Group #20275455 — subgroups

Subgroups — #20275455

Order 6784.

240 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 99 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×#13133150	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133148	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133128	C₂	1	1	yes	maximal, characteristic
4	#16810521	C₂	1	1	yes	maximal, characteristic
5	#16810385	C₂	1	1	yes	maximal, characteristic
6	#16810385	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#389	C₂	1	1	yes	maximal, characteristic
8	C₂²×#88322	C₂²	1	1	yes	characteristic
9	C₂²×#88322	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
12	C₂×#11719919	—	2	1	no	—
13	#13133150	—	2	2	no	—
14	#13133148	C₄	1	4	yes	complemented
15	#13133128	—	2	2	no	—
16	#13133124	—	2	1	no	—
17	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
18	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	#95×C₅₃	—	2	1	no	—
22	C₂²×C₄×C₅₃	—	2	1	no	—
23	C₂²×C₄×C₅₃	—	2	1	no	—
24	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
27	C₂²×Dic₅₃	—	2	1	no	—
28	C₂²×Dic₅₃	—	2	1	no	—
29	C₄⋊C₄×C₅₃	—	2	1	no	—
30	C₄⋊C₄×C₅₃	—	2	1	no	—
31	C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₄⋊C₄×C₅₃	—	4	2	no	—
33	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
34	C₄²×C₅₃	—	2	1	no	—
35	C₄²×C₅₃	—	2	1	no	—
36	C₄⋊Dic₅₃	—	2	1	no	—
37	C₄⋊Dic₅₃	—	2	1	no	—
38	C₄⋊Dic₅₃	—	2	1	no	—
39	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
40	#11719919	—	4	2	no	—
41	C₂×#88322	—	2	1	no	—
42	C₂×#88322	—	2	1	no	—
43	C₂×#88322	C₂×C₄	1	2	yes	—
44	C₂×#88322	D₄	1	2	yes	—
45	C₂×#88322	Q₈	1	2	yes	—
46	C₂×#88322	Q₈	1	2	yes	—
47	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	—	2	1	no	—
50	C₂×C₄×C₅₃	—	2	1	no	—
51	C₂×C₄×C₅₃	—	2	1	no	—
52	C₂×C₄×C₅₃	—	2	1	no	—
53	C₂×C₄×C₅₃	—	2	1	no	—
54	C₂×C₄×C₅₃	—	2	1	no	—
55	C₂×C₄×C₅₃	—	4	1	no	—
56	C₂×C₄×C₅₃	—	4	1	no	—
57	C₂×C₄×C₅₃	—	4	1	no	—
58	C₂×C₄×C₅₃	—	4	1	no	—
59	C₂×C₄×C₅₃	—	4	1	no	—
60	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
66	C₂×Dic₅₃	—	2	2	no	—
67	C₂×Dic₅₃	—	4	1	no	—
68	C₂×Dic₅₃	—	4	1	no	—
69	C₂×Dic₅₃	—	4	1	no	—
70	C₂×Dic₅₃	—	4	1	no	—
71	C₂×Dic₅₃	—	4	1	no	—
72	#88322	—	2	2	no	—
73	#88322	C₄⋊C₄	1	4	yes	complemented
74	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
75	C₂²×C₅₃	#121	1	1	yes	characteristic
76	C₂²×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
77	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂²×C₅₃	#137	1	1	yes	characteristic
80	C₂²×C₅₃	#137	1	1	yes	characteristic
81	C₄×C₅₃	—	4	2	no	—
82	C₄×C₅₃	—	4	1	no	—
83	C₄×C₅₃	—	4	1	no	—
84	C₄×C₅₃	—	8	1	no	—
85	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
87	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
88	C₄×C₅₃	#125	1	1	yes	characteristic
89	Dic₅₃	—	4	2	no	—
90	Dic₅₃	—	8	1	no	—
91	#1944	—	53	1	no	maximal, Sylow
92	C₂×C₅₃	#389	1	1	yes	characteristic
93	C₂×C₅₃	#440	1	1	yes	characteristic
94	C₂×C₅₃	#440	1	1	yes	characteristic
95	C₂×C₅₃	#469	1	1	yes	characteristic
96	C₂×C₅₃	#470	1	1	yes	characteristic
97	C₂×C₅₃	#501	1	1	yes	characteristic
98	C₂×C₅₃	#501	1	1	yes	characteristic
99	C₂×#107	—	53	1	no	—
100	C₂×#106	—	53	1	no	—
101	C₂×#97	—	53	1	no	—
102	#389	—	53	1	no	—
103	#389	D₅₃	1	1	yes	characteristic
104	#340	—	53	1	no	—
105	#340	—	53	1	no	—
106	C₅₃	#1944	1	1	yes	characteristic, complemented, Sylow
107	C₂²×C₈	—	53	1	no	—
108	C₂²×C₈	—	53	1	no	—
109	C₂×C₄⋊C₄	—	2	1	no	—
110	C₂×C₄⋊C₄	—	53	1	no	—
111	C₂×C₄⋊C₄	—	53	1	no	—
112	C₂×C₄⋊C₄	—	106	1	no	—
113	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
114	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
115	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
116	#107	—	106	2	no	—
117	#106	—	53	4	no	—
118	#97	—	106	2	no	—
119	#95	—	2	1	no	—
120	#95	—	106	1	no	—
121	C₂²×C₄	—	2	1	no	—
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	—	106	1	no	—
124	C₂²×C₄	—	106	1	no	—
125	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
126	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
128	C₂×C₈	—	53	2	no	—
129	C₂×C₈	—	53	2	no	—
130	C₂×C₈	—	53	2	no	—
131	C₂×C₈	—	53	2	no	—
132	C₂×C₈	—	106	1	no	—
133	C₂×C₈	—	106	1	no	—
134	C₄⋊C₄	—	2	1	no	—
135	C₄⋊C₄	—	2	1	no	—
136	C₄⋊C₄	—	2	1	no	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄⋊C₄	—	53	2	no	—
139	C₄⋊C₄	—	106	1	no	—
140	C₄⋊C₄	—	106	1	no	—
141	C₄⋊C₄	—	106	1	no	—
142	C₄⋊C₄	—	212	2	no	—
143	C₄⋊C₄	C₄×D₅₃	1	2	yes	—
144	C₄²	—	2	1	no	—
145	C₄²	—	2	1	no	—
146	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	106	2	no	—
160	C₂×C₄	—	212	1	no	—
161	C₂×C₄	—	212	1	no	—
162	C₂×C₄	—	212	1	no	—
163	C₂×C₄	—	212	1	no	—
164	C₂×C₄	—	212	1	no	—
165	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
166	C₂×C₄	#11719945	1	1	yes	characteristic
167	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
168	C₂×C₄	#11719947	1	1	yes	characteristic
169	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
170	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
171	C₈	—	53	4	no	—
172	C₈	—	106	2	no	—
173	C₂²	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
174	C₂²	#13133189	1	1	yes	characteristic
175	C₂²	#13133215	1	1	yes	characteristic
176	C₂²	#13133242	1	1	yes	characteristic
177	C₂²	#13133243	1	1	yes	characteristic
178	C₂²	#13133247	1	1	yes	characteristic
179	C₂²	#13133248	1	1	yes	characteristic
180	C₄	—	4	1	no	—
181	C₄	—	4	1	no	—
182	C₄	—	4	2	no	—
183	C₄	—	8	1	no	—
184	C₄	—	424	1	no	—
185	C₄	—	212	2	no	—
186	C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
187	C₄	#13133182	1	1	yes	characteristic
188	C₄	#13133195	1	1	yes	characteristic
189	C₄	#13133207	1	1	yes	characteristic
190	C₂	#16810518	1	1	yes	characteristic
191	C₂	#16810721	1	1	yes	characteristic
192	C₂	#16810723	1	1	yes	characteristic
193	C₂	#16810874	1	1	yes	characteristic
194	C₂	#16810879	1	1	yes	characteristic
195	C₂	#16811029	1	1	yes	characteristic
196	C₂	#16811033	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133150

C₂

yes

maximal, characteristic

C₂×#13133148

C₂

yes

maximal, characteristic

C₂×#13133128

C₂

yes

maximal, characteristic

#16810521

C₂

yes

maximal, characteristic

#16810385

C₂

yes

maximal, characteristic

#16810385

C₂

yes

maximal, characteristic

C₅₃×#389

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

—

—

C₄

yes

complemented

#13133128

—

#13133124

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#95×C₅₃

—

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

—

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

—

C₂×#88322

D₄

yes

—

C₂×#88322

Q₈

yes

—

C₂×#88322

Q₈

yes

—

C₂³×C₅₃

C₂×D₄

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

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

#88322

—

#88322

C₄⋊C₄

yes

complemented

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₂²×C₅₃

#121

yes

characteristic

C₂²×C₅₃

C₄⋊Q₈

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₄×Q₈

yes

characteristic

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₄×C₅₃

#125

yes

characteristic

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

100

C₂×#106

—

101

C₂×#97

—

102

#389

—

103

#389

D₅₃

yes

characteristic

104

#340

—

105

#340

—

106

C₅₃

#1944

yes

characteristic, complemented, Sylow

107

C₂²×C₈

—

108

C₂²×C₈

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

106

—

113

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

114

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

115

C₂×C₄²

C₂×D₅₃

yes

characteristic

116

#107

—

106

—

117

#106

—

118

#97

—

106

—

119

#95

—

120

#95

—

106

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

106

—

124

C₂²×C₄

—

106

—

125

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

126

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

127

C₂²×C₄

C₂²×D₅₃

yes

characteristic

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

106

—

133

C₂×C₈

—

106

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

106

—

140

C₄⋊C₄

—

106

—

141

C₄⋊C₄

—

106

—

142

C₄⋊C₄

—

212

—

143

C₄⋊C₄

C₄×D₅₃

yes

—

144

C₄²

—

145

C₄²

—

146

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

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

—

106

—

160

C₂×C₄

—

212

—

161

C₂×C₄

—

212

—

162

C₂×C₄

—

212

—

163

C₂×C₄

—

212

—

164

C₂×C₄

—

212

—

165

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

166

C₂×C₄

#11719945

yes

characteristic

167

C₂×C₄

D₄×D₅₃

yes

characteristic

168

C₂×C₄

#11719947

yes

characteristic

169

C₂×C₄

Q₈×D₅₃

yes

characteristic

170

C₂×C₄

Q₈×D₅₃

yes

characteristic

171

C₈

—

172

C₈

—

106

—

173

C₂²

C₄×C₂²⋊D₅₃

yes

characteristic

174

C₂²

#13133189

yes

characteristic

175

C₂²

#13133215

yes

characteristic

176

C₂²

#13133242

yes

characteristic

177

C₂²

#13133243

yes

characteristic

178

C₂²

#13133247

yes

characteristic

179

C₂²

#13133248

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

424

—

185

C₄

—

212

—

186

C₄

C₄⋊C₄×D₅₃

yes

characteristic

187

C₄

#13133182

yes

characteristic

188

C₄

#13133195

yes

characteristic

189

C₄

#13133207

yes

characteristic

190

C₂

#16810518

yes

characteristic

191

C₂

#16810721

yes

characteristic

192

C₂

#16810723

yes

characteristic

193

C₂

#16810874

yes

characteristic

194

C₂

#16810879

yes

characteristic

195

C₂

#16811029

yes

characteristic

196

C₂

#16811033

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#389	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	#389	D₅₃
hypocenter	C₅₃	#1944
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#389

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

N	H ≅ G/N	type
14. #13133148	C₄	semidirect
73. #88322	C₄⋊C₄	semidirect
106. C₅₃	#1944	semidirect

H ≅ G/N

type

14. #13133148

C₄

semidirect

73. #88322

C₄⋊C₄

semidirect

106. C₅₃

#1944

semidirect

Subgroups — #20275455

Conjugacy classes of subgroups

Complements