Group #27304207 — subgroups

Subgroups — #27304207

Order 13568.

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

normal subgroups only

This group has 318 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	C₂×#16810445	C₂	1	2	yes	maximal
2	C₂×#16810439	C₂	1	2	yes	maximal
3	#20274890	C₂	1	1	yes	maximal, characteristic
4	#20272799	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1971	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	2	yes	—
7	C₂×#13133153	—	2	1	no	—
8	C₂×#13133153	C₂²	1	2	yes	—
9	#16810647	—	2	1	no	—
10	#16810445	—	2	4	no	—
11	#16810439	—	2	4	no	—
12	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#103×C₅₃	—	2	1	no	—
15	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#391	—	2	1	no	—
17	C₂×#11719917	—	2	2	no	—
18	#13133157	—	2	2	no	—
19	#13133157	D₄	1	4	yes	—
20	C₂×#11719908	—	2	2	no	—
21	#13133153	—	2	2	no	—
22	#13133153	—	2	2	no	—
23	#13133153	—	4	2	no	—
24	C₄⋊#88322	—	4	2	no	—
25	C₂²×Q₈×C₅₃	—	2	1	no	—
26	C₂×Q₁₆×C₅₃	—	2	1	no	—
27	C₂×Q₁₆×C₅₃	—	2	2	no	—
28	C₂×Q₁₆×C₅₃	—	2	2	no	—
29	C₂×Q₁₆×C₅₃	Q₈	1	2	yes	—
30	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
34	#106×C₅₃	—	2	2	no	—
35	#103×C₅₃	—	4	2	no	—
36	#97×C₅₃	—	2	2	no	—
37	#95×C₅₃	—	4	2	no	—
38	C₂×Q₈×C₅₃	—	2	2	no	—
39	C₂×Q₈×C₅₃	—	4	1	no	—
40	C₂×Q₈×C₅₃	—	4	1	no	—
41	C₂×Q₈×C₅₃	—	4	1	no	—
42	C₂×Q₈×C₅₃	—	4	1	no	—
43	C₂×Q₈×C₅₃	—	4	1	no	—
44	C₂²×C₄×C₅₃	—	4	1	no	—
45	C₂²×C₄×C₅₃	—	4	1	no	—
46	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	Q₁₆×C₅₃	—	2	4	no	—
49	Q₁₆×C₅₃	—	4	2	no	—
50	C₂×C₈×C₅₃	—	2	1	no	—
51	C₂×C₈×C₅₃	—	2	1	no	—
52	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
53	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
54	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₅₃	#41	1	2	yes	—
56	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
57	C₄⋊C₄×C₅₃	—	4	1	no	—
58	C₄⋊C₄×C₅₃	—	4	1	no	—
59	C₄²×C₅₃	—	2	1	no	—
60	C₄²×C₅₃	—	2	1	no	—
61	#11719917	—	4	2	no	—
62	#11719917	—	4	2	no	—
63	C₂×#88322	—	4	2	no	—
64	#11719908	—	2	4	no	—
65	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
66	Q₈×C₅₃	—	4	2	no	—
67	Q₈×C₅₃	—	4	2	no	—
68	Q₈×C₅₃	—	8	1	no	—
69	Q₈×C₅₃	—	8	1	no	—
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	—	4	2	no	—
72	C₂×C₄×C₅₃	—	4	2	no	—
73	C₂×C₄×C₅₃	—	8	1	no	—
74	C₂×C₄×C₅₃	—	8	1	no	—
75	C₂×C₄×C₅₃	—	8	2	no	—
76	C₂×C₄×C₅₃	—	8	1	no	—
77	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
83	C₈×C₅₃	—	2	2	no	—
84	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
85	C₈×C₅₃	#123	1	2	yes	—
86	#88322	—	8	2	no	—
87	#13536	—	53	1	no	maximal, Sylow
88	C₂²×C₅₃	—	2	2	no	—
89	C₂²×C₅₃	#476	1	1	yes	characteristic
90	C₂²×C₅₃	#481	1	1	yes	characteristic
91	C₂²×C₅₃	#492	1	1	yes	characteristic
92	C₄×C₅₃	—	4	2	no	—
93	C₄×C₅₃	—	8	2	no	—
94	C₄×C₅₃	—	16	1	no	—
95	C₄×C₅₃	#399	1	1	yes	characteristic
96	C₄×C₅₃	#476	1	1	yes	characteristic
97	C₄×C₅₃	#481	1	1	yes	characteristic
98	C₄×C₅₃	#496	1	1	yes	characteristic
99	C₂×#365	—	53	2	no	—
100	C₂×#362	—	53	2	no	—
101	#1971	D₅₃	1	1	yes	characteristic
102	#1849	—	53	1	no	—
103	#1377	—	53	1	no	—
104	C₂×C₅₃	—	2	1	no	—
105	C₂×C₅₃	—	2	1	no	—
106	C₂×C₅₃	#2094	1	1	yes	characteristic
107	C₂×C₅₃	#2239	1	2	yes	—
108	C₂²×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
109	C₂×M₃₂	—	53	2	no	—
110	C₂×#108	—	53	2	no	—
111	C₂×#108	—	106	1	no	—
112	C₂×#106	C₂×D₅₃	1	1	yes	characteristic
113	#423	—	106	1	no	—
114	C₂×#103	—	2	1	no	—
115	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
116	#391	—	2	1	no	—
117	#365	—	106	4	no	—
118	#362	—	106	4	no	—
119	C₅₃	#13536	1	1	yes	characteristic, complemented, Sylow
120	C₂²×Q₈	—	2	1	no	—
121	C₂×Q₁₆	—	2	1	no	—
122	C₂×Q₁₆	—	2	2	no	—
123	C₂×Q₁₆	—	2	2	no	—
124	C₂×Q₁₆	Dic₁₀₆	1	2	yes	—
125	C₂×M₁₆	—	106	2	no	—
126	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
128	C₂×C₄⋊C₄	—	2	1	no	—
129	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
130	M₃₂	—	53	4	no	—
131	M₃₂	—	106	2	no	—
132	C₂×C₁₆	—	106	2	no	—
133	#108	—	106	2	no	—
134	#108	—	106	2	no	—
135	#108	—	212	2	no	—
136	#106	—	2	2	no	—
137	C₄⋊C₈	—	212	2	no	—
138	#103	—	4	2	no	—
139	#97	—	2	2	no	—
140	#95	—	4	2	no	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	—	4	1	no	—
143	C₂×Q₈	—	4	1	no	—
144	C₂×Q₈	—	4	1	no	—
145	C₂×Q₈	—	4	1	no	—
146	C₂×Q₈	—	4	1	no	—
147	C₂²×C₄	—	4	1	no	—
148	C₂²×C₄	—	4	1	no	—
149	C₂²×C₄	#11719923	1	2	yes	—
150	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
151	Q₁₆	—	2	4	no	—
152	Q₁₆	—	4	2	no	—
153	M₁₆	—	212	2	no	—
154	M₁₆	—	212	2	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	212	2	no	—
158	C₂×C₈	#11719923	1	2	yes	—
159	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
160	C₂×C₈	#11719945	1	1	yes	characteristic
161	C₂×C₈	D₄×D₅₃	1	2	yes	—
162	C₂×C₈	#11719947	1	2	yes	—
163	C₄⋊C₄	—	4	1	no	—
164	C₄⋊C₄	—	4	1	no	—
165	C₄²	—	2	1	no	—
166	C₄²	—	2	1	no	—
167	C₁₆	—	106	4	no	—
168	C₂³	C₂×#11719923	1	1	yes	characteristic
169	Q₈	—	4	2	no	—
170	Q₈	—	4	2	no	—
171	Q₈	—	8	1	no	—
172	Q₈	—	8	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	8	1	no	—
177	C₂×C₄	—	8	1	no	—
178	C₂×C₄	—	8	2	no	—
179	C₂×C₄	—	8	1	no	—
180	C₂×C₄	#13133196	1	1	yes	characteristic
181	C₂×C₄	#13133198	1	1	yes	characteristic
182	C₂×C₄	#13133213	1	1	yes	characteristic
183	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
184	C₂×C₄	#13133242	1	1	yes	characteristic
185	C₂×C₄	#13133242	1	1	yes	characteristic
186	C₈	—	2	2	no	—
187	C₈	—	424	2	no	—
188	C₈	#13133194	1	2	yes	—
189	C₈	#13133199	1	2	yes	—
190	C₂²	—	2	2	no	—
191	C₂²	#16810913	1	1	yes	characteristic
192	C₂²	#16810939	1	1	yes	characteristic
193	C₂²	#16810995	1	1	yes	characteristic
194	C₄	—	4	2	no	—
195	C₄	—	8	2	no	—
196	C₄	—	16	1	no	—
197	C₄	#16810559	1	1	yes	characteristic
198	C₄	#16810913	1	1	yes	characteristic
199	C₄	#16810939	1	1	yes	characteristic
200	C₄	#16811013	1	1	yes	characteristic
201	C₂	—	2	1	no	—
202	C₂	—	2	1	no	—
203	C₂	#20276248	1	1	yes	characteristic
204	C₂	#20276932	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810445

C₂

yes

maximal

C₂×#16810439

C₂

yes

maximal

#20274890

C₂

yes

maximal, characteristic

#20272799

C₂

yes

maximal, characteristic

C₅₃×#1971

C₂

yes

maximal, characteristic

C₂×#13133157

C₂²

yes

—

C₂×#13133153

—

C₂×#13133153

C₂²

yes

—

#16810647

—

#16810445

—

#16810439

—

C₂²×Q₁₆×C₅₃

C₂²

yes

characteristic

C₂×#106×C₅₃

C₂²

yes

characteristic

C₂×#103×C₅₃

—

C₂×#97×C₅₃

C₂²

yes

characteristic

C₅₃×#391

—

C₂×#11719917

—

#13133157

—

#13133157

D₄

yes

—

—

—

—

—

—

C₂²×Q₈×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

Q₈

yes

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

D₄

yes

characteristic

#106×C₅₃

—

#103×C₅₃

—

#97×C₅₃

—

#95×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

SD₁₆

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₂×C₈×C₅₃

#41

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

—

—

—

—

C₂³×C₅₃

C₂×SD₁₆

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×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

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

C₂×SD₁₆

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₈×C₅₃

—

C₈×C₅₃

C₂²⋊Q₈

yes

—

C₈×C₅₃

#123

yes

—

#88322

—

#13536

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

100

C₂×#362

—

101

#1971

D₅₃

yes

characteristic

102

#1849

—

103

#1377

—

104

C₂×C₅₃

—

105

C₂×C₅₃

—

106

C₂×C₅₃

#2094

yes

characteristic

107

C₂×C₅₃

#2239

yes

—

108

C₂²×Q₁₆

C₂×D₅₃

yes

characteristic

109

C₂×M₃₂

—

110

C₂×#108

—

111

C₂×#108

—

106

—

112

C₂×#106

C₂×D₅₃

yes

characteristic

113

#423

—

106

—

114

C₂×#103

—

115

C₂×#97

C₂×D₅₃

yes

characteristic

116

#391

—

117

#365

—

106

—

118

#362

—

106

—

119

C₅₃

#13536

yes

characteristic, complemented, Sylow

120

C₂²×Q₈

—

121

C₂×Q₁₆

—

122

C₂×Q₁₆

—

123

C₂×Q₁₆

—

124

C₂×Q₁₆

Dic₁₀₆

yes

—

125

C₂×M₁₆

—

106

—

126

C₂²×C₈

C₂²⋊D₅₃

yes

characteristic

127

C₂²×C₈

C₂²×D₅₃

yes

characteristic

128

C₂×C₄⋊C₄

—

129

C₂×C₄²

C₂²⋊D₅₃

yes

characteristic

130

M₃₂

—

131

M₃₂

—

106

—

132

C₂×C₁₆

—

106

—

133

#108

—

106

—

134

#108

—

106

—

135

#108

—

212

—

136

#106

—

137

C₄⋊C₈

—

212

—

138

#103

—

139

#97

—

140

#95

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂×Q₈

—

146

C₂×Q₈

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

#11719923

yes

—

150

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

151

Q₁₆

—

152

Q₁₆

—

153

M₁₆

—

212

—

154

M₁₆

—

212

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

212

—

158

C₂×C₈

#11719923

yes

—

159

C₂×C₈

C₂×Dic₁₀₆

yes

characteristic

160

C₂×C₈

#11719945

yes

characteristic

161

C₂×C₈

D₄×D₅₃

yes

—

162

C₂×C₈

#11719947

yes

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄²

—

166

C₄²

—

167

C₁₆

—

106

—

168

C₂³

C₂×#11719923

yes

characteristic

169

Q₈

—

170

Q₈

—

171

Q₈

—

172

Q₈

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

#13133196

yes

characteristic

181

C₂×C₄

#13133198

yes

characteristic

182

C₂×C₄

#13133213

yes

characteristic

183

C₂×C₄

C₂×#11719923

yes

characteristic

184

C₂×C₄

#13133242

yes

characteristic

185

C₂×C₄

#13133242

yes

characteristic

186

C₈

—

187

C₈

—

424

—

188

C₈

#13133194

yes

—

189

C₈

#13133199

yes

—

190

C₂²

—

191

C₂²

#16810913

yes

characteristic

192

C₂²

#16810939

yes

characteristic

193

C₂²

#16810995

yes

characteristic

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

#16810559

yes

characteristic

198

C₄

#16810913

yes

characteristic

199

C₄

#16810939

yes

characteristic

200

C₄

#16811013

yes

characteristic

201

C₂

—

202

C₂

—

203

C₂

#20276248

yes

characteristic

204

C₂

#20276932

yes

—

	subgroup	G/H
center	C₂²	#16810939
commutator, cosocle	C₂²×C₈×C₅₃	C₂³
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1971	C₂
socle	C₂²×C₅₃	#481
prime radical	C₂³×C₅₃	C₂×SD₁₆
hypercenter	#1971	D₅₃
hypocenter	C₅₃	#13536
LCS	G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1971

subgroup

G/H

center

C₂²

#16810939

commutator, cosocle

C₂²×C₈×C₅₃

C₂³

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#1971

C₂

socle

C₂²×C₅₃

#481

prime radical

C₂³×C₅₃

C₂×SD₁₆

D₅₃

C₅₃

G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1971

N	H ≅ G/N	type
119. C₅₃	#13536	semidirect

H ≅ G/N

type

119. C₅₃

#13536

semidirect

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

subgroup	index
C₂×#16810445	2
C₂×#16810439	2
#20274890	2
#20272799	2
C₅₃×#1971	2
#13536	53

subgroup

index

C₅₃×#1971

Subgroups — #27304207

Conjugacy classes of subgroups

Complements

Maximal subgroups