Group #16785020 — subgroups

Subgroups — #16785020

Order 3328.

406 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 112 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	#13130186	C₂	1	2	yes	maximal
3	#13130185	C₂	1	2	yes	maximal
4	#13130030	C₂	1	1	yes	maximal, characteristic
5	#13125347	C₂	1	1	yes	maximal, characteristic
6	#11719433	C₂²	1	2	yes	—
7	C₂×#88103	—	2	2	no	—
8	#11719052	C₂²	1	2	yes	—
9	C₄×D₁₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₄×D₁₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₁₃×#500	C₄	1	1	yes	characteristic
12	#11718740	C₄	1	2	yes	—
13	#11718541	—	2	2	no	—
14	#11718539	—	2	2	no	—
15	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
16	#11718343	C₄	1	1	yes	characteristic
17	Q₈⋊#1057	—	2	4	no	—
18	#11718251	—	2	2	no	—
19	#11718244	—	2	4	no	—
20	C₂×#6118	—	2	2	no	—
21	C₂×D₁₃⋊C₈	—	2	1	no	—
22	C₂×D₁₃⋊C₈	—	2	1	no	—
23	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
24	#88200	—	2	2	no	—
25	Q₈×Dic₁₃	—	2	2	no	—
26	C₂×C₈×D₁₃	D₄	1	2	yes	—
27	#88151	—	2	2	no	—
28	#88146	—	2	2	no	—
29	#88144	—	2	2	no	—
30	#88141	—	2	2	no	—
31	C₄²×D₁₃	C₂³	1	1	yes	characteristic
32	#88123	C₂×C₄	1	2	yes	—
33	C₄⋊#1057	—	2	2	no	—
34	C₄⋊#1057	—	2	2	no	—
35	C₄×#1057	—	2	1	no	—
36	C₄×#1057	—	2	1	no	—
37	C₄×#1057	D₄	1	2	yes	—
38	C₄×#1057	D₄	1	2	yes	—
39	#88103	—	4	4	no	—
40	C₁₃×#107	—	2	2	no	—
41	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	#88058	—	2	2	no	—
43	C₈×Dic₁₃	D₄	1	2	yes	—
44	#88048	—	2	4	no	—
45	C₄×#1055	C₂×C₄	1	1	yes	characteristic
46	#9861	—	13	1	no	maximal, Sylow
47	C₂×Q₈×C₁₃	—	2	2	no	—
48	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
50	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
51	C₂×Dic₂₆	—	2	2	no	—
52	C₂×#1057	—	2	2	no	—
53	C₂×#1057	—	2	2	no	—
54	C₂×#1057	—	4	2	no	—
55	#6118	—	4	4	no	—
56	D₁₃⋊C₈	—	4	2	no	—
57	D₁₃⋊C₈	—	4	2	no	—
58	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
59	C₁₃×C₄⋊C₄	—	2	2	no	—
60	C₁₃×C₄⋊C₄	—	2	2	no	—
61	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	#6103	—	4	2	no	—
63	C₄⋊Dic₁₃	—	2	2	no	—
64	C₄⋊Dic₁₃	—	2	2	no	—
65	C₄⋊Dic₁₃	—	4	2	no	—
66	#6101	—	4	2	no	—
67	C₄×Dic₁₃	—	4	2	no	—
68	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
69	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
70	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂×#1055	C₂²⋊C₄	1	2	yes	—
72	C₈×D₁₃	SD₁₆	1	4	yes	—
73	C₈×D₁₃	Q₁₆	1	4	yes	—
74	#3160	—	13	1	no	—
75	#1848	—	13	1	no	—
76	#1493	—	13	2	no	—
77	#1493	—	13	2	no	—
78	C₂×#322	—	13	1	no	—
79	C₂²×D₁₃	#124	1	1	yes	characteristic
80	Q₈×C₁₃	—	4	4	no	—
81	C₂×C₄×C₁₃	—	4	2	no	—
82	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
85	C₂×Dic₁₃	—	4	2	no	—
86	C₂×Dic₁₃	#121	1	1	yes	characteristic
87	C₂×Dic₁₃	#121	1	1	yes	characteristic
88	C₂×Dic₁₃	#127	1	1	yes	characteristic
89	C₄×D₁₃	—	2	2	no	—
90	C₄×D₁₃	—	2	2	no	—
91	C₄×D₁₃	C₂×SD₁₆	1	2	yes	—
92	C₄×D₁₃	C₂×Q₁₆	1	2	yes	—
93	Dic₂₆	—	4	4	no	—
94	#1057	—	4	2	no	—
95	#1057	—	4	2	no	—
96	#1057	—	8	2	no	—
97	C₈×C₁₃	#103	1	4	yes	—
98	#1055	#103	1	4	yes	—
99	#533	—	13	2	no	—
100	#500	—	13	1	no	—
101	#500	#261	1	1	yes	characteristic
102	#499	—	26	2	no	—
103	#498	—	13	2	no	—
104	C₂×#108	—	26	2	no	—
105	#427	—	26	2	no	—
106	#423	—	13	2	no	—
107	C₂×C₄×C₈	—	13	1	no	—
108	C₂×C₄×C₈	—	13	1	no	—
109	C₂×C₄×C₈	—	13	1	no	—
110	Q₈⋊C₈	—	26	4	no	—
111	Q₈⋊C₈	—	26	4	no	—
112	#322	—	26	2	no	—
113	C₂²×C₁₃	#390	1	1	yes	characteristic
114	C₂×D₁₃	—	2	1	no	—
115	C₂×D₁₃	—	2	1	no	—
116	C₂×D₁₃	#487	1	2	yes	—
117	C₄×C₁₃	—	8	2	no	—
118	C₄×C₁₃	C₂×#103	1	2	yes	—
119	C₄×C₁₃	#444	1	2	yes	—
120	C₄×C₁₃	#444	1	2	yes	—
121	Dic₁₃	—	2	1	no	—
122	Dic₁₃	—	2	1	no	—
123	Dic₁₃	—	8	2	no	—
124	Dic₁₃	#492	1	1	yes	characteristic
125	Dic₁₃	#494	1	1	yes	characteristic
126	C₂×M₁₆	—	26	2	no	—
127	C₂²×C₈	—	13	2	no	—
128	C₂²×C₈	—	26	1	no	—
129	C₂²×C₈	—	26	1	no	—
130	C₄⋊Q₈	—	13	2	no	—
131	C₄⋊Q₈	C₂×#261	1	2	yes	—
132	#125	—	26	2	no	—
133	C₂²⋊Q₈	—	26	2	no	—
134	C₂²⋊Q₈	—	26	2	no	—
135	C₄×Q₈	—	26	2	no	—
136	C₄×Q₈	—	26	2	no	—
137	#117	—	26	2	no	—
138	C₂×C₄²	—	13	1	no	—
139	#108	—	52	4	no	—
140	#107	—	2	2	no	—
141	#107	—	26	2	no	—
142	#106	—	26	4	no	—
143	C₄⋊C₈	—	26	2	no	—
144	C₄⋊C₈	—	26	2	no	—
145	C₄×C₈	—	13	2	no	—
146	C₄×C₈	—	13	2	no	—
147	C₄×C₈	—	13	2	no	—
148	C₄×C₈	—	13	1	no	—
149	C₄×C₈	—	26	1	no	—
150	C₄×C₈	—	26	1	no	—
151	C₄×C₈	C₂×#261	1	1	yes	characteristic
152	C₂×C₁₃	#1956	1	1	yes	characteristic
153	C₂×C₁₃	#1975	1	1	yes	characteristic
154	C₂×C₁₃	#1975	1	1	yes	characteristic
155	D₁₃	—	2	2	no	—
156	C₂×Q₈	—	2	2	no	—
157	C₂×Q₈	—	26	2	no	—
158	C₂²×C₄	—	13	1	no	—
159	C₂²×C₄	—	13	1	no	—
160	C₂²×C₄	—	13	1	no	—
161	M₁₆	—	52	4	no	—
162	C₂×C₈	—	13	2	no	—
163	C₂×C₈	—	13	4	no	—
164	C₂×C₈	—	13	4	no	—
165	C₂×C₈	—	26	2	no	—
166	C₂×C₈	—	26	2	no	—
167	C₂×C₈	—	52	2	no	—
168	C₂×C₈	—	52	2	no	—
169	C₂×C₈	—	52	2	no	—
170	C₂×C₈	C₂²⋊#261	1	2	yes	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	26	2	no	—
174	C₄⋊C₄	—	26	2	no	—
175	C₄⋊C₄	—	52	2	no	—
176	C₄⋊C₄	—	52	2	no	—
177	C₂²⋊C₄	—	52	2	no	—
178	C₄²	—	13	1	no	—
179	C₄²	—	13	1	no	—
180	C₄²	—	13	1	no	—
181	C₄²	—	52	2	no	—
182	C₄²	C₂²×#261	1	1	yes	characteristic
183	C₁₃	#9861	1	1	yes	characteristic, complemented, Sylow
184	C₂³	—	13	1	no	—
185	Q₈	—	4	4	no	—
186	Q₈	—	52	4	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	13	1	no	—
189	C₂×C₄	—	13	2	no	—
190	C₂×C₄	—	13	2	no	—
191	C₂×C₄	—	13	1	no	—
192	C₂×C₄	—	13	1	no	—
193	C₂×C₄	—	26	2	no	—
194	C₂×C₄	—	26	2	no	—
195	C₂×C₄	—	52	2	no	—
196	C₂×C₄	D₄×#261	1	1	yes	characteristic
197	C₂×C₄	D₄×#261	1	1	yes	characteristic
198	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
199	C₈	—	13	4	no	—
200	C₈	—	52	2	no	—
201	C₈	—	52	2	no	—
202	C₈	—	104	2	no	—
203	C₈	#88117	1	4	yes	—
204	C₂²	—	13	2	no	—
205	C₂²	—	26	1	no	—
206	C₂²	—	26	1	no	—
207	C₂²	#11719151	1	1	yes	characteristic
208	C₄	—	8	2	no	—
209	C₄	—	13	1	no	—
210	C₄	—	13	1	no	—
211	C₄	—	26	1	no	—
212	C₄	—	26	1	no	—
213	C₄	—	104	2	no	—
214	C₄	#11719104	1	2	yes	—
215	C₄	#11719112	1	2	yes	—
216	C₄	C₂×#88117	1	2	yes	—
217	C₂	—	26	2	no	—
218	C₂	#13130963	1	1	yes	characteristic
219	C₂	#13130990	1	1	yes	characteristic
220	C₂	#13130991	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13130775

C₂

yes

maximal, characteristic

#13130186

C₂

yes

maximal

#13130185

C₂

yes

maximal

#13130030

C₂

yes

maximal, characteristic

#13125347

C₂

yes

maximal, characteristic

#11719433

C₂²

yes

—

C₂×#88103

—

#11719052

C₂²

yes

—

C₄×D₁₃⋊C₈

C₂²

yes

characteristic

C₄×D₁₃⋊C₈

C₂²

yes

characteristic

C₁₃×#500

C₄

yes

characteristic

#11718740

C₄

yes

—

#11718541

—

#11718539

—

C₄×C₈×D₁₃

C₂²

yes

characteristic

#11718343

C₄

yes

—

—

—

—

C₂×D₁₃⋊C₈

—

C₂×D₁₃⋊C₈

—

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

—

C₄×#1057

D₄

yes

—

#88103

—

C₁₃×#107

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

#88058

—

C₈×Dic₁₃

D₄

yes

—

#88048

—

C₄×#1055

C₂×C₄

yes

characteristic

#9861

—

maximal, Sylow

C₂×Q₈×C₁₃

—

C₂×C₄×D₁₃

C₂×D₄

yes

C₂×C₄×D₁₃

yes

C₂×C₄×D₁₃

yes

C₂×Dic₂₆

—

—

—

—

—

D₁₃⋊C₈

—

D₁₃⋊C₈

—

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

characteristic

C₄×Dic₁₃

C₂×D₄

yes

characteristic

C₂×#1055

C₂²⋊C₄

yes

—

C₈×D₁₃

SD₁₆

yes

—

C₈×D₁₃

Q₁₆

yes

—

—

—

—

—

—

C₂²×D₁₃

#124

yes

characteristic

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

C₂×SD₁₆

yes

—

C₄×D₁₃

C₂×Q₁₆

yes

—

Dic₂₆

—

#1057

—

#1057

—

#1057

—

C₈×C₁₃

#103

yes

—

#1055

#103

yes

—

#533

—

100

#500

—

101

yes

102

—

103

#498

—

104

C₂×#108

—

105

#427

—

106

#423

—

107

C₂×C₄×C₈

—

108

C₂×C₄×C₈

—

109

C₂×C₄×C₈

—

110

Q₈⋊C₈

—

111

Q₈⋊C₈

—

112

#322

—

113

C₂²×C₁₃

#390

yes

characteristic

114

C₂×D₁₃

—

115

C₂×D₁₃

—

116

C₂×D₁₃

#487

yes

—

117

C₄×C₁₃

—

118

C₄×C₁₃

C₂×#103

yes

—

119

C₄×C₁₃

#444

yes

—

120

C₄×C₁₃

#444

yes

—

121

Dic₁₃

—

122

Dic₁₃

—

123

Dic₁₃

—

124

Dic₁₃

#492

yes

characteristic

125

Dic₁₃

#494

yes

characteristic

126

C₂×M₁₆

—

127

C₂²×C₈

—

128

C₂²×C₈

—

129

C₂²×C₈

—

130

C₄⋊Q₈

—

131

C₄⋊Q₈

C₂×#261

yes

—

132

#125

—

133

C₂²⋊Q₈

—

134

C₂²⋊Q₈

—

135

C₄×Q₈

—

136

C₄×Q₈

—

137

#117

—

138

C₂×C₄²

—

139

#108

—

140

#107

—

141

#107

—

142

#106

—

143

C₄⋊C₈

—

144

C₄⋊C₈

—

145

C₄×C₈

—

146

C₄×C₈

—

147

C₄×C₈

—

148

C₄×C₈

—

149

C₄×C₈

—

150

C₄×C₈

—

151

C₄×C₈

C₂×#261

yes

characteristic

152

C₂×C₁₃

#1956

yes

characteristic

153

C₂×C₁₃

#1975

yes

characteristic

154

C₂×C₁₃

#1975

yes

characteristic

155

D₁₃

—

156

C₂×Q₈

—

157

C₂×Q₈

—

158

C₂²×C₄

—

159

C₂²×C₄

—

160

C₂²×C₄

—

161

M₁₆

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

C₂²⋊#261

yes

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

—

175

C₄⋊C₄

—

176

C₄⋊C₄

—

177

C₂²⋊C₄

—

178

C₄²

—

179

C₄²

—

180

C₄²

—

181

C₄²

—

182

C₄²

C₂²×#261

yes

characteristic

183

C₁₃

#9861

yes

characteristic, complemented, Sylow

184

C₂³

—

185

Q₈

—

186

Q₈

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

D₄×#261

yes

characteristic

197

C₂×C₄

D₄×#261

yes

characteristic

198

C₂×C₄

C₂×C₂²⋊#261

yes

characteristic

199

C₈

—

200

C₈

—

201

C₈

—

202

C₈

—

104

—

203

C₈

#88117

yes

—

204

C₂²

—

205

C₂²

—

206

C₂²

—

207

C₂²

#11719151

yes

characteristic

208

C₄

—

209

C₄

—

210

C₄

—

211

C₄

—

212

C₄

—

213

C₄

—

104

—

214

C₄

#11719104

yes

—

215

C₄

#11719112

yes

—

216

C₄

C₂×#88117

yes

—

217

C₂

—

218

C₂

#13130963

yes

characteristic

219

C₂

#13130990

yes

characteristic

220

C₂

#13130991

yes

characteristic

	subgroup	G/H
center	C₂²	#11719151
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₄²	C₂²×#261
Fitting	C₁₃×#500	C₄
socle	C₂²×C₁₃	#390
prime radical	C₂²×D₁₃	#124
cosocle	C₄²×D₁₃	C₂³
hypercenter	#500	#261
hypocenter	C₁₃	#9861
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

#11719151

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₄²

C₂²×#261

Fitting

C₁₃×#500

C₄

C₂²×C₁₃

C₂²×D₁₃

C₄²×D₁₃

C₂³

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
183. C₁₃	#9861	semidirect

H ≅ G/N

type

183. C₁₃

#9861

semidirect

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

subgroup	index
#13130775	2
#13130186	2
#13130185	2
#13130030	2
#13125347	2
#9861	13

subgroup

index

Subgroups — #16785020

Conjugacy classes of subgroups

Complements

Maximal subgroups