Group #15782312 — subgroups

Subgroups — #15782312

Order 3328.

436 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 97 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×#11718360	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718109	C₂	1	2	yes	maximal
3	#13116557	C₂	1	2	yes	maximal
4	#13115584	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₂	—	2	1	no	—
7	C₂²×Dic₅₂	C₂²	1	2	yes	—
8	C₂×#88123	—	2	1	no	—
9	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88059	C₂²	1	1	yes	characteristic
12	#11718767	C₂²	1	1	yes	characteristic
13	#11718763	—	2	1	no	—
14	#11718360	—	2	4	no	—
15	#11718115	—	2	1	no	—
16	#11718109	—	2	4	no	—
17	#11718095	—	2	4	no	—
18	C₂²×Dic₂₆	—	2	2	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	—	4	2	no	—
24	C₂×#6099	—	2	1	no	—
25	C₂×Dic₅₂	—	2	2	no	—
26	C₂×Dic₅₂	—	2	2	no	—
27	C₂×Dic₅₂	—	2	2	no	—
28	C₂×Dic₅₂	—	2	2	no	—
29	C₂×Dic₅₂	—	2	4	no	—
30	C₂×Dic₅₂	—	2	2	no	—
31	C₂×Dic₅₂	—	4	2	no	—
32	C₂×Dic₅₂	—	4	2	no	—
33	C₂×Dic₅₂	D₄	1	4	yes	—
34	#88123	—	2	4	no	—
35	#88123	—	4	2	no	—
36	C₁₃×M₃₂	—	2	2	no	—
37	C₁₃×M₃₂	—	2	2	no	—
38	C₂×C₁₃×C₁₆	—	2	2	no	—
39	C₄×C₈×C₁₃	—	2	2	no	—
40	#88059	—	2	2	no	—
41	#88058	D₄	1	2	yes	—
42	#88057	—	2	1	no	—
43	#88054	—	4	2	no	—
44	#13415	—	13	1	no	maximal, Sylow
45	C₂²×C₄×C₁₃	—	2	1	no	—
46	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×Dic₁₃	—	4	2	no	—
48	C₂×Dic₂₆	—	2	4	no	—
49	C₂×Dic₂₆	—	4	2	no	—
50	C₂×Dic₂₆	—	4	2	no	—
51	C₂×Dic₂₆	—	4	2	no	—
52	C₂×Dic₂₆	—	4	2	no	—
53	C₂×Dic₂₆	—	4	2	no	—
54	C₂×C₈×C₁₃	—	2	1	no	—
55	C₂×C₈×C₁₃	—	2	1	no	—
56	C₂×C₈×C₁₃	D₈	1	2	yes	—
57	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₁₃	#41	1	2	yes	—
60	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
61	C₄²×C₁₃	—	2	1	no	—
62	C₄²×C₁₃	SD₁₆	1	2	yes	—
63	C₂²⋊Dic₁₃	—	4	1	no	—
64	C₄⋊Dic₁₃	—	2	2	no	—
65	C₄⋊Dic₁₃	—	4	8	no	—
66	C₄×Dic₁₃	—	4	1	no	—
67	#6099	—	4	1	no	—
68	#6099	—	4	1	no	—
69	C₂×#1055	—	4	1	no	—
70	Dic₅₂	—	2	4	no	—
71	Dic₅₂	—	2	4	no	—
72	Dic₅₂	—	4	8	no	—
73	Dic₅₂	—	4	2	no	—
74	Dic₅₂	—	4	2	no	—
75	C₁₃×C₁₆	—	4	2	no	—
76	C₂×#494	—	13	1	no	—
77	C₂×#362	—	13	2	no	—
78	#2141	—	13	2	no	—
79	#1945	—	13	1	no	—
80	#1376	D₁₃	1	1	yes	characteristic
81	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₂×C₄×C₁₃	—	2	4	no	—
83	C₂×C₄×C₁₃	—	2	2	no	—
84	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
85	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
86	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
89	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
90	C₂×Dic₁₃	—	4	4	no	—
91	C₂×Dic₁₃	—	4	2	no	—
92	C₂×Dic₁₃	—	8	2	no	—
93	C₂×Dic₁₃	—	8	2	no	—
94	Dic₂₆	—	4	4	no	—
95	Dic₂₆	—	4	4	no	—
96	Dic₂₆	—	8	2	no	—
97	Dic₂₆	—	8	2	no	—
98	C₈×C₁₃	—	2	2	no	—
99	C₈×C₁₃	#121	1	2	yes	—
100	C₈×C₁₃	#123	1	2	yes	—
101	#1055	—	8	1	no	—
102	C₂²×Q₁₆	—	13	2	no	—
103	C₂²×Q₁₆	—	26	1	no	—
104	C₂×C₄⋊Q₈	—	26	1	no	—
105	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
106	#494	—	26	4	no	—
107	C₂×#108	—	13	1	no	—
108	#427	—	13	1	no	—
109	#424	—	26	1	no	—
110	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
111	#362	—	26	4	no	—
112	#358	—	26	4	no	—
113	#339	—	26	1	no	—
114	C₂²×C₁₃	—	2	2	no	—
115	C₂²×C₁₃	#450	1	1	yes	characteristic
116	C₂²×C₁₃	#459	1	1	yes	characteristic
117	C₂²×C₁₃	#490	1	1	yes	characteristic
118	C₄×C₁₃	—	2	4	no	—
119	C₄×C₁₃	#394	1	1	yes	characteristic
120	C₄×C₁₃	#447	1	1	yes	characteristic
121	C₄×C₁₃	#460	1	1	yes	characteristic
122	C₄×C₁₃	#486	1	1	yes	characteristic
123	Dic₁₃	—	8	2	no	—
124	Dic₁₃	—	8	4	no	—
125	C₂²×Q₈	—	26	2	no	—
126	C₂×Q₁₆	—	13	4	no	—
127	C₂×Q₁₆	—	26	4	no	—
128	C₂×Q₁₆	—	26	2	no	—
129	C₂×Q₁₆	—	26	2	no	—
130	C₂×Q₁₆	—	26	2	no	—
131	C₂×Q₁₆	—	26	2	no	—
132	C₂×Q₁₆	—	26	2	no	—
133	C₂×Q₁₆	—	52	2	no	—
134	C₂×Q₁₆	—	52	2	no	—
135	C₂×M₁₆	—	26	1	no	—
136	C₂²×C₈	D₅₂	1	1	yes	characteristic
137	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
138	C₄⋊Q₈	—	26	4	no	—
139	C₄⋊Q₈	—	52	2	no	—
140	#117	—	26	1	no	—
141	C₂×C₄⋊C₄	—	52	2	no	—
142	C₂×C₄²	D₅₂	1	1	yes	characteristic
143	M₃₂	—	2	2	no	—
144	M₃₂	—	2	2	no	—
145	C₂×C₁₆	—	2	2	no	—
146	#108	—	26	2	no	—
147	#107	—	13	2	no	—
148	#106	—	26	1	no	—
149	C₄⋊C₈	—	52	2	no	—
150	C₄×C₈	—	2	2	no	—
151	C₂×C₁₃	—	2	1	no	—
152	C₂×C₁₃	—	2	1	no	—
153	C₂×C₁₃	#2041	1	1	yes	characteristic
154	C₂×C₁₃	#2214	1	2	yes	—
155	C₂×Q₈	—	26	4	no	—
156	C₂×Q₈	—	52	2	no	—
157	C₂×Q₈	—	52	2	no	—
158	C₂×Q₈	—	52	2	no	—
159	C₂×Q₈	—	52	2	no	—
160	C₂×Q₈	—	52	2	no	—
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	52	2	no	—
163	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
164	Q₁₆	—	26	4	no	—
165	Q₁₆	—	26	4	no	—
166	Q₁₆	—	52	8	no	—
167	Q₁₆	—	52	2	no	—
168	Q₁₆	—	52	2	no	—
169	M₁₆	—	52	1	no	—
170	M₁₆	—	52	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	52	1	no	—
174	C₂×C₈	D₁₀₄	1	2	yes	—
175	C₂×C₈	#6127	1	2	yes	—
176	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
177	C₂×C₈	D₄×D₁₃	1	2	yes	—
178	C₂×C₈	#6131	1	1	yes	characteristic
179	C₄⋊C₄	—	26	2	no	—
180	C₄⋊C₄	—	52	8	no	—
181	C₂²⋊C₄	—	52	1	no	—
182	C₄²	—	2	1	no	—
183	C₄²	—	52	1	no	—
184	C₄²	#6095	1	2	yes	—
185	C₁₆	—	4	2	no	—
186	C₁₃	#13415	1	1	yes	characteristic, complemented, Sylow
187	C₂³	#88162	1	1	yes	characteristic
188	Q₈	—	52	4	no	—
189	Q₈	—	52	4	no	—
190	Q₈	—	104	2	no	—
191	Q₈	—	104	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	52	2	no	—
195	C₂×C₄	—	52	4	no	—
196	C₂×C₄	—	104	2	no	—
197	C₂×C₄	—	104	2	no	—
198	C₂×C₄	#88129	1	1	yes	characteristic
199	C₂×C₄	#88136	1	1	yes	characteristic
200	C₂×C₄	#88149	1	1	yes	characteristic
201	C₂×C₄	C₂×#6095	1	1	yes	characteristic
202	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
203	C₂×C₄	#88162	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	104	1	no	—
206	C₈	#88138	1	2	yes	—
207	C₈	#88148	1	2	yes	—
208	C₂²	—	2	2	no	—
209	C₂²	#11718374	1	1	yes	characteristic
210	C₂²	#11718392	1	1	yes	characteristic
211	C₂²	#11718504	1	1	yes	characteristic
212	C₄	—	2	4	no	—
213	C₄	—	104	4	no	—
214	C₄	—	104	2	no	—
215	C₄	#11718333	1	1	yes	characteristic
216	C₄	#11718355	1	1	yes	characteristic
217	C₄	#11718393	1	1	yes	characteristic
218	C₄	#11718509	1	1	yes	characteristic
219	C₂	—	2	1	no	—
220	C₂	—	2	1	no	—
221	C₂	#13116071	1	1	yes	characteristic
222	C₂	#13116918	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718360

C₂

yes

maximal, characteristic

C₂×#11718109

C₂

yes

maximal

#13116557

C₂

yes

maximal

#13115584

C₂

yes

maximal, characteristic

C₁₃×#1376

C₂

yes

maximal, characteristic

C₂²×Dic₅₂

—

C₂²×Dic₅₂

C₂²

yes

—

C₂×#88123

—

C₂×C₁₃×M₃₂

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×#88059

C₂²

yes

characteristic

#11718767

C₂²

yes

—

—

—

—

—

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×#6099

—

C₂×Dic₅₂

—

C₂×Dic₅₂

—

C₂×Dic₅₂

—

C₂×Dic₅₂

—

C₂×Dic₅₂

—

C₂×Dic₅₂

—

C₂×Dic₅₂

—

C₂×Dic₅₂

—

C₂×Dic₅₂

D₄

yes

—

#88123

—

#88123

—

C₁₃×M₃₂

—

C₁₃×M₃₂

—

C₂×C₁₃×C₁₆

—

C₄×C₈×C₁₃

—

#88059

—

#88058

D₄

yes

—

—

—

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₂×C₈×C₁₃

#41

yes

—

C₂×C₈×C₁₃

#41

yes

characteristic

C₄²×C₁₃

—

C₄²×C₁₃

SD₁₆

yes

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

#6099

—

#6099

—

C₂×#1055

—

Dic₅₂

—

Dic₅₂

—

Dic₅₂

—

Dic₅₂

—

Dic₅₂

—

C₁₃×C₁₆

—

—

—

—

—

D₁₃

yes

characteristic

C₂³×C₁₃

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

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

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

#121

yes

—

100

C₈×C₁₃

#123

yes

—

101

#1055

—

102

C₂²×Q₁₆

—

103

C₂²×Q₁₆

—

104

C₂×C₄⋊Q₈

—

105

C₂×M₃₂

C₂×D₁₃

yes

—

106

#494

—

107

C₂×#108

—

108

#427

—

109

#424

—

110

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

111

#362

—

112

#358

—

113

#339

—

114

C₂²×C₁₃

—

115

C₂²×C₁₃

#450

yes

characteristic

116

C₂²×C₁₃

#459

yes

characteristic

117

C₂²×C₁₃

#490

yes

characteristic

118

C₄×C₁₃

—

119

C₄×C₁₃

#394

yes

characteristic

120

C₄×C₁₃

#447

yes

characteristic

121

C₄×C₁₃

#460

yes

characteristic

122

C₄×C₁₃

#486

yes

characteristic

123

Dic₁₃

—

124

Dic₁₃

—

125

C₂²×Q₈

—

126

C₂×Q₁₆

—

127

C₂×Q₁₆

—

128

C₂×Q₁₆

—

129

C₂×Q₁₆

—

130

C₂×Q₁₆

—

131

C₂×Q₁₆

—

132

C₂×Q₁₆

—

133

C₂×Q₁₆

—

134

C₂×Q₁₆

—

135

C₂×M₁₆

—

136

C₂²×C₈

D₅₂

yes

characteristic

137

C₂²×C₈

C₂²×D₁₃

yes

characteristic

138

C₄⋊Q₈

—

139

C₄⋊Q₈

—

140

#117

—

141

C₂×C₄⋊C₄

—

142

C₂×C₄²

D₅₂

yes

characteristic

143

M₃₂

—

144

M₃₂

—

145

C₂×C₁₆

—

146

#108

—

147

#107

—

148

#106

—

149

C₄⋊C₈

—

150

C₄×C₈

—

151

C₂×C₁₃

—

152

C₂×C₁₃

—

153

C₂×C₁₃

#2041

yes

characteristic

154

C₂×C₁₃

#2214

yes

—

155

C₂×Q₈

—

156

C₂×Q₈

—

157

C₂×Q₈

—

158

C₂×Q₈

—

159

C₂×Q₈

—

160

C₂×Q₈

—

161

C₂²×C₄

—

162

C₂²×C₄

—

163

C₂²×C₄

C₂×D₅₂

yes

characteristic

164

Q₁₆

—

165

Q₁₆

—

166

Q₁₆

—

167

Q₁₆

—

168

Q₁₆

—

169

M₁₆

—

170

M₁₆

—

171

C₂×C₈

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

D₁₀₄

yes

—

175

C₂×C₈

#6127

yes

—

176

C₂×C₈

D₄×D₁₃

yes

characteristic

177

C₂×C₈

D₄×D₁₃

yes

—

178

C₂×C₈

#6131

yes

characteristic

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₂²⋊C₄

—

182

C₄²

—

183

C₄²

—

184

C₄²

#6095

yes

—

185

C₁₆

—

186

C₁₃

#13415

yes

characteristic, complemented, Sylow

187

C₂³

#88162

yes

characteristic

188

Q₈

—

189

Q₈

—

190

Q₈

—

104

—

191

Q₈

—

104

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

104

—

197

C₂×C₄

—

104

—

198

C₂×C₄

#88129

yes

characteristic

199

C₂×C₄

#88136

yes

characteristic

200

C₂×C₄

#88149

yes

characteristic

201

C₂×C₄

C₂×#6095

yes

characteristic

202

C₂×C₄

C₂×D₁₀₄

yes

characteristic

203

C₂×C₄

#88162

yes

characteristic

204

C₈

—

205

C₈

—

104

—

206

C₈

#88138

yes

—

207

C₈

#88148

yes

—

208

C₂²

—

209

C₂²

#11718374

yes

characteristic

210

C₂²

#11718392

yes

characteristic

211

C₂²

#11718504

yes

characteristic

212

C₄

—

213

C₄

—

104

—

214

C₄

—

104

—

215

C₄

#11718333

yes

characteristic

216

C₄

#11718355

yes

characteristic

217

C₄

#11718393

yes

characteristic

218

C₄

#11718509

yes

characteristic

219

C₂

—

220

C₂

—

221

C₂

#13116071

yes

characteristic

222

C₂

#13116918

yes

—

	subgroup	G/H
center	C₂²	#11718504
commutator, cosocle	C₂²×C₈×C₁₃	C₂³
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#1376	C₂
socle	C₂²×C₁₃	#459
prime radical	C₂³×C₁₃	AGL(1,ℤ/8)
hypercenter	#1376	D₁₃
hypocenter	C₁₃	#13415
LCS	G ▷ C₂²×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

subgroup

G/H

center

C₂²

#11718504

commutator, cosocle

C₂²×C₈×C₁₃

C₂³

Frattini

C₂²×C₈

C₂²×D₁₃

Fitting

C₁₃×#1376

C₂

C₂²×C₁₃

C₂³×C₁₃

D₁₃

C₁₃

G ▷ C₂²×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

N	H ≅ G/N	type
186. C₁₃	#13415	semidirect

H ≅ G/N

type

186. C₁₃

#13415

semidirect

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

subgroup	index
C₂×#11718360	2
C₂×#11718109	2
#13116557	2
#13115584	2
C₁₃×#1376	2
#13415	13

subgroup

index

C₁₃×#1376

Subgroups — #15782312

Conjugacy classes of subgroups

Complements

Maximal subgroups