Group #20274721 — subgroups

Subgroups — #20274721

Order 6784.

308 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 125 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×C₄×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133142	C₂	1	2	yes	maximal
3	#16810493	C₂	1	1	yes	maximal, characteristic
4	#16810386	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₀₆	C₂²	1	1	yes	characteristic
7	C₄×Dic₁₀₆	—	2	2	no	—
8	C₄×Dic₁₀₆	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₅₃	—	2	1	no	—
10	C₂×C₄⋊Dic₅₃	—	2	1	no	—
11	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
13	C₂×#11719919	—	2	1	no	—
14	C₂×C₄×Dic₅₃	—	2	1	no	—
15	#13133142	—	2	4	no	—
16	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₄⋊C₈×C₅₃	—	2	2	no	—
19	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₂²×Dic₅₃	—	2	1	no	—
23	C₂²×Dic₅₃	—	2	1	no	—
24	C₂²×Dic₅₃	—	2	2	no	—
25	C₂×Dic₁₀₆	—	2	1	no	—
26	C₂×Dic₁₀₆	—	2	1	no	—
27	C₂×Dic₁₀₆	—	2	1	no	—
28	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
29	C₂×Dic₁₀₆	D₄	1	2	yes	—
30	C₂×Dic₁₀₆	Q₈	1	2	yes	—
31	C₂×C₈×C₅₃	—	2	2	no	—
32	C₂×C₈×C₅₃	—	2	2	no	—
33	C₂×C₈×C₅₃	—	2	2	no	—
34	C₄²×C₅₃	C₂×C₄	1	2	yes	—
35	C₄²×C₅₃	D₄	1	2	yes	—
36	C₄⋊Dic₅₃	—	2	1	no	—
37	C₄⋊Dic₅₃	—	2	2	no	—
38	C₄⋊Dic₅₃	—	2	2	no	—
39	C₄⋊Dic₅₃	—	4	2	no	—
40	C₄⋊Dic₅₃	—	4	2	no	—
41	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
42	#11719919	—	2	4	no	—
43	C₄×Dic₅₃	—	2	4	no	—
44	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	—	2	1	no	—
46	C₂×C₄×C₅₃	—	2	1	no	—
47	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
49	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
50	C₂×C₄×C₅₃	Q₁₆	1	2	yes	—
51	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
57	C₂×Dic₅₃	—	2	2	no	—
58	C₂×Dic₅₃	—	2	2	no	—
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	2	no	—
64	C₂×Dic₅₃	—	4	2	no	—
65	C₂×Dic₅₃	—	4	2	no	—
66	Dic₁₀₆	—	2	2	no	—
67	Dic₁₀₆	—	2	2	no	—
68	Dic₁₀₆	—	2	2	no	—
69	Dic₁₀₆	C₄⋊C₄	1	4	yes	complemented
70	C₈×C₅₃	—	4	2	no	—
71	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₅₃	#121	1	1	yes	characteristic
73	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂²×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
75	C₂²×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
76	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₂²×C₅₃	#137	1	1	yes	characteristic
78	C₄×C₅₃	—	2	2	no	—
79	C₄×C₅₃	#103	1	4	yes	complemented
80	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₄×C₅₃	#117	1	1	yes	characteristic
82	C₄×C₅₃	#121	1	1	yes	characteristic
83	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
84	Dic₅₃	—	2	4	no	—
85	Dic₅₃	—	4	2	no	—
86	Dic₅₃	—	8	2	no	—
87	#1803	—	53	1	no	maximal, Sylow
88	C₂×C₅₃	#380	1	1	yes	characteristic
89	C₂×C₅₃	C₂×#103	1	1	yes	characteristic
90	C₂×C₅₃	#417	1	1	yes	characteristic
91	C₂×C₅₃	#460	1	1	yes	characteristic
92	C₂×C₅₃	#462	1	1	yes	characteristic
93	C₂×C₅₃	#475	1	1	yes	characteristic
94	C₂×C₅₃	#477	1	1	yes	characteristic
95	C₂×C₄×Q₈	—	53	1	no	—
96	C₂×C₄⋊C₈	D₅₃	1	1	yes	characteristic
97	C₂×#103	—	53	2	no	—
98	#384	—	53	1	no	—
99	#340	—	53	2	no	—
100	C₅₃	#1803	1	1	yes	characteristic, complemented, Sylow
101	C₂²×Q₈	—	53	1	no	—
102	C₂²×C₈	C₂×D₅₃	1	2	yes	—
103	C₄×Q₈	—	53	4	no	—
104	C₄×Q₈	—	106	2	no	—
105	C₂×C₄⋊C₄	—	53	1	no	—
106	C₂×C₄⋊C₄	—	53	2	no	—
107	C₂×C₄⋊C₄	—	106	1	no	—
108	C₂×C₄⋊C₄	—	106	1	no	—
109	C₂×C₄⋊C₄	—	106	1	no	—
110	C₂×C₄²	—	106	1	no	—
111	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
112	C₄⋊C₈	—	2	2	no	—
113	#103	—	106	4	no	—
114	C₂×Q₈	—	53	2	no	—
115	C₂×Q₈	—	53	2	no	—
116	C₂×Q₈	—	53	2	no	—
117	C₂×Q₈	—	106	1	no	—
118	C₂×Q₈	—	106	1	no	—
119	C₂×Q₈	—	106	1	no	—
120	C₂²×C₄	—	106	1	no	—
121	C₂²×C₄	—	106	1	no	—
122	C₂²×C₄	—	106	2	no	—
123	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
124	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₄⋊C₄	—	53	2	no	—
130	C₄⋊C₄	—	106	1	no	—
131	C₄⋊C₄	—	106	4	no	—
132	C₄⋊C₄	—	106	2	no	—
133	C₄⋊C₄	—	106	2	no	—
134	C₄⋊C₄	—	212	2	no	—
135	C₄⋊C₄	—	212	2	no	—
136	C₄²	—	106	4	no	—
137	C₄²	C₄×D₅₃	1	2	yes	—
138	C₄²	C₂²⋊D₅₃	1	2	yes	—
139	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
140	Q₈	—	53	4	no	—
141	Q₈	—	106	2	no	—
142	Q₈	—	106	2	no	—
143	Q₈	—	106	2	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	106	2	no	—
147	C₂×C₄	—	106	2	no	—
148	C₂×C₄	—	106	2	no	—
149	C₂×C₄	—	106	2	no	—
150	C₂×C₄	—	212	2	no	—
151	C₂×C₄	—	212	2	no	—
152	C₂×C₄	—	212	2	no	—
153	C₂×C₄	—	212	1	no	—
154	C₂×C₄	—	212	1	no	—
155	C₂×C₄	#11719913	1	2	yes	—
156	C₂×C₄	Dic₂₁₂	1	2	yes	—
157	C₂×C₄	#11719921	1	2	yes	—
158	C₂×C₄	#11719921	1	2	yes	—
159	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
160	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
161	C₂×C₄	#11719947	1	1	yes	characteristic
162	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
163	C₂×C₄	#11719949	1	1	yes	characteristic
164	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
165	C₈	—	4	2	no	—
166	C₂²	C₂×#11719921	1	1	yes	characteristic
167	C₂²	#13133192	1	1	yes	characteristic
168	C₂²	#13133197	1	1	yes	characteristic
169	C₂²	C₂×#11719913	1	1	yes	characteristic
170	C₂²	C₂×Dic₂₁₂	1	1	yes	characteristic
171	C₂²	#13133245	1	1	yes	characteristic
172	C₂²	#13133250	1	1	yes	characteristic
173	C₄	—	2	2	no	—
174	C₄	—	106	4	no	—
175	C₄	—	424	2	no	—
176	C₄	—	212	2	no	—
177	C₄	#13133142	1	4	yes	complemented
178	C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
179	C₄	#13133175	1	1	yes	characteristic
180	C₄	#13133188	1	1	yes	characteristic
181	C₄	#13133193	1	1	yes	characteristic
182	C₂	#16810480	1	1	yes	characteristic
183	C₂	C₂×#13133142	1	1	yes	characteristic
184	C₂	#16810625	1	1	yes	characteristic
185	C₂	#16810824	1	1	yes	characteristic
186	C₂	#16810834	1	1	yes	characteristic
187	C₂	#16810909	1	1	yes	characteristic
188	C₂	#16810919	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₀₆

C₂

yes

maximal, characteristic

C₂×#13133142

C₂

yes

maximal

#16810493

C₂

yes

maximal, characteristic

#16810386

C₂

yes

maximal

C₂×C₄⋊C₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

characteristic

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

C₄

yes

complemented

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×#11719919

—

C₂×C₄×Dic₅₃

—

#13133142

—

C₂²×C₈×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₂×Dic₁₀₆

D₄

yes

—

C₂×Dic₁₀₆

Q₈

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

D₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

—

C₄×Dic₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

Q₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

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

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

C₄⋊C₄

yes

complemented

C₈×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

#121

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₃

C₂×SD₁₆

yes

characteristic

C₂²×C₅₃

C₂×Q₁₆

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

#103

yes

complemented

C₄×C₅₃

C₂×C₄⋊C₄

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄×Q₈

—

C₂×C₄⋊C₈

D₅₃

yes

—

—

—

100

C₅₃

#1803

yes

characteristic, complemented, Sylow

101

C₂²×Q₈

—

102

C₂²×C₈

C₂×D₅₃

yes

—

103

C₄×Q₈

—

104

C₄×Q₈

—

106

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

106

—

108

C₂×C₄⋊C₄

—

106

—

109

C₂×C₄⋊C₄

—

106

—

110

C₂×C₄²

—

106

—

111

C₂×C₄²

C₂×D₅₃

yes

characteristic

112

C₄⋊C₈

—

113

#103

—

106

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×Q₈

—

106

—

118

C₂×Q₈

—

106

—

119

C₂×Q₈

—

106

—

120

C₂²×C₄

—

106

—

121

C₂²×C₄

—

106

—

122

C₂²×C₄

—

106

—

123

C₂²×C₄

D₂₁₂

yes

characteristic

124

C₂²×C₄

D₂₁₂

yes

characteristic

125

C₂²×C₄

C₂²×D₅₃

yes

characteristic

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

106

—

131

C₄⋊C₄

—

106

—

132

C₄⋊C₄

—

106

—

133

C₄⋊C₄

—

106

—

134

C₄⋊C₄

—

212

—

135

C₄⋊C₄

—

212

—

136

C₄²

—

106

—

137

C₄²

C₄×D₅₃

yes

—

138

C₄²

C₂²⋊D₅₃

yes

—

139

C₂³

C₂×D₂₁₂

yes

characteristic

140

Q₈

—

141

Q₈

—

106

—

142

Q₈

—

106

—

143

Q₈

—

106

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

106

—

147

C₂×C₄

—

106

—

148

C₂×C₄

—

106

—

149

C₂×C₄

—

106

—

150

C₂×C₄

—

212

—

151

C₂×C₄

—

212

—

152

C₂×C₄

—

212

—

153

C₂×C₄

—

212

—

154

C₂×C₄

—

212

—

155

C₂×C₄

#11719913

yes

—

156

C₂×C₄

Dic₂₁₂

yes

—

157

C₂×C₄

#11719921

yes

—

158

C₂×C₄

#11719921

yes

—

159

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

160

C₂×C₄

D₄×D₅₃

yes

characteristic

161

C₂×C₄

#11719947

yes

characteristic

162

C₂×C₄

Q₈×D₅₃

yes

characteristic

163

C₂×C₄

#11719949

yes

characteristic

164

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

165

C₈

—

166

C₂²

C₂×#11719921

yes

characteristic

167

C₂²

#13133192

yes

characteristic

168

C₂²

#13133197

yes

characteristic

169

C₂²

C₂×#11719913

yes

characteristic

170

C₂²

C₂×Dic₂₁₂

yes

characteristic

171

C₂²

#13133245

yes

characteristic

172

C₂²

#13133250

yes

characteristic

173

C₄

—

174

C₄

—

106

—

175

C₄

—

424

—

176

C₄

—

212

—

177

C₄

#13133142

yes

complemented

178

C₄

C₄⋊C₄×D₅₃

yes

characteristic

179

C₄

#13133175

yes

characteristic

180

C₄

#13133188

yes

characteristic

181

C₄

#13133193

yes

characteristic

182

C₂

#16810480

yes

characteristic

183

C₂

C₂×#13133142

yes

characteristic

184

C₂

#16810625

yes

characteristic

185

C₂

#16810824

yes

characteristic

186

C₂

#16810834

yes

characteristic

187

C₂

#16810909

yes

characteristic

188

C₂

#16810919

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	#1803
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

#1803

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
8. C₄×Dic₁₀₆	C₄	semidirect
69. Dic₁₀₆	C₄⋊C₄	semidirect
79. C₄×C₅₃	#103	semidirect
100. C₅₃	#1803	semidirect
177. C₄	#13133142	semidirect

H ≅ G/N

type

8. C₄×Dic₁₀₆

C₄

semidirect

69. Dic₁₀₆

C₄⋊C₄

semidirect

79. C₄×C₅₃

#103

semidirect

100. C₅₃

#1803

semidirect

177. C₄

#13133142

semidirect

Subgroups — #20274721

Conjugacy classes of subgroups

Complements