Group #28305595 — subgroups

Subgroups — #28305595

Order 13568.

378 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 128 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	#20290812	C₂	1	2	yes	maximal
2	C₂×#16811486	C₂	1	1	yes	maximal, characteristic
3	C₂×#88324⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#20289551	C₂	1	2	yes	maximal
5	#20274607	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88324	—	2	2	no	—
7	C₂×C₄⋊#88324	C₂²	1	2	yes	—
8	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
9	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
10	#16811489	—	2	2	no	—
11	#16811486	—	2	2	no	—
12	#88324⋊C₈	—	2	2	no	—
13	C₂×#13133150	C₄	1	2	yes	—
14	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×Dic₅₃	C₂²	1	2	yes	—
16	#16810386	C₄	1	1	yes	characteristic
17	C₅₃×#340	C₄	1	1	yes	characteristic
18	C₂²×#88324	—	2	1	no	—
19	C₂²×#88324	—	2	2	no	—
20	C₂²×#88324	D₄	1	1	yes	characteristic
21	C₂²×#88324	D₄	1	1	yes	characteristic
22	C₄⋊#88324	—	2	2	no	—
23	C₄⋊#88324	—	2	2	no	—
24	C₄⋊#88324	—	4	4	no	—
25	C₄×#88324	—	2	1	no	—
26	C₄×#88324	—	2	1	no	—
27	C₄×#88324	—	2	1	no	—
28	C₄×#88324	Q₈	1	2	yes	—
29	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₄×Dic₅₃	—	2	2	no	—
32	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
33	#13133150	—	2	4	no	—
34	C₈×Dic₅₃	D₄	1	2	yes	—
35	C₈×Dic₅₃	Q₈	1	2	yes	—
36	#13133124	—	2	2	no	—
37	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×Dic₅₃	—	2	2	no	—
42	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₅₃	#41	1	1	yes	characteristic
44	C₂×#88324	—	2	1	no	—
45	C₂×#88324	—	2	1	no	—
46	C₂×#88324	—	2	1	no	—
47	C₂×#88324	—	2	1	no	—
48	C₂×#88324	—	2	2	no	—
49	C₂×#88324	—	4	2	no	—
50	C₂×#88324	—	4	2	no	—
51	C₂×#88324	—	4	2	no	—
52	C₂×#88324	—	4	2	no	—
53	C₂×#88324	D₈	1	2	yes	—
54	C₂×#88324	Q₁₆	1	2	yes	—
55	C₂×C₈×C₅₃	—	2	2	no	—
56	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
57	C₄⋊C₄×C₅₃	—	2	2	no	—
58	C₄⋊C₄×C₅₃	M₁₆	1	4	yes	—
59	C₄⋊Dic₅₃	—	2	2	no	—
60	C₄⋊Dic₅₃	—	2	2	no	—
61	C₄×Dic₅₃	—	4	2	no	—
62	C₄×Dic₅₃	—	4	2	no	—
63	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
64	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
65	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
66	C₄×Dic₅₃	#41	1	1	yes	characteristic
67	C₂×#88322	—	2	2	no	—
68	C₂×#88322	C₄⋊C₄	1	2	yes	—
69	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	—	2	4	no	—
71	C₂×C₄×C₅₃	—	4	2	no	—
72	C₂×C₄×C₅₃	—	4	2	no	—
73	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
76	C₂×C₄×C₅₃	#131	1	2	yes	—
77	C₂×Dic₅₃	—	2	2	no	—
78	C₂×Dic₅₃	—	4	2	no	—
79	C₂×Dic₅₃	—	4	2	no	—
80	C₂×Dic₅₃	—	4	2	no	—
81	C₂×Dic₅₃	#121	1	1	yes	characteristic
82	C₂×Dic₅₃	C₂²⋊Q₈	1	1	yes	characteristic
83	C₂×Dic₅₃	#125	1	1	yes	characteristic
84	C₂×Dic₅₃	C₄⋊Q₈	1	1	yes	characteristic
85	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
86	C₂×Dic₅₃	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×Dic₅₃	#135	1	1	yes	characteristic
88	C₂×Dic₅₃	#135	1	1	yes	characteristic
89	#88324	—	2	2	no	—
90	#88324	—	2	2	no	—
91	#88324	—	4	2	no	—
92	#88324	—	8	2	no	—
93	C₈×C₅₃	—	2	2	no	—
94	#88322	—	2	2	no	—
95	#9598	—	53	1	no	maximal, Sylow
96	C₂²×C₅₃	#389	1	1	yes	characteristic
97	C₂²×C₅₃	#435	1	2	yes	—
98	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
99	C₂²×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
100	C₂²×C₅₃	#444	1	2	yes	—
101	C₄×C₅₃	—	4	4	no	—
102	C₄×C₅₃	#423	1	1	yes	characteristic
103	C₄×C₅₃	#424	1	1	yes	characteristic
104	C₄×C₅₃	#446	1	2	yes	—
105	Dic₅₃	—	2	1	no	—
106	Dic₅₃	—	2	1	no	—
107	Dic₅₃	—	8	2	no	—
108	Dic₅₃	#466	1	1	yes	characteristic
109	Dic₅₃	#467	1	1	yes	characteristic
110	Dic₅₃	#499	1	1	yes	characteristic
111	Dic₅₃	#500	1	1	yes	characteristic
112	#1939	—	53	2	no	—
113	C₄×#107	—	53	1	no	—
114	C₂×#335	—	53	1	no	—
115	C₂×#322	—	53	1	no	—
116	#1279	—	53	2	no	—
117	C₂×C₅₃	#1581	1	2	yes	—
118	C₂×C₅₃	#1584	1	2	yes	—
119	C₂×C₅₃	#1940	1	1	yes	characteristic
120	C₂×C₅₃	#1942	1	1	yes	characteristic
121	C₂×C₅₃	#1946	1	1	yes	characteristic
122	C₂×#107	—	53	2	no	—
123	C₂×C₄⋊C₈	—	53	2	no	—
124	C₂×C₄⋊C₈	—	106	2	no	—
125	C₂×C₄×C₈	—	53	1	no	—
126	C₂×C₄×C₈	—	53	1	no	—
127	C₂×C₄×C₈	—	53	1	no	—
128	C₄×C₄⋊C₄	—	53	2	no	—
129	#340	—	53	1	no	—
130	#340	#6157	1	1	yes	characteristic
131	#336	—	106	2	no	—
132	#335	—	106	2	no	—
133	#322	—	106	2	no	—
134	C₅₃	#9598	1	1	yes	characteristic, complemented, Sylow
135	C₂²×C₈	—	53	1	no	—
136	C₂²×C₈	—	53	1	no	—
137	C₂²×C₈	—	53	1	no	—
138	C₂²×C₈	—	106	1	no	—
139	C₂²×C₈	—	106	2	no	—
140	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
141	C₂×C₄⋊C₄	—	53	2	no	—
142	C₂×C₄⋊C₄	C₂×#6157	1	2	yes	—
143	C₂×C₄²	—	53	1	no	—
144	C₂×C₄²	—	106	2	no	—
145	#107	—	106	4	no	—
146	C₄⋊C₈	—	106	2	no	—
147	C₄⋊C₈	—	106	2	no	—
148	C₄⋊C₈	—	212	4	no	—
149	C₄×C₈	—	53	2	no	—
150	C₄×C₈	—	53	2	no	—
151	C₄×C₈	—	53	2	no	—
152	C₄×C₈	—	106	1	no	—
153	C₄×C₈	—	106	1	no	—
154	C₄×C₈	—	106	1	no	—
155	#95	—	106	2	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	53	1	no	—
158	C₂²×C₄	—	53	1	no	—
159	C₂²×C₄	—	106	2	no	—
160	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	53	2	no	—
163	C₂×C₈	—	53	2	no	—
164	C₂×C₈	—	53	2	no	—
165	C₂×C₈	—	106	1	no	—
166	C₂×C₈	—	106	1	no	—
167	C₂×C₈	—	106	1	no	—
168	C₂×C₈	—	106	1	no	—
169	C₂×C₈	—	106	2	no	—
170	C₂×C₈	—	106	2	no	—
171	C₂×C₈	—	212	2	no	—
172	C₂×C₈	—	212	2	no	—
173	C₂×C₈	—	212	2	no	—
174	C₂×C₈	—	212	2	no	—
175	C₂×C₈	C₄⋊#6157	1	2	yes	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	106	2	no	—
178	C₄⋊C₄	—	106	2	no	—
179	C₄⋊C₄	#11719936	1	4	yes	—
180	C₄²	—	53	1	no	—
181	C₄²	—	53	1	no	—
182	C₄²	—	53	1	no	—
183	C₄²	—	53	1	no	—
184	C₄²	—	212	2	no	—
185	C₄²	—	212	2	no	—
186	C₂³	D₄×#6157	1	1	yes	characteristic
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	53	1	no	—
191	C₂×C₄	—	53	1	no	—
192	C₂×C₄	—	53	1	no	—
193	C₂×C₄	—	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	—	106	2	no	—
199	C₂×C₄	—	212	2	no	—
200	C₂×C₄	—	212	2	no	—
201	C₂×C₄	—	212	2	no	—
202	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
203	C₂×C₄	Q₈×#6157	1	1	yes	characteristic
204	C₂×C₄	C₂×#11719936	1	2	yes	—
205	C₂×C₄	#13133304	1	2	yes	—
206	C₈	—	2	2	no	—
207	C₈	—	106	2	no	—
208	C₈	—	106	2	no	—
209	C₈	—	106	2	no	—
210	C₈	—	424	2	no	—
211	C₈	—	212	2	no	—
212	C₂²	#16811574	1	1	yes	characteristic
213	C₂²	#16811652	1	2	yes	—
214	C₂²	D₈×#6157	1	1	yes	characteristic
215	C₂²	Q₁₆×#6157	1	1	yes	characteristic
216	C₂²	#16811668	1	2	yes	—
217	C₄	—	4	4	no	—
218	C₄	—	53	1	no	—
219	C₄	—	53	1	no	—
220	C₄	—	53	1	no	—
221	C₄	—	53	1	no	—
222	C₄	—	106	1	no	—
223	C₄	—	106	1	no	—
224	C₄	—	424	2	no	—
225	C₄	#16811627	1	1	yes	characteristic
226	C₄	#16811629	1	1	yes	characteristic
227	C₄	#16811673	1	2	yes	—
228	C₂	#20290314	1	2	yes	—
229	C₂	#20290326	1	2	yes	—
230	C₂	#20290815	1	1	yes	characteristic
231	C₂	#20290819	1	1	yes	characteristic
232	C₂	#20290825	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20290812

C₂

yes

maximal

C₂×#16811486

C₂

yes

maximal, characteristic

C₂×#88324⋊C₈

C₂

yes

maximal, characteristic

#20289551

C₂

yes

maximal

#20274607

C₂

yes

maximal, characteristic

C₂×C₄⋊#88324

—

C₂×C₄⋊#88324

C₂²

yes

—

C₂×C₄×#88324

C₂²

yes

characteristic

C₂×C₄×#88324

C₂²

yes

—

—

—

C₄

yes

—

C₂×C₈×Dic₅₃

C₂²

yes

characteristic

C₄⋊C₄×Dic₅₃

C₂²

yes

—

#16810386

C₄

yes

characteristic

C₅₃×#340

C₄

yes

characteristic

C₂²×#88324

—

C₂²×#88324

—

C₂²×#88324

D₄

yes

characteristic

C₂²×#88324

D₄

yes

—

—

—

—

—

—

Q₈

yes

—

C₂²×#88322

C₂×C₄

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂×C₄

yes

—

C₂×C₄×Dic₅₃

—

C₂×C₄×Dic₅₃

C₂³

yes

characteristic

#13133150

—

C₈×Dic₅₃

D₄

yes

—

C₈×Dic₅₃

Q₈

yes

—

#13133124

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

C₂×D₄

yes

C₂²×Dic₅₃

yes

—

—

—

—

—

—

—

—

—

D₈

yes

—

C₂×#88324

Q₁₆

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

M₁₆

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×D₄

yes

characteristic

C₄×Dic₅₃

C₂×Q₈

yes

characteristic

C₄×Dic₅₃

C₂×Q₈

yes

C₄×Dic₅₃

yes

—

C₄⋊C₄

yes

—

C₂³×C₅₃

C₄×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

#121

yes

characteristic

C₂×Dic₅₃

C₂²⋊Q₈

yes

characteristic

C₂×Dic₅₃

#125

yes

characteristic

C₂×Dic₅₃

C₄⋊Q₈

yes

characteristic

C₂×Dic₅₃

C₂×D₈

yes

characteristic

C₂×Dic₅₃

C₂×Q₁₆

yes

C₂×Dic₅₃

yes

C₂×Dic₅₃

yes

—

—

—

—

C₈×C₅₃

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

C₄×D₈

yes

characteristic

C₂²×C₅₃

C₄×Q₁₆

yes

characteristic

100

C₂²×C₅₃

#444

yes

—

101

C₄×C₅₃

—

102

C₄×C₅₃

#423

yes

characteristic

103

C₄×C₅₃

#424

yes

characteristic

104

C₄×C₅₃

#446

yes

—

105

Dic₅₃

—

106

Dic₅₃

—

107

Dic₅₃

—

108

Dic₅₃

#466

yes

characteristic

109

Dic₅₃

#467

yes

characteristic

110

Dic₅₃

#499

yes

characteristic

111

Dic₅₃

#500

yes

characteristic

112

#1939

—

113

C₄×#107

—

114

C₂×#335

—

115

C₂×#322

—

116

#1279

—

117

C₂×C₅₃

#1581

yes

—

118

C₂×C₅₃

#1584

yes

—

119

C₂×C₅₃

#1940

yes

characteristic

120

C₂×C₅₃

#1942

yes

characteristic

121

C₂×C₅₃

#1946

yes

characteristic

122

C₂×#107

—

123

C₂×C₄⋊C₈

—

124

C₂×C₄⋊C₈

—

106

—

125

C₂×C₄×C₈

—

126

C₂×C₄×C₈

—

127

C₂×C₄×C₈

—

128

C₄×C₄⋊C₄

—

129

#340

—

130

yes

131

—

106

—

132

#335

—

106

—

133

#322

—

106

—

134

C₅₃

#9598

yes

characteristic, complemented, Sylow

135

C₂²×C₈

—

136

C₂²×C₈

—

137

C₂²×C₈

—

138

C₂²×C₈

—

106

—

139

C₂²×C₈

—

106

—

140

C₂²×C₈

C₂×#6157

yes

characteristic

141

C₂×C₄⋊C₄

—

142

C₂×C₄⋊C₄

C₂×#6157

yes

—

143

C₂×C₄²

—

144

C₂×C₄²

—

106

—

145

#107

—

106

—

146

C₄⋊C₈

—

106

—

147

C₄⋊C₈

—

106

—

148

C₄⋊C₈

—

212

—

149

C₄×C₈

—

150

C₄×C₈

—

151

C₄×C₈

—

152

C₄×C₈

—

106

—

153

C₄×C₈

—

106

—

154

C₄×C₈

—

106

—

155

#95

—

106

—

156

C₂²×C₄

—

157

C₂²×C₄

—

158

C₂²×C₄

—

159

C₂²×C₄

—

106

—

160

C₂²×C₄

C₂²×#6157

yes

characteristic

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

106

—

166

C₂×C₈

—

106

—

167

C₂×C₈

—

106

—

168

C₂×C₈

—

106

—

169

C₂×C₈

—

106

—

170

C₂×C₈

—

106

—

171

C₂×C₈

—

212

—

172

C₂×C₈

—

212

—

173

C₂×C₈

—

212

—

174

C₂×C₈

—

212

—

175

C₂×C₈

C₄⋊#6157

yes

—

176

C₄⋊C₄

—

177

C₄⋊C₄

—

106

—

178

C₄⋊C₄

—

106

—

179

C₄⋊C₄

#11719936

yes

—

180

C₄²

—

181

C₄²

—

182

C₄²

—

183

C₄²

—

184

C₄²

—

212

—

185

C₄²

—

212

—

186

C₂³

D₄×#6157

yes

characteristic

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

106

—

199

C₂×C₄

—

212

—

200

C₂×C₄

—

212

—

201

C₂×C₄

—

212

—

202

C₂×C₄

C₂×C₄⋊#6157

yes

characteristic

203

C₂×C₄

Q₈×#6157

yes

characteristic

204

C₂×C₄

C₂×#11719936

yes

—

205

C₂×C₄

#13133304

yes

—

206

C₈

—

207

C₈

—

106

—

208

C₈

—

106

—

209

C₈

—

106

—

210

C₈

—

424

—

211

C₈

—

212

—

212

C₂²

#16811574

yes

characteristic

213

C₂²

#16811652

yes

—

214

C₂²

D₈×#6157

yes

characteristic

215

C₂²

Q₁₆×#6157

yes

characteristic

216

C₂²

#16811668

yes

—

217

C₄

—

218

C₄

—

219

C₄

—

220

C₄

—

221

C₄

—

222

C₄

—

106

—

223

C₄

—

106

—

224

C₄

—

424

—

225

C₄

#16811627

yes

characteristic

226

C₄

#16811629

yes

characteristic

227

C₄

#16811673

yes

—

228

C₂

#20290314

yes

—

229

C₂

#20290326

yes

—

230

C₂

#20290815

yes

characteristic

231

C₂

#20290819

yes

characteristic

232

C₂

#20290825

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×#6157
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₅₃×#340	C₄
socle, prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	#340	#6157
hypocenter	C₅₃	#9598
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×#6157

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×#6157

Fitting

C₅₃×#340

C₄

socle, prime radical

C₂³×C₅₃

C₄×D₄

cosocle

C₂×C₄×Dic₅₃

C₂³

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
134. C₅₃	#9598	semidirect

H ≅ G/N

type

134. C₅₃

#9598

semidirect

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

subgroup	index
#20290812	2
C₂×#16811486	2
C₂×#88324⋊C₈	2
#20289551	2
#20274607	2
#9598	53

subgroup

index

C₂×#88324⋊C₈

Subgroups — #28305595

Conjugacy classes of subgroups

Complements

Maximal subgroups