Group #27300248 — subgroups

Subgroups — #27300248

Order 13568.

358 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 143 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	#20276488	C₂	1	2	yes	maximal, complemented
2	#20274765	C₂	1	1	yes	maximal, characteristic
3	#20274761	C₂	1	1	yes	maximal, characteristic
4	#20272811	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#504	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133153	C₂²	1	2	yes	—
7	#16810661	C₂²	1	2	yes	—
8	#16810607	—	2	1	no	—
9	#16810597	—	2	1	no	—
10	#16810446	—	2	4	no	—
11	#16810381	—	2	1	no	—
12	#16810380	—	2	1	no	—
13	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#504	—	2	4	no	—
15	C₂×M₃₂×C₅₃	C₄	1	2	yes	complemented
16	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₅₃	C₄	1	2	yes	complemented
18	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#11719917	—	2	2	no	—
20	#13133176	—	2	2	no	—
21	#13133153	—	2	2	no	—
22	#13133153	—	2	2	no	—
23	#13133151	—	2	2	no	—
24	#13133149	—	2	2	no	—
25	#13133136	—	4	1	no	—
26	#13133134	—	4	1	no	—
27	#13133132	—	4	2	no	—
28	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
29	#131×C₅₃	—	2	2	no	—
30	#131×C₅₃	—	2	2	no	—
31	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
32	C₂×M₁₆×C₅₃	Q₈	1	2	yes	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	M₃₂×C₅₃	—	2	4	no	—
36	M₃₂×C₅₃	D₄	1	4	yes	—
37	C₂×C₁₆×C₅₃	—	2	2	no	—
38	C₂×C₁₆×C₅₃	—	2	4	no	—
39	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
42	#41×C₅₃	—	2	2	no	—
43	#41×C₅₃	—	2	2	no	—
44	C₂×Q₈×C₅₃	SD₁₆	1	1	yes	characteristic
45	C₂×D₄×C₅₃	SD₁₆	1	2	yes	—
46	C₂×D₄×C₅₃	SD₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	M₁₆×C₅₃	—	2	2	no	—
50	M₁₆×C₅₃	—	2	4	no	—
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	—	2	2	no	—
53	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
54	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₅₃	#41	1	2	yes	—
58	C₂²⋊Dic₅₃	—	4	2	no	—
59	C₄⋊Dic₅₃	—	4	2	no	—
60	C₄×Dic₅₃	—	4	2	no	—
61	#11719917	—	4	2	no	—
62	#11719917	—	4	2	no	—
63	C₂×#88322	—	4	2	no	—
64	C₁₆×C₅₃	—	2	1	no	—
65	C₁₆×C₅₃	—	2	1	no	—
66	C₁₆×C₅₃	C₂²⋊C₄	1	4	yes	—
67	C₂³×C₅₃	#106	1	2	yes	—
68	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
69	Q₈×C₅₃	—	2	2	no	—
70	D₄×C₅₃	—	2	4	no	—
71	D₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	2	2	no	—
73	C₂×C₄×C₅₃	—	2	2	no	—
74	C₂×C₄×C₅₃	#106	1	2	yes	—
75	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#121	1	2	yes	—
77	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#135	1	2	yes	—
79	C₂×Dic₅₃	—	8	2	no	—
80	C₈×C₅₃	—	2	2	no	—
81	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₈×C₅₃	#117	1	1	yes	characteristic
83	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
84	#88322	—	8	2	no	—
85	#12832	—	53	1	no	maximal, Sylow
86	C₂²×C₅₃	—	2	2	no	—
87	C₂²×C₅₃	—	4	1	no	—
88	C₂²×C₅₃	—	4	2	no	—
89	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
90	C₂²×C₅₃	#465	1	2	yes	—
91	C₄×C₅₃	—	2	2	no	—
92	C₄×C₅₃	#380	1	1	yes	characteristic
93	C₄×C₅₃	#427	1	1	yes	characteristic
94	C₄×C₅₃	#465	1	2	yes	—
95	Dic₅₃	—	16	2	no	—
96	C₂×#504	D₅₃	1	1	yes	characteristic
97	#2155	—	53	2	no	—
98	#1813	—	53	1	no	—
99	#1812	—	53	1	no	—
100	#1382	—	53	2	no	—
101	C₂×C₅₃	—	2	2	no	—
102	C₂×C₅₃	—	2	1	no	—
103	C₂×C₅₃	—	4	2	no	—
104	C₂×C₅₃	#1807	1	1	yes	characteristic
105	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
106	#504	—	2	4	no	—
107	C₂×M₃₂	Dic₅₃	1	2	yes	complemented
108	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
109	C₂²×C₁₆	Dic₅₃	1	2	yes	complemented
110	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
111	C₂×#108	—	53	2	no	—
112	#427	—	53	2	no	—
113	#413	—	106	1	no	—
114	#410	—	106	1	no	—
115	#365	—	106	4	no	—
116	#338	—	106	1	no	—
117	H(3,ℤ/4)	—	106	1	no	—
118	C₅₃	#12832	1	1	yes	characteristic, complemented, Sylow
119	C₂×#41	D₂₁₂	1	1	yes	characteristic
120	#131	—	2	2	no	—
121	#131	—	2	2	no	—
122	C₂×M₁₆	—	106	2	no	—
123	C₂×M₁₆	Dic₁₀₆	1	2	yes	—
124	C₂×M₁₆	D₂₁₂	1	1	yes	characteristic
125	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
126	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
127	#117	—	106	2	no	—
128	M₃₂	—	2	4	no	—
129	M₃₂	C₂²⋊D₅₃	1	4	yes	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	2	4	no	—
132	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
133	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
134	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
135	#108	—	106	2	no	—
136	#108	—	106	2	no	—
137	#107	—	106	2	no	—
138	#106	—	106	2	no	—
139	#101	—	212	1	no	—
140	#100	—	212	1	no	—
141	#99	—	212	2	no	—
142	#41	—	2	2	no	—
143	#41	—	2	2	no	—
144	C₂×Q₈	#11719913	1	1	yes	characteristic
145	C₂×D₄	#11719913	1	2	yes	—
146	C₂×D₄	#11719913	1	1	yes	characteristic
147	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
148	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
149	M₁₆	—	2	2	no	—
150	M₁₆	—	2	4	no	—
151	M₁₆	—	212	2	no	—
152	M₁₆	—	212	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	212	2	no	—
156	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
157	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
158	C₂×C₈	#11719945	1	2	yes	—
159	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
160	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
161	C₄⋊C₄	—	212	2	no	—
162	C₂²⋊C₄	—	212	2	no	—
163	C₄²	—	212	2	no	—
164	C₁₆	—	2	1	no	—
165	C₁₆	—	2	1	no	—
166	C₁₆	C₂²⋊Dic₅₃	1	4	yes	—
167	C₂³	#13133149	1	2	yes	—
168	C₂³	C₂×#11719913	1	1	yes	characteristic
169	Q₈	—	2	2	no	—
170	D₄	—	2	4	no	—
171	D₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	424	2	no	—
175	C₂×C₄	#13133149	1	2	yes	—
176	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
177	C₂×C₄	#13133191	1	2	yes	—
178	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
179	C₂×C₄	#13133239	1	2	yes	—
180	C₈	—	2	2	no	—
181	C₈	—	424	2	no	—
182	C₈	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
183	C₈	#13133176	1	1	yes	characteristic
184	C₈	#13133196	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	—	4	1	no	—
187	C₂²	—	4	2	no	—
188	C₂²	C₂×#13133149	1	1	yes	characteristic
189	C₂²	#16810852	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	848	2	no	—
192	C₄	#16810483	1	1	yes	characteristic
193	C₄	#16810661	1	1	yes	characteristic
194	C₄	#16810852	1	2	yes	—
195	C₂	—	2	2	no	—
196	C₂	—	2	1	no	—
197	C₂	—	4	2	no	—
198	C₂	#20274740	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276488

C₂

yes

maximal, complemented

#20274765

C₂

yes

maximal, characteristic

#20274761

C₂

yes

maximal, characteristic

#20272811

C₂

yes

maximal, complemented

C₂×C₅₃×#504

C₂

yes

maximal, characteristic

C₂×#13133153

C₂²

yes

—

#16810661

C₂²

yes

—

—

—

—

—

—

C₂×#131×C₅₃

C₂²

yes

characteristic

C₅₃×#504

—

C₂×M₃₂×C₅₃

C₄

yes

complemented

C₂×M₃₂×C₅₃

C₂²

yes

characteristic

C₂²×C₁₆×C₅₃

C₄

yes

complemented

C₂²×C₁₆×C₅₃

C₂²

yes

—

—

—

—

—

—

—

—

—

C₂×#41×C₅₃

D₄

yes

characteristic

#131×C₅₃

—

#131×C₅₃

—

C₂×M₁₆×C₅₃

D₄

yes

characteristic

C₂×M₁₆×C₅₃

Q₈

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

M₃₂×C₅₃

—

M₃₂×C₅₃

D₄

yes

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

SD₁₆

yes

characteristic

C₂×D₄×C₅₃

SD₁₆

yes

—

C₂×D₄×C₅₃

SD₁₆

yes

characteristic

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

#11719917

—

#11719917

—

C₂×#88322

—

C₁₆×C₅₃

—

C₁₆×C₅₃

—

C₁₆×C₅₃

C₂²⋊C₄

yes

—

C₂³×C₅₃

#106

yes

—

C₂³×C₅₃

C₂×SD₁₆

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#106

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₃

#135

yes

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₅₃

#117

yes

characteristic

C₈×C₅₃

C₂²⋊Q₈

yes

—

#88322

—

#12832

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×#106

yes

characteristic

C₂²×C₅₃

#465

yes

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

C₂×#504

D₅₃

yes

—

—

—

100

#1382

—

101

C₂×C₅₃

—

102

C₂×C₅₃

—

103

C₂×C₅₃

—

104

C₂×C₅₃

#1807

yes

characteristic

105

C₂×#131

C₂×D₅₃

yes

characteristic

106

#504

—

107

C₂×M₃₂

Dic₅₃

yes

complemented

108

C₂×M₃₂

C₂×D₅₃

yes

characteristic

109

C₂²×C₁₆

Dic₅₃

yes

complemented

110

C₂²×C₁₆

C₂×D₅₃

yes

characteristic

111

C₂×#108

—

112

#427

—

113

#413

—

106

—

114

#410

—

106

—

115

#365

—

106

—

116

#338

—

106

—

117

H(3,ℤ/4)

—

106

—

118

C₅₃

#12832

yes

characteristic, complemented, Sylow

119

C₂×#41

D₂₁₂

yes

characteristic

120

#131

—

121

#131

—

122

C₂×M₁₆

—

106

—

123

C₂×M₁₆

Dic₁₀₆

yes

—

124

C₂×M₁₆

D₂₁₂

yes

characteristic

125

C₂²×C₈

C₂×Dic₅₃

yes

—

126

C₂²×C₈

C₂²×D₅₃

yes

characteristic

127

#117

—

106

—

128

M₃₂

—

129

M₃₂

C₂²⋊D₅₃

yes

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

132

C₂×C₁₆

C₂×Dic₅₃

yes

characteristic

133

C₂×C₁₆

C₂×Dic₅₃

yes

characteristic

134

C₂×C₁₆

C₂×Dic₅₃

yes

—

135

#108

—

106

—

136

#108

—

106

—

137

#107

—

106

—

138

#106

—

106

—

139

#101

—

212

—

140

#100

—

212

—

141

#99

—

212

—

142

#41

—

143

#41

—

144

C₂×Q₈

#11719913

yes

characteristic

145

C₂×D₄

#11719913

yes

—

146

C₂×D₄

#11719913

yes

characteristic

147

C₂²×C₄

C₄⋊Dic₅₃

yes

—

148

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

149

M₁₆

—

150

M₁₆

—

151

M₁₆

—

212

—

152

M₁₆

—

212

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

212

—

156

C₂×C₈

C₄⋊Dic₅₃

yes

—

157

C₂×C₈

C₂×Dic₁₀₆

yes

characteristic

158

C₂×C₈

#11719945

yes

—

159

C₂×C₈

C₂²×Dic₅₃

yes

characteristic

160

C₂×C₈

C₂×C₂²⋊D₅₃

yes

—

161

C₄⋊C₄

—

212

—

162

C₂²⋊C₄

—

212

—

163

C₄²

—

212

—

164

C₁₆

—

165

C₁₆

—

166

C₁₆

C₂²⋊Dic₅₃

yes

—

167

C₂³

#13133149

yes

—

168

C₂³

C₂×#11719913

yes

characteristic

169

Q₈

—

170

D₄

—

171

D₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

424

—

175

C₂×C₄

#13133149

yes

—

176

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

177

C₂×C₄

#13133191

yes

—

178

C₂×C₄

C₂×#11719913

yes

characteristic

179

C₂×C₄

#13133239

yes

—

180

C₈

—

181

C₈

—

424

—

182

C₈

C₂×C₂²⋊Dic₅₃

yes

characteristic

183

C₈

#13133176

yes

characteristic

184

C₈

#13133196

yes

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

C₂×#13133149

yes

characteristic

189

C₂²

#16810852

yes

—

190

C₄

—

191

C₄

—

848

—

192

C₄

#16810483

yes

characteristic

193

C₄

#16810661

yes

characteristic

194

C₄

#16810852

yes

—

195

C₂

—

196

C₂

—

197

C₂

—

198

C₂

#20274740

yes

characteristic

	subgroup	G/H
center	C₄	#16810661
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#504	C₂
socle	C₂×C₅₃	#1807
prime radical	C₂×#41×C₅₃	D₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#504	D₅₃
hypocenter	C₅₃	#12832
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ C₂×#504

subgroup

G/H

center

C₄

#16810661

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₂×C₅₃×#504

C₂

socle

C₂×C₅₃

#1807

prime radical

C₂×#41×C₅₃

D₄

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ C₂×#504

N	H ≅ G/N	type
1. #20276488	C₂	semidirect
4. #20272811	C₂	semidirect
15. C₂×M₃₂×C₅₃	C₄	semidirect
17. C₂²×C₁₆×C₅₃	C₄	semidirect
107. C₂×M₃₂	Dic₅₃	semidirect
109. C₂²×C₁₆	Dic₅₃	semidirect
118. C₅₃	#12832	semidirect

H ≅ G/N

type

1. #20276488

C₂

semidirect

4. #20272811

C₂

semidirect

15. C₂×M₃₂×C₅₃

C₄

semidirect

17. C₂²×C₁₆×C₅₃

C₄

semidirect

107. C₂×M₃₂

Dic₅₃

semidirect

109. C₂²×C₁₆

Dic₅₃

semidirect

118. C₅₃

#12832

semidirect

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

subgroup	index
#20276488	2
#20274765	2
#20274761	2
#20272811	2
C₂×C₅₃×#504	2
#12832	53

subgroup

index

C₂×C₅₃×#504

Subgroups — #27300248

Conjugacy classes of subgroups

Complements

Maximal subgroups