Group #15762653 — subgroups

Subgroups — #15762653

Order 3328.

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

normal subgroups only

This group has 526 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	C₂×#11718341	C₂	1	1	yes	maximal, characteristic
2	#13115797	C₂	1	2	yes	maximal
3	#13115638	C₂	1	2	yes	maximal
4	#13114993	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈²×C₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#88123	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88057	—	2	2	no	—
10	C₂×#88057	—	2	2	no	—
11	C₂×#88056	—	2	4	no	—
12	#11718596	C₂²	1	1	yes	characteristic
13	#11718596	C₂²	1	1	yes	characteristic
14	#11718341	—	2	2	no	—
15	#11718341	C₄	1	4	yes	complemented
16	C₈²×C₁₃	—	2	2	no	—
17	#11718143	—	2	4	no	—
18	C₂²×Dic₂₆	—	2	2	no	—
19	C₂²×C₈×C₁₃	D₄	1	4	yes	—
20	C₂²×C₈×C₁₃	D₄	1	2	yes	—
21	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₃	—	2	2	no	—
23	C₂×C₄⋊Dic₁₃	—	2	2	no	—
24	C₂×C₄⋊Dic₁₃	—	2	2	no	—
25	C₂×C₄⋊Dic₁₃	—	2	1	no	—
26	C₂×C₄⋊Dic₁₃	—	2	1	no	—
27	C₂×C₄⋊Dic₁₃	—	2	2	no	—
28	#88123	—	2	2	no	—
29	#88123	—	2	2	no	—
30	#88123	—	2	2	no	—
31	#88123	C₂×C₄	1	4	yes	—
32	C₄×C₈×C₁₃	—	2	2	no	—
33	C₄×C₈×C₁₃	—	2	2	no	—
34	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₁₃	D₄	1	2	yes	—
36	#88057	—	2	8	no	—
37	#88057	—	4	4	no	—
38	#88056	—	4	8	no	—
39	#9819	—	13	1	no	maximal, Sylow
40	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
42	C₂²×Dic₁₃	—	4	2	no	—
43	C₂²×Dic₁₃	—	4	1	no	—
44	C₂²×Dic₁₃	—	4	1	no	—
45	C₂×Dic₂₆	—	2	4	no	—
46	C₂×Dic₂₆	—	4	4	no	—
47	C₂×Dic₂₆	—	4	4	no	—
48	C₂×Dic₂₆	—	4	2	no	—
49	C₂×C₈×C₁₃	—	2	4	no	—
50	C₂×C₈×C₁₃	—	2	4	no	—
51	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
52	C₂×C₈×C₁₃	SD₁₆	1	8	yes	—
53	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
54	C₂×C₈×C₁₃	Q₁₆	1	4	yes	—
55	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₁₃	#41	1	2	yes	—
58	C₄⋊Dic₁₃	—	2	4	no	—
59	C₄⋊Dic₁₃	—	2	4	no	—
60	C₄⋊Dic₁₃	—	4	2	no	—
61	C₄⋊Dic₁₃	—	4	2	no	—
62	C₄⋊Dic₁₃	—	4	4	no	—
63	C₄⋊Dic₁₃	—	4	2	no	—
64	C₄⋊Dic₁₃	—	4	2	no	—
65	C₄⋊Dic₁₃	—	4	4	no	—
66	C₂×#498	—	13	1	no	—
67	#1990	—	13	2	no	—
68	#1956	—	13	2	no	—
69	#1844	—	13	1	no	—
70	C₂×C₈²	D₁₃	1	1	yes	characteristic
71	C₂³×C₁₃	#127	1	1	yes	characteristic
72	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
74	C₂×C₄×C₁₃	#121	1	2	yes	—
75	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
77	C₂×C₄×C₁₃	C₂×SD₁₆	1	4	yes	—
78	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
79	C₂×C₄×C₁₃	#135	1	4	yes	—
80	C₂×Dic₁₃	—	4	4	no	—
81	C₂×Dic₁₃	—	8	2	no	—
82	C₂×Dic₁₃	—	8	2	no	—
83	C₂×Dic₁₃	—	8	1	no	—
84	C₂×Dic₁₃	—	8	2	no	—
85	C₂×Dic₁₃	—	8	1	no	—
86	C₂×Dic₁₃	—	8	2	no	—
87	Dic₂₆	—	4	8	no	—
88	Dic₂₆	—	8	4	no	—
89	C₈×C₁₃	—	2	8	no	—
90	C₈×C₁₃	#103	1	8	yes	complemented
91	C₂×C₄⋊Q₈	—	13	2	no	—
92	#498	—	13	4	no	—
93	#498	—	26	2	no	—
94	C₂×#106	—	26	2	no	—
95	C₂×#106	—	26	2	no	—
96	C₂×#103	—	26	4	no	—
97	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
98	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
99	#384	—	13	1	no	—
100	#384	—	13	1	no	—
101	#340	—	26	4	no	—
102	C₈²	—	2	2	no	—
103	C₂²×C₁₃	#390	1	1	yes	characteristic
104	C₂²×C₁₃	#492	1	1	yes	characteristic
105	C₂²×C₁₃	#492	1	2	yes	—
106	C₂²×C₁₃	#494	1	1	yes	characteristic
107	C₂²×C₁₃	#495	1	2	yes	—
108	C₄×C₁₃	C₂×#103	1	2	yes	—
109	C₄×C₁₃	C₄×SD₁₆	1	4	yes	—
110	C₄×C₁₃	#465	1	4	yes	—
111	C₄×C₁₃	#487	1	2	yes	—
112	Dic₁₃	—	8	4	no	—
113	Dic₁₃	—	16	1	no	—
114	Dic₁₃	—	16	1	no	—
115	C₂²×Q₈	—	26	2	no	—
116	C₂²×C₈	D₅₂	1	4	yes	—
117	C₂²×C₈	D₅₂	1	2	yes	—
118	C₄⋊Q₈	—	13	4	no	—
119	C₄⋊Q₈	—	26	2	no	—
120	C₄⋊Q₈	—	26	2	no	—
121	C₄⋊Q₈	—	26	2	no	—
122	C₂×C₄⋊C₄	—	26	2	no	—
123	C₂×C₄⋊C₄	—	26	2	no	—
124	C₂×C₄⋊C₄	—	26	2	no	—
125	C₂×C₄⋊C₄	—	26	2	no	—
126	C₂×C₄⋊C₄	—	26	1	no	—
127	C₂×C₄⋊C₄	—	26	1	no	—
128	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
129	#106	—	26	8	no	—
130	#106	—	52	4	no	—
131	#103	—	52	8	no	—
132	C₄×C₈	—	2	2	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	C₄×D₁₃	1	2	yes	—
135	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
136	C₂×C₁₃	#1705	1	1	yes	characteristic
137	C₂×C₁₃	#1707	1	2	yes	—
138	C₂×C₁₃	#1708	1	1	yes	characteristic
139	C₂×C₁₃	#1956	1	1	yes	characteristic
140	C₂×C₁₃	#1968	1	2	yes	—
141	C₂×Q₈	—	26	4	no	—
142	C₂×Q₈	—	52	2	no	—
143	C₂×Q₈	—	52	4	no	—
144	C₂×Q₈	—	52	4	no	—
145	C₂²×C₄	—	52	2	no	—
146	C₂²×C₄	—	52	1	no	—
147	C₂²×C₄	—	52	1	no	—
148	C₂²×C₄	C₂×D₅₂	1	2	yes	—
149	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	#6095	1	8	yes	—
153	C₂×C₈	#6095	1	4	yes	—
154	C₂×C₈	Dic₅₂	1	4	yes	—
155	C₂×C₈	#6103	1	4	yes	—
156	C₄⋊C₄	—	26	4	no	—
157	C₄⋊C₄	—	26	4	no	—
158	C₄⋊C₄	—	52	2	no	—
159	C₄⋊C₄	—	52	2	no	—
160	C₄⋊C₄	—	52	4	no	—
161	C₄⋊C₄	—	52	2	no	—
162	C₄⋊C₄	—	52	2	no	—
163	C₄⋊C₄	—	52	4	no	—
164	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
165	C₄²	#6127	1	2	yes	—
166	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
167	C₁₃	#9819	1	1	yes	characteristic, complemented, Sylow
168	C₂³	#88128	1	1	yes	characteristic
169	Q₈	—	52	8	no	—
170	Q₈	—	104	4	no	—
171	C₂×C₄	—	52	4	no	—
172	C₂×C₄	—	104	2	no	—
173	C₂×C₄	—	104	2	no	—
174	C₂×C₄	—	104	2	no	—
175	C₂×C₄	—	104	1	no	—
176	C₂×C₄	—	104	1	no	—
177	C₂×C₄	—	104	2	no	—
178	C₂×C₄	C₄×D₅₂	1	2	yes	—
179	C₂×C₄	#88129	1	1	yes	characteristic
180	C₂×C₄	C₂×#6095	1	2	yes	—
181	C₂×C₄	C₂×#6095	1	4	yes	—
182	C₂×C₄	#88158	1	4	yes	—
183	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
184	C₂×C₄	C₂×#6103	1	1	yes	characteristic
185	C₂×C₄	#88184	1	2	yes	—
186	C₈	—	2	8	no	—
187	C₈	#88056	1	8	yes	complemented
188	C₂²	#11718354	1	2	yes	—
189	C₂²	#11718354	1	1	yes	characteristic
190	C₂²	#11718357	1	2	yes	—
191	C₂²	#11718360	1	1	yes	characteristic
192	C₂²	#11718600	1	1	yes	characteristic
193	C₄	—	104	4	no	—
194	C₄	—	208	1	no	—
195	C₄	—	208	1	no	—
196	C₄	#11718342	1	2	yes	—
197	C₄	C₄×#6095	1	4	yes	—
198	C₄	C₂×#88056	1	2	yes	—
199	C₄	#11718775	1	4	yes	—
200	C₂	#13114423	1	1	yes	characteristic
201	C₂	#13114433	1	2	yes	—
202	C₂	#13114437	1	1	yes	characteristic
203	C₂	#13115638	1	1	yes	characteristic
204	C₂	#13115690	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718341

C₂

yes

maximal, characteristic

#13115797

C₂

yes

maximal

#13115638

C₂

yes

maximal

#13114993

C₂

yes

maximal, characteristic

C₂×C₈²×C₁₃

C₂

yes

maximal, characteristic

C₂×#88123

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

—

—

—

C₂²

yes

characteristic

#11718596

C₂²

yes

characteristic

#11718341

—

#11718341

C₄

yes

complemented

C₈²×C₁₃

—

#11718143

—

C₂²×Dic₂₆

—

C₂²×C₈×C₁₃

D₄

yes

—

C₂²×C₈×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

—

—

—

C₂×C₄

yes

—

C₄×C₈×C₁₃

—

C₄×C₈×C₁₃

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

D₄

yes

—

—

—

—

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂²⋊C₄

yes

—

C₂×C₈×C₁₃

SD₁₆

yes

—

C₂×C₈×C₁₃

SD₁₆

yes

—

C₂×C₈×C₁₃

Q₁₆

yes

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

#41

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

—

—

—

—

C₂×C₈²

D₁₃

yes

characteristic

C₂³×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₄×D₄

yes

—

C₂×C₄×C₁₃

#121

yes

—

C₂×C₄×C₁₃

#124

yes

characteristic

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₂×C₄×C₁₃

C₂×Q₁₆

yes

—

C₂×C₄×C₁₃

#135

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

#103

yes

complemented

C₂×C₄⋊Q₈

—

—

—

—

—

—

C₂×C₄×C₈

C₂×D₁₃

yes

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

#384

—

100

#384

—

101

#340

—

102

C₈²

—

103

C₂²×C₁₃

#390

yes

characteristic

104

C₂²×C₁₃

#492

yes

characteristic

105

C₂²×C₁₃

#492

yes

—

106

C₂²×C₁₃

#494

yes

characteristic

107

C₂²×C₁₃

#495

yes

—

108

C₄×C₁₃

C₂×#103

yes

—

109

C₄×C₁₃

C₄×SD₁₆

yes

—

110

C₄×C₁₃

#465

yes

—

111

C₄×C₁₃

#487

yes

—

112

Dic₁₃

—

113

Dic₁₃

—

114

Dic₁₃

—

115

C₂²×Q₈

—

116

C₂²×C₈

D₅₂

yes

—

117

C₂²×C₈

D₅₂

yes

—

118

C₄⋊Q₈

—

119

C₄⋊Q₈

—

120

C₄⋊Q₈

—

121

C₄⋊Q₈

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄²

C₂²×D₁₃

yes

characteristic

129

#106

—

130

#106

—

131

#103

—

132

C₄×C₈

—

133

C₄×C₈

—

134

C₄×C₈

C₄×D₁₃

yes

—

135

C₄×C₈

C₂²⋊D₁₃

yes

—

136

C₂×C₁₃

#1705

yes

characteristic

137

C₂×C₁₃

#1707

yes

—

138

C₂×C₁₃

#1708

yes

characteristic

139

C₂×C₁₃

#1956

yes

characteristic

140

C₂×C₁₃

#1968

yes

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂²×C₄

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

C₂×D₅₂

yes

—

149

C₂²×C₄

C₂×D₅₂

yes

characteristic

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

#6095

yes

—

153

C₂×C₈

#6095

yes

—

154

C₂×C₈

Dic₅₂

yes

—

155

C₂×C₈

#6103

yes

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄²

C₂×C₄×D₁₃

yes

characteristic

165

C₄²

#6127

yes

—

166

C₄²

C₂×C₂²⋊D₁₃

yes

characteristic

167

C₁₃

#9819

yes

characteristic, complemented, Sylow

168

C₂³

#88128

yes

characteristic

169

Q₈

—

170

Q₈

—

104

—

171

C₂×C₄

—

172

C₂×C₄

—

104

—

173

C₂×C₄

—

104

—

174

C₂×C₄

—

104

—

175

C₂×C₄

—

104

—

176

C₂×C₄

—

104

—

177

C₂×C₄

—

104

—

178

C₂×C₄

C₄×D₅₂

yes

—

179

C₂×C₄

#88129

yes

characteristic

180

C₂×C₄

C₂×#6095

yes

—

181

C₂×C₄

C₂×#6095

yes

—

182

C₂×C₄

#88158

yes

—

183

C₂×C₄

C₂×Dic₅₂

yes

—

184

C₂×C₄

C₂×#6103

yes

characteristic

185

C₂×C₄

#88184

yes

—

186

C₈

—

187

C₈

#88056

yes

complemented

188

C₂²

#11718354

yes

—

189

C₂²

#11718354

yes

characteristic

190

C₂²

#11718357

yes

—

191

C₂²

#11718360

yes

characteristic

192

C₂²

#11718600

yes

characteristic

193

C₄

—

104

—

194

C₄

—

208

—

195

C₄

—

208

—

196

C₄

#11718342

yes

—

197

C₄

C₄×#6095

yes

—

198

C₄

C₂×#88056

yes

—

199

C₄

#11718775

yes

—

200

C₂

#13114423

yes

characteristic

201

C₂

#13114433

yes

—

202

C₂

#13114437

yes

characteristic

203

C₂

#13115638

yes

characteristic

204

C₂

#13115690

yes

—

	subgroup	G/H
center	C₂³	#88128
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₈²×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	#127
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×C₈²	D₁₃
hypocenter	C₁₃	#9819
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂³

#88128

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₂×C₈²×C₁₃

C₂

socle, prime radical

C₂³×C₁₃

#127

cosocle

C₂×C₄²×C₁₃

C₂³

hypercenter

C₂×C₈²

D₁₃

hypocenter

C₁₃

#9819

LCS

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
15. #11718341	C₄	semidirect
90. C₈×C₁₃	#103	semidirect
167. C₁₃	#9819	semidirect
187. C₈	#88056	semidirect

H ≅ G/N

type

15. #11718341

C₄

semidirect

90. C₈×C₁₃

#103

semidirect

167. C₁₃

#9819

semidirect

187. C₈

#88056

semidirect

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

subgroup	index
C₂×#11718341	2
#13115797	2
#13115638	2
#13114993	2
C₂×C₈²×C₁₃	2
#9819	13

subgroup

index

C₂×C₈²×C₁₃

Subgroups — #15762653

Conjugacy classes of subgroups

Complements

Maximal subgroups