Group #20285173 — subgroups

Subgroups — #20285173

Order 6784.

552 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 145 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	C₂×#13133249	C₂	1	4	yes	maximal, complemented
2	#16811174	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#16811016	C₂	1	4	yes	maximal, complemented
5	#16811002	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810997	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810996	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#11719912	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11719949	C₂²	1	2	yes	—
11	C₂×Q₈×D₅₃	C₂²	1	2	yes	—
12	#13133249	—	2	16	no	—
13	C₂×#11719925	C₂²	1	4	yes	—
14	C₂×#11719913	C₂²	1	4	yes	—
15	C₂×#11719924	C₂²	1	4	yes	—
16	C₂×#11719912	C₂²	1	2	yes	—
17	#13133217	C₂²	1	1	yes	characteristic
18	#13133216	—	2	2	no	—
19	#13133215	C₂²	1	2	yes	—
20	#13133214	C₂²	1	1	yes	characteristic
21	#13133206	C₂²	1	2	yes	—
22	#13133192	—	2	2	no	—
23	#13133179	—	2	2	no	—
24	C₄⋊C₄×D₅₃	—	2	2	no	—
25	C₄²×D₅₃	C₂²	1	1	yes	characteristic
26	#13133127	C₂²	1	2	yes	—
27	C₄×#88322	C₂²	1	1	yes	characteristic
28	C₂×Q₁₆×C₅₃	C₂²	1	4	yes	—
29	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
33	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
34	#11719949	—	2	8	no	—
35	Q₈×D₅₃	—	2	8	no	—
36	C₂×D₂₁₂	—	2	2	no	—
37	C₂×D₂₁₂	—	2	2	no	—
38	C₂×D₂₁₂	C₂³	1	2	yes	—
39	C₂×C₄×D₅₃	—	2	2	no	—
40	C₂×C₄×D₅₃	—	2	2	no	—
41	C₂×C₄×D₅₃	D₄	1	2	yes	—
42	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
43	C₂×Dic₁₀₆	—	2	2	no	—
44	C₂×Dic₁₀₆	C₂³	1	2	yes	—
45	Q₁₆×C₅₃	—	2	8	no	—
46	C₂×C₈×C₅₃	C₂³	1	2	yes	—
47	C₄⋊C₄×C₅₃	—	2	2	no	—
48	C₄⋊C₄×C₅₃	—	2	2	no	—
49	C₄²×C₅₃	C₂³	1	1	yes	characteristic
50	#11719925	—	2	8	no	—
51	#11719924	—	2	8	no	—
52	#11719921	—	2	4	no	—
53	C₄⋊Dic₅₃	—	2	2	no	—
54	#11719919	—	2	4	no	—
55	C₄×Dic₅₃	D₄	1	2	yes	—
56	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
57	C₂×#88322	C₂³	1	2	yes	—
58	#11719913	—	2	8	no	—
59	#11719912	D₄	1	8	yes	complemented
60	C₂²×D₅₃	—	2	2	no	—
61	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
62	Q₈×C₅₃	—	2	4	no	—
63	Q₈×C₅₃	—	2	4	no	—
64	C₂×C₄×C₅₃	—	2	2	no	—
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
68	C₂×Dic₅₃	—	2	2	no	—
69	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
70	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
71	D₂₁₂	—	2	4	no	—
72	D₂₁₂	—	4	4	no	—
73	D₂₁₂	—	4	4	no	—
74	C₄×D₅₃	—	2	4	no	—
75	C₄×D₅₃	—	4	4	no	—
76	C₄×D₅₃	—	4	4	no	—
77	C₄×D₅₃	C₂×D₄	1	4	yes	—
78	Dic₁₀₆	—	2	4	no	—
79	Dic₁₀₆	—	4	4	no	—
80	C₈×C₅₃	C₂×D₄	1	4	yes	—
81	#88322	C₂×D₄	1	4	yes	—
82	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
83	C₂×D₅₃	—	2	2	no	—
84	C₂×D₅₃	—	4	2	no	—
85	C₂×D₅₃	—	4	4	no	—
86	C₂×D₅₃	#127	1	2	yes	—
87	C₄×C₅₃	—	4	2	no	—
88	C₄×C₅₃	—	4	2	no	—
89	C₄×C₅₃	#137	1	4	yes	complemented
90	C₄×C₅₃	C₂²×D₄	1	2	yes	—
91	Dic₅₃	—	2	2	no	—
92	Dic₅₃	—	4	2	no	—
93	Dic₅₃	#127	1	2	yes	—
94	#3152	—	53	1	no	maximal, Sylow
95	C₂×C₅₃	C₂×#127	1	1	yes	characteristic
96	C₂×C₅₃	C₂×#137	1	2	yes	—
97	D₅₃	—	2	2	no	—
98	D₅₃	—	8	2	no	—
99	C₂×#137	—	53	4	no	—
100	#532	—	53	1	no	—
101	C₂×C₄⋊Q₈	—	53	1	no	—
102	#497	—	53	4	no	—
103	#494	—	53	1	no	—
104	#494	D₅₃	1	1	yes	characteristic, complemented
105	#492	—	53	1	no	—
106	#492	—	53	1	no	—
107	C₄×M₁₆	—	53	1	no	—
108	C₅₃	#3152	1	1	yes	characteristic, complemented, Sylow
109	C₂×#41	—	53	2	no	—
110	C₂²×Q₈	—	53	2	no	—
111	#137	—	106	16	no	—
112	C₂×Q₁₆	—	53	4	no	—
113	C₂×Q₁₆	C₂×D₅₃	1	4	yes	—
114	C₂×SD₁₆	—	53	4	no	—
115	C₂×SD₁₆	—	53	4	no	—
116	C₂×M₁₆	—	53	2	no	—
117	C₄⋊Q₈	—	53	2	no	—
118	C₄⋊Q₈	—	53	1	no	—
119	C₄⋊Q₈	—	106	2	no	—
120	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
121	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
122	#127	—	53	1	no	—
123	#124	—	53	2	no	—
124	#121	—	106	2	no	—
125	C₄×D₄	—	106	2	no	—
126	C₂×C₄⋊C₄	—	106	2	no	—
127	C₂×C₄²	—	53	1	no	—
128	#97	—	53	2	no	—
129	C₄×C₈	—	53	1	no	—
130	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
131	#41	—	106	8	no	—
132	C₂×Q₈	—	53	2	no	—
133	C₂×Q₈	—	106	2	no	—
134	C₂×Q₈	—	106	8	no	—
135	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
136	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
137	C₂×D₄	—	53	2	no	—
138	C₂×D₄	—	106	2	no	—
139	C₂×D₄	—	106	2	no	—
140	C₂²×C₄	—	53	1	no	—
141	C₂²×C₄	—	53	2	no	—
142	C₂²×C₄	—	106	2	no	—
143	C₂²×C₄	—	106	2	no	—
144	Q₁₆	—	2	8	no	—
145	Q₁₆	—	106	8	no	—
146	SD₁₆	—	106	8	no	—
147	SD₁₆	—	106	8	no	—
148	M₁₆	—	53	8	no	—
149	C₂×C₈	—	53	2	no	—
150	C₂×C₈	C₂²×D₅₃	1	2	yes	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	—	106	4	no	—
154	C₄⋊C₄	—	106	2	no	—
155	C₂²⋊C₄	—	106	4	no	—
156	C₄²	—	53	2	no	—
157	C₄²	—	53	1	no	—
158	C₄²	C₂²×D₅₃	1	1	yes	characteristic
159	C₂³	—	53	1	no	—
160	C₂³	—	106	2	no	—
161	Q₈	—	2	4	no	—
162	Q₈	—	2	4	no	—
163	Q₈	—	106	4	no	—
164	Q₈	—	212	4	no	—
165	D₄	—	106	4	no	—
166	D₄	—	212	4	no	—
167	D₄	—	212	4	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	53	1	no	—
171	C₂×C₄	—	53	4	no	—
172	C₂×C₄	—	53	2	no	—
173	C₂×C₄	—	106	4	no	—
174	C₂×C₄	—	106	2	no	—
175	C₂×C₄	—	212	4	no	—
176	C₂×C₄	—	212	4	no	—
177	C₂×C₄	D₄×D₅₃	1	2	yes	—
178	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
179	C₈	—	53	4	no	—
180	C₈	D₄×D₅₃	1	4	yes	—
181	C₂²	—	53	2	no	—
182	C₂²	—	106	2	no	—
183	C₂²	—	212	2	no	—
184	C₂²	—	212	4	no	—
185	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
186	C₄	—	4	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	53	2	no	—
189	C₄	—	106	2	no	—
190	C₄	—	212	2	no	—
191	C₄	#13133249	1	4	yes	complemented
192	C₄	C₂×D₄×D₅₃	1	2	yes	—
193	C₂	—	106	2	no	—
194	C₂	—	424	2	no	—
195	C₂	#127×D₅₃	1	1	yes	characteristic
196	C₂	C₂×#13133249	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133249

C₂

yes

maximal, complemented

#16811174

C₂

yes

maximal, characteristic

C₄⋊Q₈×D₅₃

C₂

yes

maximal, characteristic, complemented

#16811016

C₂

yes

maximal, complemented

#16811002

C₂

yes

maximal, characteristic, complemented

#16810997

C₂

yes

maximal, characteristic, complemented

#16810996

C₂

yes

maximal, characteristic, complemented

C₄×#11719912

C₂

yes

maximal, characteristic, complemented

C₅₃×#494

C₂

yes

maximal, characteristic, complemented

C₂×#11719949

C₂²

yes

—

C₂×Q₈×D₅₃

C₂²

yes

—

#13133249

—

C₂×#11719925

C₂²

yes

—

C₂×#11719913

C₂²

yes

—

C₂×#11719924

C₂²

yes

—

C₂×#11719912

C₂²

yes

—

#13133217

C₂²

yes

characteristic

#13133216

—

#13133215

C₂²

yes

—

#13133214

C₂²

yes

characteristic

#13133206

C₂²

yes

—

#13133192

—

#13133179

—

C₄⋊C₄×D₅₃

—

C₄²×D₅₃

C₂²

yes

characteristic

#13133127

C₂²

yes

—

C₄×#88322

C₂²

yes

characteristic

C₂×Q₁₆×C₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

C₂³

yes

—

C₂×Q₈×C₅₃

C₂³

yes

—

#11719949

—

Q₈×D₅₃

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

C₂³

yes

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

D₄

yes

—

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂³

yes

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

—

—

—

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

D₄

yes

—

C₄×Dic₅₃

C₂³

yes

characteristic

C₂×#88322

C₂³

yes

—

#11719913

—

#11719912

D₄

yes

complemented

C₂²×D₅₃

—

C₂²×D₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

C₂×D₄

yes

characteristic

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

C₂×D₄

yes

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

C₂×D₄

yes

—

#88322

C₂×D₄

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#127

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#137

yes

complemented

C₄×C₅₃

C₂²×D₄

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#127

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

—

D₅₃

—

C₂×#137

—

100

#532

—

101

C₂×C₄⋊Q₈

—

102

#497

—

103

#494

—

104

#494

D₅₃

yes

characteristic, complemented

105

#492

—

106

#492

—

107

C₄×M₁₆

—

108

C₅₃

#3152

yes

characteristic, complemented, Sylow

109

C₂×#41

—

110

C₂²×Q₈

—

111

#137

—

106

—

112

C₂×Q₁₆

—

113

C₂×Q₁₆

C₂×D₅₃

yes

—

114

C₂×SD₁₆

—

115

C₂×SD₁₆

—

116

C₂×M₁₆

—

117

C₄⋊Q₈

—

118

C₄⋊Q₈

—

119

C₄⋊Q₈

—

106

—

120

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

121

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

122

#127

—

123

#124

—

124

#121

—

106

—

125

C₄×D₄

—

106

—

126

C₂×C₄⋊C₄

—

106

—

127

C₂×C₄²

—

128

#97

—

129

C₄×C₈

—

130

C₄×C₈

C₂×D₅₃

yes

characteristic

131

#41

—

106

—

132

C₂×Q₈

—

133

C₂×Q₈

—

106

—

134

C₂×Q₈

—

106

—

135

C₂×Q₈

C₂²×D₅₃

yes

—

136

C₂×Q₈

C₂²×D₅₃

yes

—

137

C₂×D₄

—

138

C₂×D₄

—

106

—

139

C₂×D₄

—

106

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

—

106

—

143

C₂²×C₄

—

106

—

144

Q₁₆

—

145

Q₁₆

—

106

—

146

SD₁₆

—

106

—

147

SD₁₆

—

106

—

148

M₁₆

—

149

C₂×C₈

—

150

C₂×C₈

C₂²×D₅₃

yes

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

106

—

154

C₄⋊C₄

—

106

—

155

C₂²⋊C₄

—

106

—

156

C₄²

—

157

C₄²

—

158

C₄²

C₂²×D₅₃

yes

characteristic

159

C₂³

—

160

C₂³

—

106

—

161

Q₈

—

162

Q₈

—

163

Q₈

—

106

—

164

Q₈

—

212

—

165

D₄

—

106

—

166

D₄

—

212

—

167

D₄

—

212

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

106

—

174

C₂×C₄

—

106

—

175

C₂×C₄

—

212

—

176

C₂×C₄

—

212

—

177

C₂×C₄

D₄×D₅₃

yes

—

178

C₂×C₄

C₂³×D₅₃

yes

characteristic

179

C₈

—

180

C₈

D₄×D₅₃

yes

—

181

C₂²

—

182

C₂²

—

106

—

183

C₂²

—

212

—

184

C₂²

—

212

—

185

C₂²

C₂×D₄×D₅₃

yes

characteristic

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

106

—

190

C₄

—

212

—

191

C₄

#13133249

yes

complemented

192

C₄

C₂×D₄×D₅₃

yes

—

193

C₂

—

106

—

194

C₂

—

424

—

195

C₂

#127×D₅₃

yes

characteristic

196

C₂

C₂×#13133249

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#494	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#494	D₅₃
hypocenter	C₅₃	#3152
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

C₂×D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#494

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
1. C₂×#13133249	C₂	semidirect
3. C₄⋊Q₈×D₅₃	C₂	semidirect
4. #16811016	C₂	semidirect
5. #16811002	C₂	semidirect
6. #16810997	C₂	semidirect
7. #16810996	C₂	semidirect
8. C₄×#11719912	C₂	semidirect
9. C₅₃×#494	C₂	semidirect
59. #11719912	D₄	semidirect
89. C₄×C₅₃	#137	semidirect
104. #494	D₅₃	semidirect
108. C₅₃	#3152	semidirect
191. C₄	#13133249	semidirect

H ≅ G/N

type

1. C₂×#13133249

C₂

semidirect

3. C₄⋊Q₈×D₅₃

C₂

semidirect

4. #16811016

C₂

semidirect

5. #16811002

C₂

semidirect

6. #16810997

C₂

semidirect

7. #16810996

C₂

semidirect

8. C₄×#11719912

C₂

semidirect

9. C₅₃×#494

C₂

semidirect

59. #11719912

D₄

semidirect

89. C₄×C₅₃

#137

semidirect

104. #494

D₅₃

semidirect

108. C₅₃

#3152

semidirect

191. C₄

#13133249

semidirect

Subgroups — #20285173

Conjugacy classes of subgroups

Complements