Group #20275563 — subgroups

Subgroups — #20275563

Order 6784.

312 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 83 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	C₂×#13133206	C₂	1	1	yes	maximal, characteristic
2	#16810646	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13133136	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13133133	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133133	C₂	1	1	yes	maximal, characteristic
6	M₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#101×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₂₁₂	C₂²	1	1	yes	characteristic
9	C₂×#11719917	C₂²	1	1	yes	characteristic
10	C₂×#11719917	C₂²	1	1	yes	characteristic
11	#13133206	—	2	1	no	—
12	#13133206	—	2	1	no	—
13	#13133206	—	2	2	no	—
14	C₂×#11719921	C₄	1	4	yes	—
15	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
16	#13133146	—	2	2	no	—
17	#13133136	—	2	2	no	—
18	#13133133	—	2	2	no	—
19	#13133133	—	2	2	no	—
20	#13133127	—	2	1	no	—
21	C₈×Dic₅₃	—	2	1	no	—
22	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
25	#101×C₅₃	—	2	2	no	—
26	C₂³×D₅₃	C₂×C₄	1	2	yes	—
27	C₂×Q₈×C₅₃	—	2	1	no	—
28	C₂×Q₈×C₅₃	—	2	1	no	—
29	C₂×Q₈×C₅₃	—	2	1	no	—
30	C₂×Q₈×C₅₃	—	2	1	no	—
31	C₂×Q₈×C₅₃	—	2	1	no	—
32	C₂×Q₈×C₅₃	—	2	1	no	—
33	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
35	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
36	C₂×D₂₁₂	—	2	1	no	—
37	C₂×D₂₁₂	—	2	1	no	—
38	C₂×D₂₁₂	—	2	1	no	—
39	C₂×D₂₁₂	—	2	1	no	—
40	C₂×D₂₁₂	—	2	1	no	—
41	C₂×D₂₁₂	—	2	1	no	—
42	M₁₆×C₅₃	—	2	1	no	—
43	M₁₆×C₅₃	—	2	1	no	—
44	M₁₆×C₅₃	—	2	1	no	—
45	M₁₆×C₅₃	D₄	1	2	yes	—
46	C₂×C₈×C₅₃	—	2	1	no	—
47	C₂×C₈×C₅₃	—	2	1	no	—
48	#11719921	—	2	4	no	—
49	#11719921	—	2	4	no	—
50	C₄×Dic₅₃	D₄	1	1	yes	characteristic
51	C₄×Dic₅₃	D₄	1	1	yes	characteristic
52	C₄×Dic₅₃	D₄	1	2	yes	—
53	#11719917	—	2	1	no	—
54	#11719917	—	2	1	no	—
55	#11719917	—	2	1	no	—
56	#11719917	D₄	1	2	yes	—
57	C₂×#88322	—	2	1	no	—
58	C₂×#88322	—	2	1	no	—
59	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×D₅₃	—	2	2	no	—
61	C₂²×D₅₃	—	2	2	no	—
62	C₂²×D₅₃	—	2	2	no	—
63	C₂²×D₅₃	—	4	4	no	—
64	Q₈×C₅₃	—	4	1	no	—
65	Q₈×C₅₃	—	4	1	no	—
66	Q₈×C₅₃	—	4	1	no	—
67	Q₈×C₅₃	—	4	1	no	—
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	—	2	2	no	—
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
77	C₂×Dic₅₃	—	4	2	no	—
78	C₂×Dic₅₃	C₂²⋊C₄	1	2	yes	—
79	C₂×Dic₅₃	C₂²⋊C₄	1	2	yes	—
80	D₂₁₂	—	4	1	no	—
81	D₂₁₂	—	4	1	no	—
82	D₂₁₂	—	4	1	no	—
83	D₂₁₂	—	4	1	no	—
84	C₈×C₅₃	—	2	2	no	—
85	C₈×C₅₃	—	4	1	no	—
86	#88322	—	2	2	no	—
87	#88322	—	4	1	no	—
88	C₂²×C₅₃	—	2	2	no	—
89	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
91	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
92	C₂×D₅₃	—	4	2	no	—
93	C₂×D₅₃	—	4	4	no	—
94	C₂×D₅₃	—	8	2	no	—
95	C₂×D₅₃	—	8	2	no	—
96	C₄×C₅₃	—	4	2	no	—
97	C₄×C₅₃	#121	1	1	yes	characteristic
98	C₄×C₅₃	#121	1	1	yes	characteristic
99	C₄×C₅₃	#124	1	1	yes	characteristic
100	C₄×C₅₃	#127	1	1	yes	characteristic
101	Dic₅₃	—	2	2	no	—
102	Dic₅₃	—	2	2	no	—
103	#1967	—	53	1	no	maximal, Sylow
104	C₂×C₅₃	—	2	1	no	—
105	C₂×C₅₃	—	2	1	no	—
106	C₂×C₅₃	#390	1	1	yes	characteristic
107	C₂×C₅₃	#413	1	2	yes	—
108	D₅₃	—	8	2	no	—
109	C₂×#124	—	53	1	no	—
110	#423	—	53	1	no	—
111	C₂×#101	—	53	1	no	—
112	C₂×#101	D₅₃	1	1	yes	characteristic, complemented
113	C₂×#100	—	53	1	no	—
114	C₂×#100	—	53	1	no	—
115	C₄×M₁₆	—	53	1	no	—
116	C₅₃	#1967	1	1	yes	characteristic, complemented, Sylow
117	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
118	C₂²×D₄	—	53	1	no	—
119	C₂×M₁₆	—	53	1	no	—
120	C₂×M₁₆	—	53	1	no	—
121	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
122	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
123	#124	—	106	1	no	—
124	#124	—	106	1	no	—
125	#124	—	106	2	no	—
126	C₂×C₂²⋊C₄	—	53	4	no	—
127	C₂×C₄²	—	53	1	no	—
128	C₄⋊C₈	—	106	2	no	—
129	#101	—	2	2	no	—
130	#101	—	106	2	no	—
131	#100	—	106	2	no	—
132	#100	—	106	2	no	—
133	#97	—	106	1	no	—
134	C₄×C₈	—	106	1	no	—
135	C₂⁴	—	53	2	no	—
136	C₂×Q₈	—	2	1	no	—
137	C₂×Q₈	—	2	1	no	—
138	C₂×Q₈	—	2	1	no	—
139	C₂×Q₈	—	2	1	no	—
140	C₂×Q₈	—	2	1	no	—
141	C₂×Q₈	—	2	1	no	—
142	C₂×D₄	—	106	1	no	—
143	C₂×D₄	—	106	1	no	—
144	C₂×D₄	—	106	1	no	—
145	C₂×D₄	—	106	1	no	—
146	C₂×D₄	—	106	1	no	—
147	C₂×D₄	—	106	1	no	—
148	C₂²×C₄	—	53	2	no	—
149	C₂²×C₄	C₄×D₅₃	1	2	yes	—
150	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
151	M₁₆	—	2	1	no	—
152	M₁₆	—	2	1	no	—
153	M₁₆	—	2	1	no	—
154	M₁₆	—	53	2	no	—
155	M₁₆	—	106	1	no	—
156	M₁₆	—	106	1	no	—
157	M₁₆	—	106	1	no	—
158	M₁₆	C₂²⋊D₅₃	1	2	yes	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	106	1	no	—
162	C₂×C₈	—	106	1	no	—
163	C₂²⋊C₄	—	106	4	no	—
164	C₂²⋊C₄	—	106	4	no	—
165	C₄²	—	53	1	no	—
166	C₄²	—	53	1	no	—
167	C₄²	—	53	2	no	—
168	C₂³	—	106	2	no	—
169	C₂³	—	106	2	no	—
170	C₂³	—	106	2	no	—
171	C₂³	—	212	4	no	—
172	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
173	Q₈	—	4	1	no	—
174	Q₈	—	4	1	no	—
175	Q₈	—	4	1	no	—
176	Q₈	—	4	1	no	—
177	D₄	—	212	1	no	—
178	D₄	—	212	1	no	—
179	D₄	—	212	1	no	—
180	D₄	—	212	1	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	53	2	no	—
185	C₂×C₄	—	53	2	no	—
186	C₂×C₄	—	212	2	no	—
187	C₂×C₄	#11719945	1	1	yes	characteristic
188	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
189	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
190	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
191	C₂×C₄	#11719947	1	1	yes	characteristic
192	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
193	C₈	—	2	2	no	—
194	C₈	—	4	1	no	—
195	C₈	—	106	2	no	—
196	C₈	—	212	1	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	424	2	no	—
199	C₂²	—	424	2	no	—
200	C₂²	—	212	2	no	—
201	C₂²	—	212	4	no	—
202	C₂²	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
203	C₂²	#13133177	1	1	yes	characteristic
204	C₂²	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
205	C₄	—	4	2	no	—
206	C₄	—	106	2	no	—
207	C₄	—	106	2	no	—
208	C₄	#13133189	1	1	yes	characteristic
209	C₄	#13133190	1	1	yes	characteristic
210	C₄	#13133203	1	1	yes	characteristic
211	C₄	#13133213	1	1	yes	characteristic
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	—	424	2	no	—
215	C₂	#16810523	1	1	yes	characteristic
216	C₂	#16810606	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133206

C₂

yes

maximal, characteristic

#16810646

C₂

yes

maximal, characteristic, complemented

C₂×#13133136

C₂

yes

maximal, characteristic, complemented

C₂×#13133133

C₂

yes

maximal, characteristic

C₂×#13133133

C₂

yes

maximal, characteristic

M₁₆×Dic₅₃

C₂

yes

maximal, characteristic, complemented

C₂×#101×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₁₂

C₂²

yes

characteristic

C₂×#11719917

C₂²

yes

characteristic

C₂×#11719917

C₂²

yes

—

—

—

C₄

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

—

—

—

—

C₈×Dic₅₃

—

C₂²×Q₈×C₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

#101×C₅₃

—

C₂³×D₅₃

C₂×C₄

yes

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

#11719921

—

#11719921

—

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

—

—

—

—

D₄

yes

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂²⋊C₄

yes

—

C₂×Dic₅₃

C₂²⋊C₄

yes

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

100

C₄×C₅₃

#127

yes

characteristic

101

Dic₅₃

—

102

Dic₅₃

—

103

#1967

—

maximal, Sylow

104

C₂×C₅₃

—

105

C₂×C₅₃

—

106

C₂×C₅₃

#390

yes

characteristic

107

C₂×C₅₃

#413

yes

—

108

D₅₃

—

109

C₂×#124

—

110

#423

—

111

C₂×#101

—

112

C₂×#101

D₅₃

yes

characteristic, complemented

113

C₂×#100

—

114

C₂×#100

—

115

C₄×M₁₆

—

116

C₅₃

#1967

yes

characteristic, complemented, Sylow

117

C₂²×Q₈

C₂×D₅₃

yes

characteristic

118

C₂²×D₄

—

119

C₂×M₁₆

—

120

C₂×M₁₆

—

121

C₂×M₁₆

C₂×D₅₃

yes

characteristic

122

C₂×M₁₆

C₂×D₅₃

yes

characteristic

123

#124

—

106

—

124

#124

—

106

—

125

#124

—

106

—

126

C₂×C₂²⋊C₄

—

127

C₂×C₄²

—

128

C₄⋊C₈

—

106

—

129

#101

—

130

#101

—

106

—

131

#100

—

106

—

132

#100

—

106

—

133

#97

—

106

—

134

C₄×C₈

—

106

—

135

C₂⁴

—

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×D₄

—

106

—

143

C₂×D₄

—

106

—

144

C₂×D₄

—

106

—

145

C₂×D₄

—

106

—

146

C₂×D₄

—

106

—

147

C₂×D₄

—

106

—

148

C₂²×C₄

—

149

C₂²×C₄

C₄×D₅₃

yes

—

150

C₂²×C₄

C₂²×D₅₃

yes

characteristic

151

M₁₆

—

152

M₁₆

—

153

M₁₆

—

154

M₁₆

—

155

M₁₆

—

106

—

156

M₁₆

—

106

—

157

M₁₆

—

106

—

158

M₁₆

C₂²⋊D₅₃

yes

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

106

—

162

C₂×C₈

—

106

—

163

C₂²⋊C₄

—

106

—

164

C₂²⋊C₄

—

106

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₂³

—

106

—

169

C₂³

—

106

—

170

C₂³

—

106

—

171

C₂³

—

212

—

172

C₂³

C₂×C₄×D₅₃

yes

characteristic

173

Q₈

—

174

Q₈

—

175

Q₈

—

176

Q₈

—

177

D₄

—

212

—

178

D₄

—

212

—

179

D₄

—

212

—

180

D₄

—

212

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

212

—

187

C₂×C₄

#11719945

yes

characteristic

188

C₂×C₄

D₄×D₅₃

yes

characteristic

189

C₂×C₄

D₄×D₅₃

yes

characteristic

190

C₂×C₄

D₄×D₅₃

yes

characteristic

191

C₂×C₄

#11719947

yes

characteristic

192

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

193

C₈

—

194

C₈

—

195

C₈

—

106

—

196

C₈

—

212

—

197

C₂²

—

198

C₂²

—

424

—

199

C₂²

—

424

—

200

C₂²

—

212

—

201

C₂²

—

212

—

202

C₂²

C₂²⋊C₄×D₅₃

yes

characteristic

203

C₂²

#13133177

yes

characteristic

204

C₂²

C₄×C₂²⋊D₅₃

yes

characteristic

205

C₄

—

206

C₄

—

106

—

207

C₄

—

106

—

208

C₄

#13133189

yes

characteristic

209

C₄

#13133190

yes

characteristic

210

C₄

#13133203

yes

characteristic

211

C₄

#13133213

yes

characteristic

212

C₂

—

213

C₂

—

214

C₂

—

424

—

215

C₂

#16810523

yes

characteristic

216

C₂

#16810606

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₅₃
commutator	C₂³×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#101×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	C₂×#101	D₅₃
hypocenter	C₅₃	#1967
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₅₃

commutator

C₂³×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×#101×C₅₃

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
2. #16810646	C₂	semidirect
3. C₂×#13133136	C₂	semidirect
6. M₁₆×Dic₅₃	C₂	semidirect
7. C₂×#101×C₅₃	C₂	semidirect
112. C₂×#101	D₅₃	semidirect
116. C₅₃	#1967	semidirect

H ≅ G/N

type

2. #16810646

C₂

semidirect

3. C₂×#13133136

C₂

semidirect

6. M₁₆×Dic₅₃

C₂

semidirect

7. C₂×#101×C₅₃

C₂

semidirect

112. C₂×#101

D₅₃

semidirect

116. C₅₃

#1967

semidirect

Subgroups — #20275563

Conjugacy classes of subgroups

Complements