Group #20286181 — subgroups

Subgroups — #20286181

Order 6784.

346 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 109 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	#16811246	C₂	1	2	yes	maximal, complemented
2	#16810912	C₂	1	2	yes	maximal, complemented
3	#16810906	C₂	1	2	yes	maximal, complemented
4	#16810876	C₂	1	1	yes	maximal, characteristic
5	#16810854	C₂	1	2	yes	maximal, complemented
6	#16810847	C₂	1	2	yes	maximal, complemented
7	#16810841	C₂	1	2	yes	maximal, complemented
8	#16810651	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#470	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11719924	C₂²	1	2	yes	—
11	C₂×D₄₂₄	C₂²	1	1	yes	characteristic
12	C₂²×#88322	C₂²	1	1	yes	characteristic
13	#13133216	C₂²	1	2	yes	—
14	#13133212	—	2	2	no	—
15	#13133206	C₂²	1	2	yes	—
16	#13133203	—	2	2	no	—
17	#13133199	—	2	2	no	—
18	#13133197	—	2	2	no	—
19	#13133191	C₂²	1	2	yes	—
20	Q₈×Dic₅₃	C₂²	1	2	yes	—
21	#13133179	C₂²	1	2	yes	—
22	#13133176	C₂²	1	1	yes	characteristic
23	#13133149	C₂²	1	1	yes	characteristic
24	#13133148	C₂²	1	2	yes	—
25	#13133146	C₂²	1	2	yes	—
26	#13133146	C₂²	1	2	yes	—
27	#13133141	C₂²	1	2	yes	—
28	#13133139	C₂²	1	2	yes	—
29	#13133137	C₂²	1	2	yes	—
30	C₂×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
32	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
33	#106×C₅₃	C₂²	1	1	yes	characteristic
34	#103×C₅₃	C₂²	1	2	yes	—
35	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
36	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
37	C₂×C₂²⋊D₅₃	—	2	2	no	—
38	C₂×D₂₁₂	C₂³	1	2	yes	—
39	C₂×C₄×D₅₃	—	2	2	no	—
40	C₂×Dic₁₀₆	—	2	2	no	—
41	Q₁₆×C₅₃	—	4	1	no	—
42	M₁₆×C₅₃	—	4	1	no	—
43	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₅₃	—	2	2	no	—
46	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
47	C₂²⋊C₄×C₅₃	—	2	2	no	—
48	C₂²⋊Dic₅₃	D₄	1	1	yes	characteristic
49	C₂²⋊Dic₅₃	D₄	1	1	yes	characteristic
50	#11719924	—	4	2	no	—
51	#11719921	—	2	2	no	—
52	#11719921	—	2	2	no	—
53	#11719921	—	2	2	no	—
54	#11719921	—	2	2	no	—
55	C₄⋊Dic₅₃	—	2	2	no	—
56	C₄⋊Dic₅₃	D₄	1	2	yes	—
57	C₄⋊Dic₅₃	C₂³	1	2	yes	—
58	#11719919	—	2	2	no	—
59	C₄×Dic₅₃	—	2	2	no	—
60	C₄×Dic₅₃	C₂³	1	2	yes	—
61	C₂×#88322	—	4	1	no	—
62	C₂×#88322	C₂³	1	2	yes	—
63	D₄₂₄	—	4	1	no	—
64	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×D₅₃	—	2	2	no	—
66	Q₈×C₅₃	—	2	2	no	—
67	Q₈×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	2	1	no	—
69	C₂×C₄×C₅₃	—	2	1	no	—
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
74	C₂²⋊D₅₃	—	8	2	no	—
75	C₂×Dic₅₃	—	2	2	no	—
76	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
77	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
78	D₂₁₂	—	2	2	no	—
79	D₂₁₂	—	2	2	no	—
80	C₄×D₅₃	—	8	2	no	—
81	Dic₁₀₆	—	8	2	no	—
82	C₈×C₅₃	—	2	1	no	—
83	C₈×C₅₃	—	2	1	no	—
84	#88322	—	2	2	no	—
85	C₂²×C₅₃	—	2	1	no	—
86	C₂²×C₅₃	—	2	2	no	—
87	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
88	C₂×D₅₃	—	4	2	no	—
89	C₂×D₅₃	—	4	2	no	—
90	C₂×D₅₃	—	4	2	no	—
91	C₄×C₅₃	—	2	1	no	—
92	C₄×C₅₃	—	4	2	no	—
93	C₄×C₅₃	—	4	2	no	—
94	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
95	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
96	Dic₅₃	—	2	2	no	—
97	Dic₅₃	—	2	2	no	—
98	Dic₅₃	—	4	2	no	—
99	#3263	—	53	1	no	maximal, Sylow
100	C₂×C₅₃	—	4	1	no	—
101	C₂×C₅₃	#537	1	1	yes	characteristic
102	C₂×C₅₃	#577	1	2	yes	—
103	D₅₃	—	8	2	no	—
104	#542	—	53	2	no	—
105	#476	—	53	2	no	—
106	#475	—	53	2	no	—
107	#470	D₅₃	1	1	yes	characteristic, complemented
108	#469	—	53	1	no	—
109	#465	—	53	2	no	—
110	#464	—	53	2	no	—
111	#463	—	53	2	no	—
112	#424	—	53	1	no	—
113	C₅₃	#3263	1	1	yes	characteristic, complemented, Sylow
114	C₂×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
115	C₂×SD₁₆	—	53	2	no	—
116	C₂×D₈	—	53	1	no	—
117	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
118	C₂²×C₈	—	53	1	no	—
119	C₄⋊Q₈	—	53	2	no	—
120	#126	—	106	2	no	—
121	#124	—	53	2	no	—
122	#124	—	106	2	no	—
123	#123	—	106	2	no	—
124	C₂²⋊Q₈	—	106	2	no	—
125	C₂²⋊Q₈	C₂×D₅₃	1	2	yes	—
126	#121	—	53	2	no	—
127	C₄×Q₈	—	53	2	no	—
128	C₄×D₄	—	53	2	no	—
129	#117	—	53	1	no	—
130	#106	—	53	1	no	—
131	#106	—	53	2	no	—
132	#106	C₂×D₅₃	1	1	yes	characteristic
133	C₄⋊C₈	—	53	2	no	—
134	C₄⋊C₈	—	53	2	no	—
135	#103	—	53	2	no	—
136	#103	C₂×D₅₃	1	2	yes	—
137	#102	—	53	2	no	—
138	#102	—	53	2	no	—
139	C₂×Q₈	—	106	2	no	—
140	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
141	C₂×D₄	—	53	2	no	—
142	C₂×D₄	—	106	2	no	—
143	C₂²×C₄	—	106	2	no	—
144	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
145	Q₁₆	—	4	1	no	—
146	SD₁₆	—	212	2	no	—
147	D₈	—	212	1	no	—
148	M₁₆	—	4	1	no	—
149	C₂×C₈	—	53	2	no	—
150	C₂×C₈	—	212	1	no	—
151	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
152	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	53	2	no	—
155	C₄⋊C₄	—	53	2	no	—
156	C₄⋊C₄	—	106	2	no	—
157	C₄⋊C₄	—	106	2	no	—
158	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
159	C₂²⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	53	1	no	—
161	C₂²⋊C₄	—	53	1	no	—
162	C₂²⋊C₄	—	106	2	no	—
163	C₂²⋊C₄	—	106	2	no	—
164	C₂²⋊C₄	—	106	2	no	—
165	C₂²⋊C₄	—	106	2	no	—
166	C₄²	—	53	2	no	—
167	C₄²	—	106	2	no	—
168	C₂³	—	106	2	no	—
169	C₂³	D₄×D₅₃	1	1	yes	characteristic
170	Q₈	—	2	2	no	—
171	Q₈	—	2	2	no	—
172	Q₈	—	424	2	no	—
173	D₄	—	106	2	no	—
174	D₄	—	106	2	no	—
175	D₄	—	424	2	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	53	2	no	—
181	C₂×C₄	—	53	2	no	—
182	C₂×C₄	—	106	2	no	—
183	C₂×C₄	—	424	2	no	—
184	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
185	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
186	C₈	—	2	1	no	—
187	C₈	—	2	1	no	—
188	C₈	—	106	2	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	212	2	no	—
192	C₂²	—	212	2	no	—
193	C₂²	—	212	2	no	—
194	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	4	2	no	—
197	C₄	—	4	2	no	—
198	C₄	—	106	2	no	—
199	C₄	—	106	2	no	—
200	C₄	—	212	2	no	—
201	C₄	#13133262	1	1	yes	characteristic
202	C₄	#13133264	1	1	yes	characteristic
203	C₂	—	4	1	no	—
204	C₂	—	424	2	no	—
205	C₂	#16811200	1	1	yes	characteristic
206	C₂	#16811432	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811246

C₂

yes

maximal, complemented

#16810912

C₂

yes

maximal, complemented

#16810906

C₂

yes

maximal, complemented

#16810876

C₂

yes

maximal, characteristic

#16810854

C₂

yes

maximal, complemented

#16810847

C₂

yes

maximal, complemented

#16810841

C₂

yes

maximal, complemented

#16810651

C₂

yes

maximal, characteristic, complemented

C₅₃×#470

C₂

yes

maximal, characteristic, complemented

C₂×#11719924

C₂²

yes

—

C₂×D₄₂₄

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

#13133216

C₂²

yes

—

#13133212

—

#13133206

C₂²

yes

—

—

—

—

C₂²

yes

—

Q₈×Dic₅₃

C₂²

yes

—

#13133179

C₂²

yes

—

#13133176

C₂²

yes

characteristic

#13133149

C₂²

yes

characteristic

#13133148

C₂²

yes

—

#13133146

C₂²

yes

—

#13133146

C₂²

yes

—

#13133141

C₂²

yes

—

#13133139

C₂²

yes

—

#13133137

C₂²

yes

—

C₂×Q₁₆×C₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂²⋊Q₈×C₅₃

C₂²

yes

—

#106×C₅₃

C₂²

yes

characteristic

#103×C₅₃

C₂²

yes

—

C₂×Q₈×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₂²⋊D₅₃

—

C₂×D₂₁₂

C₂³

yes

—

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

—

Q₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₂²⋊C₄×C₅₃

—

C₂²⋊Dic₅₃

D₄

yes

characteristic

C₂²⋊Dic₅₃

D₄

yes

—

—

—

—

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂³

yes

—

C₂×#88322

—

C₂×#88322

C₂³

yes

—

D₄₂₄

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂²×D₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

C₂×D₄

yes

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

H(5,𝔽₂)

yes

characteristic

C₄×C₅₃

2₋¹⁺⁴

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#3263

—

maximal, Sylow

100

C₂×C₅₃

—

101

C₂×C₅₃

#537

yes

characteristic

102

C₂×C₅₃

#577

yes

—

103

D₅₃

—

104

#542

—

105

#476

—

106

#475

—

107

#470

D₅₃

yes

characteristic, complemented

108

#469

—

109

#465

—

110

#464

—

111

#463

—

112

#424

—

113

C₅₃

#3263

yes

characteristic, complemented, Sylow

114

C₂×Q₁₆

C₂×D₅₃

yes

characteristic

115

C₂×SD₁₆

—

116

C₂×D₈

—

117

C₂×M₁₆

C₂×D₅₃

yes

characteristic

118

C₂²×C₈

—

119

C₄⋊Q₈

—

120

#126

—

106

—

121

#124

—

122

#124

—

106

—

123

#123

—

106

—

124

C₂²⋊Q₈

—

106

—

125

C₂²⋊Q₈

C₂×D₅₃

yes

—

126

#121

—

127

C₄×Q₈

—

128

C₄×D₄

—

129

#117

—

130

#106

—

131

#106

—

132

#106

C₂×D₅₃

yes

characteristic

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

#103

—

136

#103

C₂×D₅₃

yes

—

137

#102

—

138

#102

—

139

C₂×Q₈

—

106

—

140

C₂×Q₈

C₂²×D₅₃

yes

—

141

C₂×D₄

—

142

C₂×D₄

—

106

—

143

C₂²×C₄

—

106

—

144

C₂²×C₄

C₂²×D₅₃

yes

characteristic

145

Q₁₆

—

146

SD₁₆

—

212

—

147

D₈

—

212

—

148

M₁₆

—

149

C₂×C₈

—

150

C₂×C₈

—

212

—

151

C₂×C₈

C₂²×D₅₃

yes

characteristic

152

C₂×C₈

C₂²×D₅₃

yes

characteristic

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

106

—

157

C₄⋊C₄

—

106

—

158

C₄⋊C₄

C₂²×D₅₃

yes

—

159

C₂²⋊C₄

—

160

C₂²⋊C₄

—

161

C₂²⋊C₄

—

162

C₂²⋊C₄

—

106

—

163

C₂²⋊C₄

—

106

—

164

C₂²⋊C₄

—

106

—

165

C₂²⋊C₄

—

106

—

166

C₄²

—

167

C₄²

—

106

—

168

C₂³

—

106

—

169

C₂³

D₄×D₅₃

yes

characteristic

170

Q₈

—

171

Q₈

—

172

Q₈

—

424

—

173

D₄

—

106

—

174

D₄

—

106

—

175

D₄

—

424

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

106

—

183

C₂×C₄

—

424

—

184

C₂×C₄

D₄×D₅₃

yes

characteristic

185

C₂×C₄

C₂³×D₅₃

yes

characteristic

186

C₈

—

187

C₈

—

188

C₈

—

106

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

212

—

192

C₂²

—

212

—

193

C₂²

—

212

—

194

C₂²

C₂×D₄×D₅₃

yes

characteristic

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

106

—

199

C₄

—

106

—

200

C₄

—

212

—

201

C₄

#13133262

yes

characteristic

202

C₄

#13133264

yes

characteristic

203

C₂

—

204

C₂

—

424

—

205

C₂

#16811200

yes

characteristic

206

C₂

#16811432

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#470	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#470	D₅₃
hypocenter	C₅₃	#3263
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #470

subgroup

G/H

center

C₂²

C₂×D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#470

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #470

N	H ≅ G/N	type
1. #16811246	C₂	semidirect
2. #16810912	C₂	semidirect
3. #16810906	C₂	semidirect
5. #16810854	C₂	semidirect
6. #16810847	C₂	semidirect
7. #16810841	C₂	semidirect
8. #16810651	C₂	semidirect
9. C₅₃×#470	C₂	semidirect
107. #470	D₅₃	semidirect
113. C₅₃	#3263	semidirect

H ≅ G/N

type

1. #16811246

C₂

semidirect

2. #16810912

C₂

semidirect

3. #16810906

C₂

semidirect

5. #16810854

C₂

semidirect

6. #16810847

C₂

semidirect

7. #16810841

C₂

semidirect

8. #16810651

C₂

semidirect

9. C₅₃×#470

C₂

semidirect

107. #470

D₅₃

semidirect

113. C₅₃

#3263

semidirect

Subgroups — #20286181

Conjugacy classes of subgroups

Complements