Group #15762746 — subgroups

Subgroups — #15762746

Order 3328.

526 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 163 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	C₂×#11718360	C₂	1	1	yes	maximal, characteristic
2	#13115796	C₂	1	2	yes	maximal
3	#13115796	C₂	1	2	yes	maximal
4	C₂×#11718046	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1834	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₂	—	2	2	no	—
7	C₂×#88123	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₁₃×#384	C₂²	1	2	yes	—
10	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
11	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
12	#11718360	—	2	2	no	—
13	#11718360	C₄	1	4	yes	complemented
14	C₁₃×#340	—	2	2	no	—
15	#11718121	—	2	4	no	—
16	#11718121	—	2	4	no	—
17	#11718046	—	2	2	no	—
18	C₂²×Dic₂₆	—	2	2	no	—
19	C₂²×C₈×C₁₃	D₄	1	2	yes	—
20	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
21	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
22	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
23	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₃	—	2	2	no	—
25	C₂×C₄⋊Dic₁₃	—	2	2	no	—
26	C₂²×#1055	D₄	1	2	yes	—
27	C₂²×#1055	D₄	1	2	yes	—
28	C₂×Dic₅₂	—	2	8	no	—
29	C₂×Dic₅₂	—	4	8	no	—
30	#88123	—	2	2	no	—
31	#88123	—	2	2	no	—
32	#88123	—	2	2	no	—
33	#88123	C₂×C₄	1	4	yes	—
34	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₁₃	D₄	1	2	yes	—
36	C₄×#1055	—	2	1	no	—
37	C₄×#1055	—	2	1	no	—
38	C₄×#1055	—	2	1	no	—
39	C₄×#1055	—	2	1	no	—
40	#9835	—	13	1	no	maximal, Sylow
41	C₂²×C₄×C₁₃	—	4	2	no	—
42	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₁₃	—	4	2	no	—
46	C₂×Dic₂₆	—	2	4	no	—
47	C₂×Dic₂₆	—	4	8	no	—
48	C₂×Dic₂₆	—	4	2	no	—
49	C₂×C₈×C₁₃	—	2	4	no	—
50	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
51	C₁₃×C₄⋊C₄	—	4	4	no	—
52	C₁₃×C₄⋊C₄	—	4	4	no	—
53	C₁₃×C₄⋊C₄	—	4	4	no	—
54	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₁₃	#41	1	1	yes	characteristic
57	C₄²×C₁₃	#41	1	1	yes	characteristic
58	C₄⋊Dic₁₃	—	2	4	no	—
59	C₄⋊Dic₁₃	—	2	4	no	—
60	C₄⋊Dic₁₃	—	4	2	no	—
61	C₄⋊Dic₁₃	—	4	2	no	—
62	C₂×#1055	—	2	2	no	—
63	C₂×#1055	—	2	2	no	—
64	C₂×#1055	—	2	2	no	—
65	C₂×#1055	—	2	4	no	—
66	C₂×#1055	Q₁₆	1	4	yes	complemented
67	Dic₅₂	—	4	16	no	—
68	C₂×#494	—	13	1	no	—
69	#1990	—	13	2	no	—
70	#1990	—	13	2	no	—
71	#1834	D₁₃	1	1	yes	characteristic
72	C₂×#322	—	13	1	no	—
73	C₂³×C₁₃	#127	1	1	yes	characteristic
74	C₂×C₄×C₁₃	—	8	2	no	—
75	C₂×C₄×C₁₃	—	8	2	no	—
76	C₂×C₄×C₁₃	—	8	2	no	—
77	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
81	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
82	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
84	C₂×C₄×C₁₃	#135	1	2	yes	—
85	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
86	C₂×C₄×C₁₃	#137	1	4	yes	—
87	C₂×Dic₁₃	—	4	4	no	—
88	C₂×Dic₁₃	—	8	4	no	—
89	Dic₂₆	—	4	8	no	—
90	Dic₂₆	—	8	4	no	—
91	C₈×C₁₃	—	2	4	no	—
92	#1055	—	2	4	no	—
93	#1055	—	4	2	no	—
94	C₂²×Q₁₆	—	26	2	no	—
95	C₂×C₄⋊Q₈	—	13	2	no	—
96	#494	—	13	4	no	—
97	#494	—	26	2	no	—
98	C₂×C₄×C₈	—	13	1	no	—
99	C₂×C₄×C₈	—	13	1	no	—
100	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
101	#384	C₂×D₁₃	1	2	yes	—
102	#340	—	2	2	no	—
103	#340	—	26	4	no	—
104	#340	—	26	4	no	—
105	#322	—	26	2	no	—
106	C₂²×C₁₃	#390	1	1	yes	characteristic
107	C₂²×C₁₃	#494	1	1	yes	characteristic
108	C₂²×C₁₃	#495	1	1	yes	characteristic
109	C₂²×C₁₃	#496	1	2	yes	—
110	C₂²×C₁₃	#497	1	2	yes	—
111	C₄×C₁₃	—	16	2	no	—
112	C₄×C₁₃	#418	1	2	yes	—
113	C₄×C₁₃	C₄×Q₁₆	1	2	yes	—
114	C₄×C₁₃	#441	1	2	yes	—
115	C₄×C₁₃	#467	1	2	yes	—
116	C₄×C₁₃	#470	1	2	yes	—
117	C₄×C₁₃	#490	1	2	yes	—
118	Dic₁₃	—	8	4	no	—
119	C₂²×Q₈	—	26	2	no	—
120	C₂×Q₁₆	—	26	8	no	—
121	C₂×Q₁₆	—	52	8	no	—
122	C₂²×C₈	—	13	2	no	—
123	C₂²×C₈	—	13	2	no	—
124	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
125	C₄⋊Q₈	—	13	4	no	—
126	C₄⋊Q₈	—	26	2	no	—
127	C₄⋊Q₈	—	26	2	no	—
128	C₄⋊Q₈	—	26	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	—	26	2	no	—
133	C₂×C₄⋊C₄	—	26	2	no	—
134	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
135	C₄×C₈	—	26	1	no	—
136	C₄×C₈	—	26	1	no	—
137	C₄×C₈	—	26	1	no	—
138	C₄×C₈	—	26	1	no	—
139	C₄×C₈	C₄×D₁₃	1	2	yes	—
140	C₄×C₈	D₅₂	1	2	yes	—
141	C₂×C₁₃	#1720	1	2	yes	—
142	C₂×C₁₃	#1723	1	2	yes	—
143	C₂×C₁₃	#1961	1	1	yes	characteristic
144	C₂×C₁₃	#1969	1	1	yes	characteristic
145	C₂×C₁₃	#1971	1	1	yes	characteristic
146	C₂×Q₈	—	26	4	no	—
147	C₂×Q₈	—	52	2	no	—
148	C₂×Q₈	—	52	8	no	—
149	C₂²×C₄	—	4	2	no	—
150	C₂²×C₄	—	52	2	no	—
151	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
152	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
154	Q₁₆	—	52	16	no	—
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	—	13	4	no	—
157	C₂×C₈	—	26	2	no	—
158	C₂×C₈	—	26	2	no	—
159	C₂×C₈	—	26	2	no	—
160	C₂×C₈	—	26	4	no	—
161	C₂×C₈	#6103	1	4	yes	—
162	C₄⋊C₄	—	4	4	no	—
163	C₄⋊C₄	—	4	4	no	—
164	C₄⋊C₄	—	4	4	no	—
165	C₄⋊C₄	—	26	4	no	—
166	C₄⋊C₄	—	26	4	no	—
167	C₄⋊C₄	—	52	2	no	—
168	C₄⋊C₄	—	52	2	no	—
169	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
170	C₄²	C₂×D₅₂	1	1	yes	characteristic
171	C₄²	#6131	1	1	yes	characteristic
172	C₄²	#6131	1	1	yes	characteristic
173	C₁₃	#9835	1	1	yes	characteristic, complemented, Sylow
174	C₂³	#88194	1	1	yes	characteristic
175	Q₈	—	52	8	no	—
176	Q₈	—	104	4	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	8	2	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	—	52	4	no	—
181	C₂×C₄	—	104	4	no	—
182	C₂×C₄	#88147	1	1	yes	characteristic
183	C₂×C₄	#88147	1	1	yes	characteristic
184	C₂×C₄	#88149	1	1	yes	characteristic
185	C₂×C₄	#88149	1	1	yes	characteristic
186	C₂×C₄	#88166	1	2	yes	—
187	C₂×C₄	#88169	1	4	yes	—
188	C₂×C₄	Q₁₆×D₁₃	1	2	yes	—
189	C₂×C₄	C₂×#6103	1	1	yes	characteristic
190	C₂×C₄	#88201	1	1	yes	characteristic
191	C₂×C₄	#88203	1	4	yes	—
192	C₈	—	2	4	no	—
193	C₈	—	26	4	no	—
194	C₈	—	52	2	no	—
195	C₂²	#11718649	1	1	yes	characteristic
196	C₂²	#11718719	1	2	yes	—
197	C₂²	#11718816	1	1	yes	characteristic
198	C₂²	#11718828	1	2	yes	—
199	C₂²	#11718842	1	1	yes	characteristic
200	C₄	—	16	2	no	—
201	C₄	—	104	4	no	—
202	C₄	#11718514	1	2	yes	—
203	C₄	#11718529	1	2	yes	—
204	C₄	#11718534	1	2	yes	—
205	C₄	#11718548	1	2	yes	—
206	C₄	#11718662	1	2	yes	—
207	C₄	#11718728	1	2	yes	—
208	C₂	#13114495	1	2	yes	—
209	C₂	#13114515	1	2	yes	—
210	C₂	#13115659	1	1	yes	characteristic
211	C₂	#13115695	1	1	yes	characteristic
212	C₂	#13115705	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718360

C₂

yes

maximal, characteristic

#13115796

C₂

yes

maximal

#13115796

C₂

yes

maximal

C₂×#11718046

C₂

yes

maximal, characteristic

C₁₃×#1834

C₂

yes

maximal, characteristic

C₂²×Dic₅₂

—

C₂×#88123

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₁₃×#384

C₂²

yes

—

C₂×C₄×#1055

C₂²

yes

characteristic

C₂×C₄×#1055

C₂²

yes

characteristic

#11718360

—

#11718360

C₄

yes

complemented

C₁₃×#340

—

#11718121

—

#11718121

—

#11718046

—

C₂²×Dic₂₆

—

C₂²×C₈×C₁₃

D₄

yes

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂²×#1055

D₄

yes

—

C₂²×#1055

D₄

yes

—

C₂×Dic₅₂

—

C₂×Dic₅₂

—

—

—

—

C₂×C₄

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

D₄

yes

—

—

—

—

—

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂²⋊C₄

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

C₄²×C₁₃

yes

C₄²×C₁₃

yes

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

—

—

—

—

Q₁₆

yes

complemented

Dic₅₂

—

—

—

—

D₁₃

yes

characteristic

C₂×#322

—

C₂³×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×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

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

C₂×Q₁₆

yes

—

C₂×C₄×C₁₃

#135

yes

—

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₁₃

#137

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

#1055

—

#1055

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

#494

—

#494

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

100

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

101

#384

C₂×D₁₃

yes

—

102

#340

—

103

#340

—

104

#340

—

105

#322

—

106

C₂²×C₁₃

#390

yes

characteristic

107

C₂²×C₁₃

#494

yes

characteristic

108

C₂²×C₁₃

#495

yes

characteristic

109

C₂²×C₁₃

#496

yes

—

110

C₂²×C₁₃

#497

yes

—

111

C₄×C₁₃

—

112

C₄×C₁₃

#418

yes

—

113

C₄×C₁₃

C₄×Q₁₆

yes

—

114

C₄×C₁₃

#441

yes

—

115

C₄×C₁₃

#467

yes

—

116

C₄×C₁₃

#470

yes

—

117

C₄×C₁₃

#490

yes

—

118

Dic₁₃

—

119

C₂²×Q₈

—

120

C₂×Q₁₆

—

121

C₂×Q₁₆

—

122

C₂²×C₈

—

123

C₂²×C₈

—

124

C₂²×C₈

C₂²⋊D₁₃

yes

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

134

C₂×C₄²

C₂²×D₁₃

yes

characteristic

135

C₄×C₈

—

136

C₄×C₈

—

137

C₄×C₈

—

138

C₄×C₈

—

139

C₄×C₈

C₄×D₁₃

yes

—

140

C₄×C₈

D₅₂

yes

—

141

C₂×C₁₃

#1720

yes

—

142

C₂×C₁₃

#1723

yes

—

143

C₂×C₁₃

#1961

yes

characteristic

144

C₂×C₁₃

#1969

yes

characteristic

145

C₂×C₁₃

#1971

yes

characteristic

146

C₂×Q₈

—

147

C₂×Q₈

—

148

C₂×Q₈

—

149

C₂²×C₄

—

150

C₂²×C₄

—

151

C₂²×C₄

D₄×D₁₃

yes

characteristic

152

C₂²×C₄

D₄×D₁₃

yes

characteristic

153

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

154

Q₁₆

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

#6103

yes

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄²

C₂×C₄×D₁₃

yes

characteristic

170

C₄²

C₂×D₅₂

yes

characteristic

171

C₄²

#6131

yes

characteristic

172

C₄²

#6131

yes

characteristic

173

C₁₃

#9835

yes

characteristic, complemented, Sylow

174

C₂³

#88194

yes

characteristic

175

Q₈

—

176

Q₈

—

104

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

104

—

182

C₂×C₄

#88147

yes

characteristic

183

C₂×C₄

#88147

yes

characteristic

184

C₂×C₄

#88149

yes

characteristic

185

C₂×C₄

#88149

yes

characteristic

186

C₂×C₄

#88166

yes

—

187

C₂×C₄

#88169

yes

—

188

C₂×C₄

Q₁₆×D₁₃

yes

—

189

C₂×C₄

C₂×#6103

yes

characteristic

190

C₂×C₄

#88201

yes

characteristic

191

C₂×C₄

#88203

yes

—

192

C₈

—

193

C₈

—

194

C₈

—

195

C₂²

#11718649

yes

characteristic

196

C₂²

#11718719

yes

—

197

C₂²

#11718816

yes

characteristic

198

C₂²

#11718828

yes

—

199

C₂²

#11718842

yes

characteristic

200

C₄

—

201

C₄

—

104

—

202

C₄

#11718514

yes

—

203

C₄

#11718529

yes

—

204

C₄

#11718534

yes

—

205

C₄

#11718548

yes

—

206

C₄

#11718662

yes

—

207

C₄

#11718728

yes

—

208

C₂

#13114495

yes

—

209

C₂

#13114515

yes

—

210

C₂

#13115659

yes

characteristic

211

C₂

#13115695

yes

characteristic

212

C₂

#13115705

yes

characteristic

	subgroup	G/H
center	C₂³	#88194
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1834	C₂
socle, prime radical	C₂³×C₁₃	#127
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1834	D₁₃
hypocenter	C₁₃	#9835
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

subgroup

G/H

center

C₂³

#88194

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1834

C₂

socle, prime radical

C₂³×C₁₃

#127

cosocle

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

N	H ≅ G/N	type
13. #11718360	C₄	semidirect
66. C₂×#1055	Q₁₆	semidirect
173. C₁₃	#9835	semidirect

H ≅ G/N

type

13. #11718360

C₄

semidirect

66. C₂×#1055

Q₁₆

semidirect

173. C₁₃

#9835

semidirect

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

subgroup	index
C₂×#11718360	2
#13115796	2
C₂×#11718046	2
C₁₃×#1834	2
#9835	13

subgroup

index

C₁₃×#1834

Subgroups — #15762746

Conjugacy classes of subgroups

Complements

Maximal subgroups