Group #27302259 — subgroups

Subgroups — #27302259

Order 13568.

442 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 139 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	#20283428	C₂	1	1	yes	maximal, characteristic
2	#20276471	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16810406	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13133223	C₂²	1	2	yes	—
6	#16811173	C₄	1	1	yes	characteristic
7	C₂×#13133157	C₂²	1	2	yes	—
8	#16810702	—	2	2	no	—
9	C₈×D₂₁₂	—	2	2	no	—
10	#16810647	C₄	1	1	yes	characteristic
11	#16810589	C₄	1	2	yes	—
12	C₄⋊#11719908	—	2	2	no	—
13	#16810406	—	2	8	no	—
14	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#346	—	2	2	no	—
17	C₂×#11719945	C₂×C₄	1	2	yes	—
18	#13133223	—	2	4	no	—
19	#13133223	—	2	4	no	—
20	C₂×#11719917	C₂×C₄	1	2	yes	—
21	C₂×#11719912	—	2	2	no	—
22	C₂×C₈×D₅₃	—	2	2	no	—
23	#13133217	C₈	1	1	yes	characteristic
24	#13133214	C₈	1	1	yes	characteristic
25	#13133205	C₈	1	2	yes	—
26	#13133191	—	2	2	no	—
27	C₄×D₂₁₂	—	2	2	no	—
28	#13133157	—	2	4	no	—
29	C₂×#11719908	—	2	2	no	—
30	C₄⋊#88322	—	2	2	no	—
31	#13133129	—	2	4	no	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₆×C₅₃	—	2	2	no	—
36	C₂×C₁₆×C₅₃	—	2	2	no	—
37	C₂×C₁₆×C₅₃	D₄	1	4	yes	complemented
38	C₄×C₈×C₅₃	D₄	1	2	yes	—
39	C₄×C₈×C₅₃	D₄	1	2	yes	—
40	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₂²⋊D₅₃	—	2	2	no	—
43	#11719945	—	2	4	no	—
44	#11719945	—	2	4	no	—
45	C₂×D₂₁₂	—	2	2	no	—
46	C₂×D₂₁₂	C₂×C₈	1	2	yes	—
47	C₂×C₄×D₅₃	—	2	2	no	—
48	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
49	C₂×C₈×C₅₃	—	2	2	no	—
50	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
53	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₅₃	#41	1	2	yes	—
55	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
56	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
57	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
58	#11719921	—	2	4	no	—
59	C₄⋊Dic₅₃	—	2	2	no	—
60	#11719917	—	2	4	no	—
61	C₂×#88322	—	2	2	no	—
62	#11719912	—	4	4	no	—
63	C₈×D₅₃	—	4	2	no	—
64	C₈×D₅₃	—	4	2	no	—
65	C₁₆×C₅₃	—	2	4	no	—
66	#11719908	—	4	2	no	—
67	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
68	C₂²×D₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	—	4	2	no	—
70	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
71	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
72	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
74	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#131	1	2	yes	—
76	C₂²⋊D₅₃	—	4	4	no	—
77	C₂×Dic₅₃	—	2	2	no	—
78	D₂₁₂	—	2	4	no	—
79	D₂₁₂	—	4	4	no	—
80	C₄×D₅₃	—	4	2	no	—
81	C₄×D₅₃	—	4	2	no	—
82	Dic₁₀₆	—	2	4	no	—
83	C₈×C₅₃	—	2	2	no	—
84	C₈×C₅₃	#121	1	2	yes	—
85	C₈×C₅₃	#124	1	1	yes	characteristic
86	C₈×C₅₃	#127	1	1	yes	characteristic
87	#88322	—	4	2	no	—
88	#13192	—	53	1	no	maximal, Sylow
89	C₂²×C₅₃	—	2	2	no	—
90	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
91	C₂²×C₅₃	#436	1	2	yes	—
92	C₂×D₅₃	—	4	4	no	—
93	C₂×D₅₃	—	4	2	no	—
94	C₄×C₅₃	—	2	2	no	—
95	C₄×C₅₃	—	2	2	no	—
96	C₄×C₅₃	#390	1	1	yes	characteristic
97	C₄×C₅₃	#408	1	1	yes	characteristic
98	C₄×C₅₃	C₈×D₄	1	2	yes	—
99	Dic₅₃	—	4	2	no	—
100	#2931	—	53	1	no	—
101	#2150	—	53	1	no	—
102	C₂×#350	—	53	4	no	—
103	C₂×#346	D₅₃	1	1	yes	characteristic, complemented
104	C₂×C₅₃	—	2	2	no	—
105	C₂×C₅₃	#1954	1	1	yes	characteristic
106	C₂×C₅₃	#2117	1	2	yes	—
107	D₅₃	—	8	2	no	—
108	C₂×#131	—	53	2	no	—
109	#532	—	53	1	no	—
110	C₂×M₃₂	—	53	2	no	—
111	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
112	#436	—	106	2	no	—
113	C₈×D₄	—	106	2	no	—
114	#423	—	53	1	no	—
115	#408	—	53	2	no	—
116	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
117	C₄⋊C₁₆	—	106	2	no	—
118	#350	—	106	8	no	—
119	#346	—	2	2	no	—
120	C₅₃	#13192	1	1	yes	characteristic, complemented, Sylow
121	C₂×#41	—	53	2	no	—
122	#131	—	106	4	no	—
123	#131	—	106	4	no	—
124	C₂×M₁₆	—	53	2	no	—
125	C₂×M₁₆	—	106	2	no	—
126	C₂²×C₈	—	106	2	no	—
127	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
128	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
129	C₄⋊Q₈	—	53	1	no	—
130	#127	—	53	1	no	—
131	#124	—	53	2	no	—
132	#121	—	106	2	no	—
133	C₄×D₄	—	106	2	no	—
134	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
135	M₃₂	—	106	4	no	—
136	C₂×C₁₆	—	2	2	no	—
137	C₂×C₁₆	—	2	2	no	—
138	C₂×C₁₆	—	106	2	no	—
139	C₂×C₁₆	D₂₁₂	1	4	yes	complemented
140	C₄⋊C₈	—	106	2	no	—
141	C₂²⋊C₈	—	106	4	no	—
142	C₄×C₈	D₂₁₂	1	2	yes	—
143	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
144	#41	—	106	4	no	—
145	#41	—	106	4	no	—
146	C₂×Q₈	—	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	#11719912	1	2	yes	—
152	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
153	M₁₆	—	106	4	no	—
154	M₁₆	—	212	4	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	106	2	no	—
157	C₂×C₈	—	212	2	no	—
158	C₂×C₈	—	212	2	no	—
159	C₂×C₈	#11719921	1	2	yes	—
160	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
161	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
162	C₂×C₈	#11719945	1	2	yes	—
163	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
164	C₄⋊C₄	—	106	2	no	—
165	C₂²⋊C₄	—	106	4	no	—
166	C₄²	C₈×D₅₃	1	1	yes	characteristic
167	C₄²	C₈×D₅₃	1	1	yes	characteristic
168	C₄²	#11719921	1	2	yes	—
169	C₁₆	—	2	4	no	—
170	C₁₆	—	212	2	no	—
171	C₂³	—	106	2	no	—
172	C₂³	C₂×#11719912	1	1	yes	characteristic
173	Q₈	—	106	4	no	—
174	D₄	—	106	4	no	—
175	D₄	—	212	4	no	—
176	D₄	—	212	4	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	106	2	no	—
179	C₂×C₄	—	212	2	no	—
180	C₂×C₄	—	212	2	no	—
181	C₂×C₄	#13133129	1	2	yes	—
182	C₂×C₄	#13133129	1	2	yes	—
183	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
184	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
185	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
186	C₂×C₄	#13133223	1	2	yes	—
187	C₈	—	2	2	no	—
188	C₈	—	212	2	no	—
189	C₈	#13133191	1	2	yes	—
190	C₈	#13133205	1	1	yes	characteristic
191	C₈	#13133214	1	1	yes	characteristic
192	C₂²	—	2	2	no	—
193	C₂²	—	212	2	no	—
194	C₂²	—	212	4	no	—
195	C₂²	C₂×#13133129	1	1	yes	characteristic
196	C₂²	#16810702	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	2	2	no	—
199	C₄	—	212	2	no	—
200	C₄	#16810525	1	1	yes	characteristic
201	C₄	#16810589	1	1	yes	characteristic
202	C₄	C₈×D₂₁₂	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	—	424	2	no	—
205	C₂	#20275509	1	1	yes	characteristic
206	C₂	#20276342	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20283428

C₂

yes

maximal, characteristic

#20276471

C₂

yes

maximal, characteristic, complemented

C₂×#16810406

C₂

yes

maximal, complemented

C₂×C₅₃×#346

C₂

yes

maximal, characteristic, complemented

C₂×#13133223

C₂²

yes

—

#16811173

C₄

yes

characteristic

C₂×#13133157

C₂²

yes

—

#16810702

—

C₈×D₂₁₂

—

#16810647

C₄

yes

characteristic

#16810589

C₄

yes

—

C₄⋊#11719908

—

#16810406

—

C₂²×C₁₆×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#346

—

C₂×#11719945

C₂×C₄

yes

—

#13133223

—

#13133223

—

C₂×#11719917

C₂×C₄

yes

—

C₂×#11719912

—

C₂×C₈×D₅₃

—

#13133217

C₈

yes

characteristic

#13133214

C₈

yes

characteristic

#13133205

C₈

yes

—

#13133191

—

C₄×D₂₁₂

—

—

—

—

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

D₄

yes

complemented

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

M₁₆

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₂²⋊D₅₃

—

#11719945

—

#11719945

—

C₂×D₂₁₂

—

C₂×D₂₁₂

C₂×C₈

yes

—

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

C₂×C₈

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₄²×C₅₃

C₂²⋊C₄

yes

—

C₄²×C₅₃

C₂×C₈

yes

characteristic

C₄²×C₅₃

C₂×C₈

yes

characteristic

#11719921

—

C₄⋊Dic₅₃

—

#11719917

—

C₂×#88322

—

#11719912

—

C₈×D₅₃

—

C₈×D₅₃

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

C₂×M₁₆

yes

characteristic

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₈

yes

—

C₂×C₄×C₅₃

C₂²⋊C₈

yes

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

#131

yes

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

#121

yes

—

C₈×C₅₃

yes

C₈×C₅₃

yes

—

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₅₃

#436

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₈×D₄

yes

—

Dic₅₃

—

100

#2931

—

101

#2150

—

102

C₂×#350

—

103

C₂×#346

D₅₃

yes

characteristic, complemented

104

C₂×C₅₃

—

105

C₂×C₅₃

#1954

yes

characteristic

106

C₂×C₅₃

#2117

yes

—

107

D₅₃

—

108

C₂×#131

—

109

#532

—

110

C₂×M₃₂

—

111

C₂²×C₁₆

C₂×D₅₃

yes

—

112

#436

—

106

—

113

C₈×D₄

—

106

—

114

#423

—

115

#408

—

116

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

117

C₄⋊C₁₆

—

106

—

118

#350

—

106

—

119

#346

—

120

C₅₃

#13192

yes

characteristic, complemented, Sylow

121

C₂×#41

—

122

#131

—

106

—

123

#131

—

106

—

124

C₂×M₁₆

—

125

C₂×M₁₆

—

106

—

126

C₂²×C₈

—

106

—

127

C₂²×C₈

C₄×D₅₃

yes

characteristic

128

C₂²×C₈

C₂²×D₅₃

yes

characteristic

129

C₄⋊Q₈

—

130

#127

—

131

#124

—

132

#121

—

106

—

133

C₄×D₄

—

106

—

134

C₂×C₄²

C₄×D₅₃

yes

characteristic

135

M₃₂

—

106

—

136

C₂×C₁₆

—

137

C₂×C₁₆

—

138

C₂×C₁₆

—

106

—

139

C₂×C₁₆

D₂₁₂

yes

complemented

140

C₄⋊C₈

—

106

—

141

C₂²⋊C₈

—

106

—

142

C₄×C₈

D₂₁₂

yes

—

143

C₄×C₈

C₂²⋊D₅₃

yes

—

144

#41

—

106

—

145

#41

—

106

—

146

C₂×Q₈

—

147

C₂×D₄

—

148

C₂×D₄

—

106

—

149

C₂×D₄

—

106

—

150

C₂²×C₄

—

106

—

151

C₂²×C₄

#11719912

yes

—

152

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

153

M₁₆

—

106

—

154

M₁₆

—

212

—

155

C₂×C₈

—

156

C₂×C₈

—

106

—

157

C₂×C₈

—

212

—

158

C₂×C₈

—

212

—

159

C₂×C₈

#11719921

yes

—

160

C₂×C₈

C₂×D₂₁₂

yes

—

161

C₂×C₈

C₂×D₂₁₂

yes

characteristic

162

C₂×C₈

#11719945

yes

—

163

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

164

C₄⋊C₄

—

106

—

165

C₂²⋊C₄

—

106

—

166

C₄²

C₈×D₅₃

yes

characteristic

167

C₄²

C₈×D₅₃

yes

characteristic

168

C₄²

#11719921

yes

—

169

C₁₆

—

170

C₁₆

—

212

—

171

C₂³

—

106

—

172

C₂³

C₂×#11719912

yes

characteristic

173

Q₈

—

106

—

174

D₄

—

106

—

175

D₄

—

212

—

176

D₄

—

212

—

177

C₂×C₄

—

178

C₂×C₄

—

106

—

179

C₂×C₄

—

212

—

180

C₂×C₄

—

212

—

181

C₂×C₄

#13133129

yes

—

182

C₂×C₄

#13133129

yes

—

183

C₂×C₄

C₂×#11719921

yes

characteristic

184

C₂×C₄

C₄×D₂₁₂

yes

—

185

C₂×C₄

C₂×C₈×D₅₃

yes

characteristic

186

C₂×C₄

#13133223

yes

—

187

C₈

—

188

C₈

—

212

—

189

C₈

#13133191

yes

—

190

C₈

#13133205

yes

characteristic

191

C₈

#13133214

yes

characteristic

192

C₂²

—

193

C₂²

—

212

—

194

C₂²

—

212

—

195

C₂²

C₂×#13133129

yes

characteristic

196

C₂²

#16810702

yes

—

197

C₄

—

198

C₄

—

199

C₄

—

212

—

200

C₄

#16810525

yes

characteristic

201

C₄

#16810589

yes

characteristic

202

C₄

C₈×D₂₁₂

yes

—

203

C₂

—

204

C₂

—

424

—

205

C₂

#20275509

yes

characteristic

206

C₂

#20276342

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#346	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	#16811173	C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#346	D₅₃
hypocenter	C₅₃	#13192
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₈

C₂×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₂×C₅₃×#346

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₈

prime radical

#16811173

C₄

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
2. #20276471	C₂	semidirect
3. C₂×#16810406	C₂	semidirect
4. C₂×C₅₃×#346	C₂	semidirect
37. C₂×C₁₆×C₅₃	D₄	semidirect
103. C₂×#346	D₅₃	semidirect
120. C₅₃	#13192	semidirect
139. C₂×C₁₆	D₂₁₂	semidirect

H ≅ G/N

type

2. #20276471

C₂

semidirect

3. C₂×#16810406

C₂

semidirect

4. C₂×C₅₃×#346

C₂

semidirect

37. C₂×C₁₆×C₅₃

D₄

semidirect

103. C₂×#346

D₅₃

semidirect

120. C₅₃

#13192

semidirect

139. C₂×C₁₆

D₂₁₂

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20283428	2
#20276471	2
C₂×#16810406	2
C₂×C₅₃×#346	2
#13192	53

subgroup

index

C₂×C₅₃×#346

Subgroups — #27302259

Conjugacy classes of subgroups

Complements

Maximal subgroups