Group #27277764 — subgroups

Subgroups — #27277764

Order 13568.

322 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 131 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	#20274831	C₂	1	2	yes	maximal
2	#20274460	C₂	1	1	yes	maximal, characteristic
3	#20272550	C₂	1	1	yes	maximal, characteristic
4	#20272549	C₂	1	2	yes	maximal
5	C₅₃×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
8	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
9	#16810377	—	2	2	no	—
10	#16810377	—	2	2	no	—
11	#16810377	C₄	1	2	yes	—
12	#16810377	C₄	1	2	yes	—
13	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×#88322	—	2	2	no	—
16	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
17	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
20	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
21	C₂×C₄×Dic₅₃	Q₈	1	2	yes	—
22	C₄⋊#88322	—	2	2	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄×#88322	—	2	2	no	—
25	C₂²×C₈×C₅₃	—	2	2	no	—
26	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₅₃	—	2	2	no	—
29	C₄⋊C₈×C₅₃	—	2	2	no	—
30	C₄×C₈×C₅₃	—	2	2	no	—
31	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
34	C₂²×Dic₅₃	C₄⋊C₄	1	2	yes	—
35	C₂²×Dic₅₃	C₄⋊C₄	1	2	yes	—
36	C₂×C₈×C₅₃	—	2	2	no	—
37	C₂×C₈×C₅₃	—	2	2	no	—
38	C₂×C₈×C₅₃	—	2	2	no	—
39	C₂×C₈×C₅₃	—	4	2	no	—
40	C₂×C₈×C₅₃	—	4	2	no	—
41	C₂×C₈×C₅₃	—	4	2	no	—
42	C₄²×C₅₃	#41	1	2	yes	—
43	C₄²×C₅₃	#41	1	2	yes	—
44	C₄×Dic₅₃	—	2	2	no	—
45	C₄×Dic₅₃	—	2	2	no	—
46	C₄×Dic₅₃	—	2	1	no	—
47	C₄×Dic₅₃	—	2	2	no	—
48	C₄×Dic₅₃	—	2	1	no	—
49	C₄×Dic₅₃	—	4	2	no	—
50	C₂×#88322	—	2	1	no	—
51	C₂×#88322	—	2	1	no	—
52	C₂×#88322	—	2	2	no	—
53	C₂×#88322	—	2	2	no	—
54	C₂×#88322	—	4	2	no	—
55	C₂×#88322	—	4	2	no	—
56	C₂×#88322	—	4	2	no	—
57	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	—	2	2	no	—
59	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
60	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
61	C₂×C₄×C₅₃	#124	1	2	yes	—
62	C₂×C₄×C₅₃	#125	1	2	yes	—
63	C₂×C₄×C₅₃	#131	1	2	yes	—
64	C₂×C₄×C₅₃	#131	1	2	yes	—
65	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
66	C₂×C₄×C₅₃	#135	1	2	yes	—
67	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
68	C₂×Dic₅₃	—	2	2	no	—
69	C₂×Dic₅₃	—	2	2	no	—
70	C₂×Dic₅₃	—	2	2	no	—
71	C₂×Dic₅₃	—	2	2	no	—
72	C₂×Dic₅₃	—	2	2	no	—
73	C₂×Dic₅₃	—	2	2	no	—
74	C₈×C₅₃	—	4	2	no	—
75	C₈×C₅₃	—	8	2	no	—
76	#88322	—	4	1	no	—
77	#88322	—	4	1	no	—
78	#88322	—	8	2	no	—
79	#8788	—	53	1	no	maximal, Sylow
80	C₂²×C₅₃	#382	1	1	yes	characteristic
81	C₂²×C₅₃	#424	1	2	yes	—
82	C₂²×C₅₃	#427	1	1	yes	characteristic
83	C₂²×C₅₃	#427	1	1	yes	characteristic
84	C₂²×C₅₃	#430	1	2	yes	—
85	C₄×C₅₃	—	2	2	no	—
86	C₄×C₅₃	#433	1	2	yes	—
87	C₄×C₅₃	#433	1	1	yes	characteristic
88	C₄×C₅₃	#433	1	1	yes	characteristic
89	C₄×C₅₃	#488	1	2	yes	—
90	C₄×C₅₃	#488	1	2	yes	—
91	Dic₅₃	—	4	2	no	—
92	Dic₅₃	—	4	2	no	—
93	#1831	—	53	2	no	—
94	#1744	—	53	1	no	—
95	#1277	—	53	2	no	—
96	#1277	—	53	1	no	—
97	#1277	D₅₃	1	1	yes	characteristic
98	C₂×C₅₃	#1572	1	2	yes	—
99	C₂×C₅₃	#1572	1	2	yes	—
100	C₂×C₅₃	#1832	1	1	yes	characteristic
101	C₂×C₅₃	#1835	1	1	yes	characteristic
102	C₂×C₅₃	#1838	1	1	yes	characteristic
103	C₂×C₄⋊C₈	—	53	2	no	—
104	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
105	C₂×C₄×C₈	—	53	1	no	—
106	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
107	C₄³	—	53	1	no	—
108	#336	—	53	2	no	—
109	#336	—	53	2	no	—
110	#336	—	106	2	no	—
111	#336	—	106	2	no	—
112	C₅₃	#8788	1	1	yes	characteristic, complemented, Sylow
113	C₂²×C₈	—	2	2	no	—
114	C₂²×C₈	—	53	1	no	—
115	C₂²×C₈	—	53	1	no	—
116	C₂²×C₈	—	106	2	no	—
117	C₂²×C₈	C₄×D₅₃	1	2	yes	—
118	C₂×C₄²	—	53	2	no	—
119	C₂×C₄²	—	53	1	no	—
120	C₂×C₄²	—	53	1	no	—
121	C₂×C₄²	—	53	2	no	—
122	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	106	2	no	—
126	C₄⋊C₈	—	106	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	106	2	no	—
129	C₂²×C₄	—	53	2	no	—
130	C₂²×C₄	—	53	2	no	—
131	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
132	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
133	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	106	1	no	—
141	C₂×C₈	—	106	1	no	—
142	C₂×C₈	—	106	2	no	—
143	C₂×C₈	—	106	2	no	—
144	C₂×C₈	—	212	2	no	—
145	C₂×C₈	—	212	2	no	—
146	C₂×C₈	—	212	2	no	—
147	C₄²	—	106	2	no	—
148	C₄²	—	106	2	no	—
149	C₄²	—	106	1	no	—
150	C₄²	—	106	2	no	—
151	C₄²	—	106	1	no	—
152	C₄²	—	212	2	no	—
153	C₄²	#11719945	1	2	yes	—
154	C₄²	#11719947	1	2	yes	—
155	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	106	2	no	—
158	C₂×C₄	—	106	2	no	—
159	C₂×C₄	—	106	2	no	—
160	C₂×C₄	—	106	2	no	—
161	C₂×C₄	—	106	2	no	—
162	C₂×C₄	—	106	2	no	—
163	C₂×C₄	#13133173	1	1	yes	characteristic
164	C₂×C₄	#13133174	1	1	yes	characteristic
165	C₂×C₄	#13133203	1	2	yes	—
166	C₂×C₄	#13133207	1	2	yes	—
167	C₂×C₄	#13133223	1	2	yes	—
168	C₂×C₄	#13133225	1	2	yes	—
169	C₂×C₄	#13133236	1	1	yes	characteristic
170	C₂×C₄	#13133238	1	2	yes	—
171	C₂×C₄	#13133240	1	1	yes	characteristic
172	C₈	—	4	2	no	—
173	C₈	—	8	2	no	—
174	C₈	—	424	2	no	—
175	C₈	—	212	1	no	—
176	C₈	—	212	1	no	—
177	C₂²	#16810487	1	1	yes	characteristic
178	C₂²	#16810648	1	2	yes	—
179	C₂²	#16810658	1	1	yes	characteristic
180	C₂²	#16810660	1	1	yes	characteristic
181	C₂²	#16810670	1	2	yes	—
182	C₄	—	2	2	no	—
183	C₄	—	212	2	no	—
184	C₄	—	212	2	no	—
185	C₄	#16810681	1	2	yes	—
186	C₄	#16810682	1	1	yes	characteristic
187	C₄	#16810685	1	1	yes	characteristic
188	C₄	#16810976	1	2	yes	—
189	C₄	#16810979	1	2	yes	—
190	C₂	#20273492	1	2	yes	—
191	C₂	#20273494	1	2	yes	—
192	C₂	#20274833	1	1	yes	characteristic
193	C₂	#20274844	1	1	yes	characteristic
194	C₂	#20274854	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274831

C₂

yes

maximal

#20274460

C₂

yes

maximal, characteristic

#20272550

C₂

yes

maximal, characteristic

#20272549

C₂

yes

maximal

C₅₃×#1277

C₂

yes

maximal, characteristic

C₂×C₄⋊#88322

C₂²

yes

—

C₂×C₄×#88322

C₂²

yes

characteristic

C₄²×Dic₅₃

C₂²

yes

—

—

C₄

yes

—

#16810377

C₄

yes

—

C₂×C₄⋊C₈×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂²×#88322

—

C₂²×#88322

C₂×C₄

yes

characteristic

C₂²×#88322

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

C₂×C₄×Dic₅₃

D₄

yes

characteristic

C₂×C₄×Dic₅₃

D₄

yes

characteristic

C₂×C₄×Dic₅₃

Q₈

yes

—

C₄⋊#88322

—

C₄⋊#88322

—

C₄×#88322

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂²×Dic₅₃

C₄⋊C₄

yes

—

C₂²×Dic₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

#41

yes

—

C₄²×C₅₃

#41

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

—

—

—

—

—

—

—

C₂³×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

#125

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₂×C₄×C₅₃

#135

yes

characteristic

C₂×C₄×C₅₃

#135

yes

—

C₂×C₄×C₅₃

#135

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₄×C₅₃

—

C₄×C₅₃

#433

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₄×C₅₃

#488

yes

—

Dic₅₃

—

Dic₅₃

—

—

—

—

—

D₅₃

yes

characteristic

C₂×C₅₃

#1572

yes

—

C₂×C₅₃

#1572

yes

—

100

C₂×C₅₃

#1832

yes

characteristic

101

C₂×C₅₃

#1835

yes

characteristic

102

C₂×C₅₃

#1838

yes

characteristic

103

C₂×C₄⋊C₈

—

104

C₂×C₄⋊C₈

C₂×D₅₃

yes

—

105

C₂×C₄×C₈

—

106

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

107

C₄³

—

108

#336

—

109

#336

—

110

#336

—

106

—

111

#336

—

106

—

112

C₅₃

#8788

yes

characteristic, complemented, Sylow

113

C₂²×C₈

—

114

C₂²×C₈

—

115

C₂²×C₈

—

116

C₂²×C₈

—

106

—

117

C₂²×C₈

C₄×D₅₃

yes

—

118

C₂×C₄²

—

119

C₂×C₄²

—

120

C₂×C₄²

—

121

C₂×C₄²

—

122

C₂×C₄²

C₂²×D₅₃

yes

characteristic

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

C₄⋊C₈

—

106

—

126

C₄⋊C₈

—

106

—

127

C₄×C₈

—

128

C₄×C₈

—

106

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

132

C₂²×C₄

D₄×D₅₃

yes

characteristic

133

C₂²×C₄

Q₈×D₅₃

yes

characteristic

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

106

—

141

C₂×C₈

—

106

—

142

C₂×C₈

—

106

—

143

C₂×C₈

—

106

—

144

C₂×C₈

—

212

—

145

C₂×C₈

—

212

—

146

C₂×C₈

—

212

—

147

C₄²

—

106

—

148

C₄²

—

106

—

149

C₄²

—

106

—

150

C₄²

—

106

—

151

C₄²

—

106

—

152

C₄²

—

212

—

153

C₄²

#11719945

yes

—

154

C₄²

#11719947

yes

—

155

C₂³

C₄⋊C₄×D₅₃

yes

characteristic

156

C₂×C₄

—

157

C₂×C₄

—

106

—

158

C₂×C₄

—

106

—

159

C₂×C₄

—

106

—

160

C₂×C₄

—

106

—

161

C₂×C₄

—

106

—

162

C₂×C₄

—

106

—

163

C₂×C₄

#13133173

yes

characteristic

164

C₂×C₄

#13133174

yes

characteristic

165

C₂×C₄

#13133203

yes

—

166

C₂×C₄

#13133207

yes

—

167

C₂×C₄

#13133223

yes

—

168

C₂×C₄

#13133225

yes

—

169

C₂×C₄

#13133236

yes

characteristic

170

C₂×C₄

#13133238

yes

—

171

C₂×C₄

#13133240

yes

characteristic

172

C₈

—

173

C₈

—

174

C₈

—

424

—

175

C₈

—

212

—

176

C₈

—

212

—

177

C₂²

#16810487

yes

characteristic

178

C₂²

#16810648

yes

—

179

C₂²

#16810658

yes

characteristic

180

C₂²

#16810660

yes

characteristic

181

C₂²

#16810670

yes

—

182

C₄

—

183

C₄

—

212

—

184

C₄

—

212

—

185

C₄

#16810681

yes

—

186

C₄

#16810682

yes

characteristic

187

C₄

#16810685

yes

characteristic

188

C₄

#16810976

yes

—

189

C₄

#16810979

yes

—

190

C₂

#20273492

yes

—

191

C₂

#20273494

yes

—

192

C₂

#20274833

yes

characteristic

193

C₂

#20274844

yes

characteristic

194

C₂

#20274854

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1277	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1277	D₅₃
hypocenter	C₅₃	#8788
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

subgroup

G/H

center

C₂²×C₄

D₄×D₅₃

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1277

C₂

socle, prime radical

C₂³×C₅₃

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

N	H ≅ G/N	type
112. C₅₃	#8788	semidirect

H ≅ G/N

type

112. C₅₃

#8788

semidirect

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

subgroup	index
#20274831	2
#20274460	2
#20272550	2
#20272549	2
C₅₃×#1277	2
#8788	53

subgroup

index

C₅₃×#1277

Subgroups — #27277764

Conjugacy classes of subgroups

Complements

Maximal subgroups