Group #27272356 — subgroups

Subgroups — #27272356

Order 13568.

436 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 135 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	#20274568	C₂	1	2	yes	maximal
2	#20272538	C₂	1	4	yes	maximal
3	C₄×#125×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#88322	—	2	2	no	—
5	C₂×#13133128	—	2	2	no	—
6	#16810378	C₂²	1	4	yes	—
7	C₂×#125×C₅₃	C₄	1	1	yes	characteristic
8	C₅₃×#389	C₂²	1	2	yes	—
9	C₅₃×#387	—	2	2	no	—
10	C₅₃×#382	C₄	1	1	yes	characteristic
11	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
12	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
13	C₄³×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×#88322	—	2	4	no	—
15	C₄⋊#88322	—	2	8	no	—
16	#13133128	—	4	4	no	—
17	#125×C₅₃	—	2	4	no	—
18	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
19	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
20	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
21	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₅₃	—	2	2	no	—
23	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	D₄	1	2	yes	—
26	C₂×C₄²×C₅₃	D₄	1	2	yes	—
27	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
28	#95×C₅₃	—	2	2	no	—
29	#95×C₅₃	—	2	2	no	—
30	#95×C₅₃	—	2	2	no	—
31	C₂²×C₄×C₅₃	—	2	2	no	—
32	C₂²×C₄×C₅₃	—	2	2	no	—
33	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
38	C₄⋊C₄×C₅₃	—	2	8	no	—
39	C₄⋊C₄×C₅₃	—	2	2	no	—
40	C₄⋊C₄×C₅₃	—	4	2	no	—
41	C₄⋊C₄×C₅₃	—	4	2	no	—
42	C₄⋊C₄×C₅₃	—	4	2	no	—
43	C₄⋊C₄×C₅₃	M₁₆	1	4	yes	—
44	C₄²×C₅₃	—	2	2	no	—
45	C₄²×C₅₃	—	2	4	no	—
46	C₄²×C₅₃	—	2	2	no	—
47	C₄²×C₅₃	—	2	2	no	—
48	C₄²×C₅₃	—	2	2	no	—
49	C₄²×C₅₃	—	4	1	no	—
50	C₄²×C₅₃	—	4	4	no	—
51	C₂×#88322	—	2	8	no	—
52	C₂×#88322	—	4	4	no	—
53	C₂×#88322	—	4	4	no	—
54	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	—	2	1	no	—
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	—	2	1	no	—
58	C₂×C₄×C₅₃	—	2	2	no	—
59	C₂×C₄×C₅₃	—	2	4	no	—
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	—	2	1	no	—
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	2	2	no	—
65	C₂×C₄×C₅₃	—	4	2	no	—
66	C₂×C₄×C₅₃	—	4	2	no	—
67	C₂×C₄×C₅₃	—	4	2	no	—
68	C₂×C₄×C₅₃	—	4	2	no	—
69	C₂×C₄×C₅₃	—	4	2	no	—
70	C₂×C₄×C₅₃	#101	1	2	yes	—
71	C₂×C₄×C₅₃	#104	1	4	yes	—
72	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
73	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
74	C₂×C₄×C₅₃	#137	1	2	yes	—
75	#88322	—	4	8	no	—
76	#7636	—	53	1	no	maximal, Sylow
77	C₂²×C₅₃	#407	1	1	yes	characteristic
78	C₂²×C₅₃	#410	1	1	yes	characteristic
79	C₂²×C₅₃	C₂×#101	1	1	yes	characteristic
80	C₂²×C₅₃	#417	1	2	yes	—
81	C₂²×C₅₃	C₂×#104	1	2	yes	—
82	C₄×C₅₃	—	2	2	no	—
83	C₄×C₅₃	—	2	2	no	—
84	C₄×C₅₃	—	2	2	no	—
85	C₄×C₅₃	—	4	1	no	—
86	C₄×C₅₃	—	4	2	no	—
87	C₄×C₅₃	—	4	4	no	—
88	C₄×C₅₃	—	8	2	no	—
89	C₄×C₅₃	#330	1	4	yes	—
90	C₄×#125	D₅₃	1	1	yes	characteristic
91	#1767	—	53	2	no	—
92	#1271	—	53	4	no	—
93	C₂×C₅₃	#1467	1	1	yes	characteristic
94	C₂×C₅₃	#1489	1	2	yes	—
95	C₂×C₅₃	#1509	1	2	yes	—
96	C₂×C₅₃	C₂×#330	1	1	yes	characteristic
97	C₂×C₅₃	#1561	1	1	yes	characteristic
98	C₂×#125	Dic₅₃	1	1	yes	characteristic
99	C₂×C₄⋊C₈	—	106	2	no	—
100	C₂×#97	—	106	2	no	—
101	#389	C₂×D₅₃	1	2	yes	—
102	#387	—	2	2	no	—
103	#382	Dic₅₃	1	1	yes	characteristic
104	C₄×C₄⋊C₄	—	2	2	no	—
105	C₄×C₄⋊C₄	Dic₅₃	1	2	yes	—
106	C₄³	C₂×D₅₃	1	1	yes	characteristic
107	#336	—	53	4	no	—
108	C₅₃	#7636	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₈	—	106	4	no	—
110	#125	—	2	4	no	—
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄⋊C₄	—	2	2	no	—
113	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
114	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
115	C₂×C₄²	—	2	2	no	—
116	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
117	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
118	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
119	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
120	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
121	C₄⋊C₈	—	106	8	no	—
122	#97	—	212	4	no	—
123	#95	—	2	2	no	—
124	#95	—	2	2	no	—
125	#95	—	2	2	no	—
126	C₂²×C₄	—	2	2	no	—
127	C₂²×C₄	—	2	2	no	—
128	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
129	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
130	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
131	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
133	C₂×C₈	—	106	8	no	—
134	C₂×C₈	—	212	4	no	—
135	C₂×C₈	—	212	4	no	—
136	C₄⋊C₄	—	2	8	no	—
137	C₄⋊C₄	—	2	2	no	—
138	C₄⋊C₄	—	4	2	no	—
139	C₄⋊C₄	—	4	2	no	—
140	C₄⋊C₄	—	4	2	no	—
141	C₄⋊C₄	#11719917	1	4	yes	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	—	2	2	no	—
147	C₄²	—	4	1	no	—
148	C₄²	—	4	4	no	—
149	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	#13133136	1	2	yes	—
166	C₂×C₄	#13133144	1	4	yes	—
167	C₂×C₄	C₂×#11719917	1	2	yes	—
168	C₂×C₄	#13133242	1	2	yes	—
169	C₂×C₄	#13133247	1	2	yes	—
170	C₈	—	212	8	no	—
171	C₂²	#16810587	1	1	yes	characteristic
172	C₂²	#16810597	1	1	yes	characteristic
173	C₂²	C₂×#13133136	1	1	yes	characteristic
174	C₂²	#16810624	1	2	yes	—
175	C₂²	C₂×#13133144	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	2	2	no	—
179	C₄	—	4	2	no	—
180	C₄	—	4	4	no	—
181	C₄	—	4	1	no	—
182	C₄	—	8	2	no	—
183	C₄	#16810365	1	4	yes	—
184	C₂	#20273059	1	1	yes	characteristic
185	C₂	#20273154	1	2	yes	—
186	C₂	#20273240	1	2	yes	—
187	C₂	C₂×#16810365	1	1	yes	characteristic
188	C₂	#20273453	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274568

C₂

yes

maximal

#20272538

C₂

yes

maximal

C₄×#125×C₅₃

C₂

yes

maximal, characteristic

C₂×C₄⋊#88322

—

C₂×#13133128

—

#16810378

C₂²

yes

—

C₂×#125×C₅₃

C₄

yes

characteristic

C₅₃×#389

C₂²

yes

—

C₅₃×#387

—

C₅₃×#382

C₄

yes

characteristic

C₄×C₄⋊C₄×C₅₃

—

C₄×C₄⋊C₄×C₅₃

C₄

yes

—

C₄³×C₅₃

C₂²

yes

characteristic

C₂²×#88322

—

C₄⋊#88322

—

#13133128

—

#125×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#95×C₅₃

—

#95×C₅₃

—

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

M₁₆

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#101

yes

—

C₂×C₄×C₅₃

#104

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅₃

#137

yes

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

C₂×#104

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#330

yes

—

C₄×#125

D₅₃

yes

characteristic

#1767

—

#1271

—

C₂×C₅₃

#1467

yes

characteristic

C₂×C₅₃

#1489

yes

—

C₂×C₅₃

#1509

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

Dic₅₃

yes

characteristic

C₂×C₄⋊C₈

—

106

—

100

C₂×#97

—

106

—

101

#389

C₂×D₅₃

yes

—

102

#387

—

103

#382

Dic₅₃

yes

characteristic

104

C₄×C₄⋊C₄

—

105

C₄×C₄⋊C₄

Dic₅₃

yes

—

106

C₄³

C₂×D₅₃

yes

characteristic

107

#336

—

108

C₅₃

#7636

yes

characteristic, complemented, Sylow

109

C₂²×C₈

—

106

—

110

#125

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

C₂×Dic₅₃

yes

—

114

C₂×C₄⋊C₄

C₂×Dic₅₃

yes

—

115

C₂×C₄²

—

116

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

117

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

118

C₂×C₄²

C₂²⋊D₅₃

yes

—

119

C₂×C₄²

C₂²⋊D₅₃

yes

—

120

C₂×C₄²

C₂²×D₅₃

yes

characteristic

121

C₄⋊C₈

—

106

—

122

#97

—

212

—

123

#95

—

124

#95

—

125

#95

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

C₂²⋊Dic₅₃

yes

characteristic

129

C₂²×C₄

C₂²⋊Dic₅₃

yes

—

130

C₂²×C₄

C₂²⋊Dic₅₃

yes

characteristic

131

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

132

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

—

133

C₂×C₈

—

106

—

134

C₂×C₈

—

212

—

135

C₂×C₈

—

212

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

#11719917

yes

—

142

C₄²

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₄²

—

147

C₄²

—

148

C₄²

—

149

C₂³

C₂×C₂²⋊Dic₅₃

yes

characteristic

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

#13133136

yes

—

166

C₂×C₄

#13133144

yes

—

167

C₂×C₄

C₂×#11719917

yes

—

168

C₂×C₄

#13133242

yes

—

169

C₂×C₄

#13133247

yes

—

170

C₈

—

212

—

171

C₂²

#16810587

yes

characteristic

172

C₂²

#16810597

yes

characteristic

173

C₂²

C₂×#13133136

yes

characteristic

174

C₂²

#16810624

yes

—

175

C₂²

C₂×#13133144

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

#16810365

yes

—

184

C₂

#20273059

yes

characteristic

185

C₂

#20273154

yes

—

186

C₂

#20273240

yes

—

187

C₂

C₂×#16810365

yes

characteristic

188

C₂

#20273453

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×#125×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×#125	D₅₃
hypocenter	C₅₃	#7636
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₅₃

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₄×#125×C₅₃

C₂

socle, prime radical

C₂³×C₅₃

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

N	H ≅ G/N	type
108. C₅₃	#7636	semidirect

H ≅ G/N

type

108. C₅₃

#7636

semidirect

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

subgroup	index
#20274568	2
#20272538	2
C₄×#125×C₅₃	2
#7636	53

subgroup

index

#20274568

#20272538

C₄×#125×C₅₃

#7636

Subgroups — #27272356

Conjugacy classes of subgroups

Complements

Maximal subgroups