Group #15765610 — subgroups

Subgroups — #15765610

Order 3328.

912 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 912 conjugacy classes of subgroups — too many to draw. Only its 113 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	#13125777	C₂	1	2	yes	maximal, complemented
2	#13115813	C₂	1	1	yes	maximal, characteristic, complemented
3	#13113504	C₂	1	2	yes	maximal, complemented
4	#13112669	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#2867	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#127	—	2	4	no	—
7	C₂×C₁₃×#127	—	2	2	no	—
8	C₂×C₁₃×#127	C₂²	1	2	yes	—
9	#11719175	C₂²	1	1	yes	characteristic
10	C₄³×C₁₃	C₂²	1	1	yes	characteristic
11	#11718879	—	2	4	no	—
12	#11718737	—	2	2	no	—
13	#11718736	—	2	4	no	—
14	#11718731	—	2	2	no	—
15	#11718586	C₂²	1	2	yes	—
16	#11718585	C₂²	1	1	yes	characteristic
17	#11718136	—	2	4	no	—
18	#11718115	—	2	2	no	—
19	C₂²×D₄×C₁₃	—	2	8	no	—
20	C₂²×D₄×C₁₃	—	2	2	no	—
21	C₂²×D₄×C₁₃	—	2	2	no	—
22	C₂²×D₄×C₁₃	D₄	1	4	yes	—
23	C₁₃×#127	—	2	2	no	—
24	C₁₃×#127	—	2	2	no	—
25	C₁₃×#127	—	2	2	no	—
26	C₁₃×#127	—	2	2	no	—
27	C₁₃×#127	—	4	16	no	—
28	C₁₃×#127	—	4	4	no	—
29	C₁₃×#127	—	4	4	no	—
30	C₂×C₄²×C₁₃	—	2	2	no	—
31	C₂×C₄²×C₁₃	D₄	1	2	yes	—
32	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×#6105	—	2	8	no	—
34	#88180	—	2	1	no	—
35	#88178	—	2	2	no	—
36	C₂×#6099	—	2	1	no	—
37	#88124	—	2	1	no	—
38	#88123	D₄	1	2	yes	complemented
39	C₄×Dic₂₆	—	2	2	no	—
40	#88072	—	2	8	no	—
41	C₂²⋊#1055	—	2	2	no	—
42	C₄⋊#1055	—	2	2	no	—
43	C₄⋊#1055	D₄	1	4	yes	complemented
44	C₄×#1055	—	2	2	no	—
45	#10290	—	13	1	no	maximal, Sylow
46	C₂⁴×C₁₃	—	2	4	no	—
47	C₂×D₄×C₁₃	—	2	4	no	—
48	C₂×D₄×C₁₃	—	2	4	no	—
49	C₂×D₄×C₁₃	—	4	8	no	—
50	C₂×D₄×C₁₃	—	4	32	no	—
51	C₂×D₄×C₁₃	—	4	16	no	—
52	C₂×D₄×C₁₃	—	4	8	no	—
53	C₂×D₄×C₁₃	—	4	2	no	—
54	C₂×D₄×C₁₃	—	4	2	no	—
55	C₂×D₄×C₁₃	—	4	2	no	—
56	C₂×D₄×C₁₃	—	4	2	no	—
57	C₂×D₄×C₁₃	—	4	4	no	—
58	C₂×D₄×C₁₃	—	4	4	no	—
59	C₂²×C₄×C₁₃	—	2	2	no	—
60	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂×Dic₂₆	—	2	2	no	—
63	C₄²×C₁₃	—	2	8	no	—
64	C₄²×C₁₃	—	2	2	no	—
65	C₄²×C₁₃	SD₁₆	1	4	yes	complemented
66	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₁₃	C₂×D₄	1	2	yes	—
68	C₄²×C₁₃	#41	1	1	yes	characteristic
69	C₂²⋊Dic₁₃	—	4	1	no	—
70	#6105	—	8	8	no	—
71	C₄⋊Dic₁₃	—	2	2	no	—
72	C₄⋊Dic₁₃	—	2	2	no	—
73	#6101	—	4	2	no	—
74	C₄×Dic₁₃	—	4	1	no	—
75	#6099	—	4	2	no	—
76	C₂×#1055	—	2	4	no	—
77	C₂×#1055	—	4	1	no	—
78	#3208	—	13	2	no	—
79	#2867	D₁₃	1	1	yes	characteristic
80	#1994	—	13	1	no	—
81	#1545	—	13	2	no	—
82	#1276	—	13	1	no	—
83	C₂³×C₁₃	—	4	8	no	—
84	C₂³×C₁₃	—	4	8	no	—
85	C₂³×C₁₃	—	4	4	no	—
86	C₂³×C₁₃	—	4	8	no	—
87	C₂³×C₁₃	#120	1	1	yes	characteristic
88	D₄×C₁₃	—	4	8	no	—
89	D₄×C₁₃	—	8	32	no	—
90	D₄×C₁₃	—	8	4	no	—
91	D₄×C₁₃	—	8	4	no	—
92	D₄×C₁₃	—	8	4	no	—
93	D₄×C₁₃	—	8	4	no	—
94	D₄×C₁₃	—	8	2	no	—
95	D₄×C₁₃	—	8	2	no	—
96	C₂×C₄×C₁₃	—	2	8	no	—
97	C₂×C₄×C₁₃	—	2	4	no	—
98	C₂×C₄×C₁₃	—	2	4	no	—
99	C₂×C₄×C₁₃	#120	1	2	yes	—
100	C₂×C₄×C₁₃	#121	1	2	yes	—
101	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
102	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
104	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
105	C₂×Dic₁₃	—	4	2	no	—
106	Dic₂₆	—	4	4	no	—
107	#1055	—	4	4	no	—
108	#1055	—	8	1	no	—
109	#533	—	13	1	no	—
110	C₂×#127	—	2	4	no	—
111	C₂×#127	—	2	2	no	—
112	C₂×#127	C₂×D₁₃	1	2	yes	—
113	#492	—	26	2	no	—
114	#486	—	26	2	no	—
115	#460	—	26	4	no	—
116	#450	—	26	4	no	—
117	#431	—	13	2	no	—
118	#423	—	13	1	no	—
119	C₄³	C₂×D₁₃	1	1	yes	characteristic
120	#339	—	26	2	no	—
121	#331	—	26	4	no	—
122	C₂²×C₁₃	—	2	2	no	—
123	C₂²×C₁₃	—	8	16	no	—
124	C₂²×C₁₃	—	8	4	no	—
125	C₂²×C₁₃	—	8	4	no	—
126	C₂²×C₁₃	—	8	4	no	—
127	C₂²×C₁₃	#392	1	1	yes	characteristic
128	C₂²×C₁₃	#450	1	2	yes	—
129	C₄×C₁₃	—	2	8	no	—
130	C₄×C₁₃	—	2	2	no	—
131	C₄×C₁₃	#453	1	4	yes	—
132	C₄×C₁₃	#460	1	2	yes	—
133	C₄×C₁₃	#492	1	1	yes	characteristic
134	C₄×C₁₃	#496	1	1	yes	characteristic
135	Dic₁₃	—	8	2	no	—
136	C₂²×D₄	—	2	8	no	—
137	C₂²×D₄	—	2	2	no	—
138	C₂²×D₄	—	2	2	no	—
139	C₂²×D₄	C₂²⋊D₁₃	1	4	yes	—
140	C₂×SD₁₆	—	26	8	no	—
141	C₂×M₁₆	—	26	1	no	—
142	C₄⋊Q₈	—	13	2	no	—
143	#127	—	2	2	no	—
144	#127	—	2	2	no	—
145	#127	—	2	2	no	—
146	#127	—	2	2	no	—
147	#127	—	4	16	no	—
148	#127	—	4	4	no	—
149	#127	—	4	4	no	—
150	#125	—	26	1	no	—
151	C₂²⋊Q₈	—	26	2	no	—
152	C₄×Q₈	—	26	2	no	—
153	#117	—	26	1	no	—
154	C₂×C₄²	—	2	2	no	—
155	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
156	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
157	C₄⋊C₈	—	13	4	no	—
158	C₄⋊C₈	—	26	2	no	—
159	#102	—	26	8	no	—
160	C₂²⋊C₈	—	26	2	no	—
161	C₄×C₈	—	26	2	no	—
162	C₂×C₁₃	—	2	2	no	—
163	C₂×C₁₃	—	16	4	no	—
164	C₂×C₁₃	#2002	1	2	yes	—
165	C₂×C₁₃	#2014	1	1	yes	characteristic
166	C₂⁴	—	2	4	no	—
167	C₂×Q₈	—	26	2	no	—
168	C₂×D₄	—	2	4	no	—
169	C₂×D₄	—	2	4	no	—
170	C₂×D₄	—	4	8	no	—
171	C₂×D₄	—	4	32	no	—
172	C₂×D₄	—	4	16	no	—
173	C₂×D₄	—	4	8	no	—
174	C₂×D₄	—	4	4	no	—
175	C₂×D₄	—	4	4	no	—
176	C₂×D₄	—	4	2	no	—
177	C₂×D₄	—	4	2	no	—
178	C₂×D₄	—	4	2	no	—
179	C₂×D₄	—	4	2	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
182	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
183	SD₁₆	—	104	8	no	—
184	M₁₆	—	52	2	no	—
185	C₂×C₈	—	26	4	no	—
186	C₂×C₈	—	52	1	no	—
187	C₄⋊C₄	—	26	2	no	—
188	C₄⋊C₄	—	26	2	no	—
189	C₄⋊C₄	—	52	2	no	—
190	C₂²⋊C₄	—	52	1	no	—
191	C₄²	—	2	8	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	52	1	no	—
194	C₄²	#6105	1	4	yes	complemented
195	C₄²	D₄×D₁₃	1	1	yes	characteristic
196	C₄²	D₄×D₁₃	1	2	yes	—
197	C₄²	#6129	1	1	yes	characteristic
198	C₁₃	#10290	1	1	yes	characteristic, complemented, Sylow
199	C₂³	—	4	8	no	—
200	C₂³	—	4	8	no	—
201	C₂³	—	4	8	no	—
202	C₂³	—	4	4	no	—
203	C₂³	#88207	1	1	yes	characteristic
204	Q₈	—	52	4	no	—
205	D₄	—	4	8	no	—
206	D₄	—	8	4	no	—
207	D₄	—	8	32	no	—
208	D₄	—	8	4	no	—
209	D₄	—	8	2	no	—
210	D₄	—	8	4	no	—
211	D₄	—	8	2	no	—
212	D₄	—	8	4	no	—
213	C₂×C₄	—	2	8	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	52	2	no	—
217	C₂×C₄	#88186	1	2	yes	—
218	C₂×C₄	C₂×#6105	1	2	yes	—
219	C₂×C₄	#88191	1	2	yes	—
220	C₂×C₄	#88192	1	1	yes	characteristic
221	C₂×C₄	#88193	1	2	yes	—
222	C₂×C₄	#88194	1	1	yes	characteristic
223	C₈	—	52	4	no	—
224	C₈	—	104	1	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	8	16	no	—
227	C₂²	—	8	4	no	—
228	C₂²	—	8	4	no	—
229	C₂²	—	8	4	no	—
230	C₂²	#11718879	1	2	yes	—
231	C₂²	#11718884	1	1	yes	characteristic
232	C₄	—	2	8	no	—
233	C₄	—	2	2	no	—
234	C₄	—	104	2	no	—
235	C₄	#11718735	1	1	yes	characteristic
236	C₄	#11718736	1	2	yes	—
237	C₄	#11718737	1	1	yes	characteristic
238	C₄	#11718738	1	4	yes	—
239	C₂	—	2	2	no	—
240	C₂	—	16	4	no	—
241	C₂	#13115851	1	2	yes	—
242	C₂	#13115924	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13125777

C₂

yes

maximal, complemented

#13115813

C₂

yes

maximal, characteristic, complemented

#13113504

C₂

yes

maximal, complemented

#13112669

C₂

yes

maximal, characteristic, complemented

C₁₃×#2867

C₂

yes

maximal, characteristic

C₂×C₁₃×#127

—

C₂×C₁₃×#127

—

C₂×C₁₃×#127

C₂²

yes

—

#11719175

C₂²

yes

characteristic

C₄³×C₁₃

C₂²

yes

—

—

—

—

C₂²

yes

—

#11718585

C₂²

yes

characteristic

#11718136

—

#11718115

—

C₂²×D₄×C₁₃

—

C₂²×D₄×C₁₃

—

C₂²×D₄×C₁₃

—

C₂²×D₄×C₁₃

D₄

yes

—

C₁₃×#127

—

C₁₃×#127

—

C₁₃×#127

—

C₁₃×#127

—

C₁₃×#127

—

C₁₃×#127

—

C₁₃×#127

—

C₂×C₄²×C₁₃

—

C₂×C₄²×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

—

—

—

—

—

D₄

yes

complemented

C₄×Dic₂₆

—

#88072

—

C₂²⋊#1055

—

C₄⋊#1055

—

C₄⋊#1055

D₄

yes

—

—

C₂⁴×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×Dic₂₆

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

SD₁₆

yes

complemented

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

#41

yes

characteristic

C₂²⋊Dic₁₃

—

#6105

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

—

—

—

—

D₁₃

yes

—

—

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

#120

yes

characteristic

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#120

yes

—

100

C₂×C₄×C₁₃

#121

yes

—

101

C₂×C₄×C₁₃

#121

yes

characteristic

102

C₂×C₄×C₁₃

#127

yes

characteristic

103

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

104

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

105

C₂×Dic₁₃

—

106

Dic₂₆

—

107

#1055

—

108

#1055

—

109

#533

—

110

C₂×#127

—

111

C₂×#127

—

112

C₂×#127

C₂×D₁₃

yes

—

113

#492

—

114

#486

—

115

#460

—

116

#450

—

117

#431

—

118

#423

—

119

C₄³

C₂×D₁₃

yes

characteristic

120

#339

—

121

#331

—

122

C₂²×C₁₃

—

123

C₂²×C₁₃

—

124

C₂²×C₁₃

—

125

C₂²×C₁₃

—

126

C₂²×C₁₃

—

127

C₂²×C₁₃

#392

yes

characteristic

128

C₂²×C₁₃

#450

yes

—

129

C₄×C₁₃

—

130

C₄×C₁₃

—

131

C₄×C₁₃

#453

yes

—

132

C₄×C₁₃

#460

yes

—

133

C₄×C₁₃

#492

yes

characteristic

134

C₄×C₁₃

#496

yes

characteristic

135

Dic₁₃

—

136

C₂²×D₄

—

137

C₂²×D₄

—

138

C₂²×D₄

—

139

C₂²×D₄

C₂²⋊D₁₃

yes

—

140

C₂×SD₁₆

—

141

C₂×M₁₆

—

142

C₄⋊Q₈

—

143

#127

—

144

#127

—

145

#127

—

146

#127

—

147

#127

—

148

#127

—

149

#127

—

150

#125

—

151

C₂²⋊Q₈

—

152

C₄×Q₈

—

153

#117

—

154

C₂×C₄²

—

155

C₂×C₄²

C₂²⋊D₁₃

yes

—

156

C₂×C₄²

C₂²×D₁₃

yes

characteristic

157

C₄⋊C₈

—

158

C₄⋊C₈

—

159

#102

—

160

C₂²⋊C₈

—

161

C₄×C₈

—

162

C₂×C₁₃

—

163

C₂×C₁₃

—

164

C₂×C₁₃

#2002

yes

—

165

C₂×C₁₃

#2014

yes

characteristic

166

C₂⁴

—

167

C₂×Q₈

—

168

C₂×D₄

—

169

C₂×D₄

—

170

C₂×D₄

—

171

C₂×D₄

—

172

C₂×D₄

—

173

C₂×D₄

—

174

C₂×D₄

—

175

C₂×D₄

—

176

C₂×D₄

—

177

C₂×D₄

—

178

C₂×D₄

—

179

C₂×D₄

—

180

C₂²×C₄

—

181

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

—

182

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

183

SD₁₆

—

104

—

184

M₁₆

—

185

C₂×C₈

—

186

C₂×C₈

—

187

C₄⋊C₄

—

188

C₄⋊C₄

—

189

C₄⋊C₄

—

190

C₂²⋊C₄

—

191

C₄²

—

192

C₄²

—

193

C₄²

—

194

C₄²

#6105

yes

complemented

195

C₄²

D₄×D₁₃

yes

characteristic

196

C₄²

D₄×D₁₃

yes

—

197

C₄²

#6129

yes

characteristic

198

C₁₃

#10290

yes

characteristic, complemented, Sylow

199

C₂³

—

200

C₂³

—

201

C₂³

—

202

C₂³

—

203

C₂³

#88207

yes

characteristic

204

Q₈

—

205

D₄

—

206

D₄

—

207

D₄

—

208

D₄

—

209

D₄

—

210

D₄

—

211

D₄

—

212

D₄

—

213

C₂×C₄

—

214

C₂×C₄

—

215

C₂×C₄

—

216

C₂×C₄

—

217

C₂×C₄

#88186

yes

—

218

C₂×C₄

C₂×#6105

yes

—

219

C₂×C₄

#88191

yes

—

220

C₂×C₄

#88192

yes

characteristic

221

C₂×C₄

#88193

yes

—

222

C₂×C₄

#88194

yes

characteristic

223

C₈

—

224

C₈

—

104

—

225

C₂²

—

226

C₂²

—

227

C₂²

—

228

C₂²

—

229

C₂²

—

230

C₂²

#11718879

yes

—

231

C₂²

#11718884

yes

characteristic

232

C₄

—

233

C₄

—

234

C₄

—

104

—

235

C₄

#11718735

yes

characteristic

236

C₄

#11718736

yes

—

237

C₄

#11718737

yes

characteristic

238

C₄

#11718738

yes

—

239

C₂

—

240

C₂

—

241

C₂

#13115851

yes

—

242

C₂

#13115924

yes

characteristic

	subgroup	G/H
center	C₂²	#11718884
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting, prime radical	C₁₃×#2867	C₂
socle	C₂²×C₁₃	#392
hypercenter	#2867	D₁₃
hypocenter	C₁₃	#10290
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2867

subgroup

G/H

center

C₂²

#11718884

commutator, cosocle

C₂×C₄²×C₁₃

C₂³

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting, prime radical

C₁₃×#2867

C₂

C₂²×C₁₃

D₁₃

C₁₃

G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2867

N	H ≅ G/N	type
1. #13125777	C₂	semidirect
2. #13115813	C₂	semidirect
3. #13113504	C₂	semidirect
4. #13112669	C₂	semidirect
38. #88123	D₄	semidirect
43. C₄⋊#1055	D₄	semidirect
65. C₄²×C₁₃	SD₁₆	semidirect
194. C₄²	#6105	semidirect
198. C₁₃	#10290	semidirect

H ≅ G/N

type

1. #13125777

C₂

semidirect

2. #13115813

C₂

semidirect

3. #13113504

C₂

semidirect

4. #13112669

C₂

semidirect

38. #88123

D₄

semidirect

43. C₄⋊#1055

D₄

semidirect

65. C₄²×C₁₃

SD₁₆

semidirect

194. C₄²

#6105

semidirect

198. C₁₃

#10290

semidirect

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

subgroup	index
#13125777	2
#13115813	2
#13113504	2
#13112669	2
C₁₃×#2867	2
#10290	13

subgroup

index

C₁₃×#2867

Subgroups — #15765610

Conjugacy classes of subgroups

Complements

Maximal subgroups