Group #15850722 — subgroups

Subgroups — #15850722

Order 3328.

690 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 211 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	#13125418	C₂	1	4	yes	maximal
2	#13125350	C₂	1	1	yes	maximal, characteristic
3	#13114514	C₂	1	4	yes	maximal, complemented
4	#13114510	C₂	1	4	yes	maximal, complemented
5	C₂×#11718047	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#3160	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#533	C₂²	1	2	yes	—
8	#11719175	C₂²	1	2	yes	—
9	C₁₃×#500	C₂²	1	2	yes	—
10	C₁₃×#499	—	2	2	no	—
11	C₁₃×#498	C₂²	1	2	yes	—
12	C₁₃×#427	—	2	2	no	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	#11718767	—	2	2	no	—
15	#11718754	C₂²	1	4	yes	—
16	#11718753	C₂²	1	4	yes	—
17	#11718742	C₂²	1	4	yes	—
18	#11718739	C₂²	1	4	yes	—
19	#11718712	—	2	8	no	—
20	C₄×#6107	—	2	8	no	—
21	C₂×C₄×#1055	C₂²	1	2	yes	—
22	#11718344	—	2	2	no	—
23	#11718343	C₂²	1	2	yes	—
24	#11718341	C₂²	1	2	yes	—
25	#11718047	C₂²	1	2	yes	—
26	#11718047	C₂²	1	2	yes	—
27	C₂²×C₈×C₁₃	D₄	1	2	yes	—
28	C₁₃×C₄⋊Q₈	C₂³	1	4	yes	—
29	C₁₃×#125	—	2	2	no	—
30	C₁₃×C₂²⋊Q₈	—	2	4	no	—
31	C₄×Q₈×C₁₃	—	2	4	no	—
32	C₁₃×#117	—	2	2	no	—
33	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×#6107	—	2	8	no	—
35	#88180	—	2	2	no	—
36	#88178	—	2	4	no	—
37	C₂²×#1055	D₄	1	4	yes	—
38	#88124	—	2	2	no	—
39	#88123	C₂³	1	4	yes	—
40	C₄×Dic₂₆	—	2	4	no	—
41	C₁₃×#107	—	2	4	no	—
42	C₁₃×#106	—	2	4	no	—
43	C₄×C₈×C₁₃	D₄	1	2	yes	—
44	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
45	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
46	#88075	—	2	8	no	—
47	#88058	—	2	4	no	—
48	#88057	—	2	4	no	—
49	#88050	—	2	8	no	—
50	#88047	—	2	8	no	—
51	C₄×#1055	D₄	1	4	yes	—
52	C₄×#1055	C₂³	1	4	yes	—
53	#19618	—	13	1	no	maximal, Sylow
54	C₂×Q₈×C₁₃	—	2	4	no	—
55	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂×Dic₂₆	—	2	4	no	—
58	C₂×C₈×C₁₃	Q₁₆	1	8	yes	complemented
59	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
61	C₁₃×C₄⋊C₄	—	2	4	no	—
62	C₁₃×C₄⋊C₄	—	2	4	no	—
63	C₁₃×C₄⋊C₄	—	4	4	no	—
64	C₁₃×C₂²⋊C₄	—	4	2	no	—
65	C₄²×C₁₃	—	4	2	no	—
66	C₄²×C₁₃	C₂×D₄	1	2	yes	—
67	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
69	C₂²⋊Dic₁₃	—	4	2	no	—
70	#6107	—	4	16	no	—
71	C₄⋊Dic₁₃	—	2	4	no	—
72	C₄⋊Dic₁₃	—	2	4	no	—
73	#6101	—	4	4	no	—
74	C₄×Dic₁₃	—	4	2	no	—
75	C₂×#1055	—	4	4	no	—
76	C₂×#1055	C₂×D₄	1	8	yes	—
77	#3170	—	13	4	no	—
78	#3160	—	13	1	no	—
79	#3160	D₁₃	1	1	yes	characteristic
80	#1723	—	13	4	no	—
81	#1723	—	13	4	no	—
82	C₂×#322	—	13	1	no	—
83	C₂³×C₁₃	#127	1	1	yes	characteristic
84	Q₈×C₁₃	—	4	8	no	—
85	C₂×C₄×C₁₃	—	2	4	no	—
86	C₂×C₄×C₁₃	—	4	4	no	—
87	C₂×C₄×C₁₃	#127	1	2	yes	—
88	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂×Q₁₆	1	4	yes	—
90	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
92	C₂×Dic₁₃	—	4	4	no	—
93	Dic₂₆	—	4	8	no	—
94	C₈×C₁₃	C₂×Q₁₆	1	4	yes	—
95	C₈×C₁₃	C₂×Q₁₆	1	4	yes	—
96	#1055	—	2	8	no	—
97	#533	—	13	2	no	—
98	#533	C₂×D₁₃	1	2	yes	—
99	#500	—	13	4	no	—
100	#500	—	13	2	no	—
101	#500	C₂×D₁₃	1	2	yes	—
102	#499	—	2	2	no	—
103	#499	—	26	2	no	—
104	#498	—	13	2	no	—
105	#498	C₂×D₁₃	1	2	yes	—
106	#494	—	13	4	no	—
107	#487	—	13	4	no	—
108	#487	—	13	4	no	—
109	#467	—	26	8	no	—
110	C₄×Q₁₆	—	26	8	no	—
111	#427	—	2	2	no	—
112	#427	—	26	2	no	—
113	C₂×C₄×C₈	—	13	2	no	—
114	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
115	#322	—	13	2	no	—
116	#322	—	13	2	no	—
117	C₂²×C₁₃	—	2	2	no	—
118	C₂²×C₁₃	#494	1	2	yes	—
119	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
120	C₄×C₁₃	—	2	2	no	—
121	C₄×C₁₃	—	8	4	no	—
122	C₄×C₁₃	#494	1	2	yes	—
123	C₄×C₁₃	C₂²×Q₁₆	1	2	yes	—
124	C₄×C₁₃	#575	1	4	yes	—
125	Dic₁₃	—	8	4	no	—
126	C₂×Q₁₆	—	26	8	no	—
127	C₂²×C₈	—	13	4	no	—
128	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
129	C₄⋊Q₈	—	13	4	no	—
130	C₄⋊Q₈	C₂²×D₁₃	1	4	yes	—
131	#125	—	2	2	no	—
132	#125	—	26	2	no	—
133	C₂²⋊Q₈	—	2	4	no	—
134	C₂²⋊Q₈	—	26	4	no	—
135	C₄×Q₈	—	2	4	no	—
136	C₄×Q₈	—	26	4	no	—
137	#117	—	2	2	no	—
138	#117	—	26	2	no	—
139	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
140	#107	—	2	4	no	—
141	#107	—	26	8	no	—
142	#107	—	26	4	no	—
143	#106	—	2	4	no	—
144	#106	—	26	4	no	—
145	#103	—	26	8	no	—
146	#103	—	26	8	no	—
147	C₄×C₈	—	13	4	no	—
148	C₄×C₈	—	13	4	no	—
149	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
150	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
151	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
152	C₂×C₁₃	—	2	2	no	—
153	C₂×C₁₃	C₂×#494	1	1	yes	characteristic
154	C₂×C₁₃	#3150	1	2	yes	—
155	C₂×Q₈	—	2	4	no	—
156	C₂×Q₈	—	26	4	no	—
157	C₂²×C₄	D₄×D₁₃	1	2	yes	—
158	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
159	Q₁₆	—	52	16	no	—
160	C₂×C₈	—	13	8	no	—
161	C₂×C₈	—	52	4	no	—
162	C₂×C₈	#6107	1	8	yes	complemented
163	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
164	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	4	4	no	—
168	C₄⋊C₄	—	26	4	no	—
169	C₄⋊C₄	—	26	4	no	—
170	C₄⋊C₄	—	52	4	no	—
171	C₂²⋊C₄	—	4	2	no	—
172	C₂²⋊C₄	—	52	2	no	—
173	C₄²	—	4	2	no	—
174	C₄²	—	52	2	no	—
175	C₄²	D₄×D₁₃	1	2	yes	—
176	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
177	C₄²	C₂³×D₁₃	1	1	yes	characteristic
178	C₁₃	#19618	1	1	yes	characteristic, complemented, Sylow
179	C₂³	#88194	1	1	yes	characteristic
180	Q₈	—	4	8	no	—
181	Q₈	—	52	8	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	52	4	no	—
185	C₂×C₄	#88194	1	2	yes	—
186	C₂×C₄	#88194	1	1	yes	characteristic
187	C₂×C₄	C₂×#6107	1	4	yes	—
188	C₂×C₄	C₂×D₄×D₁₃	1	2	yes	—
189	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
190	C₈	—	26	8	no	—
191	C₈	C₂×#6107	1	4	yes	—
192	C₈	C₂×#6107	1	4	yes	—
193	C₂²	—	2	2	no	—
194	C₂²	#11718753	1	2	yes	—
195	C₂²	C₂×#88194	1	1	yes	characteristic
196	C₄	—	2	2	no	—
197	C₄	—	8	4	no	—
198	C₄	—	104	4	no	—
199	C₄	#11718753	1	2	yes	—
200	C₄	#11719473	1	4	yes	—
201	C₄	C₂²×#6107	1	2	yes	—
202	C₂	—	2	2	no	—
203	C₂	C₂×#11718753	1	1	yes	characteristic
204	C₂	#13125279	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13125418

C₂

yes

maximal

#13125350

C₂

yes

maximal, characteristic

#13114514

C₂

yes

maximal, complemented

#13114510

C₂

yes

maximal, complemented

C₂×#11718047

C₂

yes

maximal, characteristic

C₁₃×#3160

C₂

yes

maximal, characteristic

C₁₃×#533

C₂²

yes

—

#11719175

C₂²

yes

—

C₁₃×#500

C₂²

yes

—

C₁₃×#499

—

C₁₃×#498

C₂²

yes

—

C₁₃×#427

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

#11718767

—

#11718754

C₂²

yes

—

#11718753

C₂²

yes

—

#11718742

C₂²

yes

—

#11718739

C₂²

yes

—

#11718712

—

C₄×#6107

—

C₂×C₄×#1055

C₂²

yes

—

#11718344

—

#11718343

C₂²

yes

—

#11718341

C₂²

yes

—

#11718047

C₂²

yes

—

#11718047

C₂²

yes

—

C₂²×C₈×C₁₃

D₄

yes

—

C₁₃×C₄⋊Q₈

C₂³

yes

—

C₁₃×#125

—

C₁₃×C₂²⋊Q₈

—

C₄×Q₈×C₁₃

—

C₁₃×#117

—

C₂×C₄²×C₁₃

C₂³

yes

—

—

—

C₂²×#1055

D₄

yes

—

#88124

—

#88123

C₂³

yes

—

C₄×Dic₂₆

—

C₁₃×#107

—

C₁₃×#106

—

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

C₂³

yes

characteristic

C₄×C₈×C₁₃

C₂³

yes

—

—

—

—

—

D₄

yes

—

C₄×#1055

C₂³

yes

—

#19618

—

maximal, Sylow

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×Dic₂₆

—

C₂×C₈×C₁₃

Q₁₆

yes

complemented

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

C₂⁴

yes

characteristic

C₂²⋊Dic₁₃

—

#6107

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

C₂×#1055

—

C₂×#1055

C₂×D₄

yes

—

#3170

—

#3160

—

#3160

D₁₃

yes

—

—

—

C₂³×C₁₃

#127

yes

characteristic

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#127

yes

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×Q₁₆

yes

—

C₂×C₄×C₁₃

C₂²×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×D₄

yes

—

C₂×Dic₁₃

—

Dic₂₆

—

C₈×C₁₃

C₂×Q₁₆

yes

—

C₈×C₁₃

C₂×Q₁₆

yes

—

#1055

—

#533

—

#533

C₂×D₁₃

yes

—

#500

—

100

#500

—

101

#500

C₂×D₁₃

yes

—

102

#499

—

103

#499

—

104

#498

—

105

#498

C₂×D₁₃

yes

—

106

#494

—

107

#487

—

108

#487

—

109

#467

—

110

C₄×Q₁₆

—

111

#427

—

112

#427

—

113

C₂×C₄×C₈

—

114

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

115

#322

—

116

#322

—

117

C₂²×C₁₃

—

118

C₂²×C₁₃

#494

yes

—

119

C₂²×C₁₃

C₂×#127

yes

characteristic

120

C₄×C₁₃

—

121

C₄×C₁₃

—

122

C₄×C₁₃

#494

yes

—

123

C₄×C₁₃

C₂²×Q₁₆

yes

—

124

C₄×C₁₃

#575

yes

—

125

Dic₁₃

—

126

C₂×Q₁₆

—

127

C₂²×C₈

—

128

C₂²×C₈

C₂²⋊D₁₃

yes

—

129

C₄⋊Q₈

—

130

C₄⋊Q₈

C₂²×D₁₃

yes

—

131

#125

—

132

#125

—

133

C₂²⋊Q₈

—

134

C₂²⋊Q₈

—

135

C₄×Q₈

—

136

C₄×Q₈

—

137

#117

—

138

#117

—

139

C₂×C₄²

C₂²×D₁₃

yes

characteristic

140

#107

—

141

#107

—

142

#107

—

143

#106

—

144

#106

—

145

#103

—

146

#103

—

147

C₄×C₈

—

148

C₄×C₈

—

149

C₄×C₈

C₂²⋊D₁₃

yes

—

150

C₄×C₈

C₂²×D₁₃

yes

characteristic

151

C₄×C₈

C₂²×D₁₃

yes

characteristic

152

C₂×C₁₃

—

153

C₂×C₁₃

C₂×#494

yes

characteristic

154

C₂×C₁₃

#3150

yes

—

155

C₂×Q₈

—

156

C₂×Q₈

—

157

C₂²×C₄

D₄×D₁₃

yes

—

158

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

159

Q₁₆

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

#6107

yes

complemented

163

C₂×C₈

C₂×C₂²⋊D₁₃

yes

—

164

C₂×C₈

C₂×C₂²⋊D₁₃

yes

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₂²⋊C₄

—

172

C₂²⋊C₄

—

173

C₄²

—

174

C₄²

—

175

C₄²

D₄×D₁₃

yes

—

176

C₄²

C₂×C₂²⋊D₁₃

yes

characteristic

177

C₄²

C₂³×D₁₃

yes

characteristic

178

C₁₃

#19618

yes

characteristic, complemented, Sylow

179

C₂³

#88194

yes

characteristic

180

Q₈

—

181

Q₈

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

#88194

yes

—

186

C₂×C₄

#88194

yes

characteristic

187

C₂×C₄

C₂×#6107

yes

—

188

C₂×C₄

C₂×D₄×D₁₃

yes

—

189

C₂×C₄

C₂²×C₂²⋊D₁₃

yes

characteristic

190

C₈

—

191

C₈

C₂×#6107

yes

—

192

C₈

C₂×#6107

yes

—

193

C₂²

—

194

C₂²

#11718753

yes

—

195

C₂²

C₂×#88194

yes

characteristic

196

C₄

—

197

C₄

—

198

C₄

—

104

—

199

C₄

#11718753

yes

—

200

C₄

#11719473

yes

—

201

C₄

C₂²×#6107

yes

—

202

C₂

—

203

C₂

C₂×#11718753

yes

characteristic

204

C₂

#13125279

yes

—

	subgroup	G/H
center	C₂×C₄	#88194
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#3160	C₂
socle	C₂²×C₁₃	C₂×#127
prime radical	C₂³×C₁₃	#127
hypercenter	#3160	D₁₃
hypocenter	C₁₃	#19618
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

subgroup

G/H

center

C₂×C₄

#88194

commutator, cosocle

C₄²×C₁₃

C₂⁴

Frattini

C₄²

C₂³×D₁₃

Fitting

C₁₃×#3160

C₂

C₂²×C₁₃

C₂³×C₁₃

D₁₃

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

N	H ≅ G/N	type
3. #13114514	C₂	semidirect
4. #13114510	C₂	semidirect
58. C₂×C₈×C₁₃	Q₁₆	semidirect
162. C₂×C₈	#6107	semidirect
178. C₁₃	#19618	semidirect

H ≅ G/N

type

3. #13114514

C₂

semidirect

4. #13114510

C₂

semidirect

58. C₂×C₈×C₁₃

Q₁₆

semidirect

162. C₂×C₈

#6107

semidirect

178. C₁₃

#19618

semidirect

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

subgroup	index
#13125418	2
#13125350	2
#13114514	2
#13114510	2
C₂×#11718047	2
C₁₃×#3160	2
#19618	13

subgroup

index

C₁₃×#3160

Subgroups — #15850722

Conjugacy classes of subgroups

Complements

Maximal subgroups