Group #20286170 — subgroups

Subgroups — #20286170

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	#16811240	C₂	1	2	yes	maximal, complemented
2	#16810917	C₂	1	2	yes	maximal, complemented
3	#16810901	C₂	1	2	yes	maximal, complemented
4	#16810867	C₂	1	2	yes	maximal, complemented
5	#16810864	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810844	C₂	1	2	yes	maximal, complemented
7	#16810836	C₂	1	2	yes	maximal, complemented
8	#16810649	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#466	C₂	1	1	yes	maximal, characteristic
10	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic
11	C₂×#11719923	C₂²	1	2	yes	—
12	C₂×#11719917	C₂²	1	1	yes	characteristic
13	#13133216	C₂²	1	2	yes	—
14	#13133210	—	2	2	no	—
15	#13133204	—	2	2	no	—
16	#13133204	C₂²	1	2	yes	—
17	#13133198	—	2	2	no	—
18	#13133196	C₂²	1	2	yes	—
19	#13133194	—	2	2	no	—
20	#13133182	C₂²	1	2	yes	—
21	D₄×Dic₅₃	C₂²	1	2	yes	—
22	#13133176	C₂²	1	1	yes	characteristic
23	#13133151	C₂²	1	1	yes	characteristic
24	#13133150	C₂²	1	2	yes	—
25	#13133146	C₂²	1	2	yes	—
26	#13133146	C₂²	1	2	yes	—
27	#13133142	C₂²	1	2	yes	—
28	#13133140	C₂²	1	2	yes	—
29	#13133138	C₂²	1	2	yes	—
30	C₂×D₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
32	#121×C₅₃	C₂²	1	2	yes	—
33	#107×C₅₃	C₂²	1	1	yes	characteristic
34	#102×C₅₃	C₂²	1	2	yes	—
35	C₂×D₄×C₅₃	—	2	2	no	—
36	C₂×D₄×C₅₃	C₂³	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
38	C₂²×Dic₅₃	—	2	2	no	—
39	C₂×Dic₁₀₆	—	2	2	no	—
40	C₂×Dic₁₀₆	C₂³	1	2	yes	—
41	D₈×C₅₃	—	4	1	no	—
42	C₂×C₈×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	C₂³	1	2	yes	—
46	C₂²⋊C₄×C₅₃	—	2	2	no	—
47	C₂²⋊Dic₅₃	—	2	2	no	—
48	C₂²⋊Dic₅₃	—	2	2	no	—
49	C₂²⋊Dic₅₃	—	2	2	no	—
50	C₂²⋊Dic₅₃	—	2	2	no	—
51	C₂²⋊Dic₅₃	D₄	1	2	yes	—
52	#11719923	—	4	2	no	—
53	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
54	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
55	C₄⋊Dic₅₃	C₂³	1	2	yes	—
56	#11719919	—	2	2	no	—
57	#11719919	—	2	2	no	—
58	#11719919	—	2	2	no	—
59	C₄×Dic₅₃	—	2	2	no	—
60	C₄×Dic₅₃	C₂³	1	2	yes	—
61	#11719917	—	4	1	no	—
62	C₂×#88322	C₂³	1	2	yes	—
63	Dic₂₁₂	—	4	1	no	—
64	C₂³×C₅₃	—	2	2	no	—
65	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
66	D₄×C₅₃	—	2	2	no	—
67	D₄×C₅₃	—	2	2	no	—
68	D₄×C₅₃	—	8	2	no	—
69	C₂×C₄×C₅₃	—	2	1	no	—
70	C₂×C₄×C₅₃	—	2	1	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	C₂×Dic₅₃	—	2	2	no	—
75	C₂×Dic₅₃	—	2	2	no	—
76	C₂×Dic₅₃	—	8	2	no	—
77	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
78	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
79	Dic₁₀₆	—	2	2	no	—
80	Dic₁₀₆	—	2	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	1	no	—
87	C₂²×C₅₃	—	2	1	no	—
88	C₂²×C₅₃	—	4	2	no	—
89	C₂²×C₅₃	—	4	2	no	—
90	C₂²×C₅₃	—	4	2	no	—
91	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
92	C₄×C₅₃	—	2	1	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	Dic₅₃	—	4	2	no	—
100	#3262	—	53	1	no	maximal, Sylow
101	C₂×C₅₃	—	4	1	no	—
102	C₂×C₅₃	—	8	2	no	—
103	C₂×C₅₃	#537	1	1	yes	characteristic
104	C₂×C₅₃	#576	1	1	yes	characteristic
105	C₂×C₅₃	#578	1	1	yes	characteristic
106	#542	—	53	2	no	—
107	#477	—	53	2	no	—
108	#474	—	53	2	no	—
109	#468	—	53	2	no	—
110	#467	—	53	1	no	—
111	#466	D₅₃	1	1	yes	characteristic
112	#464	—	53	2	no	—
113	#463	—	53	2	no	—
114	#424	—	53	1	no	—
115	C₅₃	#3262	1	1	yes	characteristic, complemented, Sylow
116	C₂×Q₁₆	—	53	1	no	—
117	C₂×SD₁₆	—	53	2	no	—
118	C₂×D₈	C₂×D₅₃	1	1	yes	characteristic
119	C₂×M₁₆	—	53	1	no	—
120	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
121	C₄⋊Q₈	—	53	2	no	—
122	#126	—	106	2	no	—
123	#124	—	53	2	no	—
124	#124	—	106	2	no	—
125	#123	—	106	2	no	—
126	C₂²⋊Q₈	—	53	2	no	—
127	C₂²⋊Q₈	—	106	2	no	—
128	#121	C₂×D₅₃	1	2	yes	—
129	C₄×Q₈	—	53	2	no	—
130	C₄×D₄	—	53	2	no	—
131	#117	—	53	1	no	—
132	#107	—	53	1	no	—
133	#107	—	53	2	no	—
134	#107	C₂×D₅₃	1	1	yes	characteristic
135	C₄⋊C₈	—	53	2	no	—
136	C₄⋊C₈	—	53	2	no	—
137	#103	—	53	2	no	—
138	#103	—	53	2	no	—
139	#102	—	53	2	no	—
140	#102	C₂×D₅₃	1	2	yes	—
141	C₂×Q₈	—	53	2	no	—
142	C₂×Q₈	—	106	2	no	—
143	C₂×D₄	—	2	2	no	—
144	C₂×D₄	C₂²×D₅₃	1	2	yes	—
145	C₂²×C₄	—	106	2	no	—
146	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
147	Q₁₆	—	212	1	no	—
148	SD₁₆	—	212	2	no	—
149	D₈	—	4	1	no	—
150	M₁₆	—	212	1	no	—
151	C₂×C₈	—	4	1	no	—
152	C₂×C₈	—	53	2	no	—
153	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
154	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
155	C₄⋊C₄	—	53	1	no	—
156	C₄⋊C₄	—	53	1	no	—
157	C₄⋊C₄	—	53	2	no	—
158	C₄⋊C₄	—	106	2	no	—
159	C₄⋊C₄	—	106	2	no	—
160	C₄⋊C₄	—	106	2	no	—
161	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
162	C₂²⋊C₄	—	2	2	no	—
163	C₂²⋊C₄	—	53	2	no	—
164	C₂²⋊C₄	—	106	2	no	—
165	C₂²⋊C₄	—	106	2	no	—
166	C₂²⋊C₄	—	106	2	no	—
167	C₂²⋊C₄	—	106	2	no	—
168	C₄²	—	53	2	no	—
169	C₄²	—	106	2	no	—
170	C₂³	—	2	2	no	—
171	C₂³	D₄×D₅₃	1	1	yes	characteristic
172	Q₈	—	106	2	no	—
173	Q₈	—	106	2	no	—
174	Q₈	—	424	2	no	—
175	D₄	—	2	2	no	—
176	D₄	—	2	2	no	—
177	D₄	—	8	2	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	53	2	no	—
182	C₂×C₄	—	53	2	no	—
183	C₂×C₄	—	106	2	no	—
184	C₂×C₄	—	106	2	no	—
185	C₂×C₄	—	424	2	no	—
186	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
187	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	—	106	2	no	—
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	4	2	no	—
195	C₂²	—	4	2	no	—
196	C₂²	—	4	2	no	—
197	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
198	C₄	—	2	1	no	—
199	C₄	—	4	2	no	—
200	C₄	—	106	2	no	—
201	C₄	—	106	2	no	—
202	C₄	—	212	2	no	—
203	C₄	—	212	2	no	—
204	C₄	#13133262	1	1	yes	characteristic
205	C₄	#13133264	1	1	yes	characteristic
206	C₂	—	4	1	no	—
207	C₂	—	8	2	no	—
208	C₂	#16811200	1	1	yes	characteristic
209	C₂	#16811424	1	1	yes	characteristic
210	C₂	#16811437	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811240

C₂

yes

maximal, complemented

#16810917

C₂

yes

maximal, complemented

#16810901

C₂

yes

maximal, complemented

#16810867

C₂

yes

maximal, complemented

#16810864

C₂

yes

maximal, characteristic, complemented

#16810844

C₂

yes

maximal, complemented

#16810836

C₂

yes

maximal, complemented

#16810649

C₂

yes

maximal, characteristic, complemented

C₅₃×#466

C₂

yes

maximal, characteristic

C₂×Dic₂₁₂

C₂²

yes

characteristic

C₂×#11719923

C₂²

yes

—

C₂×#11719917

C₂²

yes

characteristic

#13133216

C₂²

yes

—

#13133210

—

#13133204

—

#13133204

C₂²

yes

—

#13133198

—

#13133196

C₂²

yes

—

#13133194

—

#13133182

C₂²

yes

—

D₄×Dic₅₃

C₂²

yes

—

#13133176

C₂²

yes

characteristic

#13133151

C₂²

yes

characteristic

#13133150

C₂²

yes

—

#13133146

C₂²

yes

—

#13133146

C₂²

yes

—

#13133142

C₂²

yes

—

#13133140

C₂²

yes

—

#13133138

C₂²

yes

—

C₂×D₈×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#121×C₅₃

C₂²

yes

—

#107×C₅₃

C₂²

yes

characteristic

#102×C₅₃

C₂²

yes

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂³

yes

—

D₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

C₂³

yes

—

C₂²⋊C₄×C₅₃

—

C₂²⋊Dic₅₃

—

C₂²⋊Dic₅₃

—

C₂²⋊Dic₅₃

—

C₂²⋊Dic₅₃

—

C₂²⋊Dic₅₃

D₄

yes

—

#11719923

—

C₄⋊Dic₅₃

D₄

yes

characteristic

C₄⋊Dic₅₃

D₄

yes

characteristic

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂³

yes

—

#11719917

—

C₂×#88322

C₂³

yes

—

Dic₂₁₂

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

C₂×D₄

yes

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

H(5,𝔽₂)

yes

characteristic

C₄×C₅₃

2₋¹⁺⁴

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

100

#3262

—

maximal, Sylow

101

C₂×C₅₃

—

102

C₂×C₅₃

—

103

C₂×C₅₃

#537

yes

characteristic

104

C₂×C₅₃

#576

yes

characteristic

105

C₂×C₅₃

#578

yes

characteristic

106

#542

—

107

#477

—

108

#474

—

109

#468

—

110

#467

—

111

#466

D₅₃

yes

characteristic

112

#464

—

113

#463

—

114

#424

—

115

C₅₃

#3262

yes

characteristic, complemented, Sylow

116

C₂×Q₁₆

—

117

C₂×SD₁₆

—

118

C₂×D₈

C₂×D₅₃

yes

characteristic

119

C₂×M₁₆

—

120

C₂²×C₈

C₂×D₅₃

yes

characteristic

121

C₄⋊Q₈

—

122

#126

—

106

—

123

#124

—

124

#124

—

106

—

125

#123

—

106

—

126

C₂²⋊Q₈

—

127

C₂²⋊Q₈

—

106

—

128

#121

C₂×D₅₃

yes

—

129

C₄×Q₈

—

130

C₄×D₄

—

131

#117

—

132

#107

—

133

#107

—

134

#107

C₂×D₅₃

yes

characteristic

135

C₄⋊C₈

—

136

C₄⋊C₈

—

137

#103

—

138

#103

—

139

#102

—

140

#102

C₂×D₅₃

yes

—

141

C₂×Q₈

—

142

C₂×Q₈

—

106

—

143

C₂×D₄

—

144

C₂×D₄

C₂²×D₅₃

yes

—

145

C₂²×C₄

—

106

—

146

C₂²×C₄

C₂²×D₅₃

yes

characteristic

147

Q₁₆

—

212

—

148

SD₁₆

—

212

—

149

D₈

—

150

M₁₆

—

212

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

C₂²×D₅₃

yes

characteristic

154

C₂×C₈

C₂²×D₅₃

yes

characteristic

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

106

—

159

C₄⋊C₄

—

106

—

160

C₄⋊C₄

—

106

—

161

C₄⋊C₄

C₂²×D₅₃

yes

—

162

C₂²⋊C₄

—

163

C₂²⋊C₄

—

164

C₂²⋊C₄

—

106

—

165

C₂²⋊C₄

—

106

—

166

C₂²⋊C₄

—

106

—

167

C₂²⋊C₄

—

106

—

168

C₄²

—

169

C₄²

—

106

—

170

C₂³

—

171

C₂³

D₄×D₅₃

yes

characteristic

172

Q₈

—

106

—

173

Q₈

—

106

—

174

Q₈

—

424

—

175

D₄

—

176

D₄

—

177

D₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

106

—

184

C₂×C₄

—

106

—

185

C₂×C₄

—

424

—

186

C₂×C₄

D₄×D₅₃

yes

characteristic

187

C₂×C₄

C₂³×D₅₃

yes

characteristic

188

C₈

—

189

C₈

—

190

C₈

—

106

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

194

C₂²

—

195

C₂²

—

196

C₂²

—

197

C₂²

C₂×D₄×D₅₃

yes

characteristic

198

C₄

—

199

C₄

—

200

C₄

—

106

—

201

C₄

—

106

—

202

C₄

—

212

—

203

C₄

—

212

—

204

C₄

#13133262

yes

characteristic

205

C₄

#13133264

yes

characteristic

206

C₂

—

207

C₂

—

208

C₂

#16811200

yes

characteristic

209

C₂

#16811424

yes

characteristic

210

C₂

#16811437

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#466	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#466	D₅₃
hypocenter	C₅₃	#3262
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #466

subgroup

G/H

center

C₂²

C₂×D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#466

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #466

N	H ≅ G/N	type
1. #16811240	C₂	semidirect
2. #16810917	C₂	semidirect
3. #16810901	C₂	semidirect
4. #16810867	C₂	semidirect
5. #16810864	C₂	semidirect
6. #16810844	C₂	semidirect
7. #16810836	C₂	semidirect
8. #16810649	C₂	semidirect
115. C₅₃	#3262	semidirect

H ≅ G/N

type

1. #16811240

C₂

semidirect

2. #16810917

C₂

semidirect

3. #16810901

C₂

semidirect

4. #16810867

C₂

semidirect

5. #16810864

C₂

semidirect

6. #16810844

C₂

semidirect

7. #16810836

C₂

semidirect

8. #16810649

C₂

semidirect

115. C₅₃

#3262

semidirect

Subgroups — #20286170

Conjugacy classes of subgroups

Complements