Group #20287805 — subgroups

Subgroups — #20287805

Order 6784.

380 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 121 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	#16811358	C₂	1	2	yes	maximal, complemented
2	#16811012	C₂	1	2	yes	maximal, complemented
3	#16811007	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810846	C₂	1	2	yes	maximal, complemented
5	#16810839	C₂	1	2	yes	maximal, complemented
6	#16810819	C₂	1	2	yes	maximal, complemented
7	#16810725	C₂	1	2	yes	maximal, complemented
8	C₄×D₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic
11	C₂×#11719913	C₂²	1	2	yes	—
12	C₂×#11719913	C₂²	1	2	yes	—
13	C₂×D₄₂₄	C₂²	1	1	yes	characteristic
14	C₂×D₄₂₄	C₂²	1	2	yes	—
15	C₂×D₄₂₄	C₂²	1	1	yes	characteristic
16	#13133214	C₂²	1	2	yes	—
17	#13133206	—	2	2	no	—
18	#13133205	C₂²	1	2	yes	—
19	#13133205	C₂²	1	2	yes	—
20	#13133202	—	2	2	no	—
21	#13133202	—	2	2	no	—
22	C₄×Dic₁₀₆	C₂²	1	2	yes	—
23	C₄×D₂₁₂	C₂²	1	2	yes	—
24	C₄×D₂₁₂	C₂²	1	2	yes	—
25	#13133179	—	2	2	no	—
26	#13133151	C₂²	1	1	yes	characteristic
27	#13133142	C₂²	1	2	yes	—
28	#13133139	C₂²	1	2	yes	—
29	#13133139	C₂²	1	2	yes	—
30	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
32	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
33	#97×C₅₃	C₂²	1	2	yes	—
34	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
36	C₂×D₂₁₂	—	2	2	no	—
37	C₂×D₂₁₂	—	2	2	no	—
38	C₂×D₂₁₂	C₂³	1	2	yes	—
39	C₂×D₂₁₂	C₂³	1	2	yes	—
40	C₂×C₄×D₅₃	—	2	2	no	—
41	C₂×C₄×D₅₃	—	2	2	no	—
42	C₂×Dic₁₀₆	C₂³	1	2	yes	—
43	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₂³	1	2	yes	—
45	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
47	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
48	C₄²×C₅₃	C₂³	1	1	yes	characteristic
49	C₄²×C₅₃	C₂³	1	2	yes	—
50	#11719921	—	2	2	no	—
51	#11719921	—	2	2	no	—
52	#11719921	—	2	2	no	—
53	#11719921	—	2	2	no	—
54	#11719921	—	2	2	no	—
55	#11719921	—	2	2	no	—
56	C₄⋊Dic₅₃	C₂³	1	2	yes	—
57	#11719919	—	2	2	no	—
58	C₄×Dic₅₃	—	2	2	no	—
59	Dic₂₁₂	—	4	1	no	—
60	D₄₂₄	—	2	2	no	—
61	D₄₂₄	—	4	2	no	—
62	D₄₂₄	—	4	1	no	—
63	#11719913	—	2	4	no	—
64	#11719913	—	4	2	no	—
65	C₂²×D₅₃	—	2	2	no	—
66	C₂²×D₅₃	—	2	2	no	—
67	Q₈×C₅₃	—	4	1	no	—
68	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
73	C₂×Dic₅₃	—	2	2	no	—
74	C₂×Dic₅₃	—	2	2	no	—
75	D₂₁₂	—	2	2	no	—
76	D₂₁₂	—	2	2	no	—
77	D₂₁₂	—	2	2	no	—
78	D₂₁₂	—	2	2	no	—
79	D₂₁₂	—	4	4	no	—
80	D₂₁₂	—	8	2	no	—
81	C₄×D₅₃	—	8	2	no	—
82	C₄×D₅₃	—	8	2	no	—
83	Dic₁₀₆	—	2	2	no	—
84	Dic₁₀₆	—	2	2	no	—
85	C₈×C₅₃	—	2	2	no	—
86	C₈×C₅₃	—	2	1	no	—
87	C₈×C₅₃	#41	1	2	yes	—
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	C₂×D₅₃	—	4	2	no	—
93	C₂×D₅₃	—	4	2	no	—
94	C₂×D₅₃	—	4	2	no	—
95	C₄×C₅₃	—	2	2	no	—
96	C₄×C₅₃	—	2	1	no	—
97	C₄×C₅₃	—	2	1	no	—
98	C₄×C₅₃	—	2	2	no	—
99	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
100	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
101	Dic₅₃	—	4	2	no	—
102	Dic₅₃	—	4	2	no	—
103	#3403	—	53	1	no	maximal, Sylow
104	C₂×C₅₃	#550	1	1	yes	characteristic
105	C₂×C₅₃	#575	1	1	yes	characteristic
106	C₂×C₅₃	#576	1	1	yes	characteristic
107	D₅₃	—	8	2	no	—
108	D₅₃	—	8	2	no	—
109	#559	—	53	2	no	—
110	#496	—	53	2	no	—
111	#495	—	53	1	no	—
112	#464	—	53	2	no	—
113	#463	—	53	2	no	—
114	#459	—	53	2	no	—
115	#446	D₅₃	1	1	yes	characteristic, complemented
116	#440	—	53	2	no	—
117	C₄×D₈	—	53	1	no	—
118	C₅₃	#3403	1	1	yes	characteristic, complemented, Sylow
119	C₂×Q₁₆	—	53	1	no	—
120	C₂×SD₁₆	—	53	2	no	—
121	C₂×SD₁₆	—	53	2	no	—
122	C₂×D₈	—	53	1	no	—
123	C₂×D₈	—	53	1	no	—
124	C₂×D₈	—	53	2	no	—
125	#127	—	53	2	no	—
126	#124	—	53	2	no	—
127	#124	—	53	2	no	—
128	#124	—	106	2	no	—
129	#123	—	106	2	no	—
130	#123	—	106	2	no	—
131	C₄×Q₈	—	53	2	no	—
132	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
133	C₄×D₄	—	53	2	no	—
134	C₄×D₄	—	53	2	no	—
135	C₄×D₄	—	106	2	no	—
136	#107	—	53	1	no	—
137	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
138	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
139	#103	—	53	2	no	—
140	#102	—	53	2	no	—
141	#102	—	53	2	no	—
142	#97	C₂×D₅₃	1	2	yes	—
143	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
144	C₂×Q₈	—	53	2	no	—
145	C₂×Q₈	D₂₁₂	1	1	yes	characteristic
146	C₂×D₄	—	53	2	no	—
147	C₂×D₄	—	53	2	no	—
148	C₂×D₄	—	106	2	no	—
149	C₂×D₄	—	106	2	no	—
150	C₂²×C₄	—	106	2	no	—
151	C₂²×C₄	—	106	2	no	—
152	Q₁₆	—	212	1	no	—
153	SD₁₆	—	106	4	no	—
154	SD₁₆	—	212	2	no	—
155	D₈	—	106	2	no	—
156	D₈	—	212	2	no	—
157	D₈	—	212	1	no	—
158	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
159	C₂×C₈	C₂²×D₅₃	1	2	yes	—
160	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
161	C₄⋊C₄	—	53	2	no	—
162	C₄⋊C₄	—	106	2	no	—
163	C₄⋊C₄	D₂₁₂	1	2	yes	—
164	C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
165	C₂²⋊C₄	—	106	2	no	—
166	C₂²⋊C₄	—	106	2	no	—
167	C₂²⋊C₄	—	106	2	no	—
168	C₂²⋊C₄	—	106	2	no	—
169	C₂²⋊C₄	—	106	2	no	—
170	C₂²⋊C₄	—	106	2	no	—
171	C₄²	—	106	2	no	—
172	C₄²	C₂²×D₅₃	1	1	yes	characteristic
173	C₄²	C₂²×D₅₃	1	2	yes	—
174	C₂³	—	106	2	no	—
175	C₂³	—	106	2	no	—
176	Q₈	—	4	1	no	—
177	Q₈	—	106	2	no	—
178	Q₈	—	106	2	no	—
179	D₄	—	106	2	no	—
180	D₄	—	106	2	no	—
181	D₄	—	106	2	no	—
182	D₄	—	106	2	no	—
183	D₄	—	424	2	no	—
184	D₄	—	212	4	no	—
185	C₂×C₄	—	106	2	no	—
186	C₂×C₄	—	106	2	no	—
187	C₂×C₄	—	424	2	no	—
188	C₂×C₄	—	424	2	no	—
189	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
190	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
191	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
192	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
193	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
194	C₈	—	2	2	no	—
195	C₈	—	2	1	no	—
196	C₈	#11719949	1	2	yes	—
197	C₂²	—	212	2	no	—
198	C₂²	—	212	2	no	—
199	C₂²	—	212	2	no	—
200	C₂²	—	212	2	no	—
201	C₂²	—	212	2	no	—
202	C₂²	—	212	2	no	—
203	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
204	C₄	—	2	2	no	—
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	2	2	no	—
208	C₄	—	212	2	no	—
209	C₄	—	212	2	no	—
210	C₄	C₂×#11719949	1	1	yes	characteristic
211	C₄	#13133263	1	1	yes	characteristic
212	C₂	—	424	2	no	—
213	C₂	—	424	2	no	—
214	C₂	#16811295	1	1	yes	characteristic
215	C₂	#16811423	1	1	yes	characteristic
216	C₂	#16811428	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811358

C₂

yes

maximal, complemented

#16811012

C₂

yes

maximal, complemented

#16811007

C₂

yes

maximal, characteristic, complemented

#16810846

C₂

yes

maximal, complemented

#16810839

C₂

yes

maximal, complemented

#16810819

C₂

yes

maximal, complemented

#16810725

C₂

yes

maximal, complemented

C₄×D₄₂₄

C₂

yes

maximal, characteristic, complemented

C₅₃×#446

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₁₂

C₂²

yes

characteristic

C₂×#11719913

C₂²

yes

—

C₂×#11719913

C₂²

yes

—

C₂×D₄₂₄

C₂²

yes

characteristic

C₂×D₄₂₄

C₂²

yes

—

C₂×D₄₂₄

C₂²

yes

characteristic

#13133214

C₂²

yes

—

#13133206

—

#13133205

C₂²

yes

—

#13133205

C₂²

yes

—

#13133202

—

#13133202

—

C₄×Dic₁₀₆

C₂²

yes

—

C₄×D₂₁₂

C₂²

yes

—

C₄×D₂₁₂

C₂²

yes

—

#13133179

—

#13133151

C₂²

yes

characteristic

#13133142

C₂²

yes

—

#13133139

C₂²

yes

—

#13133139

C₂²

yes

—

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

—

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

#97×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

D₄

yes

characteristic

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

C₂³

yes

—

C₂×D₂₁₂

C₂³

yes

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

C₂³

yes

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

C₂³

yes

—

—

—

—

—

—

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

C₄×Dic₅₃

—

Dic₂₁₂

—

D₄₂₄

—

D₄₂₄

—

D₄₂₄

—

#11719913

—

#11719913

—

C₂²×D₅₃

—

C₂²×D₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#41

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#41

yes

characteristic

100

C₄×C₅₃

H(5,𝔽₂)

yes

characteristic

101

Dic₅₃

—

102

Dic₅₃

—

103

#3403

—

maximal, Sylow

104

C₂×C₅₃

#550

yes

characteristic

105

C₂×C₅₃

#575

yes

characteristic

106

C₂×C₅₃

#576

yes

characteristic

107

D₅₃

—

108

D₅₃

—

109

#559

—

110

#496

—

111

#495

—

112

#464

—

113

#463

—

114

#459

—

115

#446

D₅₃

yes

characteristic, complemented

116

#440

—

117

C₄×D₈

—

118

C₅₃

#3403

yes

characteristic, complemented, Sylow

119

C₂×Q₁₆

—

120

C₂×SD₁₆

—

121

C₂×SD₁₆

—

122

C₂×D₈

—

123

C₂×D₈

—

124

C₂×D₈

—

125

#127

—

126

#124

—

127

#124

—

128

#124

—

106

—

129

#123

—

106

—

130

#123

—

106

—

131

C₄×Q₈

—

132

C₄×Q₈

C₂×D₅₃

yes

characteristic

133

C₄×D₄

—

134

C₄×D₄

—

135

C₄×D₄

—

106

—

136

#107

—

137

C₄⋊C₈

C₂×D₅₃

yes

—

138

C₄⋊C₈

C₂×D₅₃

yes

characteristic

139

#103

—

140

#102

—

141

#102

—

142

#97

C₂×D₅₃

yes

—

143

C₄×C₈

C₂×D₅₃

yes

characteristic

144

C₂×Q₈

—

145

C₂×Q₈

D₂₁₂

yes

characteristic

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

106

—

149

C₂×D₄

—

106

—

150

C₂²×C₄

—

106

—

151

C₂²×C₄

—

106

—

152

Q₁₆

—

212

—

153

SD₁₆

—

106

—

154

SD₁₆

—

212

—

155

D₈

—

106

—

156

D₈

—

212

—

157

D₈

—

212

—

158

C₂×C₈

C₂²×D₅₃

yes

characteristic

159

C₂×C₈

C₂²×D₅₃

yes

—

160

C₂×C₈

C₂²×D₅₃

yes

characteristic

161

C₄⋊C₄

—

162

C₄⋊C₄

—

106

—

163

C₄⋊C₄

D₂₁₂

yes

—

164

C₄⋊C₄

D₂₁₂

yes

characteristic

165

C₂²⋊C₄

—

106

—

166

C₂²⋊C₄

—

106

—

167

C₂²⋊C₄

—

106

—

168

C₂²⋊C₄

—

106

—

169

C₂²⋊C₄

—

106

—

170

C₂²⋊C₄

—

106

—

171

C₄²

—

106

—

172

C₄²

C₂²×D₅₃

yes

characteristic

173

C₄²

C₂²×D₅₃

yes

—

174

C₂³

—

106

—

175

C₂³

—

106

—

176

Q₈

—

177

Q₈

—

106

—

178

Q₈

—

106

—

179

D₄

—

106

—

180

D₄

—

106

—

181

D₄

—

106

—

182

D₄

—

106

—

183

D₄

—

424

—

184

D₄

—

212

—

185

C₂×C₄

—

106

—

186

C₂×C₄

—

106

—

187

C₂×C₄

—

424

—

188

C₂×C₄

—

424

—

189

C₂×C₄

C₂×D₂₁₂

yes

—

190

C₂×C₄

C₂×D₂₁₂

yes

characteristic

191

C₂×C₄

C₂×D₂₁₂

yes

characteristic

192

C₂×C₄

C₂×D₂₁₂

yes

—

193

C₂×C₄

C₂³×D₅₃

yes

characteristic

194

C₈

—

195

C₈

—

196

C₈

#11719949

yes

—

197

C₂²

—

212

—

198

C₂²

—

212

—

199

C₂²

—

212

—

200

C₂²

—

212

—

201

C₂²

—

212

—

202

C₂²

—

212

—

203

C₂²

C₂²×D₂₁₂

yes

characteristic

204

C₄

—

205

C₄

—

206

C₄

—

207

C₄

—

208

C₄

—

212

—

209

C₄

—

212

—

210

C₄

C₂×#11719949

yes

characteristic

211

C₄

#13133263

yes

characteristic

212

C₂

—

424

—

213

C₂

—

424

—

214

C₂

#16811295

yes

characteristic

215

C₂

#16811423

yes

characteristic

216

C₂

#16811428

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#446	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#446	D₅₃
hypocenter	C₅₃	#3403
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

subgroup

G/H

center

C₂²

C₂²×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#446

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

N	H ≅ G/N	type
1. #16811358	C₂	semidirect
2. #16811012	C₂	semidirect
3. #16811007	C₂	semidirect
4. #16810846	C₂	semidirect
5. #16810839	C₂	semidirect
6. #16810819	C₂	semidirect
7. #16810725	C₂	semidirect
8. C₄×D₄₂₄	C₂	semidirect
9. C₅₃×#446	C₂	semidirect
115. #446	D₅₃	semidirect
118. C₅₃	#3403	semidirect

H ≅ G/N

type

1. #16811358

C₂

semidirect

2. #16811012

C₂

semidirect

3. #16811007

C₂

semidirect

4. #16810846

C₂

semidirect

5. #16810839

C₂

semidirect

6. #16810819

C₂

semidirect

7. #16810725

C₂

semidirect

8. C₄×D₄₂₄

C₂

semidirect

9. C₅₃×#446

C₂

semidirect

115. #446

D₅₃

semidirect

118. C₅₃

#3403

semidirect

Subgroups — #20287805

Conjugacy classes of subgroups

Complements