Group #16784495 — subgroups

Subgroups — #16784495

Order 3328.

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

normal subgroups only

This group has 310 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	#13130775	C₂	1	1	yes	maximal, characteristic
2	#13130630	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×#88103	—	2	1	no	—
7	C₂×#88103	—	2	1	no	—
8	C₂×#88100	—	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	2	yes	—
32	#88103	—	4	2	no	—
33	#88103	—	4	2	no	—
34	#88100	—	4	2	no	—
35	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
37	#88060	—	2	2	no	—
38	C₈×Dic₁₃	—	2	2	no	—
39	C₈×Dic₁₃	Q₈	1	1	yes	characteristic
40	C₈×Dic₁₃	Q₈	1	1	yes	characteristic
41	C₄⋊#1055	C₂×C₄	1	2	yes	—
42	C₄×#1055	C₂×C₄	1	1	yes	characteristic
43	#9631	—	13	1	no	maximal, Sylow
44	C₂²×#261	—	2	2	no	—
45	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
48	C₂×#1057	—	2	2	no	—
49	C₂×#1057	—	4	2	no	—
50	C₄⋊#261	—	4	1	no	—
51	C₄⋊#261	—	4	1	no	—
52	C₄×#261	—	4	1	no	—
53	C₄×#261	—	4	1	no	—
54	#6118	—	4	2	no	—
55	#6118	—	4	2	no	—
56	D₁₃⋊C₈	—	4	1	no	—
57	D₁₃⋊C₈	—	4	1	no	—
58	C₂×C₈×C₁₃	—	2	2	no	—
59	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
61	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
63	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
64	C₄×Dic₁₃	#41	1	1	yes	characteristic
65	C₂×#1055	—	2	2	no	—
66	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
67	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
68	C₈×D₁₃	—	2	1	no	—
69	C₈×D₁₃	—	2	1	no	—
70	C₈×D₁₃	SD₁₆	1	2	yes	—
71	C₈×D₁₃	SD₁₆	1	2	yes	—
72	#1848	—	13	1	no	—
73	#1770	—	13	1	no	—
74	#1570	—	13	2	no	—
75	#1564	—	13	1	no	—
76	#1303	—	13	2	no	—
77	C₂²×D₁₃	C₂²⋊Q₈	1	1	yes	characteristic
78	C₂×#261	—	4	2	no	—
79	C₂×#261	—	4	2	no	—
80	C₂×#261	—	4	2	no	—
81	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
84	C₂×Dic₁₃	#121	1	1	yes	characteristic
85	C₂×Dic₁₃	#125	1	1	yes	characteristic
86	C₂×Dic₁₃	C₄⋊Q₈	1	1	yes	characteristic
87	C₄×D₁₃	—	2	1	no	—
88	C₄×D₁₃	—	2	1	no	—
89	C₄×D₁₃	—	4	1	no	—
90	C₄×D₁₃	C₂×SD₁₆	1	1	yes	characteristic
91	C₄×D₁₃	C₂×SD₁₆	1	1	yes	characteristic
92	C₄×D₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
93	C₄×D₁₃	#137	1	1	yes	characteristic
94	#1057	—	4	2	no	—
95	#1057	—	8	2	no	—
96	C₈×C₁₃	—	2	1	no	—
97	C₈×C₁₃	—	4	2	no	—
98	C₈×C₁₃	#106	1	2	yes	—
99	#1055	—	2	1	no	—
100	#1055	—	4	2	no	—
101	#1055	#106	1	2	yes	—
102	#431	—	13	2	no	—
103	C₂×#108	—	26	1	no	—
104	C₂×#108	—	26	1	no	—
105	#423	—	13	2	no	—
106	C₂×C₄⋊C₈	—	13	1	no	—
107	C₂×#97	—	26	1	no	—
108	C₂×C₄×C₈	—	13	1	no	—
109	C₄×C₄⋊C₄	—	13	1	no	—
110	#336	—	26	1	no	—
111	#335	—	13	1	no	—
112	#335	#261	1	1	yes	characteristic, complemented
113	#334	—	13	2	no	—
114	#334	—	26	2	no	—
115	#334	—	26	2	no	—
116	C₂²×C₁₃	#389	1	1	yes	characteristic
117	C₂×D₁₃	—	2	1	no	—
118	C₂×D₁₃	—	2	1	no	—
119	C₂×D₁₃	#475	1	1	yes	characteristic
120	C₂×D₁₃	#476	1	1	yes	characteristic
121	#261	—	8	2	no	—
122	C₄×C₁₃	—	2	1	no	—
123	C₄×C₁₃	C₂×#106	1	1	yes	characteristic
124	C₄×C₁₃	#428	1	1	yes	characteristic
125	C₄×C₁₃	#444	1	1	yes	characteristic
126	C₄×C₁₃	#444	1	1	yes	characteristic
127	Dic₁₃	—	2	1	no	—
128	Dic₁₃	—	2	1	no	—
129	Dic₁₃	#460	1	1	yes	characteristic
130	Dic₁₃	#461	1	1	yes	characteristic
131	C₂×M₁₆	—	26	2	no	—
132	C₂²×C₈	—	13	1	no	—
133	C₂²×C₈	—	13	1	no	—
134	C₂²×C₈	—	26	1	no	—
135	C₂×C₄⋊C₄	—	26	1	no	—
136	C₂×C₄²	—	13	1	no	—
137	C₂×C₄²	—	26	1	no	—
138	#108	—	52	2	no	—
139	#108	—	52	2	no	—
140	C₄⋊C₈	—	13	2	no	—
141	C₄⋊C₈	—	13	2	no	—
142	C₄⋊C₈	—	26	1	no	—
143	C₄⋊C₈	—	26	2	no	—
144	C₄⋊C₈	—	26	2	no	—
145	C₄⋊C₈	C₂×#261	1	2	yes	—
146	C₂²⋊C₈	—	26	2	no	—
147	#97	—	52	2	no	—
148	C₄×C₈	—	13	1	no	—
149	C₄×C₈	—	13	1	no	—
150	C₄×C₈	—	13	1	no	—
151	C₄×C₈	—	26	2	no	—
152	C₄×C₈	C₂×#261	1	1	yes	characteristic
153	#95	—	13	2	no	—
154	C₂×C₁₃	#1946	1	1	yes	characteristic
155	C₂×C₁₃	#1948	1	1	yes	characteristic
156	C₂×C₁₃	#1953	1	1	yes	characteristic
157	D₁₃	—	2	1	no	—
158	D₁₃	—	2	1	no	—
159	C₂²×C₄	—	13	1	no	—
160	C₂²×C₄	—	13	1	no	—
161	C₂²×C₄	—	13	1	no	—
162	C₂²×C₄	—	26	2	no	—
163	M₁₆	—	52	2	no	—
164	M₁₆	—	52	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	13	1	no	—
167	C₂×C₈	—	13	2	no	—
168	C₂×C₈	—	13	2	no	—
169	C₂×C₈	—	13	1	no	—
170	C₂×C₈	—	26	1	no	—
171	C₂×C₈	—	26	1	no	—
172	C₂×C₈	—	26	2	no	—
173	C₂×C₈	—	26	2	no	—
174	C₂×C₈	—	52	1	no	—
175	C₂×C₈	—	52	1	no	—
176	C₂×C₈	—	52	2	no	—
177	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
178	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
179	C₄⋊C₄	—	52	1	no	—
180	C₄⋊C₄	—	52	1	no	—
181	C₄²	—	13	1	no	—
182	C₄²	—	13	1	no	—
183	C₄²	—	13	1	no	—
184	C₄²	—	52	1	no	—
185	C₄²	—	52	1	no	—
186	C₄²	C₂²×#261	1	1	yes	characteristic
187	C₁₃	#9631	1	1	yes	characteristic, complemented, Sylow
188	C₂³	—	13	1	no	—
189	C₂×C₄	—	13	1	no	—
190	C₂×C₄	—	13	1	no	—
191	C₂×C₄	—	13	1	no	—
192	C₂×C₄	—	13	1	no	—
193	C₂×C₄	—	13	1	no	—
194	C₂×C₄	—	13	1	no	—
195	C₂×C₄	—	13	1	no	—
196	C₂×C₄	—	26	1	no	—
197	C₂×C₄	—	26	1	no	—
198	C₂×C₄	—	52	1	no	—
199	C₂×C₄	—	52	2	no	—
200	C₂×C₄	—	52	2	no	—
201	C₂×C₄	—	52	2	no	—
202	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
203	C₂×C₄	D₄×#261	1	1	yes	characteristic
204	C₂×C₄	Q₈×#261	1	1	yes	characteristic
205	C₈	—	2	1	no	—
206	C₈	—	4	2	no	—
207	C₈	—	13	2	no	—
208	C₈	—	26	1	no	—
209	C₈	—	52	2	no	—
210	C₈	—	52	2	no	—
211	C₈	—	104	2	no	—
212	C₈	#88101	1	2	yes	—
213	C₂²	—	13	1	no	—
214	C₂²	—	13	1	no	—
215	C₂²	—	26	1	no	—
216	C₂²	—	26	1	no	—
217	C₂²	#11719083	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	13	1	no	—
220	C₄	—	13	1	no	—
221	C₄	—	26	1	no	—
222	C₄	—	26	1	no	—
223	C₄	—	104	2	no	—
224	C₄	C₂×#88101	1	1	yes	characteristic
225	C₄	#11719098	1	1	yes	characteristic
226	C₄	#11719104	1	1	yes	characteristic
227	C₄	#11719112	1	1	yes	characteristic
228	C₂	—	26	1	no	—
229	C₂	—	26	1	no	—
230	C₂	#13130945	1	1	yes	characteristic
231	C₂	#13130948	1	1	yes	characteristic
232	C₂	#13130955	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13130775

C₂

yes

maximal, characteristic

#13130630

C₂

yes

maximal, characteristic

#13130404

C₂

yes

maximal

#13129733

C₂

yes

maximal

#13113583

C₂

yes

maximal, characteristic

C₂×#88103

—

C₂×#88103

—

C₂×#88100

—

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

—

#88103

—

#88103

—

#88100

—

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

#9631

—

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

—

—

—

—

—

—

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

—

C₈×C₁₃

—

C₈×C₁₃

#106

yes

—

#1055

—

100

#1055

—

101

#1055

#106

yes

—

102

#431

—

103

C₂×#108

—

104

C₂×#108

—

105

#423

—

106

C₂×C₄⋊C₈

—

107

C₂×#97

—

108

C₂×C₄×C₈

—

109

C₄×C₄⋊C₄

—

110

#336

—

111

#335

—

112

#335

#261

yes

characteristic, complemented

113

#334

—

114

#334

—

115

#334

—

116

C₂²×C₁₃

#389

yes

characteristic

117

C₂×D₁₃

—

118

C₂×D₁₃

—

119

C₂×D₁₃

#475

yes

characteristic

120

C₂×D₁₃

#476

yes

characteristic

121

#261

—

122

C₄×C₁₃

—

123

C₄×C₁₃

C₂×#106

yes

characteristic

124

C₄×C₁₃

#428

yes

characteristic

125

C₄×C₁₃

#444

yes

characteristic

126

C₄×C₁₃

#444

yes

characteristic

127

Dic₁₃

—

128

Dic₁₃

—

129

Dic₁₃

#460

yes

characteristic

130

Dic₁₃

#461

yes

characteristic

131

C₂×M₁₆

—

132

C₂²×C₈

—

133

C₂²×C₈

—

134

C₂²×C₈

—

135

C₂×C₄⋊C₄

—

136

C₂×C₄²

—

137

C₂×C₄²

—

138

#108

—

139

#108

—

140

C₄⋊C₈

—

141

C₄⋊C₈

—

142

C₄⋊C₈

—

143

C₄⋊C₈

—

144

C₄⋊C₈

—

145

C₄⋊C₈

C₂×#261

yes

—

146

C₂²⋊C₈

—

147

#97

—

148

C₄×C₈

—

149

C₄×C₈

—

150

C₄×C₈

—

151

C₄×C₈

—

152

C₄×C₈

C₂×#261

yes

characteristic

153

#95

—

154

C₂×C₁₃

#1946

yes

characteristic

155

C₂×C₁₃

#1948

yes

characteristic

156

C₂×C₁₃

#1953

yes

characteristic

157

D₁₃

—

158

D₁₃

—

159

C₂²×C₄

—

160

C₂²×C₄

—

161

C₂²×C₄

—

162

C₂²×C₄

—

163

M₁₆

—

164

M₁₆

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

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

C₄⋊#261

yes

characteristic

178

C₂×C₈

C₄⋊#261

yes

characteristic

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₄²

—

182

C₄²

—

183

C₄²

—

184

C₄²

—

185

C₄²

—

186

C₄²

C₂²×#261

yes

characteristic

187

C₁₃

#9631

yes

characteristic, complemented, Sylow

188

C₂³

—

189

C₂×C₄

—

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

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

C₂×C₄⋊#261

yes

characteristic

203

C₂×C₄

D₄×#261

yes

characteristic

204

C₂×C₄

Q₈×#261

yes

characteristic

205

C₈

—

206

C₈

—

207

C₈

—

208

C₈

—

209

C₈

—

210

C₈

—

211

C₈

—

104

—

212

C₈

#88101

yes

—

213

C₂²

—

214

C₂²

—

215

C₂²

—

216

C₂²

—

217

C₂²

#11719083

yes

characteristic

218

C₄

—

219

C₄

—

220

C₄

—

221

C₄

—

222

C₄

—

223

C₄

—

104

—

224

C₄

C₂×#88101

yes

characteristic

225

C₄

#11719098

yes

characteristic

226

C₄

#11719104

yes

characteristic

227

C₄

#11719112

yes

characteristic

228

C₂

—

229

C₂

—

230

C₂

#13130945

yes

characteristic

231

C₂

#13130948

yes

characteristic

232

C₂

#13130955

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	#9631
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
112. #335	#261	semidirect
187. C₁₃	#9631	semidirect

H ≅ G/N

type

17. C₁₃×#335

C₄

semidirect

18. #11718072

C₄

semidirect

19. #11718050

C₄

semidirect

112. #335

#261

semidirect

187. C₁₃

#9631

semidirect

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

subgroup	index
#13130775	2
#13130630	2
#13130404	2
#13129733	2
#13113583	2
#9631	13

subgroup

index

Subgroups — #16784495

Conjugacy classes of subgroups

Complements

Maximal subgroups