Group #27304208 — subgroups

Subgroups — #27304208

Order 13568.

318 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 97 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×#16810447	C₂	1	2	yes	maximal
2	C₂×#16810438	C₂	1	2	yes	maximal
3	#20275586	C₂	1	1	yes	maximal, characteristic
4	#20272799	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1849	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₂	C₂²	1	1	yes	characteristic
7	C₂×#13133157	C₂²	1	2	yes	—
8	C₂×#13133149	C₂²	1	1	yes	characteristic
9	C₂×#13133142	—	2	1	no	—
10	#16810529	—	2	1	no	—
11	#16810447	—	2	4	no	—
12	#16810438	—	2	4	no	—
13	C₂×#108×C₅₃	—	2	1	no	—
14	C₂×#108×C₅₃	C₂²	1	2	yes	—
15	C₅₃×#423	—	2	1	no	—
16	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₀₆	—	2	1	no	—
18	C₂×Dic₂₁₂	—	2	1	no	—
19	C₂×Dic₂₁₂	—	2	2	no	—
20	C₂×Dic₂₁₂	—	2	2	no	—
21	C₂×Dic₂₁₂	Q₈	1	2	yes	—
22	C₂×C₄⋊Dic₅₃	—	2	1	no	—
23	#13133157	—	2	2	no	—
24	#13133157	D₄	1	4	yes	—
25	C₂×#11719908	—	2	2	no	—
26	#13133149	—	2	2	no	—
27	#13133142	—	4	2	no	—
28	#13133124	—	4	2	no	—
29	C₂×M₁₆×C₅₃	—	2	2	no	—
30	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
33	#108×C₅₃	—	2	2	no	—
34	#108×C₅₃	—	2	2	no	—
35	#108×C₅₃	—	4	2	no	—
36	C₄⋊C₈×C₅₃	—	4	2	no	—
37	#97×C₅₃	—	2	2	no	—
38	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×Dic₅₃	—	4	1	no	—
41	C₂²×Dic₅₃	—	4	1	no	—
42	C₂×Dic₁₀₆	—	2	2	no	—
43	C₂×Dic₁₀₆	—	4	1	no	—
44	C₂×Dic₁₀₆	—	4	1	no	—
45	C₂×Dic₁₀₆	—	4	1	no	—
46	C₂×Dic₁₀₆	—	4	1	no	—
47	C₂×Dic₁₀₆	—	4	1	no	—
48	M₁₆×C₅₃	—	4	2	no	—
49	M₁₆×C₅₃	—	4	2	no	—
50	C₂×C₈×C₅₃	—	2	1	no	—
51	C₂×C₈×C₅₃	—	2	1	no	—
52	C₂×C₈×C₅₃	—	4	2	no	—
53	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
54	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
55	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈×C₅₃	#41	1	2	yes	—
57	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
58	C₄²×C₅₃	—	2	1	no	—
59	C₄²×C₅₃	—	2	1	no	—
60	C₄⋊Dic₅₃	—	4	1	no	—
61	C₄⋊Dic₅₃	—	4	1	no	—
62	Dic₂₁₂	—	2	4	no	—
63	Dic₂₁₂	—	4	2	no	—
64	#11719908	—	2	4	no	—
65	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	4	2	no	—
68	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
70	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
74	C₂×Dic₅₃	—	4	2	no	—
75	C₂×Dic₅₃	—	8	1	no	—
76	C₂×Dic₅₃	—	8	1	no	—
77	C₂×Dic₅₃	—	8	2	no	—
78	C₂×Dic₅₃	—	8	1	no	—
79	Dic₁₀₆	—	4	2	no	—
80	Dic₁₀₆	—	4	2	no	—
81	Dic₁₀₆	—	8	1	no	—
82	Dic₁₀₆	—	8	1	no	—
83	C₈×C₅₃	—	2	2	no	—
84	C₈×C₅₃	—	8	2	no	—
85	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
86	C₈×C₅₃	#123	1	2	yes	—
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	#399	1	1	yes	characteristic
94	C₄×C₅₃	#476	1	1	yes	characteristic
95	C₄×C₅₃	#481	1	1	yes	characteristic
96	C₄×C₅₃	#496	1	1	yes	characteristic
97	Dic₅₃	—	8	2	no	—
98	Dic₅₃	—	16	1	no	—
99	C₂×#365	—	53	2	no	—
100	C₂×#362	—	53	2	no	—
101	#1971	—	53	1	no	—
102	#1849	D₅₃	1	1	yes	characteristic
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	—	53	1	no	—
109	C₂×M₃₂	—	53	2	no	—
110	C₂×#108	—	2	1	no	—
111	C₂×#108	C₂×D₅₃	1	2	yes	—
112	C₂×#106	—	53	1	no	—
113	#423	—	2	1	no	—
114	C₂×#103	—	106	1	no	—
115	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
116	#391	—	106	1	no	—
117	#365	—	106	4	no	—
118	#362	—	106	4	no	—
119	C₅₃	#13536	1	1	yes	characteristic, complemented, Sylow
120	C₂²×Q₈	—	106	1	no	—
121	C₂×Q₁₆	—	53	2	no	—
122	C₂×Q₁₆	—	106	2	no	—
123	C₂×Q₁₆	—	106	2	no	—
124	C₂×Q₁₆	—	106	1	no	—
125	C₂×M₁₆	—	2	2	no	—
126	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
128	C₂×C₄⋊C₄	—	106	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	—	2	2	no	—
134	#108	—	2	2	no	—
135	#108	—	4	2	no	—
136	#106	—	106	2	no	—
137	C₄⋊C₈	—	4	2	no	—
138	#103	—	212	2	no	—
139	#97	—	2	2	no	—
140	#95	—	212	2	no	—
141	C₂×Q₈	—	106	2	no	—
142	C₂×Q₈	—	212	1	no	—
143	C₂×Q₈	—	212	1	no	—
144	C₂×Q₈	—	212	1	no	—
145	C₂×Q₈	—	212	1	no	—
146	C₂×Q₈	—	212	1	no	—
147	C₂²×C₄	—	212	1	no	—
148	C₂²×C₄	—	212	1	no	—
149	C₂²×C₄	#11719924	1	2	yes	—
150	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
151	Q₁₆	—	106	4	no	—
152	Q₁₆	—	212	2	no	—
153	M₁₆	—	4	2	no	—
154	M₁₆	—	4	2	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	#11719924	1	2	yes	—
159	C₂×C₈	#11719945	1	2	yes	—
160	C₂×C₈	D₄×D₅₃	1	2	yes	—
161	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
162	C₂×C₈	#11719949	1	1	yes	characteristic
163	C₄⋊C₄	—	212	1	no	—
164	C₄⋊C₄	—	212	1	no	—
165	C₄²	—	2	1	no	—
166	C₄²	—	2	1	no	—
167	C₁₆	—	106	4	no	—
168	C₂³	C₂×#11719924	1	1	yes	characteristic
169	Q₈	—	424	1	no	—
170	Q₈	—	424	1	no	—
171	Q₈	—	212	2	no	—
172	Q₈	—	212	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	424	2	no	—
176	C₂×C₄	—	424	1	no	—
177	C₂×C₄	—	424	1	no	—
178	C₂×C₄	—	424	1	no	—
179	C₂×C₄	—	212	2	no	—
180	C₂×C₄	#13133193	1	1	yes	characteristic
181	C₂×C₄	#13133200	1	1	yes	characteristic
182	C₂×C₄	#13133213	1	1	yes	characteristic
183	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
184	C₂×C₄	#13133243	1	1	yes	characteristic
185	C₂×C₄	#13133243	1	1	yes	characteristic
186	C₈	—	2	2	no	—
187	C₈	—	8	2	no	—
188	C₈	#13133195	1	2	yes	—
189	C₈	#13133202	1	2	yes	—
190	C₂²	—	2	2	no	—
191	C₂²	#16810910	1	1	yes	characteristic
192	C₂²	#16810941	1	1	yes	characteristic
193	C₂²	#16810997	1	1	yes	characteristic
194	C₄	—	4	2	no	—
195	C₄	—	848	1	no	—
196	C₄	—	424	2	no	—
197	C₄	#16810558	1	1	yes	characteristic
198	C₄	#16810910	1	1	yes	characteristic
199	C₄	#16810941	1	1	yes	characteristic
200	C₄	#16811010	1	1	yes	characteristic
201	C₂	—	2	1	no	—
202	C₂	—	2	1	no	—
203	C₂	#20276249	1	1	yes	characteristic
204	C₂	#20276934	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810447

C₂

yes

maximal

C₂×#16810438

C₂

yes

maximal

#20275586

C₂

yes

maximal, characteristic

#20272799

C₂

yes

maximal, characteristic

C₅₃×#1849

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₂

C₂²

yes

characteristic

C₂×#13133157

C₂²

yes

—

C₂×#13133149

C₂²

yes

—

—

—

—

C₂×#108×C₅₃

—

C₂×#108×C₅₃

C₂²

yes

—

C₅₃×#423

—

C₂×#97×C₅₃

C₂²

yes

characteristic

C₂²×Dic₁₀₆

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

Q₈

yes

—

C₂×C₄⋊Dic₅₃

—

#13133157

—

#13133157

D₄

yes

—

—

—

—

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

characteristic

#108×C₅₃

—

#108×C₅₃

—

#108×C₅₃

—

C₄⋊C₈×C₅₃

—

#97×C₅₃

—

C₂²×C₄×C₅₃

SD₁₆

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×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

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

Dic₂₁₂

—

Dic₂₁₂

—

#11719908

—

C₂³×C₅₃

C₂×SD₁₆

yes

characteristic

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

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₂²⋊Q₈

yes

—

C₈×C₅₃

#123

yes

—

#13536

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

C₂×#365

—

100

C₂×#362

—

101

#1971

—

102

#1849

D₅₃

yes

characteristic

103

#1377

—

104

C₂×C₅₃

—

105

C₂×C₅₃

—

106

C₂×C₅₃

#2094

yes

characteristic

107

C₂×C₅₃

#2239

yes

—

108

C₂²×Q₁₆

—

109

C₂×M₃₂

—

110

C₂×#108

—

111

C₂×#108

C₂×D₅₃

yes

—

112

C₂×#106

—

113

#423

—

114

C₂×#103

—

106

—

115

C₂×#97

C₂×D₅₃

yes

characteristic

116

#391

—

106

—

117

#365

—

106

—

118

#362

—

106

—

119

C₅₃

#13536

yes

characteristic, complemented, Sylow

120

C₂²×Q₈

—

106

—

121

C₂×Q₁₆

—

122

C₂×Q₁₆

—

106

—

123

C₂×Q₁₆

—

106

—

124

C₂×Q₁₆

—

106

—

125

C₂×M₁₆

—

126

C₂²×C₈

C₂²⋊D₅₃

yes

characteristic

127

C₂²×C₈

C₂²×D₅₃

yes

characteristic

128

C₂×C₄⋊C₄

—

106

—

129

C₂×C₄²

C₂²⋊D₅₃

yes

characteristic

130

M₃₂

—

131

M₃₂

—

106

—

132

C₂×C₁₆

—

106

—

133

#108

—

134

#108

—

135

#108

—

136

#106

—

106

—

137

C₄⋊C₈

—

138

#103

—

212

—

139

#97

—

140

#95

—

212

—

141

C₂×Q₈

—

106

—

142

C₂×Q₈

—

212

—

143

C₂×Q₈

—

212

—

144

C₂×Q₈

—

212

—

145

C₂×Q₈

—

212

—

146

C₂×Q₈

—

212

—

147

C₂²×C₄

—

212

—

148

C₂²×C₄

—

212

—

149

C₂²×C₄

#11719924

yes

—

150

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

151

Q₁₆

—

106

—

152

Q₁₆

—

212

—

153

M₁₆

—

154

M₁₆

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

#11719924

yes

—

159

C₂×C₈

#11719945

yes

—

160

C₂×C₈

D₄×D₅₃

yes

—

161

C₂×C₈

Q₈×D₅₃

yes

characteristic

162

C₂×C₈

#11719949

yes

characteristic

163

C₄⋊C₄

—

212

—

164

C₄⋊C₄

—

212

—

165

C₄²

—

166

C₄²

—

167

C₁₆

—

106

—

168

C₂³

C₂×#11719924

yes

characteristic

169

Q₈

—

424

—

170

Q₈

—

424

—

171

Q₈

—

212

—

172

Q₈

—

212

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

424

—

176

C₂×C₄

—

424

—

177

C₂×C₄

—

424

—

178

C₂×C₄

—

424

—

179

C₂×C₄

—

212

—

180

C₂×C₄

#13133193

yes

characteristic

181

C₂×C₄

#13133200

yes

characteristic

182

C₂×C₄

#13133213

yes

characteristic

183

C₂×C₄

C₂×#11719924

yes

characteristic

184

C₂×C₄

#13133243

yes

characteristic

185

C₂×C₄

#13133243

yes

characteristic

186

C₈

—

187

C₈

—

188

C₈

#13133195

yes

—

189

C₈

#13133202

yes

—

190

C₂²

—

191

C₂²

#16810910

yes

characteristic

192

C₂²

#16810941

yes

characteristic

193

C₂²

#16810997

yes

characteristic

194

C₄

—

195

C₄

—

848

—

196

C₄

—

424

—

197

C₄

#16810558

yes

characteristic

198

C₄

#16810910

yes

characteristic

199

C₄

#16810941

yes

characteristic

200

C₄

#16811010

yes

characteristic

201

C₂

—

202

C₂

—

203

C₂

#20276249

yes

characteristic

204

C₂

#20276934

yes

—

	subgroup	G/H
center	C₂²	#16810941
commutator, cosocle	C₂²×C₈×C₅₃	C₂³
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1849	C₂
socle	C₂²×C₅₃	#481
prime radical	C₂³×C₅₃	C₂×SD₁₆
hypercenter	#1849	D₅₃
hypocenter	C₅₃	#13536
LCS	G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1849

subgroup

G/H

center

C₂²

#16810941

commutator, cosocle

C₂²×C₈×C₅₃

C₂³

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#1849

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 ◁ #1849

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×#16810447	2
C₂×#16810438	2
#20275586	2
#20272799	2
C₅₃×#1849	2
#13536	53

subgroup

index

C₅₃×#1849

Subgroups — #27304208

Conjugacy classes of subgroups

Complements

Maximal subgroups