

—

106

—

171

C₄⋊C₄

—

106

—

172

C₂²⋊C₄

—

106

—

173

C₂²⋊C₄

—

106

—

174

C₂²⋊C₄

—

106

—

175

C₄²

—

176

C₄²

—

177

C₄²

—

178

C₄²

—

106

—

179

C₄²

C₂²×D₅₃

yes

characteristic

180

C₂³

—

181

C₂³

—

182

Q₈

—

183

Q₈

—

212

—

184

Q₈

—

212

—

185

D₄

—

186

D₄

—

187

D₄

—

188

D₄

—

212

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

106

—

194

C₂×C₄

—

106

—

195

C₂×C₄

—

106

—

196

C₂×C₄

—

106

—

197

C₂×C₄

—

106

—

198

C₂×C₄

—

212

—

199

C₂×C₄

D₄×D₅₃

yes

characteristic

200

C₂×C₄

D₄×D₅₃

yes

characteristic

201

C₂×C₄

C₂³×D₅₃

yes

characteristic

202

C₈

—

106

—

203

C₈

#11719947

yes

complemented

204

C₂²

—

205

C₂²

—

206

C₂²

—

207

C₂²

—

106

—

208

C₂²

—

106

—

209

C₂²

—

106

—

210

C₂²

C₂×D₄×D₅₃

yes

characteristic

211

C₄

—

212

C₄

—

106

—

213

C₄

—

106

—

214

C₄

—

106

—

215

C₄

—

212

—

216

C₄

—

212

—

217

C₄

#13133244

yes

—

218

C₄

#13133246

yes

—

219

C₄

C₂×#11719947

yes

—

220

C₂

—

221

C₂

—

212

—

222

C₂

#16811171

yes

characteristic

223

C₂

C₂×#13133244

yes

characteristic

224

C₂

C₂×#13133246

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#492	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#492	D₅₃
hypocenter	C₅₃	#3182
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

C₂×D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#492

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #16811178	C₂	semidirect
3. #16811028	C₂	semidirect
4. #16810973	C₂	semidirect
5. #16810969	C₂	semidirect
6. #16810866	C₂	semidirect
7. #16810724	C₂	semidirect
8. C₄×#11719912	C₂	semidirect
9. C₅₃×#492	C₂	semidirect
89. C₈×C₅₃	#41	semidirect
116. #492	D₅₃	semidirect
122. C₅₃	#3182	semidirect
203. C₈	#11719947	semidirect

H ≅ G/N

type

1. #16811178

C₂

semidirect

3. #16811028

C₂

semidirect

4. #16810973

C₂

semidirect

5. #16810969

C₂

semidirect

6. #16810866

C₂

semidirect

7. #16810724

C₂

semidirect

8. C₄×#11719912

C₂

semidirect

9. C₅₃×#492

C₂

semidirect

89. C₈×C₅₃

#41

semidirect

116. #492

D₅₃

semidirect

122. C₅₃

#3182

semidirect

203. C₈

#11719947

semidirect

Subgroups — #20285431

Conjugacy classes of subgroups

Complements