Group #20284485 — subgroups

Subgroups — C₂×#16810917 #20284485

Order 6784.

416 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 157 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×#13133216	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133182	C₂	1	1	yes	maximal, characteristic
3	#16810917	C₂	1	8	yes	maximal, direct product factor
4	C₂×#13133150	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133146	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133142	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133140	C₂	1	1	yes	maximal, characteristic
8	C₂×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₀₆	—	2	1	no	—
10	C₂²×Dic₁₀₆	C₂²	1	1	yes	characteristic
11	C₂²×#88322	C₂²	1	1	yes	characteristic
12	#13133216	—	2	2	no	—
13	#13133216	C₂²	1	4	yes	—
14	#13133182	—	2	2	no	—
15	#13133182	C₂²	1	4	yes	—
16	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
17	C₂×#11719919	—	2	1	no	—
18	C₂×#11719919	—	2	1	no	—
19	C₂×C₄×Dic₅₃	—	2	1	no	—
20	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
21	#13133150	C₂²	1	4	yes	—
22	#13133146	C₂²	1	4	yes	—
23	#13133142	C₂²	1	4	yes	—
24	#13133140	C₂²	1	4	yes	—
25	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
28	#107×C₅₃	C₂²	1	4	yes	—
29	C₂²×C₄×C₅₃	—	2	1	no	—
30	C₂²×C₄×C₅₃	—	2	1	no	—
31	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
32	C₂²×Dic₅₃	—	2	1	no	—
33	C₂²×Dic₅₃	—	2	1	no	—
34	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
35	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
36	C₂×Dic₁₀₆	—	2	1	no	—
37	C₂×Dic₁₀₆	—	2	2	no	—
38	C₂×Dic₁₀₆	—	2	4	no	—
39	C₂×Dic₁₀₆	—	4	4	no	—
40	C₂×Dic₁₀₆	Q₈	1	4	yes	complemented
41	C₂×Dic₁₀₆	C₂³	1	2	yes	—
42	C₂×C₈×C₅₃	—	2	2	no	—
43	C₂×C₈×C₅₃	C₂³	1	2	yes	—
44	C₄⋊C₄×C₅₃	—	2	1	no	—
45	C₄⋊C₄×C₅₃	—	2	1	no	—
46	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
47	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
48	C₄⋊Dic₅₃	—	2	1	no	—
49	C₄⋊Dic₅₃	C₂³	1	2	yes	—
50	#11719919	—	2	4	no	—
51	#11719919	—	2	4	no	—
52	C₄×Dic₅₃	—	2	4	no	—
53	C₄×Dic₅₃	D₄	1	2	yes	—
54	C₄×Dic₅₃	C₂³	1	2	yes	—
55	C₂×#88322	—	2	2	no	—
56	C₂×#88322	C₂³	1	2	yes	—
57	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	—	2	2	no	—
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	—	4	2	no	—
61	C₂×C₄×C₅₃	—	4	2	no	—
62	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
64	C₂×C₄×C₅₃	#41	1	2	yes	—
65	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
66	C₂×Dic₅₃	—	2	4	no	—
67	C₂×Dic₅₃	—	2	2	no	—
68	C₂×Dic₅₃	—	2	2	no	—
69	C₂×Dic₅₃	—	2	2	no	—
70	C₂×Dic₅₃	—	4	2	no	—
71	C₂×Dic₅₃	Q₁₆	1	4	yes	—
72	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
73	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
74	Dic₁₀₆	—	2	4	no	—
75	Dic₁₀₆	—	2	2	no	—
76	Dic₁₀₆	—	4	8	no	—
77	Dic₁₀₆	C₂×Q₈	1	4	yes	complemented
78	C₈×C₅₃	—	2	2	no	—
79	#88322	—	2	2	no	—
80	C₂²×C₅₃	C₂²⋊Q₈	1	2	yes	—
81	C₂²×C₅₃	C₂×Q₁₆	1	2	yes	—
82	C₂²×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
83	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
84	C₄×C₅₃	—	4	2	no	—
85	C₄×C₅₃	—	4	2	no	—
86	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
87	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
88	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
89	Dic₅₃	—	2	4	no	—
90	Dic₅₃	—	2	2	no	—
91	Dic₅₃	—	4	2	no	—
92	Dic₅₃	C₂×Q₁₆	1	4	yes	—
93	C₂×#477	—	53	1	no	maximal, Sylow
94	C₂×C₅₃	#477	1	4	yes	complemented
95	C₂×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
97	C₂×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
98	C₂×C₄⋊Q₈	—	53	1	no	—
99	C₂×C₄×Q₈	—	53	1	no	—
100	#477	—	53	8	no	—
101	C₂×#107	—	53	1	no	—
102	C₂×#107	D₅₃	1	1	yes	characteristic
103	C₂×C₄⋊C₈	—	53	1	no	—
104	C₂×#103	—	53	1	no	—
105	C₂×#103	—	53	1	no	—
106	C₅₃	C₂×#477	1	1	yes	characteristic, complemented, Sylow
107	C₂²×Q₈	—	53	1	no	—
108	C₂²×Q₈	—	106	1	no	—
109	C₂²×C₈	—	53	1	no	—
110	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
111	C₄⋊Q₈	—	53	4	no	—
112	C₄⋊Q₈	—	106	2	no	—
113	C₄×Q₈	—	53	4	no	—
114	C₄×Q₈	—	106	2	no	—
115	C₂×C₄⋊C₄	—	53	1	no	—
116	C₂×C₄⋊C₄	—	106	1	no	—
117	C₂×C₄⋊C₄	—	106	1	no	—
118	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
120	C₂×C₄²	—	53	1	no	—
121	C₂×C₄²	—	106	1	no	—
122	#107	—	53	4	no	—
123	#107	C₂×D₅₃	1	4	yes	—
124	C₄⋊C₈	—	53	4	no	—
125	#103	—	53	4	no	—
126	#103	—	53	4	no	—
127	C₂×Q₈	—	53	4	no	—
128	C₂×Q₈	—	53	2	no	—
129	C₂×Q₈	—	106	4	no	—
130	C₂×Q₈	—	106	2	no	—
131	C₂×Q₈	—	106	1	no	—
132	C₂×Q₈	—	212	4	no	—
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	53	1	no	—
136	C₂²×C₄	—	53	1	no	—
137	C₂²×C₄	—	106	1	no	—
138	C₂²×C₄	—	106	1	no	—
139	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	53	2	no	—
142	C₂×C₈	—	106	2	no	—
143	C₂×C₈	C₂²×D₅₃	1	2	yes	—
144	C₄⋊C₄	—	2	1	no	—
145	C₄⋊C₄	—	2	1	no	—
146	C₄⋊C₄	—	53	2	no	—
147	C₄⋊C₄	—	106	4	no	—
148	C₄⋊C₄	—	106	4	no	—
149	C₄⋊C₄	—	106	1	no	—
150	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
151	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
152	C₄²	—	53	2	no	—
153	C₄²	—	53	2	no	—
154	C₄²	—	106	4	no	—
155	C₂³	D₄×D₅₃	1	1	yes	characteristic
156	Q₈	—	53	4	no	—
157	Q₈	—	106	4	no	—
158	Q₈	—	106	2	no	—
159	Q₈	—	212	8	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	53	4	no	—
165	C₂×C₄	—	53	2	no	—
166	C₂×C₄	—	53	2	no	—
167	C₂×C₄	—	106	4	no	—
168	C₂×C₄	—	106	2	no	—
169	C₂×C₄	—	106	2	no	—
170	C₂×C₄	—	106	2	no	—
171	C₂×C₄	—	212	2	no	—
172	C₂×C₄	#11719945	1	2	yes	—
173	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
174	C₂×C₄	Q₈×D₅₃	1	2	yes	—
175	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	106	2	no	—
178	C₂²	#13133195	1	2	yes	—
179	C₂²	Q₁₆×D₅₃	1	2	yes	—
180	C₂²	#13133241	1	2	yes	—
181	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
182	C₄	—	4	2	no	—
183	C₄	—	4	2	no	—
184	C₄	—	53	4	no	—
185	C₄	—	106	4	no	—
186	C₄	—	106	2	no	—
187	C₄	—	212	2	no	—
188	C₄	#13133195	1	2	yes	—
189	C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
190	C₄	C₂×#11719945	1	1	yes	characteristic
191	C₂	#16810917	1	4	yes	direct product factor
192	C₂	C₂×#13133195	1	1	yes	characteristic
193	C₂	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
194	C₂	C₂×#13133241	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133216

C₂

yes

maximal, characteristic

C₂×#13133182

C₂

yes

maximal, characteristic

#16810917

C₂

yes

maximal, direct product factor

C₂×#13133150

C₂

yes

maximal, characteristic

C₂×#13133146

C₂

yes

maximal, characteristic

C₂×#13133142

C₂

yes

maximal, characteristic

C₂×#13133140

C₂

yes

maximal, characteristic

C₂×#107×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

—

C₂²×Dic₁₀₆

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

#13133216

—

#13133216

C₂²

yes

—

#13133182

—

#13133182

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×#11719919

—

C₂×#11719919

—

C₂×C₄×Dic₅₃

—

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

#13133150

C₂²

yes

—

#13133146

C₂²

yes

—

#13133142

C₂²

yes

—

#13133140

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

#107×C₅₃

C₂²

yes

—

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×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

Q₈

yes

complemented

C₂×Dic₁₀₆

C₂³

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

D₄

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₂×#88322

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

Q₁₆

yes

—

C₂×Dic₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

C₂×D₄

yes

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

C₂×Q₈

yes

complemented

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

C₂²⋊Q₈

yes

—

C₂²×C₅₃

C₂×Q₁₆

yes

—

C₂²×C₅₃

AGL(1,ℤ/8)

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

C₂²×Q₈

yes

characteristic

C₄×C₅₃

C₂×#41

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₂×Q₁₆

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₅₃

C₂²×Q₁₆

yes

characteristic

C₂×C₅₃

C₂×AGL(1,ℤ/8)

yes

characteristic

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

—

100

#477

—

101

C₂×#107

—

102

C₂×#107

D₅₃

yes

characteristic

103

C₂×C₄⋊C₈

—

104

C₂×#103

—

105

C₂×#103

—

106

C₅₃

C₂×#477

yes

characteristic, complemented, Sylow

107

C₂²×Q₈

—

108

C₂²×Q₈

—

106

—

109

C₂²×C₈

—

110

C₂²×C₈

C₂×D₅₃

yes

characteristic

111

C₄⋊Q₈

—

112

C₄⋊Q₈

—

106

—

113

C₄×Q₈

—

114

C₄×Q₈

—

106

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

106

—

117

C₂×C₄⋊C₄

—

106

—

118

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

119

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

120

C₂×C₄²

—

121

C₂×C₄²

—

106

—

122

#107

—

123

#107

C₂×D₅₃

yes

—

124

C₄⋊C₈

—

125

#103

—

126

#103

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂×Q₈

—

106

—

130

C₂×Q₈

—

106

—

131

C₂×Q₈

—

106

—

132

C₂×Q₈

—

212

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

106

—

138

C₂²×C₄

—

106

—

139

C₂²×C₄

C₂²×D₅₃

yes

characteristic

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

106

—

143

C₂×C₈

C₂²×D₅₃

yes

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

106

—

148

C₄⋊C₄

—

106

—

149

C₄⋊C₄

—

106

—

150

C₄⋊C₄

C₂²×D₅₃

yes

—

151

C₄⋊C₄

C₂²×D₅₃

yes

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

106

—

155

C₂³

D₄×D₅₃

yes

characteristic

156

Q₈

—

157

Q₈

—

106

—

158

Q₈

—

106

—

159

Q₈

—

212

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

106

—

168

C₂×C₄

—

106

—

169

C₂×C₄

—

106

—

170

C₂×C₄

—

106

—

171

C₂×C₄

—

212

—

172

C₂×C₄

#11719945

yes

—

173

C₂×C₄

D₄×D₅₃

yes

characteristic

174

C₂×C₄

Q₈×D₅₃

yes

—

175

C₂×C₄

C₂³×D₅₃

yes

characteristic

176

C₈

—

177

C₈

—

106

—

178

C₂²

#13133195

yes

—

179

C₂²

Q₁₆×D₅₃

yes

—

180

C₂²

#13133241

yes

—

181

C₂²

C₂×D₄×D₅₃

yes

characteristic

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

106

—

186

C₄

—

106

—

187

C₄

—

212

—

188

C₄

#13133195

yes

—

189

C₄

C₂×Q₈×D₅₃

yes

characteristic

190

C₄

C₂×#11719945

yes

characteristic

191

C₂

#16810917

yes

direct product factor

192

C₂

C₂×#13133195

yes

characteristic

193

C₂

C₂×Q₁₆×D₅₃

yes

characteristic

194

C₂

C₂×#13133241

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂×#107×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×#107	D₅₃
hypocenter	C₅₃	C₂×#477
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₂×#107×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
3. #16810917	C₂	direct (N × H)
40. C₂×Dic₁₀₆	Q₈	semidirect
77. Dic₁₀₆	C₂×Q₈	semidirect
94. C₂×C₅₃	#477	semidirect
106. C₅₃	C₂×#477	semidirect
191. C₂	#16810917	direct (N × H)

H ≅ G/N

type

3. #16810917

C₂

direct (N × H)

40. C₂×Dic₁₀₆

Q₈

semidirect

77. Dic₁₀₆

C₂×Q₈

semidirect

94. C₂×C₅₃

#477

semidirect

106. C₅₃

C₂×#477

semidirect

191. C₂

#16810917

direct (N × H)

Subgroups — C₂×#16810917 #20284485

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16810917 #20284485

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810917 #20284485