Group #16784984 — subgroups

Subgroups — #16784984

Order 3328.

530 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 100 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	C₄×#88100	C₂	1	1	yes	maximal, characteristic, complemented
3	#13130391	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	C₂×#88100	—	2	2	no	—
9	C₄²×#261	C₂²	1	1	yes	characteristic
10	#11719052	C₂²	1	2	yes	—
11	C₄×D₁₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₁₃×#493	C₄	1	1	yes	characteristic, complemented
13	#11718818	—	2	2	no	—
14	D₈×Dic₁₃	—	2	2	no	—
15	#11718731	C₄	1	2	yes	complemented
16	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
17	#11718343	C₄	1	1	yes	characteristic, complemented
18	D₄⋊#1057	—	2	4	no	—
19	#11718253	—	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	#88100	—	2	8	no	—
42	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
43	#88072	—	2	4	no	—
44	#88058	—	2	2	no	—
45	C₈×Dic₁₃	D₄	1	2	yes	—
46	C₄×#1055	C₂×C₄	1	1	yes	characteristic
47	#9849	—	13	1	no	maximal, Sylow
48	C₂²×#261	—	2	2	no	—
49	C₂×D₄×C₁₃	—	2	2	no	—
50	C₂×D₄×C₁₃	—	2	2	no	—
51	C₂×D₄×C₁₃	—	2	2	no	—
52	C₂×C₂²⋊D₁₃	—	4	2	no	—
53	C₂²×Dic₁₃	—	4	2	no	—
54	#6129	—	4	4	no	—
55	#6129	—	4	4	no	—
56	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
59	C₂×Dic₂₆	—	2	2	no	—
60	C₂×#1057	—	2	2	no	—
61	C₂×#1057	—	4	2	no	—
62	C₄×#261	—	2	4	no	—
63	C₄×#261	—	2	4	no	—
64	C₄×#261	—	4	2	no	—
65	#6118	—	4	4	no	—
66	D₁₃⋊C₈	—	2	4	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	—	2	4	no	—
79	#3169	—	13	1	no	—
80	#1962	—	13	2	no	—
81	C₄×#97	—	13	1	no	—
82	#1566	—	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	AGL(1,ℤ/8)	1	4	yes	—
105	Dic₂₆	—	4	4	no	—
106	#1057	—	2	4	no	—
107	#1057	—	8	2	no	—
108	C₈×C₁₃	#104	1	4	yes	complemented
109	#1055	—	2	2	no	—
110	#532	—	13	2	no	—
111	#500	—	13	1	no	—
112	#493	#261	1	1	yes	characteristic, complemented
113	#486	—	13	2	no	—
114	#466	—	26	2	no	—
115	C₄×D₈	—	26	2	no	—
116	#423	—	13	2	no	—
117	C₂×#104	—	26	4	no	—
118	C₂×#97	—	26	2	no	—
119	C₂×C₄×C₈	—	13	1	no	—
120	C₂×C₄×C₈	—	13	1	no	—
121	C₄³	—	13	1	no	—
122	#334	—	26	2	no	—
123	D₄⋊C₈	—	26	4	no	—
124	C₂²×C₁₃	—	8	2	no	—
125	C₂²×C₁₃	—	8	2	no	—
126	C₂²×C₁₃	—	8	2	no	—
127	C₂²×C₁₃	#390	1	1	yes	characteristic
128	C₂×D₁₃	—	2	1	no	—
129	C₂×D₁₃	—	2	1	no	—
130	C₂×D₁₃	#469	1	2	yes	—
131	#261	—	4	4	no	—
132	C₄×C₁₃	C₂×#104	1	2	yes	—
133	C₄×C₁₃	#442	1	2	yes	—
134	C₄×C₁₃	#444	1	2	yes	—
135	Dic₁₃	—	2	1	no	—
136	Dic₁₃	—	2	1	no	—
137	Dic₁₃	—	8	2	no	—
138	Dic₁₃	#496	1	2	yes	—
139	C₂×#41	—	26	2	no	—
140	C₂×D₈	—	2	2	no	—
141	C₂×M₁₆	—	26	2	no	—
142	C₂²×C₈	—	13	2	no	—
143	C₂²×C₈	—	26	1	no	—
144	C₄⋊Q₈	—	13	2	no	—
145	#127	C₂×#261	1	2	yes	—
146	#124	—	26	2	no	—
147	#121	—	26	2	no	—
148	#121	—	26	2	no	—
149	C₄×D₄	—	26	2	no	—
150	C₄×D₄	—	26	2	no	—
151	C₂×C₄²	—	13	1	no	—
152	C₂×C₄²	—	13	2	no	—
153	C₂×C₄²	—	26	1	no	—
154	C₂×C₄²	—	26	1	no	—
155	#107	—	26	2	no	—
156	C₄⋊C₈	—	26	2	no	—
157	C₄⋊C₈	—	26	2	no	—
158	#104	—	52	8	no	—
159	#102	—	26	4	no	—
160	#97	—	26	8	no	—
161	C₄×C₈	—	13	2	no	—
162	C₄×C₈	—	13	2	no	—
163	C₄×C₈	—	13	1	no	—
164	C₄×C₈	—	26	1	no	—
165	C₄×C₈	C₂×#261	1	1	yes	characteristic
166	C₂×C₁₃	—	16	2	no	—
167	C₂×C₁₃	#1962	1	1	yes	characteristic
168	C₂×C₁₃	#1972	1	1	yes	characteristic
169	C₂×C₁₃	#1975	1	1	yes	characteristic
170	D₁₃	—	2	2	no	—
171	#41	—	52	4	no	—
172	#41	—	52	4	no	—
173	C₂×Q₈	—	26	2	no	—
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	—	2	2	no	—
176	C₂×D₄	—	2	2	no	—
177	C₂×D₄	—	52	2	no	—
178	C₂²×C₄	—	13	1	no	—
179	C₂²×C₄	—	13	1	no	—
180	C₂²×C₄	—	13	1	no	—
181	C₂²×C₄	—	26	2	no	—
182	C₂²×C₄	—	52	2	no	—
183	D₈	—	4	4	no	—
184	M₁₆	—	52	4	no	—
185	C₂×C₈	—	13	2	no	—
186	C₂×C₈	—	26	2	no	—
187	C₂×C₈	—	26	4	no	—
188	C₂×C₈	—	26	4	no	—
189	C₂×C₈	—	52	2	no	—
190	C₂×C₈	C₂²⋊#261	1	2	yes	—
191	C₄⋊C₄	—	26	2	no	—
192	C₄⋊C₄	—	26	2	no	—
193	C₂²⋊C₄	—	52	2	no	—
194	C₂²⋊C₄	—	52	2	no	—
195	C₄²	—	13	1	no	—
196	C₄²	—	13	1	no	—
197	C₄²	—	13	1	no	—
198	C₄²	—	26	4	no	—
199	C₄²	—	26	4	no	—
200	C₄²	—	52	2	no	—
201	C₄²	C₂²×#261	1	1	yes	characteristic
202	C₁₃	#9849	1	1	yes	characteristic, complemented, Sylow
203	C₂³	—	4	2	no	—
204	C₂³	—	13	1	no	—
205	Q₈	—	52	4	no	—
206	D₄	—	4	4	no	—
207	D₄	—	4	4	no	—
208	D₄	—	4	4	no	—
209	D₄	—	104	4	no	—
210	C₂×C₄	—	13	1	no	—
211	C₂×C₄	—	13	4	no	—
212	C₂×C₄	—	13	1	no	—
213	C₂×C₄	—	13	1	no	—
214	C₂×C₄	—	26	2	no	—
215	C₂×C₄	—	26	2	no	—
216	C₂×C₄	—	26	4	no	—
217	C₂×C₄	—	52	2	no	—
218	C₂×C₄	—	52	2	no	—
219	C₂×C₄	—	52	2	no	—
220	C₂×C₄	—	104	2	no	—
221	C₂×C₄	—	104	2	no	—
222	C₂×C₄	D₄×#261	1	1	yes	characteristic
223	C₂×C₄	D₄×#261	1	1	yes	characteristic
224	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
225	C₈	—	26	4	no	—
226	C₈	—	26	2	no	—
227	C₈	—	104	2	no	—
228	C₈	#88116	1	4	yes	complemented
229	C₂²	—	8	2	no	—
230	C₂²	—	8	2	no	—
231	C₂²	—	8	2	no	—
232	C₂²	—	13	2	no	—
233	C₂²	—	26	1	no	—
234	C₂²	—	26	1	no	—
235	C₂²	#11719151	1	1	yes	characteristic
236	C₄	—	13	2	no	—
237	C₄	—	26	1	no	—
238	C₄	—	26	1	no	—
239	C₄	—	52	4	no	—
240	C₄	—	104	2	no	—
241	C₄	#11719102	1	2	yes	—
242	C₄	#11719104	1	2	yes	—
243	C₄	C₂×#88116	1	2	yes	—
244	C₂	—	16	2	no	—
245	C₂	—	26	2	no	—
246	C₂	#13130970	1	1	yes	characteristic
247	C₂	#13130985	1	1	yes	characteristic
248	C₂	#13130991	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13130970

C₂

yes

maximal, complemented

C₄×#88100

C₂

yes

maximal, characteristic, complemented

#13130391

C₂

yes

maximal, characteristic, complemented

#13130182

C₂

yes

maximal, complemented

#13125413

C₂

yes

maximal, characteristic

#11719421

C₂²

yes

—

C₂×#88116

—

C₂×#88100

—

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

—

#11718253

—

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

—

#88100

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

#88072

—

#88058

—

C₈×Dic₁₃

D₄

yes

—

C₄×#1055

C₂×C₄

yes

characteristic

#9849

—

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

—

—

—

—

—

—

—

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

—

#3169

—

#1962

—

C₄×#97

—

#1566

—

#1492

—

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

AGL(1,ℤ/8)

yes

—

105

Dic₂₆

—

106

#1057

—

107

#1057

—

108

C₈×C₁₃

#104

yes

complemented

109

#1055

—

110

#532

—

111

#500

—

112

#493

#261

yes

characteristic, complemented

113

#486

—

114

#466

—

115

C₄×D₈

—

116

#423

—

117

C₂×#104

—

118

C₂×#97

—

119

C₂×C₄×C₈

—

120

C₂×C₄×C₈

—

121

C₄³

—

122

#334

—

123

D₄⋊C₈

—

124

C₂²×C₁₃

—

125

C₂²×C₁₃

—

126

C₂²×C₁₃

—

127

C₂²×C₁₃

#390

yes

characteristic

128

C₂×D₁₃

—

129

C₂×D₁₃

—

130

C₂×D₁₃

#469

yes

—

131

#261

—

132

C₄×C₁₃

C₂×#104

yes

—

133

C₄×C₁₃

#442

yes

—

134

C₄×C₁₃

#444

yes

—

135

Dic₁₃

—

136

Dic₁₃

—

137

Dic₁₃

—

138

Dic₁₃

#496

yes

—

139

C₂×#41

—

140

C₂×D₈

—

141

C₂×M₁₆

—

142

C₂²×C₈

—

143

C₂²×C₈

—

144

C₄⋊Q₈

—

145

#127

C₂×#261

yes

—

146

#124

—

147

#121

—

148

#121

—

149

C₄×D₄

—

150

C₄×D₄

—

151

C₂×C₄²

—

152

C₂×C₄²

—

153

C₂×C₄²

—

154

C₂×C₄²

—

155

#107

—

156

C₄⋊C₈

—

157

C₄⋊C₈

—

158

#104

—

159

#102

—

160

#97

—

161

C₄×C₈

—

162

C₄×C₈

—

163

C₄×C₈

—

164

C₄×C₈

—

165

C₄×C₈

C₂×#261

yes

characteristic

166

C₂×C₁₃

—

167

C₂×C₁₃

#1962

yes

characteristic

168

C₂×C₁₃

#1972

yes

characteristic

169

C₂×C₁₃

#1975

yes

characteristic

170

D₁₃

—

171

#41

—

172

#41

—

173

C₂×Q₈

—

174

C₂×D₄

—

175

C₂×D₄

—

176

C₂×D₄

—

177

C₂×D₄

—

178

C₂²×C₄

—

179

C₂²×C₄

—

180

C₂²×C₄

—

181

C₂²×C₄

—

182

C₂²×C₄

—

183

D₈

—

184

M₁₆

—

185

C₂×C₈

—

186

C₂×C₈

—

187

C₂×C₈

—

188

C₂×C₈

—

189

C₂×C₈

—

190

C₂×C₈

C₂²⋊#261

yes

—

191

C₄⋊C₄

—

192

C₄⋊C₄

—

193

C₂²⋊C₄

—

194

C₂²⋊C₄

—

195

C₄²

—

196

C₄²

—

197

C₄²

—

198

C₄²

—

199

C₄²

—

200

C₄²

—

201

C₄²

C₂²×#261

yes

characteristic

202

C₁₃

#9849

yes

characteristic, complemented, Sylow

203

C₂³

—

204

C₂³

—

205

Q₈

—

206

D₄

—

207

D₄

—

208

D₄

—

209

D₄

—

104

—

210

C₂×C₄

—

211

C₂×C₄

—

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

—

104

—

221

C₂×C₄

—

104

—

222

C₂×C₄

D₄×#261

yes

characteristic

223

C₂×C₄

D₄×#261

yes

characteristic

224

C₂×C₄

C₂×C₂²⋊#261

yes

characteristic

225

C₈

—

226

C₈

—

227

C₈

—

104

—

228

C₈

#88116

yes

complemented

229

C₂²

—

230

C₂²

—

231

C₂²

—

232

C₂²

—

233

C₂²

—

234

C₂²

—

235

C₂²

#11719151

yes

characteristic

236

C₄

—

237

C₄

—

238

C₄

—

239

C₄

—

240

C₄

—

104

—

241

C₄

#11719102

yes

—

242

C₄

#11719104

yes

—

243

C₄

C₂×#88116

yes

—

244

C₂

—

245

C₂

—

246

C₂

#13130970

yes

characteristic

247

C₂

#13130985

yes

characteristic

248

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	#9849
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. C₄×#88100	C₂	semidirect
3. #13130391	C₂	semidirect
4. #13130182	C₂	semidirect
12. C₁₃×#493	C₄	semidirect
15. #11718731	C₄	semidirect
17. #11718343	C₄	semidirect
108. C₈×C₁₃	#104	semidirect
112. #493	#261	semidirect
202. C₁₃	#9849	semidirect
228. C₈	#88116	semidirect

H ≅ G/N

type

1. #13130970

C₂

semidirect

2. C₄×#88100

C₂

semidirect

3. #13130391

C₂

semidirect

4. #13130182

C₂

semidirect

12. C₁₃×#493

C₄

semidirect

15. #11718731

C₄

semidirect

17. #11718343

C₄

semidirect

108. C₈×C₁₃

#104

semidirect

112. #493

#261

semidirect

202. C₁₃

#9849

semidirect

228. C₈

#88116

semidirect

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

subgroup	index
#13130970	2
C₄×#88100	2
#13130391	2
#13130182	2
#13125413	2
#9849	13

subgroup

index

Subgroups — #16784984

Conjugacy classes of subgroups

Complements

Maximal subgroups