Group #15696869 — subgroups

Subgroups — C₁₃×#10530 #15696869

Order 3328.

402 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 98 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	C₁₃×#3253	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#1560	C₂	1	2	yes	maximal, complemented
4	C₁₃×#1492	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1309	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#532	C₂²	1	2	yes	—
7	C₁₃×#476	—	2	2	no	—
8	C₁₃×#469	—	2	2	no	—
9	C₁₃×#459	—	2	2	no	—
10	C₂×C₁₃×#108	—	2	2	no	—
11	C₁₃×#423	C₂²	1	2	yes	—
12	C₁₃×#423	C₂²	1	1	yes	characteristic
13	C₁₃×#423	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₁₃×#333	—	2	2	no	—
16	C₁₃×#331	—	2	2	no	—
17	C₁₃×D₄⋊C₈	—	2	4	no	—
18	C₂×C₁₃×#41	—	2	2	no	—
19	C₂×C₁₃×D₈	—	2	2	no	—
20	C₂×C₁₃×M₁₆	—	2	1	no	—
21	C₂×C₁₃×M₁₆	—	2	2	no	—
22	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
23	C₂²×C₈×C₁₃	—	2	1	no	—
24	C₁₃×C₄⋊Q₈	D₄	1	2	yes	—
25	C₁₃×#127	D₄	1	2	yes	—
26	C₁₃×#124	—	2	2	no	—
27	C₁₃×#121	—	2	2	no	—
28	C₁₃×#121	—	2	2	no	—
29	C₄×D₄×C₁₃	—	2	2	no	—
30	C₄×D₄×C₁₃	—	2	2	no	—
31	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
32	C₁₃×#108	—	4	4	no	—
33	C₁₃×#106	—	2	2	no	—
34	C₁₃×C₄⋊C₈	—	2	2	no	—
35	C₁₃×C₄⋊C₈	—	2	2	no	—
36	C₁₃×C₄⋊C₈	—	2	2	no	—
37	C₁₃×C₄⋊C₈	—	2	2	no	—
38	C₁₃×#102	—	2	4	no	—
39	C₄×C₈×C₁₃	—	2	1	no	—
40	C₄×C₈×C₁₃	D₄	1	2	yes	—
41	#10530	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₁₃×#41	—	8	2	no	—
43	C₁₃×#41	—	8	2	no	—
44	C₂×Q₈×C₁₃	—	2	2	no	—
45	C₂×D₄×C₁₃	—	2	2	no	—
46	C₂×D₄×C₁₃	—	2	2	no	—
47	C₂×D₄×C₁₃	—	2	2	no	—
48	C₂×D₄×C₁₃	—	4	2	no	—
49	C₂²×C₄×C₁₃	—	4	2	no	—
50	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	#41	1	2	yes	—
52	C₁₃×D₈	—	8	2	no	—
53	C₁₃×M₁₆	—	4	2	no	—
54	C₁₃×M₁₆	—	4	2	no	—
55	C₁₃×M₁₆	—	8	2	no	—
56	C₂×C₈×C₁₃	—	2	2	no	—
57	C₂×C₈×C₁₃	—	2	2	no	—
58	C₂×C₈×C₁₃	—	2	2	no	—
59	C₂×C₈×C₁₃	—	2	2	no	—
60	C₂×C₈×C₁₃	—	4	1	no	—
61	C₂×C₈×C₁₃	—	4	2	no	—
62	C₁₃×C₄⋊C₄	—	2	2	no	—
63	C₁₃×C₄⋊C₄	—	2	2	no	—
64	C₁₃×C₂²⋊C₄	—	4	2	no	—
65	C₁₃×C₂²⋊C₄	—	4	2	no	—
66	C₄²×C₁₃	C₂×D₄	1	2	yes	—
67	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₁₃	#41	1	1	yes	characteristic
69	#3253	C₂×C₁₃	1	1	yes	characteristic
70	#1848	C₂×C₁₃	1	1	yes	characteristic, complemented
71	#1560	C₂×C₁₃	1	2	yes	complemented
72	#1492	C₂×C₁₃	1	2	yes	complemented
73	#1309	C₂×C₁₃	1	1	yes	characteristic, complemented
74	C₂³×C₁₃	—	4	2	no	—
75	C₂³×C₁₃	#124	1	1	yes	characteristic
76	Q₈×C₁₃	—	8	2	no	—
77	D₄×C₁₃	—	4	2	no	—
78	D₄×C₁₃	—	4	2	no	—
79	D₄×C₁₃	—	4	2	no	—
80	D₄×C₁₃	—	4	2	no	—
81	D₄×C₁₃	—	8	2	no	—
82	D₄×C₁₃	—	8	4	no	—
83	C₂×C₄×C₁₃	—	2	1	no	—
84	C₂×C₄×C₁₃	—	2	1	no	—
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	—	4	2	no	—
88	C₂×C₄×C₁₃	—	8	2	no	—
89	C₂×C₄×C₁₃	—	8	2	no	—
90	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
91	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
92	C₂×C₄×C₁₃	#121	1	2	yes	—
93	C₂×C₄×C₁₃	#123	1	2	yes	—
94	C₈×C₁₃	—	2	4	no	—
95	C₈×C₁₃	—	4	2	no	—
96	C₈×C₁₃	—	8	1	no	—
97	C₈×C₁₃	—	8	2	no	—
98	#532	C₂²×C₁₃	1	2	yes	—
99	#476	—	2	2	no	—
100	#469	—	2	2	no	—
101	#459	—	2	2	no	—
102	C₂×#108	—	2	2	no	—
103	#423	C₂²×C₁₃	1	1	yes	characteristic
104	#423	C₂²×C₁₃	1	2	yes	—
105	#423	C₂²×C₁₃	1	1	yes	characteristic
106	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
107	#333	—	2	2	no	—
108	#331	—	2	2	no	—
109	D₄⋊C₈	—	2	4	no	—
110	C₂²×C₁₃	—	2	2	no	—
111	C₂²×C₁₃	—	2	1	no	—
112	C₂²×C₁₃	—	8	2	no	—
113	C₂²×C₁₃	—	8	2	no	—
114	C₂²×C₁₃	—	8	2	no	—
115	C₂²×C₁₃	#394	1	1	yes	characteristic
116	C₄×C₁₃	—	2	2	no	—
117	C₄×C₁₃	—	2	1	no	—
118	C₄×C₁₃	—	8	2	no	—
119	C₄×C₁₃	#453	1	1	yes	characteristic
120	C₄×C₁₃	#456	1	1	yes	characteristic
121	C₄×C₁₃	#471	1	2	yes	—
122	C₄×C₁₃	#472	1	2	yes	—
123	C₂×#41	—	2	2	no	—
124	C₂×D₈	—	2	2	no	—
125	C₂×M₁₆	—	2	1	no	—
126	C₂×M₁₆	—	2	2	no	—
127	C₂×M₁₆	D₄×C₁₃	1	2	yes	—
128	C₂²×C₈	—	2	1	no	—
129	C₄⋊Q₈	D₄×C₁₃	1	2	yes	—
130	#127	D₄×C₁₃	1	2	yes	—
131	#124	—	2	2	no	—
132	#121	—	2	2	no	—
133	#121	—	2	2	no	—
134	C₄×D₄	—	2	2	no	—
135	C₄×D₄	—	2	2	no	—
136	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
137	#108	—	4	4	no	—
138	#106	—	2	2	no	—
139	C₄⋊C₈	—	2	2	no	—
140	C₄⋊C₈	—	2	2	no	—
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	—	2	2	no	—
143	#102	—	2	4	no	—
144	C₄×C₈	—	2	1	no	—
145	C₄×C₈	D₄×C₁₃	1	2	yes	—
146	C₂×C₁₃	—	4	1	no	—
147	C₂×C₁₃	—	16	2	no	—
148	C₂×C₁₃	#2039	1	1	yes	characteristic
149	C₂×C₁₃	#2051	1	2	yes	—
150	#41	—	8	2	no	—
151	#41	—	8	2	no	—
152	C₂×Q₈	—	2	2	no	—
153	C₂×D₄	—	2	2	no	—
154	C₂×D₄	—	2	2	no	—
155	C₂×D₄	—	2	2	no	—
156	C₂×D₄	—	4	2	no	—
157	C₂²×C₄	—	4	2	no	—
158	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
159	C₂²×C₄	C₁₃×#41	1	2	yes	—
160	D₈	—	8	2	no	—
161	M₁₆	—	4	2	no	—
162	M₁₆	—	4	2	no	—
163	M₁₆	—	8	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	4	1	no	—
169	C₂×C₈	—	4	2	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₂²⋊C₄	—	4	2	no	—
173	C₂²⋊C₄	—	4	2	no	—
174	C₄²	C₂×D₄×C₁₃	1	2	yes	—
175	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
176	C₄²	C₁₃×#41	1	1	yes	characteristic
177	C₁₃	#10530	1	1	yes	characteristic, direct product factor, Sylow
178	C₂³	—	4	2	no	—
179	C₂³	C₁₃×#124	1	1	yes	characteristic
180	Q₈	—	8	2	no	—
181	D₄	—	4	2	no	—
182	D₄	—	4	2	no	—
183	D₄	—	4	2	no	—
184	D₄	—	4	2	no	—
185	D₄	—	8	2	no	—
186	D₄	—	8	4	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	8	2	no	—
193	C₂×C₄	—	8	2	no	—
194	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
195	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
196	C₂×C₄	C₁₃×#121	1	2	yes	—
197	C₂×C₄	C₁₃×#123	1	2	yes	—
198	C₈	—	2	4	no	—
199	C₈	—	4	2	no	—
200	C₈	—	8	1	no	—
201	C₈	—	8	2	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	2	2	no	—
204	C₂²	—	8	2	no	—
205	C₂²	—	8	2	no	—
206	C₂²	—	8	2	no	—
207	C₂²	C₁₃×#394	1	1	yes	characteristic
208	C₄	—	2	1	no	—
209	C₄	—	2	2	no	—
210	C₄	—	8	2	no	—
211	C₄	C₁₃×#453	1	1	yes	characteristic
212	C₄	C₁₃×#456	1	1	yes	characteristic
213	C₄	C₁₃×#471	1	2	yes	—
214	C₄	C₁₃×#472	1	2	yes	—
215	C₂	—	4	1	no	—
216	C₂	—	16	2	no	—
217	C₂	C₁₃×#2039	1	1	yes	characteristic
218	C₂	C₁₃×#2051	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#3253

C₂

yes

maximal, characteristic

C₁₃×#1848

C₂

yes

maximal, characteristic, complemented

C₁₃×#1560

C₂

yes

maximal, complemented

C₁₃×#1492

C₂

yes

maximal, complemented

C₁₃×#1309

C₂

yes

maximal, characteristic, complemented

C₁₃×#532

C₂²

yes

—

C₁₃×#476

—

C₁₃×#469

—

C₁₃×#459

—

C₂×C₁₃×#108

—

C₁₃×#423

C₂²

yes

—

C₁₃×#423

C₂²

yes

characteristic

C₁₃×#423

C₂²

yes

characteristic

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₁₃×#333

—

C₁₃×#331

—

C₁₃×D₄⋊C₈

—

C₂×C₁₃×#41

—

C₂×C₁₃×D₈

—

C₂×C₁₃×M₁₆

—

C₂×C₁₃×M₁₆

—

C₂×C₁₃×M₁₆

D₄

yes

—

C₂²×C₈×C₁₃

—

C₁₃×C₄⋊Q₈

D₄

yes

—

C₁₃×#127

D₄

yes

—

C₁₃×#124

—

C₁₃×#121

—

C₁₃×#121

—

C₄×D₄×C₁₃

—

C₄×D₄×C₁₃

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₁₃×#108

—

C₁₃×#106

—

C₁₃×C₄⋊C₈

—

C₁₃×C₄⋊C₈

—

C₁₃×C₄⋊C₈

—

C₁₃×C₄⋊C₈

—

C₁₃×#102

—

C₄×C₈×C₁₃

—

C₄×C₈×C₁₃

D₄

yes

—

#10530

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₁₃×#41

—

C₁₃×#41

—

C₂×Q₈×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

#41

yes

—

C₁₃×D₈

—

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

C₂×D₄

yes

C₄²×C₁₃

yes

C₂×C₁₃

yes

characteristic

#1848

C₂×C₁₃

yes

characteristic, complemented

#1560

C₂×C₁₃

yes

complemented

#1492

C₂×C₁₃

yes

complemented

#1309

C₂×C₁₃

yes

characteristic, complemented

C₂³×C₁₃

—

C₂³×C₁₃

#124

yes

characteristic

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

—

C₂×C₄×C₁₃

#123

yes

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

#532

C₂²×C₁₃

yes

—

#476

—

100

#469

—

101

#459

—

102

C₂×#108

—

103

#423

C₂²×C₁₃

yes

characteristic

104

#423

C₂²×C₁₃

yes

—

105

#423

C₂²×C₁₃

yes

characteristic

106

C₂×C₄×C₈

C₂²×C₁₃

yes

characteristic

107

#333

—

108

#331

—

109

D₄⋊C₈

—

110

C₂²×C₁₃

—

111

C₂²×C₁₃

—

112

C₂²×C₁₃

—

113

C₂²×C₁₃

—

114

C₂²×C₁₃

—

115

C₂²×C₁₃

#394

yes

characteristic

116

C₄×C₁₃

—

117

C₄×C₁₃

—

118

C₄×C₁₃

—

119

C₄×C₁₃

#453

yes

characteristic

120

C₄×C₁₃

#456

yes

characteristic

121

C₄×C₁₃

#471

yes

—

122

C₄×C₁₃

#472

yes

—

123

C₂×#41

—

124

C₂×D₈

—

125

C₂×M₁₆

—

126

C₂×M₁₆

—

127

C₂×M₁₆

D₄×C₁₃

yes

—

128

C₂²×C₈

—

129

C₄⋊Q₈

D₄×C₁₃

yes

—

130

#127

D₄×C₁₃

yes

—

131

#124

—

132

#121

—

133

#121

—

134

C₄×D₄

—

135

C₄×D₄

—

136

C₂×C₄²

C₂³×C₁₃

yes

characteristic

137

#108

—

138

#106

—

139

C₄⋊C₈

—

140

C₄⋊C₈

—

141

C₄⋊C₈

—

142

C₄⋊C₈

—

143

#102

—

144

C₄×C₈

—

145

C₄×C₈

D₄×C₁₃

yes

—

146

C₂×C₁₃

—

147

C₂×C₁₃

—

148

C₂×C₁₃

#2039

yes

characteristic

149

C₂×C₁₃

#2051

yes

—

150

#41

—

151

#41

—

152

C₂×Q₈

—

153

C₂×D₄

—

154

C₂×D₄

—

155

C₂×D₄

—

156

C₂×D₄

—

157

C₂²×C₄

—

158

C₂²×C₄

C₂×D₄×C₁₃

yes

characteristic

159

C₂²×C₄

C₁₃×#41

yes

—

160

D₈

—

161

M₁₆

—

162

M₁₆

—

163

M₁₆

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₂²⋊C₄

—

173

C₂²⋊C₄

—

174

C₄²

C₂×D₄×C₁₃

yes

—

175

C₄²

C₂×D₄×C₁₃

yes

characteristic

176

C₄²

C₁₃×#41

yes

characteristic

177

C₁₃

#10530

yes

characteristic, direct product factor, Sylow

178

C₂³

—

179

C₂³

C₁₃×#124

yes

characteristic

180

Q₈

—

181

D₄

—

182

D₄

—

183

D₄

—

184

D₄

—

185

D₄

—

186

D₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

C₁₃×#120

yes

characteristic

195

C₂×C₄

C₁₃×#121

yes

characteristic

196

C₂×C₄

C₁₃×#121

yes

—

197

C₂×C₄

C₁₃×#123

yes

—

198

C₈

—

199

C₈

—

200

C₈

—

201

C₈

—

202

C₂²

—

203

C₂²

—

204

C₂²

—

205

C₂²

—

206

C₂²

—

207

C₂²

C₁₃×#394

yes

characteristic

208

C₄

—

209

C₄

—

210

C₄

—

211

C₄

C₁₃×#453

yes

characteristic

212

C₄

C₁₃×#456

yes

characteristic

213

C₄

C₁₃×#471

yes

—

214

C₄

C₁₃×#472

yes

—

215

C₂

—

216

C₂

—

217

C₂

C₁₃×#2039

yes

characteristic

218

C₂

C₁₃×#2051

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₃	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
prime radical	C₁₃×#3253	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₃

#394

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₁₃

prime radical

C₁₃×#3253

C₂

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

N	H ≅ G/N	type
2. C₁₃×#1848	C₂	semidirect
3. C₁₃×#1560	C₂	semidirect
4. C₁₃×#1492	C₂	semidirect
5. C₁₃×#1309	C₂	semidirect
41. #10530	C₁₃	direct (N × H)
70. #1848	C₂×C₁₃	semidirect
71. #1560	C₂×C₁₃	semidirect
72. #1492	C₂×C₁₃	semidirect
73. #1309	C₂×C₁₃	semidirect
177. C₁₃	#10530	direct (N × H)

H ≅ G/N

type

2. C₁₃×#1848

C₂

semidirect

3. C₁₃×#1560

C₂

semidirect

4. C₁₃×#1492

C₂

semidirect

5. C₁₃×#1309

C₂

semidirect

41. #10530

C₁₃

direct (N × H)

70. #1848

C₂×C₁₃

semidirect

71. #1560

C₂×C₁₃

semidirect

72. #1492

C₂×C₁₃

semidirect

73. #1309

C₂×C₁₃

semidirect

177. C₁₃

#10530

direct (N × H)

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

subgroup	index
C₁₃×#3253	2
C₁₃×#1848	2
C₁₃×#1560	2
C₁₃×#1492	2
C₁₃×#1309	2
#10530	13

subgroup

index

C₁₃×#3253

C₁₃×#1848

C₁₃×#1560

C₁₃×#1492

C₁₃×#1309

#10530

Subgroups — C₁₃×#10530 #15696869

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C13×#10530 #15696869

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10530 #15696869