Group #20273389 — subgroups

Subgroups — #20273389

Order 6784.

260 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 73 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	#16811171	C₂	1	1	yes	maximal, characteristic
2	#16810645	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₈×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810370	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810368	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810367	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#331	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719947	—	2	1	no	—
9	C₂×#11719912	—	2	1	no	—
10	C₂×C₈×D₅₃	—	2	1	no	—
11	#13133217	C₂²	1	1	yes	characteristic
12	#13133204	—	2	1	no	—
13	#13133188	—	2	1	no	—
14	#13133188	C₄	1	2	yes	—
15	D₄×Dic₅₃	—	2	1	no	—
16	D₄×Dic₅₃	C₄	1	2	yes	—
17	C₄²×D₅₃	C₂²	1	1	yes	characteristic
18	C₄⋊#88322	C₂²	1	1	yes	characteristic
19	C₄⋊#88322	C₂²	1	1	yes	characteristic
20	#13133146	—	2	1	no	—
21	#13133146	—	2	1	no	—
22	#127×C₅₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×D₄×C₅₃	—	2	1	no	—
26	C₂×D₄×C₅₃	—	2	1	no	—
27	C₂×D₄×C₅₃	C₂×C₄	1	2	yes	—
28	C₂×C₂²⋊D₅₃	—	2	2	no	—
29	C₂²×Dic₅₃	—	2	2	no	—
30	#11719947	—	4	2	no	—
31	#11719947	—	4	2	no	—
32	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
34	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
35	C₂×Dic₁₀₆	—	2	1	no	—
36	C₂×C₈×C₅₃	—	2	1	no	—
37	C₂×C₈×C₅₃	—	2	1	no	—
38	C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	C₂²⋊Dic₅₃	—	2	2	no	—
40	C₂²⋊Dic₅₃	—	2	2	no	—
41	C₄⋊Dic₅₃	—	2	1	no	—
42	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
43	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
44	C₄×Dic₅₃	D₄	1	1	yes	characteristic
45	C₄×Dic₅₃	D₄	1	1	yes	characteristic
46	C₂×#88322	—	2	1	no	—
47	C₂×#88322	—	2	1	no	—
48	#11719912	—	4	2	no	—
49	C₈×D₅₃	—	4	2	no	—
50	C₂³×C₅₃	—	2	2	no	—
51	C₂²×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
52	D₄×C₅₃	—	4	2	no	—
53	D₄×C₅₃	—	4	1	no	—
54	D₄×C₅₃	—	4	1	no	—
55	D₄×C₅₃	—	4	1	no	—
56	D₄×C₅₃	—	4	1	no	—
57	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²⋊D₅₃	—	4	4	no	—
61	C₂×Dic₅₃	—	2	2	no	—
62	C₂×Dic₅₃	—	4	4	no	—
63	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₄×D₅₃	—	2	1	no	—
67	C₄×D₅₃	—	2	1	no	—
68	C₄×D₅₃	—	4	1	no	—
69	C₄×D₅₃	D₈	1	2	yes	—
70	C₄×D₅₃	SD₁₆	1	2	yes	—
71	Dic₁₀₆	—	4	1	no	—
72	Dic₁₀₆	—	4	1	no	—
73	C₈×C₅₃	—	4	1	no	—
74	C₈×C₅₃	—	4	1	no	—
75	#88322	—	4	1	no	—
76	#88322	—	4	1	no	—
77	C₂²×C₅₃	—	4	2	no	—
78	C₂²×C₅₃	—	4	2	no	—
79	C₂²×C₅₃	—	4	2	no	—
80	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×D₅₃	—	2	1	no	—
82	C₂×D₅₃	—	2	1	no	—
83	C₂×D₅₃	#102	1	2	yes	—
84	C₄×C₅₃	—	2	1	no	—
85	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
86	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
87	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
88	C₄×C₅₃	#137	1	1	yes	characteristic
89	Dic₅₃	—	2	1	no	—
90	Dic₅₃	—	2	1	no	—
91	Dic₅₃	—	4	2	no	—
92	Dic₅₃	#102	1	2	yes	—
93	#1547	—	53	1	no	maximal, Sylow
94	C₂×C₅₃	—	8	2	no	—
95	C₂×C₅₃	#413	1	1	yes	characteristic
96	C₂×C₅₃	C₂×#102	1	1	yes	characteristic
97	C₂×C₅₃	#417	1	1	yes	characteristic
98	D₅₃	—	2	2	no	—
99	#532	—	53	1	no	—
100	#423	—	53	1	no	—
101	C₂×C₄⋊C₈	—	53	1	no	—
102	#332	—	53	1	no	—
103	#332	—	53	1	no	—
104	#331	—	53	1	no	—
105	#331	D₅₃	1	1	yes	characteristic, complemented
106	C₅₃	#1547	1	1	yes	characteristic, complemented, Sylow
107	C₂×#41	—	106	1	no	—
108	C₂×M₁₆	—	106	1	no	—
109	C₂²×C₈	—	106	1	no	—
110	C₄⋊Q₈	—	53	1	no	—
111	#127	C₂×D₅₃	1	1	yes	characteristic
112	#124	—	106	1	no	—
113	#121	—	53	2	no	—
114	#121	—	106	1	no	—
115	C₄×D₄	—	53	2	no	—
116	C₄×D₄	—	106	1	no	—
117	C₂×C₄²	—	53	1	no	—
118	C₄⋊C₈	—	53	1	no	—
119	C₄⋊C₈	—	53	1	no	—
120	C₄⋊C₈	—	106	1	no	—
121	C₄⋊C₈	—	106	1	no	—
122	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
123	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
124	#41	—	212	2	no	—
125	#41	—	212	2	no	—
126	C₂×Q₈	—	106	1	no	—
127	C₂×D₄	—	2	1	no	—
128	C₂×D₄	—	2	1	no	—
129	C₂×D₄	—	106	2	no	—
130	C₂×D₄	C₄×D₅₃	1	2	yes	—
131	C₂²×C₄	—	53	1	no	—
132	C₂²×C₄	—	53	1	no	—
133	C₂²×C₄	—	53	1	no	—
134	C₂²×C₄	—	106	2	no	—
135	M₁₆	—	212	2	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	106	1	no	—
139	C₂×C₈	—	106	1	no	—
140	C₂×C₈	—	212	2	no	—
141	C₄⋊C₄	—	53	2	no	—
142	C₄⋊C₄	—	106	1	no	—
143	C₂²⋊C₄	—	106	2	no	—
144	C₂²⋊C₄	—	106	2	no	—
145	C₄²	—	53	1	no	—
146	C₄²	—	53	1	no	—
147	C₄²	—	53	1	no	—
148	C₄²	C₂²×D₅₃	1	1	yes	characteristic
149	C₂³	—	2	2	no	—
150	C₂³	—	53	1	no	—
151	Q₈	—	212	1	no	—
152	Q₈	—	212	1	no	—
153	D₄	—	4	2	no	—
154	D₄	—	4	1	no	—
155	D₄	—	4	1	no	—
156	D₄	—	4	1	no	—
157	D₄	—	4	1	no	—
158	D₄	—	212	4	no	—
159	C₂×C₄	—	53	1	no	—
160	C₂×C₄	—	53	2	no	—
161	C₂×C₄	—	53	2	no	—
162	C₂×C₄	—	53	1	no	—
163	C₂×C₄	—	53	1	no	—
164	C₂×C₄	—	106	1	no	—
165	C₂×C₄	—	106	1	no	—
166	C₂×C₄	—	106	2	no	—
167	C₂×C₄	—	212	1	no	—
168	C₂×C₄	—	212	4	no	—
169	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
170	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
171	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
172	C₈	—	4	1	no	—
173	C₈	—	4	1	no	—
174	C₈	—	212	1	no	—
175	C₈	—	212	1	no	—
176	C₂²	—	4	2	no	—
177	C₂²	—	4	2	no	—
178	C₂²	—	4	2	no	—
179	C₂²	—	53	2	no	—
180	C₂²	—	106	1	no	—
181	C₂²	—	106	1	no	—
182	C₂²	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	53	2	no	—
185	C₄	—	106	1	no	—
186	C₄	—	106	1	no	—
187	C₄	—	212	2	no	—
188	C₄	D₈×D₅₃	1	1	yes	characteristic
189	C₄	SD₁₆×D₅₃	1	1	yes	characteristic
190	C₄	#13133241	1	1	yes	characteristic
191	C₄	#13133246	1	1	yes	characteristic
192	C₂	—	8	2	no	—
193	C₂	—	106	2	no	—
194	C₂	#16810604	1	1	yes	characteristic
195	C₂	#102×D₅₃	1	1	yes	characteristic
196	C₂	#16810621	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811171

C₂

yes

maximal, characteristic

#16810645

C₂

yes

maximal, characteristic, complemented

C₄⋊C₈×D₅₃

C₂

yes

maximal, characteristic, complemented

#16810370

C₂

yes

maximal, characteristic, complemented

#16810368

C₂

yes

maximal, characteristic, complemented

#16810367

C₂

yes

maximal, characteristic, complemented

C₅₃×#331

C₂

yes

maximal, characteristic, complemented

C₂×#11719947

—

C₂×#11719912

—

C₂×C₈×D₅₃

—

#13133217

C₂²

yes

—

—

C₄

yes

—

D₄×Dic₅₃

—

D₄×Dic₅₃

C₄

yes

—

C₄²×D₅₃

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

characteristic

#13133146

—

#13133146

—

#127×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂×C₄

yes

—

C₂×C₂²⋊D₅₃

—

C₂²×Dic₅₃

—

#11719947

—

#11719947

—

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

—

—

—

C₈×D₅₃

—

C₂³×C₅₃

—

C₂²×D₅₃

C₂²⋊C₄

yes

characteristic

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

D₈

yes

—

C₄×D₅₃

SD₁₆

yes

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#102

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×D₈

yes

characteristic

C₄×C₅₃

C₂×SD₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#102

yes

—

#1547

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#532

—

100

#423

—

101

C₂×C₄⋊C₈

—

102

#332

—

103

#332

—

104

#331

—

105

#331

D₅₃

yes

characteristic, complemented

106

C₅₃

#1547

yes

characteristic, complemented, Sylow

107

C₂×#41

—

106

—

108

C₂×M₁₆

—

106

—

109

C₂²×C₈

—

106

—

110

C₄⋊Q₈

—

111

#127

C₂×D₅₃

yes

characteristic

112

#124

—

106

—

113

#121

—

114

#121

—

106

—

115

C₄×D₄

—

116

C₄×D₄

—

106

—

117

C₂×C₄²

—

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₄⋊C₈

—

106

—

121

C₄⋊C₈

—

106

—

122

C₄⋊C₈

C₂×D₅₃

yes

characteristic

123

C₄⋊C₈

C₂×D₅₃

yes

characteristic

124

#41

—

212

—

125

#41

—

212

—

126

C₂×Q₈

—

106

—

127

C₂×D₄

—

128

C₂×D₄

—

129

C₂×D₄

—

106

—

130

C₂×D₄

C₄×D₅₃

yes

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

—

106

—

135

M₁₆

—

212

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

106

—

139

C₂×C₈

—

106

—

140

C₂×C₈

—

212

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

106

—

143

C₂²⋊C₄

—

106

—

144

C₂²⋊C₄

—

106

—

145

C₄²

—

146

C₄²

—

147

C₄²

—

148

C₄²

C₂²×D₅₃

yes

characteristic

149

C₂³

—

150

C₂³

—

151

Q₈

—

212

—

152

Q₈

—

212

—

153

D₄

—

154

D₄

—

155

D₄

—

156

D₄

—

157

D₄

—

158

D₄

—

212

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

106

—

165

C₂×C₄

—

106

—

166

C₂×C₄

—

106

—

167

C₂×C₄

—

212

—

168

C₂×C₄

—

212

—

169

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

170

C₂×C₄

D₄×D₅₃

yes

characteristic

171

C₂×C₄

D₄×D₅₃

yes

characteristic

172

C₈

—

173

C₈

—

174

C₈

—

212

—

175

C₈

—

212

—

176

C₂²

—

177

C₂²

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

106

—

181

C₂²

—

106

—

182

C₂²

C₂²⋊C₄×D₅₃

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

106

—

186

C₄

—

106

—

187

C₄

—

212

—

188

C₄

D₈×D₅₃

yes

characteristic

189

C₄

SD₁₆×D₅₃

yes

characteristic

190

C₄

#13133241

yes

characteristic

191

C₄

#13133246

yes

characteristic

192

C₂

—

193

C₂

—

106

—

194

C₂

#16810604

yes

characteristic

195

C₂

#102×D₅₃

yes

characteristic

196

C₂

#16810621

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#331	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#331	D₅₃
hypocenter	C₅₃	#1547
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#331

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
2. #16810645	C₂	semidirect
3. C₄⋊C₈×D₅₃	C₂	semidirect
4. #16810370	C₂	semidirect
5. #16810368	C₂	semidirect
6. #16810367	C₂	semidirect
7. C₅₃×#331	C₂	semidirect
105. #331	D₅₃	semidirect
106. C₅₃	#1547	semidirect

H ≅ G/N

type

2. #16810645

C₂

semidirect

3. C₄⋊C₈×D₅₃

C₂

semidirect

4. #16810370

C₂

semidirect

5. #16810368

C₂

semidirect

6. #16810367

C₂

semidirect

7. C₅₃×#331

C₂

semidirect

105. #331

D₅₃

semidirect

106. C₅₃

#1547

semidirect

Subgroups — #20273389

Conjugacy classes of subgroups

Complements