Group #16785017 — subgroups

Subgroups — #16785017

Order 3328.

498 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 92 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	#13130760	C₂	1	1	yes	maximal, characteristic, complemented
2	#13130182	C₂	1	2	yes	maximal, complemented
3	#13130182	C₂	1	2	yes	maximal, complemented
4	#13130032	C₂	1	1	yes	maximal, characteristic, complemented
5	#13125263	C₂	1	1	yes	maximal, characteristic
6	C₂×#88166	—	2	2	no	—
7	#11719421	C₂²	1	2	yes	—
8	#11719052	C₂²	1	2	yes	—
9	C₄×D₁₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₄×D₁₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₁₃×#493	C₄	1	1	yes	characteristic
12	#11718816	—	2	2	no	—
13	#11718737	C₄	1	2	yes	—
14	#11718360	C₄	1	1	yes	characteristic
15	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
16	D₄⋊#1057	—	2	4	no	—
17	D₄⋊#1057	—	2	4	no	—
18	#11718268	—	2	2	no	—
19	#1057⋊C₈	—	2	2	no	—
20	C₂×#6129	—	2	2	no	—
21	C₂×#6118	—	2	2	no	—
22	C₂×D₁₃⋊C₈	—	2	1	no	—
23	C₂×D₁₃⋊C₈	—	2	1	no	—
24	C₂×C₁₃×D₈	—	2	2	no	—
25	C₁₃×#127	C₂×C₄	1	2	yes	—
26	#88192	—	2	2	no	—
27	#88192	—	2	2	no	—
28	#88190	—	2	2	no	—
29	D₄×Dic₁₃	—	2	2	no	—
30	D₄×Dic₁₃	—	2	2	no	—
31	C₂×#6105	—	2	4	no	—
32	#88166	—	4	8	no	—
33	C₂×Dic₅₂	—	2	2	no	—
34	C₂×C₈×D₁₃	D₄	1	2	yes	—
35	C₄²×D₁₃	C₂³	1	1	yes	characteristic
36	#88123	C₂×C₄	1	2	yes	—
37	C₄⋊#1057	—	2	2	no	—
38	C₄⋊#1057	—	2	2	no	—
39	C₄×#1057	—	2	1	no	—
40	C₄×#1057	—	2	1	no	—
41	C₄×#1057	D₄	1	2	yes	—
42	C₄×#1057	D₄	1	2	yes	—
43	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₈×Dic₁₃	D₄	1	2	yes	—
45	C₄×#1055	C₂×C₄	1	1	yes	characteristic
46	#9860	—	13	1	no	maximal, Sylow
47	C₂×D₄×C₁₃	—	2	2	no	—
48	C₂×D₄×C₁₃	—	2	2	no	—
49	C₂×D₄×C₁₃	—	2	2	no	—
50	C₂×C₂²⋊D₁₃	—	4	2	no	—
51	C₂²×Dic₁₃	—	4	2	no	—
52	#6129	—	4	8	no	—
53	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
55	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
56	C₂×Dic₂₆	—	2	2	no	—
57	C₂×#1057	—	2	2	no	—
58	C₂×#1057	—	2	2	no	—
59	C₂×#1057	—	4	2	no	—
60	#6118	—	4	4	no	—
61	D₁₃⋊C₈	—	2	4	no	—
62	D₁₃⋊C₈	—	4	2	no	—
63	C₁₃×D₈	—	4	4	no	—
64	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²⋊Dic₁₃	—	4	2	no	—
67	C₂²⋊Dic₁₃	—	4	2	no	—
68	#6105	—	4	8	no	—
69	C₄⋊Dic₁₃	—	2	2	no	—
70	C₄⋊Dic₁₃	—	2	2	no	—
71	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
72	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
73	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂×#1055	C₂²⋊C₄	1	2	yes	—
75	Dic₅₂	—	4	4	no	—
76	C₈×D₁₃	—	2	4	no	—
77	#3147	—	13	1	no	—
78	#1837	—	13	1	no	—
79	#1492	—	13	2	no	—
80	#1492	—	13	2	no	—
81	C₂×#322	—	13	1	no	—
82	C₂³×C₁₃	—	4	2	no	—
83	C₂²×D₁₃	#124	1	1	yes	characteristic
84	D₄×C₁₃	—	4	4	no	—
85	D₄×C₁₃	—	4	4	no	—
86	D₄×C₁₃	—	4	4	no	—
87	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
90	C₂²⋊D₁₃	—	8	4	no	—
91	C₂×Dic₁₃	—	4	2	no	—
92	C₂×Dic₁₃	—	8	4	no	—
93	C₂×Dic₁₃	#121	1	1	yes	characteristic
94	C₂×Dic₁₃	#121	1	1	yes	characteristic
95	C₂×Dic₁₃	#127	1	1	yes	characteristic
96	C₄×D₁₃	—	2	2	no	—
97	C₄×D₁₃	—	2	2	no	—
98	C₄×D₁₃	AGL(1,ℤ/8)	1	4	yes	—
99	Dic₂₆	—	4	4	no	—
100	#1057	—	2	4	no	—
101	#1057	—	4	2	no	—
102	#1057	—	8	2	no	—
103	C₈×C₁₃	—	2	2	no	—
104	#1055	—	2	2	no	—
105	C₂×#135	—	26	2	no	—
106	#532	—	13	2	no	—
107	#495	—	26	2	no	—
108	#494	—	13	1	no	—
109	#493	#261	1	1	yes	characteristic
110	#492	—	13	2	no	—
111	#423	—	13	2	no	—
112	C₂×C₄×C₈	—	13	1	no	—
113	C₂×C₄×C₈	—	13	1	no	—
114	C₂×C₄×C₈	—	13	1	no	—
115	#341	—	26	2	no	—
116	D₄⋊C₈	—	26	4	no	—
117	D₄⋊C₈	—	26	4	no	—
118	#322	—	26	2	no	—
119	C₂²×C₁₃	—	8	2	no	—
120	C₂²×C₁₃	—	8	2	no	—
121	C₂²×C₁₃	—	8	2	no	—
122	C₂²×C₁₃	#390	1	1	yes	characteristic
123	C₂×D₁₃	—	2	1	no	—
124	C₂×D₁₃	—	2	1	no	—
125	C₂×D₁₃	#490	1	2	yes	—
126	C₄×C₁₃	#418	1	2	yes	—
127	C₄×C₁₃	#443	1	2	yes	—
128	C₄×C₁₃	#444	1	2	yes	—
129	Dic₁₃	—	2	1	no	—
130	Dic₁₃	—	2	1	no	—
131	Dic₁₃	—	8	2	no	—
132	Dic₁₃	#496	1	2	yes	—
133	C₂×#41	—	26	2	no	—
134	#135	—	52	8	no	—
135	C₂×Q₁₆	—	26	2	no	—
136	C₂×SD₁₆	—	26	4	no	—
137	C₂×D₈	—	2	2	no	—
138	C₂×M₁₆	—	26	2	no	—
139	C₂²×C₈	—	13	2	no	—
140	C₂²×C₈	—	26	1	no	—
141	C₂²×C₈	—	26	1	no	—
142	C₄⋊Q₈	—	13	2	no	—
143	#127	C₂×#261	1	2	yes	—
144	#124	—	26	2	no	—
145	#121	—	26	2	no	—
146	#121	—	26	2	no	—
147	C₄×D₄	—	26	2	no	—
148	C₄×D₄	—	26	2	no	—
149	C₂×C₄²	—	13	1	no	—
150	C₄⋊C₈	—	26	2	no	—
151	C₄⋊C₈	—	26	2	no	—
152	C₄×C₈	—	13	2	no	—
153	C₄×C₈	—	13	2	no	—
154	C₄×C₈	—	13	2	no	—
155	C₄×C₈	—	13	1	no	—
156	C₄×C₈	—	26	1	no	—
157	C₄×C₈	—	26	1	no	—
158	C₄×C₈	C₂×#261	1	1	yes	characteristic
159	C₂×C₁₃	—	16	2	no	—
160	C₂×C₁₃	#1961	1	1	yes	characteristic
161	C₂×C₁₃	#1974	1	1	yes	characteristic
162	C₂×C₁₃	#1975	1	1	yes	characteristic
163	D₁₃	—	2	2	no	—
164	#41	—	52	8	no	—
165	C₂×Q₈	—	26	2	no	—
166	C₂×D₄	—	2	2	no	—
167	C₂×D₄	—	2	2	no	—
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	—	52	2	no	—
170	C₂²×C₄	—	13	1	no	—
171	C₂²×C₄	—	13	1	no	—
172	C₂²×C₄	—	13	1	no	—
173	C₂²×C₄	—	52	2	no	—
174	Q₁₆	—	52	4	no	—
175	SD₁₆	—	52	8	no	—
176	D₈	—	4	4	no	—
177	M₁₆	—	52	4	no	—
178	C₂×C₈	—	13	2	no	—
179	C₂×C₈	—	26	4	no	—
180	C₂×C₈	—	26	2	no	—
181	C₂×C₈	—	26	4	no	—
182	C₂×C₈	—	26	2	no	—
183	C₂×C₈	—	52	2	no	—
184	C₂×C₈	—	52	2	no	—
185	C₂×C₈	C₂²⋊#261	1	2	yes	—
186	C₄⋊C₄	—	26	2	no	—
187	C₄⋊C₄	—	26	2	no	—
188	C₂²⋊C₄	—	52	2	no	—
189	C₂²⋊C₄	—	52	2	no	—
190	C₄²	—	13	1	no	—
191	C₄²	—	13	1	no	—
192	C₄²	—	13	1	no	—
193	C₄²	C₂²×#261	1	1	yes	characteristic
194	C₁₃	#9860	1	1	yes	characteristic, complemented, Sylow
195	C₂³	—	4	2	no	—
196	C₂³	—	13	1	no	—
197	Q₈	—	52	4	no	—
198	D₄	—	4	4	no	—
199	D₄	—	4	4	no	—
200	D₄	—	4	4	no	—
201	D₄	—	104	4	no	—
202	C₂×C₄	—	13	1	no	—
203	C₂×C₄	—	13	4	no	—
204	C₂×C₄	—	13	1	no	—
205	C₂×C₄	—	13	1	no	—
206	C₂×C₄	—	26	2	no	—
207	C₂×C₄	—	26	2	no	—
208	C₂×C₄	—	52	2	no	—
209	C₂×C₄	—	104	4	no	—
210	C₂×C₄	D₄×#261	1	1	yes	characteristic
211	C₂×C₄	D₄×#261	1	1	yes	characteristic
212	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
213	C₈	—	2	2	no	—
214	C₈	—	26	4	no	—
215	C₈	—	26	2	no	—
216	C₈	—	52	2	no	—
217	C₈	—	104	2	no	—
218	C₂²	—	8	2	no	—
219	C₂²	—	8	2	no	—
220	C₂²	—	8	2	no	—
221	C₂²	—	13	2	no	—
222	C₂²	—	26	1	no	—
223	C₂²	—	26	1	no	—
224	C₂²	#11719151	1	1	yes	characteristic
225	C₄	—	13	2	no	—
226	C₄	—	26	1	no	—
227	C₄	—	26	1	no	—
228	C₄	—	104	2	no	—
229	C₄	#11719103	1	2	yes	—
230	C₄	#11719104	1	2	yes	—
231	C₄	#11719138	1	2	yes	—
232	C₂	—	16	2	no	—
233	C₂	—	26	2	no	—
234	C₂	#13130968	1	1	yes	characteristic
235	C₂	#13130989	1	1	yes	characteristic
236	C₂	#13130991	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13130760

C₂

yes

maximal, characteristic, complemented

#13130182

C₂

yes

maximal, complemented

#13130182

C₂

yes

maximal, complemented

#13130032

C₂

yes

maximal, characteristic, complemented

#13125263

C₂

yes

maximal, characteristic

C₂×#88166

—

#11719421

C₂²

yes

—

#11719052

C₂²

yes

—

C₄×D₁₃⋊C₈

C₂²

yes

characteristic

C₄×D₁₃⋊C₈

C₂²

yes

characteristic

C₁₃×#493

C₄

yes

characteristic

#11718816

—

#11718737

C₄

yes

—

#11718360

C₄

yes

characteristic

C₄×C₈×D₁₃

C₂²

yes

—

—

—

—

—

—

C₂×D₁₃⋊C₈

—

C₂×D₁₃⋊C₈

—

C₂×C₁₃×D₈

—

C₁₃×#127

C₂×C₄

yes

—

#88192

—

#88192

—

#88190

—

D₄×Dic₁₃

—

D₄×Dic₁₃

—

C₂×#6105

—

#88166

—

C₂×Dic₅₂

—

C₂×C₈×D₁₃

D₄

yes

—

C₄²×D₁₃

C₂³

yes

characteristic

#88123

C₂×C₄

yes

—

—

—

—

—

D₄

yes

—

C₄×#1057

D₄

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₈×Dic₁₃

D₄

yes

—

C₄×#1055

C₂×C₄

yes

characteristic

#9860

—

maximal, Sylow

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×C₂²⋊D₁₃

—

C₂²×Dic₁₃

—

#6129

—

C₂×C₄×D₁₃

C₂×D₄

yes

C₂×C₄×D₁₃

yes

C₂×C₄×D₁₃

yes

C₂×Dic₂₆

—

—

—

—

—

D₁₃⋊C₈

—

D₁₃⋊C₈

—

C₁₃×D₈

—

C₂×C₈×C₁₃

C₂²⋊C₄

yes

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₂²⋊Dic₁₃

—

C₂²⋊Dic₁₃

—

#6105

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

C₂×D₄

yes

characteristic

C₄×Dic₁₃

C₂×D₄

yes

characteristic

C₄×Dic₁₃

C₂×D₄

yes

characteristic

C₂×#1055

C₂²⋊C₄

yes

—

Dic₅₂

—

C₈×D₁₃

—

—

—

—

—

—

C₂³×C₁₃

—

C₂²×D₁₃

#124

yes

characteristic

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₄×D₄

yes

characteristic

C₂×C₄×C₁₃

C₄×D₄

yes

characteristic

C₂²⋊D₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

yes

C₂×Dic₁₃

yes

C₂×Dic₁₃

yes

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

AGL(1,ℤ/8)

yes

—

Dic₂₆

—

100

#1057

—

101

#1057

—

102

#1057

—

103

C₈×C₁₃

—

104

#1055

—

105

C₂×#135

—

106

#532

—

107

#495

—

108

#494

—

109

yes

110

—

111

#423

—

112

C₂×C₄×C₈

—

113

C₂×C₄×C₈

—

114

C₂×C₄×C₈

—

115

#341

—

116

D₄⋊C₈

—

117

D₄⋊C₈

—

118

#322

—

119

C₂²×C₁₃

—

120

C₂²×C₁₃

—

121

C₂²×C₁₃

—

122

C₂²×C₁₃

#390

yes

characteristic

123

C₂×D₁₃

—

124

C₂×D₁₃

—

125

C₂×D₁₃

#490

yes

—

126

C₄×C₁₃

#418

yes

—

127

C₄×C₁₃

#443

yes

—

128

C₄×C₁₃

#444

yes

—

129

Dic₁₃

—

130

Dic₁₃

—

131

Dic₁₃

—

132

Dic₁₃

#496

yes

—

133

C₂×#41

—

134

#135

—

135

C₂×Q₁₆

—

136

C₂×SD₁₆

—

137

C₂×D₈

—

138

C₂×M₁₆

—

139

C₂²×C₈

—

140

C₂²×C₈

—

141

C₂²×C₈

—

142

C₄⋊Q₈

—

143

#127

C₂×#261

yes

—

144

#124

—

145

#121

—

146

#121

—

147

C₄×D₄

—

148

C₄×D₄

—

149

C₂×C₄²

—

150

C₄⋊C₈

—

151

C₄⋊C₈

—

152

C₄×C₈

—

153

C₄×C₈

—

154

C₄×C₈

—

155

C₄×C₈

—

156

C₄×C₈

—

157

C₄×C₈

—

158

C₄×C₈

C₂×#261

yes

characteristic

159

C₂×C₁₃

—

160

C₂×C₁₃

#1961

yes

characteristic

161

C₂×C₁₃

#1974

yes

characteristic

162

C₂×C₁₃

#1975

yes

characteristic

163

D₁₃

—

164

#41

—

165

C₂×Q₈

—

166

C₂×D₄

—

167

C₂×D₄

—

168

C₂×D₄

—

169

C₂×D₄

—

170

C₂²×C₄

—

171

C₂²×C₄

—

172

C₂²×C₄

—

173

C₂²×C₄

—

174

Q₁₆

—

175

SD₁₆

—

176

D₈

—

177

M₁₆

—

178

C₂×C₈

—

179

C₂×C₈

—

180

C₂×C₈

—

181

C₂×C₈

—

182

C₂×C₈

—

183

C₂×C₈

—

184

C₂×C₈

—

185

C₂×C₈

C₂²⋊#261

yes

—

186

C₄⋊C₄

—

187

C₄⋊C₄

—

188

C₂²⋊C₄

—

189

C₂²⋊C₄

—

190

C₄²

—

191

C₄²

—

192

C₄²

—

193

C₄²

C₂²×#261

yes

characteristic

194

C₁₃

#9860

yes

characteristic, complemented, Sylow

195

C₂³

—

196

C₂³

—

197

Q₈

—

198

D₄

—

199

D₄

—

200

D₄

—

201

D₄

—

104

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

205

C₂×C₄

—

206

C₂×C₄

—

207

C₂×C₄

—

208

C₂×C₄

—

209

C₂×C₄

—

104

—

210

C₂×C₄

D₄×#261

yes

characteristic

211

C₂×C₄

D₄×#261

yes

characteristic

212

C₂×C₄

C₂×C₂²⋊#261

yes

characteristic

213

C₈

—

214

C₈

—

215

C₈

—

216

C₈

—

217

C₈

—

104

—

218

C₂²

—

219

C₂²

—

220

C₂²

—

221

C₂²

—

222

C₂²

—

223

C₂²

—

224

C₂²

#11719151

yes

characteristic

225

C₄

—

226

C₄

—

227

C₄

—

228

C₄

—

104

—

229

C₄

#11719103

yes

—

230

C₄

#11719104

yes

—

231

C₄

#11719138

yes

—

232

C₂

—

233

C₂

—

234

C₂

#13130968

yes

characteristic

235

C₂

#13130989

yes

characteristic

236

C₂

#13130991

yes

characteristic

	subgroup	G/H
center	C₂²	#11719151
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₄²	C₂²×#261
Fitting	C₁₃×#493	C₄
socle	C₂²×C₁₃	#390
prime radical	#13125263	C₂
cosocle	C₄²×D₁₃	C₂³
hypercenter	#493	#261
hypocenter	C₁₃	#9860
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#11719151

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₄²

C₂²×#261

Fitting

C₁₃×#493

C₄

C₂²×C₁₃

C₂

C₄²×D₁₃

C₂³

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #13130760	C₂	semidirect
2. #13130182	C₂	semidirect
3. #13130182	C₂	semidirect
4. #13130032	C₂	semidirect
194. C₁₃	#9860	semidirect

H ≅ G/N

type

1. #13130760

C₂

semidirect

2. #13130182

C₂

semidirect

3. #13130182

C₂

semidirect

4. #13130032

C₂

semidirect

194. C₁₃

#9860

semidirect

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

subgroup	index
#13130760	2
#13130182	2
#13130032	2
#13125263	2
#9860	13

subgroup

index

Subgroups — #16785017

Conjugacy classes of subgroups

Complements

Maximal subgroups