Group #16785053 — subgroups

Subgroups — #16785053

Order 3328.

458 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 148 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	#13130768	C₂	1	1	yes	maximal, characteristic
2	#13130550	C₂	1	1	yes	maximal, characteristic
3	#13129672	C₂	1	4	yes	maximal
4	#125×Dic₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#125	C₄	1	1	yes	characteristic
6	C₂×C₄⋊#1057	—	2	2	no	—
7	C₂×C₄⋊#1057	C₂²	1	2	yes	—
8	C₂×C₄×#1057	C₂²	1	2	yes	—
9	#11718646	C₄	1	2	yes	—
10	#11718642	—	2	2	no	—
11	C₄⋊C₄×Dic₁₃	—	2	2	no	—
12	C₄⋊C₄×Dic₁₃	C₂²	1	2	yes	—
13	#11718597	C₄	1	1	yes	characteristic
14	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
15	#11718290	—	2	2	no	—
16	C₂²×#1057	—	2	2	no	—
17	C₂²×#1057	—	2	2	no	—
18	C₁₃×#125	—	2	2	no	—
19	C₁₃×#125	C₈	1	4	yes	complemented
20	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
21	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄⋊Dic₁₃	—	2	2	no	—
24	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
25	C₂×C₄×Dic₁₃	—	2	2	no	—
26	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
28	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
29	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
30	C₄⋊#1057	—	2	4	no	—
31	C₄⋊#1057	—	4	4	no	—
32	C₄×#1057	—	2	2	no	—
33	C₄×#1057	Q₈	1	4	yes	—
34	#88071	—	2	2	no	—
35	#88071	—	2	2	no	—
36	#88071	—	2	2	no	—
37	#9872	—	13	1	no	maximal, Sylow
38	C₂²×C₄×C₁₃	—	2	2	no	—
39	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×Dic₁₃	—	2	2	no	—
42	C₂²×Dic₁₃	C₂²⋊C₄	1	2	yes	—
43	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂×#1057	—	2	4	no	—
46	C₂×#1057	—	4	4	no	—
47	C₂×#1057	—	4	2	no	—
48	C₂×#1057	—	4	4	no	—
49	C₁₃×C₄⋊C₄	—	2	8	no	—
50	C₁₃×C₄⋊C₄	—	2	2	no	—
51	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
52	C₄²×C₁₃	C₂×C₈	1	2	yes	—
53	C₄²×C₁₃	M₁₆	1	2	yes	—
54	C₄⋊Dic₁₃	—	2	4	no	—
55	C₄⋊Dic₁₃	—	4	4	no	—
56	C₄×Dic₁₃	—	2	2	no	—
57	C₄×Dic₁₃	—	2	2	no	—
58	C₄×Dic₁₃	—	2	2	no	—
59	C₄×Dic₁₃	—	4	2	no	—
60	C₄×Dic₁₃	—	4	2	no	—
61	C₄×Dic₁₃	—	4	2	no	—
62	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
63	C₄×Dic₁₃	#41	1	2	yes	—
64	C₄×#125	—	13	1	no	—
65	#1842	—	13	1	no	—
66	#1744	—	13	1	no	—
67	#1279	—	13	4	no	—
68	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₃	—	2	2	no	—
70	C₂×C₄×C₁₃	—	2	2	no	—
71	C₂×C₄×C₁₃	—	2	4	no	—
72	C₂×C₄×C₁₃	—	4	2	no	—
73	C₂×C₄×C₁₃	—	4	2	no	—
74	C₂×C₄×C₁₃	C₂²⋊C₈	1	4	yes	—
75	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
76	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₃	#131	1	2	yes	—
79	C₂×Dic₁₃	—	2	2	no	—
80	C₂×Dic₁₃	—	2	2	no	—
81	C₂×Dic₁₃	—	2	2	no	—
82	C₂×Dic₁₃	—	2	2	no	—
83	C₂×Dic₁₃	—	4	2	no	—
84	C₂×Dic₁₃	—	4	2	no	—
85	C₂×Dic₁₃	—	4	2	no	—
86	C₂×Dic₁₃	C₂²⋊Q₈	1	2	yes	—
87	C₂×Dic₁₃	#124	1	1	yes	characteristic
88	C₂×Dic₁₃	C₄⋊Q₈	1	1	yes	characteristic
89	C₂×Dic₁₃	AGL(1,ℤ/8)	1	2	yes	—
90	C₂×Dic₁₃	#137	1	2	yes	—
91	#1057	—	4	4	no	—
92	#1057	—	8	2	no	—
93	C₂×#125	#261	1	1	yes	characteristic
94	C₂×C₄⋊C₈	—	13	2	no	—
95	C₂×C₄⋊C₈	—	26	2	no	—
96	C₂×C₄×C₈	—	13	2	no	—
97	#389	—	13	2	no	—
98	#387	—	26	2	no	—
99	#382	—	13	1	no	—
100	C₄×C₄⋊C₄	—	13	2	no	—
101	C₄×C₄⋊C₄	—	26	2	no	—
102	C₄³	—	13	1	no	—
103	#336	—	26	2	no	—
104	C₂²×C₁₃	#391	1	1	yes	characteristic
105	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
106	C₂²×C₁₃	#408	1	1	yes	characteristic
107	C₂²×C₁₃	#417	1	2	yes	—
108	C₂²×C₁₃	#421	1	2	yes	—
109	C₄×C₁₃	—	2	4	no	—
110	C₄×C₁₃	—	4	4	no	—
111	C₄×C₁₃	C₈×Q₈	1	2	yes	—
112	C₄×C₁₃	#446	1	2	yes	—
113	Dic₁₃	—	2	2	no	—
114	Dic₁₃	—	4	2	no	—
115	Dic₁₃	—	8	2	no	—
116	Dic₁₃	#490	1	1	yes	characteristic
117	Dic₁₃	#491	1	1	yes	characteristic
118	Dic₁₃	#501	1	2	yes	—
119	C₂²×C₈	—	26	2	no	—
120	C₂²×C₈	—	26	2	no	—
121	#125	—	2	2	no	—
122	#125	#1057	1	4	yes	complemented
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	13	2	no	—
125	C₂×C₄⋊C₄	—	26	2	no	—
126	C₂×C₄⋊C₄	C₂×#261	1	2	yes	—
127	C₂×C₄²	—	13	1	no	—
128	C₂×C₄²	—	13	2	no	—
129	C₂×C₄²	—	13	1	no	—
130	C₂×C₄²	—	13	2	no	—
131	C₂×C₄²	—	26	2	no	—
132	C₂×C₄²	C₂×#261	1	1	yes	characteristic
133	C₄⋊C₈	—	26	4	no	—
134	C₄⋊C₈	—	52	4	no	—
135	C₄×C₈	—	13	4	no	—
136	C₄×C₈	—	26	2	no	—
137	#95	—	26	2	no	—
138	#95	—	26	2	no	—
139	#95	—	26	2	no	—
140	C₂×C₁₃	#1477	1	2	yes	—
141	C₂×C₁₃	#1495	1	2	yes	—
142	C₂×C₁₃	#1984	1	1	yes	characteristic
143	C₂×C₁₃	#1992	1	1	yes	characteristic
144	C₂×C₁₃	#1995	1	1	yes	characteristic
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	—	13	1	no	—
147	C₂²×C₄	—	13	1	no	—
148	C₂²×C₄	—	13	2	no	—
149	C₂²×C₄	—	26	2	no	—
150	C₂²×C₄	C₂²⋊#261	1	2	yes	—
151	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
152	C₂×C₈	—	26	4	no	—
153	C₂×C₈	—	52	4	no	—
154	C₂×C₈	—	52	2	no	—
155	C₂×C₈	—	52	4	no	—
156	C₄⋊C₄	—	2	8	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	26	4	no	—
159	C₄⋊C₄	—	52	4	no	—
160	C₄⋊C₄	C₂×#1057	1	4	yes	—
161	C₄²	—	13	2	no	—
162	C₄²	—	13	2	no	—
163	C₄²	—	26	2	no	—
164	C₄²	—	26	2	no	—
165	C₄²	—	26	2	no	—
166	C₄²	—	52	2	no	—
167	C₄²	—	52	2	no	—
168	C₄²	—	52	2	no	—
169	C₄²	C₂×#1057	1	2	yes	—
170	C₄²	#6122	1	2	yes	—
171	C₁₃	#9872	1	1	yes	characteristic, complemented, Sylow
172	C₂³	C₂×C₂²⋊#261	1	1	yes	characteristic
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	13	1	no	—
179	C₂×C₄	—	13	1	no	—
180	C₂×C₄	—	13	2	no	—
181	C₂×C₄	—	13	2	no	—
182	C₂×C₄	—	13	2	no	—
183	C₂×C₄	—	26	2	no	—
184	C₂×C₄	—	26	2	no	—
185	C₂×C₄	—	26	2	no	—
186	C₂×C₄	—	26	2	no	—
187	C₂×C₄	—	52	2	no	—
188	C₂×C₄	—	52	2	no	—
189	C₂×C₄	—	52	2	no	—
190	C₂×C₄	C₂²⋊#1057	1	4	yes	—
191	C₂×C₄	#88240	1	2	yes	—
192	C₂×C₄	Q₈×#261	1	2	yes	—
193	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
194	C₂×C₄	C₂×#6122	1	1	yes	characteristic
195	C₈	—	52	4	no	—
196	C₈	—	104	2	no	—
197	C₂²	#11719159	1	1	yes	characteristic
198	C₂²	(C₂×#261)⋊Q₈	1	1	yes	characteristic
199	C₂²	#11719164	1	2	yes	—
200	C₂²	#11719166	1	2	yes	—
201	C₂²	C₂×C₂²⋊#1057	1	1	yes	characteristic
202	C₄	—	2	4	no	—
203	C₄	—	4	4	no	—
204	C₄	—	13	1	no	—
205	C₄	—	13	1	no	—
206	C₄	—	13	2	no	—
207	C₄	—	26	2	no	—
208	C₄	—	52	2	no	—
209	C₄	—	104	2	no	—
210	C₄	Q₈×#1057	1	2	yes	—
211	C₄	#11719158	1	2	yes	—
212	C₂	#13130131	1	2	yes	—
213	C₂	#13130194	1	2	yes	—
214	C₂	#13131005	1	1	yes	characteristic
215	C₂	#13131019	1	1	yes	characteristic
216	C₂	#13131023	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13130768

C₂

yes

maximal, characteristic

#13130550

C₂

yes

maximal, characteristic

#13129672

C₂

yes

maximal

#125×Dic₁₃

C₂

yes

maximal, characteristic

C₂×C₁₃×#125

C₄

yes

characteristic

C₂×C₄⋊#1057

—

C₂×C₄⋊#1057

C₂²

yes

—

C₂×C₄×#1057

C₂²

yes

—

#11718646

C₄

yes

—

#11718642

—

C₄⋊C₄×Dic₁₃

—

C₄⋊C₄×Dic₁₃

C₂²

yes

—

#11718597

C₄

yes

characteristic

C₄²×Dic₁₃

C₂²

yes

characteristic

#11718290

—

C₂²×#1057

—

C₂²×#1057

—

C₁₃×#125

—

C₁₃×#125

C₈

yes

complemented

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

C₂×C₄

yes

—

C₂×C₄×Dic₁₃

—

C₂×C₄×Dic₁₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₃

D₄

yes

—

C₂×C₄×Dic₁₃

D₄

yes

—

C₂×C₄×Dic₁₃

C₂³

yes

—

—

—

Q₈

yes

—

—

—

—

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

C₂²⋊C₄

yes

—

C₂²×Dic₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

C₂×D₄

yes

—

—

—

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

C₂×C₈

yes

—

C₄²×C₁₃

C₂×C₈

yes

—

C₄²×C₁₃

M₁₆

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

C₂×Q₈

yes

—

C₄×Dic₁₃

#41

yes

—

—

—

—

—

C₂³×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₈

yes

—

C₂×C₄×C₁₃

C₄×Q₈

yes

—

C₂×C₄×C₁₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₁₃

#131

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂²⋊Q₈

yes

—

C₂×Dic₁₃

#124

yes

characteristic

C₂×Dic₁₃

C₄⋊Q₈

yes

characteristic

C₂×Dic₁₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₁₃

#137

yes

—

—

—

yes

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

#389

—

#387

—

#382

—

100

C₄×C₄⋊C₄

—

101

C₄×C₄⋊C₄

—

102

C₄³

—

103

#336

—

104

C₂²×C₁₃

#391

yes

characteristic

105

C₂²×C₁₃

C₂×C₂²⋊C₈

yes

characteristic

106

C₂²×C₁₃

#408

yes

characteristic

107

C₂²×C₁₃

#417

yes

—

108

C₂²×C₁₃

#421

yes

—

109

C₄×C₁₃

—

110

C₄×C₁₃

—

111

C₄×C₁₃

C₈×Q₈

yes

—

112

C₄×C₁₃

#446

yes

—

113

Dic₁₃

—

114

Dic₁₃

—

115

Dic₁₃

—

116

Dic₁₃

#490

yes

characteristic

117

Dic₁₃

#491

yes

characteristic

118

Dic₁₃

#501

yes

—

119

C₂²×C₈

—

120

C₂²×C₈

—

121

#125

—

122

#125

#1057

yes

complemented

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

C₂×#261

yes

—

127

C₂×C₄²

—

128

C₂×C₄²

—

129

C₂×C₄²

—

130

C₂×C₄²

—

131

C₂×C₄²

—

132

C₂×C₄²

C₂×#261

yes

characteristic

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

C₄×C₈

—

136

C₄×C₈

—

137

#95

—

138

#95

—

139

#95

—

140

C₂×C₁₃

#1477

yes

—

141

C₂×C₁₃

#1495

yes

—

142

C₂×C₁₃

#1984

yes

characteristic

143

C₂×C₁₃

#1992

yes

characteristic

144

C₂×C₁₃

#1995

yes

characteristic

145

C₂²×C₄

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

—

150

C₂²×C₄

C₂²⋊#261

yes

—

151

C₂²×C₄

C₂²×#261

yes

characteristic

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

C₂×#1057

yes

—

161

C₄²

—

162

C₄²

—

163

C₄²

—

164

C₄²

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₄²

—

169

C₄²

C₂×#1057

yes

—

170

C₄²

#6122

yes

—

171

C₁₃

#9872

yes

characteristic, complemented, Sylow

172

C₂³

C₂×C₂²⋊#261

yes

characteristic

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

C₂²⋊#1057

yes

—

191

C₂×C₄

#88240

yes

—

192

C₂×C₄

Q₈×#261

yes

—

193

C₂×C₄

C₂²×#1057

yes

characteristic

194

C₂×C₄

C₂×#6122

yes

characteristic

195

C₈

—

196

C₈

—

104

—

197

C₂²

#11719159

yes

characteristic

198

C₂²

(C₂×#261)⋊Q₈

yes

characteristic

199

C₂²

#11719164

yes

—

200

C₂²

#11719166

yes

—

201

C₂²

C₂×C₂²⋊#1057

yes

characteristic

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

—

206

C₄

—

207

C₄

—

208

C₄

—

209

C₄

—

104

—

210

C₄

Q₈×#1057

yes

—

211

C₄

#11719158

yes

—

212

C₂

#13130131

yes

—

213

C₂

#13130194

yes

—

214

C₂

#13131005

yes

characteristic

215

C₂

#13131019

yes

characteristic

216

C₂

#13131023

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊#261
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₁₃×#125	C₄
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	C₂×#125	#261
hypocenter	C₁₃	#9872
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊#261

commutator

C₂×C₄×C₁₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#261

Fitting

C₂×C₁₃×#125

C₄

socle, prime radical

C₂³×C₁₃

C₂×C₂²⋊C₄

cosocle

C₂×C₄×Dic₁₃

C₂³

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
19. C₁₃×#125	C₈	semidirect
122. #125	#1057	semidirect
171. C₁₃	#9872	semidirect

H ≅ G/N

type

19. C₁₃×#125

C₈

semidirect

122. #125

#1057

semidirect

171. C₁₃

#9872

semidirect

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

subgroup	index
#13130768	2
#13130550	2
#13129672	2
#125×Dic₁₃	2
#9872	13

subgroup

index

#125×Dic₁₃

Subgroups — #16785053

Conjugacy classes of subgroups

Complements

Maximal subgroups