Group #20287796 — subgroups

Subgroups — #20287796

Order 6784.

356 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 121 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	#16811346	C₂	1	2	yes	maximal, complemented
2	#16811017	C₂	1	2	yes	maximal, complemented
3	#16811007	C₂	1	1	yes	maximal, characteristic
4	#16810846	C₂	1	2	yes	maximal, complemented
5	#16810839	C₂	1	2	yes	maximal
6	#16810834	C₂	1	2	yes	maximal, complemented
7	#16810725	C₂	1	2	yes	maximal, complemented
8	C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₁₂	C₂²	1	2	yes	—
12	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic
13	C₂×#11719913	C₂²	1	2	yes	—
14	C₂×#11719913	C₂²	1	2	yes	—
15	C₂×D₄₂₄	C₂²	1	1	yes	characteristic
16	#13133217	C₂²	1	2	yes	—
17	#13133212	—	2	2	no	—
18	#13133212	—	2	2	no	—
19	#13133205	C₂²	1	2	yes	—
20	#13133205	C₂²	1	2	yes	—
21	#13133193	—	2	2	no	—
22	C₄×Dic₁₀₆	C₂²	1	2	yes	—
23	C₄×Dic₁₀₆	C₂²	1	2	yes	—
24	C₄×D₂₁₂	C₂²	1	2	yes	—
25	#13133175	—	2	2	no	—
26	#13133151	C₂²	1	1	yes	characteristic
27	#13133142	C₂²	1	2	yes	—
28	#13133142	C₂²	1	2	yes	—
29	#13133139	C₂²	1	2	yes	—
30	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
32	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
33	#97×C₅₃	C₂²	1	2	yes	—
34	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
36	C₂×D₂₁₂	C₂³	1	2	yes	—
37	C₂×C₄×D₅₃	—	2	2	no	—
38	C₂×Dic₁₀₆	C₂³	1	2	yes	—
39	C₂×Dic₁₀₆	C₂³	1	2	yes	—
40	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₂³	1	2	yes	—
42	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
43	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
44	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
45	C₄²×C₅₃	C₂³	1	1	yes	characteristic
46	C₄²×C₅₃	C₂³	1	2	yes	—
47	#11719921	—	2	2	no	—
48	#11719921	—	2	2	no	—
49	#11719921	—	2	2	no	—
50	#11719921	—	2	2	no	—
51	#11719921	—	2	2	no	—
52	C₄⋊Dic₅₃	—	2	2	no	—
53	C₄⋊Dic₅₃	—	2	2	no	—
54	C₄⋊Dic₅₃	C₂³	1	2	yes	—
55	#11719919	—	2	2	no	—
56	#11719919	—	2	2	no	—
57	C₄×Dic₅₃	—	2	2	no	—
58	C₄×Dic₅₃	—	2	2	no	—
59	Dic₂₁₂	—	2	2	no	—
60	Dic₂₁₂	—	4	2	no	—
61	Dic₂₁₂	—	4	1	no	—
62	D₄₂₄	—	4	1	no	—
63	#11719913	—	2	4	no	—
64	#11719913	—	4	2	no	—
65	C₂²×D₅₃	—	2	2	no	—
66	Q₈×C₅₃	—	4	1	no	—
67	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
72	C₂×Dic₅₃	—	2	2	no	—
73	C₂×Dic₅₃	—	2	2	no	—
74	C₂×Dic₅₃	—	2	2	no	—
75	D₂₁₂	—	2	2	no	—
76	D₂₁₂	—	2	2	no	—
77	C₄×D₅₃	—	4	4	no	—
78	Dic₁₀₆	—	2	2	no	—
79	Dic₁₀₆	—	2	2	no	—
80	Dic₁₀₆	—	2	2	no	—
81	Dic₁₀₆	—	2	2	no	—
82	C₈×C₅₃	—	2	2	no	—
83	C₈×C₅₃	—	2	1	no	—
84	C₈×C₅₃	#41	1	2	yes	—
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	2	no	—
90	C₄×C₅₃	—	2	1	no	—
91	C₄×C₅₃	—	2	1	no	—
92	C₄×C₅₃	—	2	2	no	—
93	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
94	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
95	Dic₅₃	—	4	2	no	—
96	Dic₅₃	—	4	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	D₅₃	1	1	yes	characteristic, complemented
110	#440	—	53	2	no	—
111	C₄×Q₁₆	—	53	1	no	—
112	C₅₃	#3402	1	1	yes	characteristic, complemented, Sylow
113	C₂×Q₁₆	—	53	1	no	—
114	C₂×Q₁₆	—	53	1	no	—
115	C₂×Q₁₆	—	53	2	no	—
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	2	no	—
126	C₄×Q₈	—	53	2	no	—
127	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
128	C₄×D₄	—	53	2	no	—
129	#117	—	106	2	no	—
130	#107	—	53	1	no	—
131	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
132	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
133	#103	—	53	2	no	—
134	#103	—	53	2	no	—
135	#102	—	53	2	no	—
136	#97	C₂×D₅₃	1	2	yes	—
137	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
138	C₂×Q₈	—	53	2	no	—
139	C₂×Q₈	—	53	2	no	—
140	C₂×Q₈	D₂₁₂	1	1	yes	characteristic
141	C₂×D₄	—	53	2	no	—
142	C₂²×C₄	—	106	2	no	—
143	Q₁₆	—	106	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	C₂²×D₅₃	1	1	yes	characteristic
150	C₂×C₈	C₂²×D₅₃	1	2	yes	—
151	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
152	C₄⋊C₄	—	53	2	no	—
153	C₄⋊C₄	—	106	2	no	—
154	C₄⋊C₄	—	106	2	no	—
155	C₄⋊C₄	—	106	2	no	—
156	C₄⋊C₄	—	106	2	no	—
157	C₄⋊C₄	D₂₁₂	1	2	yes	—
158	C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
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	—	106	2	no	—
165	C₄²	—	106	2	no	—
166	C₄²	C₂²×D₅₃	1	1	yes	characteristic
167	C₄²	C₂²×D₅₃	1	2	yes	—
168	C₂³	—	106	2	no	—
169	Q₈	—	4	1	no	—
170	Q₈	—	106	2	no	—
171	Q₈	—	106	2	no	—
172	Q₈	—	106	2	no	—
173	Q₈	—	106	2	no	—
174	D₄	—	106	2	no	—
175	D₄	—	106	2	no	—
176	C₂×C₄	—	106	2	no	—
177	C₂×C₄	—	106	2	no	—
178	C₂×C₄	—	106	2	no	—
179	C₂×C₄	—	212	4	no	—
180	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
181	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
182	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
183	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
184	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
185	C₈	—	2	1	no	—
186	C₈	—	2	2	no	—
187	C₈	#11719949	1	2	yes	—
188	C₂²	—	212	2	no	—
189	C₂²	—	212	2	no	—
190	C₂²	—	212	2	no	—
191	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	2	2	no	—
196	C₄	—	212	2	no	—
197	C₄	—	212	2	no	—
198	C₄	—	212	2	no	—
199	C₄	C₂×#11719949	1	1	yes	characteristic
200	C₄	#13133263	1	1	yes	characteristic
201	C₂	—	424	2	no	—
202	C₂	#16811295	1	1	yes	characteristic
203	C₂	#16811423	1	1	yes	characteristic
204	C₂	#16811440	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811346

C₂

yes

maximal, complemented

#16811017

C₂

yes

maximal, complemented

#16811007

C₂

yes

maximal, characteristic

#16810846

C₂

yes

maximal, complemented

#16810839

C₂

yes

maximal

#16810834

C₂

yes

maximal, complemented

#16810725

C₂

yes

maximal, complemented

C₄×Dic₂₁₂

C₂

yes

maximal, characteristic, complemented

C₅₃×#446

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₁₂

C₂²

yes

characteristic

C₂×Dic₂₁₂

C₂²

yes

—

C₂×Dic₂₁₂

C₂²

yes

characteristic

C₂×#11719913

C₂²

yes

—

C₂×#11719913

C₂²

yes

—

C₂×D₄₂₄

C₂²

yes

characteristic

#13133217

C₂²

yes

—

#13133212

—

#13133212

—

#13133205

C₂²

yes

—

#13133205

C₂²

yes

—

#13133193

—

C₄×Dic₁₀₆

C₂²

yes

—

C₄×Dic₁₀₆

C₂²

yes

—

C₄×D₂₁₂

C₂²

yes

—

#13133175

—

#13133151

C₂²

yes

characteristic

#13133142

C₂²

yes

—

#13133142

C₂²

yes

—

#13133139

C₂²

yes

—

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

—

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

#97×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

D₄

yes

characteristic

C₂×D₂₁₂

C₂³

yes

—

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

C₂³

yes

—

C₂×Dic₁₀₆

C₂³

yes

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

C₂³

yes

—

—

—

—

—

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

Dic₂₁₂

—

Dic₂₁₂

—

Dic₂₁₂

—

D₄₂₄

—

#11719913

—

#11719913

—

C₂²×D₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#41

yes

—

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

D₅₃

yes

characteristic, complemented

110

#440

—

111

C₄×Q₁₆

—

112

C₅₃

#3402

yes

characteristic, complemented, Sylow

113

C₂×Q₁₆

—

114

C₂×Q₁₆

—

115

C₂×Q₁₆

—

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

characteristic

128

C₄×D₄

—

129

#117

—

106

—

130

#107

—

131

C₄⋊C₈

C₂×D₅₃

yes

—

132

C₄⋊C₈

C₂×D₅₃

yes

characteristic

133

#103

—

134

#103

—

135

#102

—

136

#97

C₂×D₅₃

yes

—

137

C₄×C₈

C₂×D₅₃

yes

characteristic

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

D₂₁₂

yes

characteristic

141

C₂×D₄

—

142

C₂²×C₄

—

106

—

143

Q₁₆

—

106

—

144

Q₁₆

—

212

—

145

Q₁₆

—

212

—

146

SD₁₆

—

106

—

147

SD₁₆

—

212

—

148

D₈

—

212

—

149

C₂×C₈

C₂²×D₅₃

yes

characteristic

150

C₂×C₈

C₂²×D₅₃

yes

—

151

C₂×C₈

C₂²×D₅₃

yes

characteristic

152

C₄⋊C₄

—

153

C₄⋊C₄

—

106

—

154

C₄⋊C₄

—

106

—

155

C₄⋊C₄

—

106

—

156

C₄⋊C₄

—

106

—

157

C₄⋊C₄

D₂₁₂

yes

—

158

C₄⋊C₄

D₂₁₂

yes

characteristic

159

C₂²⋊C₄

—

106

—

160

C₂²⋊C₄

—

106

—

161

C₂²⋊C₄

—

106

—

162

C₂²⋊C₄

—

106

—

163

C₂²⋊C₄

—

106

—

164

C₄²

—

106

—

165

C₄²

—

106

—

166

C₄²

C₂²×D₅₃

yes

characteristic

167

C₄²

C₂²×D₅₃

yes

—

168

C₂³

—

106

—

169

Q₈

—

170

Q₈

—

106

—

171

Q₈

—

106

—

172

Q₈

—

106

—

173

Q₈

—

106

—

174

D₄

—

106

—

175

D₄

—

106

—

176

C₂×C₄

—

106

—

177

C₂×C₄

—

106

—

178

C₂×C₄

—

106

—

179

C₂×C₄

—

212

—

180

C₂×C₄

C₂×D₂₁₂

yes

characteristic

181

C₂×C₄

C₂×D₂₁₂

yes

—

182

C₂×C₄

C₂×D₂₁₂

yes

characteristic

183

C₂×C₄

C₂×D₂₁₂

yes

—

184

C₂×C₄

C₂³×D₅₃

yes

characteristic

185

C₈

—

186

C₈

—

187

C₈

#11719949

yes

—

188

C₂²

—

212

—

189

C₂²

—

212

—

190

C₂²

—

212

—

191

C₂²

C₂²×D₂₁₂

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

212

—

197

C₄

—

212

—

198

C₄

—

212

—

199

C₄

C₂×#11719949

yes

characteristic

200

C₄

#13133263

yes

characteristic

201

C₂

—

424

—

202

C₂

#16811295

yes

characteristic

203

C₂

#16811423

yes

characteristic

204

C₂

#16811440

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#446	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#446	D₅₃
hypocenter	C₅₃	#3402
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

subgroup

G/H

center

C₂²

C₂²×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#446

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

N	H ≅ G/N	type
1. #16811346	C₂	semidirect
2. #16811017	C₂	semidirect
4. #16810846	C₂	semidirect
6. #16810834	C₂	semidirect
7. #16810725	C₂	semidirect
8. C₄×Dic₂₁₂	C₂	semidirect
9. C₅₃×#446	C₂	semidirect
109. #446	D₅₃	semidirect
112. C₅₃	#3402	semidirect

H ≅ G/N

type

1. #16811346

C₂

semidirect

2. #16811017

C₂

semidirect

4. #16810846

C₂

semidirect

6. #16810834

C₂

semidirect

7. #16810725

C₂

semidirect

8. C₄×Dic₂₁₂

C₂

semidirect

9. C₅₃×#446

C₂

semidirect

109. #446

D₅₃

semidirect

112. C₅₃

#3402

semidirect

Subgroups — #20287796

Conjugacy classes of subgroups

Complements