Group #20273095 — subgroups

Subgroups — #20273095

Order 6784.

276 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 101 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	C₄×#11719945	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#88322	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810353	C₂	1	2	yes	maximal, complemented
4	#16810350	C₂	1	2	yes	maximal, complemented
5	C₅₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11719945	C₄	1	1	yes	characteristic
7	C₂²×#88322	—	2	1	no	—
8	C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
9	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
10	C₄×D₂₁₂	C₄	1	1	yes	characteristic
11	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
12	C₄×C₂²⋊D₅₃	—	2	1	no	—
13	C₄×C₂²⋊D₅₃	—	2	1	no	—
14	#13133176	C₄	1	1	yes	characteristic
15	#13133174	—	2	1	no	—
16	C₄²×D₅₃	—	2	1	no	—
17	C₄×#88322	—	2	1	no	—
18	C₄×#88322	C₂²	1	2	yes	—
19	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
21	C₂²⋊C₈×C₅₃	—	2	1	no	—
22	C₄×C₈×C₅₃	—	2	1	no	—
23	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
26	C₂×C₂²⋊D₅₃	—	2	1	no	—
27	#11719945	—	2	2	no	—
28	#11719945	C₈	1	4	yes	complemented
29	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₅₃	—	2	1	no	—
31	C₂×C₄×D₅₃	—	2	1	no	—
32	C₂×C₄×D₅₃	—	2	1	no	—
33	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×C₅₃	—	2	2	no	—
35	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₅₃	D₄	1	1	yes	characteristic
37	C₄²×C₅₃	D₄	1	1	yes	characteristic
38	C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	C₂²⋊Dic₅₃	—	2	1	no	—
40	#11719921	—	2	1	no	—
41	#11719921	—	2	1	no	—
42	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
44	#11719919	—	2	1	no	—
45	#11719919	—	2	1	no	—
46	C₄×Dic₅₃	—	2	1	no	—
47	C₄×Dic₅₃	—	2	1	no	—
48	C₄×Dic₅₃	—	2	1	no	—
49	C₂×#88322	—	2	4	no	—
50	C₂×#88322	—	2	2	no	—
51	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×D₅₃	—	2	1	no	—
53	C₂×C₄×C₅₃	—	2	2	no	—
54	C₂×C₄×C₅₃	—	2	2	no	—
55	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
59	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
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×D₄	1	1	yes	characteristic
63	C₂²⋊D₅₃	—	2	4	no	—
64	C₂×Dic₅₃	—	2	1	no	—
65	C₂×Dic₅₃	—	2	1	no	—
66	C₂×Dic₅₃	—	2	1	no	—
67	D₂₁₂	—	2	1	no	—
68	D₂₁₂	C₂×C₈	1	2	yes	complemented
69	C₄×D₅₃	—	2	4	no	—
70	C₄×D₅₃	—	4	2	no	—
71	C₄×D₅₃	—	4	2	no	—
72	Dic₁₀₆	—	2	1	no	—
73	Dic₁₀₆	C₂×C₈	1	2	yes	complemented
74	C₈×C₅₃	—	4	2	no	—
75	#88322	—	2	4	no	—
76	C₂²×C₅₃	—	2	1	no	—
77	C₂²×C₅₃	—	2	1	no	—
78	C₂²×C₅₃	C₂²⋊C₈	1	2	yes	—
79	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×D₅₃	—	2	2	no	—
81	C₂×D₅₃	—	4	1	no	—
82	C₂×D₅₃	—	4	1	no	—
83	C₄×C₅₃	—	2	1	no	—
84	C₄×C₅₃	—	2	1	no	—
85	C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
86	C₄×C₅₃	#104	1	2	yes	—
87	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
88	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
89	C₄×C₅₃	#135	1	2	yes	—
90	Dic₅₃	—	2	2	no	—
91	Dic₅₃	—	4	1	no	—
92	Dic₅₃	—	4	1	no	—
93	#1476	—	53	1	no	maximal, Sylow
94	C₂×C₅₃	—	2	2	no	—
95	C₂×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
96	C₂×C₅₃	#416	1	1	yes	characteristic
97	C₂×C₅₃	C₂×#104	1	1	yes	characteristic
98	D₅₃	—	4	2	no	—
99	C₄×#41	—	53	1	no	—
100	#431	D₅₃	1	1	yes	characteristic, complemented
101	C₂×C₄×C₈	—	53	1	no	—
102	Q₈⋊C₈	—	53	2	no	—
103	D₄⋊C₈	—	53	2	no	—
104	C₅₃	#1476	1	1	yes	characteristic, complemented, Sylow
105	C₂×#41	—	53	1	no	—
106	C₂²×C₈	—	106	1	no	—
107	C₄×Q₈	—	53	1	no	—
108	C₄×Q₈	—	53	1	no	—
109	C₄×D₄	—	53	1	no	—
110	C₄×D₄	—	53	1	no	—
111	C₄×D₄	—	106	1	no	—
112	C₄×D₄	—	106	1	no	—
113	#117	—	53	1	no	—
114	#117	—	106	1	no	—
115	C₂×C₄²	—	106	1	no	—
116	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
117	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
118	C₂²⋊C₈	—	2	1	no	—
119	C₄×C₈	—	2	1	no	—
120	C₄×C₈	—	53	2	no	—
121	C₄×C₈	—	106	1	no	—
122	#41	—	53	4	no	—
123	#41	—	106	2	no	—
124	C₂×Q₈	—	53	1	no	—
125	C₂×D₄	—	53	1	no	—
126	C₂×D₄	—	106	1	no	—
127	C₂²×C₄	—	106	1	no	—
128	C₂²×C₄	—	106	1	no	—
129	C₂²×C₄	—	106	1	no	—
130	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
131	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
132	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	106	4	no	—
135	C₂×C₈	—	106	2	no	—
136	C₄⋊C₄	—	53	1	no	—
137	C₄⋊C₄	—	53	1	no	—
138	C₄⋊C₄	—	106	1	no	—
139	C₄⋊C₄	—	106	1	no	—
140	C₂²⋊C₄	—	106	1	no	—
141	C₂²⋊C₄	—	106	1	no	—
142	C₂²⋊C₄	—	106	1	no	—
143	C₄²	—	106	1	no	—
144	C₄²	—	106	1	no	—
145	C₄²	—	106	1	no	—
146	C₄²	C₄×D₅₃	1	1	yes	characteristic
147	C₄²	D₂₁₂	1	1	yes	characteristic
148	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
149	C₄²	C₂²×D₅₃	1	1	yes	characteristic
150	C₂³	—	106	1	no	—
151	C₂³	#11719921	1	1	yes	characteristic
152	Q₈	—	53	2	no	—
153	Q₈	—	106	1	no	—
154	D₄	—	53	2	no	—
155	D₄	—	106	4	no	—
156	D₄	—	106	1	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	106	1	no	—
160	C₂×C₄	—	106	4	no	—
161	C₂×C₄	—	106	1	no	—
162	C₂×C₄	—	106	1	no	—
163	C₂×C₄	—	212	2	no	—
164	C₂×C₄	—	212	2	no	—
165	C₂×C₄	C₈×D₅₃	1	2	yes	—
166	C₂×C₄	#11719912	1	2	yes	—
167	C₂×C₄	#11719921	1	1	yes	characteristic
168	C₂×C₄	#11719921	1	1	yes	characteristic
169	C₂×C₄	#11719921	1	1	yes	characteristic
170	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
171	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
172	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
173	C₈	—	4	2	no	—
174	C₈	—	106	4	no	—
175	C₂²	—	2	1	no	—
176	C₂²	—	2	1	no	—
177	C₂²	—	106	2	no	—
178	C₂²	—	212	1	no	—
179	C₂²	—	212	1	no	—
180	C₂²	#13133129	1	2	yes	—
181	C₂²	C₂×#11719921	1	1	yes	characteristic
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	106	2	no	—
185	C₄	—	212	1	no	—
186	C₄	—	212	1	no	—
187	C₄	#13133129	1	2	yes	—
188	C₄	#13133143	1	2	yes	—
189	C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
190	C₄	C₂×#11719912	1	1	yes	characteristic
191	C₄	#13133237	1	2	yes	—
192	C₂	—	2	2	no	—
193	C₂	—	212	2	no	—
194	C₂	C₂×#13133129	1	1	yes	characteristic
195	C₂	#16810617	1	1	yes	characteristic
196	C₂	C₂×#13133143	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#11719945

C₂

yes

maximal, characteristic

C₂×C₄×#88322

C₂

yes

maximal, characteristic, complemented

#16810353

C₂

yes

maximal, complemented

#16810350

C₂

yes

maximal, complemented

C₅₃×#431

C₂

yes

maximal, characteristic, complemented

C₂×#11719945

C₄

yes

characteristic

C₂²×#88322

—

C₄×Dic₁₀₆

C₄

yes

characteristic

C₄×Dic₁₀₆

C₂²

yes

characteristic

C₄×D₂₁₂

C₄

yes

characteristic

C₄×D₂₁₂

C₂²

yes

characteristic

C₄×C₂²⋊D₅₃

—

C₄×C₂²⋊D₅₃

—

#13133176

C₄

yes

characteristic

#13133174

—

C₄²×D₅₃

—

C₄×#88322

—

C₄×#88322

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂×C₂²⋊D₅₃

—

#11719945

—

#11719945

C₈

yes

complemented

C₂×D₂₁₂

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

—

#11719921

—

#11719921

—

C₄⋊Dic₅₃

C₂×C₄

yes

characteristic

C₄⋊Dic₅₃

C₂×C₄

yes

characteristic

#11719919

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

C₂×C₈

yes

complemented

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

C₂×C₈

yes

complemented

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²⋊C₈

yes

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊C₈

yes

—

C₄×C₅₃

#104

yes

—

C₄×C₅₃

C₂²×C₈

yes

characteristic

C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₄×C₅₃

#135

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#1476

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×C₂²⋊C₈

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

C₄×#41

—

100

#431

D₅₃

yes

characteristic, complemented

101

C₂×C₄×C₈

—

102

Q₈⋊C₈

—

103

D₄⋊C₈

—

104

C₅₃

#1476

yes

characteristic, complemented, Sylow

105

C₂×#41

—

106

C₂²×C₈

—

106

—

107

C₄×Q₈

—

108

C₄×Q₈

—

109

C₄×D₄

—

110

C₄×D₄

—

111

C₄×D₄

—

106

—

112

C₄×D₄

—

106

—

113

#117

—

114

#117

—

106

—

115

C₂×C₄²

—

106

—

116

C₂×C₄²

C₂×D₅₃

yes

characteristic

117

C₄⋊C₈

C₂×D₅₃

yes

—

118

C₂²⋊C₈

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

—

106

—

122

#41

—

123

#41

—

106

—

124

C₂×Q₈

—

125

C₂×D₄

—

126

C₂×D₄

—

106

—

127

C₂²×C₄

—

106

—

128

C₂²×C₄

—

106

—

129

C₂²×C₄

—

106

—

130

C₂²×C₄

C₄×D₅₃

yes

characteristic

131

C₂²×C₄

D₂₁₂

yes

characteristic

132

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

133

C₂×C₈

—

134

C₂×C₈

—

106

—

135

C₂×C₈

—

106

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

106

—

139

C₄⋊C₄

—

106

—

140

C₂²⋊C₄

—

106

—

141

C₂²⋊C₄

—

106

—

142

C₂²⋊C₄

—

106

—

143

C₄²

—

106

—

144

C₄²

—

106

—

145

C₄²

—

106

—

146

C₄²

C₄×D₅₃

yes

characteristic

147

C₄²

D₂₁₂

yes

characteristic

148

C₄²

C₂²⋊D₅₃

yes

characteristic

149

C₄²

C₂²×D₅₃

yes

characteristic

150

C₂³

—

106

—

151

C₂³

#11719921

yes

characteristic

152

Q₈

—

153

Q₈

—

106

—

154

D₄

—

155

D₄

—

106

—

156

D₄

—

106

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

106

—

160

C₂×C₄

—

106

—

161

C₂×C₄

—

106

—

162

C₂×C₄

—

106

—

163

C₂×C₄

—

212

—

164

C₂×C₄

—

212

—

165

C₂×C₄

C₈×D₅₃

yes

—

166

C₂×C₄

#11719912

yes

—

167

C₂×C₄

#11719921

yes

characteristic

168

C₂×C₄

#11719921

yes

characteristic

169

C₂×C₄

#11719921

yes

characteristic

170

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

171

C₂×C₄

C₂×D₂₁₂

yes

characteristic

172

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

173

C₈

—

174

C₈

—

106

—

175

C₂²

—

176

C₂²

—

177

C₂²

—

106

—

178

C₂²

—

212

—

179

C₂²

—

212

—

180

C₂²

#13133129

yes

—

181

C₂²

C₂×#11719921

yes

characteristic

182

C₄

—

183

C₄

—

184

C₄

—

106

—

185

C₄

—

212

—

186

C₄

—

212

—

187

C₄

#13133129

yes

—

188

C₄

#13133143

yes

—

189

C₄

C₂×C₈×D₅₃

yes

characteristic

190

C₄

C₂×#11719912

yes

characteristic

191

C₄

#13133237

yes

—

192

C₂

—

193

C₂

—

212

—

194

C₂

C₂×#13133129

yes

characteristic

195

C₂

#16810617

yes

characteristic

196

C₂

C₂×#13133143

yes

characteristic

	subgroup	G/H
center	C₄²	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂²×C₈
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#431	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#431	D₅₃
hypocenter	C₅₃	#1476
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₄²

C₂²⋊D₅₃

commutator

C₄×C₅₃

C₂²×C₈

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#431

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
2. C₂×C₄×#88322	C₂	semidirect
3. #16810353	C₂	semidirect
4. #16810350	C₂	semidirect
5. C₅₃×#431	C₂	semidirect
28. #11719945	C₈	semidirect
68. D₂₁₂	C₂×C₈	semidirect
73. Dic₁₀₆	C₂×C₈	semidirect
100. #431	D₅₃	semidirect
104. C₅₃	#1476	semidirect

H ≅ G/N

type

2. C₂×C₄×#88322

C₂

semidirect

3. #16810353

C₂

semidirect

4. #16810350

C₂

semidirect

5. C₅₃×#431

C₂

semidirect

28. #11719945

C₈

semidirect

68. D₂₁₂

C₂×C₈

semidirect

73. Dic₁₀₆

C₂×C₈

semidirect

100. #431

D₅₃

semidirect

104. C₅₃

#1476

semidirect

Subgroups — #20273095

Conjugacy classes of subgroups

Complements