Group #27299214 — subgroups

Subgroups — #27299214

Order 13568.

382 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 151 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	#20276336	C₂	1	1	yes	maximal, characteristic
3	#20276320	C₂	1	2	yes	maximal, complemented
4	#20272804	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1757	C₂	1	1	yes	maximal, characteristic
6	#16811040	—	2	4	no	—
7	C₂×#13133157	C₄	1	2	yes	complemented
8	C₂×#13133157	C₂²	1	2	yes	—
9	C₂×#13133157	C₂²	1	1	yes	characteristic
10	C₂²×#11719908	C₄	1	2	yes	complemented
11	C₂²×#11719908	C₂²	1	1	yes	characteristic
12	#16810405	C₄	1	4	yes	complemented
13	#16810401	—	2	4	no	—
14	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#413	C₄	1	1	yes	characteristic
16	C₅₃×#410	C₄	1	1	yes	characteristic
17	C₅₃×#405	C₂²	1	2	yes	—
18	C₅₃×#344	C₄	1	2	yes	—
19	#13133157	—	2	2	no	—
20	#13133157	—	2	4	no	—
21	#13133157	C₂×C₄	1	4	yes	—
22	#13133157	D₄	1	4	yes	—
23	C₂×#11719908	—	2	2	no	—
24	C₂×#11719908	—	2	2	no	—
25	C₂×#11719908	—	2	4	no	—
26	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
27	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
28	C₂×#11719908	C₂×C₄	1	2	yes	—
29	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
30	#131×C₅₃	—	2	2	no	—
31	#131×C₅₃	—	2	2	no	—
32	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
37	#117×C₅₃	C₂×C₄	1	2	yes	—
38	#101×C₅₃	—	2	1	no	—
39	#100×C₅₃	—	2	1	no	—
40	#99×C₅₃	—	2	2	no	—
41	#97×C₅₃	—	2	2	no	—
42	C₄×C₈×C₅₃	—	2	2	no	—
43	#41×C₅₃	—	2	2	no	—
44	#41×C₅₃	—	2	2	no	—
45	C₂×Q₈×C₅₃	M₁₆	1	1	yes	characteristic
46	C₂×D₄×C₅₃	M₁₆	1	2	yes	—
47	C₂×D₄×C₅₃	M₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	M₁₆×C₅₃	—	2	2	no	—
51	M₁₆×C₅₃	—	2	4	no	—
52	M₁₆×C₅₃	—	2	2	no	—
53	M₁₆×C₅₃	—	2	2	no	—
54	C₂×C₈×C₅₃	—	2	2	no	—
55	C₂×C₈×C₅₃	—	2	2	no	—
56	C₂×C₈×C₅₃	—	2	2	no	—
57	C₂×C₈×C₅₃	C₄²	1	2	yes	—
58	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₅₃	#41	1	2	yes	—
62	C₄⋊C₄×C₅₃	—	2	2	no	—
63	C₂²⋊C₄×C₅₃	—	2	2	no	—
64	C₄²×C₅₃	—	2	2	no	—
65	#11719908	—	2	1	no	—
66	#11719908	—	2	1	no	—
67	#11719908	—	2	4	no	—
68	#11719908	C₂²⋊C₄	1	4	yes	—
69	C₂³×C₅₃	#97	1	2	yes	—
70	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
71	Q₈×C₅₃	—	2	2	no	—
72	D₄×C₅₃	—	2	4	no	—
73	D₄×C₅₃	—	2	2	no	—
74	C₂×C₄×C₅₃	—	2	2	no	—
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	4	2	no	—
77	C₂×C₄×C₅₃	#97	1	2	yes	—
78	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
80	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#131	1	2	yes	—
82	C₈×C₅₃	—	2	2	no	—
83	C₈×C₅₃	—	4	2	no	—
84	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₈×C₅₃	#117	1	1	yes	characteristic
86	C₈×C₅₃	C₄×D₄	1	2	yes	—
87	#12617	—	53	1	no	maximal, Sylow
88	C₂²×C₅₃	—	2	2	no	—
89	C₂²×C₅₃	—	4	1	no	—
90	C₂²×C₅₃	—	4	2	no	—
91	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
92	C₂²×C₅₃	#435	1	2	yes	—
93	C₄×C₅₃	—	2	2	no	—
94	C₄×C₅₃	—	8	2	no	—
95	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
96	C₄×C₅₃	#405	1	1	yes	characteristic
97	C₄×C₅₃	#435	1	2	yes	—
98	C₂×#504	—	53	1	no	—
99	#2115	—	53	1	no	—
100	#2110	—	53	2	no	—
101	#1757	D₅₃	1	1	yes	characteristic
102	#1379	—	53	2	no	—
103	C₂×C₅₃	—	2	2	no	—
104	C₂×C₅₃	—	2	1	no	—
105	C₂×C₅₃	—	4	2	no	—
106	C₂×C₅₃	#1752	1	1	yes	characteristic
107	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
108	#504	—	106	4	no	—
109	C₂×M₃₂	—	53	1	no	—
110	C₂×M₃₂	—	53	2	no	—
111	C₂×M₃₂	—	53	2	no	—
112	C₂²×C₁₆	—	53	2	no	—
113	C₂²×C₁₆	—	53	1	no	—
114	#413	Dic₅₃	1	1	yes	characteristic
115	#410	Dic₅₃	1	1	yes	characteristic
116	#405	C₂×D₅₃	1	2	yes	—
117	#349	—	53	4	no	—
118	#347	—	106	4	no	—
119	#344	Dic₅₃	1	2	yes	—
120	C₅₃	#12617	1	1	yes	characteristic, complemented, Sylow
121	C₂×#41	C₂×Dic₅₃	1	1	yes	characteristic
122	#131	—	2	2	no	—
123	#131	—	2	2	no	—
124	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
125	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
126	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
127	C₂²×C₈	C₄×D₅₃	1	2	yes	—
128	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
129	#117	C₂×Dic₅₃	1	2	yes	—
130	M₃₂	—	53	4	no	—
131	M₃₂	—	53	4	no	—
132	M₃₂	—	106	4	no	—
133	M₃₂	—	106	2	no	—
134	C₂×C₁₆	—	53	2	no	—
135	C₂×C₁₆	—	53	1	no	—
136	C₂×C₁₆	—	53	1	no	—
137	C₂×C₁₆	—	106	4	no	—
138	C₂×C₁₆	—	106	2	no	—
139	C₂×C₁₆	—	106	2	no	—
140	#101	—	2	1	no	—
141	#100	—	2	1	no	—
142	#99	—	2	2	no	—
143	#97	—	2	2	no	—
144	C₄×C₈	—	2	2	no	—
145	#41	—	2	2	no	—
146	#41	—	2	2	no	—
147	C₂×Q₈	#11719917	1	1	yes	characteristic
148	C₂×D₄	#11719917	1	2	yes	—
149	C₂×D₄	#11719917	1	1	yes	characteristic
150	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
151	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
152	M₁₆	—	2	2	no	—
153	M₁₆	—	2	4	no	—
154	M₁₆	—	2	2	no	—
155	M₁₆	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
160	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
161	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
162	C₂×C₈	D₄×D₅₃	1	2	yes	—
163	C₂×C₈	#11719947	1	2	yes	—
164	C₄⋊C₄	—	2	2	no	—
165	C₂²⋊C₄	—	2	2	no	—
166	C₄²	—	2	2	no	—
167	C₁₆	—	53	4	no	—
168	C₁₆	—	106	1	no	—
169	C₁₆	—	106	1	no	—
170	C₁₆	—	106	4	no	—
171	C₂³	#13133128	1	2	yes	—
172	C₂³	C₂×#11719917	1	1	yes	characteristic
173	Q₈	—	2	2	no	—
174	D₄	—	2	4	no	—
175	D₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	#13133128	1	2	yes	—
180	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
181	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
182	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
183	C₂×C₄	#13133224	1	2	yes	—
184	C₈	—	2	2	no	—
185	C₈	—	4	2	no	—
186	C₈	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
187	C₈	#13133173	1	1	yes	characteristic
188	C₈	#13133177	1	2	yes	—
189	C₂²	—	2	2	no	—
190	C₂²	—	4	1	no	—
191	C₂²	—	4	2	no	—
192	C₂²	C₂×#13133128	1	1	yes	characteristic
193	C₂²	#16810695	1	2	yes	—
194	C₄	—	2	2	no	—
195	C₄	—	8	2	no	—
196	C₄	C₂²⋊C₄×Dic₅₃	1	1	yes	characteristic
197	C₄	#16810579	1	1	yes	characteristic
198	C₄	#16810695	1	2	yes	—
199	C₂	—	2	2	no	—
200	C₂	—	2	1	no	—
201	C₂	—	4	2	no	—
202	C₂	#20274493	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811040

C₂

yes

maximal, characteristic

#20276336

C₂

yes

maximal, characteristic

#20276320

C₂

yes

maximal, complemented

#20272804

C₂

yes

maximal, complemented

C₅₃×#1757

C₂

yes

maximal, characteristic

#16811040

—

C₂×#13133157

C₄

yes

complemented

C₂×#13133157

C₂²

yes

—

C₂×#13133157

C₂²

yes

characteristic

C₂²×#11719908

C₄

yes

complemented

C₂²×#11719908

C₂²

yes

characteristic

#16810405

C₄

yes

complemented

#16810401

—

C₂×#131×C₅₃

C₂²

yes

characteristic

C₅₃×#413

C₄

yes

characteristic

C₅₃×#410

C₄

yes

characteristic

C₅₃×#405

C₂²

yes

—

C₅₃×#344

C₄

yes

—

#13133157

—

#13133157

—

#13133157

C₂×C₄

yes

—

#13133157

D₄

yes

—

—

—

—

C₂×C₄

yes

characteristic

C₂×#11719908

C₂×C₄

yes

characteristic

C₂×#11719908

C₂×C₄

yes

—

C₂×#41×C₅₃

C₂×C₄

yes

characteristic

#131×C₅₃

—

#131×C₅₃

—

C₂×M₁₆×C₅₃

C₂×C₄

yes

—

C₂×M₁₆×C₅₃

C₂×C₄

yes

—

C₂×M₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

#117×C₅₃

C₂×C₄

yes

—

#101×C₅₃

—

#100×C₅₃

—

#99×C₅₃

—

#97×C₅₃

—

C₄×C₈×C₅₃

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

M₁₆

yes

characteristic

C₂×D₄×C₅₃

M₁₆

yes

—

C₂×D₄×C₅₃

M₁₆

yes

characteristic

C₂²×C₄×C₅₃

C₄²

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

#41

yes

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

—

—

—

C₂²⋊C₄

yes

—

C₂³×C₅₃

#97

yes

—

C₂³×C₅₃

C₂×M₁₆

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#97

yes

—

C₂×C₄×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₃

#131

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₅₃

#117

yes

characteristic

C₈×C₅₃

C₄×D₄

yes

—

#12617

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×#97

yes

characteristic

C₂²×C₅₃

#435

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×C₂²⋊C₄

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×#504

—

#2115

—

100

#2110

—

101

#1757

D₅₃

yes

characteristic

102

#1379

—

103

C₂×C₅₃

—

104

C₂×C₅₃

—

105

C₂×C₅₃

—

106

C₂×C₅₃

#1752

yes

characteristic

107

C₂×#131

C₂×D₅₃

yes

characteristic

108

#504

—

106

—

109

C₂×M₃₂

—

110

C₂×M₃₂

—

111

C₂×M₃₂

—

112

C₂²×C₁₆

—

113

C₂²×C₁₆

—

114

#413

Dic₅₃

yes

characteristic

115

#410

Dic₅₃

yes

characteristic

116

#405

C₂×D₅₃

yes

—

117

#349

—

118

#347

—

106

—

119

#344

Dic₅₃

yes

—

120

C₅₃

#12617

yes

characteristic, complemented, Sylow

121

C₂×#41

C₂×Dic₅₃

yes

characteristic

122

#131

—

123

#131

—

124

C₂×M₁₆

C₄×D₅₃

yes

—

125

C₂×M₁₆

C₂×Dic₅₃

yes

—

126

C₂×M₁₆

C₂×Dic₅₃

yes

characteristic

127

C₂²×C₈

C₄×D₅₃

yes

—

128

C₂²×C₈

C₂²×D₅₃

yes

characteristic

129

#117

C₂×Dic₅₃

yes

—

130

M₃₂

—

131

M₃₂

—

132

M₃₂

—

106

—

133

M₃₂

—

106

—

134

C₂×C₁₆

—

135

C₂×C₁₆

—

136

C₂×C₁₆

—

137

C₂×C₁₆

—

106

—

138

C₂×C₁₆

—

106

—

139

C₂×C₁₆

—

106

—

140

#101

—

141

#100

—

142

#99

—

143

#97

—

144

C₄×C₈

—

145

#41

—

146

#41

—

147

C₂×Q₈

#11719917

yes

characteristic

148

C₂×D₄

#11719917

yes

—

149

C₂×D₄

#11719917

yes

characteristic

150

C₂²×C₄

C₄×Dic₅₃

yes

—

151

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

152

M₁₆

—

153

M₁₆

—

154

M₁₆

—

155

M₁₆

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

C₄×Dic₅₃

yes

—

160

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

161

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

162

C₂×C₈

D₄×D₅₃

yes

—

163

C₂×C₈

#11719947

yes

—

164

C₄⋊C₄

—

165

C₂²⋊C₄

—

166

C₄²

—

167

C₁₆

—

168

C₁₆

—

106

—

169

C₁₆

—

106

—

170

C₁₆

—

106

—

171

C₂³

#13133128

yes

—

172

C₂³

C₂×#11719917

yes

characteristic

173

Q₈

—

174

D₄

—

175

D₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

#13133128

yes

—

180

C₂×C₄

C₂×C₄×Dic₅₃

yes

characteristic

181

C₂×C₄

D₄×Dic₅₃

yes

—

182

C₂×C₄

C₂×#11719917

yes

characteristic

183

C₂×C₄

#13133224

yes

—

184

C₈

—

185

C₈

—

186

C₈

C₂²⋊C₄×D₅₃

yes

characteristic

187

C₈

#13133173

yes

characteristic

188

C₈

#13133177

yes

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

192

C₂²

C₂×#13133128

yes

characteristic

193

C₂²

#16810695

yes

—

194

C₄

—

195

C₄

—

196

C₄

C₂²⋊C₄×Dic₅₃

yes

characteristic

197

C₄

#16810579

yes

characteristic

198

C₄

#16810695

yes

—

199

C₂

—

200

C₂

—

201

C₂

—

202

C₂

#20274493

yes

characteristic

	subgroup	G/H
center	C₈	#13133173
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1757	C₂
socle	C₂×C₅₃	#1752
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1757	D₅₃
hypocenter	C₅₃	#12617
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1757

subgroup

G/H

center

C₈

#13133173

commutator

C₂×C₄×C₅₃

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#1757

C₂

socle

C₂×C₅₃

#1752

prime radical

C₂×#41×C₅₃

C₂×C₄

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1757

N	H ≅ G/N	type
3. #20276320	C₂	semidirect
4. #20272804	C₂	semidirect
7. C₂×#13133157	C₄	semidirect
10. C₂²×#11719908	C₄	semidirect
12. #16810405	C₄	semidirect
120. C₅₃	#12617	semidirect

H ≅ G/N

type

3. #20276320

C₂

semidirect

4. #20272804

C₂

semidirect

7. C₂×#13133157

C₄

semidirect

10. C₂²×#11719908

C₄

semidirect

12. #16810405

C₄

semidirect

120. C₅₃

#12617

semidirect

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

subgroup	index
C₂×#16811040	2
#20276336	2
#20276320	2
#20272804	2
C₅₃×#1757	2
#12617	53

subgroup

index

C₅₃×#1757

Subgroups — #27299214

Conjugacy classes of subgroups

Complements

Maximal subgroups