Group #16784956 — subgroups

Subgroups — #16784956

Order 3328.

510 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 132 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	#13130771	C₂	1	1	yes	maximal, characteristic
4	#13130030	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	C₂×#88101	—	2	2	no	—
9	#11719059	C₂²	1	2	yes	—
10	C₄×D₁₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₄×D₁₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₁₃×#500	C₄	1	1	yes	characteristic, complemented
13	#11718739	C₄	1	2	yes	complemented
14	D₁₃×#107	—	2	2	no	—
15	#11718535	—	2	2	no	—
16	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
17	#11718343	C₄	1	1	yes	characteristic, complemented
18	#11718251	—	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	#88101	—	4	4	no	—
41	C₁₃×#107	—	2	2	no	—
42	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
43	#88058	—	2	2	no	—
44	C₈×Dic₁₃	D₄	1	2	yes	—
45	#88047	—	2	4	no	—
46	C₄×#1055	C₂×C₄	1	1	yes	characteristic
47	#9836	—	13	1	no	maximal, Sylow
48	C₂²×#261	—	4	2	no	—
49	C₂×Q₈×C₁₃	—	2	2	no	—
50	Q₈×D₁₃	—	4	4	no	—
51	Q₈×D₁₃	—	4	4	no	—
52	C₂×C₄×D₁₃	—	4	2	no	—
53	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
56	C₂×Dic₂₆	—	2	2	no	—
57	C₂×Dic₂₆	—	4	2	no	—
58	C₂×#1057	—	2	2	no	—
59	C₂×#1057	—	2	2	no	—
60	C₄⋊#261	—	4	4	no	—
61	C₄⋊#261	—	4	4	no	—
62	C₄⋊#261	—	4	4	no	—
63	D₁₃⋊C₈	—	2	4	no	—
64	D₁₃⋊C₈	—	2	4	no	—
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	SD₁₆	1	4	yes	—
78	C₈×D₁₃	Q₁₆	1	4	yes	—
79	C₂×#500	—	13	1	no	—
80	#1990	—	13	2	no	—
81	#1956	—	13	2	no	—
82	#1844	—	13	1	no	—
83	C₂×#322	—	13	1	no	—
84	C₂²×D₁₃	#127	1	1	yes	characteristic
85	C₂×#261	—	8	2	no	—
86	C₂×#261	—	8	2	no	—
87	C₂×#261	—	8	2	no	—
88	Q₈×C₁₃	—	4	4	no	—
89	C₂×C₄×C₁₃	—	4	2	no	—
90	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
93	C₂×Dic₁₃	—	4	2	no	—
94	C₂×Dic₁₃	#121	1	1	yes	characteristic
95	C₂×Dic₁₃	#121	1	1	yes	characteristic
96	C₂×Dic₁₃	#124	1	1	yes	characteristic
97	C₄×D₁₃	—	8	2	no	—
98	C₄×D₁₃	—	8	2	no	—
99	C₄×D₁₃	C₂×SD₁₆	1	2	yes	—
100	C₄×D₁₃	C₂×Q₁₆	1	2	yes	—
101	C₄×D₁₃	AGL(1,ℤ/8)	1	4	yes	—
102	C₄×D₁₃	#137	1	4	yes	—
103	Dic₂₆	—	4	4	no	—
104	Dic₂₆	—	8	4	no	—
105	#1057	—	4	2	no	—
106	#1057	—	4	2	no	—
107	C₈×C₁₃	#103	1	4	yes	complemented
108	#1055	#103	1	4	yes	complemented
109	C₂×C₄⋊Q₈	—	13	2	no	—
110	#500	—	13	2	no	—
111	#500	—	13	1	no	—
112	#500	—	26	2	no	—
113	#500	#261	1	1	yes	characteristic, complemented
114	C₂×#107	—	26	2	no	—
115	C₂×#106	—	26	2	no	—
116	C₂×#103	—	26	4	no	—
117	C₂×C₄×C₈	—	13	1	no	—
118	C₂×C₄×C₈	—	13	1	no	—
119	C₂×C₄×C₈	—	13	1	no	—
120	#384	—	13	2	no	—
121	#340	—	26	4	no	—
122	#322	—	26	2	no	—
123	C₂²×C₁₃	#390	1	1	yes	characteristic
124	C₂×D₁₃	#492	1	1	yes	characteristic
125	C₂×D₁₃	#494	1	1	yes	characteristic
126	C₂×D₁₃	#496	1	2	yes	—
127	C₂×D₁₃	#497	1	2	yes	—
128	#261	—	16	2	no	—
129	C₄×C₁₃	—	8	2	no	—
130	C₄×C₁₃	C₂×#103	1	2	yes	—
131	C₄×C₁₃	#441	1	2	yes	—
132	C₄×C₁₃	#442	1	2	yes	—
133	Dic₁₃	—	8	2	no	—
134	Dic₁₃	#469	1	2	yes	—
135	Dic₁₃	#470	1	2	yes	—
136	Dic₁₃	#487	1	2	yes	—
137	C₂²×Q₈	—	26	2	no	—
138	C₂²×C₈	—	13	2	no	—
139	C₂²×C₈	—	13	2	no	—
140	C₂²×C₈	—	13	2	no	—
141	C₄⋊Q₈	—	13	2	no	—
142	C₄⋊Q₈	—	26	2	no	—
143	C₄⋊Q₈	—	26	2	no	—
144	C₄⋊Q₈	—	26	2	no	—
145	C₄⋊Q₈	C₂×#261	1	2	yes	—
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×C₄⋊C₄	—	26	2	no	—
151	C₂×C₄²	—	13	1	no	—
152	#107	—	2	2	no	—
153	#107	—	26	4	no	—
154	#107	—	26	2	no	—
155	#106	—	52	4	no	—
156	#103	—	52	8	no	—
157	C₄×C₈	—	13	2	no	—
158	C₄×C₈	—	13	1	no	—
159	C₄×C₈	—	26	1	no	—
160	C₄×C₈	—	26	1	no	—
161	C₄×C₈	—	26	1	no	—
162	C₄×C₈	—	26	1	no	—
163	C₄×C₈	C₂×#261	1	1	yes	characteristic
164	C₂×C₁₃	#1956	1	1	yes	characteristic
165	C₂×C₁₃	#1971	1	1	yes	characteristic
166	C₂×C₁₃	#1972	1	1	yes	characteristic
167	D₁₃	#1714	1	2	yes	—
168	D₁₃	#1723	1	2	yes	—
169	C₂×Q₈	—	2	2	no	—
170	C₂×Q₈	—	26	2	no	—
171	C₂×Q₈	—	52	2	no	—
172	C₂×Q₈	—	52	4	no	—
173	C₂×Q₈	—	52	4	no	—
174	C₂²×C₄	—	13	1	no	—
175	C₂²×C₄	—	13	1	no	—
176	C₂²×C₄	—	13	1	no	—
177	C₂²×C₄	—	52	2	no	—
178	C₂²×C₄	—	52	2	no	—
179	C₂×C₈	—	13	2	no	—
180	C₂×C₈	—	13	4	no	—
181	C₂×C₈	—	13	4	no	—
182	C₂×C₈	—	26	2	no	—
183	C₂×C₈	—	26	2	no	—
184	C₂×C₈	—	26	4	no	—
185	C₂×C₈	—	26	4	no	—
186	C₂×C₈	C₂²⋊#261	1	2	yes	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	26	2	no	—
190	C₄⋊C₄	—	26	2	no	—
191	C₄⋊C₄	—	52	2	no	—
192	C₄⋊C₄	—	52	2	no	—
193	C₄⋊C₄	—	52	4	no	—
194	C₄⋊C₄	—	52	4	no	—
195	C₄⋊C₄	—	52	4	no	—
196	C₄²	—	13	1	no	—
197	C₄²	—	13	1	no	—
198	C₄²	—	13	1	no	—
199	C₄²	C₂²×#261	1	1	yes	characteristic
200	C₁₃	#9836	1	1	yes	characteristic, complemented, Sylow
201	C₂³	—	13	1	no	—
202	Q₈	—	4	4	no	—
203	Q₈	—	52	4	no	—
204	Q₈	—	104	4	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	13	1	no	—
207	C₂×C₄	—	13	1	no	—
208	C₂×C₄	—	13	1	no	—
209	C₂×C₄	—	13	4	no	—
210	C₂×C₄	—	13	4	no	—
211	C₂×C₄	—	13	2	no	—
212	C₂×C₄	—	13	2	no	—
213	C₂×C₄	—	52	2	no	—
214	C₂×C₄	—	104	2	no	—
215	C₂×C₄	—	104	2	no	—
216	C₂×C₄	—	104	2	no	—
217	C₂×C₄	—	104	2	no	—
218	C₂×C₄	—	104	2	no	—
219	C₂×C₄	D₄×#261	1	1	yes	characteristic
220	C₂×C₄	D₄×#261	1	1	yes	characteristic
221	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
222	C₈	—	13	4	no	—
223	C₈	—	52	2	no	—
224	C₈	—	52	2	no	—
225	C₈	#88117	1	4	yes	complemented
226	C₂²	—	13	1	no	—
227	C₂²	—	13	1	no	—
228	C₂²	—	13	2	no	—
229	C₂²	—	13	2	no	—
230	C₂²	#11719151	1	1	yes	characteristic
231	C₄	—	8	2	no	—
232	C₄	—	13	2	no	—
233	C₄	—	13	2	no	—
234	C₄	—	13	2	no	—
235	C₄	—	104	2	no	—
236	C₄	—	208	2	no	—
237	C₄	#11719102	1	2	yes	—
238	C₄	#11719110	1	2	yes	—
239	C₄	C₂×#88117	1	2	yes	—
240	C₂	—	13	2	no	—
241	C₂	—	13	2	no	—
242	C₂	#13130963	1	1	yes	characteristic
243	C₂	#13130984	1	1	yes	characteristic
244	C₂	#13130985	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13131017

C₂

yes

maximal

#13130963

C₂

yes

maximal

#13130771

C₂

yes

maximal, characteristic

#13130030

C₂

yes

maximal, characteristic

D₁₃×#500

C₂

yes

maximal, characteristic

D₁₃×C₄⋊Q₈

C₂²

yes

—

C₂×#88117

—

C₂×#88101

—

#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

#11718251

—

#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

#9836

—

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

—

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

SD₁₆

yes

—

C₈×D₁₃

Q₁₆

yes

—

—

—

—

—

—

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×SD₁₆

yes

—

100

C₄×D₁₃

C₂×Q₁₆

yes

—

101

C₄×D₁₃

AGL(1,ℤ/8)

yes

—

102

C₄×D₁₃

#137

yes

—

103

Dic₂₆

—

104

Dic₂₆

—

105

#1057

—

106

#1057

—

107

C₈×C₁₃

yes

108

yes

109

C₂×C₄⋊Q₈

—

110

#500

—

111

#500

—

112

#500

—

113

#500

#261

yes

characteristic, complemented

114

C₂×#107

—

115

C₂×#106

—

116

C₂×#103

—

117

C₂×C₄×C₈

—

118

C₂×C₄×C₈

—

119

C₂×C₄×C₈

—

120

#384

—

121

#340

—

122

#322

—

123

C₂²×C₁₃

#390

yes

characteristic

124

C₂×D₁₃

#492

yes

characteristic

125

C₂×D₁₃

#494

yes

characteristic

126

C₂×D₁₃

#496

yes

—

127

C₂×D₁₃

#497

yes

—

128

#261

—

129

C₄×C₁₃

—

130

C₄×C₁₃

C₂×#103

yes

—

131

C₄×C₁₃

#441

yes

—

132

C₄×C₁₃

#442

yes

—

133

Dic₁₃

—

134

Dic₁₃

#469

yes

—

135

Dic₁₃

#470

yes

—

136

Dic₁₃

#487

yes

—

137

C₂²×Q₈

—

138

C₂²×C₈

—

139

C₂²×C₈

—

140

C₂²×C₈

—

141

C₄⋊Q₈

—

142

C₄⋊Q₈

—

143

C₄⋊Q₈

—

144

C₄⋊Q₈

—

145

C₄⋊Q₈

C₂×#261

yes

—

146

C₂×C₄⋊C₄

—

147

C₂×C₄⋊C₄

—

148

C₂×C₄⋊C₄

—

149

C₂×C₄⋊C₄

—

150

C₂×C₄⋊C₄

—

151

C₂×C₄²

—

152

#107

—

153

#107

—

154

#107

—

155

#106

—

156

#103

—

157

C₄×C₈

—

158

C₄×C₈

—

159

C₄×C₈

—

160

C₄×C₈

—

161

C₄×C₈

—

162

C₄×C₈

—

163

C₄×C₈

C₂×#261

yes

characteristic

164

C₂×C₁₃

#1956

yes

characteristic

165

C₂×C₁₃

#1971

yes

characteristic

166

C₂×C₁₃

#1972

yes

characteristic

167

D₁₃

#1714

yes

—

168

D₁₃

#1723

yes

—

169

C₂×Q₈

—

170

C₂×Q₈

—

171

C₂×Q₈

—

172

C₂×Q₈

—

173

C₂×Q₈

—

174

C₂²×C₄

—

175

C₂²×C₄

—

176

C₂²×C₄

—

177

C₂²×C₄

—

178

C₂²×C₄

—

179

C₂×C₈

—

180

C₂×C₈

—

181

C₂×C₈

—

182

C₂×C₈

—

183

C₂×C₈

—

184

C₂×C₈

—

185

C₂×C₈

—

186

C₂×C₈

C₂²⋊#261

yes

—

187

C₄⋊C₄

—

188

C₄⋊C₄

—

189

C₄⋊C₄

—

190

C₄⋊C₄

—

191

C₄⋊C₄

—

192

C₄⋊C₄

—

193

C₄⋊C₄

—

194

C₄⋊C₄

—

195

C₄⋊C₄

—

196

C₄²

—

197

C₄²

—

198

C₄²

—

199

C₄²

C₂²×#261

yes

characteristic

200

C₁₃

#9836

yes

characteristic, complemented, Sylow

201

C₂³

—

202

Q₈

—

203

Q₈

—

204

Q₈

—

104

—

205

C₂×C₄

—

206

C₂×C₄

—

207

C₂×C₄

—

208

C₂×C₄

—

209

C₂×C₄

—

210

C₂×C₄

—

211

C₂×C₄

—

212

C₂×C₄

—

213

C₂×C₄

—

214

C₂×C₄

—

104

—

215

C₂×C₄

—

104

—

216

C₂×C₄

—

104

—

217

C₂×C₄

—

104

—

218

C₂×C₄

—

104

—

219

C₂×C₄

D₄×#261

yes

characteristic

220

C₂×C₄

D₄×#261

yes

characteristic

221

C₂×C₄

C₂×C₂²⋊#261

yes

characteristic

222

C₈

—

223

C₈

—

224

C₈

—

225

C₈

#88117

yes

complemented

226

C₂²

—

227

C₂²

—

228

C₂²

—

229

C₂²

—

230

C₂²

#11719151

yes

characteristic

231

C₄

—

232

C₄

—

233

C₄

—

234

C₄

—

235

C₄

—

104

—

236

C₄

—

208

—

237

C₄

#11719102

yes

—

238

C₄

#11719110

yes

—

239

C₄

C₂×#88117

yes

—

240

C₂

—

241

C₂

—

242

C₂

#13130963

yes

characteristic

243

C₂

#13130984

yes

characteristic

244

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	#9836
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
12. C₁₃×#500	C₄	semidirect
13. #11718739	C₄	semidirect
17. #11718343	C₄	semidirect
107. C₈×C₁₃	#103	semidirect
108. #1055	#103	semidirect
113. #500	#261	semidirect
200. C₁₃	#9836	semidirect
225. C₈	#88117	semidirect

H ≅ G/N

type

12. C₁₃×#500

C₄

semidirect

13. #11718739

C₄

semidirect

17. #11718343

C₄

semidirect

107. C₈×C₁₃

#103

semidirect

108. #1055

#103

semidirect

113. #500

#261

semidirect

200. C₁₃

#9836

semidirect

225. C₈

#88117

semidirect

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

subgroup	index
#13131017	2
#13130963	2
#13130771	2
#13130030	2
D₁₃×#500	2
#9836	13

subgroup

index

D₁₃×#500

Subgroups — #16784956

Conjugacy classes of subgroups

Complements

Maximal subgroups