Group #27274523 — subgroups

Subgroups — #27274523

Order 13568.

414 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 179 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	#20274460	C₂	1	2	yes	maximal
2	C₄×#13133127	C₂	1	1	yes	maximal, characteristic
3	#20272542	C₂	1	1	yes	maximal, characteristic
4	#20272542	C₂	1	2	yes	maximal
5	C₅₃×#1273	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133127	C₄	1	2	yes	—
7	C₂×#13133127	C₄	1	2	yes	—
8	C₂×C₄×#88322	C₂²	1	2	yes	—
9	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
10	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
11	#16810377	C₄	1	4	yes	—
12	#16810377	C₄	1	4	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×#88322	C₂×C₄	1	4	yes	—
16	C₂²×#88322	C₂×C₄	1	2	yes	—
17	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
21	#13133127	—	2	4	no	—
22	#13133127	—	2	4	no	—
23	C₄×#88322	—	2	1	no	—
24	C₄×#88322	—	2	2	no	—
25	C₄×#88322	—	2	1	no	—
26	C₄×#88322	—	2	2	no	—
27	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
30	C₄×C₈×C₅₃	—	2	2	no	—
31	C₄×C₈×C₅₃	—	2	2	no	—
32	C₄×C₈×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×Dic₅₃	C₄²	1	1	yes	characteristic
36	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
37	C₂²×Dic₅₃	C₄²	1	2	yes	—
38	C₂×C₈×C₅₃	—	2	2	no	—
39	C₂×C₈×C₅₃	—	2	2	no	—
40	C₂×C₈×C₅₃	—	2	4	no	—
41	C₂×C₈×C₅₃	—	2	4	no	—
42	C₂×C₈×C₅₃	—	2	4	no	—
43	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
44	C₄²×C₅₃	#41	1	2	yes	—
45	C₄²×C₅₃	#41	1	1	yes	characteristic
46	C₄²×C₅₃	#41	1	1	yes	characteristic
47	C₄×Dic₅₃	—	2	2	no	—
48	C₄×Dic₅₃	—	2	2	no	—
49	C₄×Dic₅₃	—	2	4	no	—
50	C₄×Dic₅₃	—	2	4	no	—
51	C₂×#88322	—	2	2	no	—
52	C₂×#88322	—	2	2	no	—
53	C₂×#88322	—	2	4	no	—
54	C₂×#88322	—	2	4	no	—
55	C₂×#88322	—	2	4	no	—
56	C₂×#88322	M₁₆	1	4	yes	—
57	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
58	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
59	C₂×C₄×C₅₃	#117	1	2	yes	—
60	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
61	C₂×C₄×C₅₃	#117	1	2	yes	—
62	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
63	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
64	C₂×C₄×C₅₃	#131	1	4	yes	—
65	C₂×C₄×C₅₃	#131	1	4	yes	—
66	C₂×Dic₅₃	—	2	1	no	—
67	C₂×Dic₅₃	—	2	1	no	—
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	C₈×C₅₃	—	2	4	no	—
74	C₈×C₅₃	—	4	4	no	—
75	#88322	—	2	4	no	—
76	#88322	—	4	4	no	—
77	#8134	—	53	1	no	maximal, Sylow
78	C₂²×C₅₃	#376	1	1	yes	characteristic
79	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
80	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
81	C₂²×C₅₃	#405	1	2	yes	—
82	C₂²×C₅₃	#405	1	2	yes	—
83	C₄×C₅₃	#432	1	2	yes	—
84	C₄×C₅₃	#432	1	2	yes	—
85	C₄×C₅₃	#433	1	4	yes	—
86	C₄×C₅₃	#433	1	4	yes	—
87	Dic₅₃	—	4	1	no	—
88	Dic₅₃	—	4	2	no	—
89	Dic₅₃	—	4	1	no	—
90	#1744	—	53	2	no	—
91	C₄×#97	—	53	1	no	—
92	#1273	—	53	2	no	—
93	#1273	—	53	1	no	—
94	#1273	D₅₃	1	1	yes	characteristic
95	C₂×C₅₃	#1453	1	1	yes	characteristic
96	C₂×C₅₃	#1453	1	2	yes	—
97	C₂×C₅₃	#1453	1	1	yes	characteristic
98	C₂×C₅₃	#1745	1	1	yes	characteristic
99	C₂×C₅₃	#1746	1	2	yes	—
100	C₂×#97	—	53	2	no	—
101	C₂×#97	—	53	2	no	—
102	C₂×C₄×C₈	—	53	1	no	—
103	C₂×C₄×C₈	—	53	2	no	—
104	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
105	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
106	C₄³	—	53	1	no	—
107	#336	—	53	4	no	—
108	#336	—	53	4	no	—
109	C₅₃	#8134	1	1	yes	characteristic, complemented, Sylow
110	C₂²×C₈	—	53	2	no	—
111	C₂²×C₈	—	53	4	no	—
112	C₂²×C₈	C₄×D₅₃	1	4	yes	—
113	C₂²×C₈	C₄×D₅₃	1	2	yes	—
114	C₂×C₄²	—	53	2	no	—
115	C₂×C₄²	—	53	2	no	—
116	C₂×C₄²	—	53	1	no	—
117	C₂×C₄²	—	53	1	no	—
118	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
119	#97	—	106	4	no	—
120	#97	—	106	4	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	—	2	2	no	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	—	106	1	no	—
125	C₄×C₈	—	106	2	no	—
126	C₄×C₈	—	106	1	no	—
127	C₄×C₈	—	106	2	no	—
128	C₂²×C₄	—	53	1	no	—
129	C₂²×C₄	—	53	1	no	—
130	C₂²×C₄	—	53	2	no	—
131	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	2	4	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	53	4	no	—
139	C₂×C₈	—	106	2	no	—
140	C₂×C₈	—	106	2	no	—
141	C₂×C₈	—	106	4	no	—
142	C₂×C₈	—	106	4	no	—
143	C₂×C₈	—	106	4	no	—
144	C₂×C₈	#11719912	1	4	yes	—
145	C₄²	—	106	2	no	—
146	C₄²	—	106	2	no	—
147	C₄²	—	106	4	no	—
148	C₄²	—	106	4	no	—
149	C₄²	#11719947	1	2	yes	—
150	C₄²	#11719947	1	1	yes	characteristic
151	C₄²	#11719949	1	1	yes	characteristic
152	C₂³	C₄²×D₅₃	1	1	yes	characteristic
153	C₂×C₄	—	106	1	no	—
154	C₂×C₄	—	106	1	no	—
155	C₂×C₄	—	106	2	no	—
156	C₂×C₄	—	106	2	no	—
157	C₂×C₄	—	106	2	no	—
158	C₂×C₄	—	106	2	no	—
159	C₂×C₄	—	106	2	no	—
160	C₂×C₄	#13133173	1	1	yes	characteristic
161	C₂×C₄	#13133173	1	2	yes	—
162	C₂×C₄	#13133175	1	1	yes	characteristic
163	C₂×C₄	#13133175	1	2	yes	—
164	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
165	C₂×C₄	C₂×#11719912	1	2	yes	—
166	C₂×C₄	#13133223	1	4	yes	—
167	C₂×C₄	#13133225	1	4	yes	—
168	C₈	—	2	4	no	—
169	C₈	—	4	4	no	—
170	C₈	—	106	4	no	—
171	C₈	—	212	4	no	—
172	C₂²	#16810468	1	1	yes	characteristic
173	C₂²	C₄×#11719912	1	1	yes	characteristic
174	C₂²	#16810575	1	1	yes	characteristic
175	C₂²	#16810577	1	2	yes	—
176	C₂²	#16810580	1	2	yes	—
177	C₄	—	212	1	no	—
178	C₄	—	212	2	no	—
179	C₄	—	212	1	no	—
180	C₄	#16810677	1	2	yes	—
181	C₄	#16810679	1	2	yes	—
182	C₄	#16810681	1	4	yes	—
183	C₄	#16810683	1	4	yes	—
184	C₂	#20273009	1	1	yes	characteristic
185	C₂	#20273011	1	2	yes	—
186	C₂	#20273013	1	1	yes	characteristic
187	C₂	#20274462	1	1	yes	characteristic
188	C₂	#20274465	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20274460

C₂

yes

maximal

C₄×#13133127

C₂

yes

maximal, characteristic

#20272542

C₂

yes

maximal, characteristic

#20272542

C₂

yes

maximal

C₅₃×#1273

C₂

yes

maximal, characteristic

C₂×#13133127

C₄

yes

—

C₂×#13133127

C₄

yes

—

C₂×C₄×#88322

C₂²

yes

—

C₂×C₄×#88322

C₂²

yes

characteristic

C₄²×Dic₅₃

C₂²

yes

characteristic

#16810377

C₄

yes

—

#16810377

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂²×#88322

C₂×C₄

yes

—

C₂²×#88322

C₂×C₄

yes

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

—

—

—

—

—

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

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

C₂²×C₄

yes

—

C₂²×Dic₅₃

C₄²

yes

characteristic

C₂²×Dic₅₃

C₄²

yes

characteristic

C₂²×Dic₅₃

C₄²

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

M₁₆

yes

—

C₄²×C₅₃

#41

yes

—

C₄²×C₅₃

yes

C₄²×C₅₃

yes

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

—

—

—

—

—

M₁₆

yes

—

C₂³×C₅₃

C₂×C₄²

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

#117

yes

characteristic

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

C₄×M₁₆

yes

characteristic

C₂²×C₅₃

C₄×M₁₆

yes

characteristic

C₂²×C₅₃

#405

yes

—

C₂²×C₅₃

#405

yes

—

C₄×C₅₃

#432

yes

—

C₄×C₅₃

#432

yes

—

C₄×C₅₃

#433

yes

—

C₄×C₅₃

#433

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#1744

—

C₄×#97

—

#1273

—

#1273

—

#1273

D₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

100

C₂×#97

—

101

C₂×#97

—

102

C₂×C₄×C₈

—

103

C₂×C₄×C₈

—

104

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

105

C₂×C₄×C₈

C₂×D₅₃

yes

—

106

C₄³

—

107

#336

—

108

#336

—

109

C₅₃

#8134

yes

characteristic, complemented, Sylow

110

C₂²×C₈

—

111

C₂²×C₈

—

112

C₂²×C₈

C₄×D₅₃

yes

—

113

C₂²×C₈

C₄×D₅₃

yes

—

114

C₂×C₄²

—

115

C₂×C₄²

—

116

C₂×C₄²

—

117

C₂×C₄²

—

118

C₂×C₄²

C₂²×D₅₃

yes

characteristic

119

#97

—

106

—

120

#97

—

106

—

121

C₄×C₈

—

122

C₄×C₈

—

123

C₄×C₈

—

124

C₄×C₈

—

106

—

125

C₄×C₈

—

106

—

126

C₄×C₈

—

106

—

127

C₄×C₈

—

106

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

132

C₂²×C₄

C₂×C₄×D₅₃

yes

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

106

—

140

C₂×C₈

—

106

—

141

C₂×C₈

—

106

—

142

C₂×C₈

—

106

—

143

C₂×C₈

—

106

—

144

C₂×C₈

#11719912

yes

—

145

C₄²

—

106

—

146

C₄²

—

106

—

147

C₄²

—

106

—

148

C₄²

—

106

—

149

C₄²

#11719947

yes

—

150

C₄²

#11719947

yes

characteristic

151

C₄²

#11719949

yes

characteristic

152

C₂³

C₄²×D₅₃

yes

characteristic

153

C₂×C₄

—

106

—

154

C₂×C₄

—

106

—

155

C₂×C₄

—

106

—

156

C₂×C₄

—

106

—

157

C₂×C₄

—

106

—

158

C₂×C₄

—

106

—

159

C₂×C₄

—

106

—

160

C₂×C₄

#13133173

yes

characteristic

161

C₂×C₄

#13133173

yes

—

162

C₂×C₄

#13133175

yes

characteristic

163

C₂×C₄

#13133175

yes

—

164

C₂×C₄

M₁₆×D₅₃

yes

—

165

C₂×C₄

C₂×#11719912

yes

—

166

C₂×C₄

#13133223

yes

—

167

C₂×C₄

#13133225

yes

—

168

C₈

—

169

C₈

—

170

C₈

—

106

—

171

C₈

—

212

—

172

C₂²

#16810468

yes

characteristic

173

C₂²

C₄×#11719912

yes

characteristic

174

C₂²

#16810575

yes

characteristic

175

C₂²

#16810577

yes

—

176

C₂²

#16810580

yes

—

177

C₄

—

212

—

178

C₄

—

212

—

179

C₄

—

212

—

180

C₄

#16810677

yes

—

181

C₄

#16810679

yes

—

182

C₄

#16810681

yes

—

183

C₄

#16810683

yes

—

184

C₂

#20273009

yes

characteristic

185

C₂

#20273011

yes

—

186

C₂

#20273013

yes

characteristic

187

C₂

#20274462

yes

characteristic

188

C₂

#20274465

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₃
commutator, socle, prime radical	C₂³×C₅₃	C₂×C₄²
Fitting	C₅₃×#1273	C₂
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1273	D₅₃
hypocenter	C₅₃	#8134
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₅₃

commutator, socle, prime radical

C₂³×C₅₃

C₂×C₄²

Fitting

C₅₃×#1273

C₂

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
109. C₅₃	#8134	semidirect

H ≅ G/N

type

109. C₅₃

#8134

semidirect

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

subgroup	index
#20274460	2
C₄×#13133127	2
#20272542	2
C₅₃×#1273	2
#8134	53

subgroup

index

C₅₃×#1273

Subgroups — #27274523

Conjugacy classes of subgroups

Complements

Maximal subgroups