Group #20285909 — subgroups

Subgroups — #20285909

Order 6784.

366 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 121 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	#16811180	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811029	C₂	1	2	yes	maximal, complemented
3	#16810987	C₂	1	2	yes	maximal, complemented
4	#16810944	C₂	1	4	yes	maximal, complemented
5	#16810913	C₂	1	2	yes	maximal, complemented
6	#16810903	C₂	1	2	yes	maximal, complemented
7	#16810680	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#532	C₂	1	1	yes	maximal, characteristic
9	#13133217	C₂²	1	1	yes	characteristic
10	#13133217	C₂²	1	1	yes	characteristic
11	#13133209	C₂²	1	2	yes	—
12	#13133196	—	2	1	no	—
13	#13133196	—	2	1	no	—
14	C₄×Dic₁₀₆	—	2	1	no	—
15	C₄×Dic₁₀₆	—	2	1	no	—
16	#13133176	C₂²	1	2	yes	—
17	#13133150	C₂²	1	4	yes	—
18	#13133148	C₂²	1	4	yes	—
19	C₄⋊#88322	C₂²	1	2	yes	—
20	#13133138	C₂²	1	4	yes	—
21	#13133138	C₂²	1	4	yes	—
22	C₂²⋊#88322	C₂²	1	2	yes	—
23	#13133128	C₂²	1	2	yes	—
24	C₂×#41×C₅₃	—	2	1	no	—
25	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
26	#127×C₅₃	C₂²	1	1	yes	characteristic
27	#124×C₅₃	—	2	1	no	—
28	#121×C₅₃	—	2	1	no	—
29	#121×C₅₃	C₂²	1	2	yes	—
30	C₄×D₄×C₅₃	—	2	1	no	—
31	C₄×D₄×C₅₃	C₂²	1	2	yes	—
32	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
33	#41×C₅₃	—	4	4	no	—
34	C₂×Q₈×C₅₃	—	2	1	no	—
35	C₂×D₄×C₅₃	—	2	2	no	—
36	C₂×D₄×C₅₃	—	2	1	no	—
37	C₂×D₄×C₅₃	—	2	1	no	—
38	C₂×D₄×C₅₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
43	C₂×Dic₁₀₆	—	2	1	no	—
44	C₂×Dic₁₀₆	—	2	1	no	—
45	C₄⋊C₄×C₅₃	—	2	1	no	—
46	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
47	C₂²⋊C₄×C₅₃	—	2	2	no	—
48	C₂²⋊C₄×C₅₃	—	2	2	no	—
49	C₄²×C₅₃	D₄	1	1	yes	characteristic
50	C₄²×C₅₃	D₄	1	1	yes	characteristic
51	C₄²×C₅₃	C₂³	1	1	yes	characteristic
52	C₄²×C₅₃	C₂³	1	1	yes	characteristic
53	C₂²⋊Dic₅₃	—	2	2	no	—
54	C₄⋊Dic₅₃	—	2	1	no	—
55	C₄⋊Dic₅₃	—	2	1	no	—
56	C₄⋊Dic₅₃	C₂³	1	2	yes	—
57	C₄⋊Dic₅₃	C₂³	1	2	yes	—
58	#11719919	—	2	2	no	—
59	#11719919	—	2	2	no	—
60	C₄×Dic₅₃	—	2	2	no	—
61	C₂×#88322	C₂³	1	4	yes	—
62	C₂³×C₅₃	—	2	2	no	—
63	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
64	Q₈×C₅₃	—	4	2	no	—
65	D₄×C₅₃	—	2	4	no	—
66	D₄×C₅₃	—	4	2	no	—
67	D₄×C₅₃	—	4	2	no	—
68	D₄×C₅₃	—	4	4	no	—
69	C₂×C₄×C₅₃	—	2	2	no	—
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	4	1	no	—
73	C₂×C₄×C₅₃	—	4	4	no	—
74	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
80	C₂×Dic₅₃	—	2	2	no	—
81	C₂×Dic₅₃	—	2	2	no	—
82	Dic₁₀₆	—	4	2	no	—
83	Dic₁₀₆	—	4	2	no	—
84	#88322	—	2	4	no	—
85	C₂²×C₅₃	—	2	1	no	—
86	C₂²×C₅₃	—	2	1	no	—
87	C₂²×C₅₃	—	2	1	no	—
88	C₂²×C₅₃	—	4	2	no	—
89	C₂²×C₅₃	—	4	2	no	—
90	C₂²×C₅₃	—	4	2	no	—
91	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
92	C₄×C₅₃	—	2	1	no	—
93	C₄×C₅₃	—	2	1	no	—
94	C₄×C₅₃	—	2	1	no	—
95	C₄×C₅₃	—	2	1	no	—
96	C₄×C₅₃	—	4	2	no	—
97	C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
98	C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
99	Dic₅₃	—	4	2	no	—
100	Dic₅₃	—	4	2	no	—
101	#3235	—	53	1	no	maximal, Sylow
102	C₂×C₅₃	—	4	1	no	—
103	C₂×C₅₃	—	8	2	no	—
104	C₂×C₅₃	#536	1	1	yes	characteristic
105	C₂×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
106	C₂×C₅₃	#575	1	1	yes	characteristic
107	#533	—	53	1	no	—
108	#532	D₅₃	1	1	yes	characteristic
109	#501	—	53	2	no	—
110	#490	—	53	2	no	—
111	#482	—	53	4	no	—
112	#476	—	53	2	no	—
113	#474	—	53	2	no	—
114	#432	—	53	1	no	—
115	C₅₃	#3235	1	1	yes	characteristic, complemented, Sylow
116	C₂×#41	—	2	1	no	—
117	C₄⋊Q₈	—	53	1	no	—
118	C₄⋊Q₈	—	53	1	no	—
119	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
120	#127	C₂×D₅₃	1	1	yes	characteristic
121	#125	—	53	2	no	—
122	#124	—	2	1	no	—
123	C₂²⋊Q₈	—	106	1	no	—
124	C₂²⋊Q₈	—	106	1	no	—
125	#121	—	2	1	no	—
126	#121	C₂×D₅₃	1	2	yes	—
127	C₄×Q₈	—	106	1	no	—
128	C₄×Q₈	—	106	1	no	—
129	C₄×D₄	—	2	1	no	—
130	C₄×D₄	C₂×D₅₃	1	2	yes	—
131	#117	—	53	2	no	—
132	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
133	#107	—	53	4	no	—
134	#106	—	53	4	no	—
135	C₄⋊C₈	—	53	2	no	—
136	#102	—	53	4	no	—
137	#102	—	53	4	no	—
138	C₂²⋊C₈	—	53	2	no	—
139	#97	—	53	2	no	—
140	#41	—	4	4	no	—
141	C₂×Q₈	—	2	1	no	—
142	C₂×Q₈	—	106	1	no	—
143	C₂×Q₈	—	106	1	no	—
144	C₂×D₄	—	2	2	no	—
145	C₂×D₄	—	2	1	no	—
146	C₂×D₄	—	2	1	no	—
147	C₂×D₄	C₂²×D₅₃	1	2	yes	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
150	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
151	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
152	C₂×C₈	—	53	4	no	—
153	C₄⋊C₄	—	2	1	no	—
154	C₄⋊C₄	—	53	2	no	—
155	C₄⋊C₄	—	53	2	no	—
156	C₄⋊C₄	—	106	2	no	—
157	C₄⋊C₄	—	106	2	no	—
158	C₄⋊C₄	—	106	1	no	—
159	C₄⋊C₄	—	106	1	no	—
160	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
161	C₂²⋊C₄	—	2	2	no	—
162	C₂²⋊C₄	—	2	2	no	—
163	C₂²⋊C₄	—	106	2	no	—
164	C₄²	—	106	2	no	—
165	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
166	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
167	C₄²	C₂²×D₅₃	1	1	yes	characteristic
168	C₄²	C₂²×D₅₃	1	1	yes	characteristic
169	C₂³	—	2	2	no	—
170	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
171	Q₈	—	4	2	no	—
172	Q₈	—	212	2	no	—
173	Q₈	—	212	2	no	—
174	D₄	—	2	4	no	—
175	D₄	—	4	2	no	—
176	D₄	—	4	2	no	—
177	D₄	—	4	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	4	1	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	106	2	no	—
184	C₂×C₄	—	106	2	no	—
185	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
186	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
187	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
188	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
189	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
190	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
191	C₈	—	106	4	no	—
192	C₂²	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	4	2	no	—
196	C₂²	—	4	2	no	—
197	C₂²	—	4	2	no	—
198	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
199	C₄	—	2	1	no	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	4	2	no	—
204	C₄	—	212	2	no	—
205	C₄	—	212	2	no	—
206	C₄	#13133243	1	2	yes	—
207	C₄	#13133264	1	2	yes	—
208	C₂	—	4	1	no	—
209	C₂	—	8	2	no	—
210	C₂	#16811194	1	1	yes	characteristic
211	C₂	C₂×#13133243	1	1	yes	characteristic
212	C₂	#16811421	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811180

C₂

yes

maximal, characteristic, complemented

#16811029

C₂

yes

maximal, complemented

#16810987

C₂

yes

maximal, complemented

#16810944

C₂

yes

maximal, complemented

#16810913

C₂

yes

maximal, complemented

#16810903

C₂

yes

maximal, complemented

#16810680

C₂

yes

maximal, characteristic, complemented

C₅₃×#532

C₂

yes

maximal, characteristic

#13133217

C₂²

yes

characteristic

#13133217

C₂²

yes

characteristic

#13133209

C₂²

yes

—

#13133196

—

#13133196

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

—

#13133176

C₂²

yes

—

#13133150

C₂²

yes

—

#13133148

C₂²

yes

—

C₄⋊#88322

C₂²

yes

—

#13133138

C₂²

yes

—

#13133138

C₂²

yes

—

C₂²⋊#88322

C₂²

yes

—

#13133128

C₂²

yes

—

C₂×#41×C₅₃

—

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

#127×C₅₃

C₂²

yes

characteristic

#124×C₅₃

—

#121×C₅₃

—

#121×C₅₃

C₂²

yes

—

C₄×D₄×C₅₃

—

C₄×D₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#41×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₄×C₅₃

2₋¹⁺⁴

yes

—

Dic₅₃

—

100

Dic₅₃

—

101

#3235

—

maximal, Sylow

102

C₂×C₅₃

—

103

C₂×C₅₃

—

104

C₂×C₅₃

#536

yes

characteristic

105

C₂×C₅₃

C₂×AGL(1,ℤ/8)

yes

characteristic

106

C₂×C₅₃

#575

yes

characteristic

107

#533

—

108

#532

D₅₃

yes

characteristic

109

#501

—

110

#490

—

111

#482

—

112

#476

—

113

#474

—

114

#432

—

115

C₅₃

#3235

yes

characteristic, complemented, Sylow

116

C₂×#41

—

117

C₄⋊Q₈

—

118

C₄⋊Q₈

—

119

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

120

#127

C₂×D₅₃

yes

characteristic

121

#125

—

122

#124

—

123

C₂²⋊Q₈

—

106

—

124

C₂²⋊Q₈

—

106

—

125

#121

—

126

#121

C₂×D₅₃

yes

—

127

C₄×Q₈

—

106

—

128

C₄×Q₈

—

106

—

129

C₄×D₄

—

130

C₄×D₄

C₂×D₅₃

yes

—

131

#117

—

132

C₂×C₄²

C₂×D₅₃

yes

characteristic

133

#107

—

134

#106

—

135

C₄⋊C₈

—

136

#102

—

137

#102

—

138

C₂²⋊C₈

—

139

#97

—

140

#41

—

141

C₂×Q₈

—

142

C₂×Q₈

—

106

—

143

C₂×Q₈

—

106

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

C₂²×D₅₃

yes

—

148

C₂²×C₄

—

149

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

150

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

151

C₂²×C₄

C₂²×D₅₃

yes

characteristic

152

C₂×C₈

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

106

—

157

C₄⋊C₄

—

106

—

158

C₄⋊C₄

—

106

—

159

C₄⋊C₄

—

106

—

160

C₄⋊C₄

C₂²×D₅₃

yes

—

161

C₂²⋊C₄

—

162

C₂²⋊C₄

—

163

C₂²⋊C₄

—

106

—

164

C₄²

—

106

—

165

C₄²

C₂²⋊D₅₃

yes

characteristic

166

C₄²

C₂²⋊D₅₃

yes

characteristic

167

C₄²

C₂²×D₅₃

yes

characteristic

168

C₄²

C₂²×D₅₃

yes

characteristic

169

C₂³

—

170

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

171

Q₈

—

172

Q₈

—

212

—

173

Q₈

—

212

—

174

D₄

—

175

D₄

—

176

D₄

—

177

D₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

106

—

184

C₂×C₄

—

106

—

185

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

186

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

187

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

188

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

189

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

190

C₂×C₄

C₂³×D₅₃

yes

characteristic

191

C₈

—

106

—

192

C₂²

—

193

C₂²

—

194

C₂²

—

195

C₂²

—

196

C₂²

—

197

C₂²

—

198

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

—

212

—

205

C₄

—

212

—

206

C₄

#13133243

yes

—

207

C₄

#13133264

yes

—

208

C₂

—

209

C₂

—

210

C₂

#16811194

yes

characteristic

211

C₂

C₂×#13133243

yes

characteristic

212

C₂

#16811421

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#532	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#532	D₅₃
hypocenter	C₅₃	#3235
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#532

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #16811180	C₂	semidirect
2. #16811029	C₂	semidirect
3. #16810987	C₂	semidirect
4. #16810944	C₂	semidirect
5. #16810913	C₂	semidirect
6. #16810903	C₂	semidirect
7. #16810680	C₂	semidirect
115. C₅₃	#3235	semidirect

H ≅ G/N

type

1. #16811180

C₂

semidirect

2. #16811029

C₂

semidirect

3. #16810987

C₂

semidirect

4. #16810944

C₂

semidirect

5. #16810913

C₂

semidirect

6. #16810903

C₂

semidirect

7. #16810680

C₂

semidirect

115. C₅₃

#3235

semidirect

Subgroups — #20285909

Conjugacy classes of subgroups

Complements