Group #15757424 — subgroups

Subgroups — #15757424

Order 3328.

412 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 111 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	#13123903	C₂	1	1	yes	maximal, characteristic
2	#13115665	C₂	1	1	yes	maximal, characteristic
3	#13113625	C₂	1	2	yes	maximal, complemented
4	#13112668	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1543	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#532	C₂²	1	1	yes	characteristic
7	C₁₃×#431	C₂²	1	2	yes	—
8	C₂×#88077	—	2	2	no	—
9	#11718895	—	2	2	no	—
10	#11718673	—	2	2	no	—
11	D₄×#1055	—	2	2	no	—
12	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
13	#11718586	C₂²	1	2	yes	—
14	#11718585	C₂²	1	1	yes	characteristic
15	C₁₃×#333	C₄	1	2	yes	complemented
16	C₁₃×#331	C₄	1	1	yes	characteristic, complemented
17	C₁₃×#331	C₄	1	1	yes	characteristic, complemented
18	#11718050	—	2	2	no	—
19	#11718049	—	2	2	no	—
20	C₂×C₁₃×#41	D₄	1	2	yes	—
21	C₁₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
22	C₁₃×#127	C₂×C₄	1	1	yes	characteristic
23	C₁₃×#124	Q₈	1	2	yes	—
24	C₁₃×#121	—	2	2	no	—
25	C₄×D₄×C₁₃	—	2	2	no	—
26	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
27	C₂×C₄×Dic₁₃	—	2	1	no	—
28	C₂×C₄×Dic₁₃	—	2	2	no	—
29	C₂×#6099	—	2	2	no	—
30	C₂×#6099	—	2	1	no	—
31	C₂²×#1055	—	2	2	no	—
32	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
33	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
34	C₁₃×C₂²⋊C₈	—	2	2	no	—
35	C₄×C₈×C₁₃	—	2	2	no	—
36	#88077	—	4	4	no	—
37	C₂²⋊#1055	—	2	4	no	—
38	C₂²⋊#1055	—	2	2	no	—
39	C₄⋊#1055	—	2	2	no	—
40	C₄⋊#1055	D₄	1	2	yes	—
41	C₄⋊#1055	D₄	1	2	yes	—
42	C₄×#1055	—	2	2	no	—
43	#8934	—	13	1	no	maximal, Sylow
44	C₁₃×#41	—	4	2	no	—
45	C₁₃×#41	—	4	2	no	—
46	C₂×Q₈×C₁₃	C₄⋊C₄	1	2	yes	—
47	C₂×D₄×C₁₃	—	2	2	no	—
48	C₂×D₄×C₁₃	—	2	2	no	—
49	C₂×D₄×C₁₃	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₁₃	—	2	2	no	—
51	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	#41	1	2	yes	—
53	C₂²×Dic₁₃	—	4	1	no	—
54	C₂×C₈×C₁₃	—	2	2	no	—
55	C₂×C₈×C₁₃	—	2	2	no	—
56	C₁₃×C₄⋊C₄	—	2	2	no	—
57	C₁₃×C₂²⋊C₄	—	2	4	no	—
58	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₁₃	C₂×D₄	1	2	yes	—
60	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
61	C₄×Dic₁₃	—	2	2	no	—
62	C₄×Dic₁₃	—	2	4	no	—
63	C₄×Dic₁₃	—	4	2	no	—
64	C₄×Dic₁₃	—	4	1	no	—
65	#6099	—	4	2	no	—
66	#6099	—	8	2	no	—
67	C₂×#1055	—	2	2	no	—
68	C₂×#1055	—	2	2	no	—
69	C₂×#1055	—	4	1	no	—
70	C₂×#1055	—	8	2	no	—
71	#2979	—	13	1	no	—
72	#1962	—	13	1	no	—
73	#1574	—	13	2	no	—
74	#1543	D₁₃	1	1	yes	characteristic
75	#1276	—	13	2	no	—
76	C₂³×C₁₃	—	2	2	no	—
77	C₂³×C₁₃	#123	1	1	yes	characteristic
78	Q₈×C₁₃	—	4	2	no	—
79	D₄×C₁₃	—	4	2	no	—
80	D₄×C₁₃	—	4	2	no	—
81	D₄×C₁₃	—	4	2	no	—
82	D₄×C₁₃	—	8	2	no	—
83	C₂×C₄×C₁₃	—	2	2	no	—
84	C₂×C₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	—	2	1	no	—
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	—	2	2	no	—
88	C₂×C₄×C₁₃	—	4	2	no	—
89	C₂×C₄×C₁₃	—	4	2	no	—
90	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
92	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
94	C₂×Dic₁₃	—	4	2	no	—
95	C₂×Dic₁₃	—	4	4	no	—
96	C₈×C₁₃	—	4	2	no	—
97	C₈×C₁₃	—	4	2	no	—
98	#1055	—	4	2	no	—
99	#1055	—	4	2	no	—
100	#1055	—	8	1	no	—
101	#532	C₂×D₁₃	1	1	yes	characteristic
102	#436	—	26	2	no	—
103	C₈×D₄	—	26	2	no	—
104	#431	—	13	2	no	—
105	#431	C₂×D₁₃	1	2	yes	—
106	#423	—	13	1	no	—
107	C₂×#104	—	26	2	no	—
108	#408	—	26	2	no	—
109	C₄³	—	13	1	no	—
110	#335	—	26	2	no	—
111	#334	—	26	2	no	—
112	#333	Dic₁₃	1	2	yes	complemented
113	#331	Dic₁₃	1	1	yes	characteristic, complemented
114	#331	Dic₁₃	1	1	yes	characteristic, complemented
115	C₂²×C₁₃	—	2	1	no	—
116	C₂²×C₁₃	—	2	2	no	—
117	C₂²×C₁₃	—	4	2	no	—
118	C₂²×C₁₃	—	4	4	no	—
119	C₂²×C₁₃	#385	1	1	yes	characteristic
120	C₄×C₁₃	—	2	1	no	—
121	C₄×C₁₃	—	2	2	no	—
122	C₄×C₁₃	—	4	2	no	—
123	C₄×C₁₃	#443	1	2	yes	—
124	C₄×C₁₃	#444	1	2	yes	—
125	C₄×C₁₃	#453	1	1	yes	characteristic
126	C₄×C₁₃	#455	1	1	yes	characteristic
127	Dic₁₃	—	4	4	no	—
128	C₂×#41	D₅₂	1	2	yes	—
129	C₂×M₁₆	—	26	2	no	—
130	C₂×M₁₆	—	26	1	no	—
131	C₂²×C₈	—	26	2	no	—
132	C₄⋊Q₈	C₂×Dic₁₃	1	1	yes	characteristic
133	#127	C₂×Dic₁₃	1	1	yes	characteristic
134	#124	Dic₂₆	1	2	yes	—
135	#121	—	2	2	no	—
136	C₄×D₄	—	2	2	no	—
137	C₂×C₄²	—	26	1	no	—
138	C₂×C₄²	—	26	2	no	—
139	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
140	C₄⋊C₈	—	13	2	no	—
141	C₄⋊C₈	—	13	2	no	—
142	C₄⋊C₈	—	26	2	no	—
143	C₄⋊C₈	C₂×Dic₁₃	1	2	yes	—
144	C₄⋊C₈	C₂×Dic₁₃	1	2	yes	—
145	#104	—	52	4	no	—
146	C₂²⋊C₈	—	2	2	no	—
147	C₂²⋊C₈	—	26	4	no	—
148	C₂²⋊C₈	—	26	2	no	—
149	C₄×C₈	—	2	2	no	—
150	C₄×C₈	—	26	2	no	—
151	C₂×C₁₃	—	4	1	no	—
152	C₂×C₁₃	—	8	2	no	—
153	C₂×C₁₃	#1856	1	1	yes	characteristic
154	C₂×C₁₃	#1866	1	2	yes	—
155	#41	—	4	2	no	—
156	#41	—	4	2	no	—
157	C₂×Q₈	C₄⋊Dic₁₃	1	2	yes	—
158	C₂×D₄	—	2	2	no	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	C₄⋊Dic₁₃	1	2	yes	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	—	52	1	no	—
163	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
164	C₂²×C₄	#6129	1	2	yes	—
165	M₁₆	—	52	2	no	—
166	M₁₆	—	104	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	26	2	no	—
170	C₂×C₈	—	26	2	no	—
171	C₂×C₈	—	52	1	no	—
172	C₂×C₈	—	104	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₂²⋊C₄	—	2	4	no	—
175	C₄²	—	26	2	no	—
176	C₄²	—	26	4	no	—
177	C₄²	—	52	1	no	—
178	C₄²	—	52	2	no	—
179	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
180	C₄²	D₄×D₁₃	1	2	yes	—
181	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
182	C₁₃	#8934	1	1	yes	characteristic, complemented, Sylow
183	C₂³	—	2	2	no	—
184	C₂³	#88140	1	1	yes	characteristic
185	Q₈	—	4	2	no	—
186	D₄	—	4	2	no	—
187	D₄	—	4	2	no	—
188	D₄	—	4	2	no	—
189	D₄	—	8	2	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	52	2	no	—
198	C₂×C₄	—	52	4	no	—
199	C₂×C₄	#88132	1	2	yes	—
200	C₂×C₄	#88136	1	1	yes	characteristic
201	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
202	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
203	C₈	—	4	2	no	—
204	C₈	—	4	2	no	—
205	C₈	—	52	2	no	—
206	C₈	—	52	2	no	—
207	C₈	—	104	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	4	2	no	—
211	C₂²	—	4	4	no	—
212	C₂²	#11718611	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	2	2	no	—
215	C₄	—	4	2	no	—
216	C₄	—	52	4	no	—
217	C₄	#11718407	1	1	yes	characteristic
218	C₄	#11718408	1	1	yes	characteristic
219	C₄	#11718857	1	2	yes	—
220	C₄	#11718858	1	2	yes	—
221	C₂	—	4	1	no	—
222	C₂	—	8	2	no	—
223	C₂	#13115048	1	1	yes	characteristic
224	C₂	#13115113	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13123903

C₂

yes

maximal, characteristic

#13115665

C₂

yes

maximal, characteristic

#13113625

C₂

yes

maximal, complemented

#13112668

C₂

yes

maximal, complemented

C₁₃×#1543

C₂

yes

maximal, characteristic

C₁₃×#532

C₂²

yes

characteristic

C₁₃×#431

C₂²

yes

—

—

—

—

—

C₄²×Dic₁₃

C₂²

yes

characteristic

#11718586

C₂²

yes

—

#11718585

C₂²

yes

characteristic

C₁₃×#333

C₄

yes

complemented

C₁₃×#331

C₄

yes

characteristic, complemented

C₁₃×#331

C₄

yes

characteristic, complemented

#11718050

—

#11718049

—

C₂×C₁₃×#41

D₄

yes

—

C₁₃×C₄⋊Q₈

C₂×C₄

yes

characteristic

C₁₃×#127

C₂×C₄

yes

characteristic

C₁₃×#124

Q₈

yes

—

C₁₃×#121

—

C₄×D₄×C₁₃

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄×Dic₁₃

—

C₂×C₄×Dic₁₃

—

C₂×#6099

—

C₂×#6099

—

C₂²×#1055

—

C₁₃×C₄⋊C₈

C₂×C₄

yes

—

C₁₃×C₄⋊C₈

C₂×C₄

yes

—

C₁₃×C₂²⋊C₈

—

C₄×C₈×C₁₃

—

#88077

—

C₂²⋊#1055

—

C₂²⋊#1055

—

C₄⋊#1055

—

C₄⋊#1055

D₄

yes

—

C₄⋊#1055

D₄

yes

—

C₄×#1055

—

#8934

—

maximal, Sylow

C₁₃×#41

—

C₁₃×#41

—

C₂×Q₈×C₁₃

C₄⋊C₄

yes

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

C₄⋊C₄

yes

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

#41

yes

—

C₂²×Dic₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

C₂×Q₈

yes

characteristic

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

—

—

—

—

—

—

—

—

—

D₁₃

yes

characteristic

#1276

—

C₂³×C₁₃

—

C₂³×C₁₃

#123

yes

characteristic

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₄×D₄

yes

—

C₂×C₄×C₁₃

#120

yes

characteristic

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

#1055

—

#1055

—

100

#1055

—

101

#532

C₂×D₁₃

yes

characteristic

102

#436

—

103

C₈×D₄

—

104

#431

—

105

#431

C₂×D₁₃

yes

—

106

#423

—

107

C₂×#104

—

108

#408

—

109

C₄³

—

110

#335

—

111

#334

—

112

#333

Dic₁₃

yes

complemented

113

#331

Dic₁₃

yes

characteristic, complemented

114

#331

Dic₁₃

yes

characteristic, complemented

115

C₂²×C₁₃

—

116

C₂²×C₁₃

—

117

C₂²×C₁₃

—

118

C₂²×C₁₃

—

119

C₂²×C₁₃

#385

yes

characteristic

120

C₄×C₁₃

—

121

C₄×C₁₃

—

122

C₄×C₁₃

—

123

C₄×C₁₃

#443

yes

—

124

C₄×C₁₃

#444

yes

—

125

C₄×C₁₃

#453

yes

characteristic

126

C₄×C₁₃

#455

yes

characteristic

127

Dic₁₃

—

128

C₂×#41

D₅₂

yes

—

129

C₂×M₁₆

—

130

C₂×M₁₆

—

131

C₂²×C₈

—

132

C₄⋊Q₈

C₂×Dic₁₃

yes

characteristic

133

#127

C₂×Dic₁₃

yes

characteristic

134

#124

Dic₂₆

yes

—

135

#121

—

136

C₄×D₄

—

137

C₂×C₄²

—

138

C₂×C₄²

—

139

C₂×C₄²

C₂²×D₁₃

yes

characteristic

140

C₄⋊C₈

—

141

C₄⋊C₈

—

142

C₄⋊C₈

—

143

C₄⋊C₈

C₂×Dic₁₃

yes

—

144

C₄⋊C₈

C₂×Dic₁₃

yes

—

145

#104

—

146

C₂²⋊C₈

—

147

C₂²⋊C₈

—

148

C₂²⋊C₈

—

149

C₄×C₈

—

150

C₄×C₈

—

151

C₂×C₁₃

—

152

C₂×C₁₃

—

153

C₂×C₁₃

#1856

yes

characteristic

154

C₂×C₁₃

#1866

yes

—

155

#41

—

156

#41

—

157

C₂×Q₈

C₄⋊Dic₁₃

yes

—

158

C₂×D₄

—

159

C₂×D₄

—

160

C₂×D₄

C₄⋊Dic₁₃

yes

—

161

C₂²×C₄

—

162

C₂²×C₄

—

163

C₂²×C₄

C₂×D₅₂

yes

characteristic

164

C₂²×C₄

#6129

yes

—

165

M₁₆

—

166

M₁₆

—

104

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

—

172

C₂×C₈

—

104

—

173

C₄⋊C₄

—

174

C₂²⋊C₄

—

175

C₄²

—

176

C₄²

—

177

C₄²

—

178

C₄²

—

179

C₄²

C₂×Dic₂₆

yes

characteristic

180

C₄²

D₄×D₁₃

yes

—

181

C₄²

C₂²×Dic₁₃

yes

characteristic

182

C₁₃

#8934

yes

characteristic, complemented, Sylow

183

C₂³

—

184

C₂³

#88140

yes

characteristic

185

Q₈

—

186

D₄

—

187

D₄

—

188

D₄

—

189

D₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

#88132

yes

—

200

C₂×C₄

#88136

yes

characteristic

201

C₂×C₄

C₂×C₄⋊Dic₁₃

yes

characteristic

202

C₂×C₄

D₄×Dic₁₃

yes

—

203

C₈

—

204

C₈

—

205

C₈

—

206

C₈

—

207

C₈

—

104

—

208

C₂²

—

209

C₂²

—

210

C₂²

—

211

C₂²

—

212

C₂²

#11718611

yes

characteristic

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

—

217

C₄

#11718407

yes

characteristic

218

C₄

#11718408

yes

characteristic

219

C₄

#11718857

yes

—

220

C₄

#11718858

yes

—

221

C₂

—

222

C₂

—

223

C₂

#13115048

yes

characteristic

224

C₂

#13115113

yes

—

	subgroup	G/H
center	C₂×C₄	#88136
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1543	C₂
socle	C₂²×C₁₃	#385
prime radical	C₁₃×#532	C₂²
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1543	D₁₃
hypocenter	C₁₃	#8934
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1543

subgroup

G/H

center

C₂×C₄

#88136

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1543

C₂

socle

C₂²×C₁₃

#385

prime radical

C₁₃×#532

C₂²

cosocle

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1543

N	H ≅ G/N	type
3. #13113625	C₂	semidirect
4. #13112668	C₂	semidirect
15. C₁₃×#333	C₄	semidirect
16. C₁₃×#331	C₄	semidirect
17. C₁₃×#331	C₄	semidirect
112. #333	Dic₁₃	semidirect
113. #331	Dic₁₃	semidirect
114. #331	Dic₁₃	semidirect
182. C₁₃	#8934	semidirect

H ≅ G/N

type

3. #13113625

C₂

semidirect

4. #13112668

C₂

semidirect

15. C₁₃×#333

C₄

semidirect

16. C₁₃×#331

C₄

semidirect

17. C₁₃×#331

C₄

semidirect

112. #333

Dic₁₃

semidirect

113. #331

Dic₁₃

semidirect

114. #331

Dic₁₃

semidirect

182. C₁₃

#8934

semidirect

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

subgroup	index
#13123903	2
#13115665	2
#13113625	2
#13112668	2
C₁₃×#1543	2
#8934	13

subgroup

index

C₁₃×#1543

Subgroups — #15757424

Conjugacy classes of subgroups

Complements

Maximal subgroups