Group #20275485 — subgroups

Subgroups — #20275485

Order 6784.

240 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 113 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×#13133151	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133149	C₂	1	1	yes	maximal, characteristic
3	#16810489	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	#16810386	C₂	1	1	yes	maximal, characteristic
6	#16810386	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₅₃	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×C₄×Dic₅₃	—	2	1	no	—
13	#13133151	—	2	2	no	—
14	#13133149	—	2	2	no	—
15	#13133124	—	2	1	no	—
16	#13133124	—	2	1	no	—
17	#13133124	—	2	1	no	—
18	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₅₃	C₄	1	4	yes	complemented
22	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
25	C₂²×Dic₅₃	—	2	1	no	—
26	C₂²×Dic₅₃	—	2	1	no	—
27	C₂²×Dic₅₃	—	2	1	no	—
28	C₂²×Dic₅₃	—	2	1	no	—
29	C₂×C₈×C₅₃	—	2	1	no	—
30	C₂×C₈×C₅₃	—	2	1	no	—
31	C₂×C₈×C₅₃	—	2	1	no	—
32	C₂×C₈×C₅₃	—	2	1	no	—
33	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₄²×C₅₃	C₂×C₄	1	2	yes	—
36	C₄²×C₅₃	Q₈	1	2	yes	—
37	C₄⋊Dic₅₃	—	2	1	no	—
38	C₄⋊Dic₅₃	—	2	1	no	—
39	C₄⋊Dic₅₃	—	2	1	no	—
40	C₄⋊Dic₅₃	—	2	1	no	—
41	C₄⋊Dic₅₃	—	2	1	no	—
42	C₄⋊Dic₅₃	—	2	1	no	—
43	C₄⋊Dic₅₃	D₄	1	2	yes	—
44	C₄⋊Dic₅₃	Q₈	1	2	yes	—
45	C₄×Dic₅₃	—	4	2	no	—
46	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	—	2	1	no	—
48	C₂×C₄×C₅₃	—	2	1	no	—
49	C₂×C₄×C₅₃	—	2	1	no	—
50	C₂×C₄×C₅₃	—	2	1	no	—
51	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
52	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
53	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
58	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
59	C₂×Dic₅₃	—	2	2	no	—
60	C₂×Dic₅₃	—	2	2	no	—
61	C₂×Dic₅₃	—	4	1	no	—
62	C₂×Dic₅₃	—	4	1	no	—
63	C₂×Dic₅₃	—	4	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	—	4	1	no	—
70	C₂×Dic₅₃	—	4	1	no	—
71	C₈×C₅₃	—	2	2	no	—
72	C₈×C₅₃	—	2	2	no	—
73	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₅₃	#121	1	1	yes	characteristic
75	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂²×C₅₃	#135	1	1	yes	characteristic
77	C₂²×C₅₃	#135	1	1	yes	characteristic
78	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂²×C₅₃	#137	1	1	yes	characteristic
80	C₄×C₅₃	—	2	2	no	—
81	C₄×C₅₃	—	2	2	no	—
82	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
83	C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
84	C₄×C₅₃	#125	1	1	yes	characteristic
85	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
86	Dic₅₃	—	4	2	no	—
87	Dic₅₃	—	4	2	no	—
88	Dic₅₃	—	8	1	no	—
89	Dic₅₃	—	8	1	no	—
90	#1949	—	53	1	no	maximal, Sylow
91	C₂×C₅₃	#389	1	1	yes	characteristic
92	C₂×C₅₃	#427	1	1	yes	characteristic
93	C₂×C₅₃	#428	1	1	yes	characteristic
94	C₂×C₅₃	#463	1	1	yes	characteristic
95	C₂×C₅₃	#464	1	1	yes	characteristic
96	C₂×C₅₃	#478	1	1	yes	characteristic
97	C₂×C₅₃	#479	1	1	yes	characteristic
98	C₂×#107	—	53	1	no	—
99	C₂×#106	—	53	1	no	—
100	C₂×C₄⋊C₈	D₅₃	1	1	yes	characteristic
101	#382	—	53	1	no	—
102	C₄×C₄⋊C₄	—	53	1	no	—
103	#340	—	53	1	no	—
104	#340	—	53	1	no	—
105	C₅₃	#1949	1	1	yes	characteristic, complemented, Sylow
106	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
107	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
108	C₂×C₄⋊C₄	—	53	1	no	—
109	C₂×C₄⋊C₄	—	53	1	no	—
110	C₂×C₄⋊C₄	—	53	1	no	—
111	C₂×C₄⋊C₄	—	53	1	no	—
112	C₂×C₄²	—	106	1	no	—
113	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
114	#107	—	106	2	no	—
115	#106	—	106	2	no	—
116	C₄⋊C₈	Dic₅₃	1	4	yes	complemented
117	#95	—	106	1	no	—
118	#95	—	106	1	no	—
119	#95	—	106	1	no	—
120	C₂²×C₄	—	106	1	no	—
121	C₂²×C₄	—	106	1	no	—
122	C₂²×C₄	—	106	1	no	—
123	C₂²×C₄	—	106	1	no	—
124	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
125	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
126	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
132	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
133	C₄⋊C₄	—	53	2	no	—
134	C₄⋊C₄	—	53	2	no	—
135	C₄⋊C₄	—	106	1	no	—
136	C₄⋊C₄	—	106	1	no	—
137	C₄⋊C₄	—	106	1	no	—
138	C₄⋊C₄	—	106	1	no	—
139	C₄⋊C₄	—	106	1	no	—
140	C₄⋊C₄	—	106	1	no	—
141	C₄²	—	212	2	no	—
142	C₄²	Dic₁₀₆	1	2	yes	—
143	C₄²	C₂×Dic₅₃	1	2	yes	—
144	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	106	2	no	—
150	C₂×C₄	—	106	2	no	—
151	C₂×C₄	—	212	1	no	—
152	C₂×C₄	—	212	1	no	—
153	C₂×C₄	—	212	1	no	—
154	C₂×C₄	—	212	1	no	—
155	C₂×C₄	—	212	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	C₄⋊Dic₅₃	1	2	yes	—
162	C₂×C₄	C₄⋊Dic₅₃	1	2	yes	—
163	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
164	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
165	C₂×C₄	#11719947	1	1	yes	characteristic
166	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
167	C₂×C₄	#11719949	1	1	yes	characteristic
168	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	—	2	2	no	—
171	C₂²	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
172	C₂²	#13133192	1	1	yes	characteristic
173	C₂²	#13133197	1	1	yes	characteristic
174	C₂²	#13133239	1	1	yes	characteristic
175	C₂²	#13133239	1	1	yes	characteristic
176	C₂²	#13133245	1	1	yes	characteristic
177	C₂²	#13133250	1	1	yes	characteristic
178	C₄	—	2	2	no	—
179	C₄	—	2	2	no	—
180	C₄	—	424	1	no	—
181	C₄	—	424	1	no	—
182	C₄	—	212	2	no	—
183	C₄	—	212	2	no	—
184	C₄	D₄×Dic₅₃	1	1	yes	characteristic
185	C₄	Q₈×Dic₅₃	1	1	yes	characteristic
186	C₄	#13133208	1	1	yes	characteristic
187	C₄	#13133216	1	1	yes	characteristic
188	C₂	#16810522	1	1	yes	characteristic
189	C₂	#16810661	1	1	yes	characteristic
190	C₂	#16810665	1	1	yes	characteristic
191	C₂	#16810839	1	1	yes	characteristic
192	C₂	#16810846	1	1	yes	characteristic
193	C₂	#16810924	1	1	yes	characteristic
194	C₂	#16810931	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133151

C₂

yes

maximal, characteristic

C₂×#13133149

C₂

yes

maximal, characteristic

#16810489

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₅₃

C₂

yes

maximal, characteristic

#16810386

C₂

yes

maximal, characteristic

#16810386

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₅₃

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×C₄×Dic₅₃

—

—

—

—

—

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₄

yes

complemented

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

Q₈

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

—

C₄⋊Dic₅₃

Q₈

yes

—

C₄×Dic₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

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

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

#121

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

characteristic

C₄×C₅₃

C₄×Q₈

yes

characteristic

C₄×C₅₃

#125

yes

characteristic

C₄×C₅₃

C₄⋊Q₈

yes

characteristic

Dic₅₃

—

Dic₅₃

—

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×C₄⋊C₈

D₅₃

yes

characteristic

101

#382

—

102

C₄×C₄⋊C₄

—

103

#340

—

104

#340

—

105

C₅₃

#1949

yes

characteristic, complemented, Sylow

106

C₂²×C₈

C₂×D₅₃

yes

characteristic

107

C₂²×C₈

C₂×D₅₃

yes

characteristic

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄²

—

106

—

113

C₂×C₄²

C₂×D₅₃

yes

characteristic

114

#107

—

106

—

115

#106

—

106

—

116

C₄⋊C₈

Dic₅₃

yes

complemented

117

#95

—

106

—

118

#95

—

106

—

119

#95

—

106

—

120

C₂²×C₄

—

106

—

121

C₂²×C₄

—

106

—

122

C₂²×C₄

—

106

—

123

C₂²×C₄

—

106

—

124

C₂²×C₄

D₂₁₂

yes

characteristic

125

C₂²×C₄

D₂₁₂

yes

characteristic

126

C₂²×C₄

C₂²×D₅₃

yes

characteristic

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

C₂×Dic₅₃

yes

—

132

C₂×C₈

C₂×Dic₅₃

yes

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

106

—

136

C₄⋊C₄

—

106

—

137

C₄⋊C₄

—

106

—

138

C₄⋊C₄

—

106

—

139

C₄⋊C₄

—

106

—

140

C₄⋊C₄

—

106

—

141

C₄²

—

212

—

142

C₄²

Dic₁₀₆

yes

—

143

C₄²

C₂×Dic₅₃

yes

—

144

C₂³

C₂×D₂₁₂

yes

characteristic

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

106

—

150

C₂×C₄

—

106

—

151

C₂×C₄

—

212

—

152

C₂×C₄

—

212

—

153

C₂×C₄

—

212

—

154

C₂×C₄

—

212

—

155

C₂×C₄

—

212

—

156

C₂×C₄

—

212

—

157

C₂×C₄

—

212

—

158

C₂×C₄

—

212

—

159

C₂×C₄

—

212

—

160

C₂×C₄

—

212

—

161

C₂×C₄

C₄⋊Dic₅₃

yes

—

162

C₂×C₄

C₄⋊Dic₅₃

yes

—

163

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

164

C₂×C₄

D₄×D₅₃

yes

characteristic

165

C₂×C₄

#11719947

yes

characteristic

166

C₂×C₄

Q₈×D₅₃

yes

characteristic

167

C₂×C₄

#11719949

yes

characteristic

168

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

169

C₈

—

170

C₈

—

171

C₂²

C₂×C₄⋊Dic₅₃

yes

characteristic

172

C₂²

#13133192

yes

characteristic

173

C₂²

#13133197

yes

characteristic

174

C₂²

#13133239

yes

characteristic

175

C₂²

#13133239

yes

characteristic

176

C₂²

#13133245

yes

characteristic

177

C₂²

#13133250

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

—

424

—

181

C₄

—

424

—

182

C₄

—

212

—

183

C₄

—

212

—

184

C₄

D₄×Dic₅₃

yes

characteristic

185

C₄

Q₈×Dic₅₃

yes

characteristic

186

C₄

#13133208

yes

characteristic

187

C₄

#13133216

yes

characteristic

188

C₂

#16810522

yes

characteristic

189

C₂

#16810661

yes

characteristic

190

C₂

#16810665

yes

characteristic

191

C₂

#16810839

yes

characteristic

192

C₂

#16810846

yes

characteristic

193

C₂

#16810924

yes

characteristic

194

C₂

#16810931

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×C₄⋊C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₅₃
hypocenter	C₅₃	#1949
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×C₄⋊C₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₅₃

hypocenter

C₅₃

#1949

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
21. C₄⋊C₈×C₅₃	C₄	semidirect
105. C₅₃	#1949	semidirect
116. C₄⋊C₈	Dic₅₃	semidirect

H ≅ G/N

type

21. C₄⋊C₈×C₅₃

C₄

semidirect

105. C₅₃

#1949

semidirect

116. C₄⋊C₈

Dic₅₃

semidirect

Subgroups — #20275485

Conjugacy classes of subgroups

Complements