Group #20274720 — subgroups

Subgroups — #20274720

Order 6784.

308 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 141 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	C₂×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133141	C₂	1	2	yes	maximal
3	#16810493	C₂	1	1	yes	maximal, characteristic
4	#16810385	C₂	1	2	yes	maximal
5	C₂×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#88322	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₅₃	—	2	1	no	—
8	C₂×C₄⋊Dic₅₃	—	2	1	no	—
9	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
10	C₄⋊#88322	—	2	2	no	—
11	#13133141	—	2	4	no	—
12	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₄×Q₈×C₅₃	—	2	2	no	—
14	C₄×Q₈×C₅₃	C₄	1	4	yes	complemented
15	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
16	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₅₃	—	2	1	no	—
18	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×Q₈×C₅₃	—	2	1	no	—
20	C₂×Q₈×C₅₃	—	2	1	no	—
21	C₂×Q₈×C₅₃	—	2	1	no	—
22	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₂×Q₈×C₅₃	D₄	1	2	yes	—
24	C₂×Q₈×C₅₃	Q₈	1	2	yes	—
25	C₂²×C₄×C₅₃	—	2	1	no	—
26	C₂²×C₄×C₅₃	—	2	1	no	—
27	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
30	C₂²×Dic₅₃	—	2	2	no	—
31	C₄⋊C₄×C₅₃	—	2	4	no	—
32	C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
34	C₄²×C₅₃	—	2	4	no	—
35	C₄²×C₅₃	C₂×C₄	1	2	yes	—
36	C₄²×C₅₃	D₄	1	2	yes	—
37	C₄⋊Dic₅₃	—	2	2	no	—
38	C₄⋊Dic₅₃	—	2	2	no	—
39	C₄⋊Dic₅₃	—	4	2	no	—
40	C₄⋊Dic₅₃	—	4	2	no	—
41	C₂×#88322	—	2	2	no	—
42	C₂×#88322	—	2	2	no	—
43	C₂×#88322	—	2	2	no	—
44	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
45	Q₈×C₅₃	—	2	2	no	—
46	Q₈×C₅₃	—	2	2	no	—
47	Q₈×C₅₃	—	2	2	no	—
48	Q₈×C₅₃	C₄⋊C₄	1	4	yes	complemented
49	C₂×C₄×C₅₃	—	2	2	no	—
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	—	2	2	no	—
52	C₂×C₄×C₅₃	—	2	1	no	—
53	C₂×C₄×C₅₃	—	2	1	no	—
54	C₂×C₄×C₅₃	—	2	2	no	—
55	C₂×C₄×C₅₃	—	4	1	no	—
56	C₂×C₄×C₅₃	—	4	1	no	—
57	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
60	C₂×C₄×C₅₃	Q₁₆	1	2	yes	—
61	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
66	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
67	C₂×Dic₅₃	—	4	2	no	—
68	C₂×Dic₅₃	—	4	2	no	—
69	C₂×Dic₅₃	—	4	2	no	—
70	#88322	—	4	2	no	—
71	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₅₃	#121	1	1	yes	characteristic
73	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂²×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
75	C₂²×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
76	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₂²×C₅₃	#137	1	1	yes	characteristic
78	C₄×C₅₃	—	2	4	no	—
79	C₄×C₅₃	—	2	2	no	—
80	C₄×C₅₃	—	4	2	no	—
81	C₄×C₅₃	#103	1	4	yes	complemented
82	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₄×C₅₃	#117	1	1	yes	characteristic
84	C₄×C₅₃	#121	1	1	yes	characteristic
85	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
86	Dic₅₃	—	8	2	no	—
87	#1803	—	53	1	no	maximal, Sylow
88	C₂×C₅₃	#380	1	1	yes	characteristic
89	C₂×C₅₃	C₂×#103	1	1	yes	characteristic
90	C₂×C₅₃	#417	1	1	yes	characteristic
91	C₂×C₅₃	#460	1	1	yes	characteristic
92	C₂×C₅₃	#462	1	1	yes	characteristic
93	C₂×C₅₃	#475	1	1	yes	characteristic
94	C₂×C₅₃	#477	1	1	yes	characteristic
95	C₂×C₄×Q₈	D₅₃	1	1	yes	characteristic
96	C₂×C₄⋊C₈	—	53	1	no	—
97	C₂×#103	—	53	2	no	—
98	#384	—	53	1	no	—
99	#340	—	53	2	no	—
100	C₅₃	#1803	1	1	yes	characteristic, complemented, Sylow
101	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
102	C₂²×C₈	—	53	2	no	—
103	C₄×Q₈	—	2	2	no	—
104	C₄×Q₈	Dic₅₃	1	4	yes	complemented
105	C₂×C₄⋊C₄	—	2	1	no	—
106	C₂×C₄⋊C₄	—	53	2	no	—
107	C₂×C₄⋊C₄	—	106	1	no	—
108	C₂×C₄⋊C₄	—	106	1	no	—
109	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
110	C₂×C₄²	—	2	1	no	—
111	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
112	C₄⋊C₈	—	106	2	no	—
113	#103	—	106	4	no	—
114	C₂×Q₈	—	2	1	no	—
115	C₂×Q₈	—	2	1	no	—
116	C₂×Q₈	—	2	1	no	—
117	C₂×Q₈	Dic₁₀₆	1	2	yes	—
118	C₂×Q₈	D₂₁₂	1	2	yes	—
119	C₂×Q₈	C₂×Dic₅₃	1	2	yes	—
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	—	2	1	no	—
122	C₂²×C₄	—	106	2	no	—
123	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
124	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
126	C₂×C₈	—	106	2	no	—
127	C₂×C₈	—	106	2	no	—
128	C₂×C₈	—	106	2	no	—
129	C₄⋊C₄	—	2	1	no	—
130	C₄⋊C₄	—	2	4	no	—
131	C₄⋊C₄	—	106	2	no	—
132	C₄⋊C₄	—	106	2	no	—
133	C₄⋊C₄	—	212	2	no	—
134	C₄⋊C₄	—	212	2	no	—
135	C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
136	C₄²	—	2	4	no	—
137	C₄²	C₂×Dic₅₃	1	2	yes	—
138	C₄²	C₂²⋊D₅₃	1	2	yes	—
139	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
140	Q₈	—	2	2	no	—
141	Q₈	—	2	2	no	—
142	Q₈	—	2	2	no	—
143	Q₈	C₄⋊Dic₅₃	1	4	yes	complemented
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	212	2	no	—
153	C₂×C₄	—	212	2	no	—
154	C₂×C₄	—	212	2	no	—
155	C₂×C₄	#11719924	1	2	yes	—
156	C₂×C₄	#11719925	1	2	yes	—
157	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
158	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
159	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
160	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
161	C₂×C₄	#11719945	1	1	yes	characteristic
162	C₂×C₄	#11719945	1	1	yes	characteristic
163	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
164	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
165	C₈	—	212	2	no	—
166	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
167	C₂²	#13133191	1	1	yes	characteristic
168	C₂²	#13133196	1	1	yes	characteristic
169	C₂²	C₂×#11719924	1	1	yes	characteristic
170	C₂²	C₂×#11719925	1	1	yes	characteristic
171	C₂²	#13133242	1	1	yes	characteristic
172	C₂²	#13133247	1	1	yes	characteristic
173	C₄	—	2	2	no	—
174	C₄	—	2	4	no	—
175	C₄	—	4	2	no	—
176	C₄	—	424	2	no	—
177	C₄	#13133141	1	4	yes	complemented
178	C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
179	C₄	#13133176	1	1	yes	characteristic
180	C₄	#13133191	1	1	yes	characteristic
181	C₄	#13133196	1	1	yes	characteristic
182	C₂	#16810483	1	1	yes	characteristic
183	C₂	C₂×#13133141	1	1	yes	characteristic
184	C₂	#16810624	1	1	yes	characteristic
185	C₂	#16810823	1	1	yes	characteristic
186	C₂	#16810833	1	1	yes	characteristic
187	C₂	#16810908	1	1	yes	characteristic
188	C₂	#16810918	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#88322

C₂

yes

maximal, characteristic

C₂×#13133141

C₂

yes

maximal

#16810493

C₂

yes

maximal, characteristic

#16810385

C₂

yes

maximal

C₂×C₄×Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₄⋊#88322

—

#13133141

—

C₂²×Q₈×C₅₃

C₂²

yes

characteristic

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

C₄

yes

complemented

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂×Q₈×C₅₃

D₄

yes

—

C₂×Q₈×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

D₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

C₄⋊C₄

yes

complemented

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

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

Q₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

#88322

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

#121

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₃

C₂×SD₁₆

yes

characteristic

C₂²×C₅₃

C₂×Q₁₆

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#103

yes

complemented

C₄×C₅₃

C₂×C₄⋊C₄

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄×Q₈

D₅₃

yes

characteristic

C₂×C₄⋊C₈

—

C₂×#103

—

#384

—

#340

—

100

C₅₃

#1803

yes

characteristic, complemented, Sylow

101

C₂²×Q₈

C₂×D₅₃

yes

characteristic

102

C₂²×C₈

—

103

C₄×Q₈

—

104

C₄×Q₈

Dic₅₃

yes

complemented

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

106

—

108

C₂×C₄⋊C₄

—

106

—

109

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

110

C₂×C₄²

—

111

C₂×C₄²

C₂×D₅₃

yes

characteristic

112

C₄⋊C₈

—

106

—

113

#103

—

106

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×Q₈

Dic₁₀₆

yes

—

118

C₂×Q₈

D₂₁₂

yes

—

119

C₂×Q₈

C₂×Dic₅₃

yes

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

106

—

123

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

124

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

125

C₂²×C₄

C₂²×D₅₃

yes

characteristic

126

C₂×C₈

—

106

—

127

C₂×C₈

—

106

—

128

C₂×C₈

—

106

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

106

—

132

C₄⋊C₄

—

106

—

133

C₄⋊C₄

—

212

—

134

C₄⋊C₄

—

212

—

135

C₄⋊C₄

C₂×Dic₅₃

yes

—

136

C₄²

—

137

C₄²

C₂×Dic₅₃

yes

—

138

C₄²

C₂²⋊D₅₃

yes

—

139

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

140

Q₈

—

141

Q₈

—

142

Q₈

—

143

Q₈

C₄⋊Dic₅₃

yes

complemented

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

212

—

153

C₂×C₄

—

212

—

154

C₂×C₄

—

212

—

155

C₂×C₄

#11719924

yes

—

156

C₂×C₄

#11719925

yes

—

157

C₂×C₄

C₂²⋊Dic₅₃

yes

—

158

C₂×C₄

C₂²⋊Dic₅₃

yes

—

159

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

160

C₂×C₄

C₂×D₂₁₂

yes

characteristic

161

C₂×C₄

#11719945

yes

characteristic

162

C₂×C₄

#11719945

yes

characteristic

163

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

164

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

165

C₈

—

212

—

166

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

167

C₂²

#13133191

yes

characteristic

168

C₂²

#13133196

yes

characteristic

169

C₂²

C₂×#11719924

yes

characteristic

170

C₂²

C₂×#11719925

yes

characteristic

171

C₂²

#13133242

yes

characteristic

172

C₂²

#13133247

yes

characteristic

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

424

—

177

C₄

#13133141

yes

complemented

178

C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

179

C₄

#13133176

yes

characteristic

180

C₄

#13133191

yes

characteristic

181

C₄

#13133196

yes

characteristic

182

C₂

#16810483

yes

characteristic

183

C₂

C₂×#13133141

yes

characteristic

184

C₂

#16810624

yes

characteristic

185

C₂

#16810823

yes

characteristic

186

C₂

#16810833

yes

characteristic

187

C₂

#16810908

yes

characteristic

188

C₂

#16810918

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×C₄×Q₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×C₄×Q₈	D₅₃
hypocenter	C₅₃	#1803
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×C₄×Q₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

hypercenter

C₂×C₄×Q₈

D₅₃

hypocenter

C₅₃

#1803

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
14. C₄×Q₈×C₅₃	C₄	semidirect
48. Q₈×C₅₃	C₄⋊C₄	semidirect
81. C₄×C₅₃	#103	semidirect
100. C₅₃	#1803	semidirect
104. C₄×Q₈	Dic₅₃	semidirect
143. Q₈	C₄⋊Dic₅₃	semidirect
177. C₄	#13133141	semidirect

H ≅ G/N

type

14. C₄×Q₈×C₅₃

C₄

semidirect

48. Q₈×C₅₃

C₄⋊C₄

semidirect

81. C₄×C₅₃

#103

semidirect

100. C₅₃

#1803

semidirect

104. C₄×Q₈

Dic₅₃

semidirect

143. Q₈

C₄⋊Dic₅₃

semidirect

177. C₄

#13133141

semidirect

Subgroups — #20274720

Conjugacy classes of subgroups

Complements