Group #27302654 — subgroups

Subgroups — #27302654

Order 13568.

334 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 127 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×#16811025	C₂	1	1	yes	maximal, characteristic
2	#20274594	C₂	1	1	yes	maximal, characteristic
3	#20272821	C₂	1	2	yes	maximal
4	#20272814	C₂	1	2	yes	maximal
5	C₂×C₅₃×#364	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133209	—	2	1	no	—
7	#16811025	—	2	2	no	—
8	#16811025	C₄	1	4	yes	complemented
9	C₂×#13133153	C₂²	1	1	yes	characteristic
10	#16810661	C₂²	1	1	yes	characteristic
11	C₂×#13133151	—	2	1	no	—
12	C₂×#13133149	C₂²	1	2	yes	—
13	#16810578	—	2	1	no	—
14	#16810383	—	2	1	no	—
15	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#364	—	2	2	no	—
18	C₂×#11719917	—	2	1	no	—
19	#13133209	—	2	4	no	—
20	#13133209	—	4	2	no	—
21	#13133176	—	2	1	no	—
22	C₂×C₄⋊Dic₅₃	—	2	2	no	—
23	C₂×#11719919	—	4	2	no	—
24	#13133153	—	2	1	no	—
25	#13133153	Q₈	1	2	yes	—
26	#13133151	—	2	4	no	—
27	#13133151	Q₈	1	2	yes	—
28	#13133149	—	2	2	no	—
29	#13133149	—	2	1	no	—
30	#13133149	C₂×C₄	1	4	yes	—
31	#13133127	—	4	1	no	—
32	C₈×Dic₅₃	—	4	1	no	—
33	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
36	M₃₂×C₅₃	—	2	2	no	—
37	M₃₂×C₅₃	—	2	2	no	—
38	C₂×C₁₆×C₅₃	—	2	2	no	—
39	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₅₃	D₄	1	2	yes	—
41	C₂²×C₄×C₅₃	—	2	1	no	—
42	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₅₃	—	4	2	no	—
44	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
45	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
46	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
47	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
48	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
53	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
54	C₄²×C₅₃	—	2	1	no	—
55	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
56	C₂²⋊Dic₅₃	—	4	1	no	—
57	C₄⋊Dic₅₃	—	2	4	no	—
58	C₄⋊Dic₅₃	—	2	2	no	—
59	C₄⋊Dic₅₃	—	4	2	no	—
60	#11719919	—	4	8	no	—
61	C₄×Dic₅₃	—	4	1	no	—
62	#11719917	—	2	2	no	—
63	#11719917	—	4	1	no	—
64	C₂×#88322	—	4	1	no	—
65	C₁₆×C₅₃	—	4	2	no	—
66	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	4	2	no	—
69	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
74	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₂×Dic₅₃	—	4	4	no	—
76	C₂×Dic₅₃	—	4	2	no	—
77	C₂×Dic₅₃	—	8	4	no	—
78	C₈×C₅₃	#103	1	2	yes	—
79	C₈×C₅₃	#103	1	2	yes	—
80	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
81	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
82	C₈×C₅₃	#124	1	1	yes	characteristic
83	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
84	#88322	—	4	2	no	—
85	#13273	—	53	1	no	maximal, Sylow
86	C₂²×C₅₃	—	2	2	no	—
87	C₂²×C₅₃	#418	1	1	yes	characteristic
88	C₂²×C₅₃	#476	1	1	yes	characteristic
89	C₂²×C₅₃	#477	1	1	yes	characteristic
90	C₄×C₅₃	—	4	2	no	—
91	C₄×C₅₃	#391	1	1	yes	characteristic
92	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
93	C₄×C₅₃	#476	1	1	yes	characteristic
94	C₄×C₅₃	#477	1	1	yes	characteristic
95	Dic₅₃	—	8	4	no	—
96	Dic₅₃	—	8	2	no	—
97	C₂×#499	—	53	1	no	—
98	C₂×#364	D₅₃	1	1	yes	characteristic
99	#1772	—	53	1	no	—
100	#1386	—	53	2	no	—
101	#1383	—	53	2	no	—
102	C₂×C₅₃	—	2	1	no	—
103	C₂×C₅₃	—	2	1	no	—
104	C₂×C₅₃	#1987	1	1	yes	characteristic
105	C₂×C₅₃	#2231	1	2	yes	—
106	C₂×#125	—	106	1	no	—
107	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
108	#499	—	53	4	no	—
109	#499	—	106	2	no	—
110	C₂×#108	—	53	1	no	—
111	#427	—	53	1	no	—
112	C₂×#107	—	106	1	no	—
113	C₂×#106	—	53	2	no	—
114	#405	—	106	1	no	—
115	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
116	#364	—	2	2	no	—
117	#339	—	106	1	no	—
118	C₅₃	#13273	1	1	yes	characteristic, complemented, Sylow
119	C₂×M₁₆	—	106	1	no	—
120	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
121	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
122	#125	—	106	4	no	—
123	#125	—	212	2	no	—
124	#117	—	106	1	no	—
125	C₂×C₄⋊C₄	—	106	2	no	—
126	C₂×C₄⋊C₄	—	212	2	no	—
127	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
128	M₃₂	—	2	2	no	—
129	M₃₂	—	2	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	#108	—	53	2	no	—
132	#108	—	106	1	no	—
133	#107	—	53	2	no	—
134	#107	—	106	4	no	—
135	#106	—	53	4	no	—
136	#106	—	106	1	no	—
137	#106	—	106	2	no	—
138	#97	—	212	1	no	—
139	C₄×C₈	—	212	1	no	—
140	C₄×C₈	C₄×D₅₃	1	2	yes	—
141	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	—	212	2	no	—
144	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
145	M₁₆	—	106	2	no	—
146	M₁₆	—	212	1	no	—
147	C₂×C₈	—	212	1	no	—
148	C₂×C₈	#11719913	1	1	yes	characteristic
149	C₂×C₈	#11719913	1	1	yes	characteristic
150	C₂×C₈	Dic₂₁₂	1	2	yes	—
151	C₂×C₈	#11719921	1	2	yes	—
152	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
153	C₂×C₈	#11719947	1	1	yes	characteristic
154	C₂×C₈	#11719947	1	1	yes	characteristic
155	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
156	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
157	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
158	C₄⋊C₄	—	106	4	no	—
159	C₄⋊C₄	—	106	2	no	—
160	C₄⋊C₄	—	212	8	no	—
161	C₄⋊C₄	—	212	2	no	—
162	C₂²⋊C₄	—	212	1	no	—
163	C₄²	—	2	1	no	—
164	C₄²	—	212	1	no	—
165	C₄²	#11719921	1	2	yes	—
166	C₁₆	—	4	2	no	—
167	C₂³	#13133245	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	424	4	no	—
171	C₂×C₄	—	212	4	no	—
172	C₂×C₄	—	212	2	no	—
173	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
174	C₂×C₄	#13133197	1	1	yes	characteristic
175	C₂×C₄	#13133197	1	1	yes	characteristic
176	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
177	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
178	C₂×C₄	#13133245	1	1	yes	characteristic
179	C₈	—	212	2	no	—
180	C₈	#13133142	1	2	yes	—
181	C₈	#13133142	1	2	yes	—
182	C₈	#13133182	1	1	yes	characteristic
183	C₈	#13133182	1	1	yes	characteristic
184	C₈	#13133204	1	1	yes	characteristic
185	C₈	#13133215	1	1	yes	characteristic
186	C₂²	—	2	2	no	—
187	C₂²	#16810629	1	1	yes	characteristic
188	C₂²	#16810914	1	1	yes	characteristic
189	C₂²	#16810919	1	1	yes	characteristic
190	C₄	—	4	2	no	—
191	C₄	—	424	4	no	—
192	C₄	—	424	2	no	—
193	C₄	#16810528	1	1	yes	characteristic
194	C₄	C₂×#13133142	1	1	yes	characteristic
195	C₄	#16810914	1	1	yes	characteristic
196	C₄	#16810919	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	#20275662	1	1	yes	characteristic
200	C₂	#20276892	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811025

C₂

yes

maximal, characteristic

#20274594

C₂

yes

maximal, characteristic

#20272821

C₂

yes

maximal

#20272814

C₂

yes

maximal

C₂×C₅₃×#364

C₂

yes

maximal, characteristic

C₂×#13133209

—

#16811025

—

#16811025

C₄

yes

complemented

C₂×#13133153

C₂²

yes

characteristic

#16810661

C₂²

yes

characteristic

C₂×#13133151

—

C₂×#13133149

C₂²

yes

—

#16810578

—

#16810383

—

C₂×M₃₂×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#364

—

—

—

—

—

C₂×C₄⋊Dic₅₃

—

C₂×#11719919

—

#13133153

—

#13133153

Q₈

yes

—

#13133151

—

#13133151

Q₈

yes

—

#13133149

—

#13133149

—

#13133149

C₂×C₄

yes

—

#13133127

—

C₈×Dic₅₃

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

characteristic

M₃₂×C₅₃

—

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

characteristic

C₂×C₈×C₅₃

SD₁₆

yes

characteristic

C₂×C₈×C₅₃

Q₁₆

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

C₂×Q₈

yes

C₂×C₈×C₅₃

yes

C₂×C₈×C₅₃

yes

C₄²×C₅₃

—

C₄²×C₅₃

C₂²⋊C₄

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

—

#11719917

—

#11719917

—

C₂×#88322

—

C₁₆×C₅₃

—

C₂³×C₅₃

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

#103

yes

—

C₈×C₅₃

#103

yes

—

C₈×C₅₃

C₄×Q₈

yes

characteristic

C₈×C₅₃

C₄×Q₈

yes

characteristic

C₈×C₅₃

#124

yes

characteristic

C₈×C₅₃

C₄⋊Q₈

yes

—

—

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×#499

—

C₂×#364

D₅₃

yes

characteristic

#1772

—

100

#1386

—

101

#1383

—

102

C₂×C₅₃

—

103

C₂×C₅₃

—

104

C₂×C₅₃

#1987

yes

characteristic

105

C₂×C₅₃

#2231

yes

—

106

C₂×#125

—

106

—

107

C₂×M₃₂

C₂×D₅₃

yes

—

108

#499

—

109

#499

—

106

—

110

C₂×#108

—

111

#427

—

112

C₂×#107

—

106

—

113

C₂×#106

—

114

#405

—

106

—

115

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

116

#364

—

117

#339

—

106

—

118

C₅₃

#13273

yes

characteristic, complemented, Sylow

119

C₂×M₁₆

—

106

—

120

C₂²×C₈

D₂₁₂

yes

characteristic

121

C₂²×C₈

C₂²×D₅₃

yes

characteristic

122

#125

—

106

—

123

#125

—

212

—

124

#117

—

106

—

125

C₂×C₄⋊C₄

—

106

—

126

C₂×C₄⋊C₄

—

212

—

127

C₂×C₄²

D₂₁₂

yes

characteristic

128

M₃₂

—

129

M₃₂

—

130

C₂×C₁₆

—

131

#108

—

132

#108

—

106

—

133

#107

—

134

#107

—

106

—

135

#106

—

136

#106

—

106

—

137

#106

—

106

—

138

#97

—

212

—

139

C₄×C₈

—

212

—

140

C₄×C₈

C₄×D₅₃

yes

—

141

C₄×C₈

C₂²⋊D₅₃

yes

—

142

C₂²×C₄

—

143

C₂²×C₄

—

212

—

144

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

145

M₁₆

—

106

—

146

M₁₆

—

212

—

147

C₂×C₈

—

212

—

148

C₂×C₈

#11719913

yes

characteristic

149

C₂×C₈

#11719913

yes

characteristic

150

C₂×C₈

Dic₂₁₂

yes

—

151

C₂×C₈

#11719921

yes

—

152

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

153

C₂×C₈

#11719947

yes

characteristic

154

C₂×C₈

#11719947

yes

characteristic

155

C₂×C₈

Q₈×D₅₃

yes

characteristic

156

C₂×C₈

Q₈×D₅₃

yes

characteristic

157

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

158

C₄⋊C₄

—

106

—

159

C₄⋊C₄

—

106

—

160

C₄⋊C₄

—

212

—

161

C₄⋊C₄

—

212

—

162

C₂²⋊C₄

—

212

—

163

C₄²

—

164

C₄²

—

212

—

165

C₄²

#11719921

yes

—

166

C₁₆

—

167

C₂³

#13133245

yes

characteristic

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

424

—

171

C₂×C₄

—

212

—

172

C₂×C₄

—

212

—

173

C₂×C₄

C₂×#11719921

yes

characteristic

174

C₂×C₄

#13133197

yes

characteristic

175

C₂×C₄

#13133197

yes

characteristic

176

C₂×C₄

C₂×#11719913

yes

characteristic

177

C₂×C₄

C₂×Dic₂₁₂

yes

characteristic

178

C₂×C₄

#13133245

yes

characteristic

179

C₈

—

212

—

180

C₈

#13133142

yes

—

181

C₈

#13133142

yes

—

182

C₈

#13133182

yes

characteristic

183

C₈

#13133182

yes

characteristic

184

C₈

#13133204

yes

characteristic

185

C₈

#13133215

yes

characteristic

186

C₂²

—

187

C₂²

#16810629

yes

characteristic

188

C₂²

#16810914

yes

characteristic

189

C₂²

#16810919

yes

characteristic

190

C₄

—

191

C₄

—

424

—

192

C₄

—

424

—

193

C₄

#16810528

yes

characteristic

194

C₄

C₂×#13133142

yes

characteristic

195

C₄

#16810914

yes

characteristic

196

C₄

#16810919

yes

characteristic

197

C₂

—

198

C₂

—

199

C₂

#20275662

yes

characteristic

200

C₂

#20276892

yes

—

	subgroup	G/H
center	C₂²	#16810914
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#364	C₂
socle	C₂²×C₅₃	#476
prime radical	C₂³×C₅₃	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#364	D₅₃
hypocenter	C₅₃	#13273
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

subgroup

G/H

center

C₂²

#16810914

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₂×C₅₃×#364

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

N	H ≅ G/N	type
8. #16811025	C₄	semidirect
118. C₅₃	#13273	semidirect

H ≅ G/N

type

8. #16811025

C₄

semidirect

118. C₅₃

#13273

semidirect

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

subgroup	index
C₂×#16811025	2
#20274594	2
#20272821	2
#20272814	2
C₂×C₅₃×#364	2
#13273	53

subgroup

index

C₂×C₅₃×#364

Subgroups — #27302654

Conjugacy classes of subgroups

Complements

Maximal subgroups