Group #28305663 — subgroups

Subgroups — #28305663

Order 13568.

322 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 98 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	#20290656	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊#6157	C₂	1	1	yes	maximal, characteristic
3	#20290283	C₂	1	2	yes	maximal
4	#20289612	C₂	1	2	yes	maximal
5	#20273462	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133288	—	2	1	no	—
7	C₂×#13133288	—	2	1	no	—
8	#16811627	C₂²	1	2	yes	—
9	D₅₃⋊(C₄⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×C₈×#6157	—	2	1	no	—
11	C₄×C₄⋊#6157	C₂²	1	1	yes	characteristic
12	(C₂×#6157)⋊C₈	—	2	1	no	—
13	#16811485	—	2	2	no	—
14	#16811484	—	2	2	no	—
15	#16810675	C₂²	1	2	yes	—
16	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
17	#16810376	C₄	1	1	yes	characteristic, complemented
18	#16810373	C₄	1	2	yes	complemented
19	C₅₃×#335	C₄	1	1	yes	characteristic, complemented
20	C₂×#11719936	—	2	2	no	—
21	C₂×D₅₃⋊C₈	—	2	1	no	—
22	C₂×C₄⋊#6157	—	2	1	no	—
23	C₂×C₄×#6157	—	2	1	no	—
24	#13133288	—	4	2	no	—
25	#13133288	—	4	2	no	—
26	#13133284	—	2	1	no	—
27	C₄⋊#88324	—	2	2	no	—
28	C₄⋊#88324	—	2	2	no	—
29	C₄⋊#88324	D₄	1	1	yes	characteristic
30	C₄⋊#88324	D₄	1	1	yes	characteristic
31	C₈×#6157	—	4	2	no	—
32	(C₂×#6157)⋊C₄	Q₈	1	2	yes	—
33	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
34	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
35	C₄²×D₅₃	C₂³	1	1	yes	characteristic
36	C₄⋊#88322	C₂×C₄	1	2	yes	—
37	#13133129	—	2	2	no	—
38	C₄×#88322	C₂×C₄	1	1	yes	characteristic
39	C₈×Dic₅₃	—	2	2	no	—
40	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
41	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
42	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×#6157	—	2	2	no	—
45	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
48	C₂×#88324	—	2	1	no	—
49	C₂×#88324	—	2	1	no	—
50	C₂×#88324	—	4	2	no	—
51	C₄⋊#6157	—	2	2	no	—
52	C₄⋊#6157	—	2	2	no	—
53	C₄×#6157	—	4	1	no	—
54	C₄×#6157	—	4	1	no	—
55	#11719936	—	4	2	no	—
56	#11719936	—	4	2	no	—
57	D₅₃⋊C₈	—	4	1	no	—
58	D₅₃⋊C₈	—	4	1	no	—
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
62	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
64	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
65	C₄×Dic₅₃	#41	1	1	yes	characteristic
66	C₂×#88322	—	2	2	no	—
67	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
68	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
69	C₈×D₅₃	—	2	1	no	—
70	C₈×D₅₃	—	2	1	no	—
71	C₈×D₅₃	SD₁₆	1	2	yes	—
72	C₈×D₅₃	SD₁₆	1	2	yes	—
73	C₂²×D₅₃	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂×#6157	—	2	4	no	—
75	C₂×#6157	—	4	2	no	—
76	C₂×#6157	—	4	2	no	—
77	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
80	C₂×Dic₅₃	#121	1	1	yes	characteristic
81	C₂×Dic₅₃	#125	1	1	yes	characteristic
82	C₂×Dic₅₃	C₄⋊Q₈	1	1	yes	characteristic
83	C₄×D₅₃	—	2	1	no	—
84	C₄×D₅₃	—	2	1	no	—
85	C₄×D₅₃	—	4	1	no	—
86	C₄×D₅₃	C₂×SD₁₆	1	1	yes	characteristic
87	C₄×D₅₃	C₂×SD₁₆	1	1	yes	characteristic
88	C₄×D₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
89	C₄×D₅₃	#137	1	1	yes	characteristic
90	#88324	—	4	1	no	—
91	#88324	—	4	1	no	—
92	#88324	—	8	2	no	—
93	C₈×C₅₃	—	2	1	no	—
94	C₈×C₅₃	—	4	2	no	—
95	C₈×C₅₃	#106	1	2	yes	—
96	#88322	—	2	1	no	—
97	#88322	—	4	2	no	—
98	#88322	#106	1	2	yes	—
99	#9628	—	53	1	no	maximal, Sylow
100	C₂²×C₅₃	#389	1	1	yes	characteristic
101	C₂×D₅₃	—	2	1	no	—
102	C₂×D₅₃	—	2	1	no	—
103	C₂×D₅₃	#475	1	1	yes	characteristic
104	C₂×D₅₃	#476	1	1	yes	characteristic
105	#6157	—	4	4	no	—
106	C₄×C₅₃	—	2	1	no	—
107	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
108	C₄×C₅₃	#428	1	1	yes	characteristic
109	C₄×C₅₃	#443	1	1	yes	characteristic
110	C₄×C₅₃	#443	1	1	yes	characteristic
111	Dic₅₃	—	2	1	no	—
112	Dic₅₃	—	2	1	no	—
113	Dic₅₃	#460	1	1	yes	characteristic
114	Dic₅₃	#461	1	1	yes	characteristic
115	#1848	—	53	1	no	—
116	C₈×C₄⋊C₄	—	53	1	no	—
117	#1570	—	53	2	no	—
118	#1564	—	53	1	no	—
119	#1303	—	53	2	no	—
120	C₂×C₅₃	#1945	1	1	yes	characteristic
121	C₂×C₅₃	#1948	1	1	yes	characteristic
122	C₂×C₅₃	#1953	1	1	yes	characteristic
123	D₅₃	—	2	1	no	—
124	D₅₃	—	2	1	no	—
125	#431	—	53	2	no	—
126	C₂×#108	—	106	1	no	—
127	C₂×#108	—	106	1	no	—
128	#423	—	53	2	no	—
129	C₂×C₄⋊C₈	—	53	1	no	—
130	C₂×C₄×C₈	—	53	1	no	—
131	C₂×C₄×C₈	—	106	1	no	—
132	C₄×C₄⋊C₄	—	53	1	no	—
133	#336	—	106	1	no	—
134	#335	—	53	1	no	—
135	#335	#6157	1	1	yes	characteristic, complemented
136	#334	—	53	2	no	—
137	#334	—	106	2	no	—
138	#334	—	106	2	no	—
139	C₅₃	#9628	1	1	yes	characteristic, complemented, Sylow
140	C₂×M₁₆	—	106	2	no	—
141	C₂²×C₈	—	53	1	no	—
142	C₂²×C₈	—	53	1	no	—
143	C₂²×C₈	—	106	1	no	—
144	C₂×C₄⋊C₄	—	106	1	no	—
145	C₂×C₄²	—	53	1	no	—
146	C₂×C₄²	—	106	1	no	—
147	#108	—	212	2	no	—
148	#108	—	212	2	no	—
149	C₄⋊C₈	—	53	1	no	—
150	C₄⋊C₈	—	53	1	no	—
151	C₄⋊C₈	—	53	2	no	—
152	C₄⋊C₈	—	106	1	no	—
153	C₄⋊C₈	—	106	2	no	—
154	C₄⋊C₈	—	106	2	no	—
155	C₄⋊C₈	C₂×#6157	1	2	yes	—
156	C₂²⋊C₈	—	106	2	no	—
157	C₄×C₈	—	53	1	no	—
158	C₄×C₈	—	53	1	no	—
159	C₄×C₈	—	53	1	no	—
160	C₄×C₈	—	106	2	no	—
161	C₄×C₈	—	212	2	no	—
162	C₄×C₈	C₂×#6157	1	1	yes	characteristic
163	#95	—	53	2	no	—
164	C₂²×C₄	—	53	1	no	—
165	C₂²×C₄	—	53	1	no	—
166	C₂²×C₄	—	53	1	no	—
167	C₂²×C₄	—	106	2	no	—
168	M₁₆	—	212	2	no	—
169	M₁₆	—	212	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	53	1	no	—
172	C₂×C₈	—	53	2	no	—
173	C₂×C₈	—	53	2	no	—
174	C₂×C₈	—	53	1	no	—
175	C₂×C₈	—	106	1	no	—
176	C₂×C₈	—	106	1	no	—
177	C₂×C₈	—	106	1	no	—
178	C₂×C₈	—	106	1	no	—
179	C₂×C₈	—	106	2	no	—
180	C₂×C₈	—	212	2	no	—
181	C₂×C₈	—	212	1	no	—
182	C₂×C₈	—	212	1	no	—
183	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
184	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
185	C₄⋊C₄	—	106	2	no	—
186	C₄⋊C₄	—	106	2	no	—
187	C₄²	—	53	1	no	—
188	C₄²	—	53	1	no	—
189	C₄²	—	53	1	no	—
190	C₄²	—	212	1	no	—
191	C₄²	—	212	1	no	—
192	C₄²	C₂²×#6157	1	1	yes	characteristic
193	C₂³	—	53	1	no	—
194	C₂×C₄	—	53	1	no	—
195	C₂×C₄	—	53	1	no	—
196	C₂×C₄	—	53	1	no	—
197	C₂×C₄	—	53	1	no	—
198	C₂×C₄	—	53	1	no	—
199	C₂×C₄	—	53	1	no	—
200	C₂×C₄	—	53	1	no	—
201	C₂×C₄	—	106	1	no	—
202	C₂×C₄	—	106	1	no	—
203	C₂×C₄	—	106	4	no	—
204	C₂×C₄	—	212	1	no	—
205	C₂×C₄	—	212	2	no	—
206	C₂×C₄	—	212	2	no	—
207	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
208	C₂×C₄	D₄×#6157	1	1	yes	characteristic
209	C₂×C₄	Q₈×#6157	1	1	yes	characteristic
210	C₈	—	2	1	no	—
211	C₈	—	4	2	no	—
212	C₈	—	53	2	no	—
213	C₈	—	106	1	no	—
214	C₈	—	424	2	no	—
215	C₈	—	212	1	no	—
216	C₈	—	212	1	no	—
217	C₈	—	212	2	no	—
218	C₈	#13133285	1	2	yes	—
219	C₂²	—	53	1	no	—
220	C₂²	—	53	1	no	—
221	C₂²	—	106	1	no	—
222	C₂²	—	106	1	no	—
223	C₂²	#16811574	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	53	1	no	—
226	C₄	—	53	1	no	—
227	C₄	—	106	1	no	—
228	C₄	—	106	1	no	—
229	C₄	—	212	4	no	—
230	C₄	C₂×#13133285	1	1	yes	characteristic
231	C₄	#16811634	1	1	yes	characteristic
232	C₄	#16811665	1	1	yes	characteristic
233	C₄	#16811667	1	1	yes	characteristic
234	C₂	—	106	1	no	—
235	C₂	—	106	1	no	—
236	C₂	#20290823	1	1	yes	characteristic
237	C₂	#20290827	1	1	yes	characteristic
238	C₂	#20290835	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20290656

C₂

yes

maximal, characteristic

C₈×C₄⋊#6157

C₂

yes

maximal, characteristic

#20290283

C₂

yes

maximal

#20289612

C₂

yes

maximal

#20273462

C₂

yes

maximal, characteristic

C₂×#13133288

—

C₂×#13133288

—

#16811627

C₂²

yes

—

D₅₃⋊(C₄⋊C₈)

C₂²

yes

characteristic

C₂×C₈×#6157

—

C₄×C₄⋊#6157

C₂²

yes

—

—

—

C₂²

yes

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810376

C₄

yes

characteristic, complemented

#16810373

C₄

yes

complemented

C₅₃×#335

C₄

yes

characteristic, complemented

C₂×#11719936

—

C₂×D₅₃⋊C₈

—

C₂×C₄⋊#6157

—

C₂×C₄×#6157

—

—

—

—

—

—

D₄

yes

characteristic

C₄⋊#88324

D₄

yes

characteristic

C₈×#6157

—

(C₂×#6157)⋊C₄

Q₈

yes

—

C₂×C₈×D₅₃

D₄

yes

characteristic

C₂×C₈×D₅₃

D₄

yes

characteristic

C₄²×D₅₃

C₂³

yes

characteristic

C₄⋊#88322

C₂×C₄

yes

—

#13133129

—

C₄×#88322

C₂×C₄

yes

characteristic

C₈×Dic₅₃

—

C₈×Dic₅₃

Q₈

yes

characteristic

C₈×Dic₅₃

Q₈

yes

characteristic

C₄⋊C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×#6157

—

C₂×C₄×D₅₃

C₂×D₄

yes

characteristic

C₂×C₄×D₅₃

C₂×Q₈

yes

C₂×C₄×D₅₃

yes

—

—

—

—

—

—

—

—

—

D₅₃⋊C₈

—

D₅₃⋊C₈

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄×Dic₅₃

C₂×D₄

yes

characteristic

C₄×Dic₅₃

C₂×Q₈

yes

C₄×Dic₅₃

yes

—

C₄⋊C₄

yes

characteristic

C₂×#88322

C₄⋊C₄

yes

characteristic

C₈×D₅₃

—

C₈×D₅₃

—

C₈×D₅₃

SD₁₆

yes

—

C₈×D₅₃

SD₁₆

yes

—

C₂²×D₅₃

C₂²⋊Q₈

yes

—

—

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂×C₄×C₅₃

C₄×Q₈

yes

C₂×Dic₅₃

yes

C₂×Dic₅₃

yes

C₂×Dic₅₃

C₄⋊Q₈

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

C₂×SD₁₆

yes

characteristic

C₄×D₅₃

C₂×SD₁₆

yes

C₄×D₅₃

yes

C₄×D₅₃

yes

—

—

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#106

yes

—

—

—

yes

—

#9628

—

maximal, Sylow

100

C₂²×C₅₃

#389

yes

characteristic

101

C₂×D₅₃

—

102

C₂×D₅₃

—

103

C₂×D₅₃

#475

yes

characteristic

104

C₂×D₅₃

#476

yes

characteristic

105

#6157

—

106

C₄×C₅₃

—

107

C₄×C₅₃

C₂×#106

yes

characteristic

108

C₄×C₅₃

#428

yes

characteristic

109

C₄×C₅₃

#443

yes

characteristic

110

C₄×C₅₃

#443

yes

characteristic

111

Dic₅₃

—

112

Dic₅₃

—

113

Dic₅₃

#460

yes

characteristic

114

Dic₅₃

#461

yes

characteristic

115

#1848

—

116

C₈×C₄⋊C₄

—

117

#1570

—

118

#1564

—

119

#1303

—

120

C₂×C₅₃

#1945

yes

characteristic

121

C₂×C₅₃

#1948

yes

characteristic

122

C₂×C₅₃

#1953

yes

characteristic

123

D₅₃

—

124

D₅₃

—

125

#431

—

126

C₂×#108

—

106

—

127

C₂×#108

—

106

—

128

#423

—

129

C₂×C₄⋊C₈

—

130

C₂×C₄×C₈

—

131

C₂×C₄×C₈

—

106

—

132

C₄×C₄⋊C₄

—

133

#336

—

106

—

134

#335

—

135

#335

#6157

yes

characteristic, complemented

136

#334

—

137

#334

—

106

—

138

#334

—

106

—

139

C₅₃

#9628

yes

characteristic, complemented, Sylow

140

C₂×M₁₆

—

106

—

141

C₂²×C₈

—

142

C₂²×C₈

—

143

C₂²×C₈

—

106

—

144

C₂×C₄⋊C₄

—

106

—

145

C₂×C₄²

—

146

C₂×C₄²

—

106

—

147

#108

—

212

—

148

#108

—

212

—

149

C₄⋊C₈

—

150

C₄⋊C₈

—

151

C₄⋊C₈

—

152

C₄⋊C₈

—

106

—

153

C₄⋊C₈

—

106

—

154

C₄⋊C₈

—

106

—

155

C₄⋊C₈

C₂×#6157

yes

—

156

C₂²⋊C₈

—

106

—

157

C₄×C₈

—

158

C₄×C₈

—

159

C₄×C₈

—

160

C₄×C₈

—

106

—

161

C₄×C₈

—

212

—

162

C₄×C₈

C₂×#6157

yes

characteristic

163

#95

—

164

C₂²×C₄

—

165

C₂²×C₄

—

166

C₂²×C₄

—

167

C₂²×C₄

—

106

—

168

M₁₆

—

212

—

169

M₁₆

—

212

—

170

C₂×C₈

—

171

C₂×C₈

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

106

—

176

C₂×C₈

—

106

—

177

C₂×C₈

—

106

—

178

C₂×C₈

—

106

—

179

C₂×C₈

—

106

—

180

C₂×C₈

—

212

—

181

C₂×C₈

—

212

—

182

C₂×C₈

—

212

—

183

C₂×C₈

C₄⋊#6157

yes

characteristic

184

C₂×C₈

C₄⋊#6157

yes

characteristic

185

C₄⋊C₄

—

106

—

186

C₄⋊C₄

—

106

—

187

C₄²

—

188

C₄²

—

189

C₄²

—

190

C₄²

—

212

—

191

C₄²

—

212

—

192

C₄²

C₂²×#6157

yes

characteristic

193

C₂³

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

106

—

202

C₂×C₄

—

106

—

203

C₂×C₄

—

106

—

204

C₂×C₄

—

212

—

205

C₂×C₄

—

212

—

206

C₂×C₄

—

212

—

207

C₂×C₄

C₂×C₄⋊#6157

yes

characteristic

208

C₂×C₄

D₄×#6157

yes

characteristic

209

C₂×C₄

Q₈×#6157

yes

characteristic

210

C₈

—

211

C₈

—

212

C₈

—

213

C₈

—

106

—

214

C₈

—

424

—

215

C₈

—

212

—

216

C₈

—

212

—

217

C₈

—

212

—

218

C₈

#13133285

yes

—

219

C₂²

—

220

C₂²

—

221

C₂²

—

106

—

222

C₂²

—

106

—

223

C₂²

#16811574

yes

characteristic

224

C₄

—

225

C₄

—

226

C₄

—

227

C₄

—

106

—

228

C₄

—

106

—

229

C₄

—

212

—

230

C₄

C₂×#13133285

yes

characteristic

231

C₄

#16811634

yes

characteristic

232

C₄

#16811665

yes

characteristic

233

C₄

#16811667

yes

characteristic

234

C₂

—

106

—

235

C₂

—

106

—

236

C₂

#20290823

yes

characteristic

237

C₂

#20290827

yes

characteristic

238

C₂

#20290835

yes

characteristic

	subgroup	G/H
center	C₂²	#16811574
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×#6157
Fitting	C₅₃×#335	C₄
socle	C₂²×C₅₃	#389
prime radical	C₂²×D₅₃	C₂²⋊Q₈
cosocle	C₄²×D₅₃	C₂³
hypercenter	#335	#6157
hypocenter	C₅₃	#9628
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

subgroup

G/H

center

C₂²

#16811574

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×#6157

Fitting

C₅₃×#335

C₄

socle

C₂²×C₅₃

#389

prime radical

C₂²×D₅₃

C₂²⋊Q₈

cosocle

C₄²×D₅₃

C₂³

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #335

N	H ≅ G/N	type
17. #16810376	C₄	semidirect
18. #16810373	C₄	semidirect
19. C₅₃×#335	C₄	semidirect
135. #335	#6157	semidirect
139. C₅₃	#9628	semidirect

H ≅ G/N

type

17. #16810376

C₄

semidirect

18. #16810373

C₄

semidirect

19. C₅₃×#335

C₄

semidirect

135. #335

#6157

semidirect

139. C₅₃

#9628

semidirect

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

subgroup	index
#20290656	2
C₈×C₄⋊#6157	2
#20290283	2
#20289612	2
#20273462	2
#9628	53

subgroup

index

C₈×C₄⋊#6157

Subgroups — #28305663

Conjugacy classes of subgroups

Complements

Maximal subgroups