Group #15767664 — subgroups

Subgroups — #15767664

Order 3328.

350 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 119 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	#13115772	C₂	1	1	yes	maximal, characteristic
2	#13114838	C₂	1	2	yes	maximal
3	#13112719	C₂	1	2	yes	maximal
4	#13112671	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1834	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₁₃×#384	C₂²	1	2	yes	—
9	C₂×#88050	—	2	2	no	—
10	C₂×C₄⋊#1055	C₂²	1	2	yes	—
11	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
12	C₁₃×#340	—	2	2	no	—
13	#11718142	—	2	2	no	—
14	#11718121	—	2	2	no	—
15	C₂²×Dic₂₆	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₃	D₄	1	2	yes	—
17	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
18	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
19	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
20	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
21	C₂×C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
22	C₂×#6101	—	2	1	no	—
23	C₂×C₄×Dic₁₃	—	2	1	no	—
24	C₂²×#1055	—	2	2	no	—
25	C₂²×#1055	—	2	1	no	—
26	C₄×Dic₂₆	—	2	2	no	—
27	C₄×Dic₂₆	—	2	2	no	—
28	C₄×C₈×C₁₃	—	2	2	no	—
29	#88050	—	4	4	no	—
30	C₄⋊#1055	—	2	2	no	—
31	C₄⋊#1055	—	2	2	no	—
32	C₄⋊#1055	—	2	2	no	—
33	#10623	—	13	1	no	maximal, Sylow
34	C₂²×C₄×C₁₃	—	4	2	no	—
35	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
38	C₂²×Dic₁₃	—	2	1	no	—
39	C₂²×Dic₁₃	—	2	1	no	—
40	C₂×Dic₂₆	—	2	2	no	—
41	C₂×Dic₂₆	—	2	2	no	—
42	C₂×Dic₂₆	—	2	1	no	—
43	C₂×Dic₂₆	D₈	1	2	yes	—
44	C₂×C₈×C₁₃	—	2	2	no	—
45	C₂×C₈×C₁₃	—	2	2	no	—
46	C₂×C₈×C₁₃	—	2	2	no	—
47	C₁₃×C₄⋊C₄	—	4	2	no	—
48	C₁₃×C₄⋊C₄	—	4	2	no	—
49	C₁₃×C₄⋊C₄	—	4	2	no	—
50	C₁₃×C₄⋊C₄	—	4	2	no	—
51	C₁₃×C₄⋊C₄	—	8	2	no	—
52	C₄²×C₁₃	#41	1	2	yes	—
53	C₄²×C₁₃	#41	1	2	yes	—
54	C₄⋊Dic₁₃	—	2	2	no	—
55	#6101	—	4	2	no	—
56	C₄×Dic₁₃	—	4	1	no	—
57	C₄×Dic₁₃	—	4	1	no	—
58	C₂×#1055	—	4	2	no	—
59	C₂×#1055	—	4	1	no	—
60	C₂×#1055	—	4	2	no	—
61	C₂×#1055	—	4	2	no	—
62	C₂×#1055	—	4	2	no	—
63	#1984	—	13	1	no	—
64	#1834	D₁₃	1	1	yes	characteristic
65	#1803	—	13	2	no	—
66	#1296	—	13	2	no	—
67	#1277	—	13	1	no	—
68	C₂³×C₁₃	#121	1	1	yes	characteristic
69	C₂×C₄×C₁₃	—	2	2	no	—
70	C₂×C₄×C₁₃	—	8	2	no	—
71	C₂×C₄×C₁₃	—	8	2	no	—
72	C₂×C₄×C₁₃	—	8	2	no	—
73	C₂×C₄×C₁₃	#123	1	2	yes	—
74	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
75	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
76	C₂×C₄×C₁₃	#126	1	2	yes	—
77	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
79	C₂×C₄×C₁₃	#135	1	2	yes	—
80	C₂×C₄×C₁₃	#137	1	2	yes	—
81	C₂×C₄×C₁₃	#137	1	2	yes	—
82	C₂×Dic₁₃	—	2	2	no	—
83	C₂×Dic₁₃	—	4	1	no	—
84	C₂×Dic₁₃	—	4	1	no	—
85	C₂×Dic₁₃	—	4	1	no	—
86	C₂×Dic₁₃	—	4	2	no	—
87	Dic₂₆	—	2	4	no	—
88	Dic₂₆	—	4	2	no	—
89	C₈×C₁₃	—	4	2	no	—
90	#1055	—	8	2	no	—
91	#1055	—	8	1	no	—
92	C₂×C₄×Q₈	—	13	1	no	—
93	C₂×C₄⋊C₈	—	13	2	no	—
94	C₂×C₄⋊C₈	—	13	1	no	—
95	C₂×#103	—	26	2	no	—
96	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
97	#384	C₂×D₁₃	1	2	yes	—
98	#340	—	2	2	no	—
99	#340	—	26	2	no	—
100	#336	—	26	2	no	—
101	C₂²×C₁₃	#397	1	1	yes	characteristic
102	C₂²×C₁₃	#464	1	2	yes	—
103	C₂²×C₁₃	#466	1	1	yes	characteristic
104	C₂²×C₁₃	#470	1	1	yes	characteristic
105	C₂²×C₁₃	#471	1	2	yes	—
106	C₄×C₁₃	—	2	2	no	—
107	C₄×C₁₃	—	16	2	no	—
108	C₄×C₁₃	#480	1	2	yes	—
109	C₄×C₁₃	#485	1	2	yes	—
110	C₄×C₁₃	#485	1	2	yes	—
111	C₄×C₁₃	#488	1	1	yes	characteristic
112	C₄×C₁₃	#491	1	1	yes	characteristic
113	Dic₁₃	—	4	2	no	—
114	Dic₁₃	—	8	1	no	—
115	C₂²×Q₈	—	13	1	no	—
116	C₂²×C₈	—	26	2	no	—
117	C₂²×C₈	—	26	1	no	—
118	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
119	C₄×Q₈	—	26	2	no	—
120	C₄×Q₈	—	26	2	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	13	1	no	—
125	C₂×C₄⋊C₄	—	26	1	no	—
126	C₂×C₄²	—	26	1	no	—
127	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
128	C₄⋊C₈	—	26	2	no	—
129	C₄⋊C₈	—	26	2	no	—
130	C₄⋊C₈	—	26	2	no	—
131	#103	—	52	4	no	—
132	C₄×C₈	—	2	2	no	—
133	C₂×C₁₃	#1682	1	2	yes	—
134	C₂×C₁₃	#1704	1	2	yes	—
135	C₂×C₁₃	#2074	1	1	yes	characteristic
136	C₂×C₁₃	#2075	1	1	yes	characteristic
137	C₂×C₁₃	#2079	1	1	yes	characteristic
138	C₂×Q₈	—	13	2	no	—
139	C₂×Q₈	—	26	1	no	—
140	C₂×Q₈	—	26	2	no	—
141	C₂×Q₈	—	26	2	no	—
142	C₂²×C₄	—	4	2	no	—
143	C₂²×C₄	—	26	1	no	—
144	C₂²×C₄	—	26	1	no	—
145	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
146	C₂²×C₄	#6129	1	1	yes	characteristic
147	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	52	2	no	—
152	C₂×C₈	—	52	2	no	—
153	C₂×C₈	—	52	2	no	—
154	C₂×C₈	—	52	1	no	—
155	C₂×C₈	—	52	2	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	8	2	no	—
161	C₄⋊C₄	—	26	2	no	—
162	C₄⋊C₄	—	52	2	no	—
163	C₄²	—	52	1	no	—
164	C₄²	—	52	1	no	—
165	C₄²	#6127	1	2	yes	—
166	C₄²	#6131	1	2	yes	—
167	C₁₃	#10623	1	1	yes	characteristic, complemented, Sylow
168	C₂³	#88192	1	1	yes	characteristic
169	Q₈	—	26	4	no	—
170	Q₈	—	52	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	8	2	no	—
175	C₂×C₄	—	26	2	no	—
176	C₂×C₄	—	52	1	no	—
177	C₂×C₄	—	52	1	no	—
178	C₂×C₄	—	52	2	no	—
179	C₂×C₄	—	52	1	no	—
180	C₂×C₄	#88148	1	2	yes	—
181	C₂×C₄	#88152	1	2	yes	—
182	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
183	C₂×C₄	#88166	1	1	yes	characteristic
184	C₂×C₄	#88172	1	2	yes	—
185	C₂×C₄	#88183	1	1	yes	characteristic
186	C₂×C₄	#88201	1	1	yes	characteristic
187	C₂×C₄	#88204	1	2	yes	—
188	C₂×C₄	#88205	1	2	yes	—
189	C₈	—	4	2	no	—
190	C₈	—	104	1	no	—
191	C₈	—	104	2	no	—
192	C₂²	#11718654	1	1	yes	characteristic
193	C₂²	#11718715	1	2	yes	—
194	C₂²	#11718818	1	1	yes	characteristic
195	C₂²	#11718838	1	2	yes	—
196	C₂²	#11718850	1	1	yes	characteristic
197	C₄	—	2	2	no	—
198	C₄	—	16	2	no	—
199	C₄	—	52	2	no	—
200	C₄	—	104	1	no	—
201	C₄	#11718543	1	2	yes	—
202	C₄	#11718549	1	2	yes	—
203	C₄	#11718657	1	2	yes	—
204	C₄	#11718729	1	1	yes	characteristic
205	C₄	#11718730	1	1	yes	characteristic
206	C₂	#13114277	1	2	yes	—
207	C₂	#13114415	1	2	yes	—
208	C₂	#13116255	1	1	yes	characteristic
209	C₂	#13116260	1	1	yes	characteristic
210	C₂	#13116282	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13115772

C₂

yes

maximal, characteristic

#13114838

C₂

yes

maximal

#13112719

C₂

yes

maximal

#13112671

C₂

yes

maximal, characteristic

C₁₃×#1834

C₂

yes

maximal, characteristic

C₂×C₄×Dic₂₆

C₂²

yes

characteristic

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₁₃×#384

C₂²

yes

—

C₂×#88050

—

C₂×C₄⋊#1055

C₂²

yes

—

C₂×C₄⋊#1055

C₂²

yes

characteristic

C₁₃×#340

—

#11718142

—

#11718121

—

C₂²×Dic₂₆

D₄

yes

characteristic

C₂²×C₈×C₁₃

D₄

yes

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₃

D₄

yes

characteristic

C₂×#6101

—

C₂×C₄×Dic₁₃

—

C₂²×#1055

—

C₂²×#1055

—

C₄×Dic₂₆

—

C₄×Dic₂₆

—

C₄×C₈×C₁₃

—

—

—

—

—

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

#41

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

D₈

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

#41

yes

—

C₄²×C₁₃

#41

yes

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

—

—

—

—

—

—

D₁₃

yes

—

—

—

C₂³×C₁₃

#121

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#123

yes

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

—

C₂×C₄×C₁₃

C₂×D₈

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

—

C₂×C₄×C₁₃

#137

yes

—

C₂×C₄×C₁₃

#137

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

#1055

—

#1055

—

C₂×C₄×Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#103

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

#384

C₂×D₁₃

yes

—

#340

—

#340

—

100

#336

—

101

C₂²×C₁₃

#397

yes

characteristic

102

C₂²×C₁₃

#464

yes

—

103

C₂²×C₁₃

#466

yes

characteristic

104

C₂²×C₁₃

#470

yes

characteristic

105

C₂²×C₁₃

#471

yes

—

106

C₄×C₁₃

—

107

C₄×C₁₃

—

108

C₄×C₁₃

#480

yes

—

109

C₄×C₁₃

#485

yes

—

110

C₄×C₁₃

#485

yes

—

111

C₄×C₁₃

#488

yes

characteristic

112

C₄×C₁₃

#491

yes

characteristic

113

Dic₁₃

—

114

Dic₁₃

—

115

C₂²×Q₈

—

116

C₂²×C₈

—

117

C₂²×C₈

—

118

C₂²×C₈

C₂²⋊D₁₃

yes

—

119

C₄×Q₈

—

120

C₄×Q₈

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄²

—

127

C₂×C₄²

C₂²×D₁₃

yes

characteristic

128

C₄⋊C₈

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

131

#103

—

132

C₄×C₈

—

133

C₂×C₁₃

#1682

yes

—

134

C₂×C₁₃

#1704

yes

—

135

C₂×C₁₃

#2074

yes

characteristic

136

C₂×C₁₃

#2075

yes

characteristic

137

C₂×C₁₃

#2079

yes

characteristic

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

D₄×D₁₃

yes

characteristic

146

C₂²×C₄

#6129

yes

characteristic

147

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄²

—

164

C₄²

—

165

C₄²

#6127

yes

—

166

C₄²

#6131

yes

—

167

C₁₃

#10623

yes

characteristic, complemented, Sylow

168

C₂³

#88192

yes

characteristic

169

Q₈

—

170

Q₈

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

#88148

yes

—

181

C₂×C₄

#88152

yes

—

182

C₂×C₄

D₈×D₁₃

yes

characteristic

183

C₂×C₄

#88166

yes

characteristic

184

C₂×C₄

#88172

yes

—

185

C₂×C₄

#88183

yes

characteristic

186

C₂×C₄

#88201

yes

characteristic

187

C₂×C₄

#88204

yes

—

188

C₂×C₄

#88205

yes

—

189

C₈

—

190

C₈

—

104

—

191

C₈

—

104

—

192

C₂²

#11718654

yes

characteristic

193

C₂²

#11718715

yes

—

194

C₂²

#11718818

yes

characteristic

195

C₂²

#11718838

yes

—

196

C₂²

#11718850

yes

characteristic

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

104

—

201

C₄

#11718543

yes

—

202

C₄

#11718549

yes

—

203

C₄

#11718657

yes

—

204

C₄

#11718729

yes

characteristic

205

C₄

#11718730

yes

characteristic

206

C₂

#13114277

yes

—

207

C₂

#13114415

yes

—

208

C₂

#13116255

yes

characteristic

209

C₂

#13116260

yes

characteristic

210

C₂

#13116282

yes

characteristic

	subgroup	G/H
center	C₂³	#88192
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1834	C₂
socle, prime radical	C₂³×C₁₃	#121
hypercenter	#1834	D₁₃
hypocenter	C₁₃	#10623
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

subgroup

G/H

center

C₂³

#88192

commutator, cosocle

C₂×C₄²×C₁₃

C₂³

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1834

C₂

C₂³×C₁₃

D₁₃

C₁₃

G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

N	H ≅ G/N	type
167. C₁₃	#10623	semidirect

H ≅ G/N

type

167. C₁₃

#10623

semidirect

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

subgroup	index
#13115772	2
#13114838	2
#13112719	2
#13112671	2
C₁₃×#1834	2
#10623	13

subgroup

index

C₁₃×#1834

Subgroups — #15767664

Conjugacy classes of subgroups

Complements

Maximal subgroups