Group #20287750 — subgroups

Subgroups — #20287750

Order 6784.

350 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 115 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	#16811348	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811343	C₂	1	1	yes	maximal, characteristic
3	#16811014	C₂	1	2	yes	maximal
4	#16811002	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810843	C₂	1	2	yes	maximal
6	#16810834	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810832	C₂	1	2	yes	maximal, complemented
8	#16810750	C₂	1	1	yes	maximal, characteristic, complemented
9	#16810727	C₂	1	2	yes	maximal, complemented
10	C₄×#11719923	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#461	C₂	1	1	yes	maximal, characteristic
12	C₂×Dic₂₁₂	C₂²	1	2	yes	—
13	C₂×#11719925	C₂²	1	2	yes	—
14	C₂×#11719925	C₂²	1	2	yes	—
15	C₂×#11719923	C₂²	1	1	yes	characteristic
16	#13133217	C₂²	1	1	yes	characteristic
17	#13133215	C₂²	1	2	yes	—
18	#13133210	—	2	2	no	—
19	#13133208	—	2	1	no	—
20	#13133207	—	2	2	no	—
21	#13133204	C₂²	1	2	yes	—
22	#13133196	—	2	1	no	—
23	Q₈×Dic₅₃	C₂²	1	2	yes	—
24	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
25	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
26	#13133182	—	2	1	no	—
27	#13133182	C₂²	1	2	yes	—
28	#13133176	—	2	1	no	—
29	#13133148	C₂²	1	1	yes	characteristic
30	#13133146	C₂²	1	2	yes	—
31	#13133142	C₂²	1	2	yes	—
32	#13133140	C₂²	1	1	yes	characteristic
33	#13133138	C₂²	1	1	yes	characteristic
34	#13133127	C₂²	1	2	yes	—
35	C₄×#88322	C₂²	1	1	yes	characteristic
36	C₂×SD₁₆×C₅₃	C₂²	1	2	yes	—
37	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
38	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
40	#103×C₅₃	C₂²	1	2	yes	—
41	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
42	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	—	2	1	no	—
44	C₂×Dic₁₀₆	D₄	1	1	yes	characteristic
45	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
46	C₂×Dic₁₀₆	C₂³	1	2	yes	—
47	SD₁₆×C₅₃	—	4	2	no	—
48	C₂×C₈×C₅₃	C₂³	1	2	yes	—
49	C₄⋊C₄×C₅₃	—	2	1	no	—
50	C₄⋊C₄×C₅₃	—	2	1	no	—
51	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
52	C₂²⋊C₄×C₅₃	—	2	1	no	—
53	C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	C₂²⋊Dic₅₃	—	2	2	no	—
55	C₂²⋊Dic₅₃	—	2	2	no	—
56	#11719925	—	2	4	no	—
57	#11719925	—	4	2	no	—
58	#11719923	—	2	2	no	—
59	C₄⋊Dic₅₃	—	2	1	no	—
60	C₄⋊Dic₅₃	—	2	1	no	—
61	C₄⋊Dic₅₃	—	2	2	no	—
62	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
63	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
64	#11719919	—	2	1	no	—
65	#11719919	—	2	2	no	—
66	#11719919	—	2	2	no	—
67	#11719919	—	2	2	no	—
68	#11719919	D₄	1	2	yes	—
69	C₄×Dic₅₃	—	2	1	no	—
70	C₄×Dic₅₃	—	2	2	no	—
71	C₄×Dic₅₃	—	2	2	no	—
72	C₄×Dic₅₃	C₂³	1	2	yes	—
73	C₂×#88322	C₂³	1	1	yes	characteristic
74	C₂×#88322	C₂³	1	1	yes	characteristic
75	Dic₂₁₂	—	4	2	no	—
76	C₂³×C₅₃	—	2	1	no	—
77	Q₈×C₅₃	—	2	2	no	—
78	Q₈×C₅₃	—	2	2	no	—
79	D₄×C₅₃	—	2	1	no	—
80	D₄×C₅₃	—	2	1	no	—
81	C₂×C₄×C₅₃	—	2	1	no	—
82	C₂×C₄×C₅₃	—	2	2	no	—
83	C₂×C₄×C₅₃	—	4	2	no	—
84	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
87	C₂×Dic₅₃	—	2	1	no	—
88	C₂×Dic₅₃	—	2	2	no	—
89	C₂×Dic₅₃	—	2	1	no	—
90	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
91	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
92	Dic₁₀₆	—	2	2	no	—
93	Dic₁₀₆	—	2	1	no	—
94	Dic₁₀₆	—	2	2	no	—
95	Dic₁₀₆	—	2	1	no	—
96	Dic₁₀₆	—	2	2	no	—
97	C₈×C₅₃	—	2	2	no	—
98	#88322	—	2	1	no	—
99	#88322	#41	1	2	yes	—
100	C₂²×C₅₃	—	4	1	no	—
101	C₂²×C₅₃	—	4	1	no	—
102	C₂²×C₅₃	—	4	1	no	—
103	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
104	C₄×C₅₃	—	2	1	no	—
105	C₄×C₅₃	—	4	2	no	—
106	C₄×C₅₃	—	4	1	no	—
107	C₄×C₅₃	#137	1	2	yes	—
108	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
109	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
110	Dic₅₃	—	2	2	no	—
111	Dic₅₃	—	2	2	no	—
112	Dic₅₃	—	4	1	no	—
113	Dic₅₃	—	4	2	no	—
114	Dic₅₃	—	4	1	no	—
115	#3398	—	53	1	no	maximal, Sylow
116	C₂×C₅₃	—	8	1	no	—
117	C₂×C₅₃	#550	1	1	yes	characteristic
118	C₂×C₅₃	C₂×#137	1	1	yes	characteristic
119	C₂×C₅₃	#578	1	1	yes	characteristic
120	#557	—	53	1	no	—
121	#556	—	53	1	no	—
122	#497	—	53	2	no	—
123	#494	—	53	1	no	—
124	#464	—	53	2	no	—
125	#462	—	53	1	no	—
126	#462	—	53	2	no	—
127	#461	D₅₃	1	1	yes	characteristic
128	#446	—	53	1	no	—
129	#441	—	53	2	no	—
130	C₄×SD₁₆	—	53	1	no	—
131	C₅₃	#3398	1	1	yes	characteristic, complemented, Sylow
132	C₂×Q₁₆	—	53	2	no	—
133	C₂×Q₁₆	—	53	2	no	—
134	C₂×Q₁₆	—	53	2	no	—
135	C₂×SD₁₆	—	53	1	no	—
136	C₂×SD₁₆	C₂×D₅₃	1	2	yes	—
137	C₄⋊Q₈	—	53	2	no	—
138	C₄⋊Q₈	—	53	1	no	—
139	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
140	#126	—	106	2	no	—
141	#125	—	106	2	no	—
142	#125	—	106	1	no	—
143	#124	—	53	2	no	—
144	C₂²⋊Q₈	—	106	1	no	—
145	C₄×Q₈	—	53	1	no	—
146	C₄×Q₈	—	53	2	no	—
147	C₄×Q₈	—	53	2	no	—
148	C₄×Q₈	—	53	1	no	—
149	C₄×Q₈	—	106	1	no	—
150	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
151	#117	—	106	1	no	—
152	#106	—	53	1	no	—
153	C₄⋊C₈	—	53	2	no	—
154	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
155	#103	—	53	2	no	—
156	#103	—	53	1	no	—
157	#103	C₂×D₅₃	1	2	yes	—
158	#102	—	53	1	no	—
159	#97	—	53	2	no	—
160	C₄×C₈	—	53	1	no	—
161	C₂×Q₈	—	53	1	no	—
162	C₂×Q₈	—	53	2	no	—
163	C₂×Q₈	—	53	1	no	—
164	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
165	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
166	C₂²×C₄	—	2	1	no	—
167	Q₁₆	—	106	4	no	—
168	Q₁₆	—	212	2	no	—
169	Q₁₆	—	212	2	no	—
170	SD₁₆	—	4	2	no	—
171	SD₁₆	—	106	2	no	—
172	C₂×C₈	—	53	1	no	—
173	C₂×C₈	—	53	1	no	—
174	C₂×C₈	C₂²×D₅₃	1	2	yes	—
175	C₄⋊C₄	—	2	1	no	—
176	C₄⋊C₄	—	2	1	no	—
177	C₄⋊C₄	—	53	1	no	—
178	C₄⋊C₄	—	53	2	no	—
179	C₄⋊C₄	—	53	1	no	—
180	C₄⋊C₄	—	106	2	no	—
181	C₄⋊C₄	—	106	2	no	—
182	C₄⋊C₄	—	106	2	no	—
183	C₄⋊C₄	—	106	2	no	—
184	C₄⋊C₄	—	106	1	no	—
185	C₄⋊C₄	—	106	1	no	—
186	C₄⋊C₄	—	106	1	no	—
187	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
188	C₂²⋊C₄	—	2	1	no	—
189	C₂²⋊C₄	—	106	2	no	—
190	C₂²⋊C₄	—	106	2	no	—
191	C₄²	—	53	2	no	—
192	C₄²	—	106	2	no	—
193	C₄²	—	106	2	no	—
194	C₄²	—	106	1	no	—
195	C₄²	C₂²×D₅₃	1	1	yes	characteristic
196	C₂³	—	2	1	no	—
197	Q₈	—	2	2	no	—
198	Q₈	—	2	2	no	—
199	Q₈	—	106	2	no	—
200	Q₈	—	106	1	no	—
201	Q₈	—	106	2	no	—
202	Q₈	—	106	1	no	—
203	Q₈	—	106	2	no	—
204	D₄	—	2	1	no	—
205	D₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	53	2	no	—
210	C₂×C₄	—	53	2	no	—
211	C₂×C₄	—	106	1	no	—
212	C₂×C₄	—	106	2	no	—
213	C₂×C₄	—	106	1	no	—
214	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
215	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
216	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
217	C₈	—	2	2	no	—
218	C₈	—	53	2	no	—
219	C₈	—	106	1	no	—
220	C₂²	—	4	1	no	—
221	C₂²	—	4	1	no	—
222	C₂²	—	4	1	no	—
223	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	4	2	no	—
226	C₄	—	4	1	no	—
227	C₄	—	106	2	no	—
228	C₄	—	106	2	no	—
229	C₄	—	212	2	no	—
230	C₄	—	212	1	no	—
231	C₄	—	212	1	no	—
232	C₄	#13133246	1	2	yes	—
233	C₄	#41×D₅₃	1	1	yes	characteristic
234	C₄	#13133262	1	1	yes	characteristic
235	C₂	—	8	1	no	—
236	C₂	#16811291	1	1	yes	characteristic
237	C₂	C₂×#13133246	1	1	yes	characteristic
238	C₂	#16811439	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811348

C₂

yes

maximal, characteristic, complemented

#16811343

C₂

yes

maximal, characteristic

#16811014

C₂

yes

maximal

#16811002

C₂

yes

maximal, characteristic, complemented

#16810843

C₂

yes

maximal

#16810834

C₂

yes

maximal, characteristic, complemented

#16810832

C₂

yes

maximal, complemented

#16810750

C₂

yes

maximal, characteristic, complemented

#16810727

C₂

yes

maximal, complemented

C₄×#11719923

C₂

yes

maximal, characteristic

C₅₃×#461

C₂

yes

maximal, characteristic

C₂×Dic₂₁₂

C₂²

yes

—

C₂×#11719925

C₂²

yes

—

C₂×#11719925

C₂²

yes

—

C₂×#11719923

C₂²

yes

characteristic

#13133217

C₂²

yes

characteristic

#13133215

C₂²

yes

—

—

—

—

C₂²

yes

—

#13133196

—

Q₈×Dic₅₃

C₂²

yes

—

C₄×Dic₁₀₆

C₂²

yes

characteristic

C₄×Dic₁₀₆

C₂²

yes

characteristic

#13133182

—

#13133182

C₂²

yes

—

#13133176

—

#13133148

C₂²

yes

characteristic

#13133146

C₂²

yes

—

#13133142

C₂²

yes

—

#13133140

C₂²

yes

characteristic

#13133138

C₂²

yes

characteristic

#13133127

C₂²

yes

—

C₄×#88322

C₂²

yes

characteristic

C₂×SD₁₆×C₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₄×D₄×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

#103×C₅₃

C₂²

yes

—

C₂×Q₈×C₅₃

C₂³

yes

—

C₂×D₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

—

C₂×Dic₁₀₆

D₄

yes

characteristic

C₂×Dic₁₀₆

C₂³

yes

characteristic

C₂×Dic₁₀₆

C₂³

yes

—

SD₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

characteristic

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

—

C₂²⋊Dic₅₃

—

#11719925

—

#11719925

—

#11719923

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

characteristic

C₄⋊Dic₅₃

C₂³

yes

—

—

—

—

D₄

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂³

yes

—

C₂×#88322

C₂³

yes

characteristic

C₂×#88322

C₂³

yes

characteristic

Dic₂₁₂

—

C₂³×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×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×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

C₂×D₄

yes

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

#88322

#41

yes

—

100

C₂²×C₅₃

—

101

C₂²×C₅₃

—

102

C₂²×C₅₃

—

103

C₂²×C₅₃

C₂²×D₄

yes

characteristic

104

C₄×C₅₃

—

105

C₄×C₅₃

—

106

C₄×C₅₃

—

107

C₄×C₅₃

#137

yes

—

108

C₄×C₅₃

C₂×#41

yes

characteristic

109

C₄×C₅₃

H(5,𝔽₂)

yes

characteristic

110

Dic₅₃

—

111

Dic₅₃

—

112

Dic₅₃

—

113

Dic₅₃

—

114

Dic₅₃

—

115

#3398

—

maximal, Sylow

116

C₂×C₅₃

—

117

C₂×C₅₃

#550

yes

characteristic

118

C₂×C₅₃

C₂×#137

yes

characteristic

119

C₂×C₅₃

#578

yes

characteristic

120

#557

—

121

#556

—

122

#497

—

123

#494

—

124

#464

—

125

#462

—

126

#462

—

127

#461

D₅₃

yes

characteristic

128

#446

—

129

#441

—

130

C₄×SD₁₆

—

131

C₅₃

#3398

yes

characteristic, complemented, Sylow

132

C₂×Q₁₆

—

133

C₂×Q₁₆

—

134

C₂×Q₁₆

—

135

C₂×SD₁₆

—

136

C₂×SD₁₆

C₂×D₅₃

yes

—

137

C₄⋊Q₈

—

138

C₄⋊Q₈

—

139

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

140

#126

—

106

—

141

#125

—

106

—

142

#125

—

106

—

143

#124

—

144

C₂²⋊Q₈

—

106

—

145

C₄×Q₈

—

146

C₄×Q₈

—

147

C₄×Q₈

—

148

C₄×Q₈

—

149

C₄×Q₈

—

106

—

150

C₄×D₄

C₂×D₅₃

yes

characteristic

151

#117

—

106

—

152

#106

—

153

C₄⋊C₈

—

154

C₄⋊C₈

C₂×D₅₃

yes

characteristic

155

#103

—

156

#103

—

157

#103

C₂×D₅₃

yes

—

158

#102

—

159

#97

—

160

C₄×C₈

—

161

C₂×Q₈

—

162

C₂×Q₈

—

163

C₂×Q₈

—

164

C₂×Q₈

C₂²×D₅₃

yes

—

165

C₂×D₄

C₂²×D₅₃

yes

characteristic

166

C₂²×C₄

—

167

Q₁₆

—

106

—

168

Q₁₆

—

212

—

169

Q₁₆

—

212

—

170

SD₁₆

—

171

SD₁₆

—

106

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

C₂²×D₅₃

yes

—

175

C₄⋊C₄

—

176

C₄⋊C₄

—

177

C₄⋊C₄

—

178

C₄⋊C₄

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

106

—

181

C₄⋊C₄

—

106

—

182

C₄⋊C₄

—

106

—

183

C₄⋊C₄

—

106

—

184

C₄⋊C₄

—

106

—

185

C₄⋊C₄

—

106

—

186

C₄⋊C₄

—

106

—

187

C₄⋊C₄

C₂²×D₅₃

yes

characteristic

188

C₂²⋊C₄

—

189

C₂²⋊C₄

—

106

—

190

C₂²⋊C₄

—

106

—

191

C₄²

—

192

C₄²

—

106

—

193

C₄²

—

106

—

194

C₄²

—

106

—

195

C₄²

C₂²×D₅₃

yes

characteristic

196

C₂³

—

197

Q₈

—

198

Q₈

—

199

Q₈

—

106

—

200

Q₈

—

106

—

201

Q₈

—

106

—

202

Q₈

—

106

—

203

Q₈

—

106

—

204

D₄

—

205

D₄

—

206

C₂×C₄

—

207

C₂×C₄

—

208

C₂×C₄

—

209

C₂×C₄

—

210

C₂×C₄

—

211

C₂×C₄

—

106

—

212

C₂×C₄

—

106

—

213

C₂×C₄

—

106

—

214

C₂×C₄

D₄×D₅₃

yes

characteristic

215

C₂×C₄

D₄×D₅₃

yes

characteristic

216

C₂×C₄

C₂³×D₅₃

yes

characteristic

217

C₈

—

218

C₈

—

219

C₈

—

106

—

220

C₂²

—

221

C₂²

—

222

C₂²

—

223

C₂²

C₂×D₄×D₅₃

yes

characteristic

224

C₄

—

225

C₄

—

226

C₄

—

227

C₄

—

106

—

228

C₄

—

106

—

229

C₄

—

212

—

230

C₄

—

212

—

231

C₄

—

212

—

232

C₄

#13133246

yes

—

233

C₄

#41×D₅₃

yes

characteristic

234

C₄

#13133262

yes

characteristic

235

C₂

—

236

C₂

#16811291

yes

characteristic

237

C₂

C₂×#13133246

yes

characteristic

238

C₂

#16811439

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#461	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#461	D₅₃
hypocenter	C₅₃	#3398
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #461

subgroup

G/H

center

C₂²

C₂×D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#461

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #461

N	H ≅ G/N	type
1. #16811348	C₂	semidirect
4. #16811002	C₂	semidirect
6. #16810834	C₂	semidirect
7. #16810832	C₂	semidirect
8. #16810750	C₂	semidirect
9. #16810727	C₂	semidirect
131. C₅₃	#3398	semidirect

H ≅ G/N

type

1. #16811348

C₂

semidirect

4. #16811002

C₂

semidirect

6. #16810834

C₂

semidirect

7. #16810832

C₂

semidirect

8. #16810750

C₂

semidirect

9. #16810727

C₂

semidirect

131. C₅₃

#3398

semidirect

Subgroups — #20287750

Conjugacy classes of subgroups

Complements