Group #27289143 — subgroups

Subgroups — #27289143

Order 13568.

442 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 153 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	#20275675	C₂	1	2	yes	maximal
2	#20275517	C₂	1	1	yes	maximal, characteristic
3	#20274874	C₂	1	2	yes	maximal
4	#20272542	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1845	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133217	C₂²	1	1	yes	characteristic
7	C₂×#13133150	—	2	4	no	—
8	C₂×#13133142	—	2	2	no	—
9	C₂×C₄×#88322	C₂²	1	2	yes	—
10	#16810493	C₂²	1	1	yes	characteristic
11	#16810385	—	2	4	no	—
12	C₂×#107×C₅₃	—	2	2	no	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#384	C₂²	1	2	yes	—
15	C₂²×Dic₁₀₆	—	2	1	no	—
16	C₂²×#88322	D₄	1	4	yes	—
17	#13133217	—	2	1	no	—
18	#13133217	—	2	2	no	—
19	#13133217	Q₈	1	2	yes	—
20	C₂×C₄⋊Dic₅₃	—	2	2	no	—
21	C₂×C₄⋊Dic₅₃	—	2	1	no	—
22	C₂×C₄⋊Dic₅₃	—	2	2	no	—
23	#13133150	—	4	8	no	—
24	#13133142	—	4	4	no	—
25	C₄×#88322	—	2	2	no	—
26	C₄×#88322	—	2	2	no	—
27	C₂²×C₈×C₅₃	D₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
31	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	#107×C₅₃	—	4	4	no	—
33	C₄×C₈×C₅₃	—	2	2	no	—
34	C₂²×C₄×C₅₃	—	4	2	no	—
35	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
37	C₂²×Dic₅₃	—	4	1	no	—
38	C₂²×Dic₅₃	—	4	1	no	—
39	C₂×Dic₁₀₆	—	2	2	no	—
40	C₂×Dic₁₀₆	—	4	2	no	—
41	C₂×Dic₁₀₆	—	4	2	no	—
42	C₂×Dic₁₀₆	—	4	1	no	—
43	C₂×C₈×C₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	—	2	2	no	—
45	C₂×C₈×C₅₃	D₈	1	4	yes	—
46	C₄⋊C₄×C₅₃	—	4	4	no	—
47	C₄⋊C₄×C₅₃	—	4	4	no	—
48	C₄⋊C₄×C₅₃	—	4	4	no	—
49	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₅₃	#41	1	2	yes	—
51	C₄²×C₅₃	#41	1	1	yes	characteristic
52	C₄⋊Dic₅₃	—	2	4	no	—
53	C₄⋊Dic₅₃	—	4	2	no	—
54	C₄⋊Dic₅₃	—	4	2	no	—
55	C₄⋊Dic₅₃	—	4	4	no	—
56	C₂×#88322	—	2	4	no	—
57	C₂×#88322	—	2	4	no	—
58	C₂×#88322	Q₁₆	1	8	yes	—
59	C₂³×C₅₃	#127	1	1	yes	characteristic
60	C₂×C₄×C₅₃	—	8	2	no	—
61	C₂×C₄×C₅₃	—	8	2	no	—
62	C₂×C₄×C₅₃	—	8	2	no	—
63	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
65	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
66	C₂×C₄×C₅₃	#123	1	2	yes	—
67	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
68	C₂×C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
69	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
70	C₂×C₄×C₅₃	#137	1	2	yes	—
71	C₂×Dic₅₃	—	4	2	no	—
72	C₂×Dic₅₃	—	8	1	no	—
73	C₂×Dic₅₃	—	8	1	no	—
74	C₂×Dic₅₃	—	8	1	no	—
75	C₂×Dic₅₃	—	8	2	no	—
76	Dic₁₀₆	—	4	4	no	—
77	Dic₁₀₆	—	8	2	no	—
78	C₈×C₅₃	—	2	4	no	—
79	#88322	—	2	8	no	—
80	#10674	—	53	1	no	maximal, Sylow
81	C₂²×C₅₃	#399	1	1	yes	characteristic
82	C₂²×C₅₃	#493	1	1	yes	characteristic
83	C₂²×C₅₃	#494	1	2	yes	—
84	C₂²×C₅₃	#496	1	2	yes	—
85	C₂²×C₅₃	#497	1	1	yes	characteristic
86	C₄×C₅₃	—	16	2	no	—
87	C₄×C₅₃	#474	1	2	yes	—
88	C₄×C₅₃	#477	1	4	yes	—
89	C₄×C₅₃	#480	1	2	yes	—
90	C₄×C₅₃	#484	1	4	yes	—
91	Dic₅₃	—	8	2	no	—
92	Dic₅₃	—	16	1	no	—
93	#1990	—	53	2	no	—
94	#1956	—	53	1	no	—
95	#1845	—	53	2	no	—
96	#1845	D₅₃	1	1	yes	characteristic
97	#1273	—	53	1	no	—
98	C₂×C₅₃	#1713	1	1	yes	characteristic
99	C₂×C₅₃	#1721	1	1	yes	characteristic
100	C₂×C₅₃	#1723	1	2	yes	—
101	C₂×C₅₃	#2093	1	1	yes	characteristic
102	C₂×C₅₃	#2095	1	2	yes	—
103	C₂×C₄⋊Q₈	—	53	1	no	—
104	C₂×#107	—	2	2	no	—
105	C₂×#107	—	106	4	no	—
106	C₂×#103	—	106	2	no	—
107	C₂×C₄×C₈	—	53	2	no	—
108	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
109	#384	—	53	1	no	—
110	#384	C₂×D₅₃	1	2	yes	—
111	#340	—	106	4	no	—
112	C₅₃	#10674	1	1	yes	characteristic, complemented, Sylow
113	C₂²×Q₈	—	106	1	no	—
114	C₂²×C₈	—	53	4	no	—
115	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
116	C₄⋊Q₈	—	53	2	no	—
117	C₄⋊Q₈	—	106	2	no	—
118	C₄⋊Q₈	—	106	1	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄⋊C₄	—	106	2	no	—
123	C₂×C₄⋊C₄	—	106	2	no	—
124	C₂×C₄⋊C₄	—	106	1	no	—
125	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
126	#107	—	4	4	no	—
127	#107	—	212	8	no	—
128	#103	—	212	4	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	106	2	no	—
131	C₄×C₈	—	106	2	no	—
132	C₂×Q₈	—	106	2	no	—
133	C₂×Q₈	—	212	2	no	—
134	C₂×Q₈	—	212	2	no	—
135	C₂×Q₈	—	212	1	no	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	—	212	1	no	—
138	C₂²×C₄	—	212	1	no	—
139	C₂²×C₄	D₄×D₅₃	1	2	yes	—
140	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	53	8	no	—
144	C₂×C₈	—	106	4	no	—
145	C₂×C₈	—	106	4	no	—
146	C₂×C₈	#11719922	1	4	yes	—
147	C₄⋊C₄	—	4	4	no	—
148	C₄⋊C₄	—	4	4	no	—
149	C₄⋊C₄	—	4	4	no	—
150	C₄⋊C₄	—	106	4	no	—
151	C₄⋊C₄	—	212	2	no	—
152	C₄⋊C₄	—	212	4	no	—
153	C₄⋊C₄	—	212	2	no	—
154	C₄²	#11719945	1	2	yes	—
155	C₄²	Q₈×D₅₃	1	1	yes	characteristic
156	C₄²	#11719949	1	1	yes	characteristic
157	C₂³	#13133213	1	1	yes	characteristic
158	Q₈	—	424	2	no	—
159	Q₈	—	212	4	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	—	424	2	no	—
164	C₂×C₄	—	424	1	no	—
165	C₂×C₄	—	424	1	no	—
166	C₂×C₄	—	424	1	no	—
167	C₂×C₄	—	212	2	no	—
168	C₂×C₄	#13133193	1	1	yes	characteristic
169	C₂×C₄	#13133195	1	2	yes	—
170	C₂×C₄	#13133200	1	1	yes	characteristic
171	C₂×C₄	#13133202	1	2	yes	—
172	C₂×C₄	C₂×#11719922	1	2	yes	—
173	C₂×C₄	Q₁₆×D₅₃	1	4	yes	—
174	C₂×C₄	#13133241	1	4	yes	—
175	C₂×C₄	#13133248	1	2	yes	—
176	C₈	—	2	4	no	—
177	C₈	—	106	8	no	—
178	C₂²	#16810558	1	1	yes	characteristic
179	C₂²	#16810999	1	1	yes	characteristic
180	C₂²	#16811001	1	2	yes	—
181	C₂²	#16811011	1	2	yes	—
182	C₂²	#16811015	1	1	yes	characteristic
183	C₄	—	16	2	no	—
184	C₄	—	848	1	no	—
185	C₄	—	424	2	no	—
186	C₄	#16810900	1	2	yes	—
187	C₄	#16810917	1	4	yes	—
188	C₄	#16810936	1	2	yes	—
189	C₄	#16810960	1	4	yes	—
190	C₂	#20274331	1	1	yes	characteristic
191	C₂	#20274382	1	1	yes	characteristic
192	C₂	#20274391	1	2	yes	—
193	C₂	#20276244	1	1	yes	characteristic
194	C₂	#20276250	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20275675

C₂

yes

maximal

#20275517

C₂

yes

maximal, characteristic

#20274874

C₂

yes

maximal

#20272542

C₂

yes

maximal, characteristic

C₅₃×#1845

C₂

yes

maximal, characteristic

C₂×#13133217

C₂²

yes

characteristic

C₂×#13133150

—

C₂×#13133142

—

C₂×C₄×#88322

C₂²

yes

—

#16810493

C₂²

yes

characteristic

#16810385

—

C₂×#107×C₅₃

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#384

C₂²

yes

—

C₂²×Dic₁₀₆

—

C₂²×#88322

D₄

yes

—

#13133217

—

#13133217

—

#13133217

Q₈

yes

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

—

—

—

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#107×C₅₃

—

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₈

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₄²×C₅₃

#41

yes

—

C₄²×C₅₃

#41

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

Q₁₆

yes

—

C₂³×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

#123

yes

characteristic

C₂×C₄×C₅₃

#123

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

—

C₂×C₄×C₅₃

C₂×Q₁₆

yes

—

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#496

yes

—

C₂²×C₅₃

#497

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#474

yes

—

C₄×C₅₃

#477

yes

—

C₄×C₅₃

#480

yes

—

C₄×C₅₃

#484

yes

—

Dic₅₃

—

Dic₅₃

—

—

—

—

D₅₃

yes

characteristic

#1273

—

C₂×C₅₃

yes

C₂×C₅₃

yes

100

C₂×C₅₃

#1723

yes

—

101

C₂×C₅₃

#2093

yes

characteristic

102

C₂×C₅₃

#2095

yes

—

103

C₂×C₄⋊Q₈

—

104

C₂×#107

—

105

C₂×#107

—

106

—

106

C₂×#103

—

106

—

107

C₂×C₄×C₈

—

108

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

109

#384

—

110

#384

C₂×D₅₃

yes

—

111

#340

—

106

—

112

C₅₃

#10674

yes

characteristic, complemented, Sylow

113

C₂²×Q₈

—

106

—

114

C₂²×C₈

—

115

C₂²×C₈

C₂²⋊D₅₃

yes

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

—

106

—

118

C₄⋊Q₈

—

106

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

106

—

123

C₂×C₄⋊C₄

—

106

—

124

C₂×C₄⋊C₄

—

106

—

125

C₂×C₄²

C₂²×D₅₃

yes

characteristic

126

#107

—

127

#107

—

212

—

128

#103

—

212

—

129

C₄×C₈

—

130

C₄×C₈

—

106

—

131

C₄×C₈

—

106

—

132

C₂×Q₈

—

106

—

133

C₂×Q₈

—

212

—

134

C₂×Q₈

—

212

—

135

C₂×Q₈

—

212

—

136

C₂²×C₄

—

137

C₂²×C₄

—

212

—

138

C₂²×C₄

—

212

—

139

C₂²×C₄

D₄×D₅₃

yes

—

140

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

106

—

145

C₂×C₈

—

106

—

146

C₂×C₈

#11719922

yes

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

106

—

151

C₄⋊C₄

—

212

—

152

C₄⋊C₄

—

212

—

153

C₄⋊C₄

—

212

—

154

C₄²

#11719945

yes

—

155

C₄²

Q₈×D₅₃

yes

characteristic

156

C₄²

#11719949

yes

characteristic

157

C₂³

#13133213

yes

characteristic

158

Q₈

—

424

—

159

Q₈

—

212

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

424

—

164

C₂×C₄

—

424

—

165

C₂×C₄

—

424

—

166

C₂×C₄

—

424

—

167

C₂×C₄

—

212

—

168

C₂×C₄

#13133193

yes

characteristic

169

C₂×C₄

#13133195

yes

—

170

C₂×C₄

#13133200

yes

characteristic

171

C₂×C₄

#13133202

yes

—

172

C₂×C₄

C₂×#11719922

yes

—

173

C₂×C₄

Q₁₆×D₅₃

yes

—

174

C₂×C₄

#13133241

yes

—

175

C₂×C₄

#13133248

yes

—

176

C₈

—

177

C₈

—

106

—

178

C₂²

#16810558

yes

characteristic

179

C₂²

#16810999

yes

characteristic

180

C₂²

#16811001

yes

—

181

C₂²

#16811011

yes

—

182

C₂²

#16811015

yes

characteristic

183

C₄

—

184

C₄

—

848

—

185

C₄

—

424

—

186

C₄

#16810900

yes

—

187

C₄

#16810917

yes

—

188

C₄

#16810936

yes

—

189

C₄

#16810960

yes

—

190

C₂

#20274331

yes

characteristic

191

C₂

#20274382

yes

characteristic

192

C₂

#20274391

yes

—

193

C₂

#20276244

yes

characteristic

194

C₂

#20276250

yes

—

	subgroup	G/H
center	C₂³	#13133213
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1845	C₂
socle, prime radical	C₂³×C₅₃	#127
hypercenter	#1845	D₅₃
hypocenter	C₅₃	#10674
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

subgroup

G/H

center

C₂³

#13133213

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1845

C₂

C₂³×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

N	H ≅ G/N	type
112. C₅₃	#10674	semidirect

H ≅ G/N

type

112. C₅₃

#10674

semidirect

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

subgroup	index
#20275675	2
#20275517	2
#20274874	2
#20272542	2
C₅₃×#1845	2
#10674	53

subgroup

index

C₅₃×#1845

Subgroups — #27289143

Conjugacy classes of subgroups

Complements

Maximal subgroups