Group #28302174 — subgroups

Subgroups — #28302174

Order 13568.

358 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 144 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	#20290264	C₂	1	1	yes	maximal, characteristic
2	M₁₆×#88324	C₂	1	2	yes	maximal
3	C₂×#16811464	C₂	1	1	yes	maximal, characteristic
4	#20289548	C₂	1	2	yes	maximal
5	#20274616	C₂	1	1	yes	maximal, characteristic
6	#16811639	C₂²	1	2	yes	—
7	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
8	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
9	#16811487	C₄	1	2	yes	—
10	#16811485	C₄	1	2	yes	—
11	#16811464	—	2	2	no	—
12	#16811464	C₄	1	2	yes	—
13	#16811464	C₄	1	2	yes	—
14	C₈×#88324	—	2	2	no	—
15	C₂×#13133152	C₄	1	2	yes	—
16	M₁₆×Dic₅₃	C₂²	1	2	yes	—
17	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
18	#16810388	C₄	1	1	yes	characteristic
19	C₅₃×#341	C₄	1	1	yes	characteristic
20	C₂²×#88324	—	2	1	no	—
21	C₂²×#88324	Q₈	1	2	yes	—
22	C₄⋊#88324	C₂×C₄	1	4	yes	—
23	C₂²⋊#88324	—	2	2	no	—
24	C₄×#88324	—	2	2	no	—
25	C₄×#88324	—	2	1	no	—
26	C₄×#88324	C₂×C₄	1	2	yes	—
27	C₄×#88324	C₂×C₄	1	2	yes	—
28	C₄×#88324	D₄	1	2	yes	—
29	C₂×#11719917	C₂×C₄	1	2	yes	—
30	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
32	#13133152	C₈	1	8	yes	complemented
33	#13133127	—	2	2	no	—
34	C₈×Dic₅₃	—	2	2	no	—
35	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
36	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
37	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
38	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
43	C₂²×Dic₅₃	#41	1	1	yes	characteristic
44	C₂×#88324	—	2	2	no	—
45	C₂×#88324	—	2	4	no	—
46	C₂×#88324	—	4	2	no	—
47	C₂×#88324	—	4	2	no	—
48	C₂×#88324	C₄⋊C₄	1	4	yes	—
49	M₁₆×C₅₃	—	2	2	no	—
50	M₁₆×C₅₃	C₂×C₈	1	4	yes	—
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	—	2	2	no	—
53	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
55	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
58	C₄×Dic₅₃	#41	1	1	yes	characteristic
59	#11719917	—	2	2	no	—
60	#11719917	C₂×C₈	1	4	yes	—
61	C₂×#88322	—	2	2	no	—
62	C₂×#88322	C₄²	1	1	yes	characteristic
63	C₂×#88322	C₄²	1	1	yes	characteristic
64	C₂×#88322	C₂×C₈	1	4	yes	—
65	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
69	C₂×C₄×C₅₃	#131	1	2	yes	—
70	C₂×Dic₅₃	—	2	2	no	—
71	C₂×Dic₅₃	—	2	2	no	—
72	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×Dic₅₃	#117	1	1	yes	characteristic
74	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
75	C₂×Dic₅₃	C₄×Q₈	1	1	yes	characteristic
76	#88324	—	2	4	no	—
77	#88324	—	4	2	no	—
78	#88324	—	4	4	no	—
79	C₈×C₅₃	—	2	1	no	—
80	C₈×C₅₃	—	2	1	no	—
81	C₈×C₅₃	—	2	4	no	—
82	#88322	—	2	4	no	—
83	#88322	C₄×C₈	1	2	yes	—
84	#88322	C₄×C₈	1	2	yes	—
85	#8294	—	53	1	no	maximal, Sylow
86	C₂²×C₅₃	—	2	1	no	—
87	C₂²×C₅₃	—	2	1	no	—
88	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₅₃	C₈×Q₈	1	2	yes	—
90	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
91	C₄×C₅₃	#405	1	1	yes	characteristic
92	C₄×C₅₃	C₈×D₄	1	2	yes	—
93	Dic₅₃	—	2	1	no	—
94	Dic₅₃	—	2	1	no	—
95	Dic₅₃	#430	1	2	yes	—
96	Dic₅₃	#444	1	2	yes	—
97	C₄×#108	—	53	1	no	—
98	#1564	—	53	1	no	—
99	C₈×M₁₆	—	53	2	no	—
100	C₂×#322	—	53	1	no	—
101	#1278	—	53	2	no	—
102	C₂×C₅₃	—	2	2	no	—
103	C₂×C₅₃	C₈×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₅₃	#1773	1	1	yes	characteristic
105	C₂×C₅₃	#1778	1	1	yes	characteristic
106	#431	—	53	2	no	—
107	C₂×#108	—	53	2	no	—
108	C₄×M₁₆	—	53	2	no	—
109	C₂×C₄×C₈	—	53	1	no	—
110	C₂×C₄×C₈	—	53	1	no	—
111	C₂×C₄×C₈	—	53	1	no	—
112	#341	—	53	1	no	—
113	#341	#6157	1	1	yes	characteristic
114	#335	—	53	2	no	—
115	#334	—	53	2	no	—
116	#322	—	53	2	no	—
117	#322	—	53	2	no	—
118	#322	—	106	2	no	—
119	C₈²	—	106	2	no	—
120	C₅₃	#8294	1	1	yes	characteristic, complemented, Sylow
121	C₂×M₁₆	—	53	2	no	—
122	C₂×M₁₆	C₂×#6157	1	2	yes	—
123	C₂²×C₈	—	53	1	no	—
124	C₂²×C₈	—	53	2	no	—
125	C₂²×C₈	—	106	1	no	—
126	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
127	C₂×C₄²	—	53	1	no	—
128	#108	—	53	8	no	—
129	C₄⋊C₈	—	53	4	no	—
130	C₂²⋊C₈	—	106	2	no	—
131	#97	—	106	2	no	—
132	C₄×C₈	—	53	2	no	—
133	C₄×C₈	—	53	2	no	—
134	C₄×C₈	—	53	1	no	—
135	C₄×C₈	—	53	1	no	—
136	C₄×C₈	—	53	1	no	—
137	C₄×C₈	—	53	1	no	—
138	C₄×C₈	—	53	2	no	—
139	C₄×C₈	—	106	2	no	—
140	C₄×C₈	—	106	2	no	—
141	C₄×C₈	—	106	1	no	—
142	C₂²×C₄	—	53	1	no	—
143	C₂²×C₄	—	53	1	no	—
144	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
145	M₁₆	—	2	2	no	—
146	M₁₆	—	53	4	no	—
147	M₁₆	—	106	2	no	—
148	M₁₆	D₅₃⋊C₈	1	4	yes	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	53	1	no	—
152	C₂×C₈	—	53	1	no	—
153	C₂×C₈	—	53	4	no	—
154	C₂×C₈	—	53	4	no	—
155	C₂×C₈	—	106	2	no	—
156	C₂×C₈	—	106	2	no	—
157	C₂×C₈	—	106	4	no	—
158	C₂×C₈	—	212	2	no	—
159	C₂×C₈	—	212	2	no	—
160	C₂×C₈	C₄×#6157	1	1	yes	characteristic
161	C₂×C₈	C₄×#6157	1	1	yes	characteristic
162	C₄²	—	53	1	no	—
163	C₄²	—	53	1	no	—
164	C₄²	—	53	1	no	—
165	C₄²	—	53	1	no	—
166	C₂³	Q₈×#6157	1	1	yes	characteristic
167	C₂×C₄	—	53	1	no	—
168	C₂×C₄	—	53	1	no	—
169	C₂×C₄	—	53	1	no	—
170	C₂×C₄	—	53	1	no	—
171	C₂×C₄	—	106	2	no	—
172	C₂×C₄	—	106	2	no	—
173	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
174	C₂×C₄	D₄×#6157	1	1	yes	characteristic
175	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
176	C₂×C₄	#13133304	1	2	yes	—
177	C₈	—	2	1	no	—
178	C₈	—	2	1	no	—
179	C₈	—	2	4	no	—
180	C₈	—	53	2	no	—
181	C₈	—	53	2	no	—
182	C₈	—	106	4	no	—
183	C₈	—	106	4	no	—
184	C₈	—	212	4	no	—
185	C₈	—	212	2	no	—
186	C₂²	—	2	1	no	—
187	C₂²	—	2	1	no	—
188	C₂²	C₄⋊C₄×#6157	1	1	yes	characteristic
189	C₂²	#16811672	1	2	yes	—
190	C₄	—	53	2	no	—
191	C₄	—	53	2	no	—
192	C₄	—	106	1	no	—
193	C₄	—	106	1	no	—
194	C₄	C₄×D₅₃⋊C₈	1	1	yes	characteristic
195	C₄	#16811587	1	1	yes	characteristic
196	C₄	#16811650	1	2	yes	—
197	C₂	—	2	2	no	—
198	C₂	#20290507	1	1	yes	characteristic
199	C₂	#20290524	1	1	yes	characteristic
200	C₂	#20290536	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20290264

C₂

yes

maximal, characteristic

M₁₆×#88324

C₂

yes

maximal

C₂×#16811464

C₂

yes

maximal, characteristic

#20289548

C₂

yes

maximal

#20274616

C₂

yes

maximal, characteristic

#16811639

C₂²

yes

—

C₂×C₄×#88324

C₂²

yes

characteristic

C₂×C₄×#88324

C₂²

yes

characteristic

#16811487

C₄

yes

—

#16811485

C₄

yes

—

#16811464

—

#16811464

C₄

yes

—

#16811464

C₄

yes

—

C₈×#88324

—

C₂×#13133152

C₄

yes

—

M₁₆×Dic₅₃

C₂²

yes

—

C₂×C₈×Dic₅₃

C₂²

yes

characteristic

#16810388

C₄

yes

characteristic

C₅₃×#341

C₄

yes

characteristic

C₂²×#88324

—

C₂²×#88324

Q₈

yes

—

C₄⋊#88324

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

—

C₄×#88324

C₂×C₄

yes

—

C₄×#88324

D₄

yes

—

C₂×#11719917

C₂×C₄

yes

—

C₂²×#88322

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₂³

yes

characteristic

#13133152

C₈

yes

complemented

#13133127

—

C₈×Dic₅₃

—

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₂×M₁₆×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×Q₈

yes

C₂²×Dic₅₃

yes

—

—

—

—

C₄⋊C₄

yes

—

M₁₆×C₅₃

—

M₁₆×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₄×Dic₅₃

C₂²×C₄

yes

characteristic

C₄×Dic₅₃

C₂²×C₄

yes

characteristic

C₄×Dic₅₃

C₂×D₄

yes

C₄×Dic₅₃

yes

—

C₂×C₈

yes

—

C₂×#88322

—

C₂×#88322

C₄²

yes

characteristic

C₂×#88322

C₄²

yes

characteristic

C₂×#88322

C₂×C₈

yes

—

C₂³×C₅₃

C₄×Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₈

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

#117

yes

characteristic

C₂×Dic₅₃

C₄×D₄

yes

characteristic

C₂×Dic₅₃

C₄×Q₈

yes

—

—

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

C₄×C₈

yes

—

#88322

C₄×C₈

yes

—

#8294

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

C₈×Q₈

yes

—

C₄×C₅₃

C₂×C₄×C₈

yes

characteristic

C₄×C₅₃

#405

yes

characteristic

C₄×C₅₃

C₈×D₄

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#430

yes

—

Dic₅₃

#444

yes

—

C₄×#108

—

#1564

—

C₈×M₁₆

—

100

C₂×#322

—

101

#1278

—

102

C₂×C₅₃

—

103

C₂×C₅₃

C₈×C₄⋊C₄

yes

characteristic

104

C₂×C₅₃

#1773

yes

characteristic

105

C₂×C₅₃

#1778

yes

characteristic

106

#431

—

107

C₂×#108

—

108

C₄×M₁₆

—

109

C₂×C₄×C₈

—

110

C₂×C₄×C₈

—

111

C₂×C₄×C₈

—

112

#341

—

113

yes

114

—

115

#334

—

116

#322

—

117

#322

—

118

#322

—

106

—

119

C₈²

—

106

—

120

C₅₃

#8294

yes

characteristic, complemented, Sylow

121

C₂×M₁₆

—

122

C₂×M₁₆

C₂×#6157

yes

—

123

C₂²×C₈

—

124

C₂²×C₈

—

125

C₂²×C₈

—

106

—

126

C₂²×C₈

C₂×#6157

yes

characteristic

127

C₂×C₄²

—

128

#108

—

129

C₄⋊C₈

—

130

C₂²⋊C₈

—

106

—

131

#97

—

106

—

132

C₄×C₈

—

133

C₄×C₈

—

134

C₄×C₈

—

135

C₄×C₈

—

136

C₄×C₈

—

137

C₄×C₈

—

138

C₄×C₈

—

139

C₄×C₈

—

106

—

140

C₄×C₈

—

106

—

141

C₄×C₈

—

106

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

C₂²×#6157

yes

characteristic

145

M₁₆

—

146

M₁₆

—

147

M₁₆

—

106

—

148

M₁₆

D₅₃⋊C₈

yes

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

106

—

156

C₂×C₈

—

106

—

157

C₂×C₈

—

106

—

158

C₂×C₈

—

212

—

159

C₂×C₈

—

212

—

160

C₂×C₈

C₄×#6157

yes

characteristic

161

C₂×C₈

C₄×#6157

yes

characteristic

162

C₄²

—

163

C₄²

—

164

C₄²

—

165

C₄²

—

166

C₂³

Q₈×#6157

yes

characteristic

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

106

—

172

C₂×C₄

—

106

—

173

C₂×C₄

C₂×C₄×#6157

yes

characteristic

174

C₂×C₄

D₄×#6157

yes

characteristic

175

C₂×C₄

C₂×D₅₃⋊C₈

yes

—

176

C₂×C₄

#13133304

yes

—

177

C₈

—

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

—

182

C₈

—

106

—

183

C₈

—

106

—

184

C₈

—

212

—

185

C₈

—

212

—

186

C₂²

—

187

C₂²

—

188

C₂²

C₄⋊C₄×#6157

yes

characteristic

189

C₂²

#16811672

yes

—

190

C₄

—

191

C₄

—

192

C₄

—

106

—

193

C₄

—

106

—

194

C₄

C₄×D₅₃⋊C₈

yes

characteristic

195

C₄

#16811587

yes

characteristic

196

C₄

#16811650

yes

—

197

C₂

—

198

C₂

#20290507

yes

characteristic

199

C₂

#20290524

yes

characteristic

200

C₂

#20290536

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6157
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₅₃×#341	C₄
socle	C₂²×C₅₃	C₄×C₄⋊C₄
prime radical	C₂³×C₅₃	C₄×Q₈
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	#341	#6157
hypocenter	C₅₃	#8294
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×#6157

commutator

C₄×C₅₃

C₂×C₄×C₈

Frattini

C₂²×C₄

C₂²×#6157

Fitting

C₅₃×#341

C₄

socle

C₂²×C₅₃

C₄×C₄⋊C₄

prime radical

C₂³×C₅₃

C₄×Q₈

cosocle

C₂×C₄×Dic₅₃

C₂³

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
32. #13133152	C₈	semidirect
120. C₅₃	#8294	semidirect

H ≅ G/N

type

32. #13133152

C₈

semidirect

120. C₅₃

#8294

semidirect

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

subgroup	index
#20290264	2
M₁₆×#88324	2
C₂×#16811464	2
#20289548	2
#20274616	2
#8294	53

subgroup

index

M₁₆×#88324

Subgroups — #28302174

Conjugacy classes of subgroups

Complements

Maximal subgroups