Group #15781476 — subgroups

Subgroups — #15781476

Order 3328.

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

normal subgroups only

This group has 366 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	C₂×#11718341	C₂	1	1	yes	maximal, characteristic
2	#13114715	C₂	1	1	yes	maximal, characteristic
3	#13112942	C₂	1	2	yes	maximal
4	#13112935	C₂	1	2	yes	maximal
5	C₂×C₁₃×#364	C₂	1	1	yes	maximal, characteristic
6	C₂×#88123	—	2	1	no	—
7	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88059	C₂²	1	1	yes	characteristic
10	#11718767	C₂²	1	1	yes	characteristic
11	C₂×#88057	—	2	1	no	—
12	C₂×#88057	C₂²	1	2	yes	—
13	#11718762	—	2	1	no	—
14	#11718341	—	2	2	no	—
15	#11718341	C₄	1	4	yes	complemented
16	C₁₃×#364	—	2	2	no	—
17	#11718115	—	2	1	no	—
18	C₂²×Dic₂₆	—	4	1	no	—
19	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
22	#88180	—	2	1	no	—
23	C₂×C₄⋊Dic₁₃	—	2	2	no	—
24	C₂×C₄⋊Dic₁₃	—	4	1	no	—
25	C₂×#6099	—	2	1	no	—
26	#88123	—	2	4	no	—
27	#88123	—	4	2	no	—
28	C₁₃×M₃₂	—	2	2	no	—
29	C₁₃×M₃₂	—	2	2	no	—
30	C₂×C₁₃×C₁₆	—	2	2	no	—
31	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₁₃	D₄	1	2	yes	—
33	#88059	—	2	1	no	—
34	#88059	Q₈	1	2	yes	complemented
35	#88058	Q₈	1	2	yes	complemented
36	#88057	—	2	4	no	—
37	#88057	—	2	2	no	—
38	#88057	—	2	1	no	—
39	#88057	C₂×C₄	1	4	yes	—
40	#88055	—	4	1	no	—
41	C₈×Dic₁₃	—	4	1	no	—
42	#13272	—	13	1	no	maximal, Sylow
43	C₂²×C₄×C₁₃	—	2	1	no	—
44	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₁₃	—	4	2	no	—
46	C₂×Dic₂₆	—	4	4	no	—
47	C₂×Dic₂₆	—	8	4	no	—
48	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
49	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
51	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
52	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
57	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
58	C₄²×C₁₃	—	2	1	no	—
59	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
60	C₂²⋊Dic₁₃	—	4	1	no	—
61	C₄⋊Dic₁₃	—	2	4	no	—
62	C₄⋊Dic₁₃	—	2	2	no	—
63	C₄⋊Dic₁₃	—	4	4	no	—
64	C₄⋊Dic₁₃	—	4	2	no	—
65	C₄×Dic₁₃	—	4	1	no	—
66	#6099	—	2	2	no	—
67	#6099	—	4	1	no	—
68	C₂×#1055	—	4	1	no	—
69	C₁₃×C₁₆	—	4	2	no	—
70	C₂×#498	—	13	1	no	—
71	C₂×#364	D₁₃	1	1	yes	characteristic
72	#1772	—	13	1	no	—
73	#1386	—	13	2	no	—
74	#1383	—	13	2	no	—
75	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
76	C₂×C₄×C₁₃	—	2	4	no	—
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
83	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
84	C₂×Dic₁₃	—	4	4	no	—
85	C₂×Dic₁₃	—	4	2	no	—
86	C₂×Dic₁₃	—	8	4	no	—
87	Dic₂₆	—	8	8	no	—
88	C₈×C₁₃	#103	1	2	yes	—
89	C₈×C₁₃	#103	1	2	yes	—
90	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
91	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
92	C₈×C₁₃	#124	1	1	yes	characteristic
93	C₈×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
94	#1055	—	4	2	no	—
95	C₂×C₄⋊Q₈	—	26	1	no	—
96	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
97	#498	—	13	4	no	—
98	#498	—	26	2	no	—
99	C₂×#108	—	13	1	no	—
100	#427	—	13	1	no	—
101	C₂×#106	—	13	2	no	—
102	C₂×#106	—	26	1	no	—
103	#405	—	26	1	no	—
104	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
105	#364	—	2	2	no	—
106	#339	—	26	1	no	—
107	C₂²×C₁₃	—	2	2	no	—
108	C₂²×C₁₃	#418	1	1	yes	characteristic
109	C₂²×C₁₃	#476	1	1	yes	characteristic
110	C₂²×C₁₃	#477	1	1	yes	characteristic
111	C₄×C₁₃	—	2	4	no	—
112	C₄×C₁₃	#391	1	1	yes	characteristic
113	C₄×C₁₃	C₂×#103	1	1	yes	characteristic
114	C₄×C₁₃	#476	1	1	yes	characteristic
115	C₄×C₁₃	#477	1	1	yes	characteristic
116	Dic₁₃	—	8	4	no	—
117	Dic₁₃	—	8	2	no	—
118	C₂²×Q₈	—	52	1	no	—
119	C₂×M₁₆	—	26	1	no	—
120	C₂²×C₈	D₅₂	1	1	yes	characteristic
121	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
122	C₄⋊Q₈	—	26	4	no	—
123	C₄⋊Q₈	—	52	2	no	—
124	#117	—	26	1	no	—
125	C₂×C₄⋊C₄	—	26	2	no	—
126	C₂×C₄⋊C₄	—	52	1	no	—
127	C₂×C₄²	D₅₂	1	1	yes	characteristic
128	M₃₂	—	2	2	no	—
129	M₃₂	—	2	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	#108	—	13	2	no	—
132	#108	—	26	1	no	—
133	#107	—	13	2	no	—
134	#106	—	13	4	no	—
135	#106	—	26	4	no	—
136	#106	—	26	1	no	—
137	#106	—	26	2	no	—
138	#97	—	52	1	no	—
139	C₄×C₈	—	52	1	no	—
140	C₄×C₈	C₄×D₁₃	1	2	yes	—
141	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
142	C₂×C₁₃	—	2	1	no	—
143	C₂×C₁₃	—	2	1	no	—
144	C₂×C₁₃	#1987	1	1	yes	characteristic
145	C₂×C₁₃	#2230	1	2	yes	—
146	C₂×Q₈	—	52	4	no	—
147	C₂×Q₈	—	104	4	no	—
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	—	52	2	no	—
150	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
151	M₁₆	—	26	2	no	—
152	M₁₆	—	52	1	no	—
153	C₂×C₈	—	52	1	no	—
154	C₂×C₈	#6095	1	1	yes	characteristic
155	C₂×C₈	#6095	1	1	yes	characteristic
156	C₂×C₈	Dic₅₂	1	2	yes	—
157	C₂×C₈	#6103	1	2	yes	—
158	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
159	C₂×C₈	#6129	1	1	yes	characteristic
160	C₂×C₈	#6129	1	1	yes	characteristic
161	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
162	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
163	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
164	C₄⋊C₄	—	26	2	no	—
165	C₄⋊C₄	—	26	4	no	—
166	C₄⋊C₄	—	52	4	no	—
167	C₄⋊C₄	—	52	2	no	—
168	C₂²⋊C₄	—	52	1	no	—
169	C₄²	—	2	1	no	—
170	C₄²	—	52	1	no	—
171	C₄²	#6103	1	2	yes	—
172	C₁₆	—	4	2	no	—
173	C₁₃	#13272	1	1	yes	characteristic, complemented, Sylow
174	C₂³	#88162	1	1	yes	characteristic
175	Q₈	—	104	8	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	52	2	no	—
179	C₂×C₄	—	52	4	no	—
180	C₂×C₄	—	104	4	no	—
181	C₂×C₄	#88151	1	1	yes	characteristic
182	C₂×C₄	#88151	1	1	yes	characteristic
183	C₂×C₄	C₂×#6095	1	1	yes	characteristic
184	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
185	C₂×C₄	#88162	1	1	yes	characteristic
186	C₂×C₄	C₂×#6103	1	1	yes	characteristic
187	C₈	—	52	2	no	—
188	C₈	#88056	1	2	yes	—
189	C₈	#88056	1	2	yes	—
190	C₈	#88141	1	1	yes	characteristic
191	C₈	#88141	1	1	yes	characteristic
192	C₈	#88190	1	1	yes	characteristic
193	C₈	#88198	1	1	yes	characteristic
194	C₂²	—	2	2	no	—
195	C₂²	#11718503	1	1	yes	characteristic
196	C₂²	#11718511	1	1	yes	characteristic
197	C₂²	#11718792	1	1	yes	characteristic
198	C₄	—	2	4	no	—
199	C₄	—	104	4	no	—
200	C₄	—	104	2	no	—
201	C₄	#11718503	1	1	yes	characteristic
202	C₄	#11718511	1	1	yes	characteristic
203	C₄	#11718635	1	1	yes	characteristic
204	C₄	C₂×#88056	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	#13115783	1	1	yes	characteristic
208	C₂	#13117008	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718341

C₂

yes

maximal, characteristic

#13114715

C₂

yes

maximal, characteristic

#13112942

C₂

yes

maximal

#13112935

C₂

yes

maximal

C₂×C₁₃×#364

C₂

yes

maximal, characteristic

C₂×#88123

—

C₂×C₁₃×M₃₂

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×#88059

C₂²

yes

characteristic

#11718767

C₂²

yes

characteristic

C₂×#88057

—

C₂×#88057

C₂²

yes

—

#11718762

—

#11718341

—

#11718341

C₄

yes

complemented

C₁₃×#364

—

#11718115

—

C₂²×Dic₂₆

—

C₂²×C₈×C₁₃

D₄

yes

characteristic

C₂²×C₈×C₁₃

C₂³

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

characteristic

#88180

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×#6099

—

#88123

—

#88123

—

C₁₃×M₃₂

—

C₁₃×M₃₂

—

C₂×C₁₃×C₁₆

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

D₄

yes

—

#88059

—

#88059

Q₈

yes

complemented

#88058

Q₈

yes

—

—

—

C₂×C₄

yes

—

#88055

—

C₈×Dic₁₃

—

#13272

—

maximal, Sylow

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

C₂²⋊C₄

yes

—

C₂×C₈×C₁₃

SD₁₆

yes

characteristic

C₂×C₈×C₁₃

SD₁₆

yes

characteristic

C₂×C₈×C₁₃

Q₁₆

yes

—

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₃

C₂×Q₈

yes

C₂×C₈×C₁₃

yes

C₂×C₈×C₁₃

yes

C₄²×C₁₃

—

C₄²×C₁₃

C₂²⋊C₄

yes

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

#6099

—

#6099

—

C₂×#1055

—

C₁₃×C₁₆

—

C₂×#498

—

C₂×#364

D₁₃

yes

—

—

—

C₂³×C₁₃

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

C₈×C₁₃

#103

yes

—

C₈×C₁₃

#103

yes

—

C₈×C₁₃

C₄×Q₈

yes

characteristic

C₈×C₁₃

C₄×Q₈

yes

characteristic

C₈×C₁₃

#124

yes

characteristic

C₈×C₁₃

C₄⋊Q₈

yes

characteristic

#1055

—

C₂×C₄⋊Q₈

—

C₂×M₃₂

C₂×D₁₃

yes

—

#498

—

#498

—

C₂×#108

—

100

#427

—

101

C₂×#106

—

102

C₂×#106

—

103

#405

—

104

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

105

#364

—

106

#339

—

107

C₂²×C₁₃

—

108

C₂²×C₁₃

#418

yes

characteristic

109

C₂²×C₁₃

#476

yes

characteristic

110

C₂²×C₁₃

#477

yes

characteristic

111

C₄×C₁₃

—

112

C₄×C₁₃

#391

yes

characteristic

113

C₄×C₁₃

C₂×#103

yes

characteristic

114

C₄×C₁₃

#476

yes

characteristic

115

C₄×C₁₃

#477

yes

characteristic

116

Dic₁₃

—

117

Dic₁₃

—

118

C₂²×Q₈

—

119

C₂×M₁₆

—

120

C₂²×C₈

D₅₂

yes

characteristic

121

C₂²×C₈

C₂²×D₁₃

yes

characteristic

122

C₄⋊Q₈

—

123

C₄⋊Q₈

—

124

#117

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄²

D₅₂

yes

characteristic

128

M₃₂

—

129

M₃₂

—

130

C₂×C₁₆

—

131

#108

—

132

#108

—

133

#107

—

134

#106

—

135

#106

—

136

#106

—

137

#106

—

138

#97

—

139

C₄×C₈

—

140

C₄×C₈

C₄×D₁₃

yes

—

141

C₄×C₈

C₂²⋊D₁₃

yes

—

142

C₂×C₁₃

—

143

C₂×C₁₃

—

144

C₂×C₁₃

#1987

yes

characteristic

145

C₂×C₁₃

#2230

yes

—

146

C₂×Q₈

—

147

C₂×Q₈

—

104

—

148

C₂²×C₄

—

149

C₂²×C₄

—

150

C₂²×C₄

C₂×D₅₂

yes

characteristic

151

M₁₆

—

152

M₁₆

—

153

C₂×C₈

—

154

C₂×C₈

#6095

yes

characteristic

155

C₂×C₈

#6095

yes

characteristic

156

C₂×C₈

Dic₅₂

yes

—

157

C₂×C₈

#6103

yes

—

158

C₂×C₈

C₂×C₄×D₁₃

yes

characteristic

159

C₂×C₈

#6129

yes

characteristic

160

C₂×C₈

#6129

yes

characteristic

161

C₂×C₈

Q₈×D₁₃

yes

characteristic

162

C₂×C₈

Q₈×D₁₃

yes

characteristic

163

C₂×C₈

C₂×C₂²⋊D₁₃

yes

characteristic

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₂²⋊C₄

—

169

C₄²

—

170

C₄²

—

171

C₄²

#6103

yes

—

172

C₁₆

—

173

C₁₃

#13272

yes

characteristic, complemented, Sylow

174

C₂³

#88162

yes

characteristic

175

Q₈

—

104

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

104

—

181

C₂×C₄

#88151

yes

characteristic

182

C₂×C₄

#88151

yes

characteristic

183

C₂×C₄

C₂×#6095

yes

characteristic

184

C₂×C₄

C₂×Dic₅₂

yes

characteristic

185

C₂×C₄

#88162

yes

characteristic

186

C₂×C₄

C₂×#6103

yes

characteristic

187

C₈

—

188

C₈

#88056

yes

—

189

C₈

#88056

yes

—

190

C₈

#88141

yes

characteristic

191

C₈

#88141

yes

characteristic

192

C₈

#88190

yes

characteristic

193

C₈

#88198

yes

characteristic

194

C₂²

—

195

C₂²

#11718503

yes

characteristic

196

C₂²

#11718511

yes

characteristic

197

C₂²

#11718792

yes

characteristic

198

C₄

—

199

C₄

—

104

—

200

C₄

—

104

—

201

C₄

#11718503

yes

characteristic

202

C₄

#11718511

yes

characteristic

203

C₄

#11718635

yes

characteristic

204

C₄

C₂×#88056

yes

characteristic

205

C₂

—

206

C₂

—

207

C₂

#13115783

yes

characteristic

208

C₂

#13117008

yes

—

	subgroup	G/H
center	C₂²	#11718503
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×#364	C₂
socle	C₂²×C₁₃	#476
prime radical	C₂³×C₁₃	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#364	D₁₃
hypocenter	C₁₃	#13272
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

subgroup

G/H

center

C₂²

#11718503

commutator

C₂×C₈×C₁₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₃

Fitting

C₂×C₁₃×#364

C₂

C₂²×C₁₃

C₂³×C₁₃

C₂²×C₈×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

N	H ≅ G/N	type
15. #11718341	C₄	semidirect
34. #88059	Q₈	semidirect
35. #88058	Q₈	semidirect
173. C₁₃	#13272	semidirect

H ≅ G/N

type

15. #11718341

C₄

semidirect

34. #88059

Q₈

semidirect

35. #88058

Q₈

semidirect

173. C₁₃

#13272

semidirect

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

subgroup	index
C₂×#11718341	2
#13114715	2
#13112942	2
#13112935	2
C₂×C₁₃×#364	2
#13272	13

subgroup

index

C₂×C₁₃×#364

Subgroups — #15781476

Conjugacy classes of subgroups

Complements

Maximal subgroups