Group #15780388 — subgroups

Subgroups — #15780388

Order 3328.

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

normal subgroups only

This group has 454 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	#13124491	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718109	C₂	1	2	yes	maximal, complemented
3	#13114886	C₂	1	1	yes	maximal, characteristic
4	#13112951	C₂	1	2	yes	maximal, complemented
5	C₁₃×#3407	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×#504	—	2	4	no	—
10	C₂×C₁₃×M₃₂	—	2	2	no	—
11	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
12	#11718898	—	2	1	no	—
13	#11718798	C₄	1	4	yes	—
14	#11718781	—	2	1	no	—
15	#11718777	—	2	2	no	—
16	C₂×#88059	C₂²	1	2	yes	—
17	#11718767	C₂²	1	1	yes	characteristic
18	#11718152	—	2	1	no	—
19	#11718109	—	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	#88198	—	4	1	no	—
29	#88190	—	4	1	no	—
30	#88189	—	4	2	no	—
31	#88180	—	2	1	no	—
32	#88178	—	2	4	no	—
33	C₂×#6099	—	2	2	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×M₃₂	—	2	4	no	—
39	C₁₃×M₃₂	—	2	2	no	—
40	C₁₃×M₃₂	D₄	1	4	yes	—
41	C₂×C₁₃×C₁₆	—	2	4	no	—
42	C₂×C₁₃×C₁₆	—	2	2	no	—
43	#88076	—	4	1	no	—
44	#88073	—	4	1	no	—
45	#88059	—	2	2	no	—
46	#88059	—	2	2	no	—
47	#88058	—	2	1	no	—
48	#88057	C₂×C₄	1	2	yes	—
49	#88056	—	2	4	no	—
50	#13078	—	13	1	no	maximal, Sylow
51	C₁₃×#41	—	2	2	no	—
52	C₁₃×#41	—	2	2	no	—
53	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
54	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
55	C₂×D₄×C₁₃	SD₁₆	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×Dic₁₃	—	4	1	no	—
59	C₂×Dic₂₆	—	2	2	no	—
60	C₂×Dic₂₆	—	4	2	no	—
61	C₂×Dic₂₆	—	4	2	no	—
62	C₂×Dic₂₆	—	4	1	no	—
63	C₁₃×M₁₆	—	2	2	no	—
64	C₁₃×M₁₆	—	2	4	no	—
65	C₂×C₈×C₁₃	—	2	2	no	—
66	C₂×C₈×C₁₃	—	2	2	no	—
67	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
68	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃	#41	1	2	yes	—
72	C₂²⋊Dic₁₃	—	4	4	no	—
73	C₂²⋊Dic₁₃	—	4	1	no	—
74	C₄⋊Dic₁₃	—	2	2	no	—
75	#6101	—	4	4	no	—
76	C₄×Dic₁₃	—	4	1	no	—
77	#6099	—	4	2	no	—
78	#6099	—	4	2	no	—
79	C₂×#1055	—	4	2	no	—
80	Dic₅₂	—	2	4	no	—
81	Dic₅₂	—	4	1	no	—
82	Dic₅₂	—	4	1	no	—
83	C₁₃×C₁₆	—	2	4	no	—
84	#3407	D₁₃	1	1	yes	characteristic
85	#3059	—	13	1	no	—
86	C₂×#362	—	13	2	no	—
87	#1813	—	13	1	no	—
88	#1389	—	13	2	no	—
89	C₂³×C₁₃	#102	1	2	yes	—
90	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
91	Q₈×C₁₃	—	2	2	no	—
92	D₄×C₁₃	—	2	4	no	—
93	D₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	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×D₈	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	#1055	—	8	2	no	—
112	C₂²×Q₁₆	—	13	1	no	—
113	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
114	#536	—	26	1	no	—
115	#504	—	2	4	no	—
116	C₂×M₃₂	—	2	2	no	—
117	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
118	#470	—	13	4	no	—
119	#467	—	26	2	no	—
120	C₂×#108	—	13	2	no	—
121	#427	—	13	1	no	—
122	#419	—	26	1	no	—
123	#413	—	26	1	no	—
124	#362	—	26	4	no	—
125	#338	—	26	1	no	—
126	C₂²×C₁₃	—	2	2	no	—
127	C₂²×C₁₃	—	4	1	no	—
128	C₂²×C₁₃	—	4	2	no	—
129	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
130	C₂²×C₁₃	#465	1	2	yes	—
131	C₄×C₁₃	—	2	2	no	—
132	C₄×C₁₃	#386	1	1	yes	characteristic
133	C₄×C₁₃	#416	1	1	yes	characteristic
134	C₄×C₁₃	#466	1	2	yes	—
135	Dic₁₃	—	8	2	no	—
136	Dic₁₃	—	8	2	no	—
137	C₂×#41	D₅₂	1	1	yes	characteristic
138	C₂²×Q₈	—	26	1	no	—
139	C₂×Q₁₆	—	13	2	no	—
140	C₂×Q₁₆	—	26	2	no	—
141	C₂×Q₁₆	—	26	2	no	—
142	C₂×Q₁₆	—	26	1	no	—
143	#131	—	2	2	no	—
144	#131	—	2	2	no	—
145	C₂×M₁₆	—	26	2	no	—
146	C₂×M₁₆	C₄×D₁₃	1	2	yes	—
147	C₂×M₁₆	D₅₂	1	1	yes	characteristic
148	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
149	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
150	C₄⋊Q₈	—	52	1	no	—
151	#124	—	52	1	no	—
152	#123	—	52	2	no	—
153	C₂²⋊Q₈	—	26	4	no	—
154	#117	—	26	1	no	—
155	M₃₂	—	2	4	no	—
156	M₃₂	—	2	2	no	—
157	M₃₂	C₂²⋊D₁₃	1	4	yes	—
158	C₂×C₁₆	—	2	4	no	—
159	C₂×C₁₆	—	2	2	no	—
160	#108	—	26	2	no	—
161	#108	—	26	2	no	—
162	#107	—	26	1	no	—
163	#106	—	13	2	no	—
164	#103	—	26	4	no	—
165	#101	—	52	1	no	—
166	#100	—	52	1	no	—
167	C₂×C₁₃	—	2	1	no	—
168	C₂×C₁₃	—	2	2	no	—
169	C₂×C₁₃	—	4	2	no	—
170	C₂×C₁₃	#1893	1	1	yes	characteristic
171	#41	—	2	2	no	—
172	#41	—	2	2	no	—
173	C₂×Q₈	—	26	2	no	—
174	C₂×Q₈	—	52	1	no	—
175	C₂×Q₈	—	52	2	no	—
176	C₂×Q₈	—	52	2	no	—
177	C₂×Q₈	D₁₀₄	1	1	yes	characteristic
178	C₂×D₄	#6095	1	2	yes	—
179	C₂×D₄	D₁₀₄	1	1	yes	characteristic
180	C₂²×C₄	—	52	1	no	—
181	C₂²×C₄	#6103	1	2	yes	—
182	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
183	Q₁₆	—	26	4	no	—
184	Q₁₆	—	52	1	no	—
185	Q₁₆	—	52	1	no	—
186	M₁₆	—	2	2	no	—
187	M₁₆	—	2	4	no	—
188	M₁₆	—	52	2	no	—
189	M₁₆	—	52	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	52	2	no	—
193	C₂×C₈	#6103	1	2	yes	—
194	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
195	C₂×C₈	#6127	1	2	yes	—
196	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
197	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
198	C₄⋊C₄	—	26	2	no	—
199	C₄⋊C₄	—	52	4	no	—
200	C₂²⋊C₄	—	52	1	no	—
201	C₂²⋊C₄	—	52	4	no	—
202	C₄²	—	52	1	no	—
203	C₁₆	—	2	4	no	—
204	C₁₃	#13078	1	1	yes	characteristic, complemented, Sylow
205	C₂³	#88061	1	2	yes	—
206	C₂³	C₂×#6095	1	1	yes	characteristic
207	Q₈	—	2	2	no	—
208	Q₈	—	52	2	no	—
209	Q₈	—	52	2	no	—
210	Q₈	—	104	2	no	—
211	D₄	—	2	4	no	—
212	D₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	52	2	no	—
216	C₂×C₄	—	52	2	no	—
217	C₂×C₄	—	104	2	no	—
218	C₂×C₄	#88061	1	2	yes	—
219	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
220	C₂×C₄	#88158	1	2	yes	—
221	C₂×C₄	C₂×#6103	1	1	yes	characteristic
222	C₂×C₄	#88184	1	2	yes	—
223	C₈	—	2	2	no	—
224	C₈	—	104	2	no	—
225	C₈	C₄×C₂²⋊D₁₃	1	2	yes	—
226	C₈	#88183	1	1	yes	characteristic
227	C₈	#88207	1	1	yes	characteristic
228	C₂²	—	2	2	no	—
229	C₂²	—	4	1	no	—
230	C₂²	—	4	2	no	—
231	C₂²	C₂×#88061	1	1	yes	characteristic
232	C₂²	#11718775	1	2	yes	—
233	C₄	—	2	2	no	—
234	C₄	—	104	2	no	—
235	C₄	—	104	2	no	—
236	C₄	#11718774	1	1	yes	characteristic
237	C₄	#11718776	1	2	yes	—
238	C₄	#11718874	1	1	yes	characteristic
239	C₂	—	2	1	no	—
240	C₂	—	2	2	no	—
241	C₂	—	4	2	no	—
242	C₂	#13115264	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13124491

C₂

yes

maximal, characteristic

C₂×#11718109

C₂

yes

maximal, complemented

#13114886

C₂

yes

maximal, characteristic

#13112951

C₂

yes

maximal, complemented

C₁₃×#3407

C₂

yes

maximal, characteristic

C₂×C₁₃×#131

C₂²

yes

characteristic

#11719530

—

C₂²×Dic₅₂

C₂²

yes

characteristic

C₁₃×#504

—

C₂×C₁₃×M₃₂

—

C₂×C₁₃×M₃₂

C₂²

yes

—

#11718898

—

#11718798

C₄

yes

—

#11718781

—

#11718777

—

C₂×#88059

C₂²

yes

—

#11718767

C₂²

yes

characteristic

#11718152

—

#11718109

—

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

—

—

—

—

—

—

C₂×Dic₅₂

—

C₂×Dic₅₂

—

C₂×Dic₅₂

—

C₂×Dic₅₂

C₂×C₄

yes

—

C₁₃×M₃₂

—

C₁₃×M₃₂

—

C₁₃×M₃₂

D₄

yes

—

C₂×C₁₃×C₁₆

—

C₂×C₁₃×C₁₆

—

—

—

—

—

—

C₂×C₄

yes

—

#88056

—

#13078

—

maximal, Sylow

C₁₃×#41

—

C₁₃×#41

—

C₂×Q₈×C₁₃

D₈

yes

characteristic

C₂×D₄×C₁₃

D₈

yes

characteristic

C₂×D₄×C₁₃

SD₁₆

yes

—

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

—

C₂²⋊Dic₁₃

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

#6099

—

#6099

—

C₂×#1055

—

Dic₅₂

—

Dic₅₂

—

Dic₅₂

—

C₁₃×C₁₆

—

#3407

D₁₃

yes

—

—

—

—

C₂³×C₁₃

#102

yes

—

C₂³×C₁₃

C₂×SD₁₆

yes

characteristic

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×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×D₈

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

#1055

—

112

C₂²×Q₁₆

—

113

C₂×#131

C₂×D₁₃

yes

characteristic

114

#536

—

115

#504

—

116

C₂×M₃₂

—

117

C₂×M₃₂

C₂×D₁₃

yes

—

118

#470

—

119

#467

—

120

C₂×#108

—

121

#427

—

122

#419

—

123

#413

—

124

#362

—

125

#338

—

126

C₂²×C₁₃

—

127

C₂²×C₁₃

—

128

C₂²×C₁₃

—

129

C₂²×C₁₃

C₂×#102

yes

characteristic

130

C₂²×C₁₃

#465

yes

—

131

C₄×C₁₃

—

132

C₄×C₁₃

#386

yes

characteristic

133

C₄×C₁₃

#416

yes

characteristic

134

C₄×C₁₃

#466

yes

—

135

Dic₁₃

—

136

Dic₁₃

—

137

C₂×#41

D₅₂

yes

characteristic

138

C₂²×Q₈

—

139

C₂×Q₁₆

—

140

C₂×Q₁₆

—

141

C₂×Q₁₆

—

142

C₂×Q₁₆

—

143

#131

—

144

#131

—

145

C₂×M₁₆

—

146

C₂×M₁₆

C₄×D₁₃

yes

—

147

C₂×M₁₆

D₅₂

yes

characteristic

148

C₂²×C₈

C₂²⋊D₁₃

yes

—

149

C₂²×C₈

C₂²×D₁₃

yes

characteristic

150

C₄⋊Q₈

—

151

#124

—

152

#123

—

153

C₂²⋊Q₈

—

154

#117

—

155

M₃₂

—

156

M₃₂

—

157

M₃₂

C₂²⋊D₁₃

yes

—

158

C₂×C₁₆

—

159

C₂×C₁₆

—

160

#108

—

161

#108

—

162

#107

—

163

#106

—

164

#103

—

165

#101

—

166

#100

—

167

C₂×C₁₃

—

168

C₂×C₁₃

—

169

C₂×C₁₃

—

170

C₂×C₁₃

#1893

yes

characteristic

171

#41

—

172

#41

—

173

C₂×Q₈

—

174

C₂×Q₈

—

175

C₂×Q₈

—

176

C₂×Q₈

—

177

C₂×Q₈

D₁₀₄

yes

characteristic

178

C₂×D₄

#6095

yes

—

179

C₂×D₄

D₁₀₄

yes

characteristic

180

C₂²×C₄

—

181

C₂²×C₄

#6103

yes

—

182

C₂²×C₄

C₂×D₅₂

yes

characteristic

183

Q₁₆

—

184

Q₁₆

—

185

Q₁₆

—

186

M₁₆

—

187

M₁₆

—

188

M₁₆

—

189

M₁₆

—

190

C₂×C₈

—

191

C₂×C₈

—

192

C₂×C₈

—

193

C₂×C₈

#6103

yes

—

194

C₂×C₈

C₂×C₄×D₁₃

yes

characteristic

195

C₂×C₈

#6127

yes

—

196

C₂×C₈

C₂×C₂²⋊D₁₃

yes

—

197

C₂×C₈

C₂×C₂²⋊D₁₃

yes

characteristic

198

C₄⋊C₄

—

199

C₄⋊C₄

—

200

C₂²⋊C₄

—

201

C₂²⋊C₄

—

202

C₄²

—

203

C₁₆

—

204

C₁₃

#13078

yes

characteristic, complemented, Sylow

205

C₂³

#88061

yes

—

206

C₂³

C₂×#6095

yes

characteristic

207

Q₈

—

208

Q₈

—

209

Q₈

—

210

Q₈

—

104

—

211

D₄

—

212

D₄

—

213

C₂×C₄

—

214

C₂×C₄

—

215

C₂×C₄

—

216

C₂×C₄

—

217

C₂×C₄

—

104

—

218

C₂×C₄

#88061

yes

—

219

C₂×C₄

C₂×D₁₀₄

yes

characteristic

220

C₂×C₄

#88158

yes

—

221

C₂×C₄

C₂×#6103

yes

characteristic

222

C₂×C₄

#88184

yes

—

223

C₈

—

224

C₈

—

104

—

225

C₈

C₄×C₂²⋊D₁₃

yes

—

226

C₈

#88183

yes

characteristic

227

C₈

#88207

yes

characteristic

228

C₂²

—

229

C₂²

—

230

C₂²

—

231

C₂²

C₂×#88061

yes

characteristic

232

C₂²

#11718775

yes

—

233

C₄

—

234

C₄

—

104

—

235

C₄

—

104

—

236

C₄

#11718774

yes

characteristic

237

C₄

#11718776

yes

—

238

C₄

#11718874

yes

characteristic

239

C₂

—

240

C₂

—

241

C₂

—

242

C₂

#13115264

yes

characteristic

	subgroup	G/H
center	C₂	#13115264
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#3407	C₂
socle	C₂×C₁₃	#1893
prime radical	C₂×C₁₃×#41	D₄
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#3407	D₁₃
hypocenter	C₁₃	#13078
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #3407

subgroup

G/H

center

C₂

#13115264

commutator

C₂×C₈×C₁₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₃

Fitting

C₁₃×#3407

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 ◁ #3407

N	H ≅ G/N	type
2. C₂×#11718109	C₂	semidirect
4. #13112951	C₂	semidirect
204. C₁₃	#13078	semidirect

H ≅ G/N

type

2. C₂×#11718109

C₂

semidirect

4. #13112951

C₂

semidirect

204. C₁₃

#13078

semidirect

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

subgroup	index
#13124491	2
C₂×#11718109	2
#13114886	2
#13112951	2
C₁₃×#3407	2
#13078	13

subgroup

index

C₁₃×#3407

Subgroups — #15780388

Conjugacy classes of subgroups

Complements

Maximal subgroups