Group #15782300 — subgroups

Subgroups — #15782300

Order 3328.

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

normal subgroups only

This group has 540 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×#11718356	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718108	C₂	1	2	yes	maximal, complemented
3	#13116553	C₂	1	2	yes	maximal, complemented
4	#13115584	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1376	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₀₄	—	2	1	no	—
7	C₂²×D₁₀₄	C₂²	1	2	yes	—
8	C₂×#88128	—	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	#11718356	—	2	4	no	—
15	#11718115	—	2	1	no	—
16	#11718108	—	2	4	no	—
17	#11718101	—	2	4	no	—
18	C₂²×D₅₂	—	2	2	no	—
19	C₂²×D₅₂	—	4	2	no	—
20	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
23	#88180	—	2	1	no	—
24	C₂×#6099	—	2	1	no	—
25	C₂×D₁₀₄	—	2	2	no	—
26	C₂×D₁₀₄	—	2	2	no	—
27	C₂×D₁₀₄	—	2	2	no	—
28	C₂×D₁₀₄	—	2	2	no	—
29	C₂×D₁₀₄	—	2	4	no	—
30	C₂×D₁₀₄	—	2	2	no	—
31	C₂×D₁₀₄	—	4	2	no	—
32	C₂×D₁₀₄	—	4	2	no	—
33	C₂×D₁₀₄	D₄	1	4	yes	—
34	#88128	—	2	4	no	—
35	#88128	—	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	complemented
42	#88057	—	2	1	no	—
43	#88054	—	4	2	no	—
44	#13413	—	13	1	no	maximal, Sylow
45	C₂³×D₁₃	—	4	2	no	—
46	C₂²×C₄×C₁₃	—	2	1	no	—
47	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
48	C₂×D₅₂	—	2	4	no	—
49	C₂×D₅₂	—	4	2	no	—
50	C₂×D₅₂	—	4	2	no	—
51	C₂×D₅₂	—	4	2	no	—
52	C₂×D₅₂	—	4	2	no	—
53	C₂×D₅₂	—	4	8	no	—
54	C₂×D₅₂	—	4	2	no	—
55	C₂×D₅₂	—	8	8	no	—
56	C₂×C₈×C₁₃	—	2	1	no	—
57	C₂×C₈×C₁₃	—	2	1	no	—
58	C₂×C₈×C₁₃	D₈	1	2	yes	—
59	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₁₃	#41	1	2	yes	—
62	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
63	C₄²×C₁₃	—	2	1	no	—
64	C₄²×C₁₃	SD₁₆	1	2	yes	—
65	C₂²⋊Dic₁₃	—	4	1	no	—
66	C₄⋊Dic₁₃	—	2	2	no	—
67	C₄×Dic₁₃	—	4	1	no	—
68	#6099	—	4	1	no	—
69	#6099	—	4	1	no	—
70	C₂×#1055	—	4	1	no	—
71	D₁₀₄	—	2	4	no	—
72	D₁₀₄	—	2	4	no	—
73	D₁₀₄	—	4	8	no	—
74	D₁₀₄	—	4	2	no	—
75	D₁₀₄	—	4	2	no	—
76	C₁₃×C₁₆	—	4	2	no	—
77	C₂×#493	—	13	1	no	—
78	C₂×#361	—	13	2	no	—
79	#2140	—	13	2	no	—
80	#1945	—	13	1	no	—
81	#1376	D₁₃	1	1	yes	characteristic, complemented
82	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
83	C₂²×D₁₃	—	4	4	no	—
84	C₂²×D₁₃	—	8	8	no	—
85	C₂²×D₁₃	—	8	2	no	—
86	C₂²×D₁₃	—	8	2	no	—
87	C₂×C₄×C₁₃	—	2	4	no	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
90	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
91	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
94	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
95	C₂×Dic₁₃	—	4	2	no	—
96	D₅₂	—	4	4	no	—
97	D₅₂	—	4	4	no	—
98	D₅₂	—	8	16	no	—
99	D₅₂	—	8	2	no	—
100	D₅₂	—	8	2	no	—
101	C₈×C₁₃	—	2	2	no	—
102	C₈×C₁₃	#121	1	2	yes	—
103	C₈×C₁₃	#123	1	2	yes	—
104	#1055	—	8	1	no	—
105	C₂²×D₈	—	13	2	no	—
106	C₂²×D₈	—	26	1	no	—
107	C₂×#127	—	26	1	no	—
108	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
109	#493	—	26	4	no	—
110	C₂×#108	—	13	1	no	—
111	#427	—	13	1	no	—
112	#424	—	26	1	no	—
113	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
114	#361	—	26	4	no	—
115	#357	—	26	4	no	—
116	#339	—	26	1	no	—
117	C₂²×C₁₃	—	2	2	no	—
118	C₂²×C₁₃	#450	1	1	yes	characteristic
119	C₂²×C₁₃	#459	1	1	yes	characteristic
120	C₂²×C₁₃	#490	1	1	yes	characteristic
121	C₂×D₁₃	—	8	8	no	—
122	C₂×D₁₃	—	8	4	no	—
123	C₂×D₁₃	—	16	4	no	—
124	C₂×D₁₃	—	16	4	no	—
125	C₄×C₁₃	—	2	4	no	—
126	C₄×C₁₃	#394	1	1	yes	characteristic
127	C₄×C₁₃	#447	1	1	yes	characteristic
128	C₄×C₁₃	#460	1	1	yes	characteristic
129	C₄×C₁₃	#486	1	1	yes	characteristic
130	Dic₁₃	—	8	2	no	—
131	C₂²×D₄	—	26	2	no	—
132	C₂²×D₄	—	52	2	no	—
133	C₂×D₈	—	13	4	no	—
134	C₂×D₈	—	26	4	no	—
135	C₂×D₈	—	26	2	no	—
136	C₂×D₈	—	26	2	no	—
137	C₂×D₈	—	26	2	no	—
138	C₂×D₈	—	26	2	no	—
139	C₂×D₈	—	26	2	no	—
140	C₂×D₈	—	52	2	no	—
141	C₂×D₈	—	52	2	no	—
142	C₂×M₁₆	—	26	1	no	—
143	C₂²×C₈	D₅₂	1	1	yes	characteristic
144	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
145	#127	—	26	4	no	—
146	#127	—	52	2	no	—
147	#117	—	26	1	no	—
148	C₂×C₄²	D₅₂	1	1	yes	characteristic
149	M₃₂	—	2	2	no	—
150	M₃₂	—	2	2	no	—
151	C₂×C₁₆	—	2	2	no	—
152	#108	—	26	2	no	—
153	#107	—	13	2	no	—
154	#106	—	26	1	no	—
155	C₄⋊C₈	—	52	2	no	—
156	C₄×C₈	—	2	2	no	—
157	C₂×C₁₃	—	2	1	no	—
158	C₂×C₁₃	—	2	1	no	—
159	C₂×C₁₃	#2041	1	1	yes	characteristic
160	C₂×C₁₃	#2213	1	2	yes	—
161	D₁₃	—	16	4	no	—
162	C₂⁴	—	52	2	no	—
163	C₂×D₄	—	26	4	no	—
164	C₂×D₄	—	52	8	no	—
165	C₂×D₄	—	52	2	no	—
166	C₂×D₄	—	52	2	no	—
167	C₂×D₄	—	52	2	no	—
168	C₂×D₄	—	52	2	no	—
169	C₂×D₄	—	52	2	no	—
170	C₂×D₄	—	104	8	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
173	D₈	—	26	4	no	—
174	D₈	—	26	4	no	—
175	D₈	—	52	8	no	—
176	D₈	—	52	2	no	—
177	D₈	—	52	2	no	—
178	M₁₆	—	52	1	no	—
179	M₁₆	—	52	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	52	1	no	—
183	C₂×C₈	D₁₀₄	1	2	yes	—
184	C₂×C₈	#6127	1	2	yes	—
185	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
186	C₂×C₈	D₄×D₁₃	1	2	yes	—
187	C₂×C₈	#6131	1	1	yes	characteristic
188	C₄⋊C₄	—	26	2	no	—
189	C₂²⋊C₄	—	52	1	no	—
190	C₄²	—	2	1	no	—
191	C₄²	—	52	1	no	—
192	C₄²	#6095	1	2	yes	—
193	C₁₆	—	4	2	no	—
194	C₁₃	#13413	1	1	yes	characteristic, complemented, Sylow
195	C₂³	—	52	4	no	—
196	C₂³	—	104	8	no	—
197	C₂³	—	104	2	no	—
198	C₂³	—	104	2	no	—
199	C₂³	#88162	1	1	yes	characteristic
200	D₄	—	52	4	no	—
201	D₄	—	52	4	no	—
202	D₄	—	104	16	no	—
203	D₄	—	104	2	no	—
204	D₄	—	104	2	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	52	2	no	—
208	C₂×C₄	#88129	1	1	yes	characteristic
209	C₂×C₄	#88136	1	1	yes	characteristic
210	C₂×C₄	#88149	1	1	yes	characteristic
211	C₂×C₄	C₂×#6095	1	1	yes	characteristic
212	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
213	C₂×C₄	#88162	1	1	yes	characteristic
214	C₈	—	2	2	no	—
215	C₈	—	104	1	no	—
216	C₈	#88138	1	2	yes	—
217	C₈	#88148	1	2	yes	—
218	C₂²	—	2	2	no	—
219	C₂²	—	104	8	no	—
220	C₂²	—	104	4	no	—
221	C₂²	—	208	4	no	—
222	C₂²	—	208	4	no	—
223	C₂²	#11718374	1	1	yes	characteristic
224	C₂²	#11718392	1	1	yes	characteristic
225	C₂²	#11718504	1	1	yes	characteristic
226	C₄	—	2	4	no	—
227	C₄	—	104	2	no	—
228	C₄	#11718333	1	1	yes	characteristic
229	C₄	#11718355	1	1	yes	characteristic
230	C₄	#11718393	1	1	yes	characteristic
231	C₄	#11718509	1	1	yes	characteristic
232	C₂	—	2	1	no	—
233	C₂	—	2	1	no	—
234	C₂	—	208	4	no	—
235	C₂	#13116071	1	1	yes	characteristic
236	C₂	#13116913	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718356

C₂

yes

maximal, characteristic

C₂×#11718108

C₂

yes

maximal, complemented

#13116553

C₂

yes

maximal, complemented

#13115584

C₂

yes

maximal, characteristic, complemented

C₁₃×#1376

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₀₄

—

C₂²×D₁₀₄

C₂²

yes

—

C₂×#88128

—

C₂×C₁₃×M₃₂

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×#88059

C₂²

yes

characteristic

#11718767

C₂²

yes

—

—

—

—

—

C₂²×D₅₂

—

C₂²×D₅₂

—

C₂²×C₈×C₁₃

D₄

yes

characteristic

C₂²×C₈×C₁₃

C₂³

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

characteristic

#88180

—

C₂×#6099

—

C₂×D₁₀₄

—

C₂×D₁₀₄

—

C₂×D₁₀₄

—

C₂×D₁₀₄

—

C₂×D₁₀₄

—

C₂×D₁₀₄

—

C₂×D₁₀₄

—

C₂×D₁₀₄

—

C₂×D₁₀₄

D₄

yes

—

#88128

—

#88128

—

C₁₃×M₃₂

—

C₁₃×M₃₂

—

C₂×C₁₃×C₁₆

—

C₄×C₈×C₁₃

—

#88059

—

#88058

D₄

yes

—

—

—

C₂³×D₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

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

—

#6099

—

#6099

—

C₂×#1055

—

D₁₀₄

—

D₁₀₄

—

D₁₀₄

—

D₁₀₄

—

D₁₀₄

—

C₁₃×C₁₆

—

—

—

—

—

D₁₃

yes

characteristic, complemented

C₂³×C₁₃

AGL(1,ℤ/8)

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

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

—

D₅₂

—

D₅₂

—

D₅₂

—

D₅₂

—

100

D₅₂

—

101

C₈×C₁₃

—

102

C₈×C₁₃

#121

yes

—

103

C₈×C₁₃

#123

yes

—

104

#1055

—

105

C₂²×D₈

—

106

C₂²×D₈

—

107

C₂×#127

—

108

C₂×M₃₂

C₂×D₁₃

yes

—

109

#493

—

110

C₂×#108

—

111

#427

—

112

#424

—

113

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

114

#361

—

115

#357

—

116

#339

—

117

C₂²×C₁₃

—

118

C₂²×C₁₃

#450

yes

characteristic

119

C₂²×C₁₃

#459

yes

characteristic

120

C₂²×C₁₃

#490

yes

characteristic

121

C₂×D₁₃

—

122

C₂×D₁₃

—

123

C₂×D₁₃

—

124

C₂×D₁₃

—

125

C₄×C₁₃

—

126

C₄×C₁₃

#394

yes

characteristic

127

C₄×C₁₃

#447

yes

characteristic

128

C₄×C₁₃

#460

yes

characteristic

129

C₄×C₁₃

#486

yes

characteristic

130

Dic₁₃

—

131

C₂²×D₄

—

132

C₂²×D₄

—

133

C₂×D₈

—

134

C₂×D₈

—

135

C₂×D₈

—

136

C₂×D₈

—

137

C₂×D₈

—

138

C₂×D₈

—

139

C₂×D₈

—

140

C₂×D₈

—

141

C₂×D₈

—

142

C₂×M₁₆

—

143

C₂²×C₈

D₅₂

yes

characteristic

144

C₂²×C₈

C₂²×D₁₃

yes

characteristic

145

#127

—

146

#127

—

147

#117

—

148

C₂×C₄²

D₅₂

yes

characteristic

149

M₃₂

—

150

M₃₂

—

151

C₂×C₁₆

—

152

#108

—

153

#107

—

154

#106

—

155

C₄⋊C₈

—

156

C₄×C₈

—

157

C₂×C₁₃

—

158

C₂×C₁₃

—

159

C₂×C₁₃

#2041

yes

characteristic

160

C₂×C₁₃

#2213

yes

—

161

D₁₃

—

162

C₂⁴

—

163

C₂×D₄

—

164

C₂×D₄

—

165

C₂×D₄

—

166

C₂×D₄

—

167

C₂×D₄

—

168

C₂×D₄

—

169

C₂×D₄

—

170

C₂×D₄

—

104

—

171

C₂²×C₄

—

172

C₂²×C₄

C₂×D₅₂

yes

characteristic

173

D₈

—

174

D₈

—

175

D₈

—

176

D₈

—

177

D₈

—

178

M₁₆

—

179

M₁₆

—

180

C₂×C₈

—

181

C₂×C₈

—

182

C₂×C₈

—

183

C₂×C₈

D₁₀₄

yes

—

184

C₂×C₈

#6127

yes

—

185

C₂×C₈

D₄×D₁₃

yes

characteristic

186

C₂×C₈

D₄×D₁₃

yes

—

187

C₂×C₈

#6131

yes

characteristic

188

C₄⋊C₄

—

189

C₂²⋊C₄

—

190

C₄²

—

191

C₄²

—

192

C₄²

#6095

yes

—

193

C₁₆

—

194

C₁₃

#13413

yes

characteristic, complemented, Sylow

195

C₂³

—

196

C₂³

—

104

—

197

C₂³

—

104

—

198

C₂³

—

104

—

199

C₂³

#88162

yes

characteristic

200

D₄

—

201

D₄

—

202

D₄

—

104

—

203

D₄

—

104

—

204

D₄

—

104

—

205

C₂×C₄

—

206

C₂×C₄

—

207

C₂×C₄

—

208

C₂×C₄

#88129

yes

characteristic

209

C₂×C₄

#88136

yes

characteristic

210

C₂×C₄

#88149

yes

characteristic

211

C₂×C₄

C₂×#6095

yes

characteristic

212

C₂×C₄

C₂×D₁₀₄

yes

characteristic

213

C₂×C₄

#88162

yes

characteristic

214

C₈

—

215

C₈

—

104

—

216

C₈

#88138

yes

—

217

C₈

#88148

yes

—

218

C₂²

—

219

C₂²

—

104

—

220

C₂²

—

104

—

221

C₂²

—

208

—

222

C₂²

—

208

—

223

C₂²

#11718374

yes

characteristic

224

C₂²

#11718392

yes

characteristic

225

C₂²

#11718504

yes

characteristic

226

C₄

—

227

C₄

—

104

—

228

C₄

#11718333

yes

characteristic

229

C₄

#11718355

yes

characteristic

230

C₄

#11718393

yes

characteristic

231

C₄

#11718509

yes

characteristic

232

C₂

—

233

C₂

—

234

C₂

—

208

—

235

C₂

#13116071

yes

characteristic

236

C₂

#13116913

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	C₂×#11718356	C₂
hypercenter	#1376	D₁₃
hypocenter	C₁₃	#13413
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

C₂

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
2. C₂×#11718108	C₂	semidirect
3. #13116553	C₂	semidirect
4. #13115584	C₂	semidirect
5. C₁₃×#1376	C₂	semidirect
41. #88058	D₄	semidirect
81. #1376	D₁₃	semidirect
194. C₁₃	#13413	semidirect

H ≅ G/N

type

2. C₂×#11718108

C₂

semidirect

3. #13116553

C₂

semidirect

4. #13115584

C₂

semidirect

5. C₁₃×#1376

C₂

semidirect

41. #88058

D₄

semidirect

81. #1376

D₁₃

semidirect

194. C₁₃

#13413

semidirect

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

subgroup	index
C₂×#11718356	2
C₂×#11718108	2
#13116553	2
#13115584	2
C₁₃×#1376	2
#13413	13

subgroup

index

C₁₃×#1376

Subgroups — #15782300

Conjugacy classes of subgroups

Complements

Maximal subgroups