Group #16784979 — subgroups

Subgroups — #16784979

Order 3328.

498 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 96 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	#13130970	C₂	1	2	yes	maximal, complemented
2	#13130549	C₂	1	1	yes	maximal, characteristic, complemented
3	#13130395	C₂	1	1	yes	maximal, characteristic, complemented
4	#13130182	C₂	1	2	yes	maximal, complemented
5	#13125413	C₂	1	1	yes	maximal, characteristic
6	#11719421	C₂²	1	2	yes	—
7	C₂×#88116	—	2	4	no	—
8	C₄²×#261	C₂²	1	1	yes	characteristic
9	#11719052	C₂²	1	2	yes	—
10	C₄×D₁₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₁₃×#493	C₄	1	1	yes	characteristic, complemented
12	#11718818	—	2	2	no	—
13	D₈×Dic₁₃	—	2	2	no	—
14	#11718731	C₄	1	2	yes	complemented
15	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
16	#11718343	C₄	1	1	yes	characteristic, complemented
17	D₄⋊#1057	—	2	4	no	—
18	(C₂×#261)⋊C₈	—	2	2	no	—
19	#11718256	—	2	2	no	—
20	C₂×#6129	—	2	2	no	—
21	C₂×C₄×#261	—	2	1	no	—
22	C₂×C₄×#261	—	2	1	no	—
23	C₂×C₄×#261	D₄	1	2	yes	—
24	C₂×#6118	—	2	2	no	—
25	C₂×D₁₃⋊C₈	—	2	1	no	—
26	C₂×C₁₃×D₈	—	2	2	no	—
27	C₁₃×#127	C₂×C₄	1	2	yes	—
28	#88192	—	2	2	no	—
29	#88192	—	2	2	no	—
30	#88190	—	2	2	no	—
31	D₄×Dic₁₃	—	2	2	no	—
32	D₄×Dic₁₃	—	2	2	no	—
33	C₂×C₈×D₁₃	D₄	1	2	yes	—
34	C₄²×D₁₃	C₂³	1	1	yes	characteristic
35	#88123	C₂×C₄	1	2	yes	—
36	#88116	—	4	8	no	—
37	C₄⋊#1057	—	2	2	no	—
38	C₄⋊#1057	—	2	2	no	—
39	C₄×#1057	—	2	1	no	—
40	C₄×#1057	D₄	1	2	yes	—
41	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	#88072	—	2	4	no	—
43	#88058	—	2	2	no	—
44	C₈×Dic₁₃	D₄	1	2	yes	—
45	C₄×#1055	C₂×C₄	1	1	yes	characteristic
46	#9847	—	13	1	no	maximal, Sylow
47	C₂²×#261	—	2	2	no	—
48	C₂×D₄×C₁₃	—	2	2	no	—
49	C₂×D₄×C₁₃	—	2	2	no	—
50	C₂×D₄×C₁₃	—	2	2	no	—
51	C₂×C₂²⋊D₁₃	—	4	2	no	—
52	C₂²×Dic₁₃	—	4	2	no	—
53	#6129	—	4	4	no	—
54	#6129	—	4	4	no	—
55	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
58	C₂×Dic₂₆	—	2	2	no	—
59	C₂×#1057	—	2	2	no	—
60	C₂×#1057	—	4	2	no	—
61	C₄×#261	—	2	2	no	—
62	C₄×#261	—	4	2	no	—
63	C₄×#261	—	4	2	no	—
64	C₄×#261	D₈	1	4	yes	complemented
65	#6118	—	4	4	no	—
66	D₁₃⋊C₈	—	4	2	no	—
67	C₁₃×D₈	—	4	4	no	—
68	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
69	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²⋊Dic₁₃	—	4	2	no	—
71	C₂²⋊Dic₁₃	—	4	2	no	—
72	C₄⋊Dic₁₃	—	2	2	no	—
73	C₄⋊Dic₁₃	—	2	2	no	—
74	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
75	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
76	C₄×Dic₁₃	#41	1	1	yes	characteristic
77	C₂×#1055	C₂²⋊C₄	1	2	yes	—
78	C₈×D₁₃	—	4	2	no	—
79	#3169	—	13	1	no	—
80	#1962	—	13	2	no	—
81	#1744	—	13	1	no	—
82	#1567	—	13	1	no	—
83	#1492	—	13	2	no	—
84	C₂³×C₁₃	—	4	2	no	—
85	C₂²×D₁₃	#121	1	1	yes	characteristic
86	C₂×#261	—	2	4	no	—
87	C₂×#261	—	4	2	no	—
88	C₂×#261	—	4	2	no	—
89	D₄×C₁₃	—	4	4	no	—
90	D₄×C₁₃	—	4	4	no	—
91	D₄×C₁₃	—	4	4	no	—
92	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
95	C₂²⋊D₁₃	—	8	4	no	—
96	C₂×Dic₁₃	—	4	2	no	—
97	C₂×Dic₁₃	—	8	2	no	—
98	C₂×Dic₁₃	—	8	2	no	—
99	C₂×Dic₁₃	#121	1	1	yes	characteristic
100	C₂×Dic₁₃	#124	1	1	yes	characteristic
101	C₂×Dic₁₃	#127	1	1	yes	characteristic
102	C₄×D₁₃	—	2	2	no	—
103	C₄×D₁₃	—	2	2	no	—
104	C₄×D₁₃	C₂×D₈	1	2	yes	—
105	C₄×D₁₃	#135	1	2	yes	—
106	Dic₂₆	—	4	4	no	—
107	#1057	—	4	2	no	—
108	#1057	—	8	2	no	—
109	C₈×C₁₃	—	2	2	no	—
110	#1055	—	2	2	no	—
111	#532	—	13	2	no	—
112	#500	—	13	1	no	—
113	#493	#261	1	1	yes	characteristic, complemented
114	#486	—	13	2	no	—
115	#466	—	26	2	no	—
116	C₄×D₈	—	26	2	no	—
117	#423	—	13	2	no	—
118	C₂×#104	—	26	4	no	—
119	C₂×C₄×C₈	—	13	1	no	—
120	C₂×C₄×C₈	—	13	1	no	—
121	C₄³	—	13	1	no	—
122	#336	—	26	2	no	—
123	#335	—	26	2	no	—
124	D₄⋊C₈	—	26	4	no	—
125	C₂²×C₁₃	—	8	2	no	—
126	C₂²×C₁₃	—	8	2	no	—
127	C₂²×C₁₃	—	8	2	no	—
128	C₂²×C₁₃	#390	1	1	yes	characteristic
129	C₂×D₁₃	—	2	1	no	—
130	C₂×D₁₃	—	2	1	no	—
131	C₂×D₁₃	#466	1	2	yes	—
132	#261	—	4	4	no	—
133	C₄×C₁₃	#421	1	2	yes	—
134	C₄×C₁₃	C₄×D₈	1	2	yes	—
135	C₄×C₁₃	#444	1	2	yes	—
136	Dic₁₃	—	2	1	no	—
137	Dic₁₃	—	2	1	no	—
138	Dic₁₃	—	8	2	no	—
139	Dic₁₃	#493	1	1	yes	characteristic
140	Dic₁₃	#495	1	1	yes	characteristic
141	C₂×#41	—	26	2	no	—
142	C₂×D₈	—	2	2	no	—
143	C₂×M₁₆	—	26	2	no	—
144	C₂²×C₈	—	13	2	no	—
145	C₂²×C₈	—	26	1	no	—
146	C₄⋊Q₈	—	13	2	no	—
147	#127	C₂×#261	1	2	yes	—
148	#124	—	26	2	no	—
149	#121	—	26	2	no	—
150	#121	—	26	2	no	—
151	C₄×D₄	—	26	2	no	—
152	C₄×D₄	—	26	2	no	—
153	C₂×C₄²	—	13	1	no	—
154	C₂×C₄²	—	13	2	no	—
155	C₂×C₄²	—	26	1	no	—
156	C₂×C₄²	—	26	1	no	—
157	#107	—	26	2	no	—
158	C₄⋊C₈	—	26	2	no	—
159	C₄⋊C₈	—	26	2	no	—
160	#104	—	52	8	no	—
161	#102	—	26	4	no	—
162	C₄×C₈	—	13	2	no	—
163	C₄×C₈	—	13	2	no	—
164	C₄×C₈	—	13	1	no	—
165	C₄×C₈	—	26	1	no	—
166	C₄×C₈	C₂×#261	1	1	yes	characteristic
167	C₂×C₁₃	—	16	2	no	—
168	C₂×C₁₃	#1963	1	1	yes	characteristic
169	C₂×C₁₃	#1970	1	1	yes	characteristic
170	C₂×C₁₃	#1975	1	1	yes	characteristic
171	D₁₃	—	2	2	no	—
172	#41	—	52	4	no	—
173	#41	—	52	4	no	—
174	C₂×Q₈	—	26	2	no	—
175	C₂×D₄	—	2	2	no	—
176	C₂×D₄	—	2	2	no	—
177	C₂×D₄	—	2	2	no	—
178	C₂×D₄	—	52	2	no	—
179	C₂²×C₄	—	13	1	no	—
180	C₂²×C₄	—	13	1	no	—
181	C₂²×C₄	—	13	1	no	—
182	C₂²×C₄	—	26	2	no	—
183	C₂²×C₄	—	52	2	no	—
184	D₈	—	4	4	no	—
185	M₁₆	—	52	4	no	—
186	C₂×C₈	—	13	2	no	—
187	C₂×C₈	—	26	2	no	—
188	C₂×C₈	—	52	2	no	—
189	C₂×C₈	—	52	2	no	—
190	C₂×C₈	—	52	2	no	—
191	C₂×C₈	C₂²⋊#261	1	2	yes	—
192	C₄⋊C₄	—	26	2	no	—
193	C₄⋊C₄	—	26	2	no	—
194	C₂²⋊C₄	—	52	2	no	—
195	C₂²⋊C₄	—	52	2	no	—
196	C₄²	—	13	1	no	—
197	C₄²	—	13	1	no	—
198	C₄²	—	13	1	no	—
199	C₄²	—	13	4	no	—
200	C₄²	—	26	2	no	—
201	C₄²	—	52	2	no	—
202	C₄²	—	52	2	no	—
203	C₄²	C₂²×#261	1	1	yes	characteristic
204	C₁₃	#9847	1	1	yes	characteristic, complemented, Sylow
205	C₂³	—	4	2	no	—
206	C₂³	—	13	1	no	—
207	Q₈	—	52	4	no	—
208	D₄	—	4	4	no	—
209	D₄	—	4	4	no	—
210	D₄	—	4	4	no	—
211	D₄	—	104	4	no	—
212	C₂×C₄	—	13	1	no	—
213	C₂×C₄	—	13	2	no	—
214	C₂×C₄	—	13	2	no	—
215	C₂×C₄	—	13	1	no	—
216	C₂×C₄	—	13	1	no	—
217	C₂×C₄	—	26	2	no	—
218	C₂×C₄	—	26	2	no	—
219	C₂×C₄	—	26	4	no	—
220	C₂×C₄	—	52	2	no	—
221	C₂×C₄	—	52	2	no	—
222	C₂×C₄	—	52	2	no	—
223	C₂×C₄	—	104	2	no	—
224	C₂×C₄	—	104	2	no	—
225	C₂×C₄	D₄×#261	1	1	yes	characteristic
226	C₂×C₄	D₄×#261	1	1	yes	characteristic
227	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
228	C₈	—	2	2	no	—
229	C₈	—	26	2	no	—
230	C₈	—	52	2	no	—
231	C₈	—	104	2	no	—
232	C₂²	—	8	2	no	—
233	C₂²	—	8	2	no	—
234	C₂²	—	8	2	no	—
235	C₂²	—	13	2	no	—
236	C₂²	—	26	1	no	—
237	C₂²	—	26	1	no	—
238	C₂²	#11719151	1	1	yes	characteristic
239	C₄	—	13	1	no	—
240	C₄	—	13	1	no	—
241	C₄	—	26	1	no	—
242	C₄	—	26	1	no	—
243	C₄	—	52	4	no	—
244	C₄	—	104	2	no	—
245	C₄	D₈×#261	1	2	yes	—
246	C₄	#11719104	1	2	yes	—
247	C₄	#11719140	1	2	yes	—
248	C₂	—	16	2	no	—
249	C₂	—	26	2	no	—
250	C₂	#13130974	1	1	yes	characteristic
251	C₂	#13130983	1	1	yes	characteristic
252	C₂	#13130991	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13130970

C₂

yes

maximal, complemented

#13130549

C₂

yes

maximal, characteristic, complemented

#13130395

C₂

yes

maximal, characteristic, complemented

#13130182

C₂

yes

maximal, complemented

#13125413

C₂

yes

maximal, characteristic

#11719421

C₂²

yes

—

C₂×#88116

—

C₄²×#261

C₂²

yes

characteristic

#11719052

C₂²

yes

—

C₄×D₁₃⋊C₈

C₂²

yes

characteristic

C₁₃×#493

C₄

yes

characteristic, complemented

#11718818

—

D₈×Dic₁₃

—

#11718731

C₄

yes

complemented

C₄×C₈×D₁₃

C₂²

yes

characteristic

#11718343

C₄

yes

characteristic, complemented

D₄⋊#1057

—

(C₂×#261)⋊C₈

—

#11718256

—

C₂×#6129

—

C₂×C₄×#261

—

C₂×C₄×#261

—

C₂×C₄×#261

D₄

yes

—

C₂×#6118

—

C₂×D₁₃⋊C₈

—

C₂×C₁₃×D₈

—

C₁₃×#127

C₂×C₄

yes

—

#88192

—

#88192

—

#88190

—

D₄×Dic₁₃

—

D₄×Dic₁₃

—

C₂×C₈×D₁₃

D₄

yes

—

C₄²×D₁₃

C₂³

yes

characteristic

#88123

C₂×C₄

yes

—

—

—

—

—

D₄

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

#88072

—

#88058

—

C₈×Dic₁₃

D₄

yes

—

C₄×#1055

C₂×C₄

yes

characteristic

#9847

—

maximal, Sylow

C₂²×#261

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×C₂²⋊D₁₃

—

C₂²×Dic₁₃

—

#6129

—

#6129

—

C₂×C₄×D₁₃

C₂×D₄

yes

characteristic

C₂×C₄×D₁₃

C₂×D₄

yes

characteristic

C₂×C₄×D₁₃

#41

yes

characteristic

C₂×Dic₂₆

—

—

—

—

—

—

D₈

yes

complemented

#6118

—

D₁₃⋊C₈

—

C₁₃×D₈

—

C₂×C₈×C₁₃

C₂²⋊C₄

yes

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₂²⋊Dic₁₃

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

C₂×D₄

yes

characteristic

C₄×Dic₁₃

C₂×D₄

yes

C₄×Dic₁₃

yes

C₂²⋊C₄

yes

—

C₈×D₁₃

—

—

—

—

—

—

C₂³×C₁₃

—

C₂²×D₁₃

yes

—

—

—

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

—

C₂×Dic₁₃

#121

yes

characteristic

100

C₂×Dic₁₃

#124

yes

characteristic

101

C₂×Dic₁₃

#127

yes

characteristic

102

C₄×D₁₃

—

103

C₄×D₁₃

—

104

C₄×D₁₃

C₂×D₈

yes

—

105

C₄×D₁₃

#135

yes

—

106

Dic₂₆

—

107

#1057

—

108

#1057

—

109

C₈×C₁₃

—

110

#1055

—

111

#532

—

112

#500

—

113

#493

#261

yes

characteristic, complemented

114

#486

—

115

#466

—

116

C₄×D₈

—

117

#423

—

118

C₂×#104

—

119

C₂×C₄×C₈

—

120

C₂×C₄×C₈

—

121

C₄³

—

122

#336

—

123

#335

—

124

D₄⋊C₈

—

125

C₂²×C₁₃

—

126

C₂²×C₁₃

—

127

C₂²×C₁₃

—

128

C₂²×C₁₃

#390

yes

characteristic

129

C₂×D₁₃

—

130

C₂×D₁₃

—

131

C₂×D₁₃

#466

yes

—

132

#261

—

133

C₄×C₁₃

#421

yes

—

134

C₄×C₁₃

C₄×D₈

yes

—

135

C₄×C₁₃

#444

yes

—

136

Dic₁₃

—

137

Dic₁₃

—

138

Dic₁₃

—

139

Dic₁₃

#493

yes

characteristic

140

Dic₁₃

#495

yes

characteristic

141

C₂×#41

—

142

C₂×D₈

—

143

C₂×M₁₆

—

144

C₂²×C₈

—

145

C₂²×C₈

—

146

C₄⋊Q₈

—

147

#127

C₂×#261

yes

—

148

#124

—

149

#121

—

150

#121

—

151

C₄×D₄

—

152

C₄×D₄

—

153

C₂×C₄²

—

154

C₂×C₄²

—

155

C₂×C₄²

—

156

C₂×C₄²

—

157

#107

—

158

C₄⋊C₈

—

159

C₄⋊C₈

—

160

#104

—

161

#102

—

162

C₄×C₈

—

163

C₄×C₈

—

164

C₄×C₈

—

165

C₄×C₈

—

166

C₄×C₈

C₂×#261

yes

characteristic

167

C₂×C₁₃

—

168

C₂×C₁₃

#1963

yes

characteristic

169

C₂×C₁₃

#1970

yes

characteristic

170

C₂×C₁₃

#1975

yes

characteristic

171

D₁₃

—

172

#41

—

173

#41

—

174

C₂×Q₈

—

175

C₂×D₄

—

176

C₂×D₄

—

177

C₂×D₄

—

178

C₂×D₄

—

179

C₂²×C₄

—

180

C₂²×C₄

—

181

C₂²×C₄

—

182

C₂²×C₄

—

183

C₂²×C₄

—

184

D₈

—

185

M₁₆

—

186

C₂×C₈

—

187

C₂×C₈

—

188

C₂×C₈

—

189

C₂×C₈

—

190

C₂×C₈

—

191

C₂×C₈

C₂²⋊#261

yes

—

192

C₄⋊C₄

—

193

C₄⋊C₄

—

194

C₂²⋊C₄

—

195

C₂²⋊C₄

—

196

C₄²

—

197

C₄²

—

198

C₄²

—

199

C₄²

—

200

C₄²

—

201

C₄²

—

202

C₄²

—

203

C₄²

C₂²×#261

yes

characteristic

204

C₁₃

#9847

yes

characteristic, complemented, Sylow

205

C₂³

—

206

C₂³

—

207

Q₈

—

208

D₄

—

209

D₄

—

210

D₄

—

211

D₄

—

104

—

212

C₂×C₄

—

213

C₂×C₄

—

214

C₂×C₄

—

215

C₂×C₄

—

216

C₂×C₄

—

217

C₂×C₄

—

218

C₂×C₄

—

219

C₂×C₄

—

220

C₂×C₄

—

221

C₂×C₄

—

222

C₂×C₄

—

223

C₂×C₄

—

104

—

224

C₂×C₄

—

104

—

225

C₂×C₄

D₄×#261

yes

characteristic

226

C₂×C₄

D₄×#261

yes

characteristic

227

C₂×C₄

C₂×C₂²⋊#261

yes

characteristic

228

C₈

—

229

C₈

—

230

C₈

—

231

C₈

—

104

—

232

C₂²

—

233

C₂²

—

234

C₂²

—

235

C₂²

—

236

C₂²

—

237

C₂²

—

238

C₂²

#11719151

yes

characteristic

239

C₄

—

240

C₄

—

241

C₄

—

242

C₄

—

243

C₄

—

244

C₄

—

104

—

245

C₄

D₈×#261

yes

—

246

C₄

#11719104

yes

—

247

C₄

#11719140

yes

—

248

C₂

—

249

C₂

—

250

C₂

#13130974

yes

characteristic

251

C₂

#13130983

yes

characteristic

252

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	#13125413	C₂
cosocle	C₄²×D₁₃	C₂³
hypercenter	#493	#261
hypocenter	C₁₃	#9847
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. #13130970	C₂	semidirect
2. #13130549	C₂	semidirect
3. #13130395	C₂	semidirect
4. #13130182	C₂	semidirect
11. C₁₃×#493	C₄	semidirect
14. #11718731	C₄	semidirect
16. #11718343	C₄	semidirect
64. C₄×#261	D₈	semidirect
113. #493	#261	semidirect
204. C₁₃	#9847	semidirect

H ≅ G/N

type

1. #13130970

C₂

semidirect

2. #13130549

C₂

semidirect

3. #13130395

C₂

semidirect

4. #13130182

C₂

semidirect

11. C₁₃×#493

C₄

semidirect

14. #11718731

C₄

semidirect

16. #11718343

C₄

semidirect

64. C₄×#261

D₈

semidirect

113. #493

#261

semidirect

204. C₁₃

#9847

semidirect

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

subgroup	index
#13130970	2
#13130549	2
#13130395	2
#13130182	2
#13125413	2
#9847	13

subgroup

index

Subgroups — #16784979

Conjugacy classes of subgroups

Complements

Maximal subgroups