Group #27300249 — subgroups

Subgroups — #27300249

Order 13568.

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

normal subgroups only

This group has 358 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×#16811040	C₂	1	1	yes	maximal, characteristic
2	#20276489	C₂	1	2	yes	maximal, complemented
3	#20274761	C₂	1	1	yes	maximal, characteristic
4	#20272812	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1813	C₂	1	1	yes	maximal, characteristic
6	#16811040	—	2	4	no	—
7	C₂×#13133157	C₄	1	2	yes	complemented
8	C₂×#13133157	C₂²	1	1	yes	characteristic
9	C₂²×#11719908	C₄	1	2	yes	complemented
10	C₂²×#11719908	C₂²	1	1	yes	characteristic
11	#16810661	C₂²	1	2	yes	—
12	#16810597	—	2	1	no	—
13	#16810447	—	2	4	no	—
14	#16810380	—	2	1	no	—
15	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#108×C₅₃	C₂²	1	2	yes	—
17	C₅₃×#413	—	2	1	no	—
18	C₅₃×#338	—	2	1	no	—
19	#13133176	—	2	2	no	—
20	#13133157	—	2	4	no	—
21	#13133157	D₄	1	4	yes	—
22	C₂×#11719908	—	2	2	no	—
23	C₂×#11719908	—	2	4	no	—
24	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
25	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
26	C₂×#11719908	C₂×C₄	1	2	yes	—
27	#13133151	—	2	2	no	—
28	#13133149	—	2	2	no	—
29	#13133132	—	4	2	no	—
30	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
31	#131×C₅₃	—	2	2	no	—
32	#131×C₅₃	—	2	2	no	—
33	C₂×M₁₆×C₅₃	—	2	2	no	—
34	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
35	C₂×M₁₆×C₅₃	Q₈	1	2	yes	—
36	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
38	#108×C₅₃	—	2	2	no	—
39	#108×C₅₃	—	2	2	no	—
40	#101×C₅₃	—	4	1	no	—
41	#100×C₅₃	—	4	1	no	—
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	M₁₆×C₅₃	—	4	2	no	—
52	M₁₆×C₅₃	—	4	2	no	—
53	C₂×C₈×C₅₃	—	2	2	no	—
54	C₂×C₈×C₅₃	—	2	2	no	—
55	C₂×C₈×C₅₃	—	4	2	no	—
56	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₅₃	#41	1	2	yes	—
61	C₂²⋊Dic₅₃	—	4	2	no	—
62	C₄⋊Dic₅₃	—	4	2	no	—
63	C₄×Dic₅₃	—	4	2	no	—
64	#11719908	—	2	1	no	—
65	#11719908	—	2	1	no	—
66	#11719908	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	—	8	2	no	—
82	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₈×C₅₃	#117	1	1	yes	characteristic
84	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
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	—	53	1	no	—
97	#2155	—	53	2	no	—
98	#1813	D₅₃	1	1	yes	characteristic
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	—	106	4	no	—
107	C₂×M₃₂	—	53	1	no	—
108	C₂×M₃₂	—	53	2	no	—
109	C₂²×C₁₆	—	53	2	no	—
110	C₂²×C₁₆	—	53	1	no	—
111	C₂×#108	C₂×D₅₃	1	2	yes	—
112	#427	—	53	2	no	—
113	#413	—	2	1	no	—
114	#410	—	106	1	no	—
115	#365	—	106	4	no	—
116	#338	—	2	1	no	—
117	H(3,ℤ/4)	—	106	1	no	—
118	C₅₃	#12832	1	1	yes	characteristic, complemented, Sylow
119	C₂×#41	C₂²⋊D₅₃	1	1	yes	characteristic
120	#131	—	2	2	no	—
121	#131	—	2	2	no	—
122	C₂×M₁₆	—	2	2	no	—
123	C₂×M₁₆	Dic₁₀₆	1	2	yes	—
124	C₂×M₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
125	C₂²×C₈	C₄×D₅₃	1	2	yes	—
126	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
127	#117	—	106	2	no	—
128	M₃₂	—	53	4	no	—
129	M₃₂	—	106	4	no	—
130	C₂×C₁₆	—	53	2	no	—
131	C₂×C₁₆	—	53	1	no	—
132	C₂×C₁₆	—	53	1	no	—
133	C₂×C₁₆	—	106	4	no	—
134	C₂×C₁₆	—	106	2	no	—
135	#108	—	2	2	no	—
136	#108	—	2	2	no	—
137	#107	—	106	2	no	—
138	#106	—	106	2	no	—
139	#101	—	4	1	no	—
140	#100	—	4	1	no	—
141	#99	—	212	2	no	—
142	#41	—	2	2	no	—
143	#41	—	2	2	no	—
144	C₂×Q₈	#11719923	1	1	yes	characteristic
145	C₂×D₄	#11719923	1	1	yes	characteristic
146	C₂×D₄	#11719924	1	2	yes	—
147	C₂²×C₄	#11719919	1	2	yes	—
148	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
149	M₁₆	—	2	2	no	—
150	M₁₆	—	2	4	no	—
151	M₁₆	—	4	2	no	—
152	M₁₆	—	4	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	#11719919	1	2	yes	—
157	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
158	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
159	C₂×C₈	D₄×D₅₃	1	2	yes	—
160	C₂×C₈	#11719947	1	2	yes	—
161	C₄⋊C₄	—	212	2	no	—
162	C₂²⋊C₄	—	212	2	no	—
163	C₄²	—	212	2	no	—
164	C₁₆	—	53	4	no	—
165	C₁₆	—	106	1	no	—
166	C₁₆	—	106	1	no	—
167	C₂³	#13133148	1	2	yes	—
168	C₂³	C₂×#11719924	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	#13133148	1	2	yes	—
176	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
177	C₂×C₄	#13133189	1	2	yes	—
178	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
179	C₂×C₄	#13133237	1	2	yes	—
180	C₈	—	2	2	no	—
181	C₈	—	8	2	no	—
182	C₈	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
183	C₈	#13133173	1	1	yes	characteristic
184	C₈	#13133194	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	—	4	1	no	—
187	C₂²	—	4	2	no	—
188	C₂²	C₂×#13133148	1	1	yes	characteristic
189	C₂²	#16810854	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	848	2	no	—
192	C₄	#16810481	1	1	yes	characteristic
193	C₄	#16810659	1	1	yes	characteristic
194	C₄	#16810850	1	2	yes	—
195	C₂	—	2	2	no	—
196	C₂	—	2	1	no	—
197	C₂	—	4	2	no	—
198	C₂	#20274738	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811040

C₂

yes

maximal, characteristic

#20276489

C₂

yes

maximal, complemented

#20274761

C₂

yes

maximal, characteristic

#20272812

C₂

yes

maximal, complemented

C₅₃×#1813

C₂

yes

maximal, characteristic

#16811040

—

C₂×#13133157

C₄

yes

complemented

C₂×#13133157

C₂²

yes

characteristic

C₂²×#11719908

C₄

yes

complemented

C₂²×#11719908

C₂²

yes

characteristic

#16810661

C₂²

yes

—

#16810597

—

#16810447

—

#16810380

—

C₂×#131×C₅₃

C₂²

yes

characteristic

C₂×#108×C₅₃

C₂²

yes

—

C₅₃×#413

—

C₅₃×#338

—

#13133176

—

#13133157

—

#13133157

D₄

yes

—

C₂×#11719908

—

C₂×#11719908

—

C₂×#11719908

C₂×C₄

yes

characteristic

C₂×#11719908

C₂×C₄

yes

characteristic

C₂×#11719908

C₂×C₄

yes

—

#13133151

—

#13133149

—

#13133132

—

C₂×#41×C₅₃

D₄

yes

characteristic

#131×C₅₃

—

#131×C₅₃

—

C₂×M₁₆×C₅₃

—

C₂×M₁₆×C₅₃

D₄

yes

characteristic

C₂×M₁₆×C₅₃

Q₈

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

#108×C₅₃

—

#108×C₅₃

—

#101×C₅₃

—

#100×C₅₃

—

#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

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×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

—

#11719908

—

#11719908

—

#11719908

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

—

C₈×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₅₃

#117

yes

characteristic

C₈×C₅₃

C₂²⋊Q₈

yes

—

#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

—

#2155

—

#1813

D₅₃

yes

characteristic

#1812

—

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

—

106

—

107

C₂×M₃₂

—

108

C₂×M₃₂

—

109

C₂²×C₁₆

—

110

C₂²×C₁₆

—

111

C₂×#108

C₂×D₅₃

yes

—

112

#427

—

113

#413

—

114

#410

—

106

—

115

#365

—

106

—

116

#338

—

117

H(3,ℤ/4)

—

106

—

118

C₅₃

#12832

yes

characteristic, complemented, Sylow

119

C₂×#41

C₂²⋊D₅₃

yes

characteristic

120

#131

—

121

#131

—

122

C₂×M₁₆

—

123

C₂×M₁₆

Dic₁₀₆

yes

—

124

C₂×M₁₆

C₂²⋊D₅₃

yes

characteristic

125

C₂²×C₈

C₄×D₅₃

yes

—

126

C₂²×C₈

C₂²×D₅₃

yes

characteristic

127

#117

—

106

—

128

M₃₂

—

129

M₃₂

—

106

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

132

C₂×C₁₆

—

133

C₂×C₁₆

—

106

—

134

C₂×C₁₆

—

106

—

135

#108

—

136

#108

—

137

#107

—

106

—

138

#106

—

106

—

139

#101

—

140

#100

—

141

#99

—

212

—

142

#41

—

143

#41

—

144

C₂×Q₈

#11719923

yes

characteristic

145

C₂×D₄

#11719923

yes

characteristic

146

C₂×D₄

#11719924

yes

—

147

C₂²×C₄

#11719919

yes

—

148

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

149

M₁₆

—

150

M₁₆

—

151

M₁₆

—

152

M₁₆

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

#11719919

yes

—

157

C₂×C₈

C₂×Dic₁₀₆

yes

characteristic

158

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

159

C₂×C₈

D₄×D₅₃

yes

—

160

C₂×C₈

#11719947

yes

—

161

C₄⋊C₄

—

212

—

162

C₂²⋊C₄

—

212

—

163

C₄²

—

212

—

164

C₁₆

—

165

C₁₆

—

106

—

166

C₁₆

—

106

—

167

C₂³

#13133148

yes

—

168

C₂³

C₂×#11719924

yes

characteristic

169

Q₈

—

170

D₄

—

171

D₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

424

—

175

C₂×C₄

#13133148

yes

—

176

C₂×C₄

C₂×#11719919

yes

characteristic

177

C₂×C₄

#13133189

yes

—

178

C₂×C₄

C₂×#11719923

yes

characteristic

179

C₂×C₄

#13133237

yes

—

180

C₈

—

181

C₈

—

182

C₈

C₂²⋊C₄×D₅₃

yes

characteristic

183

C₈

#13133173

yes

characteristic

184

C₈

#13133194

yes

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

C₂×#13133148

yes

characteristic

189

C₂²

#16810854

yes

—

190

C₄

—

191

C₄

—

848

—

192

C₄

#16810481

yes

characteristic

193

C₄

#16810659

yes

characteristic

194

C₄

#16810850

yes

—

195

C₂

—

196

C₂

—

197

C₂

—

198

C₂

#20274738

yes

characteristic

	subgroup	G/H
center	C₄	#16810659
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1813	C₂
socle	C₂×C₅₃	#1807
prime radical	C₂×#41×C₅₃	D₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1813	D₅₃
hypocenter	C₅₃	#12832
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ #1813

subgroup

G/H

center

C₄

#16810659

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#1813

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 ◁ #1813

N	H ≅ G/N	type
2. #20276489	C₂	semidirect
4. #20272812	C₂	semidirect
7. C₂×#13133157	C₄	semidirect
9. C₂²×#11719908	C₄	semidirect
118. C₅₃	#12832	semidirect

H ≅ G/N

type

2. #20276489

C₂

semidirect

4. #20272812

C₂

semidirect

7. C₂×#13133157

C₄

semidirect

9. C₂²×#11719908

C₄

semidirect

118. C₅₃

#12832

semidirect

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

subgroup	index
C₂×#16811040	2
#20276489	2
#20274761	2
#20272812	2
C₅₃×#1813	2
#12832	53

subgroup

index

C₅₃×#1813

Subgroups — #27300249

Conjugacy classes of subgroups

Complements

Maximal subgroups