Group #20285105 — subgroups

Subgroups — #20285105

Order 6784.

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

normal subgroups only

This group has 424 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	#16811171	C₂	1	1	yes	maximal, characteristic
2	C₄⋊Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810981	C₂	1	4	yes	maximal, complemented
4	#16810975	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810974	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810969	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810621	C₂	1	4	yes	maximal, complemented
8	C₄×#11719912	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11719947	C₂²	1	2	yes	—
11	C₂×Q₈×D₅₃	—	2	1	no	—
12	C₂×#11719912	C₂²	1	2	yes	—
13	#13133217	C₂²	1	1	yes	characteristic
14	#13133217	C₂²	1	1	yes	characteristic
15	#13133216	—	2	1	no	—
16	#13133216	C₂²	1	2	yes	—
17	#13133215	—	2	1	no	—
18	#13133204	C₂²	1	2	yes	—
19	#13133188	—	2	1	no	—
20	#13133188	—	2	1	no	—
21	D₄×Dic₅₃	—	2	1	no	—
22	D₄×Dic₅₃	—	2	1	no	—
23	C₄⋊C₄×D₅₃	—	2	1	no	—
24	C₄⋊C₄×D₅₃	C₂²	1	2	yes	—
25	C₄²×D₅₃	C₂²	1	1	yes	characteristic
26	#13133142	C₂²	1	4	yes	—
27	#13133140	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	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
32	#127×C₅₃	C₂²	1	1	yes	characteristic
33	#102×C₅₃	C₂²	1	4	yes	—
34	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×Q₈×C₅₃	—	2	1	no	—
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	Q₈×D₅₃	—	4	4	no	—
42	#11719947	—	2	8	no	—
43	C₂×C₄×D₅₃	—	2	2	no	—
44	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
45	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
46	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
47	C₂×Dic₁₀₆	—	2	1	no	—
48	C₂×Dic₁₀₆	—	2	2	no	—
49	C₂×Dic₁₀₆	C₂³	1	2	yes	—
50	C₂×C₈×C₅₃	C₂³	1	2	yes	—
51	C₄⋊C₄×C₅₃	—	2	1	no	—
52	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
53	C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	C₂²⋊Dic₅₃	—	2	2	no	—
55	C₂²⋊Dic₅₃	—	2	2	no	—
56	C₄⋊Dic₅₃	—	2	1	no	—
57	C₄⋊Dic₅₃	—	2	1	no	—
58	C₄⋊Dic₅₃	—	2	1	no	—
59	C₄⋊Dic₅₃	C₂³	1	2	yes	—
60	#11719919	—	2	2	no	—
61	#11719919	—	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	—	4	2	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	—	4	4	no	—
87	C₄×D₅₃	#41	1	4	yes	—
88	Dic₁₀₆	—	2	4	no	—
89	Dic₁₀₆	—	4	4	no	—
90	Dic₁₀₆	—	4	2	no	—
91	C₈×C₅₃	—	2	2	no	—
92	#88322	—	2	2	no	—
93	C₂²×C₅₃	—	4	2	no	—
94	C₂²×C₅₃	—	4	2	no	—
95	C₂²×C₅₃	—	4	2	no	—
96	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
97	C₂×D₅₃	—	2	1	no	—
98	C₂×D₅₃	—	2	1	no	—
99	C₂×D₅₃	#124	1	2	yes	—
100	C₄×C₅₃	—	4	2	no	—
101	C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
102	C₄×C₅₃	#137	1	2	yes	—
103	C₄×C₅₃	C₂×#41	1	2	yes	—
104	Dic₅₃	—	2	1	no	—
105	Dic₅₃	—	2	1	no	—
106	Dic₅₃	—	4	2	no	—
107	Dic₅₃	—	4	2	no	—
108	Dic₅₃	#124	1	2	yes	—
109	#3141	—	53	1	no	maximal, Sylow
110	C₂×C₅₃	—	8	2	no	—
111	C₂×C₅₃	C₂×#124	1	1	yes	characteristic
112	C₂×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
113	C₂×C₅₃	C₂×#137	1	1	yes	characteristic
114	D₅₃	—	2	2	no	—
115	#532	—	53	1	no	—
116	C₂×C₄⋊Q₈	—	53	1	no	—
117	#489	—	53	4	no	—
118	#487	—	53	1	no	—
119	#487	—	53	1	no	—
120	#486	—	53	1	no	—
121	#486	D₅₃	1	1	yes	characteristic, complemented
122	#417	—	53	4	no	—
123	C₄×M₁₆	—	53	1	no	—
124	C₅₃	#3141	1	1	yes	characteristic, complemented, Sylow
125	C₂×#41	—	53	2	no	—
126	C₂²×Q₈	—	106	1	no	—
127	C₂×M₁₆	—	53	2	no	—
128	C₄⋊Q₈	—	53	2	no	—
129	C₄⋊Q₈	—	53	1	no	—
130	C₄⋊Q₈	—	53	1	no	—
131	C₄⋊Q₈	—	106	1	no	—
132	C₄⋊Q₈	—	106	1	no	—
133	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
134	#127	C₂×D₅₃	1	1	yes	characteristic
135	#124	—	53	2	no	—
136	#121	—	106	1	no	—
137	#121	—	106	1	no	—
138	C₄×D₄	—	106	1	no	—
139	C₄×D₄	—	106	1	no	—
140	C₂×C₄⋊C₄	—	53	2	no	—
141	C₂×C₄⋊C₄	—	106	1	no	—
142	C₂×C₄²	—	53	1	no	—
143	#103	—	53	4	no	—
144	#103	—	53	4	no	—
145	#102	—	53	4	no	—
146	#102	C₂×D₅₃	1	4	yes	—
147	#97	—	53	2	no	—
148	C₄×C₈	—	53	1	no	—
149	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
150	#41	—	106	8	no	—
151	C₂×Q₈	—	2	1	no	—
152	C₂×Q₈	—	53	2	no	—
153	C₂×Q₈	—	106	2	no	—
154	C₂×Q₈	—	106	1	no	—
155	C₂×Q₈	—	212	4	no	—
156	C₂×D₄	—	2	1	no	—
157	C₂×D₄	—	2	1	no	—
158	C₂×D₄	—	106	2	no	—
159	C₂×D₄	C₂²×D₅₃	1	2	yes	—
160	C₂²×C₄	—	53	1	no	—
161	C₂²×C₄	—	53	1	no	—
162	C₂²×C₄	—	53	1	no	—
163	C₂²×C₄	—	106	2	no	—
164	C₂²×C₄	—	106	2	no	—
165	M₁₆	—	106	4	no	—
166	C₂×C₈	—	53	2	no	—
167	C₂×C₈	C₂²×D₅₃	1	2	yes	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	53	2	no	—
170	C₄⋊C₄	—	106	2	no	—
171	C₄⋊C₄	—	106	2	no	—
172	C₄⋊C₄	—	106	1	no	—
173	C₄⋊C₄	—	106	1	no	—
174	C₄⋊C₄	—	106	1	no	—
175	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
176	C₂²⋊C₄	—	106	2	no	—
177	C₂²⋊C₄	—	106	2	no	—
178	C₄²	—	53	1	no	—
179	C₄²	—	53	1	no	—
180	C₄²	—	53	1	no	—
181	C₄²	C₂²×D₅₃	1	1	yes	characteristic
182	C₂³	—	2	2	no	—
183	C₂³	—	53	1	no	—
184	Q₈	—	4	2	no	—
185	Q₈	—	106	4	no	—
186	Q₈	—	212	4	no	—
187	Q₈	—	212	2	no	—
188	D₄	—	2	4	no	—
189	D₄	—	4	2	no	—
190	D₄	—	4	2	no	—
191	D₄	—	212	4	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	53	1	no	—
194	C₂×C₄	—	53	4	no	—
195	C₂×C₄	—	53	1	no	—
196	C₂×C₄	—	53	1	no	—
197	C₂×C₄	—	106	2	no	—
198	C₂×C₄	—	106	2	no	—
199	C₂×C₄	—	106	2	no	—
200	C₂×C₄	—	106	2	no	—
201	C₂×C₄	—	212	4	no	—
202	C₂×C₄	—	212	4	no	—
203	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
204	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
205	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
206	C₈	—	2	2	no	—
207	C₈	—	106	2	no	—
208	C₂²	—	4	2	no	—
209	C₂²	—	4	2	no	—
210	C₂²	—	4	2	no	—
211	C₂²	—	53	2	no	—
212	C₂²	—	106	1	no	—
213	C₂²	—	106	1	no	—
214	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
215	C₄	—	4	2	no	—
216	C₄	—	53	2	no	—
217	C₄	—	106	1	no	—
218	C₄	—	106	1	no	—
219	C₄	—	212	2	no	—
220	C₄	—	212	2	no	—
221	C₄	#13133241	1	2	yes	—
222	C₄	#13133246	1	2	yes	—
223	C₄	#41×D₅₃	1	2	yes	—
224	C₂	—	8	2	no	—
225	C₂	—	106	2	no	—
226	C₂	#124×D₅₃	1	1	yes	characteristic
227	C₂	C₂×#13133241	1	1	yes	characteristic
228	C₂	C₂×#13133246	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811171

C₂

yes

maximal, characteristic

C₄⋊Q₈×D₅₃

C₂

yes

maximal, characteristic, complemented

#16810981

C₂

yes

maximal, complemented

#16810975

C₂

yes

maximal, characteristic, complemented

#16810974

C₂

yes

maximal, characteristic, complemented

#16810969

C₂

yes

maximal, characteristic, complemented

#16810621

C₂

yes

maximal, complemented

C₄×#11719912

C₂

yes

maximal, characteristic, complemented

C₅₃×#486

C₂

yes

maximal, characteristic, complemented

C₂×#11719947

C₂²

yes

—

C₂×Q₈×D₅₃

—

C₂×#11719912

C₂²

yes

—

#13133217

C₂²

yes

characteristic

#13133217

C₂²

yes

characteristic

#13133216

—

#13133216

C₂²

yes

—

#13133215

—

#13133204

C₂²

yes

—

#13133188

—

#13133188

—

D₄×Dic₅₃

—

D₄×Dic₅₃

—

C₄⋊C₄×D₅₃

—

C₄⋊C₄×D₅₃

C₂²

yes

—

C₄²×D₅₃

C₂²

yes

characteristic

#13133142

C₂²

yes

—

#13133140

C₂²

yes

—

#13133138

C₂²

yes

—

#13133127

C₂²

yes

—

C₄×#88322

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

#127×C₅₃

C₂²

yes

characteristic

#102×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂³

yes

—

C₂×C₂²⋊D₅₃

—

C₂²×Dic₅₃

—

Q₈×D₅₃

—

#11719947

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂³

yes

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

#11719919

—

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

—

C₄×D₅₃

#41

yes

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#124

yes

—

100

C₄×C₅₃

—

101

C₄×C₅₃

AGL(1,ℤ/8)

yes

—

102

C₄×C₅₃

#137

yes

—

103

C₄×C₅₃

C₂×#41

yes

—

104

Dic₅₃

—

105

Dic₅₃

—

106

Dic₅₃

—

107

Dic₅₃

—

108

Dic₅₃

#124

yes

—

109

#3141

—

maximal, Sylow

110

C₂×C₅₃

—

111

C₂×C₅₃

C₂×#124

yes

characteristic

112

C₂×C₅₃

C₂×AGL(1,ℤ/8)

yes

characteristic

113

C₂×C₅₃

C₂×#137

yes

characteristic

114

D₅₃

—

115

#532

—

116

C₂×C₄⋊Q₈

—

117

#489

—

118

#487

—

119

#487

—

120

#486

—

121

#486

D₅₃

yes

characteristic, complemented

122

#417

—

123

C₄×M₁₆

—

124

C₅₃

#3141

yes

characteristic, complemented, Sylow

125

C₂×#41

—

126

C₂²×Q₈

—

106

—

127

C₂×M₁₆

—

128

C₄⋊Q₈

—

129

C₄⋊Q₈

—

130

C₄⋊Q₈

—

131

C₄⋊Q₈

—

106

—

132

C₄⋊Q₈

—

106

—

133

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

134

#127

C₂×D₅₃

yes

characteristic

135

#124

—

136

#121

—

106

—

137

#121

—

106

—

138

C₄×D₄

—

106

—

139

C₄×D₄

—

106

—

140

C₂×C₄⋊C₄

—

141

C₂×C₄⋊C₄

—

106

—

142

C₂×C₄²

—

143

#103

—

144

#103

—

145

#102

—

146

#102

C₂×D₅₃

yes

—

147

#97

—

148

C₄×C₈

—

149

C₄×C₈

C₂×D₅₃

yes

characteristic

150

#41

—

106

—

151

C₂×Q₈

—

152

C₂×Q₈

—

153

C₂×Q₈

—

106

—

154

C₂×Q₈

—

106

—

155

C₂×Q₈

—

212

—

156

C₂×D₄

—

157

C₂×D₄

—

158

C₂×D₄

—

106

—

159

C₂×D₄

C₂²×D₅₃

yes

—

160

C₂²×C₄

—

161

C₂²×C₄

—

162

C₂²×C₄

—

163

C₂²×C₄

—

106

—

164

C₂²×C₄

—

106

—

165

M₁₆

—

106

—

166

C₂×C₈

—

167

C₂×C₈

C₂²×D₅₃

yes

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

106

—

171

C₄⋊C₄

—

106

—

172

C₄⋊C₄

—

106

—

173

C₄⋊C₄

—

106

—

174

C₄⋊C₄

—

106

—

175

C₄⋊C₄

C₂²×D₅₃

yes

—

176

C₂²⋊C₄

—

106

—

177

C₂²⋊C₄

—

106

—

178

C₄²

—

179

C₄²

—

180

C₄²

—

181

C₄²

C₂²×D₅₃

yes

characteristic

182

C₂³

—

183

C₂³

—

184

Q₈

—

185

Q₈

—

106

—

186

Q₈

—

212

—

187

Q₈

—

212

—

188

D₄

—

189

D₄

—

190

D₄

—

191

D₄

—

212

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

106

—

198

C₂×C₄

—

106

—

199

C₂×C₄

—

106

—

200

C₂×C₄

—

106

—

201

C₂×C₄

—

212

—

202

C₂×C₄

—

212

—

203

C₂×C₄

D₄×D₅₃

yes

characteristic

204

C₂×C₄

D₄×D₅₃

yes

characteristic

205

C₂×C₄

C₂³×D₅₃

yes

characteristic

206

C₈

—

207

C₈

—

106

—

208

C₂²

—

209

C₂²

—

210

C₂²

—

211

C₂²

—

212

C₂²

—

106

—

213

C₂²

—

106

—

214

C₂²

C₂×D₄×D₅₃

yes

characteristic

215

C₄

—

216

C₄

—

217

C₄

—

106

—

218

C₄

—

106

—

219

C₄

—

212

—

220

C₄

—

212

—

221

C₄

#13133241

yes

—

222

C₄

#13133246

yes

—

223

C₄

#41×D₅₃

yes

—

224

C₂

—

225

C₂

—

106

—

226

C₂

#124×D₅₃

yes

characteristic

227

C₂

C₂×#13133241

yes

characteristic

228

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×#486	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#486	D₅₃
hypocenter	C₅₃	#3141
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

C₂×D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#486

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
2. C₄⋊Q₈×D₅₃	C₂	semidirect
3. #16810981	C₂	semidirect
4. #16810975	C₂	semidirect
5. #16810974	C₂	semidirect
6. #16810969	C₂	semidirect
7. #16810621	C₂	semidirect
8. C₄×#11719912	C₂	semidirect
9. C₅₃×#486	C₂	semidirect
121. #486	D₅₃	semidirect
124. C₅₃	#3141	semidirect

H ≅ G/N

type

2. C₄⋊Q₈×D₅₃

C₂

semidirect

3. #16810981

C₂

semidirect

4. #16810975

C₂

semidirect

5. #16810974

C₂

semidirect

6. #16810969

C₂

semidirect

7. #16810621

C₂

semidirect

8. C₄×#11719912

C₂

semidirect

9. C₅₃×#486

C₂

semidirect

121. #486

D₅₃

semidirect

124. C₅₃

#3141

semidirect

Subgroups — #20285105

Conjugacy classes of subgroups

Complements