Group #15780371 — subgroups

Subgroups — #15780371

Order 3328.

458 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 123 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	#13127939	C₂	1	1	yes	maximal, characteristic
2	#13124491	C₂	1	1	yes	maximal, characteristic
3	#13116555	C₂	1	2	yes	maximal, complemented
4	#13112937	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1812	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
7	#11719530	—	2	1	no	—
8	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
9	C₁₃×#427	C₂²	1	2	yes	—
10	C₁₃×#410	—	2	1	no	—
11	#11718801	—	2	4	no	—
12	#11718798	C₄	1	4	yes	complemented
13	#11718781	—	2	1	no	—
14	#11718777	—	2	2	no	—
15	#11718767	C₂²	1	1	yes	characteristic
16	C₂×#88052	—	2	2	no	—
17	C₂×#88052	C₂²	1	2	yes	—
18	C₁₃×H(3,ℤ/4)	—	2	1	no	—
19	#11718084	—	2	4	no	—
20	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
21	C₂²×Dic₂₆	—	2	1	no	—
22	C₁₃×#131	—	2	2	no	—
23	C₁₃×#131	—	2	2	no	—
24	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	D₄	1	2	yes	—
27	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	C₁₃×#117	—	2	2	no	—
29	#88198	—	4	1	no	—
30	#88190	—	4	1	no	—
31	#88189	—	4	2	no	—
32	#88180	—	2	1	no	—
33	#88178	—	2	4	no	—
34	C₂×Dic₅₂	—	2	2	no	—
35	C₂×Dic₅₂	—	2	2	no	—
36	C₂×Dic₅₂	—	2	1	no	—
37	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
38	C₁₃×#107	—	2	2	no	—
39	C₁₃×#106	—	2	2	no	—
40	C₁₃×#99	—	4	2	no	—
41	#88058	—	2	1	no	—
42	#88057	C₂×C₄	1	2	yes	—
43	#88056	—	2	4	no	—
44	#88052	—	2	4	no	—
45	#88052	—	2	2	no	—
46	#88052	—	2	2	no	—
47	C₂×#6090	—	2	4	no	—
48	C₂×#6090	D₄	1	4	yes	—
49	#13075	—	13	1	no	maximal, Sylow
50	C₁₃×#41	—	2	2	no	—
51	C₁₃×#41	—	2	2	no	—
52	C₂×Q₈×C₁₃	SD₁₆	1	1	yes	characteristic
53	C₂×D₄×C₁₃	D₈	1	2	yes	—
54	C₂×D₄×C₁₃	SD₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₁₃	—	4	1	no	—
58	C₂×Dic₂₆	—	2	2	no	—
59	C₂×Dic₂₆	—	4	2	no	—
60	C₂×Dic₂₆	—	4	2	no	—
61	C₂×Dic₂₆	—	4	1	no	—
62	C₁₃×M₁₆	—	2	2	no	—
63	C₁₃×M₁₆	—	2	4	no	—
64	C₂×C₈×C₁₃	—	2	2	no	—
65	C₂×C₈×C₁₃	—	2	2	no	—
66	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₁₃	#41	1	2	yes	—
71	C₁₃×C₄⋊C₄	—	4	2	no	—
72	C₁₃×C₂²⋊C₄	—	4	2	no	—
73	C₄²×C₁₃	—	4	2	no	—
74	C₂²⋊Dic₁₃	—	4	4	no	—
75	C₂²⋊Dic₁₃	—	4	1	no	—
76	C₄⋊Dic₁₃	—	2	2	no	—
77	#6101	—	4	4	no	—
78	C₄×Dic₁₃	—	4	1	no	—
79	Dic₅₂	—	2	4	no	—
80	Dic₅₂	—	4	1	no	—
81	Dic₅₂	—	4	1	no	—
82	#6090	—	2	4	no	—
83	#3407	—	13	1	no	—
84	#3059	—	13	1	no	—
85	#2141	—	13	2	no	—
86	#1812	D₁₃	1	1	yes	characteristic
87	#1384	—	13	2	no	—
88	C₂³×C₁₃	#102	1	2	yes	—
89	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
90	Q₈×C₁₃	—	2	2	no	—
91	D₄×C₁₃	—	2	4	no	—
92	D₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	8	2	no	—
96	C₂×C₄×C₁₃	#102	1	2	yes	—
97	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃	#121	1	2	yes	—
99	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄×C₁₃	#135	1	2	yes	—
101	C₂×Dic₁₃	—	4	2	no	—
102	C₂×Dic₁₃	—	4	2	no	—
103	C₂×Dic₁₃	—	8	2	no	—
104	Dic₂₆	—	4	2	no	—
105	Dic₂₆	—	4	2	no	—
106	Dic₂₆	—	8	2	no	—
107	C₈×C₁₃	—	2	2	no	—
108	C₈×C₁₃	C₄×D₄	1	2	yes	—
109	C₈×C₁₃	#120	1	1	yes	characteristic
110	C₈×C₁₃	#123	1	1	yes	characteristic
111	C₂²×Q₁₆	—	13	1	no	—
112	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
113	#536	—	26	1	no	—
114	#504	—	26	4	no	—
115	C₂×M₃₂	—	13	2	no	—
116	C₂×M₃₂	—	26	2	no	—
117	#470	—	13	4	no	—
118	#467	—	26	2	no	—
119	#427	—	13	1	no	—
120	#427	C₂×D₁₃	1	2	yes	—
121	#419	—	26	1	no	—
122	#410	—	2	1	no	—
123	#358	—	26	4	no	—
124	H(3,ℤ/4)	—	2	1	no	—
125	C₂²×C₁₃	—	2	2	no	—
126	C₂²×C₁₃	—	4	1	no	—
127	C₂²×C₁₃	—	4	2	no	—
128	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
129	C₂²×C₁₃	#466	1	2	yes	—
130	C₄×C₁₃	—	2	2	no	—
131	C₄×C₁₃	—	16	2	no	—
132	C₄×C₁₃	#386	1	1	yes	characteristic
133	C₄×C₁₃	#416	1	1	yes	characteristic
134	C₄×C₁₃	#465	1	2	yes	—
135	Dic₁₃	—	8	2	no	—
136	Dic₁₃	—	8	2	no	—
137	C₂×#41	C₂²⋊D₁₃	1	1	yes	characteristic
138	C₂²×Q₈	—	26	1	no	—
139	C₂×Q₁₆	—	13	2	no	—
140	C₂×Q₁₆	—	26	1	no	—
141	C₂×Q₁₆	—	26	2	no	—
142	C₂×Q₁₆	—	26	2	no	—
143	#131	—	2	2	no	—
144	#131	—	2	2	no	—
145	C₂×M₁₆	C₄×D₁₃	1	2	yes	—
146	C₂×M₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
147	C₂²×C₈	D₅₂	1	2	yes	—
148	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
149	C₄⋊Q₈	—	52	1	no	—
150	#124	—	52	1	no	—
151	#123	—	52	2	no	—
152	C₂²⋊Q₈	—	26	4	no	—
153	#117	—	2	2	no	—
154	#117	—	26	1	no	—
155	M₃₂	—	26	4	no	—
156	M₃₂	—	26	2	no	—
157	M₃₂	—	26	2	no	—
158	C₂×C₁₆	—	13	4	no	—
159	C₂×C₁₆	—	26	4	no	—
160	#107	—	2	2	no	—
161	#107	—	26	1	no	—
162	#106	—	2	2	no	—
163	#106	—	13	2	no	—
164	#103	—	26	4	no	—
165	#99	—	4	2	no	—
166	C₂×C₁₃	—	2	1	no	—
167	C₂×C₁₃	—	2	2	no	—
168	C₂×C₁₃	—	4	2	no	—
169	C₂×C₁₃	#1893	1	1	yes	characteristic
170	#41	—	2	2	no	—
171	#41	—	2	2	no	—
172	C₂×Q₈	—	26	2	no	—
173	C₂×Q₈	—	52	1	no	—
174	C₂×Q₈	—	52	2	no	—
175	C₂×Q₈	—	52	2	no	—
176	C₂×Q₈	#6106	1	1	yes	characteristic
177	C₂×D₄	#6104	1	2	yes	—
178	C₂×D₄	#6106	1	1	yes	characteristic
179	C₂²×C₄	—	52	1	no	—
180	C₂²×C₄	#6103	1	2	yes	—
181	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
182	Q₁₆	—	26	4	no	—
183	Q₁₆	—	52	1	no	—
184	Q₁₆	—	52	1	no	—
185	M₁₆	—	2	2	no	—
186	M₁₆	—	2	4	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	#6103	1	2	yes	—
190	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
191	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
192	C₂×C₈	D₄×D₁₃	1	2	yes	—
193	C₂×C₈	#6129	1	2	yes	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	26	2	no	—
196	C₄⋊C₄	—	52	4	no	—
197	C₂²⋊C₄	—	4	2	no	—
198	C₂²⋊C₄	—	52	1	no	—
199	C₂²⋊C₄	—	52	4	no	—
200	C₄²	—	4	2	no	—
201	C₄²	—	52	1	no	—
202	C₁₆	—	26	4	no	—
203	C₁₃	#13075	1	1	yes	characteristic, complemented, Sylow
204	C₂³	#88049	1	2	yes	—
205	C₂³	C₂×#6104	1	1	yes	characteristic
206	Q₈	—	2	2	no	—
207	Q₈	—	52	2	no	—
208	Q₈	—	52	2	no	—
209	Q₈	—	104	2	no	—
210	D₄	—	2	4	no	—
211	D₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	8	2	no	—
215	C₂×C₄	—	52	2	no	—
216	C₂×C₄	—	52	2	no	—
217	C₂×C₄	—	104	2	no	—
218	C₂×C₄	#88049	1	2	yes	—
219	C₂×C₄	C₂×#6103	1	1	yes	characteristic
220	C₂×C₄	#88193	1	2	yes	—
221	C₂×C₄	C₂×#6106	1	1	yes	characteristic
222	C₂×C₄	#88204	1	2	yes	—
223	C₈	—	2	2	no	—
224	C₈	#88135	1	2	yes	—
225	C₈	#88136	1	1	yes	characteristic
226	C₈	#88140	1	1	yes	characteristic
227	C₂²	—	2	2	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	4	2	no	—
230	C₂²	C₂×#88049	1	1	yes	characteristic
231	C₂²	#11718699	1	2	yes	—
232	C₄	—	2	2	no	—
233	C₄	—	16	2	no	—
234	C₄	—	104	2	no	—
235	C₄	—	104	2	no	—
236	C₄	#11718619	1	1	yes	characteristic
237	C₄	#11718663	1	1	yes	characteristic
238	C₄	#11718709	1	2	yes	—
239	C₂	—	2	2	no	—
240	C₂	—	2	1	no	—
241	C₂	—	4	2	no	—
242	C₂	#13115262	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13127939

C₂

yes

maximal, characteristic

#13124491

C₂

yes

maximal, characteristic

#13116555

C₂

yes

maximal, complemented

#13112937

C₂

yes

maximal, complemented

C₁₃×#1812

C₂

yes

maximal, characteristic

C₂×C₁₃×#131

C₂²

yes

characteristic

#11719530

—

C₂²×Dic₅₂

C₂²

yes

characteristic

C₁₃×#427

C₂²

yes

—

C₁₃×#410

—

#11718801

—

#11718798

C₄

yes

—

—

C₂²

yes

characteristic

C₂×#88052

—

C₂×#88052

C₂²

yes

—

C₁₃×H(3,ℤ/4)

—

#11718084

—

C₂×C₁₃×#41

D₄

yes

characteristic

C₂²×Dic₂₆

—

C₁₃×#131

—

C₁₃×#131

—

C₂×C₁₃×M₁₆

C₂×C₄

yes

—

C₂×C₁₃×M₁₆

D₄

yes

characteristic

C₂²×C₈×C₁₃

D₄

yes

—

C₂²×C₈×C₁₃

C₂³

yes

characteristic

C₁₃×#117

—

—

—

—

—

—

C₂×Dic₅₂

—

C₂×Dic₅₂

—

C₂×Dic₅₂

—

C₂×Dic₅₂

C₂×C₄

yes

—

C₁₃×#107

—

C₁₃×#106

—

C₁₃×#99

—

#88058

—

#88057

C₂×C₄

yes

—

—

—

—

—

—

D₄

yes

—

#13075

—

maximal, Sylow

C₁₃×#41

—

C₁₃×#41

—

C₂×Q₈×C₁₃

SD₁₆

yes

characteristic

C₂×D₄×C₁₃

D₈

yes

—

C₂×D₄×C₁₃

SD₁₆

yes

characteristic

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂²⋊C₄

yes

—

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

#41

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

—

C₂²⋊Dic₁₃

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

Dic₅₂

—

Dic₅₂

—

Dic₅₂

—

—

—

—

—

D₁₃

yes

characteristic

#1384

—

C₂³×C₁₃

#102

yes

—

C₂³×C₁₃

C₂×D₈

yes

characteristic

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#102

yes

—

C₂×C₄×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

#121

yes

—

C₂×C₄×C₁₃

C₂×SD₁₆

yes

characteristic

100

C₂×C₄×C₁₃

#135

yes

—

101

C₂×Dic₁₃

—

102

C₂×Dic₁₃

—

103

C₂×Dic₁₃

—

104

Dic₂₆

—

105

Dic₂₆

—

106

Dic₂₆

—

107

C₈×C₁₃

—

108

C₈×C₁₃

C₄×D₄

yes

—

109

C₈×C₁₃

#120

yes

characteristic

110

C₈×C₁₃

#123

yes

characteristic

111

C₂²×Q₁₆

—

112

C₂×#131

C₂×D₁₃

yes

characteristic

113

#536

—

114

#504

—

115

C₂×M₃₂

—

116

C₂×M₃₂

—

117

#470

—

118

#467

—

119

#427

—

120

#427

C₂×D₁₃

yes

—

121

#419

—

122

#410

—

123

#358

—

124

H(3,ℤ/4)

—

125

C₂²×C₁₃

—

126

C₂²×C₁₃

—

127

C₂²×C₁₃

—

128

C₂²×C₁₃

C₂×#102

yes

characteristic

129

C₂²×C₁₃

#466

yes

—

130

C₄×C₁₃

—

131

C₄×C₁₃

—

132

C₄×C₁₃

#386

yes

characteristic

133

C₄×C₁₃

#416

yes

characteristic

134

C₄×C₁₃

#465

yes

—

135

Dic₁₃

—

136

Dic₁₃

—

137

C₂×#41

C₂²⋊D₁₃

yes

characteristic

138

C₂²×Q₈

—

139

C₂×Q₁₆

—

140

C₂×Q₁₆

—

141

C₂×Q₁₆

—

142

C₂×Q₁₆

—

143

#131

—

144

#131

—

145

C₂×M₁₆

C₄×D₁₃

yes

—

146

C₂×M₁₆

C₂²⋊D₁₃

yes

characteristic

147

C₂²×C₈

D₅₂

yes

—

148

C₂²×C₈

C₂²×D₁₃

yes

characteristic

149

C₄⋊Q₈

—

150

#124

—

151

#123

—

152

C₂²⋊Q₈

—

153

#117

—

154

#117

—

155

M₃₂

—

156

M₃₂

—

157

M₃₂

—

158

C₂×C₁₆

—

159

C₂×C₁₆

—

160

#107

—

161

#107

—

162

#106

—

163

#106

—

164

#103

—

165

#99

—

166

C₂×C₁₃

—

167

C₂×C₁₃

—

168

C₂×C₁₃

—

169

C₂×C₁₃

#1893

yes

characteristic

170

#41

—

171

#41

—

172

C₂×Q₈

—

173

C₂×Q₈

—

174

C₂×Q₈

—

175

C₂×Q₈

—

176

C₂×Q₈

#6106

yes

characteristic

177

C₂×D₄

#6104

yes

—

178

C₂×D₄

#6106

yes

characteristic

179

C₂²×C₄

—

180

C₂²×C₄

#6103

yes

—

181

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

182

Q₁₆

—

183

Q₁₆

—

184

Q₁₆

—

185

M₁₆

—

186

M₁₆

—

187

C₂×C₈

—

188

C₂×C₈

—

189

C₂×C₈

#6103

yes

—

190

C₂×C₈

C₂×C₄×D₁₃

yes

characteristic

191

C₂×C₈

C₂×D₅₂

yes

characteristic

192

C₂×C₈

D₄×D₁₃

yes

—

193

C₂×C₈

#6129

yes

—

194

C₄⋊C₄

—

195

C₄⋊C₄

—

196

C₄⋊C₄

—

197

C₂²⋊C₄

—

198

C₂²⋊C₄

—

199

C₂²⋊C₄

—

200

C₄²

—

201

C₄²

—

202

C₁₆

—

203

C₁₃

#13075

yes

characteristic, complemented, Sylow

204

C₂³

#88049

yes

—

205

C₂³

C₂×#6104

yes

characteristic

206

Q₈

—

207

Q₈

—

208

Q₈

—

209

Q₈

—

104

—

210

D₄

—

211

D₄

—

212

C₂×C₄

—

213

C₂×C₄

—

214

C₂×C₄

—

215

C₂×C₄

—

216

C₂×C₄

—

217

C₂×C₄

—

104

—

218

C₂×C₄

#88049

yes

—

219

C₂×C₄

C₂×#6103

yes

characteristic

220

C₂×C₄

#88193

yes

—

221

C₂×C₄

C₂×#6106

yes

characteristic

222

C₂×C₄

#88204

yes

—

223

C₈

—

224

C₈

#88135

yes

—

225

C₈

#88136

yes

characteristic

226

C₈

#88140

yes

characteristic

227

C₂²

—

228

C₂²

—

229

C₂²

—

230

C₂²

C₂×#88049

yes

characteristic

231

C₂²

#11718699

yes

—

232

C₄

—

233

C₄

—

234

C₄

—

104

—

235

C₄

—

104

—

236

C₄

#11718619

yes

characteristic

237

C₄

#11718663

yes

characteristic

238

C₄

#11718709

yes

—

239

C₂

—

240

C₂

—

241

C₂

—

242

C₂

#13115262

yes

characteristic

	subgroup	G/H
center	C₂	#13115262
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#1812	C₂
socle	C₂×C₁₃	#1893
prime radical	C₂×C₁₃×#41	D₄
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#1812	D₁₃
hypocenter	C₁₃	#13075
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #1812

subgroup

G/H

center

C₂

#13115262

commutator

C₂×C₈×C₁₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₃

Fitting

C₁₃×#1812

C₂

socle

C₂×C₁₃

#1893

prime radical

C₂×C₁₃×#41

D₄

cosocle

C₂²×C₈×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #1812

N	H ≅ G/N	type
3. #13116555	C₂	semidirect
4. #13112937	C₂	semidirect
12. #11718798	C₄	semidirect
203. C₁₃	#13075	semidirect

H ≅ G/N

type

3. #13116555

C₂

semidirect

4. #13112937

C₂

semidirect

12. #11718798

C₄

semidirect

203. C₁₃

#13075

semidirect

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

subgroup	index
#13127939	2
#13124491	2
#13116555	2
#13112937	2
C₁₃×#1812	2
#13075	13

subgroup

index

C₁₃×#1812

Subgroups — #15780371

Conjugacy classes of subgroups

Complements

Maximal subgroups