Group #16784954 — subgroups

Subgroups — #16784954

Order 3328.

494 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 116 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	#13131017	C₂	1	2	yes	maximal
2	#13130963	C₂	1	2	yes	maximal
3	#13130757	C₂	1	1	yes	maximal, characteristic
4	#13130032	C₂	1	1	yes	maximal, characteristic
5	D₁₃×#500	C₂	1	1	yes	maximal, characteristic
6	D₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂×#88117	—	2	4	no	—
8	#11719059	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, complemented
12	#11718739	C₄	1	2	yes	complemented
13	D₁₃×#107	—	2	2	no	—
14	#11718535	—	2	2	no	—
15	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
16	#11718343	C₄	1	1	yes	characteristic, complemented
17	#11718265	—	2	2	no	—
18	#1057⋊C₈	—	2	2	no	—
19	#11718246	—	2	4	no	—
20	C₂×Q₈×D₁₃	—	2	2	no	—
21	C₂×C₄⋊#261	—	2	2	no	—
22	C₂×C₄⋊#261	—	2	2	no	—
23	C₂×C₄⋊#261	—	2	2	no	—
24	C₂×D₁₃⋊C₈	D₄	1	2	yes	—
25	C₂×D₁₃⋊C₈	D₄	1	2	yes	—
26	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
27	#88198	—	2	2	no	—
28	C₂×C₈×D₁₃	D₄	1	2	yes	—
29	C₄⋊C₄×D₁₃	—	2	2	no	—
30	C₄⋊C₄×D₁₃	—	2	2	no	—
31	#88142	—	2	2	no	—
32	#88142	—	2	2	no	—
33	C₄²×D₁₃	C₂³	1	1	yes	characteristic
34	#88123	C₂×C₄	1	2	yes	—
35	#88117	—	4	8	no	—
36	C₄×#1057	—	2	1	no	—
37	C₄×#1057	—	2	1	no	—
38	C₄×#1057	—	2	1	no	—
39	C₄×#1057	—	2	1	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	#88047	—	2	4	no	—
45	C₄×#1055	C₂×C₄	1	1	yes	characteristic
46	#9834	—	13	1	no	maximal, Sylow
47	C₂²×#261	—	4	2	no	—
48	C₂×Q₈×C₁₃	—	2	2	no	—
49	Q₈×D₁₃	—	4	4	no	—
50	Q₈×D₁₃	—	4	4	no	—
51	C₂×C₄×D₁₃	—	4	2	no	—
52	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₂₆	—	2	2	no	—
56	C₂×Dic₂₆	—	4	2	no	—
57	C₂×#1057	—	2	2	no	—
58	C₂×#1057	—	2	2	no	—
59	C₄⋊#261	—	4	4	no	—
60	C₄⋊#261	—	4	4	no	—
61	C₄⋊#261	—	4	4	no	—
62	D₁₃⋊C₈	—	2	2	no	—
63	D₁₃⋊C₈	—	2	4	no	—
64	D₁₃⋊C₈	Q₁₆	1	4	yes	—
65	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
66	C₁₃×C₄⋊C₄	—	2	2	no	—
67	C₁₃×C₄⋊C₄	—	2	2	no	—
68	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₄⋊Dic₁₃	—	2	2	no	—
70	C₄⋊Dic₁₃	—	2	2	no	—
71	#6101	—	4	2	no	—
72	#6101	—	4	2	no	—
73	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
74	C₄×Dic₁₃	#41	1	1	yes	characteristic
75	C₄×Dic₁₃	#41	1	1	yes	characteristic
76	C₂×#1055	C₂²⋊C₄	1	2	yes	—
77	C₈×D₁₃	—	2	4	no	—
78	C₂×#500	—	13	1	no	—
79	#1990	—	13	2	no	—
80	#1956	—	13	2	no	—
81	#1834	—	13	1	no	—
82	C₂×#322	—	13	1	no	—
83	C₂²×D₁₃	#127	1	1	yes	characteristic
84	C₂×#261	—	8	2	no	—
85	C₂×#261	—	8	2	no	—
86	C₂×#261	—	8	2	no	—
87	Q₈×C₁₃	—	4	4	no	—
88	C₂×C₄×C₁₃	—	4	2	no	—
89	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
92	C₂×Dic₁₃	—	4	2	no	—
93	C₂×Dic₁₃	#121	1	1	yes	characteristic
94	C₂×Dic₁₃	#121	1	1	yes	characteristic
95	C₂×Dic₁₃	#124	1	1	yes	characteristic
96	C₄×D₁₃	—	8	2	no	—
97	C₄×D₁₃	—	8	2	no	—
98	C₄×D₁₃	C₂×Q₁₆	1	2	yes	—
99	C₄×D₁₃	#135	1	2	yes	—
100	C₄×D₁₃	AGL(1,ℤ/8)	1	4	yes	—
101	C₄×D₁₃	#137	1	4	yes	—
102	Dic₂₆	—	4	4	no	—
103	Dic₂₆	—	8	4	no	—
104	#1057	—	2	4	no	—
105	#1057	—	4	2	no	—
106	C₈×C₁₃	—	2	2	no	—
107	#1055	—	2	2	no	—
108	C₂×C₄⋊Q₈	—	13	2	no	—
109	#500	—	13	2	no	—
110	#500	—	13	1	no	—
111	#500	—	26	2	no	—
112	#500	#261	1	1	yes	characteristic, complemented
113	C₂×#107	—	26	2	no	—
114	C₂×#103	—	26	4	no	—
115	C₂×C₄×C₈	—	13	1	no	—
116	C₂×C₄×C₈	—	13	1	no	—
117	C₂×C₄×C₈	—	13	1	no	—
118	#384	—	13	2	no	—
119	#340	—	26	2	no	—
120	#340	—	26	4	no	—
121	#322	—	26	2	no	—
122	C₂²×C₁₃	#390	1	1	yes	characteristic
123	C₂×D₁₃	#494	1	1	yes	characteristic
124	C₂×D₁₃	#495	1	1	yes	characteristic
125	C₂×D₁₃	#496	1	2	yes	—
126	C₂×D₁₃	#497	1	2	yes	—
127	#261	—	16	2	no	—
128	C₄×C₁₃	—	8	2	no	—
129	C₄×C₁₃	#419	1	2	yes	—
130	C₄×C₁₃	C₄×Q₁₆	1	2	yes	—
131	C₄×C₁₃	#442	1	2	yes	—
132	Dic₁₃	—	8	2	no	—
133	Dic₁₃	#467	1	2	yes	—
134	Dic₁₃	#469	1	2	yes	—
135	Dic₁₃	#491	1	2	yes	—
136	C₂²×Q₈	—	26	2	no	—
137	C₂²×C₈	—	13	2	no	—
138	C₂²×C₈	—	13	2	no	—
139	C₂²×C₈	—	13	2	no	—
140	C₄⋊Q₈	—	13	2	no	—
141	C₄⋊Q₈	—	26	2	no	—
142	C₄⋊Q₈	—	26	2	no	—
143	C₄⋊Q₈	—	26	2	no	—
144	C₄⋊Q₈	C₂×#261	1	2	yes	—
145	C₂×C₄⋊C₄	—	26	2	no	—
146	C₂×C₄⋊C₄	—	26	2	no	—
147	C₂×C₄⋊C₄	—	26	2	no	—
148	C₂×C₄⋊C₄	—	26	2	no	—
149	C₂×C₄⋊C₄	—	26	2	no	—
150	C₂×C₄²	—	13	1	no	—
151	#107	—	2	2	no	—
152	#107	—	26	4	no	—
153	#107	—	26	2	no	—
154	#103	—	52	8	no	—
155	C₄×C₈	—	13	2	no	—
156	C₄×C₈	—	13	1	no	—
157	C₄×C₈	—	26	1	no	—
158	C₄×C₈	—	26	1	no	—
159	C₄×C₈	—	26	1	no	—
160	C₄×C₈	—	26	1	no	—
161	C₄×C₈	C₂×#261	1	1	yes	characteristic
162	C₂×C₁₃	#1960	1	1	yes	characteristic
163	C₂×C₁₃	#1969	1	1	yes	characteristic
164	C₂×C₁₃	#1972	1	1	yes	characteristic
165	D₁₃	#1719	1	2	yes	—
166	D₁₃	#1723	1	2	yes	—
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	—	26	2	no	—
169	C₂×Q₈	—	52	2	no	—
170	C₂×Q₈	—	52	4	no	—
171	C₂×Q₈	—	52	4	no	—
172	C₂²×C₄	—	13	1	no	—
173	C₂²×C₄	—	13	1	no	—
174	C₂²×C₄	—	13	1	no	—
175	C₂²×C₄	—	52	2	no	—
176	C₂²×C₄	—	52	2	no	—
177	C₂×C₈	—	13	4	no	—
178	C₂×C₈	—	13	2	no	—
179	C₂×C₈	—	26	4	no	—
180	C₂×C₈	—	26	2	no	—
181	C₂×C₈	—	26	2	no	—
182	C₂×C₈	—	26	2	no	—
183	C₂×C₈	—	26	4	no	—
184	C₂×C₈	C₂²⋊#261	1	2	yes	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	—	26	2	no	—
188	C₄⋊C₄	—	26	2	no	—
189	C₄⋊C₄	—	52	2	no	—
190	C₄⋊C₄	—	52	2	no	—
191	C₄⋊C₄	—	52	4	no	—
192	C₄⋊C₄	—	52	4	no	—
193	C₄⋊C₄	—	52	4	no	—
194	C₄²	—	13	1	no	—
195	C₄²	—	13	1	no	—
196	C₄²	—	13	1	no	—
197	C₄²	C₂²×#261	1	1	yes	characteristic
198	C₁₃	#9834	1	1	yes	characteristic, complemented, Sylow
199	C₂³	—	13	1	no	—
200	Q₈	—	4	4	no	—
201	Q₈	—	52	4	no	—
202	Q₈	—	104	4	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	13	1	no	—
205	C₂×C₄	—	13	1	no	—
206	C₂×C₄	—	13	1	no	—
207	C₂×C₄	—	13	4	no	—
208	C₂×C₄	—	13	2	no	—
209	C₂×C₄	—	13	2	no	—
210	C₂×C₄	—	13	4	no	—
211	C₂×C₄	—	52	2	no	—
212	C₂×C₄	—	104	2	no	—
213	C₂×C₄	—	104	2	no	—
214	C₂×C₄	—	104	2	no	—
215	C₂×C₄	—	104	2	no	—
216	C₂×C₄	—	104	2	no	—
217	C₂×C₄	D₄×#261	1	1	yes	characteristic
218	C₂×C₄	D₄×#261	1	1	yes	characteristic
219	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	—	26	4	no	—
222	C₈	—	26	2	no	—
223	C₈	—	52	2	no	—
224	C₂²	—	13	2	no	—
225	C₂²	—	13	1	no	—
226	C₂²	—	13	1	no	—
227	C₂²	—	13	2	no	—
228	C₂²	#11719151	1	1	yes	characteristic
229	C₄	—	8	2	no	—
230	C₄	—	13	2	no	—
231	C₄	—	13	2	no	—
232	C₄	—	13	2	no	—
233	C₄	—	104	2	no	—
234	C₄	—	208	2	no	—
235	C₄	#11719102	1	2	yes	—
236	C₄	Q₁₆×#261	1	2	yes	—
237	C₄	#11719153	1	2	yes	—
238	C₂	—	13	2	no	—
239	C₂	—	13	2	no	—
240	C₂	#13130967	1	1	yes	characteristic
241	C₂	#13130982	1	1	yes	characteristic
242	C₂	#13130985	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13131017

C₂

yes

maximal

#13130963

C₂

yes

maximal

#13130757

C₂

yes

maximal, characteristic

#13130032

C₂

yes

maximal, characteristic

D₁₃×#500

C₂

yes

maximal, characteristic

D₁₃×C₄⋊Q₈

C₂²

yes

—

C₂×#88117

—

#11719059

C₂²

yes

—

C₄×D₁₃⋊C₈

C₂²

yes

characteristic

C₄×D₁₃⋊C₈

C₂²

yes

characteristic

C₁₃×#500

C₄

yes

characteristic, complemented

#11718739

C₄

yes

complemented

D₁₃×#107

—

#11718535

—

C₄×C₈×D₁₃

C₂²

yes

characteristic

#11718343

C₄

yes

characteristic, complemented

#11718265

—

#1057⋊C₈

—

#11718246

—

C₂×Q₈×D₁₃

—

C₂×C₄⋊#261

—

C₂×C₄⋊#261

—

C₂×C₄⋊#261

—

C₂×D₁₃⋊C₈

D₄

yes

—

C₂×D₁₃⋊C₈

D₄

yes

—

C₁₃×C₄⋊Q₈

C₂×C₄

yes

—

#88198

—

C₂×C₈×D₁₃

D₄

yes

—

C₄⋊C₄×D₁₃

—

C₄⋊C₄×D₁₃

—

#88142

—

#88142

—

C₄²×D₁₃

C₂³

yes

characteristic

#88123

C₂×C₄

yes

—

—

—

—

—

—

C₁₃×#107

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

#88058

—

C₈×Dic₁₃

D₄

yes

—

#88047

—

C₄×#1055

C₂×C₄

yes

characteristic

#9834

—

maximal, Sylow

C₂²×#261

—

C₂×Q₈×C₁₃

—

Q₈×D₁₃

—

Q₈×D₁₃

—

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

C₂×D₄

yes

characteristic

C₂×C₄×D₁₃

C₂×D₄

yes

characteristic

C₂×C₄×D₁₃

C₂×D₄

yes

characteristic

C₂×Dic₂₆

—

C₂×Dic₂₆

—

—

—

—

—

—

D₁₃⋊C₈

—

D₁₃⋊C₈

—

D₁₃⋊C₈

Q₁₆

yes

—

C₂×C₈×C₁₃

C₂²⋊C₄

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

#6101

—

C₄×Dic₁₃

C₂×D₄

yes

C₄×Dic₁₃

yes

C₄×Dic₁₃

yes

C₂²⋊C₄

yes

—

C₈×D₁₃

—

—

—

—

—

—

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

C₂×Q₁₆

yes

—

C₄×D₁₃

#135

yes

—

100

C₄×D₁₃

AGL(1,ℤ/8)

yes

—

101

C₄×D₁₃

#137

yes

—

102

Dic₂₆

—

103

Dic₂₆

—

104

#1057

—

105

#1057

—

106

C₈×C₁₃

—

107

#1055

—

108

C₂×C₄⋊Q₈

—

109

#500

—

110

#500

—

111

#500

—

112

#500

#261

yes

characteristic, complemented

113

C₂×#107

—

114

C₂×#103

—

115

C₂×C₄×C₈

—

116

C₂×C₄×C₈

—

117

C₂×C₄×C₈

—

118

#384

—

119

#340

—

120

#340

—

121

#322

—

122

C₂²×C₁₃

#390

yes

characteristic

123

C₂×D₁₃

#494

yes

characteristic

124

C₂×D₁₃

#495

yes

characteristic

125

C₂×D₁₃

#496

yes

—

126

C₂×D₁₃

#497

yes

—

127

#261

—

128

C₄×C₁₃

—

129

C₄×C₁₃

#419

yes

—

130

C₄×C₁₃

C₄×Q₁₆

yes

—

131

C₄×C₁₃

#442

yes

—

132

Dic₁₃

—

133

Dic₁₃

#467

yes

—

134

Dic₁₃

#469

yes

—

135

Dic₁₃

#491

yes

—

136

C₂²×Q₈

—

137

C₂²×C₈

—

138

C₂²×C₈

—

139

C₂²×C₈

—

140

C₄⋊Q₈

—

141

C₄⋊Q₈

—

142

C₄⋊Q₈

—

143

C₄⋊Q₈

—

144

C₄⋊Q₈

C₂×#261

yes

—

145

C₂×C₄⋊C₄

—

146

C₂×C₄⋊C₄

—

147

C₂×C₄⋊C₄

—

148

C₂×C₄⋊C₄

—

149

C₂×C₄⋊C₄

—

150

C₂×C₄²

—

151

#107

—

152

#107

—

153

#107

—

154

#103

—

155

C₄×C₈

—

156

C₄×C₈

—

157

C₄×C₈

—

158

C₄×C₈

—

159

C₄×C₈

—

160

C₄×C₈

—

161

C₄×C₈

C₂×#261

yes

characteristic

162

C₂×C₁₃

#1960

yes

characteristic

163

C₂×C₁₃

#1969

yes

characteristic

164

C₂×C₁₃

#1972

yes

characteristic

165

D₁₃

#1719

yes

—

166

D₁₃

#1723

yes

—

167

C₂×Q₈

—

168

C₂×Q₈

—

169

C₂×Q₈

—

170

C₂×Q₈

—

171

C₂×Q₈

—

172

C₂²×C₄

—

173

C₂²×C₄

—

174

C₂²×C₄

—

175

C₂²×C₄

—

176

C₂²×C₄

—

177

C₂×C₈

—

178

C₂×C₈

—

179

C₂×C₈

—

180

C₂×C₈

—

181

C₂×C₈

—

182

C₂×C₈

—

183

C₂×C₈

—

184

C₂×C₈

C₂²⋊#261

yes

—

185

C₄⋊C₄

—

186

C₄⋊C₄

—

187

C₄⋊C₄

—

188

C₄⋊C₄

—

189

C₄⋊C₄

—

190

C₄⋊C₄

—

191

C₄⋊C₄

—

192

C₄⋊C₄

—

193

C₄⋊C₄

—

194

C₄²

—

195

C₄²

—

196

C₄²

—

197

C₄²

C₂²×#261

yes

characteristic

198

C₁₃

#9834

yes

characteristic, complemented, Sylow

199

C₂³

—

200

Q₈

—

201

Q₈

—

202

Q₈

—

104

—

203

C₂×C₄

—

204

C₂×C₄

—

205

C₂×C₄

—

206

C₂×C₄

—

207

C₂×C₄

—

208

C₂×C₄

—

209

C₂×C₄

—

210

C₂×C₄

—

211

C₂×C₄

—

212

C₂×C₄

—

104

—

213

C₂×C₄

—

104

—

214

C₂×C₄

—

104

—

215

C₂×C₄

—

104

—

216

C₂×C₄

—

104

—

217

C₂×C₄

D₄×#261

yes

characteristic

218

C₂×C₄

D₄×#261

yes

characteristic

219

C₂×C₄

C₂×C₂²⋊#261

yes

characteristic

220

C₈

—

221

C₈

—

222

C₈

—

223

C₈

—

224

C₂²

—

225

C₂²

—

226

C₂²

—

227

C₂²

—

228

C₂²

#11719151

yes

characteristic

229

C₄

—

230

C₄

—

231

C₄

—

232

C₄

—

233

C₄

—

104

—

234

C₄

—

208

—

235

C₄

#11719102

yes

—

236

C₄

Q₁₆×#261

yes

—

237

C₄

#11719153

yes

—

238

C₂

—

239

C₂

—

240

C₂

#13130967

yes

characteristic

241

C₂

#13130982

yes

characteristic

242

C₂

#13130985

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	#127
cosocle	C₄²×D₁₃	C₂³
hypercenter	#500	#261
hypocenter	C₁₃	#9834
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
11. C₁₃×#500	C₄	semidirect
12. #11718739	C₄	semidirect
16. #11718343	C₄	semidirect
112. #500	#261	semidirect
198. C₁₃	#9834	semidirect

H ≅ G/N

type

11. C₁₃×#500

C₄

semidirect

12. #11718739

C₄

semidirect

16. #11718343

C₄

semidirect

112. #500

#261

semidirect

198. C₁₃

#9834

semidirect

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

subgroup	index
#13131017	2
#13130963	2
#13130757	2
#13130032	2
D₁₃×#500	2
#9834	13

subgroup

index

D₁₃×#500

Subgroups — #16784954

Conjugacy classes of subgroups

Complements

Maximal subgroups