Group #20286161 — subgroups

Subgroups — #20286161

Order 6784.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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	#16811235	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810927	C₂	1	2	yes	maximal, complemented
3	#16810923	C₂	1	2	yes	maximal, complemented
4	#16810867	C₂	1	2	yes	maximal, complemented
5	#16810850	C₂	1	2	yes	maximal, complemented
6	#16810828	C₂	1	2	yes	maximal, complemented
7	#16810820	C₂	1	2	yes	maximal, complemented
8	#16810650	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#540	C₂	1	1	yes	maximal, characteristic
10	C₂×#11719923	C₂²	1	2	yes	—
11	C₂×#11719923	C₂²	1	2	yes	—
12	C₂×#11719917	C₂²	1	1	yes	characteristic
13	C₂²×#88322	C₂²	1	1	yes	characteristic
14	#13133217	C₂²	1	1	yes	characteristic
15	#13133210	—	2	2	no	—
16	#13133209	C₂²	1	1	yes	characteristic
17	#13133207	—	2	2	no	—
18	#13133196	C₂²	1	2	yes	—
19	C₄×Dic₁₀₆	C₂²	1	2	yes	—
20	#13133150	C₂²	1	2	yes	—
21	#13133148	C₂²	1	2	yes	—
22	C₄⋊#88322	C₂²	1	2	yes	—
23	C₄⋊#88322	C₂²	1	2	yes	—
24	#13133140	C₂²	1	2	yes	—
25	#13133140	C₂²	1	2	yes	—
26	#13133138	C₂²	1	2	yes	—
27	#13133138	C₂²	1	2	yes	—
28	#127×C₅₃	C₂²	1	1	yes	characteristic
29	#125×C₅₃	C₂²	1	1	yes	characteristic
30	#123×C₅₃	—	2	2	no	—
31	#121×C₅₃	—	2	2	no	—
32	#121×C₅₃	C₂²	1	2	yes	—
33	C₄×D₄×C₅₃	C₂²	1	2	yes	—
34	#117×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×D₄×C₅₃	—	2	2	no	—
36	C₂×D₄×C₅₃	—	2	2	no	—
37	C₂×D₄×C₅₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₅₃	—	2	2	no	—
39	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
40	C₂×Dic₁₀₆	C₂³	1	2	yes	—
41	C₄⋊C₄×C₅₃	—	2	2	no	—
42	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
43	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
44	C₂²⋊C₄×C₅₃	—	2	2	no	—
45	C₂²⋊C₄×C₅₃	—	2	2	no	—
46	C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
47	C₄²×C₅₃	C₂³	1	1	yes	characteristic
48	C₄²×C₅₃	C₂³	1	1	yes	characteristic
49	C₂²⋊Dic₅₃	—	2	2	no	—
50	#11719923	—	4	2	no	—
51	#11719923	—	4	2	no	—
52	C₄⋊Dic₅₃	—	2	2	no	—
53	C₄⋊Dic₅₃	C₂³	1	2	yes	—
54	#11719919	—	2	2	no	—
55	#11719919	—	2	2	no	—
56	#11719919	—	2	2	no	—
57	C₄×Dic₅₃	—	2	2	no	—
58	#11719917	—	4	1	no	—
59	C₂×#88322	—	4	1	no	—
60	C₂×#88322	C₂³	1	2	yes	—
61	C₂×#88322	C₂³	1	2	yes	—
62	C₂³×C₅₃	—	2	2	no	—
63	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
64	D₄×C₅₃	—	2	2	no	—
65	D₄×C₅₃	—	2	2	no	—
66	D₄×C₅₃	—	8	2	no	—
67	D₄×C₅₃	—	8	2	no	—
68	C₂×C₄×C₅₃	—	2	1	no	—
69	C₂×C₄×C₅₃	—	2	1	no	—
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	—	8	2	no	—
72	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
73	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
75	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
76	C₂×Dic₅₃	—	2	2	no	—
77	C₂×Dic₅₃	—	2	2	no	—
78	Dic₁₀₆	—	2	2	no	—
79	Dic₁₀₆	—	2	2	no	—
80	#88322	—	2	2	no	—
81	#88322	—	2	2	no	—
82	C₂²×C₅₃	—	2	1	no	—
83	C₂²×C₅₃	—	2	2	no	—
84	C₂²×C₅₃	—	4	2	no	—
85	C₂²×C₅₃	—	4	2	no	—
86	C₂²×C₅₃	—	4	2	no	—
87	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
88	C₄×C₅₃	—	2	1	no	—
89	C₄×C₅₃	—	2	2	no	—
90	C₄×C₅₃	—	2	2	no	—
91	C₄×C₅₃	—	4	2	no	—
92	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
93	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
94	Dic₅₃	—	4	2	no	—
95	Dic₅₃	—	4	2	no	—
96	#3260	—	53	1	no	maximal, Sylow
97	C₂×C₅₃	—	4	1	no	—
98	C₂×C₅₃	—	8	2	no	—
99	C₂×C₅₃	#537	1	1	yes	characteristic
100	C₂×C₅₃	#577	1	2	yes	—
101	#541	—	53	1	no	—
102	#540	D₅₃	1	1	yes	characteristic
103	#479	—	53	2	no	—
104	#478	—	53	2	no	—
105	#468	—	53	2	no	—
106	#465	—	53	2	no	—
107	#461	—	53	2	no	—
108	#460	—	53	2	no	—
109	#424	—	53	1	no	—
110	C₅₃	#3260	1	1	yes	characteristic, complemented, Sylow
111	C₂×SD₁₆	—	53	2	no	—
112	C₂×SD₁₆	—	53	2	no	—
113	C₂×M₁₆	—	53	1	no	—
114	C₂²×C₈	—	53	1	no	—
115	C₄⋊Q₈	—	53	1	no	—
116	#127	C₂×D₅₃	1	1	yes	characteristic
117	#126	—	106	2	no	—
118	#125	—	53	1	no	—
119	#125	—	106	2	no	—
120	#125	C₂×D₅₃	1	1	yes	characteristic
121	#123	—	2	2	no	—
122	C₂²⋊Q₈	—	53	2	no	—
123	#121	—	2	2	no	—
124	#121	C₂×D₅₃	1	2	yes	—
125	C₄×Q₈	—	53	2	no	—
126	C₄×D₄	C₂×D₅₃	1	2	yes	—
127	#117	C₂×D₅₃	1	1	yes	characteristic
128	#107	—	53	2	no	—
129	#106	—	53	2	no	—
130	C₄⋊C₈	—	53	2	no	—
131	C₄⋊C₈	—	53	2	no	—
132	#103	—	53	2	no	—
133	#103	—	53	2	no	—
134	#102	—	53	2	no	—
135	#102	—	53	2	no	—
136	C₂×Q₈	—	53	2	no	—
137	C₂×D₄	—	2	2	no	—
138	C₂×D₄	—	2	2	no	—
139	C₂×D₄	C₂²×D₅₃	1	2	yes	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
142	SD₁₆	—	212	2	no	—
143	SD₁₆	—	212	2	no	—
144	M₁₆	—	212	1	no	—
145	C₂×C₈	—	53	2	no	—
146	C₂×C₈	—	53	2	no	—
147	C₂×C₈	—	212	1	no	—
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	—	53	2	no	—
150	C₄⋊C₄	—	106	2	no	—
151	C₄⋊C₄	—	106	2	no	—
152	C₄⋊C₄	—	106	2	no	—
153	C₄⋊C₄	—	106	2	no	—
154	C₄⋊C₄	C₂²⋊D₅₃	1	2	yes	—
155	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
156	C₂²⋊C₄	—	2	2	no	—
157	C₂²⋊C₄	—	2	2	no	—
158	C₂²⋊C₄	—	106	2	no	—
159	C₂²⋊C₄	C₂²⋊D₅₃	1	2	yes	—
160	C₄²	—	106	2	no	—
161	C₄²	C₂²×D₅₃	1	1	yes	characteristic
162	C₄²	C₂²×D₅₃	1	1	yes	characteristic
163	C₂³	—	2	2	no	—
164	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
165	Q₈	—	106	2	no	—
166	Q₈	—	106	2	no	—
167	D₄	—	2	2	no	—
168	D₄	—	2	2	no	—
169	D₄	—	8	2	no	—
170	D₄	—	8	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	8	2	no	—
175	C₂×C₄	—	106	2	no	—
176	C₂×C₄	—	106	2	no	—
177	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
178	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
179	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
180	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
181	C₈	—	106	2	no	—
182	C₈	—	106	2	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	2	1	no	—
185	C₂²	—	4	2	no	—
186	C₂²	—	4	2	no	—
187	C₂²	—	4	2	no	—
188	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	—	2	2	no	—
191	C₄	—	2	1	no	—
192	C₄	—	4	2	no	—
193	C₄	—	212	2	no	—
194	C₄	—	212	2	no	—
195	C₄	#13133263	1	1	yes	characteristic
196	C₄	#13133264	1	1	yes	characteristic
197	C₂	—	4	1	no	—
198	C₂	—	8	2	no	—
199	C₂	#16811201	1	1	yes	characteristic
200	C₂	#16811434	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811235

C₂

yes

maximal, characteristic, complemented

#16810927

C₂

yes

maximal, complemented

#16810923

C₂

yes

maximal, complemented

#16810867

C₂

yes

maximal, complemented

#16810850

C₂

yes

maximal, complemented

#16810828

C₂

yes

maximal, complemented

#16810820

C₂

yes

maximal, complemented

#16810650

C₂

yes

maximal, characteristic, complemented

C₅₃×#540

C₂

yes

maximal, characteristic

C₂×#11719923

C₂²

yes

—

C₂×#11719923

C₂²

yes

—

C₂×#11719917

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

#13133217

C₂²

yes

characteristic

#13133210

—

#13133209

C₂²

yes

characteristic

#13133207

—

#13133196

C₂²

yes

—

C₄×Dic₁₀₆

C₂²

yes

—

#13133150

C₂²

yes

—

#13133148

C₂²

yes

—

C₄⋊#88322

C₂²

yes

—

C₄⋊#88322

C₂²

yes

—

#13133140

C₂²

yes

—

#13133140

C₂²

yes

—

#13133138

C₂²

yes

—

#13133138

C₂²

yes

—

#127×C₅₃

C₂²

yes

characteristic

#125×C₅₃

C₂²

yes

characteristic

#123×C₅₃

—

#121×C₅₃

—

#121×C₅₃

C₂²

yes

—

C₄×D₄×C₅₃

C₂²

yes

—

#117×C₅₃

C₂²

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

C₂³

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

—

#11719923

—

#11719923

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

#11719917

—

C₂×#88322

—

C₂×#88322

C₂³

yes

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

H(5,𝔽₂)

yes

characteristic

C₄×C₅₃

2₋¹⁺⁴

yes

characteristic

Dic₅₃

—

Dic₅₃

—

#3260

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#537

yes

characteristic

100

C₂×C₅₃

#577

yes

—

101

#541

—

102

#540

D₅₃

yes

characteristic

103

#479

—

104

#478

—

105

#468

—

106

#465

—

107

#461

—

108

#460

—

109

#424

—

110

C₅₃

#3260

yes

characteristic, complemented, Sylow

111

C₂×SD₁₆

—

112

C₂×SD₁₆

—

113

C₂×M₁₆

—

114

C₂²×C₈

—

115

C₄⋊Q₈

—

116

#127

C₂×D₅₃

yes

characteristic

117

#126

—

106

—

118

#125

—

119

#125

—

106

—

120

#125

C₂×D₅₃

yes

characteristic

121

#123

—

122

C₂²⋊Q₈

—

123

#121

—

124

#121

C₂×D₅₃

yes

—

125

C₄×Q₈

—

126

C₄×D₄

C₂×D₅₃

yes

—

127

#117

C₂×D₅₃

yes

characteristic

128

#107

—

129

#106

—

130

C₄⋊C₈

—

131

C₄⋊C₈

—

132

#103

—

133

#103

—

134

#102

—

135

#102

—

136

C₂×Q₈

—

137

C₂×D₄

—

138

C₂×D₄

—

139

C₂×D₄

C₂²×D₅₃

yes

—

140

C₂²×C₄

—

141

C₂²×C₄

C₂²×D₅₃

yes

characteristic

142

SD₁₆

—

212

—

143

SD₁₆

—

212

—

144

M₁₆

—

212

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

212

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

106

—

151

C₄⋊C₄

—

106

—

152

C₄⋊C₄

—

106

—

153

C₄⋊C₄

—

106

—

154

C₄⋊C₄

C₂²⋊D₅₃

yes

—

155

C₄⋊C₄

C₂²×D₅₃

yes

—

156

C₂²⋊C₄

—

157

C₂²⋊C₄

—

158

C₂²⋊C₄

—

106

—

159

C₂²⋊C₄

C₂²⋊D₅₃

yes

—

160

C₄²

—

106

—

161

C₄²

C₂²×D₅₃

yes

characteristic

162

C₄²

C₂²×D₅₃

yes

characteristic

163

C₂³

—

164

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

165

Q₈

—

106

—

166

Q₈

—

106

—

167

D₄

—

168

D₄

—

169

D₄

—

170

D₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

106

—

176

C₂×C₄

—

106

—

177

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

178

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

179

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

180

C₂×C₄

C₂³×D₅₃

yes

characteristic

181

C₈

—

106

—

182

C₈

—

106

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

212

—

194

C₄

—

212

—

195

C₄

#13133263

yes

characteristic

196

C₄

#13133264

yes

characteristic

197

C₂

—

198

C₂

—

199

C₂

#16811201

yes

characteristic

200

C₂

#16811434

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#540	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#540	D₅₃
hypocenter	C₅₃	#3260
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #540

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#540

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ #117 ◁ #540

N	H ≅ G/N	type
1. #16811235	C₂	semidirect
2. #16810927	C₂	semidirect
3. #16810923	C₂	semidirect
4. #16810867	C₂	semidirect
5. #16810850	C₂	semidirect
6. #16810828	C₂	semidirect
7. #16810820	C₂	semidirect
8. #16810650	C₂	semidirect
110. C₅₃	#3260	semidirect

H ≅ G/N

type

1. #16811235

C₂

semidirect

2. #16810927

C₂

semidirect

3. #16810923

C₂

semidirect

4. #16810867

C₂

semidirect

5. #16810850

C₂

semidirect

6. #16810828

C₂

semidirect

7. #16810820

C₂

semidirect

8. #16810650

C₂

semidirect

110. C₅₃

#3260

semidirect

Subgroups — #20286161

Conjugacy classes of subgroups

Complements