Group #13115159 — subgroups

Subgroups — #13115159

Order 1664.

412 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 113 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×C₁₃×#121	C₂	1	1	yes	maximal, characteristic
2	C₂×#88180	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88072	C₂	1	2	yes	maximal, complemented
4	C₂×C₂²⋊#1055	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718121	C₂	1	2	yes	maximal, complemented
6	C₂²×D₄×C₁₃	—	2	1	no	—
7	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
8	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
9	C₁₃×#121	—	2	2	no	—
10	C₁₃×#121	C₄	1	4	yes	complemented
11	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
13	C₂×C₂²⋊Dic₁₃	—	2	1	no	—
14	#88180	—	2	4	no	—
15	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
16	C₂×C₄×Dic₁₃	—	2	1	no	—
17	C₂²×#1055	C₂²	1	2	yes	—
18	#88072	—	2	4	no	—
19	C₂²⋊#1055	—	2	2	no	—
20	C₂⁴×C₁₃	—	2	1	no	—
21	C₂⁴×C₁₃	D₄	1	1	yes	characteristic
22	C₂×D₄×C₁₃	—	2	4	no	—
23	C₂×D₄×C₁₃	—	2	2	no	—
24	C₂×D₄×C₁₃	—	2	1	no	—
25	C₂×D₄×C₁₃	—	2	2	no	—
26	C₂×D₄×C₁₃	—	4	4	no	—
27	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₁₃	—	2	2	no	—
29	C₂²×C₄×C₁₃	—	2	2	no	—
30	C₂²×C₄×C₁₃	—	2	1	no	—
31	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₃	D₄	1	2	yes	—
34	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
35	C₂²×Dic₁₃	—	2	2	no	—
36	C₁₃×C₄⋊C₄	—	2	1	no	—
37	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₁₃×C₂²⋊C₄	—	2	4	no	—
39	C₂²⋊Dic₁₃	—	2	4	no	—
40	C₄⋊Dic₁₃	—	2	2	no	—
41	C₄⋊Dic₁₃	D₄	1	4	yes	complemented
42	C₄×Dic₁₃	—	2	4	no	—
43	C₂×#1055	—	2	2	no	—
44	C₂×#1055	—	2	2	no	—
45	C₂×#1055	—	2	2	no	—
46	#1875	—	13	1	no	maximal, Sylow
47	C₂³×C₁₃	—	2	1	no	—
48	C₂³×C₁₃	—	2	1	no	—
49	C₂³×C₁₃	—	2	2	no	—
50	C₂³×C₁₃	—	2	2	no	—
51	C₂³×C₁₃	—	2	2	no	—
52	C₂³×C₁₃	—	4	1	no	—
53	C₂³×C₁₃	—	4	1	no	—
54	C₂³×C₁₃	—	4	2	no	—
55	C₂³×C₁₃	—	4	2	no	—
56	C₂³×C₁₃	C₂²⋊C₄	1	2	yes	—
57	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
58	D₄×C₁₃	—	2	4	no	—
59	D₄×C₁₃	—	4	8	no	—
60	D₄×C₁₃	—	4	2	no	—
61	C₂×C₄×C₁₃	—	2	1	no	—
62	C₂×C₄×C₁₃	—	2	1	no	—
63	C₂×C₄×C₁₃	—	2	4	no	—
64	C₂×C₄×C₁₃	—	2	4	no	—
65	C₂×C₄×C₁₃	—	2	2	no	—
66	C₂×C₄×C₁₃	—	4	2	no	—
67	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
68	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₁₃	#41	1	2	yes	—
72	C₂×Dic₁₃	—	2	4	no	—
73	C₂×Dic₁₃	—	4	2	no	—
74	C₂×Dic₁₃	—	4	2	no	—
75	#1055	—	4	2	no	—
76	C₂×#121	D₁₃	1	1	yes	characteristic
77	C₂×#117	—	13	1	no	—
78	C₂×#102	—	13	2	no	—
79	C₂×C₂²⋊C₈	—	13	1	no	—
80	#340	—	13	2	no	—
81	C₂²×C₁₃	—	2	2	no	—
82	C₂²×C₁₃	—	2	2	no	—
83	C₂²×C₁₃	—	2	2	no	—
84	C₂²×C₁₃	—	4	2	no	—
85	C₂²×C₁₃	—	4	2	no	—
86	C₂²×C₁₃	—	4	2	no	—
87	C₂²×C₁₃	—	4	2	no	—
88	C₂²×C₁₃	—	4	2	no	—
89	C₂²×C₁₃	—	8	2	no	—
90	C₂²×C₁₃	—	8	2	no	—
91	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂²×C₁₃	#120	1	1	yes	characteristic
93	C₂²×C₁₃	#123	1	1	yes	characteristic
94	C₂²×C₁₃	#135	1	2	yes	—
95	C₂²×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
96	C₄×C₁₃	—	2	2	no	—
97	C₄×C₁₃	—	4	2	no	—
98	C₄×C₁₃	C₄×D₄	1	2	yes	—
99	C₄×C₁₃	#121	1	2	yes	—
100	Dic₁₃	—	4	4	no	—
101	C₂²×D₄	—	2	1	no	—
102	C₂²×D₄	C₂×D₁₃	1	1	yes	characteristic
103	C₂³×C₄	C₂×D₁₃	1	1	yes	characteristic
104	C₂²×C₈	—	13	2	no	—
105	#121	—	2	2	no	—
106	#121	Dic₁₃	1	4	yes	complemented
107	#117	—	26	4	no	—
108	C₂×C₄⋊C₄	—	13	2	no	—
109	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
110	C₂×C₂²⋊C₄	—	2	1	no	—
111	C₂×C₂²⋊C₄	—	26	1	no	—
112	C₂×C₄²	—	26	1	no	—
113	#102	—	26	4	no	—
114	C₂²⋊C₈	—	26	2	no	—
115	C₂×C₁₃	—	4	2	no	—
116	C₂×C₁₃	—	8	2	no	—
117	C₂×C₁₃	#386	1	1	yes	characteristic
118	C₂×C₁₃	#416	1	1	yes	characteristic
119	C₂×C₁₃	#418	1	1	yes	characteristic
120	C₂×C₁₃	#449	1	2	yes	—
121	C₂×C₁₃	#482	1	2	yes	—
122	C₂⁴	—	2	1	no	—
123	C₂⁴	C₂²⋊D₁₃	1	1	yes	characteristic
124	C₂×D₄	—	2	4	no	—
125	C₂×D₄	—	2	2	no	—
126	C₂×D₄	—	2	2	no	—
127	C₂×D₄	—	2	1	no	—
128	C₂×D₄	—	4	4	no	—
129	C₂×D₄	C₂×Dic₁₃	1	2	yes	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	—	2	2	no	—
132	C₂²×C₄	—	2	1	no	—
133	C₂²×C₄	—	26	2	no	—
134	C₂²×C₄	C₂×Dic₁₃	1	2	yes	—
135	C₂²×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
136	C₂²×C₄	C₂²⋊D₁₃	1	2	yes	—
137	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
138	C₂×C₈	—	26	2	no	—
139	C₂×C₈	—	26	2	no	—
140	C₂×C₈	—	26	2	no	—
141	C₄⋊C₄	—	2	1	no	—
142	C₄⋊C₄	—	13	4	no	—
143	C₄⋊C₄	—	26	2	no	—
144	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
145	C₂²⋊C₄	—	2	4	no	—
146	C₂²⋊C₄	—	26	4	no	—
147	C₄²	—	26	4	no	—
148	C₁₃	#1875	1	1	yes	characteristic, complemented, Sylow
149	C₂³	—	2	2	no	—
150	C₂³	—	2	1	no	—
151	C₂³	—	2	1	no	—
152	C₂³	—	2	2	no	—
153	C₂³	—	2	2	no	—
154	C₂³	—	4	2	no	—
155	C₂³	—	4	1	no	—
156	C₂³	—	4	1	no	—
157	C₂³	—	4	2	no	—
158	C₂³	C₂²⋊Dic₁₃	1	2	yes	—
159	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
160	D₄	—	2	4	no	—
161	D₄	—	4	8	no	—
162	D₄	—	4	2	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	26	4	no	—
170	C₂×C₄	—	52	2	no	—
171	C₂×C₄	—	52	2	no	—
172	C₂×C₄	C₂²⋊Dic₁₃	1	2	yes	—
173	C₂×C₄	D₄×D₁₃	1	2	yes	—
174	C₂×C₄	#6129	1	2	yes	—
175	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
176	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
177	C₈	—	52	2	no	—
178	C₂²	—	2	2	no	—
179	C₂²	—	2	2	no	—
180	C₂²	—	2	2	no	—
181	C₂²	—	4	2	no	—
182	C₂²	—	4	2	no	—
183	C₂²	—	4	2	no	—
184	C₂²	—	4	2	no	—
185	C₂²	—	4	2	no	—
186	C₂²	—	8	2	no	—
187	C₂²	—	8	2	no	—
188	C₂²	#88186	1	2	yes	—
189	C₂²	#88189	1	1	yes	characteristic
190	C₂²	#88191	1	1	yes	characteristic
191	C₂²	#88204	1	2	yes	—
192	C₂²	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
193	C₄	—	2	2	no	—
194	C₄	—	4	2	no	—
195	C₄	—	52	4	no	—
196	C₄	D₄×Dic₁₃	1	2	yes	—
197	C₄	#88193	1	2	yes	—
198	C₂	—	4	2	no	—
199	C₂	—	8	2	no	—
200	C₂	#11718696	1	2	yes	—
201	C₂	#11718698	1	2	yes	—
202	C₂	#11718876	1	1	yes	characteristic
203	C₂	#11718881	1	1	yes	characteristic
204	C₂	#11718894	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×#121

C₂

yes

maximal, characteristic

C₂×#88180

C₂

yes

maximal, characteristic, complemented

C₂×#88072

C₂

yes

maximal, complemented

C₂×C₂²⋊#1055

C₂

yes

maximal, characteristic, complemented

#11718121

C₂

yes

maximal, complemented

C₂²×D₄×C₁₃

—

C₂²×D₄×C₁₃

C₂²

yes

characteristic

C₂³×C₄×C₁₃

C₂²

yes

characteristic

C₁₃×#121

—

C₁₃×#121

C₄

yes

complemented

C₂×C₁₃×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₁₃×C₂²⋊C₄

—

C₂×C₂²⋊Dic₁₃

—

#88180

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄×Dic₁₃

—

C₂²×#1055

C₂²

yes

—

#88072

—

C₂²⋊#1055

—

C₂⁴×C₁₃

—

C₂⁴×C₁₃

D₄

yes

characteristic

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

D₄

yes

characteristic

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₁₃×C₂²⋊C₄

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

D₄

yes

complemented

C₄×Dic₁₃

—

—

—

—

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂²⋊C₄

yes

—

C₂³×C₁₃

C₂×D₄

yes

characteristic

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

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

#1055

—

C₂×#121

D₁₃

yes

characteristic

C₂×#117

—

C₂×#102

—

C₂×C₂²⋊C₈

—

#340

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₂²×C₁₃

AGL(1,ℤ/8)

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₄×D₄

yes

—

C₄×C₁₃

#121

yes

—

100

Dic₁₃

—

101

C₂²×D₄

—

102

C₂²×D₄

C₂×D₁₃

yes

characteristic

103

C₂³×C₄

C₂×D₁₃

yes

characteristic

104

C₂²×C₈

—

105

#121

—

106

#121

Dic₁₃

yes

complemented

107

#117

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

C₂×D₁₃

yes

characteristic

110

C₂×C₂²⋊C₄

—

111

C₂×C₂²⋊C₄

—

112

C₂×C₄²

—

113

#102

—

114

C₂²⋊C₈

—

115

C₂×C₁₃

—

116

C₂×C₁₃

—

117

C₂×C₁₃

#386

yes

characteristic

118

C₂×C₁₃

#416

yes

characteristic

119

C₂×C₁₃

#418

yes

characteristic

120

C₂×C₁₃

#449

yes

—

121

C₂×C₁₃

#482

yes

—

122

C₂⁴

—

123

C₂⁴

C₂²⋊D₁₃

yes

characteristic

124

C₂×D₄

—

125

C₂×D₄

—

126

C₂×D₄

—

127

C₂×D₄

—

128

C₂×D₄

—

129

C₂×D₄

C₂×Dic₁₃

yes

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

C₂×Dic₁₃

yes

—

135

C₂²×C₄

C₂²⋊D₁₃

yes

characteristic

136

C₂²×C₄

C₂²⋊D₁₃

yes

—

137

C₂²×C₄

C₂²×D₁₃

yes

characteristic

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

C₂×Dic₁₃

yes

—

145

C₂²⋊C₄

—

146

C₂²⋊C₄

—

147

C₄²

—

148

C₁₃

#1875

yes

characteristic, complemented, Sylow

149

C₂³

—

150

C₂³

—

151

C₂³

—

152

C₂³

—

153

C₂³

—

154

C₂³

—

155

C₂³

—

156

C₂³

—

157

C₂³

—

158

C₂³

C₂²⋊Dic₁₃

yes

—

159

C₂³

C₂×C₂²⋊D₁₃

yes

characteristic

160

D₄

—

161

D₄

—

162

D₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

C₂²⋊Dic₁₃

yes

—

173

C₂×C₄

D₄×D₁₃

yes

—

174

C₂×C₄

#6129

yes

—

175

C₂×C₄

C₂²×Dic₁₃

yes

characteristic

176

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

177

C₈

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

—

182

C₂²

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

#88186

yes

—

189

C₂²

#88189

yes

characteristic

190

C₂²

#88191

yes

characteristic

191

C₂²

#88204

yes

—

192

C₂²

C₂×C₂²⋊Dic₁₃

yes

characteristic

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

D₄×Dic₁₃

yes

—

197

C₄

#88193

yes

—

198

C₂

—

199

C₂

—

200

C₂

#11718696

yes

—

201

C₂

#11718698

yes

—

202

C₂

#11718876

yes

characteristic

203

C₂

#11718881

yes

characteristic

204

C₂

#11718894

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting, prime radical	C₂×C₁₃×#121	C₂
socle	C₂³×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂³
hypercenter	C₂×#121	D₁₃
hypocenter	C₁₃	#1875
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#121

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₃

commutator

C₂×C₄×C₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting, prime radical

C₂×C₁₃×#121

C₂

socle

C₂³×C₁₃

C₂×D₄

cosocle

C₂²×C₄×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#121

N	H ≅ G/N	type
2. C₂×#88180	C₂	semidirect
3. C₂×#88072	C₂	semidirect
4. C₂×C₂²⋊#1055	C₂	semidirect
5. #11718121	C₂	semidirect
10. C₁₃×#121	C₄	semidirect
41. C₄⋊Dic₁₃	D₄	semidirect
106. #121	Dic₁₃	semidirect
148. C₁₃	#1875	semidirect

H ≅ G/N

type

2. C₂×#88180

C₂

semidirect

3. C₂×#88072

C₂

semidirect

4. C₂×C₂²⋊#1055

C₂

semidirect

5. #11718121

C₂

semidirect

10. C₁₃×#121

C₄

semidirect

41. C₄⋊Dic₁₃

D₄

semidirect

106. #121

Dic₁₃

semidirect

148. C₁₃

#1875

semidirect

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

subgroup	index
C₂×C₁₃×#121	2
C₂×#88180	2
C₂×#88072	2
C₂×C₂²⋊#1055	2
#11718121	2
#1875	13

subgroup

index

C₂×C₁₃×#121

C₂×C₂²⋊#1055

Subgroups — #13115159

Conjugacy classes of subgroups

Complements

Maximal subgroups