Group #27289070 — subgroups

Subgroups — #27289070

Order 13568.

372 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 99 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	#20274890	C₂	1	1	yes	maximal, characteristic, complemented
2	#20273489	C₂	1	2	yes	maximal, complemented
3	#20273140	C₂	1	2	yes	maximal
4	#20272546	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1955	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	2	yes	—
7	C₂×#13133153	—	2	2	no	—
8	#16810647	C₂²	1	2	yes	—
9	#16810375	—	2	4	no	—
10	D₄⋊#88322	—	2	4	no	—
11	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊C₈×C₅₃	—	2	2	no	—
14	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#11719917	—	2	2	no	—
16	#13133153	—	4	4	no	—
17	C₄⋊#88322	—	2	2	no	—
18	C₄⋊#88322	—	2	2	no	—
19	C₄⋊#88322	—	2	2	no	—
20	C₂²⋊#88322	—	2	2	no	—
21	C₄×#88322	D₄	1	4	yes	—
22	C₂²×D₄×C₅₃	Q₈	1	1	yes	characteristic
23	C₂³×C₄×C₅₃	—	2	1	no	—
24	C₂²×C₈×C₅₃	—	2	1	no	—
25	C₂²×C₈×C₅₃	D₄	1	2	yes	—
26	C₄×D₄×C₅₃	—	2	2	no	—
27	C₄×D₄×C₅₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₅₃	Q₈	1	1	yes	characteristic
29	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
30	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₅₃	—	2	2	no	—
32	C₂²⋊C₈×C₅₃	—	4	4	no	—
33	#97×C₅₃	—	2	2	no	—
34	C₂⁴×C₅₃	—	2	1	no	—
35	C₂×D₄×C₅₃	—	2	2	no	—
36	C₂×D₄×C₅₃	—	4	2	no	—
37	C₂²×C₄×C₅₃	—	2	1	no	—
38	C₂²×C₄×C₅₃	—	4	1	no	—
39	C₂²×C₄×C₅₃	—	4	1	no	—
40	C₂²×C₄×C₅₃	—	4	2	no	—
41	C₂²×C₄×C₅₃	—	4	2	no	—
42	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
45	C₂×C₈×C₅₃	—	2	2	no	—
46	C₂×C₈×C₅₃	—	2	2	no	—
47	C₂×C₈×C₅₃	—	4	1	no	—
48	C₂×C₈×C₅₃	—	4	2	no	—
49	C₂×C₈×C₅₃	D₈	1	4	yes	—
50	C₄⋊C₄×C₅₃	—	2	2	no	—
51	C₂²⋊C₄×C₅₃	—	4	2	no	—
52	C₄²×C₅₃	C₂×D₄	1	2	yes	—
53	C₄²×C₅₃	#41	1	2	yes	—
54	#11719917	—	4	4	no	—
55	C₂×#88322	—	2	4	no	—
56	C₂×#88322	—	4	2	no	—
57	C₂³×C₅₃	—	2	2	no	—
58	C₂³×C₅₃	—	4	1	no	—
59	C₂³×C₅₃	—	4	2	no	—
60	C₂³×C₅₃	—	4	1	no	—
61	C₂³×C₅₃	—	4	2	no	—
62	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
63	D₄×C₅₃	—	2	4	no	—
64	D₄×C₅₃	—	4	2	no	—
65	C₂×C₄×C₅₃	—	4	1	no	—
66	C₂×C₄×C₅₃	—	4	1	no	—
67	C₂×C₄×C₅₃	—	4	1	no	—
68	C₂×C₄×C₅₃	—	4	2	no	—
69	C₂×C₄×C₅₃	—	8	2	no	—
70	C₂×C₄×C₅₃	—	8	1	no	—
71	C₂×C₄×C₅₃	—	8	1	no	—
72	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
73	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
74	C₂×C₄×C₅₃	#123	1	2	yes	—
75	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
77	C₂×C₄×C₅₃	#137	1	2	yes	—
78	C₈×C₅₃	—	2	4	no	—
79	C₈×C₅₃	—	8	1	no	—
80	#88322	—	4	4	no	—
81	#88322	—	8	2	no	—
82	#10660	—	53	1	no	maximal, Sylow
83	C₂²×C₅₃	—	2	1	no	—
84	C₂²×C₅₃	—	2	1	no	—
85	C₂²×C₅₃	—	4	4	no	—
86	C₂²×C₅₃	—	4	2	no	—
87	C₂²×C₅₃	—	8	2	no	—
88	C₂²×C₅₃	—	8	1	no	—
89	C₂²×C₅₃	—	8	1	no	—
90	C₂²×C₅₃	#399	1	1	yes	characteristic
91	C₂²×C₅₃	#474	1	2	yes	—
92	C₄×C₅₃	—	2	2	no	—
93	C₄×C₅₃	—	2	2	no	—
94	C₄×C₅₃	—	8	1	no	—
95	C₄×C₅₃	#480	1	2	yes	—
96	C₄×C₅₃	#493	1	1	yes	characteristic
97	C₄×C₅₃	#497	1	1	yes	characteristic
98	#1955	D₅₃	1	1	yes	characteristic
99	#1849	—	53	1	no	—
100	#1571	—	53	2	no	—
101	#1486	—	53	2	no	—
102	#1275	—	53	1	no	—
103	C₂×C₅₃	—	2	2	no	—
104	C₂×C₅₃	—	8	2	no	—
105	C₂×C₅₃	#2091	1	1	yes	characteristic
106	C₂×C₅₃	#2093	1	1	yes	characteristic
107	C₂×C₅₃	#2098	1	1	yes	characteristic
108	C₂×C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
109	#431	—	53	2	no	—
110	C₂×#108	—	106	2	no	—
111	#423	—	53	2	no	—
112	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
113	C₂×C₂²⋊C₈	—	2	2	no	—
114	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
115	#335	—	106	4	no	—
116	D₄⋊C₈	—	106	4	no	—
117	C₅₃	#10660	1	1	yes	characteristic, complemented, Sylow
118	C₂²×D₄	Dic₁₀₆	1	1	yes	characteristic
119	C₂³×C₄	—	2	1	no	—
120	C₂×M₁₆	—	106	2	no	—
121	C₂²×C₈	—	2	1	no	—
122	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
123	C₄×D₄	—	2	2	no	—
124	C₄×D₄	—	2	2	no	—
125	C₂×C₄⋊C₄	Dic₁₀₆	1	1	yes	characteristic
126	C₂×C₂²⋊C₄	—	2	1	no	—
127	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
128	#108	—	212	4	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	106	2	no	—
131	C₄⋊C₈	—	106	2	no	—
132	C₄⋊C₈	—	106	2	no	—
133	C₂²⋊C₈	—	4	4	no	—
134	C₂²⋊C₈	—	106	2	no	—
135	#97	—	2	2	no	—
136	C₄×C₈	—	53	4	no	—
137	C₂⁴	—	2	1	no	—
138	C₂×D₄	—	2	2	no	—
139	C₂×D₄	—	4	2	no	—
140	C₂²×C₄	—	2	1	no	—
141	C₂²×C₄	—	4	1	no	—
142	C₂²×C₄	—	4	1	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	—	4	2	no	—
145	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
146	C₂²×C₄	#11719945	1	1	yes	characteristic
147	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
148	M₁₆	—	212	4	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	4	1	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	106	4	no	—
154	C₂×C₈	—	212	2	no	—
155	C₂×C₈	#11719922	1	4	yes	—
156	C₄⋊C₄	—	2	2	no	—
157	C₂²⋊C₄	—	4	2	no	—
158	C₄²	D₄×D₅₃	1	2	yes	—
159	C₄²	#11719947	1	2	yes	—
160	C₂³	—	2	2	no	—
161	C₂³	—	4	1	no	—
162	C₂³	—	4	2	no	—
163	C₂³	—	4	1	no	—
164	C₂³	—	4	2	no	—
165	C₂³	#13133196	1	1	yes	characteristic
166	D₄	—	2	4	no	—
167	D₄	—	4	2	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	8	1	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	8	1	no	—
175	C₂×C₄	#13133194	1	2	yes	—
176	C₂×C₄	#13133198	1	1	yes	characteristic
177	C₂×C₄	#13133199	1	2	yes	—
178	C₂×C₄	#13133213	1	1	yes	characteristic
179	C₂×C₄	C₂×#11719922	1	2	yes	—
180	C₂×C₄	#13133247	1	2	yes	—
181	C₈	—	2	4	no	—
182	C₈	—	8	1	no	—
183	C₈	—	424	2	no	—
184	C₈	—	212	4	no	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	1	no	—
187	C₂²	—	4	4	no	—
188	C₂²	—	4	2	no	—
189	C₂²	—	8	2	no	—
190	C₂²	—	8	1	no	—
191	C₂²	—	8	1	no	—
192	C₂²	#16810559	1	1	yes	characteristic
193	C₂²	#16810903	1	2	yes	—
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	—	8	1	no	—
197	C₄	#16810934	1	2	yes	—
198	C₄	#16810999	1	1	yes	characteristic
199	C₄	#16811018	1	1	yes	characteristic
200	C₂	—	2	2	no	—
201	C₂	—	8	2	no	—
202	C₂	#20276231	1	1	yes	characteristic
203	C₂	#20276243	1	1	yes	characteristic
204	C₂	#20276268	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274890

C₂

yes

maximal, characteristic, complemented

#20273489

C₂

yes

maximal, complemented

#20273140

C₂

yes

maximal

#20272546

C₂

yes

maximal, characteristic, complemented

C₅₃×#1955

C₂

yes

maximal, characteristic

#16810676

C₂²

yes

—

C₂×#13133153

—

#16810647

C₂²

yes

—

#16810375

—

D₄⋊#88322

—

C₂×C₄×D₄×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₂×C₂²⋊C₈×C₅₃

—

C₂×#97×C₅₃

C₂²

yes

—

—

—

—

—

—

D₄

yes

—

C₂²×D₄×C₅₃

Q₈

yes

characteristic

C₂³×C₄×C₅₃

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₄×D₄×C₅₃

—

C₄×D₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

Q₈

yes

characteristic

C₂×C₂²⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

#97×C₅₃

—

C₂⁴×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₈

yes

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

#41

yes

—

#11719917

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²⋊Q₈

yes

characteristic

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

#123

yes

characteristic

C₂×C₄×C₅₃

#123

yes

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#399

yes

characteristic

C₂²×C₅₃

#474

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#480

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

D₅₃

yes

characteristic

#1849

—

100

#1571

—

101

#1486

—

102

#1275

—

103

C₂×C₅₃

—

104

C₂×C₅₃

—

105

C₂×C₅₃

#2091

yes

characteristic

106

C₂×C₅₃

#2093

yes

characteristic

107

C₂×C₅₃

#2098

yes

characteristic

108

C₂×C₄×D₄

C₂×D₅₃

yes

characteristic

109

#431

—

110

C₂×#108

—

106

—

111

#423

—

112

C₂×C₄⋊C₈

C₂×D₅₃

yes

characteristic

113

C₂×C₂²⋊C₈

—

114

C₂×#97

C₂×D₅₃

yes

characteristic

115

#335

—

106

—

116

D₄⋊C₈

—

106

—

117

C₅₃

#10660

yes

characteristic, complemented, Sylow

118

C₂²×D₄

Dic₁₀₆

yes

characteristic

119

C₂³×C₄

—

120

C₂×M₁₆

—

106

—

121

C₂²×C₈

—

122

C₂²×C₈

C₂²⋊D₅₃

yes

—

123

C₄×D₄

—

124

C₄×D₄

—

125

C₂×C₄⋊C₄

Dic₁₀₆

yes

characteristic

126

C₂×C₂²⋊C₄

—

127

C₂×C₄²

C₂²×D₅₃

yes

characteristic

128

#108

—

212

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

106

—

131

C₄⋊C₈

—

106

—

132

C₄⋊C₈

—

106

—

133

C₂²⋊C₈

—

134

C₂²⋊C₈

—

106

—

135

#97

—

136

C₄×C₈

—

137

C₂⁴

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

C₂×Dic₁₀₆

yes

characteristic

146

C₂²×C₄

#11719945

yes

characteristic

147

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

148

M₁₆

—

212

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

106

—

154

C₂×C₈

—

212

—

155

C₂×C₈

#11719922

yes

—

156

C₄⋊C₄

—

157

C₂²⋊C₄

—

158

C₄²

D₄×D₅₃

yes

—

159

C₄²

#11719947

yes

—

160

C₂³

—

161

C₂³

—

162

C₂³

—

163

C₂³

—

164

C₂³

—

165

C₂³

#13133196

yes

characteristic

166

D₄

—

167

D₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

#13133194

yes

—

176

C₂×C₄

#13133198

yes

characteristic

177

C₂×C₄

#13133199

yes

—

178

C₂×C₄

#13133213

yes

characteristic

179

C₂×C₄

C₂×#11719922

yes

—

180

C₂×C₄

#13133247

yes

—

181

C₈

—

182

C₈

—

183

C₈

—

424

—

184

C₈

—

212

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

192

C₂²

#16810559

yes

characteristic

193

C₂²

#16810903

yes

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

#16810934

yes

—

198

C₄

#16810999

yes

characteristic

199

C₄

#16811018

yes

characteristic

200

C₂

—

201

C₂

—

202

C₂

#20276231

yes

characteristic

203

C₂

#20276243

yes

characteristic

204

C₂

#20276268

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133213
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1955	C₂
socle	C₂²×C₅₃	#399
prime radical	C₂²×D₄×C₅₃	Q₈
hypercenter	#1955	D₅₃
hypocenter	C₅₃	#10660
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1955

subgroup

G/H

center

C₂×C₄

#13133213

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1955

C₂

socle

C₂²×C₅₃

#399

prime radical

C₂²×D₄×C₅₃

Q₈

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1955

N	H ≅ G/N	type
1. #20274890	C₂	semidirect
2. #20273489	C₂	semidirect
4. #20272546	C₂	semidirect
117. C₅₃	#10660	semidirect

H ≅ G/N

type

1. #20274890

C₂

semidirect

2. #20273489

C₂

semidirect

4. #20272546

C₂

semidirect

117. C₅₃

#10660

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20274890	2
#20273489	2
#20273140	2
#20272546	2
C₅₃×#1955	2
#10660	53

subgroup

index

C₅₃×#1955

Subgroups — #27289070

Conjugacy classes of subgroups

Complements

Maximal subgroups