Group #20286176 — subgroups

Subgroups — #20286176

Order 6784.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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	#16811244	C₂	1	2	yes	maximal, complemented
2	#16810919	C₂	1	2	yes	maximal, complemented
3	#16810904	C₂	1	2	yes	maximal, complemented
4	#16810869	C₂	1	2	yes	maximal, complemented
5	#16810864	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810859	C₂	1	1	yes	maximal, characteristic
7	#16810846	C₂	1	2	yes	maximal, complemented
8	#16810839	C₂	1	2	yes	maximal, complemented
9	C₅₃×#424	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic
11	C₂×#11719913	C₂²	1	2	yes	—
12	C₂×D₄₂₄	C₂²	1	1	yes	characteristic
13	#13133217	C₂²	1	2	yes	—
14	#13133212	—	2	2	no	—
15	#13133205	C₂²	1	2	yes	—
16	#13133203	—	2	2	no	—
17	#13133199	—	2	2	no	—
18	#13133196	C₂²	1	2	yes	—
19	#13133195	—	2	2	no	—
20	#13133191	C₂²	1	2	yes	—
21	C₄×Dic₁₀₆	C₂²	1	2	yes	—
22	C₄×D₂₁₂	C₂²	1	2	yes	—
23	#13133151	C₂²	1	1	yes	characteristic
24	#13133151	C₂²	1	1	yes	characteristic
25	#13133151	C₂²	1	2	yes	—
26	#13133142	C₂²	1	2	yes	—
27	#13133142	C₂²	1	2	yes	—
28	#13133139	C₂²	1	2	yes	—
29	#13133139	C₂²	1	2	yes	—
30	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
32	#117×C₅₃	C₂²	1	1	yes	characteristic
33	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
34	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
35	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
36	C₂×C₂²⋊D₅₃	—	2	2	no	—
37	C₂×D₂₁₂	C₂³	1	2	yes	—
38	C₂×C₄×D₅₃	—	2	2	no	—
39	C₂×Dic₁₀₆	—	2	2	no	—
40	C₂×Dic₁₀₆	C₂³	1	2	yes	—
41	M₁₆×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	C₂×C₈×C₅₃	C₂³	1	2	yes	—
46	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
47	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
48	C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
49	C₄²×C₅₃	C₂³	1	2	yes	—
50	C₂²⋊Dic₅₃	—	2	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	C₂³	1	2	yes	—
57	C₄⋊Dic₅₃	C₂³	1	2	yes	—
58	#11719919	—	2	2	no	—
59	#11719919	—	2	2	no	—
60	C₄×Dic₅₃	—	2	2	no	—
61	Dic₂₁₂	—	4	1	no	—
62	D₄₂₄	—	4	1	no	—
63	#11719913	—	4	2	no	—
64	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×D₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	2	1	no	—
67	C₂×C₄×C₅₃	—	2	1	no	—
68	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
72	C₂²⋊D₅₃	—	8	2	no	—
73	C₂×Dic₅₃	—	2	2	no	—
74	C₂×Dic₅₃	—	2	2	no	—
75	C₂×Dic₅₃	—	2	2	no	—
76	D₂₁₂	—	2	2	no	—
77	D₂₁₂	—	2	2	no	—
78	C₄×D₅₃	—	8	2	no	—
79	Dic₁₀₆	—	2	2	no	—
80	Dic₁₀₆	—	2	2	no	—
81	Dic₁₀₆	—	8	2	no	—
82	C₈×C₅₃	—	2	2	no	—
83	C₈×C₅₃	—	2	1	no	—
84	C₈×C₅₃	—	2	1	no	—
85	C₂²×C₅₃	—	2	1	no	—
86	C₂²×C₅₃	—	2	1	no	—
87	C₂²×C₅₃	—	2	1	no	—
88	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
89	C₂×D₅₃	—	4	2	no	—
90	C₂×D₅₃	—	4	2	no	—
91	C₂×D₅₃	—	4	2	no	—
92	C₄×C₅₃	—	2	1	no	—
93	C₄×C₅₃	—	2	2	no	—
94	C₄×C₅₃	—	2	2	no	—
95	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
96	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
97	Dic₅₃	—	4	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	#537	1	1	yes	characteristic
103	C₂×C₅₃	#576	1	1	yes	characteristic
104	C₂×C₅₃	#578	1	1	yes	characteristic
105	D₅₃	—	8	2	no	—
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	—	53	1	no	—
112	#464	—	53	2	no	—
113	#463	—	53	2	no	—
114	#424	D₅₃	1	1	yes	characteristic, complemented
115	C₅₃	#3262	1	1	yes	characteristic, complemented, Sylow
116	C₂×Q₁₆	—	53	1	no	—
117	C₂×SD₁₆	—	53	2	no	—
118	C₂×D₈	—	53	1	no	—
119	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
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	—	53	2	no	—
129	C₄×Q₈	—	53	2	no	—
130	C₄×D₄	—	53	2	no	—
131	#117	C₂×D₅₃	1	1	yes	characteristic
132	#107	—	53	1	no	—
133	#107	—	53	1	no	—
134	#107	—	53	2	no	—
135	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
136	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
137	#103	—	53	2	no	—
138	#103	—	53	2	no	—
139	#102	—	53	2	no	—
140	#102	—	53	2	no	—
141	C₂×Q₈	—	53	2	no	—
142	C₂×Q₈	—	106	2	no	—
143	C₂×D₄	—	53	2	no	—
144	C₂×D₄	—	106	2	no	—
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	—	212	1	no	—
150	M₁₆	—	4	1	no	—
151	C₂×C₈	—	4	1	no	—
152	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
153	C₂×C₈	C₂²×D₅₃	1	2	yes	—
154	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
155	C₄⋊C₄	—	53	2	no	—
156	C₄⋊C₄	—	53	2	no	—
157	C₄⋊C₄	—	106	2	no	—
158	C₄⋊C₄	—	106	2	no	—
159	C₄⋊C₄	—	106	2	no	—
160	C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
161	C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
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	C₂²⋊C₄	—	106	2	no	—
167	C₂²⋊C₄	D₂₁₂	1	2	yes	—
168	C₄²	—	106	2	no	—
169	C₄²	C₂²×D₅₃	1	2	yes	—
170	C₂³	—	106	2	no	—
171	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
172	Q₈	—	106	2	no	—
173	Q₈	—	106	2	no	—
174	Q₈	—	424	2	no	—
175	D₄	—	106	2	no	—
176	D₄	—	106	2	no	—
177	D₄	—	424	2	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	106	2	no	—
181	C₂×C₄	—	106	2	no	—
182	C₂×C₄	—	106	2	no	—
183	C₂×C₄	—	424	2	no	—
184	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
185	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
186	C₂×C₄	C₂×D₂₁₂	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	—	2	2	no	—
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	212	2	no	—
195	C₂²	—	212	2	no	—
196	C₂²	—	212	2	no	—
197	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	—	2	2	no	—
200	C₄	—	2	1	no	—
201	C₄	—	212	2	no	—
202	C₄	—	212	2	no	—
203	C₄	—	212	2	no	—
204	C₄	#13133262	1	1	yes	characteristic
205	C₄	#13133265	1	1	yes	characteristic
206	C₂	—	4	1	no	—
207	C₂	—	424	2	no	—
208	C₂	#16811204	1	1	yes	characteristic
209	C₂	#16811428	1	1	yes	characteristic
210	C₂	#16811440	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811244

C₂

yes

maximal, complemented

#16810919

C₂

yes

maximal, complemented

#16810904

C₂

yes

maximal, complemented

#16810869

C₂

yes

maximal, complemented

#16810864

C₂

yes

maximal, characteristic, complemented

#16810859

C₂

yes

maximal, characteristic

#16810846

C₂

yes

maximal, complemented

#16810839

C₂

yes

maximal, complemented

C₅₃×#424

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₁₂

C₂²

yes

characteristic

C₂×#11719913

C₂²

yes

—

C₂×D₄₂₄

C₂²

yes

characteristic

#13133217

C₂²

yes

—

#13133212

—

#13133205

C₂²

yes

—

#13133203

—

#13133199

—

#13133196

C₂²

yes

—

#13133195

—

#13133191

C₂²

yes

—

C₄×Dic₁₀₆

C₂²

yes

—

C₄×D₂₁₂

C₂²

yes

—

#13133151

C₂²

yes

characteristic

#13133151

C₂²

yes

characteristic

#13133151

C₂²

yes

—

#13133142

C₂²

yes

—

#13133142

C₂²

yes

—

#13133139

C₂²

yes

—

#13133139

C₂²

yes

—

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#117×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

—

C₄⋊C₈×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₂²⋊D₅₃

—

C₂×D₂₁₂

C₂³

yes

—

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂³

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₂²⋊C₄×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

—

C₂²⋊Dic₅₃

—

—

—

—

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

Dic₂₁₂

—

D₄₂₄

—

#11719913

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₂²×C₅₃

—

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

—

100

#3262

—

maximal, Sylow

101

C₂×C₅₃

—

102

C₂×C₅₃

#537

yes

characteristic

103

C₂×C₅₃

#576

yes

characteristic

104

C₂×C₅₃

#578

yes

characteristic

105

D₅₃

—

106

#542

—

107

#477

—

108

#474

—

109

#468

—

110

#467

—

111

#466

—

112

#464

—

113

#463

—

114

#424

D₅₃

yes

characteristic, complemented

115

C₅₃

#3262

yes

characteristic, complemented, Sylow

116

C₂×Q₁₆

—

117

C₂×SD₁₆

—

118

C₂×D₈

—

119

C₂×M₁₆

C₂×D₅₃

yes

characteristic

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

—

129

C₄×Q₈

—

130

C₄×D₄

—

131

#117

C₂×D₅₃

yes

characteristic

132

#107

—

133

#107

—

134

#107

—

135

C₄⋊C₈

C₂×D₅₃

yes

—

136

C₄⋊C₈

C₂×D₅₃

yes

—

137

#103

—

138

#103

—

139

#102

—

140

#102

—

141

C₂×Q₈

—

142

C₂×Q₈

—

106

—

143

C₂×D₄

—

144

C₂×D₄

—

106

—

145

C₂²×C₄

—

106

—

146

C₂²×C₄

C₂²×D₅₃

yes

characteristic

147

Q₁₆

—

212

—

148

SD₁₆

—

212

—

149

D₈

—

212

—

150

M₁₆

—

151

C₂×C₈

—

152

C₂×C₈

C₂²×D₅₃

yes

characteristic

153

C₂×C₈

C₂²×D₅₃

yes

—

154

C₂×C₈

C₂²×D₅₃

yes

characteristic

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

106

—

158

C₄⋊C₄

—

106

—

159

C₄⋊C₄

—

106

—

160

C₄⋊C₄

D₂₁₂

yes

characteristic

161

C₄⋊C₄

D₂₁₂

yes

characteristic

162

C₂²⋊C₄

—

106

—

163

C₂²⋊C₄

—

106

—

164

C₂²⋊C₄

—

106

—

165

C₂²⋊C₄

—

106

—

166

C₂²⋊C₄

—

106

—

167

C₂²⋊C₄

D₂₁₂

yes

—

168

C₄²

—

106

—

169

C₄²

C₂²×D₅₃

yes

—

170

C₂³

—

106

—

171

C₂³

C₂×D₂₁₂

yes

characteristic

172

Q₈

—

106

—

173

Q₈

—

106

—

174

Q₈

—

424

—

175

D₄

—

106

—

176

D₄

—

106

—

177

D₄

—

424

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

106

—

181

C₂×C₄

—

106

—

182

C₂×C₄

—

106

—

183

C₂×C₄

—

424

—

184

C₂×C₄

C₂×D₂₁₂

yes

—

185

C₂×C₄

C₂×D₂₁₂

yes

—

186

C₂×C₄

C₂×D₂₁₂

yes

characteristic

187

C₂×C₄

C₂³×D₅₃

yes

characteristic

188

C₈

—

189

C₈

—

190

C₈

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

194

C₂²

—

212

—

195

C₂²

—

212

—

196

C₂²

—

212

—

197

C₂²

C₂²×D₂₁₂

yes

characteristic

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

—

212

—

202

C₄

—

212

—

203

C₄

—

212

—

204

C₄

#13133262

yes

characteristic

205

C₄

#13133265

yes

characteristic

206

C₂

—

207

C₂

—

424

—

208

C₂

#16811204

yes

characteristic

209

C₂

#16811428

yes

characteristic

210

C₂

#16811440

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#424	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#424	D₅₃
hypocenter	C₅₃	#3262
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #424

subgroup

G/H

center

C₂²

C₂²×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#424

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ #117 ◁ #424

N	H ≅ G/N	type
1. #16811244	C₂	semidirect
2. #16810919	C₂	semidirect
3. #16810904	C₂	semidirect
4. #16810869	C₂	semidirect
5. #16810864	C₂	semidirect
7. #16810846	C₂	semidirect
8. #16810839	C₂	semidirect
9. C₅₃×#424	C₂	semidirect
114. #424	D₅₃	semidirect
115. C₅₃	#3262	semidirect

H ≅ G/N

type

1. #16811244

C₂

semidirect

2. #16810919

C₂

semidirect

3. #16810904

C₂

semidirect

4. #16810869

C₂

semidirect

5. #16810864

C₂

semidirect

7. #16810846

C₂

semidirect

8. #16810839

C₂

semidirect

9. C₅₃×#424

C₂

semidirect

114. #424

D₅₃

semidirect

115. C₅₃

#3262

semidirect

Subgroups — #20286176

Conjugacy classes of subgroups

Complements