Group #16784987 — subgroups

Subgroups — #16784987

Order 3328.

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

normal subgroups only

This group has 454 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	#13131021	C₂	1	2	yes	maximal
2	C₄×#88100	C₂	1	1	yes	maximal, characteristic
3	#13130391	C₂	1	1	yes	maximal, characteristic
4	#13130186	C₂	1	2	yes	maximal
5	#13125420	C₂	1	1	yes	maximal, characteristic
6	#11719433	C₂²	1	2	yes	—
7	C₂×#88100	—	2	2	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×#494	C₄	1	1	yes	characteristic, complemented
12	#11718849	—	2	2	no	—
13	Q₁₆×Dic₁₃	—	2	2	no	—
14	#11718742	C₄	1	2	yes	complemented
15	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
16	#11718343	C₄	1	1	yes	characteristic, complemented
17	#11718253	—	2	2	no	—
18	#11718247	—	2	4	no	—
19	#11718244	—	2	4	no	—
20	C₂×C₄×#261	—	2	1	no	—
21	C₂×C₄×#261	—	2	1	no	—
22	C₂×C₄×#261	D₄	1	2	yes	—
23	C₂×#6118	—	2	2	no	—
24	C₂×D₁₃⋊C₈	—	2	1	no	—
25	C₂×C₁₃×Q₁₆	—	2	2	no	—
26	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
27	#88200	—	2	2	no	—
28	Q₈×Dic₁₃	—	2	2	no	—
29	C₂×C₈×D₁₃	D₄	1	2	yes	—
30	#88151	—	2	2	no	—
31	#88146	—	2	2	no	—
32	#88144	—	2	2	no	—
33	#88141	—	2	2	no	—
34	C₄²×D₁₃	C₂³	1	1	yes	characteristic
35	#88123	C₂×C₄	1	2	yes	—
36	C₄⋊#1057	—	2	2	no	—
37	C₄⋊#1057	—	2	2	no	—
38	C₄×#1057	—	2	1	no	—
39	C₄×#1057	D₄	1	2	yes	—
40	#88100	—	2	8	no	—
41	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	#88075	—	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	#9850	—	13	1	no	maximal, Sylow
47	C₂²×#261	—	2	2	no	—
48	C₂×Q₈×C₁₃	—	2	2	no	—
49	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
52	C₂×Dic₂₆	—	2	2	no	—
53	C₂×#1057	—	2	2	no	—
54	C₂×#1057	—	4	2	no	—
55	C₄×#261	—	2	4	no	—
56	C₄×#261	—	2	4	no	—
57	C₄×#261	—	4	2	no	—
58	#6118	—	4	4	no	—
59	D₁₃⋊C₈	—	2	4	no	—
60	C₁₃×Q₁₆	—	4	4	no	—
61	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
62	C₁₃×C₄⋊C₄	—	2	2	no	—
63	C₁₃×C₄⋊C₄	—	2	2	no	—
64	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	#6103	—	4	2	no	—
66	C₄⋊Dic₁₃	—	2	2	no	—
67	C₄⋊Dic₁₃	—	2	2	no	—
68	C₄⋊Dic₁₃	—	4	2	no	—
69	#6101	—	4	2	no	—
70	C₄×Dic₁₃	—	4	2	no	—
71	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
72	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
73	C₄×Dic₁₃	#41	1	1	yes	characteristic
74	C₂×#1055	C₂²⋊C₄	1	2	yes	—
75	C₈×D₁₃	—	2	4	no	—
76	#3170	—	13	1	no	—
77	#1994	—	13	2	no	—
78	C₄×#97	—	13	1	no	—
79	#1566	—	13	1	no	—
80	#1493	—	13	2	no	—
81	C₂²×D₁₃	#121	1	1	yes	characteristic
82	C₂×#261	—	2	4	no	—
83	C₂×#261	—	4	2	no	—
84	C₂×#261	—	4	2	no	—
85	Q₈×C₁₃	—	4	4	no	—
86	C₂×C₄×C₁₃	—	4	2	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	C₂×Dic₁₃	—	4	2	no	—
91	C₂×Dic₁₃	#121	1	1	yes	characteristic
92	C₂×Dic₁₃	#124	1	1	yes	characteristic
93	C₂×Dic₁₃	#127	1	1	yes	characteristic
94	C₄×D₁₃	—	2	2	no	—
95	C₄×D₁₃	—	2	2	no	—
96	C₄×D₁₃	#137	1	4	yes	—
97	Dic₂₆	—	4	4	no	—
98	#1057	—	2	4	no	—
99	#1057	—	8	2	no	—
100	C₈×C₁₃	#104	1	4	yes	—
101	#1055	—	2	2	no	—
102	#533	—	13	2	no	—
103	#500	—	13	1	no	—
104	#494	#261	1	1	yes	characteristic, complemented
105	#487	—	13	2	no	—
106	#467	—	26	2	no	—
107	C₄×Q₁₆	—	26	2	no	—
108	#423	—	13	2	no	—
109	C₂×#97	—	26	2	no	—
110	C₂×C₄×C₈	—	13	1	no	—
111	C₂×C₄×C₈	—	13	1	no	—
112	C₄³	—	13	1	no	—
113	#339	—	26	4	no	—
114	#334	—	26	2	no	—
115	Q₈⋊C₈	—	26	4	no	—
116	C₂²×C₁₃	#390	1	1	yes	characteristic
117	C₂×D₁₃	—	2	1	no	—
118	C₂×D₁₃	—	2	1	no	—
119	C₂×D₁₃	#470	1	2	yes	—
120	#261	—	4	4	no	—
121	C₄×C₁₃	—	8	2	no	—
122	C₄×C₁₃	C₂×#104	1	2	yes	—
123	C₄×C₁₃	#441	1	2	yes	—
124	C₄×C₁₃	#444	1	2	yes	—
125	Dic₁₃	—	2	1	no	—
126	Dic₁₃	—	2	1	no	—
127	Dic₁₃	—	8	2	no	—
128	Dic₁₃	#497	1	2	yes	—
129	C₂×Q₁₆	—	2	2	no	—
130	C₂×M₁₆	—	26	2	no	—
131	C₂²×C₈	—	13	2	no	—
132	C₂²×C₈	—	26	1	no	—
133	C₄⋊Q₈	—	13	2	no	—
134	C₄⋊Q₈	C₂×#261	1	2	yes	—
135	#125	—	26	2	no	—
136	C₂²⋊Q₈	—	26	2	no	—
137	C₂²⋊Q₈	—	26	2	no	—
138	C₄×Q₈	—	26	2	no	—
139	C₄×Q₈	—	26	2	no	—
140	#117	—	26	2	no	—
141	C₂×C₄²	—	13	1	no	—
142	C₂×C₄²	—	13	2	no	—
143	C₂×C₄²	—	26	1	no	—
144	C₂×C₄²	—	26	1	no	—
145	#107	—	26	2	no	—
146	C₄⋊C₈	—	26	2	no	—
147	C₄⋊C₈	—	26	2	no	—
148	#103	—	26	4	no	—
149	#97	—	26	8	no	—
150	C₄×C₈	—	13	2	no	—
151	C₄×C₈	—	13	2	no	—
152	C₄×C₈	—	13	1	no	—
153	C₄×C₈	—	26	1	no	—
154	C₄×C₈	C₂×#261	1	1	yes	characteristic
155	C₂×C₁₃	#1962	1	1	yes	characteristic
156	C₂×C₁₃	#1971	1	1	yes	characteristic
157	C₂×C₁₃	#1975	1	1	yes	characteristic
158	D₁₃	—	2	2	no	—
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	—	26	2	no	—
161	C₂²×C₄	—	13	1	no	—
162	C₂²×C₄	—	13	1	no	—
163	C₂²×C₄	—	13	1	no	—
164	C₂²×C₄	—	26	2	no	—
165	Q₁₆	—	4	4	no	—
166	M₁₆	—	52	4	no	—
167	C₂×C₈	—	13	2	no	—
168	C₂×C₈	—	26	2	no	—
169	C₂×C₈	—	26	4	no	—
170	C₂×C₈	—	26	4	no	—
171	C₂×C₈	—	52	2	no	—
172	C₂×C₈	C₂²⋊#261	1	2	yes	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	26	2	no	—
176	C₄⋊C₄	—	26	2	no	—
177	C₄⋊C₄	—	52	2	no	—
178	C₄⋊C₄	—	52	2	no	—
179	C₂²⋊C₄	—	52	2	no	—
180	C₄²	—	13	1	no	—
181	C₄²	—	13	1	no	—
182	C₄²	—	13	1	no	—
183	C₄²	—	26	4	no	—
184	C₄²	—	26	4	no	—
185	C₄²	—	52	2	no	—
186	C₄²	—	52	2	no	—
187	C₄²	C₂²×#261	1	1	yes	characteristic
188	C₁₃	#9850	1	1	yes	characteristic, complemented, Sylow
189	C₂³	—	13	1	no	—
190	Q₈	—	4	4	no	—
191	Q₈	—	52	4	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	13	1	no	—
194	C₂×C₄	—	13	4	no	—
195	C₂×C₄	—	13	1	no	—
196	C₂×C₄	—	13	1	no	—
197	C₂×C₄	—	26	2	no	—
198	C₂×C₄	—	26	2	no	—
199	C₂×C₄	—	26	4	no	—
200	C₂×C₄	—	52	2	no	—
201	C₂×C₄	—	52	2	no	—
202	C₂×C₄	—	52	2	no	—
203	C₂×C₄	D₄×#261	1	1	yes	characteristic
204	C₂×C₄	D₄×#261	1	1	yes	characteristic
205	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
206	C₈	—	26	4	no	—
207	C₈	—	26	2	no	—
208	C₈	—	104	2	no	—
209	C₈	#88116	1	4	yes	—
210	C₂²	—	13	2	no	—
211	C₂²	—	26	1	no	—
212	C₂²	—	26	1	no	—
213	C₂²	#11719151	1	1	yes	characteristic
214	C₄	—	8	2	no	—
215	C₄	—	13	2	no	—
216	C₄	—	26	1	no	—
217	C₄	—	26	1	no	—
218	C₄	—	52	4	no	—
219	C₄	—	104	2	no	—
220	C₄	#11719110	1	2	yes	—
221	C₄	#11719112	1	2	yes	—
222	C₄	C₂×#88116	1	2	yes	—
223	C₂	—	26	2	no	—
224	C₂	#13130970	1	1	yes	characteristic
225	C₂	#13130984	1	1	yes	characteristic
226	C₂	#13130990	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13131021

C₂

yes

maximal

C₄×#88100

C₂

yes

maximal, characteristic

#13130391

C₂

yes

maximal, characteristic

#13130186

C₂

yes

maximal

#13125420

C₂

yes

maximal, characteristic

#11719433

C₂²

yes

—

C₂×#88100

—

C₄²×#261

C₂²

yes

characteristic

#11719052

C₂²

yes

—

C₄×D₁₃⋊C₈

C₂²

yes

characteristic

C₁₃×#494

C₄

yes

characteristic, complemented

#11718849

—

Q₁₆×Dic₁₃

—

#11718742

C₄

yes

complemented

C₄×C₈×D₁₃

C₂²

yes

characteristic

#11718343

C₄

yes

characteristic, complemented

#11718253

—

#11718247

—

#11718244

—

C₂×C₄×#261

—

C₂×C₄×#261

—

C₂×C₄×#261

D₄

yes

—

C₂×#6118

—

C₂×D₁₃⋊C₈

—

C₂×C₁₃×Q₁₆

—

C₁₃×C₄⋊Q₈

C₂×C₄

yes

—

#88200

—

Q₈×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

#88075

—

#88058

—

C₈×Dic₁₃

D₄

yes

—

C₄×#1055

C₂×C₄

yes

characteristic

#9850

—

maximal, Sylow

C₂²×#261

—

C₂×Q₈×C₁₃

—

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×Q₁₆

—

C₂×C₈×C₁₃

C₂²⋊C₄

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

#6103

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

C₂×D₄

yes

characteristic

C₄×Dic₁₃

C₂×D₄

yes

C₄×Dic₁₃

yes

C₂²⋊C₄

yes

—

C₈×D₁₃

—

#3170

—

#1994

—

C₄×#97

—

#1566

—

#1493

—

C₂²×D₁₃

yes

—

—

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₄×D₄

yes

characteristic

C₂×C₄×C₁₃

C₄×D₄

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

yes

C₂×Dic₁₃

yes

C₂×Dic₁₃

yes

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

#137

yes

—

Dic₂₆

—

#1057

—

#1057

—

100

C₈×C₁₃

#104

yes

—

101

#1055

—

102

#533

—

103

#500

—

104

#494

#261

yes

characteristic, complemented

105

#487

—

106

#467

—

107

C₄×Q₁₆

—

108

#423

—

109

C₂×#97

—

110

C₂×C₄×C₈

—

111

C₂×C₄×C₈

—

112

C₄³

—

113

#339

—

114

#334

—

115

Q₈⋊C₈

—

116

C₂²×C₁₃

#390

yes

characteristic

117

C₂×D₁₃

—

118

C₂×D₁₃

—

119

C₂×D₁₃

#470

yes

—

120

#261

—

121

C₄×C₁₃

—

122

C₄×C₁₃

C₂×#104

yes

—

123

C₄×C₁₃

#441

yes

—

124

C₄×C₁₃

#444

yes

—

125

Dic₁₃

—

126

Dic₁₃

—

127

Dic₁₃

—

128

Dic₁₃

#497

yes

—

129

C₂×Q₁₆

—

130

C₂×M₁₆

—

131

C₂²×C₈

—

132

C₂²×C₈

—

133

C₄⋊Q₈

—

134

C₄⋊Q₈

C₂×#261

yes

—

135

#125

—

136

C₂²⋊Q₈

—

137

C₂²⋊Q₈

—

138

C₄×Q₈

—

139

C₄×Q₈

—

140

#117

—

141

C₂×C₄²

—

142

C₂×C₄²

—

143

C₂×C₄²

—

144

C₂×C₄²

—

145

#107

—

146

C₄⋊C₈

—

147

C₄⋊C₈

—

148

#103

—

149

#97

—

150

C₄×C₈

—

151

C₄×C₈

—

152

C₄×C₈

—

153

C₄×C₈

—

154

C₄×C₈

C₂×#261

yes

characteristic

155

C₂×C₁₃

#1962

yes

characteristic

156

C₂×C₁₃

#1971

yes

characteristic

157

C₂×C₁₃

#1975

yes

characteristic

158

D₁₃

—

159

C₂×Q₈

—

160

C₂×Q₈

—

161

C₂²×C₄

—

162

C₂²×C₄

—

163

C₂²×C₄

—

164

C₂²×C₄

—

165

Q₁₆

—

166

M₁₆

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

—

172

C₂×C₈

C₂²⋊#261

yes

—

173

C₄⋊C₄

—

174

C₄⋊C₄

—

175

C₄⋊C₄

—

176

C₄⋊C₄

—

177

C₄⋊C₄

—

178

C₄⋊C₄

—

179

C₂²⋊C₄

—

180

C₄²

—

181

C₄²

—

182

C₄²

—

183

C₄²

—

184

C₄²

—

185

C₄²

—

186

C₄²

—

187

C₄²

C₂²×#261

yes

characteristic

188

C₁₃

#9850

yes

characteristic, complemented, Sylow

189

C₂³

—

190

Q₈

—

191

Q₈

—

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

—

203

C₂×C₄

D₄×#261

yes

characteristic

204

C₂×C₄

D₄×#261

yes

characteristic

205

C₂×C₄

C₂×C₂²⋊#261

yes

characteristic

206

C₈

—

207

C₈

—

208

C₈

—

104

—

209

C₈

#88116

yes

—

210

C₂²

—

211

C₂²

—

212

C₂²

—

213

C₂²

#11719151

yes

characteristic

214

C₄

—

215

C₄

—

216

C₄

—

217

C₄

—

218

C₄

—

219

C₄

—

104

—

220

C₄

#11719110

yes

—

221

C₄

#11719112

yes

—

222

C₄

C₂×#88116

yes

—

223

C₂

—

224

C₂

#13130970

yes

characteristic

225

C₂

#13130984

yes

characteristic

226

C₂

#13130990

yes

characteristic

	subgroup	G/H
center	C₂²	#11719151
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₄²	C₂²×#261
Fitting	C₁₃×#494	C₄
socle	C₂²×C₁₃	#390
prime radical	C₂²×D₁₃	#121
cosocle	C₄²×D₁₃	C₂³
hypercenter	#494	#261
hypocenter	C₁₃	#9850
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#11719151

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₄²

C₂²×#261

Fitting

C₁₃×#494

C₄

C₂²×C₁₃

C₂²×D₁₃

C₄²×D₁₃

C₂³

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
11. C₁₃×#494	C₄	semidirect
14. #11718742	C₄	semidirect
16. #11718343	C₄	semidirect
104. #494	#261	semidirect
188. C₁₃	#9850	semidirect

H ≅ G/N

type

11. C₁₃×#494

C₄

semidirect

14. #11718742

C₄

semidirect

16. #11718343

C₄

semidirect

104. #494

#261

semidirect

188. C₁₃

#9850

semidirect

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

subgroup	index
#13131021	2
C₄×#88100	2
#13130391	2
#13130186	2
#13125420	2
#9850	13

subgroup

index

Subgroups — #16784987

Conjugacy classes of subgroups

Complements

Maximal subgroups