Group #27303481 — subgroups

Subgroups — #27303481

Order 13568.

436 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 101 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×#16810439	C₂	1	2	yes	maximal
2	#20276435	C₂	1	2	yes	maximal
3	#20275463	C₂	1	1	yes	maximal, characteristic
4	#20272797	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₃×#494	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	2	yes	—
7	C₂×#13133153	C₂²	1	1	yes	characteristic
8	#16810661	C₂²	1	1	yes	characteristic
9	#16810649	—	2	1	no	—
10	#16810439	—	2	4	no	—
11	#16810427	—	2	4	no	—
12	#16810383	—	2	1	no	—
13	C₂²×Q₁₆×C₅₃	—	2	1	no	—
14	C₂²×Q₁₆×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
16	C₅₃×#494	—	2	4	no	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×#11719917	—	2	1	no	—
19	#13133176	—	2	1	no	—
20	#13133157	—	2	2	no	—
21	#13133157	—	2	2	no	—
22	C₂×#11719908	—	2	2	no	—
23	#13133153	—	2	2	no	—
24	#13133151	D₄	1	2	yes	—
25	#13133149	—	2	1	no	—
26	#13133146	—	4	2	no	—
27	C₂²×Q₈×C₅₃	—	2	2	no	—
28	C₂×Q₁₆×C₅₃	—	2	4	no	—
29	C₂×Q₁₆×C₅₃	—	2	2	no	—
30	C₂×Q₁₆×C₅₃	—	2	2	no	—
31	C₂×Q₁₆×C₅₃	—	2	2	no	—
32	C₂×Q₁₆×C₅₃	—	2	2	no	—
33	C₂×Q₁₆×C₅₃	—	2	2	no	—
34	C₂×Q₁₆×C₅₃	—	4	2	no	—
35	C₂×Q₁₆×C₅₃	—	4	2	no	—
36	C₂×Q₁₆×C₅₃	D₄	1	4	yes	—
37	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
39	C₄⋊Q₈×C₅₃	—	2	4	no	—
40	C₄⋊Q₈×C₅₃	—	4	2	no	—
41	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
42	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₅₃	—	2	2	no	—
44	C₂×Q₈×C₅₃	—	2	4	no	—
45	C₂×Q₈×C₅₃	—	4	2	no	—
46	C₂×Q₈×C₅₃	—	4	2	no	—
47	C₂×Q₈×C₅₃	—	4	2	no	—
48	C₂×Q₈×C₅₃	—	4	2	no	—
49	C₂×Q₈×C₅₃	—	4	2	no	—
50	C₂²×C₄×C₅₃	—	2	1	no	—
51	C₂²×C₄×C₅₃	—	4	2	no	—
52	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	Q₁₆×C₅₃	—	2	4	no	—
54	Q₁₆×C₅₃	—	2	4	no	—
55	Q₁₆×C₅₃	—	4	8	no	—
56	Q₁₆×C₅₃	—	4	2	no	—
57	Q₁₆×C₅₃	—	4	2	no	—
58	C₂×C₈×C₅₃	—	2	1	no	—
59	C₂×C₈×C₅₃	—	2	1	no	—
60	C₂×C₈×C₅₃	D₈	1	2	yes	—
61	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₅₃	#41	1	2	yes	—
64	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
65	C₄⋊C₄×C₅₃	—	4	8	no	—
66	C₄²×C₅₃	—	2	1	no	—
67	C₄²×C₅₃	SD₁₆	1	2	yes	—
68	C₂²⋊Dic₅₃	—	4	1	no	—
69	C₄⋊Dic₅₃	—	2	2	no	—
70	C₄×Dic₅₃	—	4	1	no	—
71	#11719917	—	4	1	no	—
72	#11719917	—	4	1	no	—
73	C₂×#88322	—	4	1	no	—
74	#11719908	—	4	2	no	—
75	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
76	Q₈×C₅₃	—	4	4	no	—
77	Q₈×C₅₃	—	4	4	no	—
78	Q₈×C₅₃	—	8	2	no	—
79	Q₈×C₅₃	—	8	2	no	—
80	C₂×C₄×C₅₃	—	2	4	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	4	4	no	—
83	C₂×C₄×C₅₃	—	8	2	no	—
84	C₂×C₄×C₅₃	—	8	2	no	—
85	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
87	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
91	C₂×Dic₅₃	—	4	2	no	—
92	C₈×C₅₃	—	2	2	no	—
93	C₈×C₅₃	#121	1	2	yes	—
94	C₈×C₅₃	#123	1	2	yes	—
95	#88322	—	8	1	no	—
96	#13415	—	53	1	no	maximal, Sylow
97	C₂²×C₅₃	—	2	2	no	—
98	C₂²×C₅₃	#450	1	1	yes	characteristic
99	C₂²×C₅₃	#459	1	1	yes	characteristic
100	C₂²×C₅₃	#490	1	1	yes	characteristic
101	C₄×C₅₃	—	2	4	no	—
102	C₄×C₅₃	—	8	4	no	—
103	C₄×C₅₃	#394	1	1	yes	characteristic
104	C₄×C₅₃	#447	1	1	yes	characteristic
105	C₄×C₅₃	#460	1	1	yes	characteristic
106	C₄×C₅₃	#486	1	1	yes	characteristic
107	Dic₅₃	—	8	2	no	—
108	C₂×#494	D₅₃	1	1	yes	characteristic
109	C₂×#362	—	53	2	no	—
110	#2141	—	53	2	no	—
111	#1945	—	53	1	no	—
112	#1376	—	53	1	no	—
113	C₂×C₅₃	—	2	1	no	—
114	C₂×C₅₃	—	2	1	no	—
115	C₂×C₅₃	#2041	1	1	yes	characteristic
116	C₂×C₅₃	#2214	1	2	yes	—
117	C₂²×Q₁₆	—	2	1	no	—
118	C₂²×Q₁₆	C₂×D₅₃	1	2	yes	—
119	C₂×C₄⋊Q₈	—	2	1	no	—
120	C₂×M₃₂	—	53	2	no	—
121	#494	—	2	4	no	—
122	C₂×#108	—	53	1	no	—
123	#427	—	53	1	no	—
124	#424	—	106	1	no	—
125	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
126	#362	—	106	4	no	—
127	#358	—	106	4	no	—
128	#339	—	106	1	no	—
129	C₅₃	#13415	1	1	yes	characteristic, complemented, Sylow
130	C₂²×Q₈	—	2	2	no	—
131	C₂×Q₁₆	—	2	4	no	—
132	C₂×Q₁₆	—	2	2	no	—
133	C₂×Q₁₆	—	2	2	no	—
134	C₂×Q₁₆	—	2	2	no	—
135	C₂×Q₁₆	—	2	2	no	—
136	C₂×Q₁₆	—	2	2	no	—
137	C₂×Q₁₆	—	4	2	no	—
138	C₂×Q₁₆	—	4	2	no	—
139	C₂×Q₁₆	C₂²⋊D₅₃	1	4	yes	—
140	C₂×M₁₆	—	106	1	no	—
141	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
142	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
143	C₄⋊Q₈	—	2	4	no	—
144	C₄⋊Q₈	—	4	2	no	—
145	#117	—	106	1	no	—
146	C₂×C₄⋊C₄	—	4	2	no	—
147	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
148	M₃₂	—	106	2	no	—
149	M₃₂	—	106	2	no	—
150	C₂×C₁₆	—	106	2	no	—
151	#108	—	106	2	no	—
152	#107	—	53	2	no	—
153	#106	—	106	1	no	—
154	C₄⋊C₈	—	212	2	no	—
155	C₄×C₈	—	2	2	no	—
156	C₂×Q₈	—	2	4	no	—
157	C₂×Q₈	—	4	2	no	—
158	C₂×Q₈	—	4	2	no	—
159	C₂×Q₈	—	4	2	no	—
160	C₂×Q₈	—	4	2	no	—
161	C₂×Q₈	—	4	2	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
165	Q₁₆	—	2	4	no	—
166	Q₁₆	—	2	4	no	—
167	Q₁₆	—	4	8	no	—
168	Q₁₆	—	4	2	no	—
169	Q₁₆	—	4	2	no	—
170	M₁₆	—	212	1	no	—
171	M₁₆	—	212	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	212	1	no	—
175	C₂×C₈	#11719922	1	2	yes	—
176	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
177	C₂×C₈	#11719947	1	2	yes	—
178	C₂×C₈	#11719947	1	1	yes	characteristic
179	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
180	C₄⋊C₄	—	4	8	no	—
181	C₄⋊C₄	—	106	2	no	—
182	C₂²⋊C₄	—	212	1	no	—
183	C₄²	—	2	1	no	—
184	C₄²	—	212	1	no	—
185	C₄²	#11719923	1	2	yes	—
186	C₁₆	—	212	2	no	—
187	C₂³	#13133243	1	1	yes	characteristic
188	Q₈	—	4	4	no	—
189	Q₈	—	4	4	no	—
190	Q₈	—	8	2	no	—
191	Q₈	—	8	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	—	8	2	no	—
196	C₂×C₄	—	8	2	no	—
197	C₂×C₄	—	212	2	no	—
198	C₂×C₄	#13133186	1	1	yes	characteristic
199	C₂×C₄	#13133188	1	1	yes	characteristic
200	C₂×C₄	#13133204	1	1	yes	characteristic
201	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
202	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
203	C₂×C₄	#13133243	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	424	1	no	—
206	C₈	#13133189	1	2	yes	—
207	C₈	#13133198	1	2	yes	—
208	C₂²	—	2	2	no	—
209	C₂²	#16810774	1	1	yes	characteristic
210	C₂²	#16810815	1	1	yes	characteristic
211	C₂²	#16810987	1	1	yes	characteristic
212	C₄	—	2	4	no	—
213	C₄	—	8	4	no	—
214	C₄	—	424	2	no	—
215	C₄	#16810537	1	1	yes	characteristic
216	C₄	#16810757	1	1	yes	characteristic
217	C₄	#16810820	1	1	yes	characteristic
218	C₄	#16810969	1	1	yes	characteristic
219	C₂	—	2	1	no	—
220	C₂	—	2	1	no	—
221	C₂	#20275946	1	1	yes	characteristic
222	C₂	#20276793	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810439

C₂

yes

maximal

#20276435

C₂

yes

maximal

#20275463

C₂

yes

maximal, characteristic

#20272797

C₂

yes

maximal, characteristic

C₂×C₅₃×#494

C₂

yes

maximal, characteristic

C₂×#13133157

C₂²

yes

—

C₂×#13133153

C₂²

yes

characteristic

#16810661

C₂²

yes

—

—

—

—

C₂²×Q₁₆×C₅₃

—

C₂²×Q₁₆×C₅₃

C₂²

yes

—

C₂×C₄⋊Q₈×C₅₃

—

C₅₃×#494

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

—

—

—

—

—

D₄

yes

—

#13133149

—

#13133146

—

C₂²×Q₈×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

D₄

yes

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

#41

yes

—

C₂×C₈×C₅₃

#41

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

SD₁₆

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

—

—

—

—

C₂³×C₅₃

AGL(1,ℤ/8)

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#121

yes

—

C₈×C₅₃

#123

yes

—

#88322

—

#13415

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

100

C₂²×C₅₃

#490

yes

characteristic

101

C₄×C₅₃

—

102

C₄×C₅₃

—

103

C₄×C₅₃

#394

yes

characteristic

104

C₄×C₅₃

#447

yes

characteristic

105

C₄×C₅₃

#460

yes

characteristic

106

C₄×C₅₃

#486

yes

characteristic

107

Dic₅₃

—

108

C₂×#494

D₅₃

yes

characteristic

109

C₂×#362

—

110

#2141

—

111

#1945

—

112

#1376

—

113

C₂×C₅₃

—

114

C₂×C₅₃

—

115

C₂×C₅₃

#2041

yes

characteristic

116

C₂×C₅₃

#2214

yes

—

117

C₂²×Q₁₆

—

118

C₂²×Q₁₆

C₂×D₅₃

yes

—

119

C₂×C₄⋊Q₈

—

120

C₂×M₃₂

—

121

#494

—

122

C₂×#108

—

123

#427

—

124

#424

—

106

—

125

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

126

#362

—

106

—

127

#358

—

106

—

128

#339

—

106

—

129

C₅₃

#13415

yes

characteristic, complemented, Sylow

130

C₂²×Q₈

—

131

C₂×Q₁₆

—

132

C₂×Q₁₆

—

133

C₂×Q₁₆

—

134

C₂×Q₁₆

—

135

C₂×Q₁₆

—

136

C₂×Q₁₆

—

137

C₂×Q₁₆

—

138

C₂×Q₁₆

—

139

C₂×Q₁₆

C₂²⋊D₅₃

yes

—

140

C₂×M₁₆

—

106

—

141

C₂²×C₈

C₂²⋊D₅₃

yes

characteristic

142

C₂²×C₈

C₂²×D₅₃

yes

characteristic

143

C₄⋊Q₈

—

144

C₄⋊Q₈

—

145

#117

—

106

—

146

C₂×C₄⋊C₄

—

147

C₂×C₄²

C₂²⋊D₅₃

yes

characteristic

148

M₃₂

—

106

—

149

M₃₂

—

106

—

150

C₂×C₁₆

—

106

—

151

#108

—

106

—

152

#107

—

153

#106

—

106

—

154

C₄⋊C₈

—

212

—

155

C₄×C₈

—

156

C₂×Q₈

—

157

C₂×Q₈

—

158

C₂×Q₈

—

159

C₂×Q₈

—

160

C₂×Q₈

—

161

C₂×Q₈

—

162

C₂²×C₄

—

163

C₂²×C₄

—

164

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

165

Q₁₆

—

166

Q₁₆

—

167

Q₁₆

—

168

Q₁₆

—

169

Q₁₆

—

170

M₁₆

—

212

—

171

M₁₆

—

212

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

212

—

175

C₂×C₈

#11719922

yes

—

176

C₂×C₈

D₄×D₅₃

yes

characteristic

177

C₂×C₈

#11719947

yes

—

178

C₂×C₈

#11719947

yes

characteristic

179

C₂×C₈

C₂×C₂²⋊D₅₃

yes

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

106

—

182

C₂²⋊C₄

—

212

—

183

C₄²

—

184

C₄²

—

212

—

185

C₄²

#11719923

yes

—

186

C₁₆

—

212

—

187

C₂³

#13133243

yes

characteristic

188

Q₈

—

189

Q₈

—

190

Q₈

—

191

Q₈

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

212

—

198

C₂×C₄

#13133186

yes

characteristic

199

C₂×C₄

#13133188

yes

characteristic

200

C₂×C₄

#13133204

yes

characteristic

201

C₂×C₄

C₂×#11719922

yes

characteristic

202

C₂×C₄

C₂×#11719923

yes

characteristic

203

C₂×C₄

#13133243

yes

characteristic

204

C₈

—

205

C₈

—

424

—

206

C₈

#13133189

yes

—

207

C₈

#13133198

yes

—

208

C₂²

—

209

C₂²

#16810774

yes

characteristic

210

C₂²

#16810815

yes

characteristic

211

C₂²

#16810987

yes

characteristic

212

C₄

—

213

C₄

—

214

C₄

—

424

—

215

C₄

#16810537

yes

characteristic

216

C₄

#16810757

yes

characteristic

217

C₄

#16810820

yes

characteristic

218

C₄

#16810969

yes

characteristic

219

C₂

—

220

C₂

—

221

C₂

#20275946

yes

characteristic

222

C₂

#20276793

yes

—

	subgroup	G/H
center	C₂²	#16810815
commutator, cosocle	C₂²×C₈×C₅₃	C₂³
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#494	C₂
socle	C₂²×C₅₃	#459
prime radical	C₂³×C₅₃	AGL(1,ℤ/8)
hypercenter	C₂×#494	D₅₃
hypocenter	C₅₃	#13415
LCS	G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#494

subgroup

G/H

center

C₂²

#16810815

commutator, cosocle

C₂²×C₈×C₅₃

C₂³

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₂×C₅₃×#494

C₂

C₂²×C₅₃

C₂³×C₅₃

D₅₃

C₅₃

G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#494

N	H ≅ G/N	type
129. C₅₃	#13415	semidirect

H ≅ G/N

type

129. C₅₃

#13415

semidirect

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

subgroup	index
C₂×#16810439	2
#20276435	2
#20275463	2
#20272797	2
C₂×C₅₃×#494	2
#13415	53

subgroup

index

C₂×C₅₃×#494

Subgroups — #27303481

Conjugacy classes of subgroups

Complements

Maximal subgroups