Group #15699778 — subgroups

Subgroups — C₁₃×#13439 #15699778

Order 3328.

418 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 110 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	C₂²×C₁₃×D₁₆	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2151	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#361	C₂	1	2	yes	maximal, complemented
4	C₁₃×#1978	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1381	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₁₃×D₈	C₂²	1	2	yes	—
7	C₂×C₁₃×D₁₆	—	2	4	no	—
8	C₂×C₁₃×D₁₆	—	2	2	no	—
9	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
10	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#108	C₂²	1	2	yes	—
12	C₁₃×#418	—	2	2	no	—
13	C₂×C₁₃×#100	—	2	2	no	—
14	C₁₃×#405	C₂²	1	1	yes	characteristic
15	C₁₃×C₄⋊C₁₆	—	2	2	no	—
16	C₁₃×#361	—	2	4	no	—
17	C₂²×D₄×C₁₃	—	2	2	no	—
18	C₂×C₁₃×D₈	—	2	2	no	—
19	C₂×C₁₃×D₈	—	2	2	no	—
20	C₂×C₁₃×D₈	—	2	2	no	—
21	C₂×C₁₃×D₈	—	2	2	no	—
22	C₂×C₁₃×D₈	—	2	2	no	—
23	C₂×C₁₃×D₈	D₄	1	4	yes	—
24	C₂×C₁₃×M₁₆	—	2	2	no	—
25	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
27	C₁₃×#117	D₄	1	1	yes	characteristic
28	C₁₃×D₁₆	—	2	8	no	—
29	C₁₃×M₃₂	—	2	2	no	—
30	C₂×C₁₃×C₁₆	—	2	2	no	—
31	C₂×C₁₃×C₁₆	—	2	1	no	—
32	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
33	C₁₃×#108	—	2	2	no	—
34	C₁₃×#108	—	2	2	no	—
35	C₁₃×#102	—	4	4	no	—
36	C₁₃×#100	—	4	4	no	—
37	C₁₃×#97	—	2	1	no	—
38	C₄×C₈×C₁₃	—	2	1	no	—
39	#13439	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂⁴×C₁₃	—	4	2	no	—
41	C₂×D₄×C₁₃	—	2	4	no	—
42	C₂×D₄×C₁₃	—	4	2	no	—
43	C₂×D₄×C₁₃	—	4	2	no	—
44	C₂×D₄×C₁₃	—	4	2	no	—
45	C₂×D₄×C₁₃	—	4	2	no	—
46	C₂×D₄×C₁₃	—	4	2	no	—
47	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
48	C₁₃×D₈	—	2	4	no	—
49	C₁₃×D₈	—	2	4	no	—
50	C₁₃×D₈	—	4	2	no	—
51	C₁₃×D₈	—	4	2	no	—
52	C₁₃×M₁₆	—	2	2	no	—
53	C₁₃×M₁₆	—	4	2	no	—
54	C₁₃×M₁₆	—	4	2	no	—
55	C₂×C₈×C₁₃	—	2	1	no	—
56	C₂×C₈×C₁₃	—	4	2	no	—
57	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
60	C₂×C₈×C₁₃	#41	1	2	yes	—
61	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
62	C₁₃×C₂²⋊C₄	—	2	1	no	—
63	C₄²×C₁₃	—	2	1	no	—
64	C₁₃×C₁₆	—	2	2	no	—
65	C₁₃×C₁₆	—	4	1	no	—
66	C₂²×D₁₆	C₂×C₁₃	1	1	yes	characteristic
67	#2151	C₂×C₁₃	1	1	yes	characteristic, complemented
68	C₂×#361	C₂×C₁₃	1	2	yes	complemented
69	#1978	C₂×C₁₃	1	2	yes	complemented
70	#1381	C₂×C₁₃	1	1	yes	characteristic, complemented
71	C₂³×C₁₃	—	4	4	no	—
72	C₂³×C₁₃	—	8	2	no	—
73	C₂³×C₁₃	—	8	2	no	—
74	C₂³×C₁₃	—	16	4	no	—
75	C₂³×C₁₃	#135	1	1	yes	characteristic
76	D₄×C₁₃	—	4	4	no	—
77	D₄×C₁₃	—	4	4	no	—
78	D₄×C₁₃	—	8	2	no	—
79	D₄×C₁₃	—	8	2	no	—
80	C₂×C₄×C₁₃	—	2	2	no	—
81	C₂×C₄×C₁₃	#121	1	2	yes	—
82	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
85	C₈×C₁₃	—	4	1	no	—
86	C₈×C₁₃	—	8	2	no	—
87	C₈×C₁₃	#120	1	1	yes	characteristic
88	C₈×C₁₃	#123	1	2	yes	—
89	C₈×C₁₃	#124	1	1	yes	characteristic
90	C₂²×D₈	C₂²×C₁₃	1	2	yes	—
91	C₂×D₁₆	—	2	4	no	—
92	C₂×D₁₆	—	2	2	no	—
93	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
94	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
95	C₂×#108	C₂²×C₁₃	1	2	yes	—
96	#418	—	2	2	no	—
97	C₂×#100	—	2	2	no	—
98	#405	C₂²×C₁₃	1	1	yes	characteristic
99	C₄⋊C₁₆	—	2	2	no	—
100	#361	—	2	4	no	—
101	C₂²×C₁₃	—	2	2	no	—
102	C₂²×C₁₃	—	8	4	no	—
103	C₂²×C₁₃	—	16	4	no	—
104	C₂²×C₁₃	—	16	4	no	—
105	C₂²×C₁₃	—	16	4	no	—
106	C₂²×C₁₃	#463	1	2	yes	—
107	C₂²×C₁₃	#466	1	1	yes	characteristic
108	C₄×C₁₃	—	4	2	no	—
109	C₄×C₁₃	#394	1	1	yes	characteristic
110	C₄×C₁₃	#459	1	2	yes	—
111	C₄×C₁₃	#469	1	1	yes	characteristic
112	C₂²×D₄	—	2	2	no	—
113	C₂×D₈	—	2	2	no	—
114	C₂×D₈	—	2	2	no	—
115	C₂×D₈	—	2	2	no	—
116	C₂×D₈	—	2	2	no	—
117	C₂×D₈	—	2	2	no	—
118	C₂×D₈	D₄×C₁₃	1	4	yes	—
119	C₂×M₁₆	—	2	2	no	—
120	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
121	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
122	#117	D₄×C₁₃	1	1	yes	characteristic
123	D₁₆	—	2	8	no	—
124	M₃₂	—	2	2	no	—
125	C₂×C₁₆	—	2	2	no	—
126	C₂×C₁₆	—	2	1	no	—
127	C₂×C₁₆	D₄×C₁₃	1	2	yes	—
128	#108	—	2	2	no	—
129	#108	—	2	2	no	—
130	#102	—	4	4	no	—
131	#100	—	4	4	no	—
132	#97	—	2	1	no	—
133	C₄×C₈	—	2	1	no	—
134	C₂×C₁₃	—	2	2	no	—
135	C₂×C₁₃	—	16	4	no	—
136	C₂×C₁₃	#2054	1	1	yes	characteristic
137	C₂×C₁₃	#2221	1	2	yes	—
138	C₂⁴	—	4	2	no	—
139	C₂×D₄	—	2	4	no	—
140	C₂×D₄	—	4	2	no	—
141	C₂×D₄	—	4	2	no	—
142	C₂×D₄	—	4	2	no	—
143	C₂×D₄	—	4	2	no	—
144	C₂×D₄	—	4	2	no	—
145	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
146	D₈	—	2	4	no	—
147	D₈	—	2	4	no	—
148	D₈	—	4	2	no	—
149	D₈	—	4	2	no	—
150	M₁₆	—	2	2	no	—
151	M₁₆	—	4	2	no	—
152	M₁₆	—	4	2	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
156	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
157	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
158	C₂×C₈	C₁₃×#41	1	2	yes	—
159	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
160	C₂²⋊C₄	—	2	1	no	—
161	C₄²	—	2	1	no	—
162	C₁₆	—	2	2	no	—
163	C₁₆	—	4	1	no	—
164	C₁₃	#13439	1	1	yes	characteristic, direct product factor, Sylow
165	C₂³	—	4	4	no	—
166	C₂³	—	8	2	no	—
167	C₂³	—	8	2	no	—
168	C₂³	—	16	4	no	—
169	C₂³	C₁₃×#135	1	1	yes	characteristic
170	D₄	—	4	4	no	—
171	D₄	—	4	4	no	—
172	D₄	—	8	2	no	—
173	D₄	—	8	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	C₁₃×#121	1	2	yes	—
176	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
177	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
178	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
179	C₈	—	4	1	no	—
180	C₈	—	8	2	no	—
181	C₈	C₁₃×#120	1	1	yes	characteristic
182	C₈	C₁₃×#123	1	2	yes	—
183	C₈	C₁₃×#124	1	1	yes	characteristic
184	C₂²	—	2	2	no	—
185	C₂²	—	8	4	no	—
186	C₂²	—	16	4	no	—
187	C₂²	—	16	4	no	—
188	C₂²	—	16	4	no	—
189	C₂²	C₁₃×#463	1	2	yes	—
190	C₂²	C₁₃×#466	1	1	yes	characteristic
191	C₄	—	4	2	no	—
192	C₄	C₁₃×#394	1	1	yes	characteristic
193	C₄	C₁₃×#459	1	2	yes	—
194	C₄	C₁₃×#469	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	16	4	no	—
197	C₂	C₁₃×#2054	1	1	yes	characteristic
198	C₂	C₁₃×#2221	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₁₃×D₁₆

C₂

yes

maximal, characteristic

C₁₃×#2151

C₂

yes

maximal, characteristic, complemented

C₂×C₁₃×#361

C₂

yes

maximal, complemented

C₁₃×#1978

C₂

yes

maximal, complemented

C₁₃×#1381

C₂

yes

maximal, characteristic, complemented

C₂²×C₁₃×D₈

C₂²

yes

—

C₂×C₁₃×D₁₆

—

C₂×C₁₃×D₁₆

—

C₂×C₁₃×M₃₂

C₂²

yes

characteristic

C₂²×C₁₃×C₁₆

C₂²

yes

characteristic

C₂×C₁₃×#108

C₂²

yes

—

C₁₃×#418

—

C₂×C₁₃×#100

—

C₁₃×#405

C₂²

yes

characteristic

C₁₃×C₄⋊C₁₆

—

C₁₃×#361

—

C₂²×D₄×C₁₃

—

C₂×C₁₃×D₈

—

C₂×C₁₃×D₈

—

C₂×C₁₃×D₈

—

C₂×C₁₃×D₈

—

C₂×C₁₃×D₈

—

C₂×C₁₃×D₈

D₄

yes

—

C₂×C₁₃×M₁₆

—

C₂×C₁₃×M₁₆

D₄

yes

characteristic

C₂²×C₈×C₁₃

C₂³

yes

characteristic

C₁₃×#117

D₄

yes

characteristic

C₁₃×D₁₆

—

C₁₃×M₃₂

—

C₂×C₁₃×C₁₆

—

C₂×C₁₃×C₁₆

—

C₂×C₁₃×C₁₆

D₄

yes

—

C₁₃×#108

—

C₁₃×#108

—

C₁₃×#102

—

C₁₃×#100

—

C₁₃×#97

—

C₄×C₈×C₁₃

—

#13439

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₂⁴×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₁₃×D₈

—

C₁₃×D₈

—

C₁₃×D₈

—

C₁₃×D₈

—

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

C₂×C₈×C₁₃

yes

C₂×C₈×C₁₃

yes

—

C₁₃×C₄⋊C₄

D₈

yes

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

—

C₁₃×C₁₆

—

C₁₃×C₁₆

—

C₂²×D₁₆

C₂×C₁₃

yes

characteristic

#2151

C₂×C₁₃

yes

characteristic, complemented

C₂×#361

C₂×C₁₃

yes

complemented

#1978

C₂×C₁₃

yes

complemented

#1381

C₂×C₁₃

yes

characteristic, complemented

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

#135

yes

characteristic

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#121

yes

—

C₂×C₄×C₁₃

#121

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

#120

yes

characteristic

C₈×C₁₃

#123

yes

—

C₈×C₁₃

#124

yes

characteristic

C₂²×D₈

C₂²×C₁₃

yes

—

C₂×D₁₆

—

C₂×D₁₆

—

C₂×M₃₂

C₂²×C₁₃

yes

characteristic

C₂²×C₁₆

C₂²×C₁₃

yes

characteristic

C₂×#108

C₂²×C₁₃

yes

—

#418

—

C₂×#100

—

#405

C₂²×C₁₃

yes

characteristic

C₄⋊C₁₆

—

100

#361

—

101

C₂²×C₁₃

—

102

C₂²×C₁₃

—

103

C₂²×C₁₃

—

104

C₂²×C₁₃

—

105

C₂²×C₁₃

—

106

C₂²×C₁₃

#463

yes

—

107

C₂²×C₁₃

#466

yes

characteristic

108

C₄×C₁₃

—

109

C₄×C₁₃

#394

yes

characteristic

110

C₄×C₁₃

#459

yes

—

111

C₄×C₁₃

#469

yes

characteristic

112

C₂²×D₄

—

113

C₂×D₈

—

114

C₂×D₈

—

115

C₂×D₈

—

116

C₂×D₈

—

117

C₂×D₈

—

118

C₂×D₈

D₄×C₁₃

yes

—

119

C₂×M₁₆

—

120

C₂×M₁₆

D₄×C₁₃

yes

characteristic

121

C₂²×C₈

C₂³×C₁₃

yes

characteristic

122

#117

D₄×C₁₃

yes

characteristic

123

D₁₆

—

124

M₃₂

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

127

C₂×C₁₆

D₄×C₁₃

yes

—

128

#108

—

129

#108

—

130

#102

—

131

#100

—

132

#97

—

133

C₄×C₈

—

134

C₂×C₁₃

—

135

C₂×C₁₃

—

136

C₂×C₁₃

#2054

yes

characteristic

137

C₂×C₁₃

#2221

yes

—

138

C₂⁴

—

139

C₂×D₄

—

140

C₂×D₄

—

141

C₂×D₄

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂²×C₄

C₂×D₄×C₁₃

yes

characteristic

146

D₈

—

147

D₈

—

148

D₈

—

149

D₈

—

150

M₁₆

—

151

M₁₆

—

152

M₁₆

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

C₂×D₄×C₁₃

yes

—

156

C₂×C₈

C₂×D₄×C₁₃

yes

characteristic

157

C₂×C₈

C₁₃×#41

yes

characteristic

158

C₂×C₈

C₁₃×#41

yes

—

159

C₄⋊C₄

C₁₃×D₈

yes

—

160

C₂²⋊C₄

—

161

C₄²

—

162

C₁₆

—

163

C₁₆

—

164

C₁₃

#13439

yes

characteristic, direct product factor, Sylow

165

C₂³

—

166

C₂³

—

167

C₂³

—

168

C₂³

—

169

C₂³

C₁₃×#135

yes

characteristic

170

D₄

—

171

D₄

—

172

D₄

—

173

D₄

—

174

C₂×C₄

—

175

C₂×C₄

C₁₃×#121

yes

—

176

C₂×C₄

C₁₃×#121

yes

characteristic

177

C₂×C₄

C₂×C₁₃×D₈

yes

characteristic

178

C₂×C₄

C₁₃×AGL(1,ℤ/8)

yes

—

179

C₈

—

180

C₈

—

181

C₈

C₁₃×#120

yes

characteristic

182

C₈

C₁₃×#123

yes

—

183

C₈

C₁₃×#124

yes

characteristic

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

C₁₃×#463

yes

—

190

C₂²

C₁₃×#466

yes

characteristic

191

C₄

—

192

C₄

C₁₃×#394

yes

characteristic

193

C₄

C₁₃×#459

yes

—

194

C₄

C₁₃×#469

yes

characteristic

195

C₂

—

196

C₂

—

197

C₂

C₁₃×#2054

yes

characteristic

198

C₂

C₁₃×#2221

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₃	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂²×C₁₃×D₁₆	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₁₃×#405 ◁ G

subgroup

G/H

center, socle

C₂²×C₁₃

#466

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₁₃

prime radical

C₂²×C₁₃×D₁₆

C₂

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₁₃×#405 ◁ G

N	H ≅ G/N	type
2. C₁₃×#2151	C₂	semidirect
3. C₂×C₁₃×#361	C₂	semidirect
4. C₁₃×#1978	C₂	semidirect
5. C₁₃×#1381	C₂	semidirect
39. #13439	C₁₃	direct (N × H)
67. #2151	C₂×C₁₃	semidirect
68. C₂×#361	C₂×C₁₃	semidirect
69. #1978	C₂×C₁₃	semidirect
70. #1381	C₂×C₁₃	semidirect
164. C₁₃	#13439	direct (N × H)

H ≅ G/N

type

2. C₁₃×#2151

C₂

semidirect

3. C₂×C₁₃×#361

C₂

semidirect

4. C₁₃×#1978

C₂

semidirect

5. C₁₃×#1381

C₂

semidirect

39. #13439

C₁₃

direct (N × H)

67. #2151

C₂×C₁₃

semidirect

68. C₂×#361

C₂×C₁₃

semidirect

69. #1978

C₂×C₁₃

semidirect

70. #1381

C₂×C₁₃

semidirect

164. C₁₃

#13439

direct (N × H)

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

subgroup	index
C₂²×C₁₃×D₁₆	2
C₁₃×#2151	2
C₂×C₁₃×#361	2
C₁₃×#1978	2
C₁₃×#1381	2
#13439	13

subgroup

index

C₂²×C₁₃×D₁₆

C₁₃×#2151

C₂×C₁₃×#361

C₁₃×#1978

C₁₃×#1381

#13439

Subgroups — C₁₃×#13439 #15699778

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C13×#13439 #15699778

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13439 #15699778