Group #20287792 — subgroups

Subgroups — #20287792

Order 6784.

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

normal subgroups only

This group has 356 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	#16811345	C₂	1	2	yes	maximal, complemented
2	#16811016	C₂	1	2	yes	maximal, complemented
3	#16811007	C₂	1	1	yes	maximal, characteristic
4	#16810845	C₂	1	2	yes	maximal, complemented
5	#16810841	C₂	1	2	yes	maximal
6	#16810832	C₂	1	2	yes	maximal, complemented
7	#16810753	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810722	C₂	1	2	yes	maximal, complemented
9	C₄×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic
11	C₂×#11719925	C₂²	1	2	yes	—
12	C₂×#11719913	C₂²	1	2	yes	—
13	C₂×#11719924	C₂²	1	2	yes	—
14	C₂×D₄₂₄	C₂²	1	1	yes	characteristic
15	#13133215	C₂²	1	2	yes	—
16	#13133212	—	2	2	no	—
17	#13133211	—	2	2	no	—
18	#13133206	C₂²	1	2	yes	—
19	#13133205	C₂²	1	2	yes	—
20	#13133195	—	2	2	no	—
21	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
22	#13133182	C₂²	1	2	yes	—
23	#13133179	C₂²	1	2	yes	—
24	#13133174	—	2	2	no	—
25	C₄⋊#88322	C₂²	1	1	yes	characteristic
26	#13133146	C₂²	1	2	yes	—
27	#13133140	C₂²	1	2	yes	—
28	#13133137	C₂²	1	2	yes	—
29	#13133127	C₂²	1	2	yes	—
30	C₂×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
31	C₄×Q₈×C₅₃	C₂²	1	2	yes	—
32	#107×C₅₃	C₂²	1	1	yes	characteristic
33	#103×C₅₃	C₂²	1	2	yes	—
34	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
36	C₂×D₂₁₂	C₂³	1	2	yes	—
37	C₂×C₄×D₅₃	—	2	2	no	—
38	C₂×Dic₁₀₆	D₄	1	1	yes	characteristic
39	C₂×Dic₁₀₆	C₂³	1	2	yes	—
40	Q₁₆×C₅₃	—	2	2	no	—
41	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
43	C₄⋊C₄×C₅₃	—	2	2	no	—
44	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
45	C₄²×C₅₃	—	2	2	no	—
46	C₄²×C₅₃	C₂³	1	1	yes	characteristic
47	#11719925	—	4	2	no	—
48	#11719924	—	4	2	no	—
49	#11719921	—	2	2	no	—
50	#11719921	—	2	2	no	—
51	#11719921	—	2	2	no	—
52	#11719921	—	2	2	no	—
53	#11719921	—	2	2	no	—
54	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
55	#11719919	—	2	2	no	—
56	#11719919	—	2	2	no	—
57	#11719919	—	2	2	no	—
58	#11719919	D₄	1	2	yes	—
59	C₄×Dic₅₃	—	2	2	no	—
60	C₄×Dic₅₃	C₂³	1	2	yes	—
61	C₂×#88322	C₂³	1	2	yes	—
62	Dic₂₁₂	—	4	1	no	—
63	D₄₂₄	—	4	1	no	—
64	#11719913	—	2	4	no	—
65	C₂²×D₅₃	—	2	2	no	—
66	Q₈×C₅₃	—	2	2	no	—
67	Q₈×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	—	2	2	no	—
70	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
73	C₂×Dic₅₃	—	2	2	no	—
74	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
75	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
76	D₂₁₂	—	2	2	no	—
77	D₂₁₂	—	2	2	no	—
78	C₄×D₅₃	—	4	4	no	—
79	Dic₁₀₆	—	2	2	no	—
80	Dic₁₀₆	—	2	2	no	—
81	Dic₁₀₆	—	4	1	no	—
82	C₈×C₅₃	—	2	1	no	—
83	C₈×C₅₃	#41	1	2	yes	—
84	#88322	—	2	2	no	—
85	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
86	C₂×D₅₃	—	4	2	no	—
87	C₂×D₅₃	—	4	2	no	—
88	C₂×D₅₃	—	4	2	no	—
89	C₄×C₅₃	—	2	1	no	—
90	C₄×C₅₃	—	2	1	no	—
91	C₄×C₅₃	—	4	2	no	—
92	C₄×C₅₃	—	4	2	no	—
93	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
94	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
95	Dic₅₃	—	2	2	no	—
96	Dic₅₃	—	2	2	no	—
97	Dic₅₃	—	4	2	no	—
98	#3402	—	53	1	no	maximal, Sylow
99	C₂×C₅₃	#550	1	1	yes	characteristic
100	C₂×C₅₃	#575	1	1	yes	characteristic
101	C₂×C₅₃	#578	1	1	yes	characteristic
102	D₅₃	—	8	2	no	—
103	#556	—	53	2	no	—
104	#497	—	53	2	no	—
105	#495	—	53	1	no	—
106	#464	—	53	2	no	—
107	#463	—	53	2	no	—
108	#462	—	53	2	no	—
109	#446	—	53	1	no	—
110	#440	—	53	2	no	—
111	C₄×Q₁₆	D₅₃	1	1	yes	characteristic, complemented
112	C₅₃	#3402	1	1	yes	characteristic, complemented, Sylow
113	C₂×Q₁₆	—	53	2	no	—
114	C₂×Q₁₆	—	53	1	no	—
115	C₂×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
116	C₂×SD₁₆	—	53	2	no	—
117	C₂×SD₁₆	—	53	2	no	—
118	C₂×D₈	—	53	1	no	—
119	C₄⋊Q₈	—	53	2	no	—
120	#126	—	106	2	no	—
121	#126	—	106	2	no	—
122	#124	—	53	2	no	—
123	#124	—	53	2	no	—
124	C₂²⋊Q₈	—	106	2	no	—
125	C₄×Q₈	—	53	1	no	—
126	C₄×Q₈	—	53	2	no	—
127	C₄×Q₈	C₂×D₅₃	1	2	yes	—
128	C₄×D₄	—	53	2	no	—
129	#117	—	106	2	no	—
130	#107	C₂×D₅₃	1	1	yes	characteristic
131	C₄⋊C₈	—	53	2	no	—
132	C₄⋊C₈	—	53	1	no	—
133	#103	—	53	2	no	—
134	#103	C₂×D₅₃	1	2	yes	—
135	#102	—	53	2	no	—
136	#97	—	53	2	no	—
137	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
138	C₂×Q₈	—	53	1	no	—
139	C₂×Q₈	—	53	2	no	—
140	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
141	C₂×D₄	—	53	2	no	—
142	C₂²×C₄	—	106	2	no	—
143	Q₁₆	—	2	2	no	—
144	Q₁₆	—	212	1	no	—
145	Q₁₆	—	212	2	no	—
146	SD₁₆	—	106	4	no	—
147	SD₁₆	—	212	2	no	—
148	D₈	—	212	1	no	—
149	C₂×C₈	—	53	2	no	—
150	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
151	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	—	53	2	no	—
154	C₄⋊C₄	—	53	1	no	—
155	C₄⋊C₄	—	106	2	no	—
156	C₄⋊C₄	—	106	2	no	—
157	C₄⋊C₄	—	106	2	no	—
158	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
159	C₂²⋊C₄	—	106	2	no	—
160	C₂²⋊C₄	—	106	2	no	—
161	C₂²⋊C₄	—	106	2	no	—
162	C₂²⋊C₄	—	106	2	no	—
163	C₂²⋊C₄	—	106	2	no	—
164	C₄²	—	2	2	no	—
165	C₄²	—	53	2	no	—
166	C₄²	—	106	2	no	—
167	C₄²	C₂²×D₅₃	1	1	yes	characteristic
168	C₂³	—	106	2	no	—
169	Q₈	—	2	2	no	—
170	Q₈	—	2	2	no	—
171	Q₈	—	106	2	no	—
172	Q₈	—	106	2	no	—
173	Q₈	—	212	1	no	—
174	D₄	—	106	2	no	—
175	D₄	—	106	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	53	2	no	—
179	C₂×C₄	—	53	2	no	—
180	C₂×C₄	—	106	2	no	—
181	C₂×C₄	—	212	4	no	—
182	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
183	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
184	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
185	C₈	—	2	1	no	—
186	C₈	—	106	2	no	—
187	C₈	#11719945	1	2	yes	—
188	C₂²	—	212	2	no	—
189	C₂²	—	212	2	no	—
190	C₂²	—	212	2	no	—
191	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	—	4	2	no	—
195	C₄	—	4	2	no	—
196	C₄	—	106	2	no	—
197	C₄	—	106	2	no	—
198	C₄	—	212	2	no	—
199	C₄	C₂×#11719945	1	1	yes	characteristic
200	C₄	#13133263	1	1	yes	characteristic
201	C₂	—	424	2	no	—
202	C₂	#16811294	1	1	yes	characteristic
203	C₂	#16811420	1	1	yes	characteristic
204	C₂	#16811436	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811345

C₂

yes

maximal, complemented

#16811016

C₂

yes

maximal, complemented

#16811007

C₂

yes

maximal, characteristic

#16810845

C₂

yes

maximal, complemented

#16810841

C₂

yes

maximal

#16810832

C₂

yes

maximal, complemented

#16810753

C₂

yes

maximal, characteristic, complemented

#16810722

C₂

yes

maximal, complemented

C₄×Q₁₆×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₁₂

C₂²

yes

characteristic

C₂×#11719925

C₂²

yes

—

C₂×#11719913

C₂²

yes

—

C₂×#11719924

C₂²

yes

—

C₂×D₄₂₄

C₂²

yes

characteristic

#13133215

C₂²

yes

—

#13133212

—

#13133211

—

#13133206

C₂²

yes

—

#13133205

C₂²

yes

—

#13133195

—

C₄×Dic₁₀₆

C₂²

yes

characteristic

#13133182

C₂²

yes

—

#13133179

C₂²

yes

—

#13133174

—

C₄⋊#88322

C₂²

yes

characteristic

#13133146

C₂²

yes

—

#13133140

C₂²

yes

—

#13133137

C₂²

yes

—

#13133127

C₂²

yes

—

C₂×Q₁₆×C₅₃

C₂²

yes

characteristic

C₄×Q₈×C₅₃

C₂²

yes

—

#107×C₅₃

C₂²

yes

characteristic

#103×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

C₂³

yes

—

C₂×D₂₁₂

C₂³

yes

—

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

D₄

yes

characteristic

C₂×Dic₁₀₆

C₂³

yes

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂³

yes

—

—

—

—

—

—

—

C₄⋊Dic₅₃

D₄

yes

—

—

—

D₄

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂³

yes

—

C₂×#88322

C₂³

yes

—

Dic₂₁₂

—

D₄₂₄

—

#11719913

—

C₂²×D₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

C₂×D₄

yes

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

#41

yes

—

#88322

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#41

yes

characteristic

C₄×C₅₃

H(5,𝔽₂)

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

100

C₂×C₅₃

#575

yes

characteristic

101

C₂×C₅₃

#578

yes

characteristic

102

D₅₃

—

103

#556

—

104

#497

—

105

#495

—

106

#464

—

107

#463

—

108

#462

—

109

#446

—

110

#440

—

111

C₄×Q₁₆

D₅₃

yes

characteristic, complemented

112

C₅₃

#3402

yes

characteristic, complemented, Sylow

113

C₂×Q₁₆

—

114

C₂×Q₁₆

—

115

C₂×Q₁₆

C₂×D₅₃

yes

characteristic

116

C₂×SD₁₆

—

117

C₂×SD₁₆

—

118

C₂×D₈

—

119

C₄⋊Q₈

—

120

#126

—

106

—

121

#126

—

106

—

122

#124

—

123

#124

—

124

C₂²⋊Q₈

—

106

—

125

C₄×Q₈

—

126

C₄×Q₈

—

127

C₄×Q₈

C₂×D₅₃

yes

—

128

C₄×D₄

—

129

#117

—

106

—

130

#107

C₂×D₅₃

yes

characteristic

131

C₄⋊C₈

—

132

C₄⋊C₈

—

133

#103

—

134

#103

C₂×D₅₃

yes

—

135

#102

—

136

#97

—

137

C₄×C₈

C₂×D₅₃

yes

characteristic

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

C₂²×D₅₃

yes

—

141

C₂×D₄

—

142

C₂²×C₄

—

106

—

143

Q₁₆

—

144

Q₁₆

—

212

—

145

Q₁₆

—

212

—

146

SD₁₆

—

106

—

147

SD₁₆

—

212

—

148

D₈

—

212

—

149

C₂×C₈

—

150

C₂×C₈

C₂²×D₅₃

yes

characteristic

151

C₂×C₈

C₂²×D₅₃

yes

characteristic

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

106

—

156

C₄⋊C₄

—

106

—

157

C₄⋊C₄

—

106

—

158

C₄⋊C₄

C₂²×D₅₃

yes

—

159

C₂²⋊C₄

—

106

—

160

C₂²⋊C₄

—

106

—

161

C₂²⋊C₄

—

106

—

162

C₂²⋊C₄

—

106

—

163

C₂²⋊C₄

—

106

—

164

C₄²

—

165

C₄²

—

166

C₄²

—

106

—

167

C₄²

C₂²×D₅₃

yes

characteristic

168

C₂³

—

106

—

169

Q₈

—

170

Q₈

—

171

Q₈

—

106

—

172

Q₈

—

106

—

173

Q₈

—

212

—

174

D₄

—

106

—

175

D₄

—

106

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

106

—

181

C₂×C₄

—

212

—

182

C₂×C₄

D₄×D₅₃

yes

characteristic

183

C₂×C₄

D₄×D₅₃

yes

characteristic

184

C₂×C₄

C₂³×D₅₃

yes

characteristic

185

C₈

—

186

C₈

—

106

—

187

C₈

#11719945

yes

—

188

C₂²

—

212

—

189

C₂²

—

212

—

190

C₂²

—

212

—

191

C₂²

C₂×D₄×D₅₃

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

106

—

197

C₄

—

106

—

198

C₄

—

212

—

199

C₄

C₂×#11719945

yes

characteristic

200

C₄

#13133263

yes

characteristic

201

C₂

—

424

—

202

C₂

#16811294

yes

characteristic

203

C₂

#16811420

yes

characteristic

204

C₂

#16811436

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₄×Q₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	C₄×Q₁₆	D₅₃
hypocenter	C₅₃	#3402
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

subgroup

G/H

center

C₂²

C₂×D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₄×Q₁₆×C₅₃

C₂

socle

C₂²×C₅₃

C₂²×D₄

hypercenter

C₄×Q₁₆

D₅₃

hypocenter

C₅₃

#3402

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

N	H ≅ G/N	type
1. #16811345	C₂	semidirect
2. #16811016	C₂	semidirect
4. #16810845	C₂	semidirect
6. #16810832	C₂	semidirect
7. #16810753	C₂	semidirect
8. #16810722	C₂	semidirect
9. C₄×Q₁₆×C₅₃	C₂	semidirect
111. C₄×Q₁₆	D₅₃	semidirect
112. C₅₃	#3402	semidirect

H ≅ G/N

type

1. #16811345

C₂

semidirect

2. #16811016

C₂

semidirect

4. #16810845

C₂

semidirect

6. #16810832

C₂

semidirect

7. #16810753

C₂

semidirect

8. #16810722

C₂

semidirect

9. C₄×Q₁₆×C₅₃

C₂

semidirect

111. C₄×Q₁₆

D₅₃

semidirect

112. C₅₃

#3402

semidirect

Subgroups — #20287792

Conjugacy classes of subgroups

Complements