Group #15762012 — subgroups

Subgroups — #15762012

Order 3328.

538 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 167 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	#13123234	C₂	1	1	yes	maximal, characteristic
2	#13114689	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718078	C₂	1	4	yes	maximal
4	C₁₃×#1831	C₂	1	1	yes	maximal, characteristic
5	C₂×#88123	—	2	1	no	—
6	C₂×#88123	—	2	1	no	—
7	C₂×#88123	C₂²	1	2	yes	—
8	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
9	C₄³×C₁₃	C₂²	1	1	yes	characteristic
10	#11718596	—	2	2	no	—
11	#11718596	C₄	1	4	yes	—
12	C₂×C₄⋊#1055	C₂²	1	2	yes	—
13	C₂×#88043	—	2	1	no	—
14	C₁₃×#336	—	2	2	no	—
15	#11718116	—	2	1	no	—
16	#11718078	—	2	8	no	—
17	C₂²×Dic₂₆	—	2	2	no	—
18	C₂²×C₈×C₁₃	—	2	2	no	—
19	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	D₄	1	2	yes	—
23	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₃	—	2	4	no	—
25	C₂×C₄⋊Dic₁₃	—	2	4	no	—
26	C₂×C₄⋊Dic₁₃	—	2	2	no	—
27	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
28	C₂²×#1055	—	2	2	no	—
29	#88123	—	2	4	no	—
30	#88123	—	2	4	no	—
31	#88123	—	4	4	no	—
32	#88123	—	4	2	no	—
33	#88123	—	4	2	no	—
34	C₁₃×C₄⋊C₈	—	2	4	no	—
35	C₄⋊#1055	—	2	2	no	—
36	C₄⋊#1055	D₄	1	4	yes	—
37	#88043	—	4	2	no	—
38	#9705	—	13	1	no	maximal, Sylow
39	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₁₃	—	2	4	no	—
45	C₂×Dic₂₆	—	4	4	no	—
46	C₂×Dic₂₆	—	4	2	no	—
47	C₂×Dic₂₆	—	4	2	no	—
48	C₂×Dic₂₆	—	4	4	no	—
49	C₂×C₈×C₁₃	—	4	2	no	—
50	C₂×C₈×C₁₃	—	4	4	no	—
51	C₄²×C₁₃	—	2	4	no	—
52	C₄²×C₁₃	—	2	2	no	—
53	C₄²×C₁₃	—	2	4	no	—
54	C₄²×C₁₃	D₈	1	2	yes	—
55	C₄²×C₁₃	SD₁₆	1	2	yes	—
56	C₄²×C₁₃	C₂×D₄	1	2	yes	—
57	C₄²×C₁₃	#41	1	2	yes	—
58	C₄⋊Dic₁₃	—	2	8	no	—
59	C₄⋊Dic₁₃	—	4	8	no	—
60	C₄⋊Dic₁₃	—	4	8	no	—
61	C₄⋊Dic₁₃	—	4	2	no	—
62	C₄⋊Dic₁₃	—	4	2	no	—
63	C₂×#1055	—	2	4	no	—
64	C₂×#1055	—	4	2	no	—
65	C₂×#1055	—	4	2	no	—
66	#2868	—	13	1	no	—
67	#1831	D₁₃	1	1	yes	characteristic
68	#1767	—	13	1	no	—
69	C₂×#332	—	13	4	no	—
70	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₃	—	2	2	no	—
72	C₂×C₄×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	#101	1	4	yes	—
77	C₂×C₄×C₁₃	#102	1	4	yes	—
78	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
79	C₂×C₄×C₁₃	#121	1	2	yes	—
80	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
81	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
84	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
85	C₂×C₄×C₁₃	#137	1	4	yes	—
86	C₂×Dic₁₃	—	4	4	no	—
87	C₂×Dic₁₃	—	4	4	no	—
88	C₂×Dic₁₃	—	4	4	no	—
89	Dic₂₆	—	8	4	no	—
90	Dic₂₆	—	8	4	no	—
91	C₈×C₁₃	—	8	2	no	—
92	#1055	—	4	4	no	—
93	C₂×C₄⋊Q₈	—	13	2	no	—
94	C₂×C₄⋊Q₈	—	26	1	no	—
95	C₂×C₄⋊Q₈	—	26	1	no	—
96	C₂×C₄⋊C₈	—	13	2	no	—
97	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
98	C₂×#97	—	26	1	no	—
99	#384	—	13	4	no	—
100	#384	—	26	2	no	—
101	C₄³	C₂×D₁₃	1	1	yes	characteristic
102	#336	—	2	2	no	—
103	#336	—	26	1	no	—
104	#332	—	26	8	no	—
105	C₂²×C₁₃	#390	1	1	yes	characteristic
106	C₂²×C₁₃	C₂×#101	1	2	yes	—
107	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
108	C₂²×C₁₃	#418	1	1	yes	characteristic
109	C₂²×C₁₃	#419	1	2	yes	—
110	C₄×C₁₃	—	2	4	no	—
111	C₄×C₁₃	—	2	4	no	—
112	C₄×C₁₃	—	2	2	no	—
113	C₄×C₁₃	#459	1	2	yes	—
114	C₄×C₁₃	#460	1	2	yes	—
115	C₄×C₁₃	#491	1	2	yes	—
116	C₄×C₁₃	#497	1	2	yes	—
117	Dic₁₃	—	8	4	no	—
118	C₂²×Q₈	—	26	2	no	—
119	C₂²×C₈	—	2	2	no	—
120	C₂²×C₈	—	26	2	no	—
121	C₄⋊Q₈	—	26	4	no	—
122	C₄⋊Q₈	—	26	4	no	—
123	C₄⋊Q₈	—	52	4	no	—
124	C₄⋊Q₈	—	52	2	no	—
125	C₄⋊Q₈	—	52	2	no	—
126	C₂×C₄⋊C₄	—	13	4	no	—
127	C₂×C₄⋊C₄	—	26	4	no	—
128	C₂×C₄⋊C₄	—	26	4	no	—
129	C₂×C₄⋊C₄	—	26	2	no	—
130	C₂×C₄²	C₄×D₁₃	1	2	yes	—
131	C₂×C₄²	D₅₂	1	1	yes	characteristic
132	C₂×C₄²	D₅₂	1	1	yes	characteristic
133	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
134	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
135	C₄⋊C₈	—	2	4	no	—
136	C₄⋊C₈	—	13	4	no	—
137	C₄⋊C₈	—	26	2	no	—
138	#97	—	52	2	no	—
139	C₂×C₁₃	#1546	1	2	yes	—
140	C₂×C₁₃	#1556	1	2	yes	—
141	C₂×C₁₃	#1960	1	1	yes	characteristic
142	C₂×C₁₃	#1966	1	1	yes	characteristic
143	C₂×C₁₃	#1982	1	1	yes	characteristic
144	C₂×Q₈	—	52	4	no	—
145	C₂×Q₈	—	52	2	no	—
146	C₂×Q₈	—	52	2	no	—
147	C₂×Q₈	—	52	4	no	—
148	C₂²×C₄	—	26	4	no	—
149	C₂²×C₄	#6103	1	2	yes	—
150	C₂²×C₄	#6103	1	2	yes	—
151	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
152	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	—	4	4	no	—
156	C₂×C₈	—	26	4	no	—
157	C₂×C₈	—	52	2	no	—
158	C₂×C₈	—	52	2	no	—
159	C₄⋊C₄	—	26	8	no	—
160	C₄⋊C₄	—	52	8	no	—
161	C₄⋊C₄	—	52	2	no	—
162	C₄⋊C₄	—	52	2	no	—
163	C₄⋊C₄	—	52	8	no	—
164	C₄²	—	2	4	no	—
165	C₄²	—	2	2	no	—
166	C₄²	—	2	4	no	—
167	C₄²	#6095	1	2	yes	—
168	C₄²	D₁₀₄	1	2	yes	—
169	C₄²	D₄×D₁₃	1	2	yes	—
170	C₄²	#6131	1	2	yes	—
171	C₁₃	#9705	1	1	yes	characteristic, complemented, Sylow
172	C₂³	C₂×#6103	1	1	yes	characteristic
173	Q₈	—	104	4	no	—
174	Q₈	—	104	4	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	52	4	no	—
181	C₂×C₄	—	52	4	no	—
182	C₂×C₄	—	52	4	no	—
183	C₂×C₄	#88061	1	4	yes	—
184	C₂×C₄	#88064	1	4	yes	—
185	C₂×C₄	#88147	1	2	yes	—
186	C₂×C₄	#88149	1	2	yes	—
187	C₂×C₄	C₂×#6095	1	1	yes	characteristic
188	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
189	C₂×C₄	#88162	1	2	yes	—
190	C₂×C₄	#88194	1	1	yes	characteristic
191	C₂×C₄	#88201	1	1	yes	characteristic
192	C₂×C₄	#88205	1	4	yes	—
193	C₈	—	8	2	no	—
194	C₈	—	52	4	no	—
195	C₂²	#11718649	1	1	yes	characteristic
196	C₂²	#11718665	1	2	yes	—
197	C₂²	C₂×#88061	1	1	yes	characteristic
198	C₂²	#11718792	1	1	yes	characteristic
199	C₂²	C₂×#88064	1	2	yes	—
200	C₄	—	2	2	no	—
201	C₄	—	2	4	no	—
202	C₄	—	2	4	no	—
203	C₄	—	104	4	no	—
204	C₄	#11718504	1	2	yes	—
205	C₄	#11718509	1	2	yes	—
206	C₄	#11718721	1	2	yes	—
207	C₄	#11718730	1	2	yes	—
208	C₂	#13113507	1	2	yes	—
209	C₂	#13113552	1	2	yes	—
210	C₂	#13115655	1	1	yes	characteristic
211	C₂	#13115679	1	1	yes	characteristic
212	C₂	#13115762	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13123234

C₂

yes

maximal, characteristic

#13114689

C₂

yes

maximal, characteristic

C₂×#11718078

C₂

yes

maximal

C₁₃×#1831

C₂

yes

maximal, characteristic

C₂×#88123

—

C₂×#88123

—

C₂×#88123

C₂²

yes

—

C₂×C₁₃×C₄⋊C₈

C₂²

yes

—

C₄³×C₁₃

C₂²

yes

characteristic

#11718596

—

#11718596

C₄

yes

—

C₂×C₄⋊#1055

C₂²

yes

—

C₂×#88043

—

C₁₃×#336

—

#11718116

—

#11718078

—

C₂²×Dic₂₆

—

C₂²×C₈×C₁₃

—

C₂×C₄²×C₁₃

C₂×C₄

yes

—

C₂×C₄²×C₁₃

D₄

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

C₂×C₄

yes

—

C₂²×#1055

—

—

—

—

—

—

C₁₃×C₄⋊C₈

—

C₄⋊#1055

—

C₄⋊#1055

D₄

yes

—

#88043

—

#9705

—

maximal, Sylow

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

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×Dic₂₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

D₈

yes

—

C₄²×C₁₃

SD₁₆

yes

—

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

#41

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

—

—

—

—

D₁₃

yes

characteristic

#1767

—

C₂×#332

—

C₂³×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#101

yes

—

C₂×C₄×C₁₃

#102

yes

—

C₂×C₄×C₁₃

C₄×D₄

yes

—

C₂×C₄×C₁₃

#121

yes

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₁₃

#137

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

#1055

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×D₁₃

yes

—

C₂×#97

—

#384

—

100

#384

—

101

C₄³

C₂×D₁₃

yes

characteristic

102

#336

—

103

#336

—

104

#332

—

105

C₂²×C₁₃

#390

yes

characteristic

106

C₂²×C₁₃

C₂×#101

yes

—

107

C₂²×C₁₃

C₂×#102

yes

characteristic

108

C₂²×C₁₃

#418

yes

characteristic

109

C₂²×C₁₃

#419

yes

—

110

C₄×C₁₃

—

111

C₄×C₁₃

—

112

C₄×C₁₃

—

113

C₄×C₁₃

#459

yes

—

114

C₄×C₁₃

#460

yes

—

115

C₄×C₁₃

#491

yes

—

116

C₄×C₁₃

#497

yes

—

117

Dic₁₃

—

118

C₂²×Q₈

—

119

C₂²×C₈

—

120

C₂²×C₈

—

121

C₄⋊Q₈

—

122

C₄⋊Q₈

—

123

C₄⋊Q₈

—

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄²

C₄×D₁₃

yes

—

131

C₂×C₄²

D₅₂

yes

characteristic

132

C₂×C₄²

D₅₂

yes

characteristic

133

C₂×C₄²

C₂²⋊D₁₃

yes

—

134

C₂×C₄²

C₂²×D₁₃

yes

characteristic

135

C₄⋊C₈

—

136

C₄⋊C₈

—

137

C₄⋊C₈

—

138

#97

—

139

C₂×C₁₃

#1546

yes

—

140

C₂×C₁₃

#1556

yes

—

141

C₂×C₁₃

#1960

yes

characteristic

142

C₂×C₁₃

#1966

yes

characteristic

143

C₂×C₁₃

#1982

yes

characteristic

144

C₂×Q₈

—

145

C₂×Q₈

—

146

C₂×Q₈

—

147

C₂×Q₈

—

148

C₂²×C₄

—

149

C₂²×C₄

#6103

yes

—

150

C₂²×C₄

#6103

yes

—

151

C₂²×C₄

C₂×C₄×D₁₃

yes

characteristic

152

C₂²×C₄

C₂×D₅₂

yes

characteristic

153

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄²

—

165

C₄²

—

166

C₄²

—

167

C₄²

#6095

yes

—

168

C₄²

D₁₀₄

yes

—

169

C₄²

D₄×D₁₃

yes

—

170

C₄²

#6131

yes

—

171

C₁₃

#9705

yes

characteristic, complemented, Sylow

172

C₂³

C₂×#6103

yes

characteristic

173

Q₈

—

104

—

174

Q₈

—

104

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

#88061

yes

—

184

C₂×C₄

#88064

yes

—

185

C₂×C₄

#88147

yes

—

186

C₂×C₄

#88149

yes

—

187

C₂×C₄

C₂×#6095

yes

characteristic

188

C₂×C₄

C₂×D₁₀₄

yes

characteristic

189

C₂×C₄

#88162

yes

—

190

C₂×C₄

#88194

yes

characteristic

191

C₂×C₄

#88201

yes

characteristic

192

C₂×C₄

#88205

yes

—

193

C₈

—

194

C₈

—

195

C₂²

#11718649

yes

characteristic

196

C₂²

#11718665

yes

—

197

C₂²

C₂×#88061

yes

characteristic

198

C₂²

#11718792

yes

characteristic

199

C₂²

C₂×#88064

yes

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

—

104

—

204

C₄

#11718504

yes

—

205

C₄

#11718509

yes

—

206

C₄

#11718721

yes

—

207

C₄

#11718730

yes

—

208

C₂

#13113507

yes

—

209

C₂

#13113552

yes

—

210

C₂

#13115655

yes

characteristic

211

C₂

#13115679

yes

characteristic

212

C₂

#13115762

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#6103
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1831	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1831	D₁₃
hypocenter	C₁₃	#9705
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂³

C₂×#6103

commutator

C₂²×C₄×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1831

C₂

socle, prime radical

C₂³×C₁₃

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

N	H ≅ G/N	type
171. C₁₃	#9705	semidirect

H ≅ G/N

type

171. C₁₃

#9705

semidirect

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

subgroup	index
#13123234	2
#13114689	2
C₂×#11718078	2
C₁₃×#1831	2
#9705	13

subgroup

index

C₁₃×#1831

Subgroups — #15762012

Conjugacy classes of subgroups

Complements

Maximal subgroups