Group #15767955 — subgroups

Subgroups — #15767955

Order 3328.

494 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 145 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	#13115642	C₂	1	1	yes	maximal, characteristic
2	#13115641	C₂	1	2	yes	maximal
3	#13114963	C₂	1	2	yes	maximal, complemented
4	#13112663	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1845	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#88128	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#107	—	2	2	no	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₁₃×#384	C₂²	1	2	yes	—
10	C₂×#88061	—	2	2	no	—
11	C₂×#88049	—	2	4	no	—
12	#11718596	C₂²	1	1	yes	characteristic
13	C₂×C₄×#1055	C₂²	1	2	yes	—
14	#11718121	—	2	4	no	—
15	C₂²×D₅₂	—	2	1	no	—
16	C₂²×D₅₂	—	2	2	no	—
17	C₂²×C₈×C₁₃	D₄	1	2	yes	—
18	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
19	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
20	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
21	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₃	—	2	1	no	—
23	C₂×C₄⋊Dic₁₃	—	2	2	no	—
24	C₂²×#1055	D₄	1	4	yes	—
25	#88128	—	2	2	no	—
26	#88128	—	2	1	no	—
27	#88128	Q₈	1	2	yes	—
28	C₁₃×#107	—	4	4	no	—
29	C₄×C₈×C₁₃	—	2	2	no	—
30	#88061	—	4	4	no	—
31	#88049	—	4	8	no	—
32	C₄×#1055	—	2	2	no	—
33	C₄×#1055	—	2	2	no	—
34	#10671	—	13	1	no	maximal, Sylow
35	C₂³×D₁₃	—	4	1	no	—
36	C₂²×C₄×C₁₃	—	4	2	no	—
37	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
38	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₁₃	—	4	1	no	—
40	C₂×D₅₂	—	2	2	no	—
41	C₂×D₅₂	—	2	4	no	—
42	C₂×D₅₂	—	4	2	no	—
43	C₂×D₅₂	—	4	4	no	—
44	C₂×D₅₂	—	4	4	no	—
45	C₂×D₅₂	—	4	2	no	—
46	C₂×D₅₂	—	4	1	no	—
47	C₂×D₅₂	—	4	2	no	—
48	C₂×C₈×C₁₃	—	2	2	no	—
49	C₂×C₈×C₁₃	—	2	2	no	—
50	C₂×C₈×C₁₃	D₈	1	4	yes	—
51	C₁₃×C₄⋊C₄	—	4	4	no	—
52	C₁₃×C₄⋊C₄	—	4	4	no	—
53	C₁₃×C₄⋊C₄	—	4	4	no	—
54	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
55	C₄²×C₁₃	#41	1	2	yes	—
56	C₄²×C₁₃	#41	1	1	yes	characteristic
57	C₄⋊Dic₁₃	—	4	2	no	—
58	C₄⋊Dic₁₃	—	4	4	no	—
59	C₂×#1055	—	2	4	no	—
60	C₂×#1055	—	2	4	no	—
61	C₂×#1055	—	2	4	no	—
62	#1957	—	13	1	no	—
63	#1957	—	13	2	no	—
64	#1845	D₁₃	1	1	yes	characteristic, complemented
65	#1834	—	13	2	no	—
66	#1273	—	13	1	no	—
67	C₂³×C₁₃	#127	1	1	yes	characteristic
68	C₂²×D₁₃	—	4	2	no	—
69	C₂²×D₁₃	—	8	1	no	—
70	C₂²×D₁₃	—	8	2	no	—
71	C₂²×D₁₃	—	8	2	no	—
72	C₂²×D₁₃	—	8	1	no	—
73	C₂×C₄×C₁₃	—	8	2	no	—
74	C₂×C₄×C₁₃	—	8	2	no	—
75	C₂×C₄×C₁₃	—	8	2	no	—
76	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
78	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
79	C₂×C₄×C₁₃	#123	1	2	yes	—
80	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
81	C₂×C₄×C₁₃	#135	1	4	yes	—
82	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
83	C₂×C₄×C₁₃	#137	1	2	yes	—
84	C₂×Dic₁₃	—	8	1	no	—
85	C₂×Dic₁₃	—	8	2	no	—
86	D₅₂	—	4	4	no	—
87	D₅₂	—	4	8	no	—
88	D₅₂	—	8	2	no	—
89	D₅₂	—	8	4	no	—
90	C₈×C₁₃	—	2	4	no	—
91	#1055	—	4	4	no	—
92	C₂×#127	—	13	1	no	—
93	C₂×#107	—	2	2	no	—
94	C₂×#102	—	26	4	no	—
95	C₂×#102	—	26	2	no	—
96	C₂×C₄×C₈	—	13	2	no	—
97	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
98	#384	—	13	1	no	—
99	#384	C₂×D₁₃	1	2	yes	—
100	#340	—	26	4	no	—
101	C₂²×C₁₃	#399	1	1	yes	characteristic
102	C₂²×C₁₃	#493	1	1	yes	characteristic
103	C₂²×C₁₃	#495	1	2	yes	—
104	C₂²×C₁₃	#496	1	2	yes	—
105	C₂²×C₁₃	#497	1	1	yes	characteristic
106	C₂×D₁₃	—	8	4	no	—
107	C₂×D₁₃	—	8	2	no	—
108	C₂×D₁₃	—	16	1	no	—
109	C₂×D₁₃	—	16	2	no	—
110	C₂×D₁₃	—	16	1	no	—
111	C₄×C₁₃	—	16	2	no	—
112	C₄×C₁₃	#474	1	2	yes	—
113	C₄×C₁₃	#478	1	4	yes	—
114	C₄×C₁₃	#480	1	2	yes	—
115	C₄×C₁₃	#482	1	4	yes	—
116	Dic₁₃	—	16	1	no	—
117	C₂²×D₄	—	26	2	no	—
118	C₂²×D₄	—	26	1	no	—
119	C₂²×C₈	—	13	4	no	—
120	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
121	#127	—	13	2	no	—
122	#127	—	26	2	no	—
123	#127	—	26	1	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄⋊C₄	—	26	2	no	—
128	C₂×C₄⋊C₄	—	26	1	no	—
129	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
130	#107	—	4	4	no	—
131	#102	—	52	8	no	—
132	#102	—	52	4	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	—	26	2	no	—
135	C₄×C₈	—	26	2	no	—
136	C₂×C₁₃	#1709	1	1	yes	characteristic
137	C₂×C₁₃	#1719	1	2	yes	—
138	C₂×C₁₃	#1721	1	1	yes	characteristic
139	C₂×C₁₃	#2092	1	2	yes	—
140	C₂×C₁₃	#2093	1	1	yes	characteristic
141	D₁₃	—	16	2	no	—
142	C₂⁴	—	52	1	no	—
143	C₂×D₄	—	26	4	no	—
144	C₂×D₄	—	26	2	no	—
145	C₂×D₄	—	52	2	no	—
146	C₂×D₄	—	52	4	no	—
147	C₂×D₄	—	52	4	no	—
148	C₂×D₄	—	52	1	no	—
149	C₂×D₄	—	52	2	no	—
150	C₂×D₄	—	52	2	no	—
151	C₂²×C₄	—	4	2	no	—
152	C₂²×C₄	—	52	1	no	—
153	C₂²×C₄	D₄×D₁₃	1	2	yes	—
154	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	26	4	no	—
158	C₂×C₈	—	26	4	no	—
159	C₂×C₈	—	26	4	no	—
160	C₂×C₈	#6104	1	4	yes	—
161	C₄⋊C₄	—	4	4	no	—
162	C₄⋊C₄	—	4	4	no	—
163	C₄⋊C₄	—	4	4	no	—
164	C₄⋊C₄	—	52	4	no	—
165	C₄⋊C₄	—	52	2	no	—
166	C₄²	#6127	1	2	yes	—
167	C₄²	Q₈×D₁₃	1	1	yes	characteristic
168	C₄²	#6131	1	1	yes	characteristic
169	C₁₃	#10671	1	1	yes	characteristic, complemented, Sylow
170	C₂³	—	52	2	no	—
171	C₂³	—	104	1	no	—
172	C₂³	—	104	1	no	—
173	C₂³	—	104	2	no	—
174	C₂³	—	104	2	no	—
175	C₂³	#88194	1	1	yes	characteristic
176	D₄	—	52	4	no	—
177	D₄	—	52	8	no	—
178	D₄	—	104	4	no	—
179	D₄	—	104	2	no	—
180	C₂×C₄	—	8	2	no	—
181	C₂×C₄	—	8	2	no	—
182	C₂×C₄	—	8	2	no	—
183	C₂×C₄	—	104	1	no	—
184	C₂×C₄	—	104	2	no	—
185	C₂×C₄	#88148	1	2	yes	—
186	C₂×C₄	#88150	1	2	yes	—
187	C₂×C₄	#88165	1	4	yes	—
188	C₂×C₄	#88173	1	4	yes	—
189	C₂×C₄	#88183	1	1	yes	characteristic
190	C₂×C₄	C₂×#6104	1	2	yes	—
191	C₂×C₄	#88196	1	2	yes	—
192	C₂×C₄	#88200	1	1	yes	characteristic
193	C₈	—	2	4	no	—
194	C₈	—	52	4	no	—
195	C₂²	—	104	4	no	—
196	C₂²	—	104	2	no	—
197	C₂²	—	208	1	no	—
198	C₂²	—	208	2	no	—
199	C₂²	—	208	1	no	—
200	C₂²	#11718653	1	1	yes	characteristic
201	C₂²	#11718734	1	1	yes	characteristic
202	C₂²	#11718747	1	1	yes	characteristic
203	C₂²	#11718815	1	2	yes	—
204	C₂²	#11718852	1	2	yes	—
205	C₄	—	16	2	no	—
206	C₄	—	208	1	no	—
207	C₄	#11718546	1	4	yes	—
208	C₄	#11718552	1	4	yes	—
209	C₄	#11718628	1	2	yes	—
210	C₄	#11718748	1	2	yes	—
211	C₂	—	208	2	no	—
212	C₂	#13114443	1	1	yes	characteristic
213	C₂	#13114492	1	2	yes	—
214	C₂	#13114503	1	1	yes	characteristic
215	C₂	#13116354	1	2	yes	—
216	C₂	#13116365	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13115642

C₂

yes

maximal, characteristic

#13115641

C₂

yes

maximal

#13114963

C₂

yes

maximal, complemented

#13112663

C₂

yes

maximal, characteristic, complemented

C₁₃×#1845

C₂

yes

maximal, characteristic, complemented

C₂×#88128

C₂²

yes

characteristic

C₂×C₁₃×#107

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₁₃×#384

C₂²

yes

—

C₂×#88061

—

C₂×#88049

—

#11718596

C₂²

yes

characteristic

C₂×C₄×#1055

C₂²

yes

—

#11718121

—

C₂²×D₅₂

—

C₂²×D₅₂

—

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

—

C₂×C₄⋊Dic₁₃

—

C₂²×#1055

D₄

yes

—

#88128

—

#88128

—

#88128

Q₈

yes

—

C₁₃×#107

—

C₄×C₈×C₁₃

—

—

—

—

—

—

C₂³×D₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

D₈

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×Q₈

yes

characteristic

C₄²×C₁₃

#41

yes

—

C₄²×C₁₃

#41

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

—

—

—

—

—

D₁₃

yes

characteristic, complemented

#1834

—

#1273

—

C₂³×C₁₃

#127

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₃

#123

yes

characteristic

C₂×C₄×C₁₃

#123

yes

—

C₂×C₄×C₁₃

C₂×D₈

yes

—

C₂×C₄×C₁₃

#135

yes

—

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₁₃

#137

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

D₅₂

—

D₅₂

—

D₅₂

—

D₅₂

—

C₈×C₁₃

—

—

—

—

—

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

#384

—

#384

C₂×D₁₃

yes

—

100

#340

—

101

C₂²×C₁₃

#399

yes

characteristic

102

C₂²×C₁₃

#493

yes

characteristic

103

C₂²×C₁₃

#495

yes

—

104

C₂²×C₁₃

#496

yes

—

105

C₂²×C₁₃

#497

yes

characteristic

106

C₂×D₁₃

—

107

C₂×D₁₃

—

108

C₂×D₁₃

—

109

C₂×D₁₃

—

110

C₂×D₁₃

—

111

C₄×C₁₃

—

112

C₄×C₁₃

#474

yes

—

113

C₄×C₁₃

#478

yes

—

114

C₄×C₁₃

#480

yes

—

115

C₄×C₁₃

#482

yes

—

116

Dic₁₃

—

117

C₂²×D₄

—

118

C₂²×D₄

—

119

C₂²×C₈

—

120

C₂²×C₈

C₂²⋊D₁₃

yes

—

121

#127

—

122

#127

—

123

#127

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄²

C₂²×D₁₃

yes

characteristic

130

#107

—

131

#102

—

132

#102

—

133

C₄×C₈

—

134

C₄×C₈

—

135

C₄×C₈

—

136

C₂×C₁₃

#1709

yes

characteristic

137

C₂×C₁₃

#1719

yes

—

138

C₂×C₁₃

#1721

yes

characteristic

139

C₂×C₁₃

#2092

yes

—

140

C₂×C₁₃

#2093

yes

characteristic

141

D₁₃

—

142

C₂⁴

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

—

150

C₂×D₄

—

151

C₂²×C₄

—

152

C₂²×C₄

—

153

C₂²×C₄

D₄×D₁₃

yes

—

154

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

#6104

yes

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄²

#6127

yes

—

167

C₄²

Q₈×D₁₃

yes

characteristic

168

C₄²

#6131

yes

characteristic

169

C₁₃

#10671

yes

characteristic, complemented, Sylow

170

C₂³

—

171

C₂³

—

104

—

172

C₂³

—

104

—

173

C₂³

—

104

—

174

C₂³

—

104

—

175

C₂³

#88194

yes

characteristic

176

D₄

—

177

D₄

—

178

D₄

—

104

—

179

D₄

—

104

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

104

—

184

C₂×C₄

—

104

—

185

C₂×C₄

#88148

yes

—

186

C₂×C₄

#88150

yes

—

187

C₂×C₄

#88165

yes

—

188

C₂×C₄

#88173

yes

—

189

C₂×C₄

#88183

yes

characteristic

190

C₂×C₄

C₂×#6104

yes

—

191

C₂×C₄

#88196

yes

—

192

C₂×C₄

#88200

yes

characteristic

193

C₈

—

194

C₈

—

195

C₂²

—

104

—

196

C₂²

—

104

—

197

C₂²

—

208

—

198

C₂²

—

208

—

199

C₂²

—

208

—

200

C₂²

#11718653

yes

characteristic

201

C₂²

#11718734

yes

characteristic

202

C₂²

#11718747

yes

characteristic

203

C₂²

#11718815

yes

—

204

C₂²

#11718852

yes

—

205

C₄

—

206

C₄

—

208

—

207

C₄

#11718546

yes

—

208

C₄

#11718552

yes

—

209

C₄

#11718628

yes

—

210

C₄

#11718748

yes

—

211

C₂

—

208

—

212

C₂

#13114443

yes

characteristic

213

C₂

#13114492

yes

—

214

C₂

#13114503

yes

characteristic

215

C₂

#13116354

yes

—

216

C₂

#13116365

yes

characteristic

	subgroup	G/H
center	C₂³	#88194
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1845	C₂
socle	C₂³×C₁₃	#127
prime radical	C₂×#88128	C₂²
hypercenter	#1845	D₁₃
hypocenter	C₁₃	#10671
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

subgroup

G/H

center

C₂³

#88194

commutator, cosocle

C₂×C₄²×C₁₃

C₂³

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1845

C₂

C₂³×C₁₃

C₂²

D₁₃

C₁₃

G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

N	H ≅ G/N	type
3. #13114963	C₂	semidirect
4. #13112663	C₂	semidirect
5. C₁₃×#1845	C₂	semidirect
64. #1845	D₁₃	semidirect
169. C₁₃	#10671	semidirect

H ≅ G/N

type

3. #13114963

C₂

semidirect

4. #13112663

C₂

semidirect

5. C₁₃×#1845

C₂

semidirect

64. #1845

D₁₃

semidirect

169. C₁₃

#10671

semidirect

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

subgroup	index
#13115642	2
#13115641	2
#13114963	2
#13112663	2
C₁₃×#1845	2
#10671	13

subgroup

index

C₁₃×#1845

Subgroups — #15767955

Conjugacy classes of subgroups

Complements

Maximal subgroups