Group #27288887 — subgroups

Subgroups — #27288887

Order 13568.

310 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 93 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	#20275535	C₂	1	1	yes	maximal, characteristic
2	#20274890	C₂	1	1	yes	maximal, characteristic
3	#20273386	C₂	1	2	yes	maximal
4	#20272574	C₂	1	2	yes	maximal
5	C₅₃×#1281	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133217	C₂²	1	1	yes	characteristic
7	C₂×#13133153	—	2	1	no	—
8	C₂×#13133153	—	2	1	no	—
9	#16810647	C₂²	1	2	yes	—
10	C₂×#13133142	—	2	1	no	—
11	C₂×#13133142	—	2	1	no	—
12	#16810489	C₂²	1	1	yes	characteristic
13	#16810368	—	2	4	no	—
14	C₅₃×#431	C₂²	1	2	yes	—
15	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×Dic₁₀₆	—	2	1	no	—
17	C₂×#11719917	—	2	2	no	—
18	#13133217	—	2	1	no	—
19	#13133217	—	2	2	no	—
20	#13133217	D₄	1	2	yes	—
21	C₂×C₄⋊Dic₅₃	—	2	1	no	—
22	C₂×C₄⋊Dic₅₃	—	2	2	no	—
23	#13133153	—	4	2	no	—
24	#13133153	—	4	2	no	—
25	C₄⋊#88322	—	2	2	no	—
26	C₄⋊#88322	—	2	2	no	—
27	#13133142	—	4	2	no	—
28	#13133142	—	4	2	no	—
29	#13133124	—	2	2	no	—
30	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₅₃	—	2	2	no	—
34	C₂²⋊C₈×C₅₃	—	2	2	no	—
35	#97×C₅₃	—	2	2	no	—
36	C₄×C₈×C₅₃	—	2	2	no	—
37	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	#41	1	2	yes	—
39	C₂²×Dic₅₃	—	4	1	no	—
40	C₂²×Dic₅₃	—	4	1	no	—
41	C₂×Dic₁₀₆	—	2	2	no	—
42	C₂×Dic₁₀₆	—	4	1	no	—
43	C₂×Dic₁₀₆	—	4	1	no	—
44	C₂×Dic₁₀₆	—	4	1	no	—
45	C₂×Dic₁₀₆	—	4	1	no	—
46	C₂×Dic₁₀₆	—	4	1	no	—
47	C₂×C₈×C₅₃	—	2	1	no	—
48	C₂×C₈×C₅₃	—	2	1	no	—
49	C₂×C₈×C₅₃	—	2	1	no	—
50	C₂×C₈×C₅₃	—	2	1	no	—
51	C₂×C₈×C₅₃	—	4	2	no	—
52	C₂×C₈×C₅₃	D₈	1	2	yes	—
53	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
54	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₅₃	#41	1	1	yes	characteristic
56	C₄²×C₅₃	#41	1	2	yes	—
57	C₄⋊Dic₅₃	—	2	4	no	—
58	C₄⋊Dic₅₃	—	4	2	no	—
59	C₄⋊Dic₅₃	—	4	1	no	—
60	C₄⋊Dic₅₃	—	4	1	no	—
61	#11719917	—	4	2	no	—
62	#11719917	—	4	2	no	—
63	C₂×#88322	—	4	2	no	—
64	C₂³×C₅₃	#124	1	1	yes	characteristic
65	C₂×C₄×C₅₃	—	4	2	no	—
66	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
67	C₂×C₄×C₅₃	#123	1	2	yes	—
68	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#126	1	2	yes	—
70	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
74	C₂×Dic₅₃	—	4	2	no	—
75	C₂×Dic₅₃	—	8	1	no	—
76	C₂×Dic₅₃	—	8	1	no	—
77	C₂×Dic₅₃	—	8	2	no	—
78	C₂×Dic₅₃	—	8	1	no	—
79	Dic₁₀₆	—	4	2	no	—
80	Dic₁₀₆	—	4	2	no	—
81	Dic₁₀₆	—	8	1	no	—
82	Dic₁₀₆	—	8	1	no	—
83	C₈×C₅₃	—	2	2	no	—
84	C₈×C₅₃	—	2	2	no	—
85	C₈×C₅₃	—	8	2	no	—
86	#88322	—	8	2	no	—
87	#10632	—	53	1	no	maximal, Sylow
88	C₂²×C₅₃	—	2	2	no	—
89	C₂²×C₅₃	#397	1	1	yes	characteristic
90	C₂²×C₅₃	#486	1	1	yes	characteristic
91	C₂²×C₅₃	#489	1	1	yes	characteristic
92	C₄×C₅₃	—	2	2	no	—
93	C₄×C₅₃	—	2	2	no	—
94	C₄×C₅₃	#459	1	1	yes	characteristic
95	C₄×C₅₃	#461	1	1	yes	characteristic
96	C₄×C₅₃	#480	1	1	yes	characteristic
97	C₄×C₅₃	#481	1	1	yes	characteristic
98	Dic₅₃	—	8	2	no	—
99	Dic₅₃	—	16	1	no	—
100	#1960	—	53	1	no	—
101	#1849	—	53	1	no	—
102	#1546	—	53	2	no	—
103	#1286	—	53	2	no	—
104	#1281	D₅₃	1	1	yes	characteristic
105	C₂×C₅₃	—	2	1	no	—
106	C₂×C₅₃	—	2	1	no	—
107	C₂×C₅₃	#2071	1	1	yes	characteristic
108	C₂×C₅₃	#2082	1	2	yes	—
109	C₂×C₄⋊Q₈	—	53	1	no	—
110	#431	C₂×D₅₃	1	2	yes	—
111	C₂×#108	—	106	1	no	—
112	C₂×#108	—	106	1	no	—
113	#423	—	53	2	no	—
114	C₂×#103	—	106	1	no	—
115	C₂×#103	—	106	1	no	—
116	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
117	#382	—	53	1	no	—
118	#332	—	106	4	no	—
119	C₅₃	#10632	1	1	yes	characteristic, complemented, Sylow
120	C₂²×Q₈	—	106	1	no	—
121	C₂×M₁₆	—	106	2	no	—
122	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
123	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
124	C₄⋊Q₈	—	53	2	no	—
125	C₄⋊Q₈	—	106	2	no	—
126	C₄⋊Q₈	—	106	1	no	—
127	C₂×C₄⋊C₄	—	106	2	no	—
128	C₂×C₄⋊C₄	—	106	1	no	—
129	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
130	#108	—	212	2	no	—
131	#108	—	212	2	no	—
132	C₄⋊C₈	—	2	2	no	—
133	C₄⋊C₈	—	106	2	no	—
134	C₄⋊C₈	—	106	2	no	—
135	#103	—	212	2	no	—
136	#103	—	212	2	no	—
137	C₂²⋊C₈	—	2	2	no	—
138	#97	—	2	2	no	—
139	C₄×C₈	—	2	2	no	—
140	#95	—	106	2	no	—
141	C₂×Q₈	—	106	2	no	—
142	C₂×Q₈	—	212	1	no	—
143	C₂×Q₈	—	212	1	no	—
144	C₂×Q₈	—	212	1	no	—
145	C₂×Q₈	—	212	1	no	—
146	C₂×Q₈	—	212	1	no	—
147	C₂²×C₄	—	212	1	no	—
148	C₂²×C₄	—	212	1	no	—
149	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
150	C₂²×C₄	#11719945	1	2	yes	—
151	M₁₆	—	212	2	no	—
152	M₁₆	—	212	2	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	212	2	no	—
159	C₂×C₈	#11719913	1	2	yes	—
160	C₂×C₈	D₄₂₄	1	2	yes	—
161	C₄⋊C₄	—	106	4	no	—
162	C₄⋊C₄	—	212	1	no	—
163	C₄⋊C₄	—	212	1	no	—
164	C₄⋊C₄	—	212	2	no	—
165	C₄²	D₄×D₅₃	1	1	yes	characteristic
166	C₄²	#11719947	1	2	yes	—
167	C₄²	#11719949	1	1	yes	characteristic
168	C₂³	#13133205	1	1	yes	characteristic
169	Q₈	—	424	1	no	—
170	Q₈	—	424	1	no	—
171	Q₈	—	212	2	no	—
172	Q₈	—	212	2	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	424	2	no	—
175	C₂×C₄	—	424	1	no	—
176	C₂×C₄	—	424	1	no	—
177	C₂×C₄	—	424	1	no	—
178	C₂×C₄	—	212	2	no	—
179	C₂×C₄	#13133192	1	1	yes	characteristic
180	C₂×C₄	#13133199	1	2	yes	—
181	C₂×C₄	#13133201	1	1	yes	characteristic
182	C₂×C₄	#13133212	1	2	yes	—
183	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
184	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
185	C₂×C₄	#13133245	1	1	yes	characteristic
186	C₂×C₄	#13133250	1	1	yes	characteristic
187	C₈	—	2	2	no	—
188	C₈	—	2	2	no	—
189	C₈	—	8	2	no	—
190	C₈	—	424	2	no	—
191	C₂²	—	2	2	no	—
192	C₂²	#16810550	1	1	yes	characteristic
193	C₂²	#16810970	1	1	yes	characteristic
194	C₂²	#16810983	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	848	1	no	—
198	C₄	—	424	2	no	—
199	C₄	#16810819	1	1	yes	characteristic
200	C₄	#16810829	1	1	yes	characteristic
201	C₄	#16810937	1	1	yes	characteristic
202	C₄	#16810942	1	1	yes	characteristic
203	C₂	—	2	1	no	—
204	C₂	—	2	1	no	—
205	C₂	#20276120	1	1	yes	characteristic
206	C₂	#20276181	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20275535

C₂

yes

maximal, characteristic

#20274890

C₂

yes

maximal, characteristic

#20273386

C₂

yes

maximal

#20272574

C₂

yes

maximal

C₅₃×#1281

C₂

yes

maximal, characteristic

C₂×#13133217

C₂²

yes

—

—

C₂²

yes

—

C₂×#13133142

—

C₂×#13133142

—

#16810489

C₂²

yes

characteristic

#16810368

—

C₅₃×#431

C₂²

yes

—

C₂×#97×C₅₃

C₂²

yes

characteristic

C₂²×Dic₁₀₆

—

—

—

—

D₄

yes

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

—

—

—

—

—

—

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

#97×C₅₃

—

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₄²×C₅₃

C₂×D₄

yes

C₄²×C₅₃

yes

C₄²×C₅₃

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719917

—

#11719917

—

C₂×#88322

—

C₂³×C₅₃

#124

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#121

yes

characteristic

C₂×C₄×C₅₃

#123

yes

—

C₂×C₄×C₅₃

#123

yes

characteristic

C₂×C₄×C₅₃

#126

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#10632

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

100

#1960

—

101

#1849

—

102

#1546

—

103

#1286

—

104

#1281

D₅₃

yes

characteristic

105

C₂×C₅₃

—

106

C₂×C₅₃

—

107

C₂×C₅₃

#2071

yes

characteristic

108

C₂×C₅₃

#2082

yes

—

109

C₂×C₄⋊Q₈

—

110

#431

C₂×D₅₃

yes

—

111

C₂×#108

—

106

—

112

C₂×#108

—

106

—

113

#423

—

114

C₂×#103

—

106

—

115

C₂×#103

—

106

—

116

C₂×#97

C₂×D₅₃

yes

characteristic

117

#382

—

118

#332

—

106

—

119

C₅₃

#10632

yes

characteristic, complemented, Sylow

120

C₂²×Q₈

—

106

—

121

C₂×M₁₆

—

106

—

122

C₂²×C₈

D₂₁₂

yes

characteristic

123

C₂²×C₈

D₂₁₂

yes

characteristic

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

106

—

126

C₄⋊Q₈

—

106

—

127

C₂×C₄⋊C₄

—

106

—

128

C₂×C₄⋊C₄

—

106

—

129

C₂×C₄²

C₂²×D₅₃

yes

characteristic

130

#108

—

212

—

131

#108

—

212

—

132

C₄⋊C₈

—

133

C₄⋊C₈

—

106

—

134

C₄⋊C₈

—

106

—

135

#103

—

212

—

136

#103

—

212

—

137

C₂²⋊C₈

—

138

#97

—

139

C₄×C₈

—

140

#95

—

106

—

141

C₂×Q₈

—

106

—

142

C₂×Q₈

—

212

—

143

C₂×Q₈

—

212

—

144

C₂×Q₈

—

212

—

145

C₂×Q₈

—

212

—

146

C₂×Q₈

—

212

—

147

C₂²×C₄

—

212

—

148

C₂²×C₄

—

212

—

149

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

150

C₂²×C₄

#11719945

yes

—

151

M₁₆

—

212

—

152

M₁₆

—

212

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

212

—

159

C₂×C₈

#11719913

yes

—

160

C₂×C₈

D₄₂₄

yes

—

161

C₄⋊C₄

—

106

—

162

C₄⋊C₄

—

212

—

163

C₄⋊C₄

—

212

—

164

C₄⋊C₄

—

212

—

165

C₄²

D₄×D₅₃

yes

characteristic

166

C₄²

#11719947

yes

—

167

C₄²

#11719949

yes

characteristic

168

C₂³

#13133205

yes

characteristic

169

Q₈

—

424

—

170

Q₈

—

424

—

171

Q₈

—

212

—

172

Q₈

—

212

—

173

C₂×C₄

—

174

C₂×C₄

—

424

—

175

C₂×C₄

—

424

—

176

C₂×C₄

—

424

—

177

C₂×C₄

—

424

—

178

C₂×C₄

—

212

—

179

C₂×C₄

#13133192

yes

characteristic

180

C₂×C₄

#13133199

yes

—

181

C₂×C₄

#13133201

yes

characteristic

182

C₂×C₄

#13133212

yes

—

183

C₂×C₄

C₂×D₄₂₄

yes

characteristic

184

C₂×C₄

C₂×#11719913

yes

characteristic

185

C₂×C₄

#13133245

yes

characteristic

186

C₂×C₄

#13133250

yes

characteristic

187

C₈

—

188

C₈

—

189

C₈

—

190

C₈

—

424

—

191

C₂²

—

192

C₂²

#16810550

yes

characteristic

193

C₂²

#16810970

yes

characteristic

194

C₂²

#16810983

yes

characteristic

195

C₄

—

196

C₄

—

197

C₄

—

848

—

198

C₄

—

424

—

199

C₄

#16810819

yes

characteristic

200

C₄

#16810829

yes

characteristic

201

C₄

#16810937

yes

characteristic

202

C₄

#16810942

yes

characteristic

203

C₂

—

204

C₂

—

205

C₂

#20276120

yes

characteristic

206

C₂

#20276181

yes

—

	subgroup	G/H
center	C₂²	#16810550
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1281	C₂
socle	C₂²×C₅₃	#397
prime radical	C₂³×C₅₃	#124
hypercenter	#1281	D₅₃
hypocenter	C₅₃	#10632
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1281

subgroup

G/H

center

C₂²

#16810550

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1281

C₂

C₂²×C₅₃

C₂³×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1281

N	H ≅ G/N	type
119. C₅₃	#10632	semidirect

H ≅ G/N

type

119. C₅₃

#10632

semidirect

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

subgroup	index
#20275535	2
#20274890	2
#20273386	2
#20272574	2
C₅₃×#1281	2
#10632	53

subgroup

index

C₅₃×#1281

Subgroups — #27288887

Conjugacy classes of subgroups

Complements

Maximal subgroups