Group #27289197 — subgroups

Subgroups — #27289197

Order 13568.

526 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 155 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	#20275522	C₂	1	2	yes	maximal
2	#20274872	C₂	1	1	yes	maximal, characteristic, complemented
3	#20274871	C₂	1	2	yes	maximal, complemented
4	#20272542	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1957	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133149	—	2	2	no	—
7	C₂×#13133148	—	2	4	no	—
8	C₂×#13133138	—	2	4	no	—
9	C₂×C₄×#88322	C₂²	1	2	yes	—
10	#16810493	C₂²	1	2	yes	—
11	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×#102×C₅₃	—	2	2	no	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#384	C₂²	1	1	yes	characteristic
15	C₂²×#88322	D₄	1	4	yes	—
16	C₂×C₄⋊Dic₅₃	—	2	2	no	—
17	C₂×C₄⋊Dic₅₃	—	2	2	no	—
18	C₂×C₄⋊Dic₅₃	—	2	2	no	—
19	#13133149	—	4	4	no	—
20	#13133148	—	4	8	no	—
21	#13133138	—	4	8	no	—
22	C₄×#88322	—	2	2	no	—
23	C₄×#88322	—	2	2	no	—
24	C₂²×D₄×C₅₃	—	2	2	no	—
25	C₂²×D₄×C₅₃	—	2	1	no	—
26	C₂²×C₈×C₅₃	D₄	1	2	yes	—
27	#127×C₅₃	—	2	1	no	—
28	#127×C₅₃	—	2	2	no	—
29	#127×C₅₃	Q₈	1	2	yes	—
30	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
31	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	#102×C₅₃	—	4	4	no	—
34	C₄×C₈×C₅₃	—	2	2	no	—
35	C₂⁴×C₅₃	—	4	1	no	—
36	C₂×D₄×C₅₃	—	2	4	no	—
37	C₂×D₄×C₅₃	—	2	2	no	—
38	C₂×D₄×C₅₃	—	4	1	no	—
39	C₂×D₄×C₅₃	—	4	2	no	—
40	C₂×D₄×C₅₃	—	4	2	no	—
41	C₂×D₄×C₅₃	—	4	4	no	—
42	C₂×D₄×C₅₃	—	4	4	no	—
43	C₂×D₄×C₅₃	—	4	2	no	—
44	C₂²×C₄×C₅₃	—	4	1	no	—
45	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×Dic₅₃	—	4	2	no	—
48	C₂×C₈×C₅₃	—	2	2	no	—
49	C₂×C₈×C₅₃	—	2	2	no	—
50	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
51	C₄⋊C₄×C₅₃	—	4	2	no	—
52	C₄⋊C₄×C₅₃	—	4	4	no	—
53	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
54	C₄²×C₅₃	#41	1	1	yes	characteristic
55	C₄²×C₅₃	#41	1	2	yes	—
56	C₄⋊Dic₅₃	—	4	4	no	—
57	C₄⋊Dic₅₃	—	4	4	no	—
58	C₄⋊Dic₅₃	—	4	4	no	—
59	C₂×#88322	—	2	4	no	—
60	C₂×#88322	—	2	4	no	—
61	C₂×#88322	SD₁₆	1	8	yes	—
62	C₂³×C₅₃	—	4	2	no	—
63	C₂³×C₅₃	—	8	2	no	—
64	C₂³×C₅₃	—	8	1	no	—
65	C₂³×C₅₃	—	8	2	no	—
66	C₂³×C₅₃	—	8	1	no	—
67	C₂³×C₅₃	#127	1	1	yes	characteristic
68	D₄×C₅₃	—	4	8	no	—
69	D₄×C₅₃	—	4	4	no	—
70	D₄×C₅₃	—	8	2	no	—
71	D₄×C₅₃	—	8	4	no	—
72	C₂×C₄×C₅₃	—	8	1	no	—
73	C₂×C₄×C₅₃	—	8	2	no	—
74	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
75	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#123	1	2	yes	—
78	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
79	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
80	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
81	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
82	C₂×Dic₅₃	—	8	2	no	—
83	C₂×Dic₅₃	—	8	2	no	—
84	C₂×Dic₅₃	—	8	2	no	—
85	C₈×C₅₃	—	2	4	no	—
86	#88322	—	2	8	no	—
87	#10683	—	53	1	no	maximal, Sylow
88	C₂²×C₅₃	—	8	4	no	—
89	C₂²×C₅₃	—	8	2	no	—
90	C₂²×C₅₃	—	16	1	no	—
91	C₂²×C₅₃	—	16	1	no	—
92	C₂²×C₅₃	—	16	2	no	—
93	C₂²×C₅₃	#399	1	1	yes	characteristic
94	C₂²×C₅₃	#492	1	2	yes	—
95	C₂²×C₅₃	#492	1	1	yes	characteristic
96	C₂²×C₅₃	#496	1	2	yes	—
97	C₂²×C₅₃	#496	1	1	yes	characteristic
98	C₄×C₅₃	—	16	1	no	—
99	C₄×C₅₃	#476	1	4	yes	—
100	C₄×C₅₃	#476	1	2	yes	—
101	C₄×C₅₃	#481	1	2	yes	—
102	C₄×C₅₃	#481	1	4	yes	—
103	Dic₅₃	—	16	2	no	—
104	#1957	—	53	2	no	—
105	#1957	D₅₃	1	1	yes	characteristic
106	#1844	—	53	1	no	—
107	#1844	—	53	2	no	—
108	#1273	—	53	1	no	—
109	C₂×C₅₃	—	16	2	no	—
110	C₂×C₅₃	#1705	1	1	yes	characteristic
111	C₂×C₅₃	#1715	1	1	yes	characteristic
112	C₂×C₅₃	#1715	1	2	yes	—
113	C₂×C₅₃	#2094	1	2	yes	—
114	C₂×C₅₃	#2094	1	1	yes	characteristic
115	C₂×#127	C₂×D₅₃	1	1	yes	characteristic
116	C₂×#106	—	106	4	no	—
117	C₂×#106	—	106	2	no	—
118	C₂×#102	—	2	2	no	—
119	C₂×#102	—	106	4	no	—
120	C₂×C₄×C₈	—	53	2	no	—
121	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
122	#384	—	53	2	no	—
123	#384	C₂×D₅₃	1	1	yes	characteristic
124	C₅₃	#10683	1	1	yes	characteristic, complemented, Sylow
125	C₂²×D₄	—	2	2	no	—
126	C₂²×D₄	—	2	1	no	—
127	C₂²×C₈	—	53	4	no	—
128	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
129	#127	—	2	1	no	—
130	#127	—	2	2	no	—
131	#127	Dic₁₀₆	1	2	yes	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	—	2	1	no	—
134	C₂×C₄⋊C₄	—	106	2	no	—
135	C₂×C₄⋊C₄	—	106	2	no	—
136	C₂×C₄⋊C₄	—	106	2	no	—
137	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
138	#106	—	212	8	no	—
139	#106	—	212	4	no	—
140	#102	—	4	4	no	—
141	#102	—	212	8	no	—
142	C₄×C₈	—	2	2	no	—
143	C₄×C₈	—	106	2	no	—
144	C₄×C₈	—	106	2	no	—
145	C₂⁴	—	4	1	no	—
146	C₂×D₄	—	2	4	no	—
147	C₂×D₄	—	2	2	no	—
148	C₂×D₄	—	4	1	no	—
149	C₂×D₄	—	4	2	no	—
150	C₂×D₄	—	4	4	no	—
151	C₂×D₄	—	4	2	no	—
152	C₂×D₄	—	4	4	no	—
153	C₂×D₄	—	4	2	no	—
154	C₂²×C₄	—	4	1	no	—
155	C₂²×C₄	—	212	2	no	—
156	C₂²×C₄	D₄×D₅₃	1	2	yes	—
157	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	53	8	no	—
161	C₂×C₈	—	106	4	no	—
162	C₂×C₈	—	106	4	no	—
163	C₂×C₈	#11719923	1	4	yes	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	4	4	no	—
166	C₄⋊C₄	—	212	4	no	—
167	C₄⋊C₄	—	212	4	no	—
168	C₄⋊C₄	—	212	4	no	—
169	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
170	C₄²	#11719945	1	1	yes	characteristic
171	C₄²	#11719947	1	2	yes	—
172	C₂³	—	4	2	no	—
173	C₂³	—	8	2	no	—
174	C₂³	—	8	1	no	—
175	C₂³	—	8	2	no	—
176	C₂³	—	8	1	no	—
177	C₂³	#13133213	1	1	yes	characteristic
178	D₄	—	4	8	no	—
179	D₄	—	4	4	no	—
180	D₄	—	8	2	no	—
181	D₄	—	8	4	no	—
182	C₂×C₄	—	8	2	no	—
183	C₂×C₄	—	8	1	no	—
184	C₂×C₄	—	424	2	no	—
185	C₂×C₄	—	424	2	no	—
186	C₂×C₄	—	424	2	no	—
187	C₂×C₄	#13133194	1	2	yes	—
188	C₂×C₄	#13133196	1	1	yes	characteristic
189	C₂×C₄	#13133198	1	1	yes	characteristic
190	C₂×C₄	#13133199	1	2	yes	—
191	C₂×C₄	SD₁₆×D₅₃	1	4	yes	—
192	C₂×C₄	C₂×#11719923	1	2	yes	—
193	C₂×C₄	#13133241	1	4	yes	—
194	C₂×C₄	#13133242	1	2	yes	—
195	C₈	—	2	4	no	—
196	C₈	—	106	8	no	—
197	C₂²	—	8	4	no	—
198	C₂²	—	8	2	no	—
199	C₂²	—	16	1	no	—
200	C₂²	—	16	1	no	—
201	C₂²	—	16	2	no	—
202	C₂²	#16810559	1	1	yes	characteristic
203	C₂²	#16810994	1	2	yes	—
204	C₂²	#16810995	1	1	yes	characteristic
205	C₂²	#16811011	1	2	yes	—
206	C₂²	#16811013	1	1	yes	characteristic
207	C₄	—	16	1	no	—
208	C₄	—	848	2	no	—
209	C₄	#16810911	1	4	yes	—
210	C₄	#16810913	1	2	yes	—
211	C₄	#16810939	1	2	yes	—
212	C₄	#16810940	1	4	yes	—
213	C₂	—	16	2	no	—
214	C₂	#20274301	1	1	yes	characteristic
215	C₂	#20274341	1	1	yes	characteristic
216	C₂	#20274345	1	2	yes	—
217	C₂	#20276246	1	2	yes	—
218	C₂	#20276248	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20275522

C₂

yes

maximal

#20274872

C₂

yes

maximal, characteristic, complemented

#20274871

C₂

yes

maximal, complemented

#20272542

C₂

yes

maximal, characteristic, complemented

C₅₃×#1957

C₂

yes

maximal, characteristic

C₂×#13133149

—

C₂×#13133148

—

C₂×#13133138

—

C₂×C₄×#88322

C₂²

yes

—

#16810493

C₂²

yes

—

C₂×#127×C₅₃

C₂²

yes

characteristic

C₂×#102×C₅₃

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#384

C₂²

yes

characteristic

C₂²×#88322

D₄

yes

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

—

—

—

—

—

C₂²×D₄×C₅₃

—

C₂²×D₄×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

#127×C₅₃

—

#127×C₅₃

—

#127×C₅₃

Q₈

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#102×C₅₃

—

C₄×C₈×C₅₃

—

C₂⁴×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×Q₈

yes

C₄²×C₅₃

yes

C₄²×C₅₃

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

SD₁₆

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

#127

yes

characteristic

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

#88322

—

#10683

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#399

yes

characteristic

C₂²×C₅₃

#492

yes

—

C₂²×C₅₃

#492

yes

characteristic

C₂²×C₅₃

#496

yes

—

C₂²×C₅₃

#496

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#476

yes

—

100

C₄×C₅₃

#476

yes

—

101

C₄×C₅₃

#481

yes

—

102

C₄×C₅₃

#481

yes

—

103

Dic₅₃

—

104

#1957

—

105

#1957

D₅₃

yes

characteristic

106

#1844

—

107

#1844

—

108

#1273

—

109

C₂×C₅₃

—

110

C₂×C₅₃

#1705

yes

characteristic

111

C₂×C₅₃

#1715

yes

characteristic

112

C₂×C₅₃

#1715

yes

—

113

C₂×C₅₃

#2094

yes

—

114

C₂×C₅₃

#2094

yes

characteristic

115

C₂×#127

C₂×D₅₃

yes

characteristic

116

C₂×#106

—

106

—

117

C₂×#106

—

106

—

118

C₂×#102

—

119

C₂×#102

—

106

—

120

C₂×C₄×C₈

—

121

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

122

#384

—

123

#384

C₂×D₅₃

yes

characteristic

124

C₅₃

#10683

yes

characteristic, complemented, Sylow

125

C₂²×D₄

—

126

C₂²×D₄

—

127

C₂²×C₈

—

128

C₂²×C₈

C₂²⋊D₅₃

yes

—

129

#127

—

130

#127

—

131

#127

Dic₁₀₆

yes

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

134

C₂×C₄⋊C₄

—

106

—

135

C₂×C₄⋊C₄

—

106

—

136

C₂×C₄⋊C₄

—

106

—

137

C₂×C₄²

C₂²×D₅₃

yes

characteristic

138

#106

—

212

—

139

#106

—

212

—

140

#102

—

141

#102

—

212

—

142

C₄×C₈

—

143

C₄×C₈

—

106

—

144

C₄×C₈

—

106

—

145

C₂⁴

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

—

150

C₂×D₄

—

151

C₂×D₄

—

152

C₂×D₄

—

153

C₂×D₄

—

154

C₂²×C₄

—

155

C₂²×C₄

—

212

—

156

C₂²×C₄

D₄×D₅₃

yes

—

157

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

106

—

162

C₂×C₈

—

106

—

163

C₂×C₈

#11719923

yes

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

212

—

167

C₄⋊C₄

—

212

—

168

C₄⋊C₄

—

212

—

169

C₄²

C₂×Dic₁₀₆

yes

characteristic

170

C₄²

#11719945

yes

characteristic

171

C₄²

#11719947

yes

—

172

C₂³

—

173

C₂³

—

174

C₂³

—

175

C₂³

—

176

C₂³

—

177

C₂³

#13133213

yes

characteristic

178

D₄

—

179

D₄

—

180

D₄

—

181

D₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

424

—

185

C₂×C₄

—

424

—

186

C₂×C₄

—

424

—

187

C₂×C₄

#13133194

yes

—

188

C₂×C₄

#13133196

yes

characteristic

189

C₂×C₄

#13133198

yes

characteristic

190

C₂×C₄

#13133199

yes

—

191

C₂×C₄

SD₁₆×D₅₃

yes

—

192

C₂×C₄

C₂×#11719923

yes

—

193

C₂×C₄

#13133241

yes

—

194

C₂×C₄

#13133242

yes

—

195

C₈

—

196

C₈

—

106

—

197

C₂²

—

198

C₂²

—

199

C₂²

—

200

C₂²

—

201

C₂²

—

202

C₂²

#16810559

yes

characteristic

203

C₂²

#16810994

yes

—

204

C₂²

#16810995

yes

characteristic

205

C₂²

#16811011

yes

—

206

C₂²

#16811013

yes

characteristic

207

C₄

—

208

C₄

—

848

—

209

C₄

#16810911

yes

—

210

C₄

#16810913

yes

—

211

C₄

#16810939

yes

—

212

C₄

#16810940

yes

—

213

C₂

—

214

C₂

#20274301

yes

characteristic

215

C₂

#20274341

yes

characteristic

216

C₂

#20274345

yes

—

217

C₂

#20276246

yes

—

218

C₂

#20276248

yes

characteristic

	subgroup	G/H
center	C₂³	#13133213
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1957	C₂
socle	C₂³×C₅₃	#127
prime radical	C₂×#127×C₅₃	C₂²
hypercenter	#1957	D₅₃
hypocenter	C₅₃	#10683
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1957

subgroup

G/H

center

C₂³

#13133213

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1957

C₂

socle

C₂³×C₅₃

#127

prime radical

C₂×#127×C₅₃

C₂²

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1957

N	H ≅ G/N	type
2. #20274872	C₂	semidirect
3. #20274871	C₂	semidirect
4. #20272542	C₂	semidirect
124. C₅₃	#10683	semidirect

H ≅ G/N

type

2. #20274872

C₂

semidirect

3. #20274871

C₂

semidirect

4. #20272542

C₂

semidirect

124. C₅₃

#10683

semidirect

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

subgroup	index
#20275522	2
#20274872	2
#20274871	2
#20272542	2
C₅₃×#1957	2
#10683	53

subgroup

index

C₅₃×#1957

Subgroups — #27289197

Conjugacy classes of subgroups

Complements

Maximal subgroups