Group #20275302 — subgroups

Subgroups — #20275302

Order 6784.

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

normal subgroups only

This group has 230 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	#16810522	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#16810386	C₂	1	1	yes	maximal, characteristic
4	#16810385	C₂	1	1	yes	maximal, characteristic
5	#16810385	C₂	1	1	yes	maximal, characteristic
6	#16810377	C₂	1	1	yes	maximal, characteristic
7	C₂×#106×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×#88322	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₅₃	—	2	1	no	—
10	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₅₃	—	2	1	no	—
13	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
14	#13133124	—	2	1	no	—
15	#13133124	—	2	1	no	—
16	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
19	#106×C₅₃	C₄	1	4	yes	complemented
20	C₂²×C₄×C₅₃	—	2	1	no	—
21	C₂²×C₄×C₅₃	—	2	1	no	—
22	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	C₂²×Dic₅₃	—	2	1	no	—
24	C₂²×Dic₅₃	—	2	1	no	—
25	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
26	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
27	C₂×C₈×C₅₃	—	2	1	no	—
28	C₂×C₈×C₅₃	—	2	1	no	—
29	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₄⋊C₄×C₅₃	—	2	1	no	—
31	C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
33	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
34	C₄⋊Dic₅₃	—	2	1	no	—
35	C₄⋊Dic₅₃	—	2	1	no	—
36	C₄⋊Dic₅₃	—	2	1	no	—
37	C₄⋊Dic₅₃	—	4	2	no	—
38	C₄⋊Dic₅₃	Q₈	1	2	yes	—
39	C₄×Dic₅₃	—	2	1	no	—
40	C₄×Dic₅₃	—	2	1	no	—
41	C₄×Dic₅₃	—	4	2	no	—
42	C₂×#88322	—	2	1	no	—
43	C₂×#88322	—	2	1	no	—
44	C₂×#88322	—	2	1	no	—
45	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	—	2	2	no	—
47	C₂×C₄×C₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	—	4	1	no	—
49	C₂×C₄×C₅₃	—	4	1	no	—
50	C₂×C₄×C₅₃	—	4	1	no	—
51	C₂×C₄×C₅₃	—	4	1	no	—
52	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
58	C₂×Dic₅₃	—	2	1	no	—
59	C₂×Dic₅₃	—	2	1	no	—
60	C₂×Dic₅₃	—	2	2	no	—
61	C₂×Dic₅₃	—	2	1	no	—
62	C₂×Dic₅₃	—	2	1	no	—
63	C₂×Dic₅₃	—	2	1	no	—
64	C₂×Dic₅₃	—	4	1	no	—
65	C₂×Dic₅₃	—	4	1	no	—
66	C₂×Dic₅₃	—	4	1	no	—
67	C₂×Dic₅₃	—	4	1	no	—
68	C₂×Dic₅₃	—	4	1	no	—
69	C₂×Dic₅₃	SD₁₆	1	2	yes	—
70	C₈×C₅₃	—	2	2	no	—
71	#88322	—	4	1	no	—
72	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
73	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂²×C₅₃	#123	1	1	yes	characteristic
75	C₂²×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
76	C₂²×C₅₃	#135	1	1	yes	characteristic
77	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂²×C₅₃	#137	1	1	yes	characteristic
79	C₄×C₅₃	—	4	2	no	—
80	C₄×C₅₃	—	4	2	no	—
81	C₄×C₅₃	#117	1	1	yes	characteristic
82	C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
83	C₄×C₅₃	#125	1	1	yes	characteristic
84	C₄×C₅₃	#126	1	1	yes	characteristic
85	Dic₅₃	—	2	2	no	—
86	Dic₅₃	—	4	2	no	—
87	Dic₅₃	—	4	1	no	—
88	Dic₅₃	—	8	1	no	—
89	#1920	—	53	1	no	maximal, Sylow
90	C₂×C₅₃	#387	1	1	yes	characteristic
91	C₂×C₅₃	C₄×SD₁₆	1	1	yes	characteristic
92	C₂×C₅₃	#440	1	1	yes	characteristic
93	C₂×C₅₃	#476	1	1	yes	characteristic
94	C₂×C₅₃	#479	1	1	yes	characteristic
95	C₂×C₅₃	#482	1	1	yes	characteristic
96	C₂×C₅₃	#483	1	1	yes	characteristic
97	C₂×#106	D₅₃	1	1	yes	characteristic
98	#389	—	53	1	no	—
99	C₄×C₄⋊C₄	—	53	1	no	—
100	#340	—	53	1	no	—
101	#340	—	53	1	no	—
102	#340	—	53	1	no	—
103	#336	—	53	1	no	—
104	C₅₃	#1920	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₈	—	53	1	no	—
106	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
107	C₂×C₄⋊C₄	—	53	1	no	—
108	C₂×C₄⋊C₄	—	53	1	no	—
109	C₂×C₄⋊C₄	—	106	1	no	—
110	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
111	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
112	C₂×C₄²	—	53	1	no	—
113	C₂×C₄²	—	106	1	no	—
114	#106	Dic₅₃	1	4	yes	complemented
115	#95	—	106	1	no	—
116	#95	—	106	1	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	53	1	no	—
120	C₂²×C₄	—	53	1	no	—
121	C₂²×C₄	—	106	1	no	—
122	C₂²×C₄	—	106	1	no	—
123	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	106	1	no	—
127	C₂×C₈	—	106	1	no	—
128	C₂×C₈	—	106	1	no	—
129	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	53	2	no	—
133	C₄⋊C₄	—	106	1	no	—
134	C₄⋊C₄	—	106	1	no	—
135	C₄⋊C₄	—	106	1	no	—
136	C₄⋊C₄	—	212	2	no	—
137	C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
138	C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
139	C₄²	—	106	1	no	—
140	C₄²	—	106	1	no	—
141	C₄²	—	212	2	no	—
142	C₂³	D₄×D₅₃	1	1	yes	characteristic
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	53	2	no	—
150	C₂×C₄	—	106	1	no	—
151	C₂×C₄	—	106	1	no	—
152	C₂×C₄	—	106	2	no	—
153	C₂×C₄	—	106	1	no	—
154	C₂×C₄	—	106	1	no	—
155	C₂×C₄	—	106	1	no	—
156	C₂×C₄	—	212	1	no	—
157	C₂×C₄	—	212	1	no	—
158	C₂×C₄	—	212	1	no	—
159	C₂×C₄	—	212	1	no	—
160	C₂×C₄	—	212	1	no	—
161	C₂×C₄	#11719945	1	1	yes	characteristic
162	C₂×C₄	#11719945	1	1	yes	characteristic
163	C₂×C₄	#11719947	1	1	yes	characteristic
164	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
165	C₂×C₄	#11719949	1	1	yes	characteristic
166	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
167	C₈	—	2	2	no	—
168	C₈	—	212	1	no	—
169	C₂²	D₄×Dic₅₃	1	1	yes	characteristic
170	C₂²	#13133195	1	1	yes	characteristic
171	C₂²	#13133202	1	1	yes	characteristic
172	C₂²	SD₁₆×D₅₃	1	1	yes	characteristic
173	C₂²	#13133238	1	1	yes	characteristic
174	C₂²	#13133244	1	1	yes	characteristic
175	C₂²	#13133246	1	1	yes	characteristic
176	C₄	—	4	2	no	—
177	C₄	—	4	2	no	—
178	C₄	—	106	2	no	—
179	C₄	—	424	1	no	—
180	C₄	—	212	2	no	—
181	C₄	—	212	1	no	—
182	C₄	#13133176	1	1	yes	characteristic
183	C₄	Q₈×Dic₅₃	1	1	yes	characteristic
184	C₄	#13133207	1	1	yes	characteristic
185	C₄	#13133212	1	1	yes	characteristic
186	C₂	#16810509	1	1	yes	characteristic
187	C₂	SD₁₆×Dic₅₃	1	1	yes	characteristic
188	C₂	#16810724	1	1	yes	characteristic
189	C₂	#16810912	1	1	yes	characteristic
190	C₂	#16810933	1	1	yes	characteristic
191	C₂	#16810949	1	1	yes	characteristic
192	C₂	#16810955	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810522

C₂

yes

maximal, characteristic

C₄⋊C₄×Dic₅₃

C₂

yes

maximal, characteristic

#16810386

C₂

yes

maximal, characteristic

#16810385

C₂

yes

maximal, characteristic

#16810385

C₂

yes

maximal, characteristic

#16810377

C₂

yes

maximal, characteristic

C₂×#106×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×C₄×Dic₅₃

—

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

#13133124

—

#13133124

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

#106×C₅₃

C₄

yes

complemented

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

D₄

yes

characteristic

C₂²×Dic₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

Q₈

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

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

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

SD₁₆

yes

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₃

#123

yes

characteristic

C₂²×C₅₃

C₂×SD₁₆

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#117

yes

characteristic

C₄×C₅₃

C₄×Q₈

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₄×SD₁₆

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

characteristic

#389

—

C₄×C₄⋊C₄

—

100

#340

—

101

#340

—

102

#340

—

103

#336

—

104

C₅₃

#1920

yes

characteristic, complemented, Sylow

105

C₂²×C₈

—

106

C₂²×C₈

C₂×D₅₃

yes

characteristic

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

106

—

110

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

111

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

112

C₂×C₄²

—

113

C₂×C₄²

—

106

—

114

#106

Dic₅₃

yes

complemented

115

#95

—

106

—

116

#95

—

106

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

106

—

122

C₂²×C₄

—

106

—

123

C₂²×C₄

C₂²×D₅₃

yes

characteristic

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

106

—

127

C₂×C₈

—

106

—

128

C₂×C₈

—

106

—

129

C₂×C₈

C₂×Dic₅₃

yes

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

106

—

134

C₄⋊C₄

—

106

—

135

C₄⋊C₄

—

106

—

136

C₄⋊C₄

—

212

—

137

C₄⋊C₄

C₂×Dic₅₃

yes

—

138

C₄⋊C₄

C₂×Dic₅₃

yes

—

139

C₄²

—

106

—

140

C₄²

—

106

—

141

C₄²

—

212

—

142

C₂³

D₄×D₅₃

yes

characteristic

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

106

—

151

C₂×C₄

—

106

—

152

C₂×C₄

—

106

—

153

C₂×C₄

—

106

—

154

C₂×C₄

—

106

—

155

C₂×C₄

—

106

—

156

C₂×C₄

—

212

—

157

C₂×C₄

—

212

—

158

C₂×C₄

—

212

—

159

C₂×C₄

—

212

—

160

C₂×C₄

—

212

—

161

C₂×C₄

#11719945

yes

characteristic

162

C₂×C₄

#11719945

yes

characteristic

163

C₂×C₄

#11719947

yes

characteristic

164

C₂×C₄

Q₈×D₅₃

yes

characteristic

165

C₂×C₄

#11719949

yes

characteristic

166

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

167

C₈

—

168

C₈

—

212

—

169

C₂²

D₄×Dic₅₃

yes

characteristic

170

C₂²

#13133195

yes

characteristic

171

C₂²

#13133202

yes

characteristic

172

C₂²

SD₁₆×D₅₃

yes

characteristic

173

C₂²

#13133238

yes

characteristic

174

C₂²

#13133244

yes

characteristic

175

C₂²

#13133246

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

106

—

179

C₄

—

424

—

180

C₄

—

212

—

181

C₄

—

212

—

182

C₄

#13133176

yes

characteristic

183

C₄

Q₈×Dic₅₃

yes

characteristic

184

C₄

#13133207

yes

characteristic

185

C₄

#13133212

yes

characteristic

186

C₂

#16810509

yes

characteristic

187

C₂

SD₁₆×Dic₅₃

yes

characteristic

188

C₂

#16810724

yes

characteristic

189

C₂

#16810912

yes

characteristic

190

C₂

#16810933

yes

characteristic

191

C₂

#16810949

yes

characteristic

192

C₂

#16810955

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#106×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×#106	D₅₃
hypocenter	C₅₃	#1920
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×#106×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
19. #106×C₅₃	C₄	semidirect
104. C₅₃	#1920	semidirect
114. #106	Dic₅₃	semidirect

H ≅ G/N

type

19. #106×C₅₃

C₄

semidirect

104. C₅₃

#1920

semidirect

114. #106

Dic₅₃

semidirect

Subgroups — #20275302

Conjugacy classes of subgroups

Complements