Group #16784490 — subgroups

Subgroups — #16784490

Order 3328.

322 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 98 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	#13130777	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊#261	C₂	1	1	yes	maximal, characteristic
3	#13130404	C₂	1	2	yes	maximal
4	#13129733	C₂	1	2	yes	maximal
5	#13113583	C₂	1	1	yes	maximal, characteristic
6	C₂×#88104	—	2	1	no	—
7	C₂×#88104	—	2	1	no	—
8	C₂×C₈×#261	—	2	1	no	—
9	C₄×C₄⋊#261	C₂²	1	1	yes	characteristic
10	#11719052	C₂²	1	2	yes	—
11	D₁₃⋊(C₄⋊C₈)	C₂²	1	1	yes	characteristic
12	#11718494	C₂²	1	2	yes	—
13	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
14	(C₂×#261)⋊C₈	—	2	1	no	—
15	#11718255	—	2	2	no	—
16	#11718253	—	2	2	no	—
17	C₁₃×#335	C₄	1	1	yes	characteristic, complemented
18	#11718072	C₄	1	2	yes	complemented
19	#11718050	C₄	1	1	yes	characteristic, complemented
20	C₂×C₄⋊#261	—	2	1	no	—
21	C₂×C₄×#261	—	2	1	no	—
22	C₂×#6118	—	2	2	no	—
23	C₂×D₁₃⋊C₈	—	2	1	no	—
24	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
25	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
26	C₄²×D₁₃	C₂³	1	1	yes	characteristic
27	(C₂×#261)⋊C₄	Q₈	1	2	yes	—
28	#88112	—	2	1	no	—
29	C₄⋊#1057	—	2	2	no	—
30	C₄⋊#1057	—	2	2	no	—
31	C₄⋊#1057	D₄	1	1	yes	characteristic
32	C₄⋊#1057	D₄	1	1	yes	characteristic
33	#88104	—	4	2	no	—
34	#88104	—	4	2	no	—
35	C₈×#261	—	4	2	no	—
36	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	#88060	—	2	2	no	—
39	C₈×Dic₁₃	—	2	2	no	—
40	C₈×Dic₁₃	Q₈	1	1	yes	characteristic
41	C₈×Dic₁₃	Q₈	1	1	yes	characteristic
42	C₄⋊#1055	C₂×C₄	1	2	yes	—
43	C₄×#1055	C₂×C₄	1	1	yes	characteristic
44	#9628	—	13	1	no	maximal, Sylow
45	C₂²×#261	—	2	2	no	—
46	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
49	C₂×#1057	—	2	1	no	—
50	C₂×#1057	—	2	1	no	—
51	C₂×#1057	—	4	2	no	—
52	C₄⋊#261	—	2	2	no	—
53	C₄⋊#261	—	2	2	no	—
54	C₄×#261	—	4	1	no	—
55	C₄×#261	—	4	1	no	—
56	#6118	—	4	2	no	—
57	#6118	—	4	2	no	—
58	D₁₃⋊C₈	—	4	1	no	—
59	D₁₃⋊C₈	—	4	1	no	—
60	C₂×C₈×C₁₃	—	2	2	no	—
61	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
63	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
65	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
66	C₄×Dic₁₃	#41	1	1	yes	characteristic
67	C₂×#1055	—	2	2	no	—
68	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
69	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
70	C₈×D₁₃	—	2	1	no	—
71	C₈×D₁₃	—	2	1	no	—
72	C₈×D₁₃	SD₁₆	1	2	yes	—
73	C₈×D₁₃	SD₁₆	1	2	yes	—
74	#1848	—	13	1	no	—
75	C₈×C₄⋊C₄	—	13	1	no	—
76	#1570	—	13	2	no	—
77	#1564	—	13	1	no	—
78	#1303	—	13	2	no	—
79	C₂²×D₁₃	C₂²⋊Q₈	1	1	yes	characteristic
80	C₂×#261	—	2	4	no	—
81	C₂×#261	—	4	2	no	—
82	C₂×#261	—	4	2	no	—
83	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
86	C₂×Dic₁₃	#121	1	1	yes	characteristic
87	C₂×Dic₁₃	#125	1	1	yes	characteristic
88	C₂×Dic₁₃	C₄⋊Q₈	1	1	yes	characteristic
89	C₄×D₁₃	—	2	1	no	—
90	C₄×D₁₃	—	2	1	no	—
91	C₄×D₁₃	—	4	1	no	—
92	C₄×D₁₃	C₂×SD₁₆	1	1	yes	characteristic
93	C₄×D₁₃	C₂×SD₁₆	1	1	yes	characteristic
94	C₄×D₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
95	C₄×D₁₃	#137	1	1	yes	characteristic
96	#1057	—	4	1	no	—
97	#1057	—	4	1	no	—
98	#1057	—	8	2	no	—
99	C₈×C₁₃	—	2	1	no	—
100	C₈×C₁₃	—	4	2	no	—
101	C₈×C₁₃	#106	1	2	yes	—
102	#1055	—	2	1	no	—
103	#1055	—	4	2	no	—
104	#1055	#106	1	2	yes	—
105	#431	—	13	2	no	—
106	C₂×#108	—	26	1	no	—
107	C₂×#108	—	26	1	no	—
108	#423	—	13	2	no	—
109	C₂×C₄⋊C₈	—	13	1	no	—
110	C₂×C₄×C₈	—	13	1	no	—
111	C₂×C₄×C₈	—	26	1	no	—
112	C₄×C₄⋊C₄	—	13	1	no	—
113	#336	—	26	1	no	—
114	#335	—	13	1	no	—
115	#335	#261	1	1	yes	characteristic, complemented
116	#334	—	13	2	no	—
117	#334	—	26	2	no	—
118	#334	—	26	2	no	—
119	C₂²×C₁₃	#389	1	1	yes	characteristic
120	C₂×D₁₃	—	2	1	no	—
121	C₂×D₁₃	—	2	1	no	—
122	C₂×D₁₃	#475	1	1	yes	characteristic
123	C₂×D₁₃	#476	1	1	yes	characteristic
124	#261	—	4	4	no	—
125	C₄×C₁₃	—	2	1	no	—
126	C₄×C₁₃	C₂×#106	1	1	yes	characteristic
127	C₄×C₁₃	#428	1	1	yes	characteristic
128	C₄×C₁₃	#443	1	1	yes	characteristic
129	C₄×C₁₃	#443	1	1	yes	characteristic
130	Dic₁₃	—	2	1	no	—
131	Dic₁₃	—	2	1	no	—
132	Dic₁₃	#460	1	1	yes	characteristic
133	Dic₁₃	#461	1	1	yes	characteristic
134	C₂×M₁₆	—	26	2	no	—
135	C₂²×C₈	—	13	1	no	—
136	C₂²×C₈	—	13	1	no	—
137	C₂²×C₈	—	26	1	no	—
138	C₂×C₄⋊C₄	—	26	1	no	—
139	C₂×C₄²	—	13	1	no	—
140	C₂×C₄²	—	26	1	no	—
141	#108	—	52	2	no	—
142	#108	—	52	2	no	—
143	C₄⋊C₈	—	13	1	no	—
144	C₄⋊C₈	—	13	1	no	—
145	C₄⋊C₈	—	13	2	no	—
146	C₄⋊C₈	—	26	1	no	—
147	C₄⋊C₈	—	26	2	no	—
148	C₄⋊C₈	—	26	2	no	—
149	C₄⋊C₈	C₂×#261	1	2	yes	—
150	C₂²⋊C₈	—	26	2	no	—
151	C₄×C₈	—	13	1	no	—
152	C₄×C₈	—	13	1	no	—
153	C₄×C₈	—	13	1	no	—
154	C₄×C₈	—	26	2	no	—
155	C₄×C₈	—	52	2	no	—
156	C₄×C₈	C₂×#261	1	1	yes	characteristic
157	#95	—	13	2	no	—
158	C₂×C₁₃	#1945	1	1	yes	characteristic
159	C₂×C₁₃	#1948	1	1	yes	characteristic
160	C₂×C₁₃	#1953	1	1	yes	characteristic
161	D₁₃	—	2	1	no	—
162	D₁₃	—	2	1	no	—
163	C₂²×C₄	—	13	1	no	—
164	C₂²×C₄	—	13	1	no	—
165	C₂²×C₄	—	13	1	no	—
166	C₂²×C₄	—	26	2	no	—
167	M₁₆	—	52	2	no	—
168	M₁₆	—	52	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	13	1	no	—
171	C₂×C₈	—	13	2	no	—
172	C₂×C₈	—	13	2	no	—
173	C₂×C₈	—	13	1	no	—
174	C₂×C₈	—	26	1	no	—
175	C₂×C₈	—	26	1	no	—
176	C₂×C₈	—	26	1	no	—
177	C₂×C₈	—	26	1	no	—
178	C₂×C₈	—	26	2	no	—
179	C₂×C₈	—	52	1	no	—
180	C₂×C₈	—	52	1	no	—
181	C₂×C₈	—	52	2	no	—
182	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
183	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
184	C₄⋊C₄	—	26	2	no	—
185	C₄⋊C₄	—	26	2	no	—
186	C₄²	—	13	1	no	—
187	C₄²	—	13	1	no	—
188	C₄²	—	13	1	no	—
189	C₄²	—	52	1	no	—
190	C₄²	—	52	1	no	—
191	C₄²	C₂²×#261	1	1	yes	characteristic
192	C₁₃	#9628	1	1	yes	characteristic, complemented, Sylow
193	C₂³	—	13	1	no	—
194	C₂×C₄	—	13	1	no	—
195	C₂×C₄	—	13	1	no	—
196	C₂×C₄	—	13	1	no	—
197	C₂×C₄	—	13	1	no	—
198	C₂×C₄	—	13	1	no	—
199	C₂×C₄	—	13	1	no	—
200	C₂×C₄	—	13	1	no	—
201	C₂×C₄	—	26	1	no	—
202	C₂×C₄	—	26	1	no	—
203	C₂×C₄	—	26	4	no	—
204	C₂×C₄	—	52	1	no	—
205	C₂×C₄	—	52	2	no	—
206	C₂×C₄	—	52	2	no	—
207	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
208	C₂×C₄	D₄×#261	1	1	yes	characteristic
209	C₂×C₄	Q₈×#261	1	1	yes	characteristic
210	C₈	—	2	1	no	—
211	C₈	—	4	2	no	—
212	C₈	—	13	2	no	—
213	C₈	—	26	1	no	—
214	C₈	—	52	1	no	—
215	C₈	—	52	1	no	—
216	C₈	—	52	2	no	—
217	C₈	—	104	2	no	—
218	C₈	#88101	1	2	yes	—
219	C₂²	—	13	1	no	—
220	C₂²	—	13	1	no	—
221	C₂²	—	26	1	no	—
222	C₂²	—	26	1	no	—
223	C₂²	#11719083	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	13	1	no	—
226	C₄	—	13	1	no	—
227	C₄	—	26	1	no	—
228	C₄	—	26	1	no	—
229	C₄	—	52	4	no	—
230	C₄	C₂×#88101	1	1	yes	characteristic
231	C₄	#11719098	1	1	yes	characteristic
232	C₄	#11719103	1	1	yes	characteristic
233	C₄	#11719111	1	1	yes	characteristic
234	C₂	—	26	1	no	—
235	C₂	—	26	1	no	—
236	C₂	#13130944	1	1	yes	characteristic
237	C₂	#13130948	1	1	yes	characteristic
238	C₂	#13130956	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13130777

C₂

yes

maximal, characteristic

C₈×C₄⋊#261

C₂

yes

maximal, characteristic

#13130404

C₂

yes

maximal

#13129733

C₂

yes

maximal

#13113583

C₂

yes

maximal, characteristic

C₂×#88104

—

C₂×#88104

—

C₂×C₈×#261

—

C₄×C₄⋊#261

C₂²

yes

characteristic

#11719052

C₂²

yes

—

D₁₃⋊(C₄⋊C₈)

C₂²

yes

characteristic

#11718494

C₂²

yes

—

C₄×C₈×D₁₃

C₂²

yes

characteristic

(C₂×#261)⋊C₈

—

#11718255

—

#11718253

—

C₁₃×#335

C₄

yes

characteristic, complemented

#11718072

C₄

yes

complemented

#11718050

C₄

yes

characteristic, complemented

C₂×C₄⋊#261

—

C₂×C₄×#261

—

C₂×#6118

—

C₂×D₁₃⋊C₈

—

C₂×C₈×D₁₃

D₄

yes

characteristic

C₂×C₈×D₁₃

D₄

yes

characteristic

C₄²×D₁₃

C₂³

yes

characteristic

(C₂×#261)⋊C₄

Q₈

yes

—

—

—

—

D₄

yes

characteristic

C₄⋊#1057

D₄

yes

—

—

—

C₁₃×C₄⋊C₈

C₂×C₄

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

#88060

—

C₈×Dic₁₃

—

C₈×Dic₁₃

Q₈

yes

characteristic

C₈×Dic₁₃

Q₈

yes

characteristic

C₄⋊#1055

C₂×C₄

yes

—

C₄×#1055

C₂×C₄

yes

characteristic

#9628

—

maximal, Sylow

C₂²×#261

—

C₂×C₄×D₁₃

C₂×D₄

yes

characteristic

C₂×C₄×D₁₃

C₂×Q₈

yes

C₂×C₄×D₁₃

yes

—

—

—

—

—

—

—

—

—

D₁₃⋊C₈

—

D₁₃⋊C₈

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₃

C₄⋊C₄

yes

characteristic

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄×Dic₁₃

C₂×D₄

yes

characteristic

C₄×Dic₁₃

C₂×Q₈

yes

C₄×Dic₁₃

yes

—

C₄⋊C₄

yes

characteristic

C₂×#1055

C₄⋊C₄

yes

characteristic

C₈×D₁₃

—

C₈×D₁₃

—

C₈×D₁₃

SD₁₆

yes

—

C₈×D₁₃

SD₁₆

yes

—

#1848

—

C₈×C₄⋊C₄

—

#1570

—

#1564

—

#1303

—

C₂²×D₁₃

C₂²⋊Q₈

yes

—

—

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₄×D₄

yes

characteristic

C₂×C₄×C₁₃

C₄×Q₈

yes

C₂×Dic₁₃

yes

C₂×Dic₁₃

yes

C₂×Dic₁₃

C₄⋊Q₈

yes

characteristic

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

C₂×SD₁₆

yes

characteristic

C₄×D₁₃

C₂×SD₁₆

yes

C₄×D₁₃

yes

C₄×D₁₃

yes

—

—

—

C₈×C₁₃

—

100

C₈×C₁₃

—

101

C₈×C₁₃

#106

yes

—

102

#1055

—

103

#1055

—

104

#1055

#106

yes

—

105

#431

—

106

C₂×#108

—

107

C₂×#108

—

108

#423

—

109

C₂×C₄⋊C₈

—

110

C₂×C₄×C₈

—

111

C₂×C₄×C₈

—

112

C₄×C₄⋊C₄

—

113

#336

—

114

#335

—

115

#335

#261

yes

characteristic, complemented

116

#334

—

117

#334

—

118

#334

—

119

C₂²×C₁₃

#389

yes

characteristic

120

C₂×D₁₃

—

121

C₂×D₁₃

—

122

C₂×D₁₃

#475

yes

characteristic

123

C₂×D₁₃

#476

yes

characteristic

124

#261

—

125

C₄×C₁₃

—

126

C₄×C₁₃

C₂×#106

yes

characteristic

127

C₄×C₁₃

#428

yes

characteristic

128

C₄×C₁₃

#443

yes

characteristic

129

C₄×C₁₃

#443

yes

characteristic

130

Dic₁₃

—

131

Dic₁₃

—

132

Dic₁₃

#460

yes

characteristic

133

Dic₁₃

#461

yes

characteristic

134

C₂×M₁₆

—

135

C₂²×C₈

—

136

C₂²×C₈

—

137

C₂²×C₈

—

138

C₂×C₄⋊C₄

—

139

C₂×C₄²

—

140

C₂×C₄²

—

141

#108

—

142

#108

—

143

C₄⋊C₈

—

144

C₄⋊C₈

—

145

C₄⋊C₈

—

146

C₄⋊C₈

—

147

C₄⋊C₈

—

148

C₄⋊C₈

—

149

C₄⋊C₈

C₂×#261

yes

—

150

C₂²⋊C₈

—

151

C₄×C₈

—

152

C₄×C₈

—

153

C₄×C₈

—

154

C₄×C₈

—

155

C₄×C₈

—

156

C₄×C₈

C₂×#261

yes

characteristic

157

#95

—

158

C₂×C₁₃

#1945

yes

characteristic

159

C₂×C₁₃

#1948

yes

characteristic

160

C₂×C₁₃

#1953

yes

characteristic

161

D₁₃

—

162

D₁₃

—

163

C₂²×C₄

—

164

C₂²×C₄

—

165

C₂²×C₄

—

166

C₂²×C₄

—

167

M₁₆

—

168

M₁₆

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

176

C₂×C₈

—

177

C₂×C₈

—

178

C₂×C₈

—

179

C₂×C₈

—

180

C₂×C₈

—

181

C₂×C₈

—

182

C₂×C₈

C₄⋊#261

yes

characteristic

183

C₂×C₈

C₄⋊#261

yes

characteristic

184

C₄⋊C₄

—

185

C₄⋊C₄

—

186

C₄²

—

187

C₄²

—

188

C₄²

—

189

C₄²

—

190

C₄²

—

191

C₄²

C₂²×#261

yes

characteristic

192

C₁₃

#9628

yes

characteristic, complemented, Sylow

193

C₂³

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

205

C₂×C₄

—

206

C₂×C₄

—

207

C₂×C₄

C₂×C₄⋊#261

yes

characteristic

208

C₂×C₄

D₄×#261

yes

characteristic

209

C₂×C₄

Q₈×#261

yes

characteristic

210

C₈

—

211

C₈

—

212

C₈

—

213

C₈

—

214

C₈

—

215

C₈

—

216

C₈

—

217

C₈

—

104

—

218

C₈

#88101

yes

—

219

C₂²

—

220

C₂²

—

221

C₂²

—

222

C₂²

—

223

C₂²

#11719083

yes

characteristic

224

C₄

—

225

C₄

—

226

C₄

—

227

C₄

—

228

C₄

—

229

C₄

—

230

C₄

C₂×#88101

yes

characteristic

231

C₄

#11719098

yes

characteristic

232

C₄

#11719103

yes

characteristic

233

C₄

#11719111

yes

characteristic

234

C₂

—

235

C₂

—

236

C₂

#13130944

yes

characteristic

237

C₂

#13130948

yes

characteristic

238

C₂

#13130956

yes

characteristic

	subgroup	G/H
center	C₂²	#11719083
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₄²	C₂²×#261
Fitting	C₁₃×#335	C₄
socle	C₂²×C₁₃	#389
prime radical	C₂²×D₁₃	C₂²⋊Q₈
cosocle	C₄²×D₁₃	C₂³
hypercenter	#335	#261
hypocenter	C₁₃	#9628
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

subgroup

G/H

center

C₂²

#11719083

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₄²

C₂²×#261

Fitting

C₁₃×#335

C₄

socle

C₂²×C₁₃

#389

prime radical

C₂²×D₁₃

C₂²⋊Q₈

cosocle

C₄²×D₁₃

C₂³

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #335

N	H ≅ G/N	type
17. C₁₃×#335	C₄	semidirect
18. #11718072	C₄	semidirect
19. #11718050	C₄	semidirect
115. #335	#261	semidirect
192. C₁₃	#9628	semidirect

H ≅ G/N

type

17. C₁₃×#335

C₄

semidirect

18. #11718072

C₄

semidirect

19. #11718050

C₄

semidirect

115. #335

#261

semidirect

192. C₁₃

#9628

semidirect

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

subgroup	index
#13130777	2
C₈×C₄⋊#261	2
#13130404	2
#13129733	2
#13113583	2
#9628	13

subgroup

index

C₈×C₄⋊#261

Subgroups — #16784490

Conjugacy classes of subgroups

Complements

Maximal subgroups