Group #15762000 — subgroups

Subgroups — #15762000

Order 3328.

538 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 171 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	#13123234	C₂	1	1	yes	maximal, characteristic
2	#13114953	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718073	C₂	1	4	yes	maximal
4	C₁₃×#1767	C₂	1	1	yes	maximal, characteristic
5	C₂×#88123	C₄	1	4	yes	—
6	C₂×#88123	C₂²	1	2	yes	—
7	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₁₃×#97	—	2	1	no	—
9	C₄³×C₁₃	C₂²	1	1	yes	characteristic
10	#11718596	—	2	2	no	—
11	#11718596	—	2	1	no	—
12	#11718596	—	2	1	no	—
13	C₂×C₄⋊#1055	C₂²	1	2	yes	—
14	C₁₃×#336	—	2	1	no	—
15	#11718116	—	2	2	no	—
16	#11718073	—	2	8	no	—
17	C₂²×Dic₂₆	C₂×C₄	1	4	yes	—
18	C₂²×C₈×C₁₃	—	2	2	no	—
19	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₁₃	D₄	1	2	yes	—
21	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₃	—	2	4	no	—
25	C₂×C₄⋊Dic₁₃	—	2	4	no	—
26	C₂×C₄⋊Dic₁₃	—	2	2	no	—
27	C₂×C₄⋊Dic₁₃	—	2	2	no	—
28	C₂²×#1055	—	2	2	no	—
29	#88123	—	2	4	no	—
30	#88123	—	2	4	no	—
31	#88123	—	4	8	no	—
32	C₁₃×C₄⋊C₈	—	2	2	no	—
33	C₁₃×C₄⋊C₈	D₄	1	4	yes	—
34	C₁₃×#97	—	4	2	no	—
35	C₄⋊#1055	—	2	4	no	—
36	#9702	—	13	1	no	maximal, Sylow
37	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
38	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×Dic₁₃	—	2	4	no	—
43	C₂×Dic₂₆	—	2	8	no	—
44	C₂×Dic₂₆	—	4	8	no	—
45	C₂×C₈×C₁₃	—	2	4	no	—
46	C₂×C₈×C₁₃	—	4	2	no	—
47	C₂×C₈×C₁₃	—	4	2	no	—
48	C₄²×C₁₃	—	2	4	no	—
49	C₄²×C₁₃	—	2	4	no	—
50	C₄²×C₁₃	—	2	2	no	—
51	C₄²×C₁₃	SD₁₆	1	2	yes	—
52	C₄²×C₁₃	Q₁₆	1	2	yes	—
53	C₄²×C₁₃	C₂×D₄	1	2	yes	—
54	C₄²×C₁₃	#41	1	2	yes	—
55	C₄⋊Dic₁₃	—	4	8	no	—
56	C₄⋊Dic₁₃	—	4	8	no	—
57	C₄⋊Dic₁₃	—	4	4	no	—
58	C₄⋊Dic₁₃	—	4	2	no	—
59	C₄⋊Dic₁₃	—	4	2	no	—
60	C₂×#1055	—	4	2	no	—
61	C₂×#1055	—	4	4	no	—
62	#2868	—	13	1	no	—
63	#1831	—	13	1	no	—
64	#1767	D₁₃	1	1	yes	characteristic
65	C₂×#333	—	13	4	no	—
66	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₃	—	2	2	no	—
68	C₂×C₄×C₁₃	—	2	2	no	—
69	C₂×C₄×C₁₃	—	2	2	no	—
70	C₂×C₄×C₁₃	—	2	2	no	—
71	C₂×C₄×C₁₃	—	2	2	no	—
72	C₂×C₄×C₁₃	#100	1	4	yes	—
73	C₂×C₄×C₁₃	#103	1	4	yes	—
74	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
75	C₂×C₄×C₁₃	#121	1	2	yes	—
76	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
77	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
80	C₂×C₄×C₁₃	#137	1	2	yes	—
81	C₂×C₄×C₁₃	#137	1	4	yes	—
82	C₂×Dic₁₃	—	4	4	no	—
83	C₂×Dic₁₃	—	4	8	no	—
84	Dic₂₆	—	8	8	no	—
85	C₈×C₁₃	—	4	4	no	—
86	#1055	—	8	2	no	—
87	C₂×C₄⋊Q₈	—	13	4	no	—
88	C₂×C₄⋊Q₈	—	13	2	no	—
89	C₂×C₄⋊C₈	—	13	2	no	—
90	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
91	C₂×#97	—	2	1	no	—
92	#384	—	26	2	no	—
93	#384	—	26	1	no	—
94	#384	—	26	1	no	—
95	C₄³	C₂×D₁₃	1	1	yes	characteristic
96	#336	—	2	1	no	—
97	#336	—	26	2	no	—
98	#333	—	26	8	no	—
99	C₂²×C₁₃	#390	1	1	yes	characteristic
100	C₂²×C₁₃	C₂×#100	1	2	yes	—
101	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
102	C₂²×C₁₃	#419	1	1	yes	characteristic
103	C₂²×C₁₃	#419	1	2	yes	—
104	C₄×C₁₃	—	2	4	no	—
105	C₄×C₁₃	—	2	4	no	—
106	C₄×C₁₃	—	2	2	no	—
107	C₄×C₁₃	#461	1	2	yes	—
108	C₄×C₁₃	#462	1	2	yes	—
109	C₄×C₁₃	#491	1	2	yes	—
110	C₄×C₁₃	#497	1	2	yes	—
111	Dic₁₃	—	8	4	no	—
112	C₂²×Q₈	—	13	4	no	—
113	C₂²×C₈	—	2	2	no	—
114	C₂²×C₈	—	26	2	no	—
115	C₄⋊Q₈	—	26	4	no	—
116	C₄⋊Q₈	—	26	4	no	—
117	C₄⋊Q₈	—	52	8	no	—
118	C₂×C₄⋊C₄	—	26	4	no	—
119	C₂×C₄⋊C₄	—	26	4	no	—
120	C₂×C₄⋊C₄	—	26	2	no	—
121	C₂×C₄⋊C₄	—	26	2	no	—
122	C₂×C₄²	C₄×D₁₃	1	2	yes	—
123	C₂×C₄²	D₅₂	1	2	yes	—
124	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
125	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
126	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
127	C₄⋊C₈	—	2	2	no	—
128	C₄⋊C₈	—	26	4	no	—
129	C₄⋊C₈	D₅₂	1	4	yes	—
130	#97	—	4	2	no	—
131	C₂×C₁₃	#1548	1	2	yes	—
132	C₂×C₁₃	#1558	1	2	yes	—
133	C₂×C₁₃	#1960	1	1	yes	characteristic
134	C₂×C₁₃	#1965	1	1	yes	characteristic
135	C₂×C₁₃	#1981	1	1	yes	characteristic
136	C₂×Q₈	—	26	8	no	—
137	C₂×Q₈	—	52	8	no	—
138	C₂²×C₄	—	26	4	no	—
139	C₂²×C₄	#6103	1	2	yes	—
140	C₂²×C₄	#6103	1	2	yes	—
141	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
142	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	—	4	2	no	—
146	C₂×C₈	—	4	2	no	—
147	C₂×C₈	—	52	2	no	—
148	C₂×C₈	—	52	4	no	—
149	C₄⋊C₄	—	52	8	no	—
150	C₄⋊C₄	—	52	2	no	—
151	C₄⋊C₄	—	52	2	no	—
152	C₄⋊C₄	—	52	4	no	—
153	C₄⋊C₄	—	52	8	no	—
154	C₄²	—	2	4	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	4	no	—
157	C₄²	#6105	1	2	yes	—
158	C₄²	#6107	1	2	yes	—
159	C₄²	C₂×D₅₂	1	2	yes	—
160	C₄²	#6127	1	2	yes	—
161	C₁₃	#9702	1	1	yes	characteristic, complemented, Sylow
162	C₂³	C₂×#6103	1	1	yes	characteristic
163	Q₈	—	104	8	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	52	8	no	—
170	C₂×C₄	—	52	4	no	—
171	C₂×C₄	#88050	1	4	yes	—
172	C₂×C₄	#88063	1	4	yes	—
173	C₂×C₄	C₄×D₅₂	1	2	yes	—
174	C₂×C₄	#88128	1	1	yes	characteristic
175	C₂×C₄	#88129	1	1	yes	characteristic
176	C₂×C₄	#88163	1	4	yes	—
177	C₂×C₄	#88184	1	2	yes	—
178	C₂×C₄	C₂×#6105	1	1	yes	characteristic
179	C₂×C₄	C₂×#6107	1	1	yes	characteristic
180	C₂×C₄	#88205	1	2	yes	—
181	C₈	—	4	4	no	—
182	C₈	—	104	2	no	—
183	C₂²	#11718600	1	1	yes	characteristic
184	C₂²	C₂×#88050	1	1	yes	characteristic
185	C₂²	#11718665	1	1	yes	characteristic
186	C₂²	#11718781	1	2	yes	—
187	C₂²	C₂×#88063	1	2	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	2	4	no	—
190	C₄	—	2	4	no	—
191	C₄	—	104	4	no	—
192	C₄	#11718364	1	2	yes	—
193	C₄	#11718376	1	2	yes	—
194	C₄	#11718680	1	2	yes	—
195	C₄	#11718692	1	2	yes	—
196	C₂	#13113518	1	2	yes	—
197	C₂	#13113561	1	2	yes	—
198	C₂	#13115656	1	1	yes	characteristic
199	C₂	#13115676	1	1	yes	characteristic
200	C₂	#13115759	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13123234

C₂

yes

maximal, characteristic

#13114953

C₂

yes

maximal, characteristic

C₂×#11718073

C₂

yes

maximal

C₁₃×#1767

C₂

yes

maximal, characteristic

C₂×#88123

C₄

yes

—

C₂×#88123

C₂²

yes

—

C₂×C₁₃×C₄⋊C₈

C₂²

yes

—

C₂×C₁₃×#97

—

C₄³×C₁₃

C₂²

yes

—

—

—

C₂×C₄⋊#1055

C₂²

yes

—

C₁₃×#336

—

#11718116

—

#11718073

—

C₂²×Dic₂₆

C₂×C₄

yes

—

C₂²×C₈×C₁₃

—

C₂×C₄²×C₁₃

C₂×C₄

yes

—

C₂×C₄²×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

D₄

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

characteristic

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂²×#1055

—

#88123

—

#88123

—

#88123

—

C₁₃×C₄⋊C₈

—

C₁₃×C₄⋊C₈

D₄

yes

—

C₁₃×#97

—

C₄⋊#1055

—

#9702

—

maximal, Sylow

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

SD₁₆

yes

—

C₄²×C₁₃

Q₁₆

yes

—

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

#41

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

—

—

—

—

D₁₃

yes

characteristic

C₂×#333

—

C₂³×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#100

yes

—

C₂×C₄×C₁₃

#103

yes

—

C₂×C₄×C₁₃

C₄×D₄

yes

—

C₂×C₄×C₁₃

#121

yes

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₁₃

#137

yes

—

C₂×C₄×C₁₃

#137

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

C₈×C₁₃

—

#1055

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×D₁₃

yes

—

C₂×#97

—

#384

—

#384

—

#384

—

C₄³

C₂×D₁₃

yes

—

—

—

C₂²×C₁₃

#390

yes

characteristic

100

C₂²×C₁₃

C₂×#100

yes

—

101

C₂²×C₁₃

C₂×#103

yes

characteristic

102

C₂²×C₁₃

#419

yes

characteristic

103

C₂²×C₁₃

#419

yes

—

104

C₄×C₁₃

—

105

C₄×C₁₃

—

106

C₄×C₁₃

—

107

C₄×C₁₃

#461

yes

—

108

C₄×C₁₃

#462

yes

—

109

C₄×C₁₃

#491

yes

—

110

C₄×C₁₃

#497

yes

—

111

Dic₁₃

—

112

C₂²×Q₈

—

113

C₂²×C₈

—

114

C₂²×C₈

—

115

C₄⋊Q₈

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄²

C₄×D₁₃

yes

—

123

C₂×C₄²

D₅₂

yes

—

124

C₂×C₄²

C₂²⋊D₁₃

yes

characteristic

125

C₂×C₄²

C₂²⋊D₁₃

yes

characteristic

126

C₂×C₄²

C₂²×D₁₃

yes

characteristic

127

C₄⋊C₈

—

128

C₄⋊C₈

—

129

C₄⋊C₈

D₅₂

yes

—

130

#97

—

131

C₂×C₁₃

#1548

yes

—

132

C₂×C₁₃

#1558

yes

—

133

C₂×C₁₃

#1960

yes

characteristic

134

C₂×C₁₃

#1965

yes

characteristic

135

C₂×C₁₃

#1981

yes

characteristic

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂²×C₄

—

139

C₂²×C₄

#6103

yes

—

140

C₂²×C₄

#6103

yes

—

141

C₂²×C₄

C₂×C₄×D₁₃

yes

characteristic

142

C₂²×C₄

C₂×D₅₂

yes

characteristic

143

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄²

—

155

C₄²

—

156

C₄²

—

157

C₄²

#6105

yes

—

158

C₄²

#6107

yes

—

159

C₄²

C₂×D₅₂

yes

—

160

C₄²

#6127

yes

—

161

C₁₃

#9702

yes

characteristic, complemented, Sylow

162

C₂³

C₂×#6103

yes

characteristic

163

Q₈

—

104

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

#88050

yes

—

172

C₂×C₄

#88063

yes

—

173

C₂×C₄

C₄×D₅₂

yes

—

174

C₂×C₄

#88128

yes

characteristic

175

C₂×C₄

#88129

yes

characteristic

176

C₂×C₄

#88163

yes

—

177

C₂×C₄

#88184

yes

—

178

C₂×C₄

C₂×#6105

yes

characteristic

179

C₂×C₄

C₂×#6107

yes

characteristic

180

C₂×C₄

#88205

yes

—

181

C₈

—

182

C₈

—

104

—

183

C₂²

#11718600

yes

characteristic

184

C₂²

C₂×#88050

yes

characteristic

185

C₂²

#11718665

yes

characteristic

186

C₂²

#11718781

yes

—

187

C₂²

C₂×#88063

yes

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

104

—

192

C₄

#11718364

yes

—

193

C₄

#11718376

yes

—

194

C₄

#11718680

yes

—

195

C₄

#11718692

yes

—

196

C₂

#13113518

yes

—

197

C₂

#13113561

yes

—

198

C₂

#13115656

yes

characteristic

199

C₂

#13115676

yes

characteristic

200

C₂

#13115759

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#6103
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1767	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1767	D₁₃
hypocenter	C₁₃	#9702
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

subgroup

G/H

center

C₂³

C₂×#6103

commutator

C₂²×C₄×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1767

C₂

socle, prime radical

C₂³×C₁₃

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

N	H ≅ G/N	type
161. C₁₃	#9702	semidirect

H ≅ G/N

type

161. C₁₃

#9702

semidirect

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

subgroup	index
#13123234	2
#13114953	2
C₂×#11718073	2
C₁₃×#1767	2
#9702	13

subgroup

index

C₁₃×#1767

Subgroups — #15762000

Conjugacy classes of subgroups

Complements

Maximal subgroups