Group #28298365 — subgroups

Subgroups — #28298365

Order 13568.

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

normal subgroups only

This group has 282 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	C₄×D₅₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133286	C₂	1	1	yes	maximal, characteristic
3	#335×D₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×D₅₃⋊C₁₆	—	2	1	no	—
5	C₂×#13133286	C₄	1	2	yes	—
6	C₄×C₄⋊#6157	—	2	1	no	—
7	#16811509	C₄	1	2	yes	complemented
8	C₄×#11719910	—	2	1	no	—
9	#16811497	—	2	1	no	—
10	C₄⋊C₈×D₅₃	—	2	1	no	—
11	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	#16810376	C₄	1	1	yes	characteristic, complemented
13	#16810375	—	2	1	no	—
14	C₅₃×#335	C₄	1	1	yes	characteristic, complemented
15	C₂×C₄⋊#6157	—	2	2	no	—
16	C₂×C₄×#6157	—	4	1	no	—
17	D₅₃⋊C₁₆	—	2	4	no	—
18	C₂×#11719910	—	2	2	no	—
19	#13133286	—	2	2	no	—
20	#13133286	C₈	1	4	yes	complemented
21	(C₂×#6157)⋊C₄	—	4	1	no	—
22	C₂×C₈×D₅₃	—	4	1	no	—
23	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
25	C₄²×D₅₃	D₄	1	1	yes	characteristic
26	C₄⋊#88322	—	2	1	no	—
27	#13133146	—	4	1	no	—
28	C₄×#88322	D₄	1	1	yes	characteristic
29	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
31	C₄⋊C₈×C₅₃	—	2	1	no	—
32	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×#6157	—	4	2	no	—
34	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
36	C₂×C₄×D₅₃	SD₁₆	1	1	yes	characteristic
37	C₄⋊#6157	—	2	4	no	—
38	C₄⋊#6157	—	4	2	no	—
39	C₄×#6157	—	8	1	no	—
40	C₄×#6157	—	8	1	no	—
41	C₂×C₈×C₅₃	—	4	1	no	—
42	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
46	C₄×Dic₅₃	SD₁₆	1	1	yes	characteristic
47	C₄×Dic₅₃	Q₁₆	1	1	yes	characteristic
48	C₂×#88322	—	4	1	no	—
49	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
51	C₈×D₅₃	—	2	2	no	—
52	C₈×D₅₃	—	8	1	no	—
53	C₈×D₅₃	—	8	1	no	—
54	C₈×D₅₃	C₂×C₈	1	2	yes	—
55	C₈×D₅₃	M₁₆	1	2	yes	—
56	#11719910	—	2	4	no	—
57	C₂²×D₅₃	#102	1	1	yes	characteristic
58	C₂×#6157	—	4	4	no	—
59	C₂×#6157	—	8	2	no	—
60	C₂×#6157	—	8	2	no	—
61	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
63	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
64	C₂×Dic₅₃	#103	1	1	yes	characteristic
65	C₂×Dic₅₃	#106	1	1	yes	characteristic
66	C₂×Dic₅₃	#107	1	1	yes	characteristic
67	C₄×D₅₃	—	2	2	no	—
68	C₄×D₅₃	C₂²⋊C₈	1	2	yes	—
69	C₄×D₅₃	#104	1	2	yes	—
70	C₄×D₅₃	D₁₆	1	1	yes	characteristic
71	C₄×D₅₃	SD₃₂	1	1	yes	characteristic
72	C₄×D₅₃	SD₃₂	1	1	yes	characteristic
73	C₄×D₅₃	Q₃₂	1	1	yes	characteristic
74	C₈×C₅₃	—	2	1	no	—
75	C₈×C₅₃	—	8	1	no	—
76	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
77	C₈×C₅₃	#97	1	1	yes	characteristic
78	#88322	—	2	1	no	—
79	#88322	—	8	1	no	—
80	#88322	C₄⋊C₈	1	2	yes	—
81	#7171	—	53	1	no	maximal, Sylow
82	C₂²×C₅₃	#340	1	1	yes	characteristic
83	C₂×D₅₃	D₄⋊C₈	1	2	yes	—
84	C₂×D₅₃	#357	1	1	yes	characteristic
85	C₂×D₅₃	#358	1	1	yes	characteristic
86	C₂×D₅₃	#359	1	2	yes	—
87	#6157	—	8	4	no	—
88	C₄×C₅₃	—	2	1	no	—
89	C₄×C₅₃	#336	1	1	yes	characteristic
90	C₄×C₅₃	#339	1	1	yes	characteristic
91	C₄×C₅₃	#357	1	1	yes	characteristic
92	C₄×C₅₃	#358	1	1	yes	characteristic
93	Dic₅₃	—	2	1	no	—
94	Dic₅₃	Q₈⋊C₈	1	2	yes	—
95	Dic₅₃	#366	1	1	yes	characteristic
96	Dic₅₃	#367	1	1	yes	characteristic
97	C₂×C₄×C₁₆	—	53	1	no	—
98	C₄×#107	—	53	1	no	—
99	C₂×#335	—	53	1	no	—
100	C₂×C₅₃	#1279	1	1	yes	characteristic
101	C₂×C₅₃	#1380	1	1	yes	characteristic
102	C₂×C₅₃	#1382	1	1	yes	characteristic
103	D₅₃	#1331	1	2	yes	—
104	D₅₃	#1332	1	2	yes	—
105	C₂²×C₁₆	—	106	1	no	—
106	C₂×#107	—	53	2	no	—
107	C₂×C₄⋊C₈	—	106	1	no	—
108	C₂×C₄×C₈	—	53	1	no	—
109	C₄×C₄⋊C₄	—	106	1	no	—
110	C₄×C₁₆	—	53	2	no	—
111	C₄×C₁₆	—	106	1	no	—
112	#340	—	106	1	no	—
113	#335	—	53	1	no	—
114	#335	—	106	1	no	—
115	#335	#6157	1	1	yes	characteristic, complemented
116	C₅₃	#7171	1	1	yes	characteristic, complemented, Sylow
117	C₂²×C₈	—	53	1	no	—
118	C₂²×C₈	—	53	1	no	—
119	C₂²×C₈	—	212	1	no	—
120	C₂×C₄⋊C₄	—	106	2	no	—
121	C₂×C₄²	—	53	1	no	—
122	C₂×C₄²	—	212	1	no	—
123	C₂×C₁₆	—	106	2	no	—
124	C₂×C₁₆	—	106	4	no	—
125	#107	—	53	4	no	—
126	#107	—	106	2	no	—
127	C₄⋊C₈	—	2	1	no	—
128	C₄⋊C₈	—	106	1	no	—
129	C₄⋊C₈	—	212	1	no	—
130	C₄×C₈	—	53	1	no	—
131	C₄×C₈	—	53	1	no	—
132	C₄×C₈	—	53	1	no	—
133	C₄×C₈	C₂×#6157	1	1	yes	characteristic
134	#95	—	212	1	no	—
135	C₂²×C₄	—	53	1	no	—
136	C₂²×C₄	—	53	1	no	—
137	C₂²×C₄	—	53	1	no	—
138	C₂²×C₄	—	212	2	no	—
139	C₂×C₈	—	4	1	no	—
140	C₂×C₈	—	53	1	no	—
141	C₂×C₈	—	53	2	no	—
142	C₂×C₈	—	53	2	no	—
143	C₂×C₈	—	53	1	no	—
144	C₂×C₈	—	106	2	no	—
145	C₂×C₈	—	424	1	no	—
146	C₂×C₈	—	424	1	no	—
147	C₂×C₈	—	212	1	no	—
148	C₂×C₈	C₄×#6157	1	1	yes	characteristic
149	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
150	C₄⋊C₄	—	106	4	no	—
151	C₄⋊C₄	—	212	2	no	—
152	C₄²	—	53	1	no	—
153	C₄²	—	53	1	no	—
154	C₄²	—	53	1	no	—
155	C₄²	—	424	1	no	—
156	C₄²	—	424	1	no	—
157	C₄²	C₂²⋊#6157	1	1	yes	characteristic
158	C₁₆	—	106	4	no	—
159	C₂³	—	53	1	no	—
160	C₂×C₄	—	53	1	no	—
161	C₂×C₄	—	53	1	no	—
162	C₂×C₄	—	53	1	no	—
163	C₂×C₄	—	53	1	no	—
164	C₂×C₄	—	53	1	no	—
165	C₂×C₄	—	53	1	no	—
166	C₂×C₄	—	53	2	no	—
167	C₂×C₄	—	53	2	no	—
168	C₂×C₄	—	53	1	no	—
169	C₂×C₄	—	106	2	no	—
170	C₂×C₄	—	424	2	no	—
171	C₂×C₄	—	424	2	no	—
172	C₂×C₄	—	212	4	no	—
173	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
174	C₂×C₄	#13133279	1	1	yes	characteristic
175	C₂×C₄	#13133280	1	1	yes	characteristic
176	C₈	—	2	1	no	—
177	C₈	—	8	1	no	—
178	C₈	—	53	2	no	—
179	C₈	—	106	1	no	—
180	C₈	—	424	1	no	—
181	C₈	C₈×#6157	1	1	yes	characteristic
182	C₈	#13133271	1	1	yes	characteristic
183	C₂²	—	53	2	no	—
184	C₂²	—	53	1	no	—
185	C₂²	—	53	1	no	—
186	C₂²	—	53	2	no	—
187	C₂²	#16811496	1	1	yes	characteristic
188	C₄	—	2	1	no	—
189	C₄	—	53	2	no	—
190	C₄	—	53	1	no	—
191	C₄	—	53	1	no	—
192	C₄	—	106	1	no	—
193	C₄	—	424	4	no	—
194	C₄	(C₂×#6157)⋊C₈	1	1	yes	characteristic
195	C₄	#16811495	1	1	yes	characteristic
196	C₄	#16811535	1	1	yes	characteristic
197	C₄	#16811536	1	1	yes	characteristic
198	C₂	—	53	2	no	—
199	C₂	—	53	2	no	—
200	C₂	#20289549	1	1	yes	characteristic
201	C₂	#20289772	1	1	yes	characteristic
202	C₂	#20289778	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₅₃⋊C₁₆

C₂

yes

maximal, characteristic

C₄×#13133286

C₂

yes

maximal, characteristic

#335×D₅₃

C₂

yes

maximal, characteristic

C₂×D₅₃⋊C₁₆

—

C₂×#13133286

C₄

yes

—

C₄×C₄⋊#6157

—

#16811509

C₄

yes

complemented

C₄×#11719910

—

#16811497

—

C₄⋊C₈×D₅₃

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810376

C₄

yes

characteristic, complemented

#16810375

—

C₅₃×#335

C₄

yes

characteristic, complemented

C₂×C₄⋊#6157

—

C₂×C₄×#6157

—

D₅₃⋊C₁₆

—

C₂×#11719910

—

#13133286

—

#13133286

C₈

yes

complemented

(C₂×#6157)⋊C₄

—

C₂×C₈×D₅₃

—

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×D₅₃

D₄

yes

characteristic

C₄²×D₅₃

D₄

yes

—

—

D₄

yes

characteristic

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

Q₈

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×#6157

—

C₂×C₄×D₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₅₃

D₈

yes

characteristic

C₂×C₄×D₅₃

SD₁₆

yes

—

—

—

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₄²×C₅₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₅₃

C₄⋊C₄

yes

characteristic

C₄×Dic₅₃

SD₁₆

yes

characteristic

C₄×Dic₅₃

Q₁₆

yes

characteristic

C₂×#88322

—

C₂×#88322

C₂²⋊C₄

yes

characteristic

C₂×#88322

C₄⋊C₄

yes

characteristic

C₈×D₅₃

—

C₈×D₅₃

—

C₈×D₅₃

—

C₈×D₅₃

C₂×C₈

yes

—

C₈×D₅₃

M₁₆

yes

—

#11719910

—

C₂²×D₅₃

yes

—

—

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

yes

C₂×Dic₅₃

yes

C₂×Dic₅₃

yes

C₄×D₅₃

—

C₄×D₅₃

C₂²⋊C₈

yes

—

C₄×D₅₃

#104

yes

—

C₄×D₅₃

D₁₆

yes

characteristic

C₄×D₅₃

SD₃₂

yes

characteristic

C₄×D₅₃

SD₃₂

yes

characteristic

C₄×D₅₃

Q₃₂

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₄×C₈

yes

C₈×C₅₃

yes

—

—

C₄⋊C₈

yes

—

—

C₂²×C₅₃

yes

C₂×D₅₃

D₄⋊C₈

yes

—

C₂×D₅₃

yes

C₂×D₅₃

yes

C₂×D₅₃

yes

—

#6157

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

Q₈⋊C₈

yes

—

Dic₅₃

yes

Dic₅₃

yes

C₂×C₄×C₁₆

—

C₄×#107

—

C₂×#335

—

100

C₂×C₅₃

#1279

yes

characteristic

101

C₂×C₅₃

#1380

yes

characteristic

102

C₂×C₅₃

#1382

yes

characteristic

103

D₅₃

#1331

yes

—

104

D₅₃

#1332

yes

—

105

C₂²×C₁₆

—

106

—

106

C₂×#107

—

107

C₂×C₄⋊C₈

—

106

—

108

C₂×C₄×C₈

—

109

C₄×C₄⋊C₄

—

106

—

110

C₄×C₁₆

—

111

C₄×C₁₆

—

106

—

112

#340

—

106

—

113

#335

—

114

#335

—

106

—

115

#335

#6157

yes

characteristic, complemented

116

C₅₃

#7171

yes

characteristic, complemented, Sylow

117

C₂²×C₈

—

118

C₂²×C₈

—

119

C₂²×C₈

—

212

—

120

C₂×C₄⋊C₄

—

106

—

121

C₂×C₄²

—

122

C₂×C₄²

—

212

—

123

C₂×C₁₆

—

106

—

124

C₂×C₁₆

—

106

—

125

#107

—

126

#107

—

106

—

127

C₄⋊C₈

—

128

C₄⋊C₈

—

106

—

129

C₄⋊C₈

—

212

—

130

C₄×C₈

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

C₂×#6157

yes

characteristic

134

#95

—

212

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

212

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

106

—

145

C₂×C₈

—

424

—

146

C₂×C₈

—

424

—

147

C₂×C₈

—

212

—

148

C₂×C₈

C₄×#6157

yes

characteristic

149

C₂×C₈

C₄⋊#6157

yes

characteristic

150

C₄⋊C₄

—

106

—

151

C₄⋊C₄

—

212

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

155

C₄²

—

424

—

156

C₄²

—

424

—

157

C₄²

C₂²⋊#6157

yes

characteristic

158

C₁₆

—

106

—

159

C₂³

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

106

—

170

C₂×C₄

—

424

—

171

C₂×C₄

—

424

—

172

C₂×C₄

—

212

—

173

C₂×C₄

(C₂×#6157)⋊C₄

yes

characteristic

174

C₂×C₄

#13133279

yes

characteristic

175

C₂×C₄

#13133280

yes

characteristic

176

C₈

—

177

C₈

—

178

C₈

—

179

C₈

—

106

—

180

C₈

—

424

—

181

C₈

C₈×#6157

yes

characteristic

182

C₈

#13133271

yes

characteristic

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

#16811496

yes

characteristic

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

106

—

193

C₄

—

424

—

194

C₄

(C₂×#6157)⋊C₈

yes

characteristic

195

C₄

#16811495

yes

characteristic

196

C₄

#16811535

yes

characteristic

197

C₄

#16811536

yes

characteristic

198

C₂

—

199

C₂

—

200

C₂

#20289549

yes

characteristic

201

C₂

#20289772

yes

characteristic

202

C₂

#20289778

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133279
commutator	C₈×C₅₃	C₄×C₈
Frattini	C₄×C₈	C₂×#6157
Fitting	C₅₃×#335	C₄
socle	C₂²×C₅₃	#340
prime radical	C₂²×D₅₃	#102
cosocle	C₄×C₈×D₅₃	C₂²
hypercenter	#335	#6157
hypocenter	C₅₃	#7171
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #335

subgroup

G/H

center

C₂×C₄

#13133279

commutator

C₈×C₅₃

C₄×C₈

Frattini

C₄×C₈

C₂×#6157

Fitting

C₅₃×#335

C₄

C₂²×C₅₃

C₂²×D₅₃

C₄×C₈×D₅₃

C₂²

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #335

N	H ≅ G/N	type
7. #16811509	C₄	semidirect
12. #16810376	C₄	semidirect
14. C₅₃×#335	C₄	semidirect
20. #13133286	C₈	semidirect
115. #335	#6157	semidirect
116. C₅₃	#7171	semidirect

H ≅ G/N

type

7. #16811509

C₄

semidirect

12. #16810376

C₄

semidirect

14. C₅₃×#335

C₄

semidirect

20. #13133286

C₈

semidirect

115. #335

#6157

semidirect

116. C₅₃

#7171

semidirect

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

subgroup	index
C₄×D₅₃⋊C₁₆	2
C₄×#13133286	2
#335×D₅₃	2
#7171	53

subgroup

index

C₄×D₅₃⋊C₁₆

C₄×#13133286

#335×D₅₃

#7171

Subgroups — #28298365

Conjugacy classes of subgroups

Complements

Maximal subgroups