Group #15767371 — subgroups

Subgroups — #15767371

Order 3328.

406 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 127 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	#13120078	C₂	1	1	yes	maximal, characteristic
2	#13115597	C₂	1	2	yes	maximal
3	#13112719	C₂	1	1	yes	maximal, characteristic
4	#13112676	C₂	1	2	yes	maximal
5	C₁₃×#1845	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#107	—	2	2	no	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₁₃×#384	C₂²	1	1	yes	characteristic
9	C₁₃×#384	C₂²	1	1	yes	characteristic
10	#11718647	—	2	2	no	—
11	#11718646	—	2	2	no	—
12	#11718638	—	2	2	no	—
13	C₂×#88048	—	2	2	no	—
14	C₄×C₄⋊Dic₁₃	C₂²	1	2	yes	—
15	C₂×C₄⋊#1055	C₂²	1	2	yes	—
16	#11718121	—	2	2	no	—
17	C₂²×C₈×C₁₃	—	2	1	no	—
18	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
19	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
20	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
21	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
22	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
23	C₂×C₄⋊Dic₁₃	—	2	2	no	—
24	C₂×C₄⋊Dic₁₃	D₄	1	2	yes	—
25	C₂×C₄⋊Dic₁₃	D₄	1	2	yes	—
26	C₂×C₄×Dic₁₃	—	2	2	no	—
27	C₂²×#1055	—	2	2	no	—
28	C₁₃×#107	—	4	4	no	—
29	C₄×C₈×C₁₃	—	2	1	no	—
30	C₄×C₈×C₁₃	Q₈	1	2	yes	—
31	#88071	—	2	2	no	—
32	#88071	—	2	4	no	—
33	#88071	—	2	2	no	—
34	#88048	—	4	4	no	—
35	C₄⋊#1055	—	2	2	no	—
36	C₄⋊#1055	Q₈	1	4	yes	—
37	#10576	—	13	1	no	maximal, Sylow
38	C₂²×C₄×C₁₃	—	2	2	no	—
39	C₂²×C₄×C₁₃	—	4	1	no	—
40	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
41	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×Dic₁₃	—	2	2	no	—
43	C₂²×Dic₁₃	—	2	2	no	—
44	C₂×C₈×C₁₃	—	2	2	no	—
45	C₂×C₈×C₁₃	—	2	2	no	—
46	C₂×C₈×C₁₃	—	2	2	no	—
47	C₁₃×C₄⋊C₄	—	4	2	no	—
48	C₁₃×C₄⋊C₄	—	4	2	no	—
49	C₁₃×C₄⋊C₄	—	4	2	no	—
50	C₁₃×C₄⋊C₄	—	4	2	no	—
51	C₁₃×C₄⋊C₄	—	4	2	no	—
52	C₁₃×C₄⋊C₄	—	4	2	no	—
53	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
54	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
55	C₄²×C₁₃	#41	1	1	yes	characteristic
56	C₄⋊Dic₁₃	—	2	2	no	—
57	C₄⋊Dic₁₃	—	2	2	no	—
58	C₄⋊Dic₁₃	—	2	2	no	—
59	C₄⋊Dic₁₃	—	2	2	no	—
60	C₄⋊Dic₁₃	—	8	2	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	#2571	—	13	1	no	—
67	#1948	—	13	2	no	—
68	#1845	D₁₃	1	1	yes	characteristic
69	#1296	—	13	1	no	—
70	#1279	—	13	2	no	—
71	C₂³×C₁₃	#120	1	1	yes	characteristic
72	C₂×C₄×C₁₃	—	2	1	no	—
73	C₂×C₄×C₁₃	—	2	1	no	—
74	C₂×C₄×C₁₃	—	4	4	no	—
75	C₂×C₄×C₁₃	—	4	2	no	—
76	C₂×C₄×C₁₃	—	8	2	no	—
77	C₂×C₄×C₁₃	—	8	1	no	—
78	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
80	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
82	C₂×C₄×C₁₃	#135	1	2	yes	—
83	C₂×C₄×C₁₃	#135	1	2	yes	—
84	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
85	C₂×C₄×C₁₃	#137	1	2	yes	—
86	C₂×Dic₁₃	—	4	2	no	—
87	C₂×Dic₁₃	—	4	2	no	—
88	C₂×Dic₁₃	—	4	2	no	—
89	C₂×Dic₁₃	—	4	2	no	—
90	C₂×Dic₁₃	—	4	2	no	—
91	C₂×Dic₁₃	—	4	2	no	—
92	C₈×C₁₃	—	2	4	no	—
93	#1055	—	4	4	no	—
94	C₂×#107	—	2	2	no	—
95	C₂×#106	—	26	2	no	—
96	C₂×C₄⋊C₈	—	13	2	no	—
97	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
98	#400	—	26	2	no	—
99	#389	—	26	2	no	—
100	#384	C₂×D₁₃	1	1	yes	characteristic
101	#384	C₂×D₁₃	1	1	yes	characteristic
102	#382	—	26	2	no	—
103	C₄×C₄⋊C₄	—	13	2	no	—
104	#340	—	26	2	no	—
105	C₂²×C₁₃	#395	1	1	yes	characteristic
106	C₂²×C₁₃	#449	1	2	yes	—
107	C₂²×C₁₃	#452	1	2	yes	—
108	C₂²×C₁₃	#453	1	1	yes	characteristic
109	C₂²×C₁₃	#456	1	1	yes	characteristic
110	C₄×C₁₃	—	2	1	no	—
111	C₄×C₁₃	—	2	1	no	—
112	C₄×C₁₃	—	8	2	no	—
113	C₄×C₁₃	—	16	1	no	—
114	C₄×C₁₃	#478	1	2	yes	—
115	C₄×C₁₃	#479	1	2	yes	—
116	C₄×C₁₃	#499	1	2	yes	—
117	C₄×C₁₃	#501	1	2	yes	—
118	Dic₁₃	—	8	2	no	—
119	Dic₁₃	—	8	2	no	—
120	C₂²×C₈	—	2	1	no	—
121	C₂²×C₈	—	26	2	no	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	2	1	no	—
125	C₂×C₄⋊C₄	—	13	2	no	—
126	C₂×C₄⋊C₄	—	13	2	no	—
127	C₂×C₄⋊C₄	—	26	2	no	—
128	C₂×C₄⋊C₄	C₂²⋊D₁₃	1	2	yes	—
129	C₂×C₄²	—	26	2	no	—
130	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
131	#107	—	4	4	no	—
132	#106	—	52	4	no	—
133	C₄⋊C₈	—	13	4	no	—
134	C₄⋊C₈	—	26	2	no	—
135	C₄×C₈	—	2	1	no	—
136	C₄×C₈	Dic₂₆	1	2	yes	—
137	#95	—	26	2	no	—
138	#95	—	26	4	no	—
139	#95	—	26	2	no	—
140	C₂×C₁₃	#1620	1	1	yes	characteristic
141	C₂×C₁₃	#1634	1	1	yes	characteristic
142	C₂×C₁₃	#1650	1	2	yes	—
143	C₂×C₁₃	#2059	1	2	yes	—
144	C₂×C₁₃	#2060	1	1	yes	characteristic
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	—	4	1	no	—
147	C₂²×C₄	—	26	2	no	—
148	C₂²×C₄	—	26	2	no	—
149	C₂²×C₄	D₄×D₁₃	1	2	yes	—
150	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	26	4	no	—
155	C₂×C₈	—	52	2	no	—
156	C₂×C₈	—	52	2	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	26	2	no	—
164	C₄⋊C₄	—	26	2	no	—
165	C₄⋊C₄	—	26	2	no	—
166	C₄⋊C₄	—	26	2	no	—
167	C₄⋊C₄	—	104	2	no	—
168	C₄²	—	52	2	no	—
169	C₄²	—	52	2	no	—
170	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
171	C₄²	#6127	1	1	yes	characteristic
172	C₄²	Q₈×D₁₃	1	2	yes	—
173	C₁₃	#10576	1	1	yes	characteristic, complemented, Sylow
174	C₂³	#88191	1	1	yes	characteristic
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	8	1	no	—
180	C₂×C₄	—	8	2	no	—
181	C₂×C₄	—	52	2	no	—
182	C₂×C₄	—	52	2	no	—
183	C₂×C₄	—	52	2	no	—
184	C₂×C₄	—	52	2	no	—
185	C₂×C₄	—	52	2	no	—
186	C₂×C₄	—	52	2	no	—
187	C₂×C₄	#88142	1	2	yes	—
188	C₂×C₄	#88150	1	2	yes	—
189	C₂×C₄	#88165	1	2	yes	—
190	C₂×C₄	#88166	1	2	yes	—
191	C₂×C₄	#88172	1	2	yes	—
192	C₂×C₄	#88173	1	2	yes	—
193	C₂×C₄	#88178	1	1	yes	characteristic
194	C₂×C₄	#88198	1	1	yes	characteristic
195	C₈	—	2	4	no	—
196	C₈	—	52	4	no	—
197	C₂²	#11718640	1	1	yes	characteristic
198	C₂²	#11718738	1	1	yes	characteristic
199	C₂²	#11718756	1	1	yes	characteristic
200	C₂²	#11718819	1	2	yes	—
201	C₂²	#11718848	1	2	yes	—
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	8	2	no	—
205	C₄	—	16	1	no	—
206	C₄	—	104	2	no	—
207	C₄	—	104	2	no	—
208	C₄	#11718537	1	2	yes	—
209	C₄	#11718538	1	2	yes	—
210	C₄	#11718539	1	2	yes	—
211	C₄	#11718552	1	2	yes	—
212	C₂	#13113893	1	1	yes	characteristic
213	C₂	#13113979	1	1	yes	characteristic
214	C₂	#13114089	1	2	yes	—
215	C₂	#13116172	1	2	yes	—
216	C₂	#13116181	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13120078

C₂

yes

maximal, characteristic

#13115597

C₂

yes

maximal

#13112719

C₂

yes

maximal, characteristic

#13112676

C₂

yes

maximal

C₁₃×#1845

C₂

yes

maximal, characteristic

C₂×C₁₃×#107

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₁₃×#384

C₂²

yes

characteristic

C₁₃×#384

C₂²

yes

—

—

—

—

C₄×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊#1055

C₂²

yes

—

#11718121

—

C₂²×C₈×C₁₃

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

D₄

yes

—

C₂×C₄⋊Dic₁₃

D₄

yes

—

C₂×C₄×Dic₁₃

—

C₂²×#1055

—

C₁₃×#107

—

C₄×C₈×C₁₃

—

C₄×C₈×C₁₃

Q₈

yes

—

—

—

—

—

—

Q₈

yes

—

#10576

—

maximal, Sylow

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×Q₈

yes

characteristic

C₄²×C₁₃

C₂×Q₈

yes

—

C₄²×C₁₃

#41

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

—

—

—

—

—

D₁₃

yes

characteristic

#1296

—

#1279

—

C₂³×C₁₃

#120

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

C₄⋊Q₈

yes

—

C₂×C₄×C₁₃

#135

yes

—

C₂×C₄×C₁₃

#135

yes

—

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₁₃

#137

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₈×C₁₃

—

#1055

—

C₂×#107

—

C₂×#106

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

#400

—

#389

—

100

#384

C₂×D₁₃

yes

characteristic

101

#384

C₂×D₁₃

yes

characteristic

102

#382

—

103

C₄×C₄⋊C₄

—

104

#340

—

105

C₂²×C₁₃

#395

yes

characteristic

106

C₂²×C₁₃

#449

yes

—

107

C₂²×C₁₃

#452

yes

—

108

C₂²×C₁₃

#453

yes

characteristic

109

C₂²×C₁₃

#456

yes

characteristic

110

C₄×C₁₃

—

111

C₄×C₁₃

—

112

C₄×C₁₃

—

113

C₄×C₁₃

—

114

C₄×C₁₃

#478

yes

—

115

C₄×C₁₃

#479

yes

—

116

C₄×C₁₃

#499

yes

—

117

C₄×C₁₃

#501

yes

—

118

Dic₁₃

—

119

Dic₁₃

—

120

C₂²×C₈

—

121

C₂²×C₈

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

C₂²⋊D₁₃

yes

—

129

C₂×C₄²

—

130

C₂×C₄²

C₂²×D₁₃

yes

characteristic

131

#107

—

132

#106

—

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

C₄×C₈

—

136

C₄×C₈

Dic₂₆

yes

—

137

#95

—

138

#95

—

139

#95

—

140

C₂×C₁₃

#1620

yes

characteristic

141

C₂×C₁₃

#1634

yes

characteristic

142

C₂×C₁₃

#1650

yes

—

143

C₂×C₁₃

#2059

yes

—

144

C₂×C₁₃

#2060

yes

characteristic

145

C₂²×C₄

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

D₄×D₁₃

yes

—

150

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

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

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

104

—

168

C₄²

—

169

C₄²

—

170

C₄²

C₂×Dic₂₆

yes

characteristic

171

C₄²

#6127

yes

characteristic

172

C₄²

Q₈×D₁₃

yes

—

173

C₁₃

#10576

yes

characteristic, complemented, Sylow

174

C₂³

#88191

yes

characteristic

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

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

#88142

yes

—

188

C₂×C₄

#88150

yes

—

189

C₂×C₄

#88165

yes

—

190

C₂×C₄

#88166

yes

—

191

C₂×C₄

#88172

yes

—

192

C₂×C₄

#88173

yes

—

193

C₂×C₄

#88178

yes

characteristic

194

C₂×C₄

#88198

yes

characteristic

195

C₈

—

196

C₈

—

197

C₂²

#11718640

yes

characteristic

198

C₂²

#11718738

yes

characteristic

199

C₂²

#11718756

yes

characteristic

200

C₂²

#11718819

yes

—

201

C₂²

#11718848

yes

—

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

—

206

C₄

—

104

—

207

C₄

—

104

—

208

C₄

#11718537

yes

—

209

C₄

#11718538

yes

—

210

C₄

#11718539

yes

—

211

C₄

#11718552

yes

—

212

C₂

#13113893

yes

characteristic

213

C₂

#13113979

yes

characteristic

214

C₂

#13114089

yes

—

215

C₂

#13116172

yes

—

216

C₂

#13116181

yes

characteristic

	subgroup	G/H
center	C₂³	#88191
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1845	C₂
socle, prime radical	C₂³×C₁₃	#120
hypercenter	#1845	D₁₃
hypocenter	C₁₃	#10576
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

subgroup

G/H

center

C₂³

#88191

commutator, cosocle

C₂×C₄²×C₁₃

C₂³

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1845

C₂

C₂³×C₁₃

D₁₃

C₁₃

G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

N	H ≅ G/N	type
173. C₁₃	#10576	semidirect

H ≅ G/N

type

173. C₁₃

#10576

semidirect

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

subgroup	index
#13120078	2
#13115597	2
#13112719	2
#13112676	2
C₁₃×#1845	2
#10576	13

subgroup

index

C₁₃×#1845

Subgroups — #15767371

Conjugacy classes of subgroups

Complements

Maximal subgroups