Group #15763070 — subgroups

Subgroups — #15763070

Order 3328.

438 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 163 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	#13123199	C₂	1	1	yes	maximal, characteristic
2	#13114988	C₂	1	1	yes	maximal, characteristic
3	#13112720	C₂	1	4	yes	maximal
4	C₁₃×#1767	C₂	1	1	yes	maximal, characteristic
5	C₂×#88124	—	2	1	no	—
6	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
7	C₂×C₁₃×#97	—	2	1	no	—
8	C₄³×C₁₃	C₂²	1	1	yes	characteristic
9	#11718597	—	2	1	no	—
10	#11718596	C₂²	1	2	yes	—
11	#11718589	—	2	2	no	—
12	#11718589	C₄	1	4	yes	—
13	C₂×C₄⋊#1055	—	2	2	no	—
14	C₂×C₄⋊#1055	C₂²	1	2	yes	—
15	C₁₃×#336	—	2	1	no	—
16	C₂²×C₈×C₁₃	—	2	2	no	—
17	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₁₃	D₄	1	2	yes	—
19	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₃	—	2	2	no	—
23	C₂×C₄⋊Dic₁₃	—	2	2	no	—
24	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
25	C₂×#6101	—	2	4	no	—
26	C₂²×#1055	—	2	2	no	—
27	#88124	—	4	2	no	—
28	#88124	—	4	2	no	—
29	C₁₃×C₄⋊C₈	—	2	2	no	—
30	C₁₃×C₄⋊C₈	Q₈	1	4	yes	—
31	C₁₃×#97	—	4	2	no	—
32	#88071	—	2	4	no	—
33	C₄⋊#1055	—	2	4	no	—
34	C₄⋊#1055	—	4	4	no	—
35	#9904	—	13	1	no	maximal, Sylow
36	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
37	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
38	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×Dic₁₃	—	2	4	no	—
42	C₂×C₈×C₁₃	—	2	4	no	—
43	C₂×C₈×C₁₃	—	4	2	no	—
44	C₂×C₈×C₁₃	—	4	2	no	—
45	C₄²×C₁₃	—	2	2	no	—
46	C₄²×C₁₃	—	2	2	no	—
47	C₄²×C₁₃	—	2	2	no	—
48	C₄²×C₁₃	—	4	2	no	—
49	C₄²×C₁₃	D₈	1	2	yes	—
50	C₄²×C₁₃	SD₁₆	1	2	yes	—
51	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
52	C₄²×C₁₃	#41	1	2	yes	—
53	C₄⋊Dic₁₃	—	4	2	no	—
54	C₄⋊Dic₁₃	—	4	2	no	—
55	C₄⋊Dic₁₃	—	4	2	no	—
56	C₄⋊Dic₁₃	—	4	2	no	—
57	C₄⋊Dic₁₃	—	4	4	no	—
58	#6101	—	4	8	no	—
59	C₂×#1055	—	4	2	no	—
60	C₂×#1055	—	4	4	no	—
61	#2859	—	13	1	no	—
62	#1842	—	13	1	no	—
63	#1767	D₁₃	1	1	yes	characteristic
64	#1296	—	13	4	no	—
65	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₃	—	2	2	no	—
67	C₂×C₄×C₁₃	—	2	2	no	—
68	C₂×C₄×C₁₃	—	2	2	no	—
69	C₂×C₄×C₁₃	—	2	2	no	—
70	C₂×C₄×C₁₃	—	2	2	no	—
71	C₂×C₄×C₁₃	—	2	2	no	—
72	C₂×C₄×C₁₃	#102	1	4	yes	—
73	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
74	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
75	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
80	C₂×C₄×C₁₃	#137	1	2	yes	—
81	C₂×C₄×C₁₃	#137	1	2	yes	—
82	C₂×Dic₁₃	—	4	4	no	—
83	C₂×Dic₁₃	—	4	4	no	—
84	C₂×Dic₁₃	—	4	4	no	—
85	C₈×C₁₃	—	4	4	no	—
86	#1055	—	8	2	no	—
87	C₂×#125	—	26	1	no	—
88	C₂×C₄⋊C₈	—	13	2	no	—
89	C₂×C₄⋊C₈	—	26	2	no	—
90	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
91	C₂×#97	—	2	1	no	—
92	#389	—	13	4	no	—
93	#389	—	26	2	no	—
94	#384	—	13	2	no	—
95	#382	—	26	1	no	—
96	C₄³	C₂×D₁₃	1	1	yes	characteristic
97	#336	—	2	1	no	—
98	C₂²×C₁₃	#391	1	1	yes	characteristic
99	C₂²×C₁₃	#413	1	2	yes	—
100	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
101	C₂²×C₁₃	#417	1	2	yes	—
102	C₂²×C₁₃	#419	1	1	yes	characteristic
103	C₄×C₁₃	—	2	4	no	—
104	C₄×C₁₃	—	2	2	no	—
105	C₄×C₁₃	—	4	2	no	—
106	C₄×C₁₃	#474	1	2	yes	—
107	C₄×C₁₃	#475	1	2	yes	—
108	C₄×C₁₃	#490	1	1	yes	characteristic
109	C₄×C₁₃	#491	1	1	yes	characteristic
110	C₄×C₁₃	#501	1	2	yes	—
111	Dic₁₃	—	8	4	no	—
112	C₂²×C₈	—	2	2	no	—
113	C₂²×C₈	—	26	2	no	—
114	#125	—	52	2	no	—
115	#125	—	52	2	no	—
116	C₂×C₄⋊C₄	—	13	4	no	—
117	C₂×C₄⋊C₄	—	26	4	no	—
118	C₂×C₄⋊C₄	—	26	2	no	—
119	C₂×C₄⋊C₄	—	26	2	no	—
120	C₂×C₄²	C₄×D₁₃	1	2	yes	—
121	C₂×C₄²	D₅₂	1	2	yes	—
122	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
123	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
124	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	26	4	no	—
127	C₄⋊C₈	—	52	4	no	—
128	C₄⋊C₈	Dic₂₆	1	4	yes	—
129	#97	—	4	2	no	—
130	#95	—	26	4	no	—
131	C₂×C₁₃	#1547	1	2	yes	—
132	C₂×C₁₃	#1559	1	2	yes	—
133	C₂×C₁₃	#1988	1	1	yes	characteristic
134	C₂×C₁₃	#1992	1	1	yes	characteristic
135	C₂×C₁₃	#1998	1	1	yes	characteristic
136	C₂²×C₄	—	26	4	no	—
137	C₂²×C₄	#6103	1	2	yes	—
138	C₂²×C₄	#6103	1	2	yes	—
139	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	4	2	no	—
144	C₂×C₈	—	4	2	no	—
145	C₂×C₈	—	52	2	no	—
146	C₂×C₈	—	52	4	no	—
147	C₄⋊C₄	—	52	4	no	—
148	C₄⋊C₄	—	52	2	no	—
149	C₄⋊C₄	—	52	2	no	—
150	C₄⋊C₄	—	52	2	no	—
151	C₄⋊C₄	—	52	2	no	—
152	C₄⋊C₄	—	52	8	no	—
153	C₄²	—	2	2	no	—
154	C₄²	—	2	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	4	2	no	—
157	C₄²	#6104	1	2	yes	—
158	C₄²	#6106	1	2	yes	—
159	C₄²	C₂×Dic₂₆	1	2	yes	—
160	C₄²	#6127	1	2	yes	—
161	C₁₃	#9904	1	1	yes	characteristic, complemented, Sylow
162	C₂³	C₂×#6103	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	52	4	no	—
170	C₂×C₄	—	52	4	no	—
171	C₂×C₄	—	52	4	no	—
172	C₂×C₄	#88049	1	4	yes	—
173	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
174	C₂×C₄	#88123	1	1	yes	characteristic
175	C₂×C₄	#88129	1	1	yes	characteristic
176	C₂×C₄	#88162	1	2	yes	—
177	C₂×C₄	#88163	1	2	yes	—
178	C₂×C₄	#88178	1	2	yes	—
179	C₂×C₄	C₂×#6104	1	1	yes	characteristic
180	C₂×C₄	C₂×#6106	1	1	yes	characteristic
181	C₂×C₄	#88205	1	2	yes	—
182	C₈	—	4	4	no	—
183	C₈	—	104	2	no	—
184	C₂²	#11718590	1	1	yes	characteristic
185	C₂²	C₂×#88049	1	1	yes	characteristic
186	C₂²	#11718665	1	1	yes	characteristic
187	C₂²	#11718793	1	2	yes	—
188	C₂²	#11718794	1	2	yes	—
189	C₄	—	2	4	no	—
190	C₄	—	2	2	no	—
191	C₄	—	4	2	no	—
192	C₄	—	104	4	no	—
193	C₄	#11718363	1	2	yes	—
194	C₄	#11718364	1	1	yes	characteristic
195	C₄	#11718374	1	1	yes	characteristic
196	C₄	#11718668	1	2	yes	—
197	C₄	#11718681	1	2	yes	—
198	C₂	#13113514	1	2	yes	—
199	C₂	#13113567	1	2	yes	—
200	C₂	#13115788	1	1	yes	characteristic
201	C₂	#13115805	1	1	yes	characteristic
202	C₂	#13115829	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13123199

C₂

yes

maximal, characteristic

#13114988

C₂

yes

maximal, characteristic

#13112720

C₂

yes

maximal

C₁₃×#1767

C₂

yes

maximal, characteristic

C₂×#88124

—

C₂×C₁₃×C₄⋊C₈

C₂²

yes

—

C₂×C₁₃×#97

—

C₄³×C₁₃

C₂²

yes

characteristic

#11718597

—

#11718596

C₂²

yes

—

#11718589

—

#11718589

C₄

yes

—

C₂×C₄⋊#1055

—

C₂×C₄⋊#1055

C₂²

yes

—

C₁₃×#336

—

C₂²×C₈×C₁₃

—

C₂×C₄²×C₁₃

C₂×C₄

yes

—

C₂×C₄²×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

D₄

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

characteristic

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

C₂×C₄

yes

—

C₂×#6101

—

C₂²×#1055

—

#88124

—

#88124

—

C₁₃×C₄⋊C₈

—

C₁₃×C₄⋊C₈

Q₈

yes

—

C₁₃×#97

—

—

—

—

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

D₈

yes

—

C₄²×C₁₃

SD₁₆

yes

—

C₄²×C₁₃

C₂×Q₈

yes

—

C₄²×C₁₃

#41

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

—

—

—

—

—

D₁₃

yes

characteristic

#1296

—

C₂³×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#102

yes

—

C₂×C₄×C₁₃

C₄×Q₈

yes

—

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₃

#124

yes

characteristic

C₂×C₄×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₁₃

#137

yes

—

C₂×C₄×C₁₃

#137

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₈×C₁₃

—

#1055

—

C₂×#125

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×D₁₃

yes

—

C₂×#97

—

—

—

—

—

C₄³

C₂×D₁₃

yes

characteristic

#336

—

C₂²×C₁₃

#391

yes

characteristic

C₂²×C₁₃

#413

yes

—

100

C₂²×C₁₃

C₂×#102

yes

characteristic

101

C₂²×C₁₃

#417

yes

—

102

C₂²×C₁₃

#419

yes

characteristic

103

C₄×C₁₃

—

104

C₄×C₁₃

—

105

C₄×C₁₃

—

106

C₄×C₁₃

#474

yes

—

107

C₄×C₁₃

#475

yes

—

108

C₄×C₁₃

#490

yes

characteristic

109

C₄×C₁₃

#491

yes

characteristic

110

C₄×C₁₃

#501

yes

—

111

Dic₁₃

—

112

C₂²×C₈

—

113

C₂²×C₈

—

114

#125

—

115

#125

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄²

C₄×D₁₃

yes

—

121

C₂×C₄²

D₅₂

yes

—

122

C₂×C₄²

C₂²⋊D₁₃

yes

characteristic

123

C₂×C₄²

C₂²⋊D₁₃

yes

characteristic

124

C₂×C₄²

C₂²×D₁₃

yes

characteristic

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₄⋊C₈

—

128

C₄⋊C₈

Dic₂₆

yes

—

129

#97

—

130

#95

—

131

C₂×C₁₃

#1547

yes

—

132

C₂×C₁₃

#1559

yes

—

133

C₂×C₁₃

#1988

yes

characteristic

134

C₂×C₁₃

#1992

yes

characteristic

135

C₂×C₁₃

#1998

yes

characteristic

136

C₂²×C₄

—

137

C₂²×C₄

#6103

yes

—

138

C₂²×C₄

#6103

yes

—

139

C₂²×C₄

C₂×C₄×D₁₃

yes

characteristic

140

C₂²×C₄

C₂×D₅₂

yes

characteristic

141

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄²

—

154

C₄²

—

155

C₄²

—

156

C₄²

—

157

C₄²

#6104

yes

—

158

C₄²

#6106

yes

—

159

C₄²

C₂×Dic₂₆

yes

—

160

C₄²

#6127

yes

—

161

C₁₃

#9904

yes

characteristic, complemented, Sylow

162

C₂³

C₂×#6103

yes

characteristic

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

#88049

yes

—

173

C₂×C₄

C₄×Dic₂₆

yes

—

174

C₂×C₄

#88123

yes

characteristic

175

C₂×C₄

#88129

yes

characteristic

176

C₂×C₄

#88162

yes

—

177

C₂×C₄

#88163

yes

—

178

C₂×C₄

#88178

yes

—

179

C₂×C₄

C₂×#6104

yes

characteristic

180

C₂×C₄

C₂×#6106

yes

characteristic

181

C₂×C₄

#88205

yes

—

182

C₈

—

183

C₈

—

104

—

184

C₂²

#11718590

yes

characteristic

185

C₂²

C₂×#88049

yes

characteristic

186

C₂²

#11718665

yes

characteristic

187

C₂²

#11718793

yes

—

188

C₂²

#11718794

yes

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

104

—

193

C₄

#11718363

yes

—

194

C₄

#11718364

yes

characteristic

195

C₄

#11718374

yes

characteristic

196

C₄

#11718668

yes

—

197

C₄

#11718681

yes

—

198

C₂

#13113514

yes

—

199

C₂

#13113567

yes

—

200

C₂

#13115788

yes

characteristic

201

C₂

#13115805

yes

characteristic

202

C₂

#13115829

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#6103
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1767	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1767	D₁₃
hypocenter	C₁₃	#9904
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

subgroup

G/H

center

C₂³

C₂×#6103

commutator

C₂²×C₄×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1767

C₂

socle, prime radical

C₂³×C₁₃

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

N	H ≅ G/N	type
161. C₁₃	#9904	semidirect

H ≅ G/N

type

161. C₁₃

#9904

semidirect

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

subgroup	index
#13123199	2
#13114988	2
#13112720	2
C₁₃×#1767	2
#9904	13

subgroup

index

C₁₃×#1767

Subgroups — #15763070

Conjugacy classes of subgroups

Complements

Maximal subgroups