Group #20275627 — subgroups

Subgroups — #20275627

Order 6784.

276 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 83 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	C₂²×#11719925	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133152	C₂	1	1	yes	maximal, characteristic
3	#16810633	C₂	1	1	yes	maximal, characteristic
4	#16810632	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133135	C₂	1	1	yes	maximal, characteristic
6	#16810581	C₂	1	1	yes	maximal, characteristic
7	C₂×#101×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₀₆	C₂²	1	1	yes	characteristic
9	C₂×#11719925	—	2	1	no	—
10	C₂×#11719925	—	2	1	no	—
11	C₂×#11719925	—	2	2	no	—
12	C₂×#11719925	C₄	1	4	yes	complemented
13	C₂×#11719917	C₂²	1	1	yes	characteristic
14	C₂²×#88322	C₂²	1	1	yes	characteristic
15	#13133176	C₂²	1	1	yes	characteristic
16	#13133152	—	2	2	no	—
17	#13133142	—	2	2	no	—
18	#13133141	—	2	2	no	—
19	#13133135	—	2	2	no	—
20	#13133127	—	2	1	no	—
21	C₈×Dic₅₃	—	2	1	no	—
22	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
25	#101×C₅₃	—	2	2	no	—
26	C₂×Q₈×C₅₃	—	2	1	no	—
27	C₂×Q₈×C₅₃	—	2	1	no	—
28	C₂×Q₈×C₅₃	—	2	1	no	—
29	C₂×Q₈×C₅₃	—	2	1	no	—
30	C₂×Q₈×C₅₃	—	2	1	no	—
31	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₅₃	—	2	1	no	—
33	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
34	C₂²×Dic₅₃	—	2	1	no	—
35	C₂×Dic₁₀₆	—	2	1	no	—
36	C₂×Dic₁₀₆	—	2	1	no	—
37	C₂×Dic₁₀₆	—	2	1	no	—
38	C₂×Dic₁₀₆	—	2	1	no	—
39	C₂×Dic₁₀₆	—	2	1	no	—
40	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
41	M₁₆×C₅₃	—	2	1	no	—
42	M₁₆×C₅₃	—	2	1	no	—
43	M₁₆×C₅₃	—	2	1	no	—
44	M₁₆×C₅₃	D₄	1	2	yes	—
45	C₂×C₈×C₅₃	—	2	1	no	—
46	C₂×C₈×C₅₃	—	2	1	no	—
47	C₂²⋊Dic₅₃	—	2	1	no	—
48	#11719925	—	2	4	no	—
49	#11719925	—	2	4	no	—
50	C₄⋊Dic₅₃	D₄	1	2	yes	—
51	C₄×Dic₅₃	—	2	1	no	—
52	#11719917	—	2	1	no	—
53	#11719917	—	2	1	no	—
54	C₂×#88322	—	2	1	no	—
55	C₂×#88322	C₂×C₄	1	2	yes	—
56	C₂×#88322	D₄	1	1	yes	characteristic
57	C₂×#88322	D₄	1	1	yes	characteristic
58	C₂×#88322	D₄	1	2	yes	—
59	C₂³×C₅₃	#41	1	1	yes	characteristic
60	Q₈×C₅₃	—	2	2	no	—
61	Q₈×C₅₃	—	2	2	no	—
62	Q₈×C₅₃	—	4	1	no	—
63	Q₈×C₅₃	—	4	1	no	—
64	C₂×C₄×C₅₃	—	2	2	no	—
65	C₂×C₄×C₅₃	—	4	1	no	—
66	C₂×C₄×C₅₃	—	4	1	no	—
67	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
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	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
73	C₂×Dic₅₃	—	2	2	no	—
74	C₂×Dic₅₃	—	2	2	no	—
75	C₂×Dic₅₃	—	4	1	no	—
76	C₂×Dic₅₃	—	4	1	no	—
77	Dic₁₀₆	—	2	2	no	—
78	Dic₁₀₆	—	2	2	no	—
79	Dic₁₀₆	—	4	1	no	—
80	Dic₁₀₆	—	4	1	no	—
81	C₈×C₅₃	—	2	2	no	—
82	C₈×C₅₃	—	4	1	no	—
83	#88322	—	4	1	no	—
84	#88322	C₂²⋊C₄	1	2	yes	—
85	#88322	C₂²⋊C₄	1	2	yes	—
86	C₂²×C₅₃	—	2	2	no	—
87	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
88	C₂²×C₅₃	#121	1	1	yes	characteristic
89	C₂²×C₅₃	#124	1	1	yes	characteristic
90	C₄×C₅₃	—	4	2	no	—
91	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
93	C₄×C₅₃	#121	1	1	yes	characteristic
94	C₄×C₅₃	#127	1	1	yes	characteristic
95	Dic₅₃	—	4	2	no	—
96	Dic₅₃	—	4	2	no	—
97	#1979	—	53	1	no	maximal, Sylow
98	C₂×C₅₃	—	2	1	no	—
99	C₂×C₅₃	—	2	1	no	—
100	C₂×C₅₃	#390	1	1	yes	characteristic
101	C₂×C₅₃	#473	1	2	yes	—
102	C₂²×Q₁₆	—	53	1	no	—
103	C₂×#108	—	53	1	no	—
104	#419	—	53	1	no	—
105	#419	—	53	1	no	—
106	C₂×#101	—	53	1	no	—
107	C₂×#101	D₅₃	1	1	yes	characteristic
108	#405	—	53	1	no	—
109	C₅₃	#1979	1	1	yes	characteristic, complemented, Sylow
110	C₂²×Q₈	—	53	1	no	—
111	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
112	C₂×Q₁₆	—	53	4	no	—
113	C₂×Q₁₆	—	106	1	no	—
114	C₂×Q₁₆	—	106	1	no	—
115	C₂×Q₁₆	—	106	2	no	—
116	C₂×M₁₆	—	53	1	no	—
117	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
118	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
119	C₂²×C₈	—	53	1	no	—
120	#117	—	53	1	no	—
121	#108	—	106	2	no	—
122	#103	—	106	2	no	—
123	#103	—	106	2	no	—
124	#101	—	2	2	no	—
125	#101	—	106	2	no	—
126	#97	—	106	1	no	—
127	C₄×C₈	—	106	1	no	—
128	C₂×Q₈	—	2	1	no	—
129	C₂×Q₈	—	2	1	no	—
130	C₂×Q₈	—	2	1	no	—
131	C₂×Q₈	—	2	1	no	—
132	C₂×Q₈	—	2	1	no	—
133	C₂×Q₈	—	53	2	no	—
134	C₂×Q₈	—	106	1	no	—
135	C₂×Q₈	—	106	1	no	—
136	C₂×Q₈	—	106	1	no	—
137	C₂×Q₈	—	106	1	no	—
138	C₂×Q₈	—	106	1	no	—
139	C₂×Q₈	C₄×D₅₃	1	2	yes	—
140	C₂²×C₄	—	2	1	no	—
141	C₂²×C₄	—	106	1	no	—
142	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
143	Q₁₆	—	106	4	no	—
144	Q₁₆	—	106	4	no	—
145	M₁₆	—	2	1	no	—
146	M₁₆	—	2	1	no	—
147	M₁₆	—	2	1	no	—
148	M₁₆	—	106	1	no	—
149	M₁₆	—	106	1	no	—
150	M₁₆	C₂²⋊D₅₃	1	2	yes	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	53	2	no	—
154	C₂×C₈	—	53	1	no	—
155	C₂×C₈	—	53	1	no	—
156	C₂×C₈	—	53	2	no	—
157	C₂×C₈	—	106	1	no	—
158	C₄⋊C₄	—	53	2	no	—
159	C₂²⋊C₄	—	106	1	no	—
160	C₄²	—	106	1	no	—
161	C₂³	#11719945	1	1	yes	characteristic
162	Q₈	—	2	2	no	—
163	Q₈	—	2	2	no	—
164	Q₈	—	4	1	no	—
165	Q₈	—	4	1	no	—
166	Q₈	—	106	2	no	—
167	Q₈	—	106	2	no	—
168	Q₈	—	212	1	no	—
169	Q₈	—	212	1	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	106	2	no	—
174	C₂×C₄	—	106	2	no	—
175	C₂×C₄	—	212	1	no	—
176	C₂×C₄	—	212	1	no	—
177	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
178	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
179	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
180	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
181	C₂×C₄	#11719947	1	1	yes	characteristic
182	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	—	4	1	no	—
185	C₈	—	53	2	no	—
186	C₈	—	53	2	no	—
187	C₈	—	212	1	no	—
188	C₂²	—	2	2	no	—
189	C₂²	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
190	C₂²	#13133190	1	1	yes	characteristic
191	C₂²	#13133203	1	1	yes	characteristic
192	C₄	—	4	2	no	—
193	C₄	—	212	2	no	—
194	C₄	—	212	2	no	—
195	C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
196	C₄	#13133177	1	1	yes	characteristic
197	C₄	#13133189	1	1	yes	characteristic
198	C₄	#13133213	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#16810523	1	1	yes	characteristic
202	C₂	#16810897	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11719925

C₂

yes

maximal, characteristic

C₂×#13133152

C₂

yes

maximal, characteristic

#16810633

C₂

yes

maximal, characteristic

#16810632

C₂

yes

maximal, characteristic

C₂×#13133135

C₂

yes

maximal, characteristic

#16810581

C₂

yes

maximal, characteristic

C₂×#101×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

—

—

—

C₄

yes

complemented

C₂×#11719917

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

#13133176

C₂²

yes

—

—

—

—

—

C₈×Dic₅₃

—

C₂²×Q₈×C₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

#101×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂²⋊Dic₅₃

—

#11719925

—

#11719925

—

C₄⋊Dic₅₃

D₄

yes

—

C₄×Dic₅₃

—

—

—

—

C₂×C₄

yes

—

C₂×#88322

D₄

yes

characteristic

C₂×#88322

D₄

yes

characteristic

C₂×#88322

D₄

yes

—

C₂³×C₅₃

#41

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

C₂²⋊C₄

yes

—

#88322

C₂²⋊C₄

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄×D₄

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

C₄×D₄

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

#1979

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

100

C₂×C₅₃

#390

yes

characteristic

101

C₂×C₅₃

#473

yes

—

102

C₂²×Q₁₆

—

103

C₂×#108

—

104

#419

—

105

#419

—

106

C₂×#101

—

107

C₂×#101

D₅₃

yes

characteristic

108

#405

—

109

C₅₃

#1979

yes

characteristic, complemented, Sylow

110

C₂²×Q₈

—

111

C₂²×Q₈

C₂×D₅₃

yes

characteristic

112

C₂×Q₁₆

—

113

C₂×Q₁₆

—

106

—

114

C₂×Q₁₆

—

106

—

115

C₂×Q₁₆

—

106

—

116

C₂×M₁₆

—

117

C₂×M₁₆

C₂×D₅₃

yes

characteristic

118

C₂×M₁₆

C₂×D₅₃

yes

characteristic

119

C₂²×C₈

—

120

#117

—

121

#108

—

106

—

122

#103

—

106

—

123

#103

—

106

—

124

#101

—

125

#101

—

106

—

126

#97

—

106

—

127

C₄×C₈

—

106

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

106

—

135

C₂×Q₈

—

106

—

136

C₂×Q₈

—

106

—

137

C₂×Q₈

—

106

—

138

C₂×Q₈

—

106

—

139

C₂×Q₈

C₄×D₅₃

yes

—

140

C₂²×C₄

—

141

C₂²×C₄

—

106

—

142

C₂²×C₄

C₂²×D₅₃

yes

characteristic

143

Q₁₆

—

106

—

144

Q₁₆

—

106

—

145

M₁₆

—

146

M₁₆

—

147

M₁₆

—

148

M₁₆

—

106

—

149

M₁₆

—

106

—

150

M₁₆

C₂²⋊D₅₃

yes

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

106

—

158

C₄⋊C₄

—

159

C₂²⋊C₄

—

106

—

160

C₄²

—

106

—

161

C₂³

#11719945

yes

characteristic

162

Q₈

—

163

Q₈

—

164

Q₈

—

165

Q₈

—

166

Q₈

—

106

—

167

Q₈

—

106

—

168

Q₈

—

212

—

169

Q₈

—

212

—

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

C₂×C₄×D₅₃

yes

characteristic

178

C₂×C₄

D₄×D₅₃

yes

characteristic

179

C₂×C₄

D₄×D₅₃

yes

characteristic

180

C₂×C₄

D₄×D₅₃

yes

characteristic

181

C₂×C₄

#11719947

yes

characteristic

182

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

183

C₈

—

184

C₈

—

185

C₈

—

186

C₈

—

187

C₈

—

212

—

188

C₂²

—

189

C₂²

C₄×C₂²⋊D₅₃

yes

characteristic

190

C₂²

#13133190

yes

characteristic

191

C₂²

#13133203

yes

characteristic

192

C₄

—

193

C₄

—

212

—

194

C₄

—

212

—

195

C₄

C₂²⋊C₄×D₅₃

yes

characteristic

196

C₄

#13133177

yes

characteristic

197

C₄

#13133189

yes

characteristic

198

C₄

#13133213

yes

characteristic

199

C₂

—

200

C₂

—

201

C₂

#16810523

yes

characteristic

202

C₂

#16810897

yes

—

	subgroup	G/H
center	C₂²	#13133190
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#101×C₅₃	C₂
socle	C₂²×C₅₃	#121
hypercenter	C₂×#101	D₅₃
hypocenter	C₅₃	#1979
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂²

#13133190

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×#101×C₅₃

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
12. C₂×#11719925	C₄	semidirect
109. C₅₃	#1979	semidirect

H ≅ G/N

type

12. C₂×#11719925

C₄

semidirect

109. C₅₃

#1979

semidirect

Subgroups — #20275627

Conjugacy classes of subgroups

Complements