Group #15766012 — subgroups

Subgroups — #15766012

Order 3328.

416 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 99 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	#13123214	C₂	1	1	yes	maximal, characteristic
2	#13113400	C₂	1	2	yes	maximal
3	#13112696	C₂	1	2	yes	maximal
4	#13112669	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1994	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#533	C₂²	1	1	yes	characteristic
7	C₂×#88123	—	2	1	no	—
8	C₁₃×#423	C₂²	1	1	yes	characteristic
9	C₄³×C₁₃	C₂²	1	1	yes	characteristic
10	#11718597	—	2	1	no	—
11	#11718597	C₂²	1	2	yes	—
12	#11718590	—	2	4	no	—
13	#11718586	C₂²	1	2	yes	—
14	C₁₃×#339	—	2	1	no	—
15	C₁₃×#339	—	2	1	no	—
16	#11718130	—	2	4	no	—
17	C₂²×Dic₂₆	—	2	2	no	—
18	C₂×C₁₃×M₁₆	—	2	1	no	—
19	C₁₃×C₄⋊Q₈	Q₈	1	2	yes	complemented
20	C₁₃×#125	—	2	1	no	—
21	C₁₃×C₂²⋊Q₈	—	2	2	no	—
22	C₄×Q₈×C₁₃	—	2	2	no	—
23	C₁₃×#117	—	2	1	no	—
24	C₂×C₄²×C₁₃	—	2	2	no	—
25	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊Dic₁₃	—	2	2	no	—
29	C₂×C₄⋊Dic₁₃	—	2	2	no	—
30	#88123	—	4	2	no	—
31	#88123	—	4	2	no	—
32	C₁₃×C₄⋊C₈	—	2	2	no	—
33	#88071	—	2	4	no	—
34	#88071	—	2	2	no	—
35	#88071	D₄	1	4	yes	—
36	C₂²⋊#1055	—	2	2	no	—
37	C₄⋊#1055	—	2	2	no	—
38	C₄×#1055	—	2	2	no	—
39	#10363	—	13	1	no	maximal, Sylow
40	C₂×Q₈×C₁₃	—	2	2	no	—
41	C₂²×C₄×C₁₃	—	2	1	no	—
42	C₂²×C₄×C₁₃	—	2	1	no	—
43	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₁₃	—	2	4	no	—
46	C₂×Dic₂₆	—	4	4	no	—
47	C₂×Dic₂₆	—	4	8	no	—
48	C₁₃×M₁₆	—	4	1	no	—
49	C₁₃×M₁₆	—	4	1	no	—
50	C₂×C₈×C₁₃	—	4	1	no	—
51	C₁₃×C₄⋊C₄	—	2	2	no	—
52	C₁₃×C₄⋊C₄	—	2	2	no	—
53	C₁₃×C₄⋊C₄	—	4	2	no	—
54	C₁₃×C₂²⋊C₄	—	4	1	no	—
55	C₄²×C₁₃	—	2	1	no	—
56	C₄²×C₁₃	—	2	1	no	—
57	C₄²×C₁₃	—	4	4	no	—
58	C₄²×C₁₃	—	4	1	no	—
59	C₄²×C₁₃	SD₁₆	1	2	yes	—
60	C₄²×C₁₃	Q₁₆	1	2	yes	—
61	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
62	C₄²×C₁₃	#41	1	1	yes	characteristic
63	C₄²×C₁₃	#41	1	2	yes	—
64	C₄⋊Dic₁₃	—	4	4	no	—
65	C₄⋊Dic₁₃	—	4	2	no	—
66	C₄⋊Dic₁₃	—	4	2	no	—
67	C₂×#1055	—	4	2	no	—
68	#2862	—	13	1	no	—
69	#1994	D₁₃	1	1	yes	characteristic
70	#1518	—	13	2	no	—
71	#1286	—	13	2	no	—
72	#1276	—	13	1	no	—
73	C₂³×C₁₃	#120	1	1	yes	characteristic
74	Q₈×C₁₃	—	8	2	no	—
75	C₂×C₄×C₁₃	—	2	4	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	—	4	2	no	—
79	C₂×C₄×C₁₃	—	4	2	no	—
80	C₂×C₄×C₁₃	—	4	2	no	—
81	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
82	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₁₃	#124	1	2	yes	—
84	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
88	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
89	C₂×Dic₁₃	—	4	8	no	—
90	C₂×Dic₁₃	—	4	4	no	—
91	Dic₂₆	—	8	8	no	—
92	C₈×C₁₃	—	8	1	no	—
93	#1055	—	8	2	no	—
94	#533	C₂×D₁₃	1	1	yes	characteristic
95	C₂×C₄⋊Q₈	—	26	1	no	—
96	#431	—	13	2	no	—
97	#423	C₂×D₁₃	1	1	yes	characteristic
98	#391	—	26	4	no	—
99	#382	—	13	2	no	—
100	#382	—	26	1	no	—
101	C₄³	C₂×D₁₃	1	1	yes	characteristic
102	#339	—	2	1	no	—
103	#339	—	2	1	no	—
104	#328	—	26	4	no	—
105	C₂²×C₁₃	—	2	2	no	—
106	C₂²×C₁₃	#393	1	1	yes	characteristic
107	C₂²×C₁₃	#450	1	1	yes	characteristic
108	C₂²×C₁₃	#451	1	1	yes	characteristic
109	C₄×C₁₃	—	2	4	no	—
110	C₄×C₁₃	—	2	2	no	—
111	C₄×C₁₃	—	2	2	no	—
112	C₄×C₁₃	—	4	2	no	—
113	C₄×C₁₃	—	8	2	no	—
114	C₄×C₁₃	#475	1	1	yes	characteristic
115	C₄×C₁₃	#477	1	1	yes	characteristic
116	C₄×C₁₃	#487	1	1	yes	characteristic
117	C₄×C₁₃	#489	1	1	yes	characteristic
118	Dic₁₃	—	8	4	no	—
119	C₂²×Q₈	—	26	2	no	—
120	C₂×M₁₆	—	2	1	no	—
121	C₄⋊Q₈	—	52	2	no	—
122	C₄⋊Q₈	—	52	2	no	—
123	C₄⋊Q₈	Dic₂₆	1	2	yes	complemented
124	#125	—	2	1	no	—
125	C₂²⋊Q₈	—	2	2	no	—
126	C₄×Q₈	—	2	2	no	—
127	#117	—	2	1	no	—
128	C₂×C₄⋊C₄	—	26	2	no	—
129	C₂×C₄⋊C₄	—	26	2	no	—
130	C₂×C₄²	—	2	2	no	—
131	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
132	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
133	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	26	2	no	—
136	C₂²⋊C₈	—	26	2	no	—
137	C₄×C₈	—	26	2	no	—
138	#95	—	13	4	no	—
139	#95	—	26	4	no	—
140	#95	—	26	2	no	—
141	C₂×C₁₃	—	2	1	no	—
142	C₂×C₁₃	—	2	1	no	—
143	C₂×C₁₃	#2024	1	1	yes	characteristic
144	C₂×C₁₃	#2031	1	2	yes	—
145	C₂×Q₈	—	2	2	no	—
146	C₂×Q₈	—	52	4	no	—
147	C₂×Q₈	—	52	8	no	—
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	26	4	no	—
151	C₂²×C₄	D₄×D₁₃	1	2	yes	—
152	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
153	M₁₆	—	4	1	no	—
154	M₁₆	—	4	1	no	—
155	C₂×C₈	—	4	1	no	—
156	C₂×C₈	—	52	2	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	52	4	no	—
161	C₄⋊C₄	—	52	2	no	—
162	C₄⋊C₄	—	52	2	no	—
163	C₂²⋊C₄	—	4	1	no	—
164	C₄²	—	2	1	no	—
165	C₄²	—	2	1	no	—
166	C₄²	—	4	4	no	—
167	C₄²	—	4	1	no	—
168	C₄²	#6106	1	2	yes	—
169	C₄²	#6107	1	2	yes	—
170	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
171	C₄²	#6127	1	1	yes	characteristic
172	C₄²	#6129	1	2	yes	—
173	C₁₃	#10363	1	1	yes	characteristic, complemented, Sylow
174	C₂³	#88191	1	1	yes	characteristic
175	Q₈	—	8	2	no	—
176	Q₈	—	104	8	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	52	8	no	—
184	C₂×C₄	—	52	4	no	—
185	C₂×C₄	#88132	1	2	yes	—
186	C₂×C₄	#88139	1	2	yes	—
187	C₂×C₄	#88178	1	1	yes	characteristic
188	C₂×C₄	#88190	1	1	yes	characteristic
189	C₂×C₄	C₂×#6106	1	1	yes	characteristic
190	C₂×C₄	C₂×#6107	1	1	yes	characteristic
191	C₂×C₄	#88203	1	1	yes	characteristic
192	C₂×C₄	#88205	1	1	yes	characteristic
193	C₈	—	8	1	no	—
194	C₈	—	104	2	no	—
195	C₂²	—	2	2	no	—
196	C₂²	#11718608	1	1	yes	characteristic
197	C₂²	#11718707	1	1	yes	characteristic
198	C₂²	#11718711	1	1	yes	characteristic
199	C₄	—	2	4	no	—
200	C₄	—	2	2	no	—
201	C₄	—	2	2	no	—
202	C₄	—	4	2	no	—
203	C₄	—	8	2	no	—
204	C₄	—	104	4	no	—
205	C₄	#11718681	1	1	yes	characteristic
206	C₄	#11718682	1	1	yes	characteristic
207	C₄	#11718716	1	1	yes	characteristic
208	C₄	#11718742	1	1	yes	characteristic
209	C₂	—	2	1	no	—
210	C₂	—	2	1	no	—
211	C₂	#13115977	1	1	yes	characteristic
212	C₂	#13116008	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13123214

C₂

yes

maximal, characteristic

#13113400

C₂

yes

maximal

#13112696

C₂

yes

maximal

#13112669

C₂

yes

maximal, characteristic

C₁₃×#1994

C₂

yes

maximal, characteristic

C₁₃×#533

C₂²

yes

characteristic

C₂×#88123

—

C₁₃×#423

C₂²

yes

characteristic

C₄³×C₁₃

C₂²

yes

characteristic

#11718597

—

#11718597

C₂²

yes

—

#11718590

—

#11718586

C₂²

yes

—

C₁₃×#339

—

C₁₃×#339

—

#11718130

—

C₂²×Dic₂₆

—

C₂×C₁₃×M₁₆

—

C₁₃×C₄⋊Q₈

Q₈

yes

complemented

C₁₃×#125

—

C₁₃×C₂²⋊Q₈

—

C₄×Q₈×C₁₃

—

C₁₃×#117

—

C₂×C₄²×C₁₃

—

C₂×C₄²×C₁₃

D₄

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

characteristic

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

#88123

—

#88123

—

C₁₃×C₄⋊C₈

—

#88071

—

#88071

—

#88071

D₄

yes

—

C₂²⋊#1055

—

—

—

—

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

SD₁₆

yes

—

C₄²×C₁₃

Q₁₆

yes

—

C₄²×C₁₃

C₂×Q₈

yes

C₄²×C₁₃

yes

C₄²×C₁₃

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₂×#1055

—

#2862

—

#1994

D₁₃

yes

—

—

—

C₂³×C₁₃

#120

yes

characteristic

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

#124

yes

—

C₂×C₄×C₁₃

#124

yes

characteristic

C₂×C₄×C₁₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₃

C₂×Q₁₆

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

C₈×C₁₃

—

#1055

—

#533

C₂×D₁₃

yes

characteristic

C₂×C₄⋊Q₈

—

#431

—

#423

C₂×D₁₃

yes

characteristic

#391

—

#382

—

100

#382

—

101

C₄³

C₂×D₁₃

yes

characteristic

102

#339

—

103

#339

—

104

#328

—

105

C₂²×C₁₃

—

106

C₂²×C₁₃

#393

yes

characteristic

107

C₂²×C₁₃

#450

yes

characteristic

108

C₂²×C₁₃

#451

yes

characteristic

109

C₄×C₁₃

—

110

C₄×C₁₃

—

111

C₄×C₁₃

—

112

C₄×C₁₃

—

113

C₄×C₁₃

—

114

C₄×C₁₃

#475

yes

characteristic

115

C₄×C₁₃

#477

yes

characteristic

116

C₄×C₁₃

#487

yes

characteristic

117

C₄×C₁₃

#489

yes

characteristic

118

Dic₁₃

—

119

C₂²×Q₈

—

120

C₂×M₁₆

—

121

C₄⋊Q₈

—

122

C₄⋊Q₈

—

123

C₄⋊Q₈

Dic₂₆

yes

complemented

124

#125

—

125

C₂²⋊Q₈

—

126

C₄×Q₈

—

127

#117

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄²

—

131

C₂×C₄²

C₂²⋊D₁₃

yes

characteristic

132

C₂×C₄²

C₂²⋊D₁₃

yes

characteristic

133

C₂×C₄²

C₂²×D₁₃

yes

characteristic

134

C₄⋊C₈

—

135

C₄⋊C₈

—

136

C₂²⋊C₈

—

137

C₄×C₈

—

138

#95

—

139

#95

—

140

#95

—

141

C₂×C₁₃

—

142

C₂×C₁₃

—

143

C₂×C₁₃

#2024

yes

characteristic

144

C₂×C₁₃

#2031

yes

—

145

C₂×Q₈

—

146

C₂×Q₈

—

147

C₂×Q₈

—

148

C₂²×C₄

—

149

C₂²×C₄

—

150

C₂²×C₄

—

151

C₂²×C₄

D₄×D₁₃

yes

—

152

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

153

M₁₆

—

154

M₁₆

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₂²⋊C₄

—

164

C₄²

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₄²

#6106

yes

—

169

C₄²

#6107

yes

—

170

C₄²

C₂×Dic₂₆

yes

characteristic

171

C₄²

#6127

yes

characteristic

172

C₄²

#6129

yes

—

173

C₁₃

#10363

yes

characteristic, complemented, Sylow

174

C₂³

#88191

yes

characteristic

175

Q₈

—

176

Q₈

—

104

—

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

#88132

yes

—

186

C₂×C₄

#88139

yes

—

187

C₂×C₄

#88178

yes

characteristic

188

C₂×C₄

#88190

yes

characteristic

189

C₂×C₄

C₂×#6106

yes

characteristic

190

C₂×C₄

C₂×#6107

yes

characteristic

191

C₂×C₄

#88203

yes

characteristic

192

C₂×C₄

#88205

yes

characteristic

193

C₈

—

194

C₈

—

104

—

195

C₂²

—

196

C₂²

#11718608

yes

characteristic

197

C₂²

#11718707

yes

characteristic

198

C₂²

#11718711

yes

characteristic

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

—

104

—

205

C₄

#11718681

yes

characteristic

206

C₄

#11718682

yes

characteristic

207

C₄

#11718716

yes

characteristic

208

C₄

#11718742

yes

characteristic

209

C₂

—

210

C₂

—

211

C₂

#13115977

yes

characteristic

212

C₂

#13116008

yes

—

	subgroup	G/H
center	C₂²	#11718608
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1994	C₂
socle	C₂²×C₁₃	#393
prime radical	C₂³×C₁₃	#120
hypercenter	#1994	D₁₃
hypocenter	C₁₃	#10363
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1994

subgroup

G/H

center

C₂²

#11718608

commutator, cosocle

C₂×C₄²×C₁₃

C₂³

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1994

C₂

C₂²×C₁₃

C₂³×C₁₃

D₁₃

C₁₃

G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1994

N	H ≅ G/N	type
19. C₁₃×C₄⋊Q₈	Q₈	semidirect
123. C₄⋊Q₈	Dic₂₆	semidirect
173. C₁₃	#10363	semidirect

H ≅ G/N

type

19. C₁₃×C₄⋊Q₈

Q₈

semidirect

123. C₄⋊Q₈

Dic₂₆

semidirect

173. C₁₃

#10363

semidirect

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

subgroup	index
#13123214	2
#13113400	2
#13112696	2
#13112669	2
C₁₃×#1994	2
#10363	13

subgroup

index

C₁₃×#1994

Subgroups — #15766012

Conjugacy classes of subgroups

Complements

Maximal subgroups