Group #27289156 — subgroups

Subgroups — #27289156

Order 13568.

410 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 147 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	#20275677	C₂	1	2	yes	maximal
2	#20274875	C₂	1	1	yes	maximal, characteristic
3	#20274842	C₂	1	2	yes	maximal
4	#20272542	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1956	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133151	—	2	2	no	—
7	C₂×C₄×#88322	C₂²	1	2	yes	—
8	#16810493	C₂²	1	2	yes	—
9	#16810385	—	2	4	no	—
10	#16810385	—	2	4	no	—
11	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×#103×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	#13133151	—	4	4	no	—
20	C₄×#88322	—	2	2	no	—
21	C₄×#88322	—	2	2	no	—
22	C₂²×Q₈×C₅₃	—	2	1	no	—
23	C₂²×C₈×C₅₃	D₄	1	2	yes	—
24	C₄⋊Q₈×C₅₃	—	2	1	no	—
25	C₄⋊Q₈×C₅₃	—	2	2	no	—
26	C₄⋊Q₈×C₅₃	Q₈	1	2	yes	—
27	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
30	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	#103×C₅₃	—	4	4	no	—
32	C₄×C₈×C₅₃	—	2	2	no	—
33	C₂×Q₈×C₅₃	—	2	2	no	—
34	C₂×Q₈×C₅₃	—	4	1	no	—
35	C₂×Q₈×C₅₃	—	4	2	no	—
36	C₂×Q₈×C₅₃	—	4	2	no	—
37	C₂²×C₄×C₅₃	—	4	1	no	—
38	C₂²×C₄×C₅₃	—	4	1	no	—
39	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
41	C₂²×Dic₅₃	—	4	2	no	—
42	C₂×C₈×C₅₃	—	2	2	no	—
43	C₂×C₈×C₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	D₈	1	4	yes	—
45	C₄⋊C₄×C₅₃	—	2	4	no	—
46	C₄⋊C₄×C₅₃	—	4	2	no	—
47	C₄⋊C₄×C₅₃	—	4	2	no	—
48	C₄⋊C₄×C₅₃	—	4	4	no	—
49	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₅₃	#41	1	1	yes	characteristic
51	C₄²×C₅₃	#41	1	2	yes	—
52	C₄⋊Dic₅₃	—	4	4	no	—
53	C₄⋊Dic₅₃	—	4	4	no	—
54	C₄⋊Dic₅₃	—	4	4	no	—
55	C₂×#88322	—	2	4	no	—
56	C₂×#88322	—	2	4	no	—
57	C₂×#88322	—	2	4	no	—
58	C₂³×C₅₃	#127	1	1	yes	characteristic
59	Q₈×C₅₃	—	4	4	no	—
60	Q₈×C₅₃	—	8	2	no	—
61	C₂×C₄×C₅₃	—	4	2	no	—
62	C₂×C₄×C₅₃	—	8	1	no	—
63	C₂×C₄×C₅₃	—	8	1	no	—
64	C₂×C₄×C₅₃	—	8	1	no	—
65	C₂×C₄×C₅₃	—	8	2	no	—
66	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
67	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#123	1	2	yes	—
70	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
71	C₂×C₄×C₅₃	#135	1	4	yes	—
72	C₂×C₄×C₅₃	#137	1	4	yes	—
73	C₂×C₄×C₅₃	#137	1	2	yes	—
74	C₂×Dic₅₃	—	8	2	no	—
75	C₂×Dic₅₃	—	8	2	no	—
76	C₂×Dic₅₃	—	8	2	no	—
77	C₈×C₅₃	—	2	4	no	—
78	#88322	—	4	4	no	—
79	#10676	—	53	1	no	maximal, Sylow
80	C₂²×C₅₃	#399	1	1	yes	characteristic
81	C₂²×C₅₃	#493	1	1	yes	characteristic
82	C₂²×C₅₃	#495	1	2	yes	—
83	C₂²×C₅₃	#497	1	2	yes	—
84	C₂²×C₅₃	#497	1	1	yes	characteristic
85	C₄×C₅₃	—	8	2	no	—
86	C₄×C₅₃	—	16	1	no	—
87	C₄×C₅₃	#474	1	2	yes	—
88	C₄×C₅₃	#479	1	4	yes	—
89	C₄×C₅₃	#480	1	2	yes	—
90	C₄×C₅₃	#485	1	4	yes	—
91	Dic₅₃	—	16	2	no	—
92	#1990	—	53	2	no	—
93	#1956	D₅₃	1	1	yes	characteristic
94	#1845	—	53	1	no	—
95	#1834	—	53	2	no	—
96	#1273	—	53	1	no	—
97	C₂×C₅₃	#1709	1	1	yes	characteristic
98	C₂×C₅₃	#1718	1	2	yes	—
99	C₂×C₅₃	#1722	1	1	yes	characteristic
100	C₂×C₅₃	#2093	1	1	yes	characteristic
101	C₂×C₅₃	#2097	1	2	yes	—
102	C₂×C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
103	C₂×#107	—	106	2	no	—
104	C₂×#103	—	2	2	no	—
105	C₂×C₄×C₈	—	53	2	no	—
106	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
107	#384	—	53	2	no	—
108	#384	C₂×D₅₃	1	1	yes	characteristic
109	#340	—	106	4	no	—
110	#340	—	106	4	no	—
111	C₅₃	#10676	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₈	—	2	1	no	—
113	C₂²×C₈	—	53	4	no	—
114	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
115	C₄⋊Q₈	—	2	1	no	—
116	C₄⋊Q₈	—	2	2	no	—
117	C₄⋊Q₈	Dic₁₀₆	1	2	yes	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	1	no	—
121	C₂×C₄⋊C₄	—	106	2	no	—
122	C₂×C₄⋊C₄	—	106	2	no	—
123	C₂×C₄⋊C₄	—	106	2	no	—
124	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
125	#107	—	212	4	no	—
126	#103	—	4	4	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	106	2	no	—
129	C₄×C₈	—	106	2	no	—
130	C₂×Q₈	—	2	2	no	—
131	C₂×Q₈	—	4	1	no	—
132	C₂×Q₈	—	4	2	no	—
133	C₂×Q₈	—	4	2	no	—
134	C₂²×C₄	—	4	1	no	—
135	C₂²×C₄	—	4	1	no	—
136	C₂²×C₄	—	212	2	no	—
137	C₂²×C₄	D₄×D₅₃	1	2	yes	—
138	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	106	4	no	—
142	C₂×C₈	—	106	4	no	—
143	C₂×C₈	—	106	4	no	—
144	C₂×C₈	#11719922	1	4	yes	—
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	4	4	no	—
149	C₄⋊C₄	—	212	4	no	—
150	C₄⋊C₄	—	212	4	no	—
151	C₄⋊C₄	—	212	4	no	—
152	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
153	C₄²	#11719945	1	1	yes	characteristic
154	C₄²	#11719947	1	2	yes	—
155	C₂³	#13133213	1	1	yes	characteristic
156	Q₈	—	4	4	no	—
157	Q₈	—	8	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	8	1	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	—	8	1	no	—
163	C₂×C₄	—	424	2	no	—
164	C₂×C₄	—	424	2	no	—
165	C₂×C₄	—	424	2	no	—
166	C₂×C₄	#13133194	1	2	yes	—
167	C₂×C₄	#13133196	1	1	yes	characteristic
168	C₂×C₄	#13133198	1	1	yes	characteristic
169	C₂×C₄	#13133199	1	2	yes	—
170	C₂×C₄	C₂×#11719922	1	2	yes	—
171	C₂×C₄	#13133240	1	4	yes	—
172	C₂×C₄	#13133246	1	4	yes	—
173	C₂×C₄	#13133247	1	2	yes	—
174	C₈	—	2	4	no	—
175	C₈	—	212	4	no	—
176	C₂²	#16810559	1	1	yes	characteristic
177	C₂²	#16810999	1	1	yes	characteristic
178	C₂²	#16811008	1	2	yes	—
179	C₂²	#16811014	1	2	yes	—
180	C₂²	#16811018	1	1	yes	characteristic
181	C₄	—	8	2	no	—
182	C₄	—	16	1	no	—
183	C₄	—	848	2	no	—
184	C₄	#16810903	1	2	yes	—
185	C₄	#16810932	1	4	yes	—
186	C₄	#16810934	1	2	yes	—
187	C₄	#16810963	1	4	yes	—
188	C₂	#20274322	1	1	yes	characteristic
189	C₂	#20274364	1	2	yes	—
190	C₂	#20274384	1	1	yes	characteristic
191	C₂	#20276243	1	1	yes	characteristic
192	C₂	#20276261	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20275677

C₂

yes

maximal

#20274875

C₂

yes

maximal, characteristic

#20274842

C₂

yes

maximal

#20272542

C₂

yes

maximal, characteristic

C₅₃×#1956

C₂

yes

maximal, characteristic

C₂×#13133151

—

C₂×C₄×#88322

C₂²

yes

—

#16810493

C₂²

yes

—

#16810385

—

#16810385

—

C₂×C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₂×#103×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

—

#13133151

—

C₄×#88322

—

C₄×#88322

—

C₂²×Q₈×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

Q₈

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#103×C₅₃

—

C₄×C₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₈

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

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

—

C₂³×C₅₃

#127

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×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×D₈

yes

—

C₂×C₄×C₅₃

#135

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#497

yes

—

C₂²×C₅₃

#497

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#474

yes

—

C₄×C₅₃

#479

yes

—

C₄×C₅₃

#480

yes

—

C₄×C₅₃

#485

yes

—

Dic₅₃

—

#1990

—

#1956

D₅₃

yes

—

—

—

C₂×C₅₃

#1709

yes

characteristic

C₂×C₅₃

#1718

yes

—

C₂×C₅₃

#1722

yes

characteristic

100

C₂×C₅₃

#2093

yes

characteristic

101

C₂×C₅₃

#2097

yes

—

102

C₂×C₄⋊Q₈

C₂×D₅₃

yes

characteristic

103

C₂×#107

—

106

—

104

C₂×#103

—

105

C₂×C₄×C₈

—

106

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

107

#384

—

108

#384

C₂×D₅₃

yes

characteristic

109

#340

—

106

—

110

#340

—

106

—

111

C₅₃

#10676

yes

characteristic, complemented, Sylow

112

C₂²×Q₈

—

113

C₂²×C₈

—

114

C₂²×C₈

C₂²⋊D₅₃

yes

—

115

C₄⋊Q₈

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

Dic₁₀₆

yes

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

106

—

122

C₂×C₄⋊C₄

—

106

—

123

C₂×C₄⋊C₄

—

106

—

124

C₂×C₄²

C₂²×D₅₃

yes

characteristic

125

#107

—

212

—

126

#103

—

127

C₄×C₈

—

128

C₄×C₈

—

106

—

129

C₄×C₈

—

106

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

212

—

137

C₂²×C₄

D₄×D₅₃

yes

—

138

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

106

—

142

C₂×C₈

—

106

—

143

C₂×C₈

—

106

—

144

C₂×C₈

#11719922

yes

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

212

—

150

C₄⋊C₄

—

212

—

151

C₄⋊C₄

—

212

—

152

C₄²

C₂×Dic₁₀₆

yes

characteristic

153

C₄²

#11719945

yes

characteristic

154

C₄²

#11719947

yes

—

155

C₂³

#13133213

yes

characteristic

156

Q₈

—

157

Q₈

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

424

—

164

C₂×C₄

—

424

—

165

C₂×C₄

—

424

—

166

C₂×C₄

#13133194

yes

—

167

C₂×C₄

#13133196

yes

characteristic

168

C₂×C₄

#13133198

yes

characteristic

169

C₂×C₄

#13133199

yes

—

170

C₂×C₄

C₂×#11719922

yes

—

171

C₂×C₄

#13133240

yes

—

172

C₂×C₄

#13133246

yes

—

173

C₂×C₄

#13133247

yes

—

174

C₈

—

175

C₈

—

212

—

176

C₂²

#16810559

yes

characteristic

177

C₂²

#16810999

yes

characteristic

178

C₂²

#16811008

yes

—

179

C₂²

#16811014

yes

—

180

C₂²

#16811018

yes

characteristic

181

C₄

—

182

C₄

—

183

C₄

—

848

—

184

C₄

#16810903

yes

—

185

C₄

#16810932

yes

—

186

C₄

#16810934

yes

—

187

C₄

#16810963

yes

—

188

C₂

#20274322

yes

characteristic

189

C₂

#20274364

yes

—

190

C₂

#20274384

yes

characteristic

191

C₂

#20276243

yes

characteristic

192

C₂

#20276261

yes

—

	subgroup	G/H
center	C₂³	#13133213
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1956	C₂
socle, prime radical	C₂³×C₅₃	#127
hypercenter	#1956	D₅₃
hypocenter	C₅₃	#10676
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

subgroup

G/H

center

C₂³

#13133213

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1956

C₂

C₂³×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

N	H ≅ G/N	type
111. C₅₃	#10676	semidirect

H ≅ G/N

type

111. C₅₃

#10676

semidirect

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

subgroup	index
#20275677	2
#20274875	2
#20274842	2
#20272542	2
C₅₃×#1956	2
#10676	53

subgroup

index

C₅₃×#1956

Subgroups — #27289156

Conjugacy classes of subgroups

Complements

Maximal subgroups