Group #28316317 — subgroups

Subgroups — #28316317

Order 13568.

612 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 234 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	#20291649	C₂	1	2	yes	maximal
2	#20290293	C₂	1	6	yes	maximal, complemented
3	#20290273	C₂	1	3	yes	maximal, complemented
4	#20290265	C₂	1	3	yes	maximal, complemented
5	C₈×#11719949	C₂	1	1	yes	maximal, characteristic
6	Q₈×#88324	—	2	2	no	—
7	#16811657	—	2	6	no	—
8	#16811642	C₂²	1	6	yes	—
9	#16811627	C₂²	1	6	yes	—
10	#16811486	C₂²	1	1	yes	characteristic
11	#16811486	C₂²	1	3	yes	—
12	#16811486	C₂²	1	6	yes	—
13	#16811485	C₂²	1	6	yes	—
14	C₄×#11719949	C₂²	1	1	yes	characteristic
15	Q₈×#88322	C₄	1	1	yes	characteristic
16	C₈×D₂₁₂	C₄	1	3	yes	—
17	#16810688	C₄	1	3	yes	—
18	#16810675	C₂²	1	3	yes	—
19	C₄×C₈×D₅₃	C₂²	1	3	yes	—
20	C₈×Q₈×C₅₃	C₄	1	1	yes	characteristic
21	C₂×#11719936	—	2	6	no	—
22	C₄⋊#88324	—	2	6	no	—
23	C₄⋊#88324	C₂³	1	6	yes	—
24	C₄⋊#88324	C₂³	1	2	yes	—
25	#13133274	—	2	6	no	—
26	C₄×#88324	—	2	6	no	—
27	C₂×#11719949	Q₈	1	1	yes	characteristic
28	C₂×C₈×D₅₃	C₂×C₄	1	3	yes	—
29	C₂×C₈×D₅₃	C₂×C₄	1	3	yes	—
30	Q₈×Dic₅₃	D₄	1	1	yes	characteristic
31	C₄×D₂₁₂	C₂×C₄	1	3	yes	—
32	#13133179	D₄	1	3	yes	—
33	#13133175	Q₈	1	3	yes	—
34	C₄²×D₅₃	C₂³	1	3	yes	—
35	C₄⋊#88322	C₂×C₄	1	3	yes	—
36	#13133129	C₂×C₄	1	6	yes	—
37	C₄×#88322	C₂×C₄	1	3	yes	—
38	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
39	C₈×Dic₅₃	C₂³	1	3	yes	—
40	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₄⋊C₈×C₅₃	C₂×C₄	1	3	yes	—
42	C₄×C₈×C₅₃	C₂×C₄	1	3	yes	—
43	C₂×Q₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	#11719949	—	2	4	no	—
45	C₂×D₂₁₂	C₄⋊C₄	1	3	yes	—
46	C₂×C₄×D₅₃	C₂²×C₄	1	3	yes	—
47	C₂×C₄×D₅₃	C₂×D₄	1	3	yes	—
48	C₂×C₄×D₅₃	C₂×Q₈	1	3	yes	—
49	C₂×#88324	—	2	2	no	—
50	C₂×#88324	—	2	6	no	—
51	#11719936	—	8	6	no	—
52	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
53	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₄⋊C₄×C₅₃	C₄⋊C₄	1	3	yes	—
55	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
56	#11719921	C₄⋊C₄	1	6	yes	—
57	C₄⋊Dic₅₃	C₄⋊C₄	1	3	yes	—
58	C₄×Dic₅₃	C₂×D₄	1	3	yes	—
59	C₄×Dic₅₃	C₂×Q₈	1	3	yes	—
60	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
61	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
62	C₂×#88322	C₂²×C₄	1	3	yes	—
63	C₈×D₅₃	—	2	6	no	—
64	C₈×D₅₃	—	4	3	no	—
65	C₂²×D₅₃	C₂×C₄⋊C₄	1	3	yes	—
66	Q₈×C₅₃	#108	1	4	yes	—
67	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
68	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
69	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
70	C₂×Dic₅₃	C₂×C₄⋊C₄	1	3	yes	—
71	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
72	C₂×Dic₅₃	C₂²×Q₈	1	1	yes	characteristic
73	D₂₁₂	—	2	6	no	—
74	C₄×D₅₃	—	2	3	no	—
75	C₄×D₅₃	—	2	3	no	—
76	C₄×D₅₃	—	2	6	no	—
77	C₄×D₅₃	—	2	3	no	—
78	C₄×D₅₃	—	2	3	no	—
79	#88324	—	4	6	no	—
80	#88324	—	4	2	no	—
81	C₈×C₅₃	—	2	3	no	—
82	C₈×C₅₃	#131	1	2	yes	—
83	#88322	—	2	3	no	—
84	#88322	—	2	1	no	—
85	#17837	—	53	1	no	maximal, Sylow
86	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
87	C₂×D₅₃	—	2	3	no	—
88	C₂×D₅₃	—	2	3	no	—
89	C₂×D₅₃	—	2	3	no	—
90	C₄×C₅₃	—	2	3	no	—
91	C₄×C₅₃	C₂×#108	1	6	yes	—
92	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
93	C₄×C₅₃	#568	1	1	yes	characteristic
94	Dic₅₃	—	2	3	no	—
95	Dic₅₃	—	2	1	no	—
96	Dic₅₃	#576	1	1	yes	characteristic
97	Dic₅₃	#578	1	1	yes	characteristic
98	#2991	—	53	2	no	—
99	C₈×#41	—	53	1	no	—
100	#1574	—	53	6	no	—
101	#1567	—	53	3	no	—
102	#1564	—	53	3	no	—
103	C₂×C₅₃	#2906	1	1	yes	characteristic
104	C₂×C₅₃	#2913	1	1	yes	characteristic
105	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
106	D₅₃	—	4	3	no	—
107	C₄×#41	—	53	1	no	—
108	C₈×Q₈	—	53	1	no	—
109	C₈×Q₈	—	106	2	no	—
110	C₈×Q₈	#6157	1	1	yes	characteristic
111	#436	—	106	6	no	—
112	C₈×D₄	—	53	3	no	—
113	C₈×D₄	—	53	3	no	—
114	#431	—	53	3	no	—
115	#431	—	53	6	no	—
116	#423	—	53	6	no	—
117	C₂×C₄×C₈	—	53	3	no	—
118	#335	—	53	1	no	—
119	#335	—	53	3	no	—
120	#335	—	53	6	no	—
121	#334	—	53	6	no	—
122	C₅₃	#17837	1	1	yes	characteristic, complemented, Sylow
123	C₂×#41	—	53	1	no	—
124	C₂×M₁₆	—	106	6	no	—
125	C₂²×C₈	—	53	3	no	—
126	C₂²×C₈	—	53	3	no	—
127	C₄×Q₈	—	53	1	no	—
128	C₄×Q₈	C₂×#6157	1	1	yes	characteristic
129	C₄×D₄	—	53	3	no	—
130	C₄×D₄	—	53	3	no	—
131	#117	—	53	3	no	—
132	C₂×C₄²	—	53	3	no	—
133	C₄⋊C₈	—	53	2	no	—
134	C₄⋊C₈	—	53	6	no	—
135	C₄⋊C₈	—	53	3	no	—
136	C₄⋊C₈	—	106	6	no	—
137	C₄⋊C₈	C₂×#6157	1	3	yes	—
138	C₂²⋊C₈	—	53	6	no	—
139	C₂²⋊C₈	—	106	6	no	—
140	C₄×C₈	—	53	3	no	—
141	C₄×C₈	—	53	1	no	—
142	C₄×C₈	—	53	3	no	—
143	C₄×C₈	—	106	6	no	—
144	C₄×C₈	C₂×#6157	1	3	yes	—
145	#41	—	106	4	no	—
146	C₂×Q₈	C₄⋊#6157	1	1	yes	characteristic
147	C₂×D₄	—	53	3	no	—
148	C₂²×C₄	—	53	3	no	—
149	C₂²×C₄	—	53	3	no	—
150	C₂²×C₄	—	53	3	no	—
151	M₁₆	—	424	6	no	—
152	C₂×C₈	—	53	1	no	—
153	C₂×C₈	—	53	3	no	—
154	C₂×C₈	—	106	6	no	—
155	C₂×C₈	—	106	2	no	—
156	C₂×C₈	—	106	6	no	—
157	C₂×C₈	—	212	3	no	—
158	C₂×C₈	C₂²×#6157	1	3	yes	—
159	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
160	C₄⋊C₄	—	53	3	no	—
161	C₄⋊C₄	C₄⋊#6157	1	3	yes	—
162	C₂²⋊C₄	—	53	6	no	—
163	C₄²	—	53	3	no	—
164	C₄²	—	53	3	no	—
165	C₄²	—	53	1	no	—
166	C₄²	C₂²×#6157	1	3	yes	—
167	C₂³	—	53	3	no	—
168	Q₈	#13133288	1	4	yes	—
169	D₄	—	106	6	no	—
170	C₂×C₄	—	53	1	no	—
171	C₂×C₄	—	53	3	no	—
172	C₂×C₄	—	53	1	no	—
173	C₂×C₄	—	106	3	no	—
174	C₂×C₄	—	106	3	no	—
175	C₂×C₄	—	106	6	no	—
176	C₂×C₄	—	106	3	no	—
177	C₂×C₄	—	106	3	no	—
178	C₂×C₄	C₂×C₄⋊#6157	1	3	yes	—
179	C₂×C₄	C₂×C₄⋊#6157	1	3	yes	—
180	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
181	C₈	—	2	3	no	—
182	C₈	—	106	3	no	—
183	C₈	—	106	1	no	—
184	C₈	—	212	2	no	—
185	C₈	—	212	6	no	—
186	C₈	#13133305	1	2	yes	—
187	C₂²	—	106	3	no	—
188	C₂²	—	106	3	no	—
189	C₂²	—	106	3	no	—
190	C₂²	C₂²×C₄⋊#6157	1	1	yes	characteristic
191	C₄	—	2	3	no	—
192	C₄	—	53	1	no	—
193	C₄	—	53	1	no	—
194	C₄	—	106	3	no	—
195	C₄	—	106	1	no	—
196	C₄	C₂×#13133288	1	6	yes	—
197	C₄	C₂×#13133305	1	1	yes	characteristic
198	C₄	#16811702	1	1	yes	characteristic
199	C₂	—	212	3	no	—
200	C₂	#20291361	1	1	yes	characteristic
201	C₂	#20291370	1	1	yes	characteristic
202	C₂	C₂²×#13133288	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20291649

C₂

yes

maximal

#20290293

C₂

yes

maximal, complemented

#20290273

C₂

yes

maximal, complemented

#20290265

C₂

yes

maximal, complemented

C₈×#11719949

C₂

yes

maximal, characteristic

Q₈×#88324

—

#16811657

—

#16811642

C₂²

yes

—

#16811627

C₂²

yes

—

#16811486

C₂²

yes

characteristic

#16811486

C₂²

yes

—

#16811486

C₂²

yes

—

#16811485

C₂²

yes

—

C₄×#11719949

C₂²

yes

characteristic

Q₈×#88322

C₄

yes

characteristic

C₈×D₂₁₂

C₄

yes

—

#16810688

C₄

yes

—

#16810675

C₂²

yes

—

C₄×C₈×D₅₃

C₂²

yes

—

C₈×Q₈×C₅₃

C₄

yes

—

—

C₂³

yes

—

C₄⋊#88324

C₂³

yes

—

#13133274

—

C₄×#88324

—

C₂×#11719949

Q₈

yes

characteristic

C₂×C₈×D₅₃

C₂×C₄

yes

—

C₂×C₈×D₅₃

C₂×C₄

yes

—

Q₈×Dic₅₃

D₄

yes

characteristic

C₄×D₂₁₂

C₂×C₄

yes

—

#13133179

D₄

yes

—

#13133175

Q₈

yes

—

C₄²×D₅₃

C₂³

yes

—

C₄⋊#88322

C₂×C₄

yes

—

#13133129

C₂×C₄

yes

—

C₄×#88322

C₂×C₄

yes

—

C₈×Dic₅₃

C₂³

yes

characteristic

C₈×Dic₅₃

C₂³

yes

—

C₄×Q₈×C₅₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₂×Q₈×C₅₃

C₄⋊C₄

yes

characteristic

#11719949

—

C₂×D₂₁₂

C₄⋊C₄

yes

—

C₂×C₄×D₅₃

C₂²×C₄

yes

—

C₂×C₄×D₅₃

C₂×D₄

yes

—

C₂×C₄×D₅₃

C₂×Q₈

yes

—

C₂×#88324

—

C₂×#88324

—

#11719936

—

C₂×C₈×C₅₃

C₂²×C₄

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₂²×C₄

yes

—

#11719921

C₄⋊C₄

yes

—

C₄⋊Dic₅₃

C₄⋊C₄

yes

—

C₄×Dic₅₃

C₂×D₄

yes

—

C₄×Dic₅₃

C₂×Q₈

yes

—

C₄×Dic₅₃

C₂⁴

yes

characteristic

C₂×#88322

C₂²×C₄

yes

characteristic

C₂×#88322

C₂²×C₄

yes

—

C₈×D₅₃

—

C₈×D₅₃

—

C₂²×D₅₃

C₂×C₄⋊C₄

yes

—

Q₈×C₅₃

#108

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂³×C₄

yes

characteristic

C₂×Dic₅₃

C₂×C₄⋊C₄

yes

—

C₂×Dic₅₃

C₂²×D₄

yes

characteristic

C₂×Dic₅₃

C₂²×Q₈

yes

characteristic

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

#88324

—

#88324

—

C₈×C₅₃

—

C₈×C₅₃

#131

yes

—

—

—

—

C₂²×C₅₃

C₂²×C₄⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#108

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

yes

Dic₅₃

yes

—

C₈×#41

—

100

#1574

—

101

#1567

—

102

#1564

—

103

C₂×C₅₃

#2906

yes

characteristic

104

C₂×C₅₃

#2913

yes

characteristic

105

C₂×C₅₃

C₂²×#108

yes

characteristic

106

D₅₃

—

107

C₄×#41

—

108

C₈×Q₈

—

109

C₈×Q₈

—

106

—

110

C₈×Q₈

#6157

yes

characteristic

111

#436

—

106

—

112

C₈×D₄

—

113

C₈×D₄

—

114

#431

—

115

#431

—

116

#423

—

117

C₂×C₄×C₈

—

118

#335

—

119

#335

—

120

#335

—

121

#334

—

122

C₅₃

#17837

yes

characteristic, complemented, Sylow

123

C₂×#41

—

124

C₂×M₁₆

—

106

—

125

C₂²×C₈

—

126

C₂²×C₈

—

127

C₄×Q₈

—

128

C₄×Q₈

C₂×#6157

yes

characteristic

129

C₄×D₄

—

130

C₄×D₄

—

131

#117

—

132

C₂×C₄²

—

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

C₄⋊C₈

—

136

C₄⋊C₈

—

106

—

137

C₄⋊C₈

C₂×#6157

yes

—

138

C₂²⋊C₈

—

139

C₂²⋊C₈

—

106

—

140

C₄×C₈

—

141

C₄×C₈

—

142

C₄×C₈

—

143

C₄×C₈

—

106

—

144

C₄×C₈

C₂×#6157

yes

—

145

#41

—

106

—

146

C₂×Q₈

C₄⋊#6157

yes

characteristic

147

C₂×D₄

—

148

C₂²×C₄

—

149

C₂²×C₄

—

150

C₂²×C₄

—

151

M₁₆

—

424

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

106

—

155

C₂×C₈

—

106

—

156

C₂×C₈

—

106

—

157

C₂×C₈

—

212

—

158

C₂×C₈

C₂²×#6157

yes

—

159

C₂×C₈

C₂²×#6157

yes

characteristic

160

C₄⋊C₄

—

161

C₄⋊C₄

C₄⋊#6157

yes

—

162

C₂²⋊C₄

—

163

C₄²

—

164

C₄²

—

165

C₄²

—

166

C₄²

C₂²×#6157

yes

—

167

C₂³

—

168

Q₈

#13133288

yes

—

169

D₄

—

106

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

106

—

174

C₂×C₄

—

106

—

175

C₂×C₄

—

106

—

176

C₂×C₄

—

106

—

177

C₂×C₄

—

106

—

178

C₂×C₄

C₂×C₄⋊#6157

yes

—

179

C₂×C₄

C₂×C₄⋊#6157

yes

—

180

C₂×C₄

C₂³×#6157

yes

characteristic

181

C₈

—

182

C₈

—

106

—

183

C₈

—

106

—

184

C₈

—

212

—

185

C₈

—

212

—

186

C₈

#13133305

yes

—

187

C₂²

—

106

—

188

C₂²

—

106

—

189

C₂²

—

106

—

190

C₂²

C₂²×C₄⋊#6157

yes

characteristic

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

106

—

195

C₄

—

106

—

196

C₄

C₂×#13133288

yes

—

197

C₄

C₂×#13133305

yes

characteristic

198

C₄

#16811702

yes

characteristic

199

C₂

—

212

—

200

C₂

#20291361

yes

characteristic

201

C₂

#20291370

yes

characteristic

202

C₂

C₂²×#13133288

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₈×Q₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂×#11719949	Q₈
cosocle	C₄×Dic₅₃	C₂⁴
hypercenter	C₈×Q₈	#6157
hypocenter	C₅₃	#17837
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×C₄⋊#6157

commutator

C₂×C₄×C₅₃

C₂³×C₄

Frattini

C₂×C₄

C₂³×#6157

Fitting

C₈×Q₈×C₅₃

C₄

socle

C₂²×C₅₃

C₂²×C₄⋊C₄

prime radical

C₂×#11719949

Q₈

cosocle

C₄×Dic₅₃

C₂⁴

C₈×Q₈

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
2. #20290293	C₂	semidirect
3. #20290273	C₂	semidirect
4. #20290265	C₂	semidirect
122. C₅₃	#17837	semidirect

H ≅ G/N

type

2. #20290293

C₂

semidirect

3. #20290273

C₂

semidirect

4. #20290265

C₂

semidirect

122. C₅₃

#17837

semidirect

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

subgroup	index
#20291649	2
#20290293	2
#20290273	2
#20290265	2
C₈×#11719949	2
#17837	53

subgroup

index

Subgroups — #28316317

Conjugacy classes of subgroups

Complements

Maximal subgroups