Group #27274786 — subgroups

Subgroups — #27274786

Order 13568.

412 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 155 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	#20273097	C₂	1	2	yes	maximal
2	#20273007	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16810344	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272553	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2922	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	2	yes	—
7	C₂×C₄×#88322	C₂²	1	2	yes	—
8	Q₈⋊#88322	C₄	1	4	yes	—
9	D₄⋊#88322	C₄	1	4	yes	—
10	#16810345	—	2	2	no	—
11	#16810344	—	2	2	no	—
12	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#433	—	2	2	no	—
14	C₅₃×#431	—	2	2	no	—
15	C₅₃×#424	C₄	1	2	yes	—
16	C₅₃×#408	C₄	1	2	yes	—
17	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²×#88322	—	2	2	no	—
20	C₄⋊#88322	C₂×C₄	1	4	yes	—
21	C₂²⋊#88322	—	2	2	no	—
22	C₄×#88322	—	2	2	no	—
23	C₄×#88322	—	2	2	no	—
24	C₄×#88322	C₂×C₄	1	4	yes	—
25	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
28	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
29	C₄×D₄×C₅₃	—	2	2	no	—
30	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
31	#117×C₅₃	—	2	1	no	—
32	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	—	2	1	no	—
34	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₅₃	—	2	4	no	—
36	C₂²⋊C₈×C₅₃	—	2	4	no	—
37	#97×C₅₃	—	2	1	no	—
38	C₄×C₈×C₅₃	—	2	1	no	—
39	C₄×C₈×C₅₃	D₄	1	2	yes	—
40	C₄×C₈×C₅₃	D₄	1	2	yes	—
41	#41×C₅₃	—	2	4	no	—
42	C₂×Q₈×C₅₃	C₄²	1	1	yes	characteristic
43	C₂×D₄×C₅₃	—	2	1	no	—
44	C₂×D₄×C₅₃	C₄²	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	—	2	1	no	—
46	C₂²×C₄×C₅₃	—	2	2	no	—
47	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	#41	1	2	yes	—
49	M₁₆×C₅₃	—	2	4	no	—
50	C₂×C₈×C₅₃	—	2	4	no	—
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
53	C₄⋊C₄×C₅₃	—	2	2	no	—
54	C₄⋊C₄×C₅₃	C₄²	1	2	yes	—
55	C₂²⋊C₄×C₅₃	—	2	1	no	—
56	C₂²⋊C₄×C₅₃	—	2	2	no	—
57	C₄²×C₅₃	—	2	1	no	—
58	C₄²×C₅₃	—	2	2	no	—
59	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
60	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂×#88322	—	2	4	no	—
63	C₂×#88322	—	2	4	no	—
64	C₂×#88322	—	4	4	no	—
65	C₂³×C₅₃	—	2	1	no	—
66	C₂³×C₅₃	#117	1	1	yes	characteristic
67	Q₈×C₅₃	—	2	2	no	—
68	D₄×C₅₃	—	2	2	no	—
69	D₄×C₅₃	—	4	2	no	—
70	C₂×C₄×C₅₃	—	2	4	no	—
71	C₂×C₄×C₅₃	—	2	1	no	—
72	C₂×C₄×C₅₃	—	2	2	no	—
73	C₂×C₄×C₅₃	—	4	2	no	—
74	C₂×C₄×C₅₃	—	4	4	no	—
75	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
78	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
79	C₂×C₄×C₅₃	#131	1	4	yes	—
80	C₈×C₅₃	—	2	4	no	—
81	C₈×C₅₃	—	4	2	no	—
82	#88322	—	4	4	no	—
83	#88322	—	4	4	no	—
84	#8188	—	53	1	no	maximal, Sylow
85	C₂²×C₅₃	—	2	2	no	—
86	C₂²×C₅₃	—	4	1	no	—
87	C₂²×C₅₃	—	4	2	no	—
88	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
89	C₂²×C₅₃	#433	1	2	yes	—
90	C₄×C₅₃	—	2	2	no	—
91	C₄×C₅₃	—	4	1	no	—
92	C₄×C₅₃	—	4	2	no	—
93	C₄×C₅₃	#405	1	2	yes	—
94	C₄×C₅₃	#408	1	2	yes	—
95	C₄×C₅₃	#444	1	4	yes	—
96	#2922	D₅₃	1	1	yes	characteristic
97	#1476	—	53	2	no	—
98	#1452	—	53	1	no	—
99	C₂×#322	—	53	1	no	—
100	#1278	—	53	2	no	—
101	C₂×C₅₃	—	2	2	no	—
102	C₂×C₅₃	—	8	1	no	—
103	C₂×C₅₃	#1750	1	1	yes	characteristic
104	C₂×C₅₃	#1765	1	2	yes	—
105	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
106	#433	—	2	2	no	—
107	#431	—	2	2	no	—
108	#431	—	53	2	no	—
109	#424	Dic₅₃	1	2	yes	—
110	#408	Dic₅₃	1	2	yes	—
111	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
112	C₂×C₄×C₈	—	53	2	no	—
113	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
114	Q₈⋊C₈	—	53	4	no	—
115	D₄⋊C₈	—	53	4	no	—
116	#322	—	106	2	no	—
117	#322	—	106	2	no	—
118	C₅₃	#8188	1	1	yes	characteristic, complemented, Sylow
119	C₂×#41	C₂×Dic₅₃	1	1	yes	characteristic
120	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
121	C₂²×C₈	—	106	2	no	—
122	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
123	C₄×Q₈	C₄×D₅₃	1	2	yes	—
124	C₄×D₄	—	2	2	no	—
125	C₄×D₄	C₄×D₅₃	1	2	yes	—
126	#117	—	2	1	no	—
127	#117	C₂×Dic₅₃	1	1	yes	characteristic
128	C₂×C₄²	—	2	1	no	—
129	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
130	C₄⋊C₈	—	2	4	no	—
131	C₄⋊C₈	—	53	4	no	—
132	C₂²⋊C₈	—	2	4	no	—
133	C₂²⋊C₈	—	106	2	no	—
134	#97	—	2	1	no	—
135	C₄×C₈	—	2	1	no	—
136	C₄×C₈	—	53	4	no	—
137	C₄×C₈	—	106	2	no	—
138	C₄×C₈	—	106	2	no	—
139	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
140	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
141	#41	—	2	4	no	—
142	C₂×Q₈	C₄×Dic₅₃	1	1	yes	characteristic
143	C₂×D₄	—	2	1	no	—
144	C₂×D₄	C₄×Dic₅₃	1	1	yes	characteristic
145	C₂²×C₄	—	2	1	no	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	#11719945	1	2	yes	—
148	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
149	M₁₆	—	2	4	no	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	106	4	no	—
153	C₂×C₈	—	106	4	no	—
154	C₂×C₈	—	212	4	no	—
155	C₂×C₈	C₂²⋊Dic₅₃	1	4	yes	—
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	C₄×Dic₅₃	1	2	yes	—
158	C₂²⋊C₄	—	2	1	no	—
159	C₂²⋊C₄	—	2	2	no	—
160	C₄²	—	2	1	no	—
161	C₄²	—	2	2	no	—
162	C₄²	C₂×C₄×D₅₃	1	2	yes	—
163	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
164	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
165	C₂³	—	2	1	no	—
166	C₂³	#13133176	1	1	yes	characteristic
167	Q₈	—	2	2	no	—
168	D₄	—	2	2	no	—
169	D₄	—	4	2	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	4	4	no	—
175	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
176	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
177	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
178	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
179	C₂×C₄	#13133224	1	4	yes	—
180	C₈	—	2	4	no	—
181	C₈	—	4	2	no	—
182	C₈	—	212	4	no	—
183	C₈	—	212	4	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	4	1	no	—
186	C₂²	—	4	2	no	—
187	C₂²	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
188	C₂²	#16810684	1	2	yes	—
189	C₄	—	2	2	no	—
190	C₄	—	4	1	no	—
191	C₄	—	4	2	no	—
192	C₄	#16810579	1	2	yes	—
193	C₄	#16810590	1	2	yes	—
194	C₄	#16810743	1	4	yes	—
195	C₂	—	2	2	no	—
196	C₂	—	8	1	no	—
197	C₂	#20274481	1	1	yes	characteristic
198	C₂	#20274557	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20273097

C₂

yes

maximal

#20273007

C₂

yes

maximal, characteristic, complemented

C₂×#16810344

C₂

yes

maximal, characteristic, complemented

#20272553

C₂

yes

maximal, complemented

C₅₃×#2922

C₂

yes

maximal, characteristic

#16810676

C₂²

yes

—

C₂×C₄×#88322

C₂²

yes

—

Q₈⋊#88322

C₄

yes

—

D₄⋊#88322

C₄

yes

—

#16810345

—

#16810344

—

C₄×#41×C₅₃

C₂²

yes

characteristic

C₅₃×#433

—

C₅₃×#431

—

C₅₃×#424

C₄

yes

—

C₅₃×#408

C₄

yes

—

C₄×M₁₆×C₅₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂²×#88322

—

C₄⋊#88322

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

—

C₂×#41×C₅₃

C₂×C₄

yes

characteristic

C₂×M₁₆×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₄×Q₈×C₅₃

C₂×C₄

yes

—

C₄×D₄×C₅₃

—

C₄×D₄×C₅₃

C₂×C₄

yes

—

#117×C₅₃

—

#117×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

#97×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

#41×C₅₃

—

C₂×Q₈×C₅₃

C₄²

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₄²

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₄²

yes

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

—

—

—

C₂³×C₅₃

—

C₂³×C₅₃

#117

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

#433

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#405

yes

—

C₄×C₅₃

#408

yes

—

C₄×C₅₃

#444

yes

—

#2922

D₅₃

yes

—

—

—

100

#1278

—

101

C₂×C₅₃

—

102

C₂×C₅₃

—

103

C₂×C₅₃

#1750

yes

characteristic

104

C₂×C₅₃

#1765

yes

—

105

C₄×#41

C₂×D₅₃

yes

characteristic

106

#433

—

107

#431

—

108

#431

—

109

#424

Dic₅₃

yes

—

110

#408

Dic₅₃

yes

—

111

C₄×M₁₆

C₂×D₅₃

yes

characteristic

112

C₂×C₄×C₈

—

113

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

114

Q₈⋊C₈

—

115

D₄⋊C₈

—

116

#322

—

106

—

117

#322

—

106

—

118

C₅₃

#8188

yes

characteristic, complemented, Sylow

119

C₂×#41

C₂×Dic₅₃

yes

characteristic

120

C₂×M₁₆

C₂×Dic₅₃

yes

—

121

C₂²×C₈

—

106

—

122

C₂²×C₈

C₂×Dic₅₃

yes

—

123

C₄×Q₈

C₄×D₅₃

yes

—

124

C₄×D₄

—

125

C₄×D₄

C₄×D₅₃

yes

—

126

#117

—

127

#117

C₂×Dic₅₃

yes

characteristic

128

C₂×C₄²

—

129

C₂×C₄²

C₂²×D₅₃

yes

characteristic

130

C₄⋊C₈

—

131

C₄⋊C₈

—

132

C₂²⋊C₈

—

133

C₂²⋊C₈

—

106

—

134

#97

—

135

C₄×C₈

—

136

C₄×C₈

—

137

C₄×C₈

—

106

—

138

C₄×C₈

—

106

—

139

C₄×C₈

C₂²⋊D₅₃

yes

—

140

C₄×C₈

C₂²⋊D₅₃

yes

—

141

#41

—

142

C₂×Q₈

C₄×Dic₅₃

yes

characteristic

143

C₂×D₄

—

144

C₂×D₄

C₄×Dic₅₃

yes

characteristic

145

C₂²×C₄

—

146

C₂²×C₄

—

147

C₂²×C₄

#11719945

yes

—

148

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

149

M₁₆

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

106

—

153

C₂×C₈

—

106

—

154

C₂×C₈

—

212

—

155

C₂×C₈

C₂²⋊Dic₅₃

yes

—

156

C₄⋊C₄

—

157

C₄⋊C₄

C₄×Dic₅₃

yes

—

158

C₂²⋊C₄

—

159

C₂²⋊C₄

—

160

C₄²

—

161

C₄²

—

162

C₄²

C₂×C₄×D₅₃

yes

—

163

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

164

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

165

C₂³

—

166

C₂³

#13133176

yes

characteristic

167

Q₈

—

168

D₄

—

169

D₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

C₂×C₄×Dic₅₃

yes

characteristic

176

C₂×C₄

C₂×C₂²⋊Dic₅₃

yes

characteristic

177

C₂×C₄

C₄×C₂²⋊D₅₃

yes

—

178

C₂×C₄

C₄×C₂²⋊D₅₃

yes

—

179

C₂×C₄

#13133224

yes

—

180

C₈

—

181

C₈

—

182

C₈

—

212

—

183

C₈

—

212

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

C₄×C₂²⋊Dic₅₃

yes

characteristic

188

C₂²

#16810684

yes

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

#16810579

yes

—

193

C₄

#16810590

yes

—

194

C₄

#16810743

yes

—

195

C₂

—

196

C₂

—

197

C₂

#20274481

yes

characteristic

198

C₂

#20274557

yes

—

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2922	C₂
socle	C₂²×C₅₃	C₄×C₂²⋊C₄
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2922	D₅₃
hypocenter	C₅₃	#8188
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #2922

subgroup

G/H

center

C₄²

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#2922

C₂

socle

C₂²×C₅₃

C₄×C₂²⋊C₄

prime radical

C₂×#41×C₅₃

C₂×C₄

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #2922

N	H ≅ G/N	type
2. #20273007	C₂	semidirect
3. C₂×#16810344	C₂	semidirect
4. #20272553	C₂	semidirect
118. C₅₃	#8188	semidirect

H ≅ G/N

type

2. #20273007

C₂

semidirect

3. C₂×#16810344

C₂

semidirect

4. #20272553

C₂

semidirect

118. C₅₃

#8188

semidirect

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

subgroup	index
#20273097	2
#20273007	2
C₂×#16810344	2
#20272553	2
C₅₃×#2922	2
#8188	53

subgroup

index

C₅₃×#2922

Subgroups — #27274786

Conjugacy classes of subgroups

Complements

Maximal subgroups