Group #16785051 — subgroups

Subgroups — #16785051

Order 3328.

494 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 144 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	#13130751	C₂	1	1	yes	maximal, characteristic
2	C₄²×#1057	C₂	1	1	yes	maximal, characteristic
3	#13129672	C₂	1	4	yes	maximal
4	#13117391	C₂	1	1	yes	maximal, characteristic
5	C₂×#88124	C₄	1	1	yes	characteristic
6	C₂×C₄⋊#1057	C₂²	1	2	yes	—
7	C₂×C₄×#1057	—	2	2	no	—
8	C₂×C₄×#1057	C₂²	1	2	yes	—
9	C₁₃×#382	C₄	1	1	yes	characteristic
10	C₄⋊C₄×Dic₁₃	C₂²	1	2	yes	—
11	#11718634	C₄	1	2	yes	—
12	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
13	#11718313	—	2	2	no	—
14	#11718310	—	2	2	no	—
15	#11718290	—	2	2	no	—
16	C₂²×#1057	—	2	2	no	—
17	C₂²×#1057	—	2	2	no	—
18	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
21	C₂×#6101	—	2	2	no	—
22	C₂×#6101	—	2	2	no	—
23	C₂×C₄×Dic₁₃	—	2	2	no	—
24	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
26	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
27	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
28	#88124	—	2	2	no	—
29	#88124	C₈	1	4	yes	complemented
30	C₄⋊#1057	—	2	4	no	—
31	C₄×#1057	—	2	8	no	—
32	C₄×#1057	—	2	2	no	—
33	C₄×#1057	Q₈	1	4	yes	—
34	C₁₃×#95	—	2	2	no	—
35	#88071	—	2	2	no	—
36	#88071	—	2	2	no	—
37	#9871	—	13	1	no	maximal, Sylow
38	C₂²×C₄×C₁₃	—	2	2	no	—
39	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×Dic₁₃	—	2	2	no	—
42	C₂²×Dic₁₃	C₂²⋊C₄	1	2	yes	—
43	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂×#1057	—	2	8	no	—
46	C₂×#1057	—	2	4	no	—
47	C₂×#1057	—	4	2	no	—
48	C₂×#1057	—	4	4	no	—
49	C₁₃×C₄⋊C₄	—	2	4	no	—
50	C₄²×C₁₃	C₂×C₈	1	2	yes	—
51	C₄²×C₁₃	M₁₆	1	2	yes	—
52	C₄⋊Dic₁₃	—	2	2	no	—
53	C₄⋊Dic₁₃	C₂×C₈	1	4	yes	—
54	#6101	—	2	8	no	—
55	#6101	—	4	4	no	—
56	C₄×Dic₁₃	—	2	2	no	—
57	C₄×Dic₁₃	—	2	4	no	—
58	C₄×Dic₁₃	—	2	4	no	—
59	C₄×Dic₁₃	—	4	4	no	—
60	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
61	C₄×Dic₁₃	#41	1	2	yes	—
62	C₄×#125	—	13	1	no	—
63	#1831	—	13	1	no	—
64	C₄²×C₈	—	13	1	no	—
65	#1279	—	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	—	4	2	no	—
70	C₂×C₄×C₁₃	—	4	2	no	—
71	C₂×C₄×C₁₃	—	4	2	no	—
72	C₂×C₄×C₁₃	C₂²⋊C₈	1	4	yes	—
73	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
74	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₁₃	#131	1	2	yes	—
77	C₂×Dic₁₃	—	2	2	no	—
78	C₂×Dic₁₃	—	2	2	no	—
79	C₂×Dic₁₃	—	2	2	no	—
80	C₂×Dic₁₃	—	2	4	no	—
81	C₂×Dic₁₃	—	4	2	no	—
82	C₂×Dic₁₃	—	4	2	no	—
83	C₂×Dic₁₃	#104	1	4	yes	—
84	C₂×Dic₁₃	C₂²⋊Q₈	1	2	yes	—
85	C₂×Dic₁₃	#124	1	1	yes	characteristic
86	C₂×Dic₁₃	C₄⋊Q₈	1	1	yes	characteristic
87	C₂×Dic₁₃	#135	1	4	yes	—
88	#1057	—	2	8	no	—
89	#1057	—	8	2	no	—
90	C₂×#125	—	13	1	no	—
91	C₂×C₄⋊C₈	—	13	2	no	—
92	C₂×C₄×C₈	—	13	2	no	—
93	C₂×C₄×C₈	—	26	2	no	—
94	#389	—	13	2	no	—
95	#387	—	26	2	no	—
96	#382	#261	1	1	yes	characteristic
97	C₄×C₄⋊C₄	—	13	2	no	—
98	C₄×C₄⋊C₄	—	26	2	no	—
99	C₄³	—	13	1	no	—
100	#336	—	26	2	no	—
101	C₂²×C₁₃	#391	1	1	yes	characteristic
102	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
103	C₂²×C₁₃	#408	1	1	yes	characteristic
104	C₂²×C₁₃	#416	1	2	yes	—
105	C₂²×C₁₃	C₂×#104	1	2	yes	—
106	C₄×C₁₃	—	2	4	no	—
107	C₄×C₁₃	—	8	2	no	—
108	C₄×C₁₃	C₈×Q₈	1	2	yes	—
109	C₄×C₁₃	#446	1	2	yes	—
110	Dic₁₃	—	2	2	no	—
111	Dic₁₃	—	2	4	no	—
112	Dic₁₃	—	4	4	no	—
113	Dic₁₃	#488	1	2	yes	—
114	Dic₁₃	#499	1	2	yes	—
115	C₂²×C₈	—	26	2	no	—
116	C₂²×C₈	—	26	2	no	—
117	#125	—	13	4	no	—
118	#125	—	26	2	no	—
119	C₂×C₄⋊C₄	—	13	2	no	—
120	C₂×C₄⋊C₄	—	26	2	no	—
121	C₂×C₄⋊C₄	—	26	2	no	—
122	C₂×C₄⋊C₄	C₂×#261	1	2	yes	—
123	C₂×C₄²	—	13	1	no	—
124	C₂×C₄²	—	13	2	no	—
125	C₂×C₄²	—	13	2	no	—
126	C₂×C₄²	—	13	1	no	—
127	C₂×C₄²	—	26	2	no	—
128	C₂×C₄²	C₂×#261	1	1	yes	characteristic
129	C₄⋊C₈	—	26	4	no	—
130	C₄×C₈	—	13	4	no	—
131	C₄×C₈	—	26	2	no	—
132	C₄×C₈	—	26	8	no	—
133	#95	—	2	2	no	—
134	#95	—	26	2	no	—
135	#95	—	26	2	no	—
136	C₂×C₁₃	#1476	1	2	yes	—
137	C₂×C₁₃	#1494	1	2	yes	—
138	C₂×C₁₃	#1984	1	1	yes	characteristic
139	C₂×C₁₃	#1991	1	1	yes	characteristic
140	C₂×C₁₃	#1994	1	1	yes	characteristic
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	—	13	1	no	—
143	C₂²×C₄	—	13	1	no	—
144	C₂²×C₄	—	13	2	no	—
145	C₂²×C₄	—	26	2	no	—
146	C₂²×C₄	C₂²⋊#261	1	2	yes	—
147	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
148	C₂×C₈	—	26	4	no	—
149	C₂×C₈	—	26	8	no	—
150	C₂×C₈	—	52	2	no	—
151	C₂×C₈	—	52	4	no	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	13	4	no	—
154	C₄⋊C₄	—	26	8	no	—
155	C₄⋊C₄	—	26	2	no	—
156	C₄⋊C₄	—	52	4	no	—
157	C₄²	—	13	2	no	—
158	C₄²	—	13	2	no	—
159	C₄²	—	26	4	no	—
160	C₄²	—	26	4	no	—
161	C₄²	—	26	2	no	—
162	C₄²	—	52	4	no	—
163	C₄²	D₁₃⋊C₈	1	2	yes	—
164	C₄²	#6118	1	2	yes	—
165	C₁₃	#9871	1	1	yes	characteristic, complemented, Sylow
166	C₂³	C₂×C₂²⋊#261	1	1	yes	characteristic
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	13	1	no	—
173	C₂×C₄	—	13	1	no	—
174	C₂×C₄	—	13	4	no	—
175	C₂×C₄	—	13	2	no	—
176	C₂×C₄	—	13	4	no	—
177	C₂×C₄	—	26	2	no	—
178	C₂×C₄	—	26	2	no	—
179	C₂×C₄	—	26	2	no	—
180	C₂×C₄	—	26	4	no	—
181	C₂×C₄	—	52	2	no	—
182	C₂×C₄	—	52	2	no	—
183	C₂×C₄	#88111	1	4	yes	—
184	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
185	C₂×C₄	C₂×#6118	1	1	yes	characteristic
186	C₂×C₄	#88238	1	2	yes	—
187	C₂×C₄	Q₈×#261	1	2	yes	—
188	C₈	—	26	8	no	—
189	C₈	—	104	2	no	—
190	C₂²	C₂×#88111	1	1	yes	characteristic
191	C₂²	#11719147	1	1	yes	characteristic
192	C₂²	(C₂×#261)⋊Q₈	1	1	yes	characteristic
193	C₂²	D₁₃⋊#104	1	2	yes	—
194	C₂²	#11719165	1	2	yes	—
195	C₄	—	2	4	no	—
196	C₄	—	8	2	no	—
197	C₄	—	13	2	no	—
198	C₄	—	13	2	no	—
199	C₄	—	26	2	no	—
200	C₄	—	26	4	no	—
201	C₄	—	52	4	no	—
202	C₄	#11719074	1	2	yes	—
203	C₄	#11719080	1	2	yes	—
204	C₂	#13130129	1	2	yes	—
205	C₂	#13130188	1	2	yes	—
206	C₂	#13131006	1	1	yes	characteristic
207	C₂	#13131018	1	1	yes	characteristic
208	C₂	#13131022	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13130751

C₂

yes

maximal, characteristic

C₄²×#1057

C₂

yes

maximal, characteristic

#13129672

C₂

yes

maximal

#13117391

C₂

yes

maximal, characteristic

C₂×#88124

C₄

yes

characteristic

C₂×C₄⋊#1057

C₂²

yes

—

C₂×C₄×#1057

—

C₂×C₄×#1057

C₂²

yes

—

C₁₃×#382

C₄

yes

characteristic

C₄⋊C₄×Dic₁₃

C₂²

yes

—

#11718634

C₄

yes

—

C₄²×Dic₁₃

C₂²

yes

—

—

—

C₂²×#1057

—

C₂²×#1057

—

C₂×C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂×C₄

yes

—

C₂×#6101

—

C₂×#6101

—

C₂×C₄×Dic₁₃

—

C₂×C₄×Dic₁₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₃

D₄

yes

—

C₂×C₄×Dic₁₃

D₄

yes

—

C₂×C₄×Dic₁₃

C₂³

yes

characteristic

#88124

—

#88124

C₈

yes

—

—

—

Q₈

yes

—

C₁₃×#95

—

—

—

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

C₂²⋊C₄

yes

—

C₂²×Dic₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

C₂×D₄

yes

—

—

—

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×C₈

yes

—

C₄²×C₁₃

M₁₆

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

C₂×C₈

yes

—

#6101

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

C₂×Q₈

yes

—

C₄×Dic₁₃

#41

yes

—

C₄×#125

—

#1831

—

C₄²×C₈

—

#1279

—

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

C₂²⋊C₈

yes

—

C₂×C₄×C₁₃

C₄×Q₈

yes

—

C₂×C₄×C₁₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₁₃

#131

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

#104

yes

—

C₂×Dic₁₃

C₂²⋊Q₈

yes

—

C₂×Dic₁₃

#124

yes

characteristic

C₂×Dic₁₃

C₄⋊Q₈

yes

characteristic

C₂×Dic₁₃

#135

yes

—

#1057

—

#1057

—

C₂×#125

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

—

—

yes

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

—

C₄³

—

100

#336

—

101

C₂²×C₁₃

#391

yes

characteristic

102

C₂²×C₁₃

C₂×C₂²⋊C₈

yes

characteristic

103

C₂²×C₁₃

#408

yes

characteristic

104

C₂²×C₁₃

#416

yes

—

105

C₂²×C₁₃

C₂×#104

yes

—

106

C₄×C₁₃

—

107

C₄×C₁₃

—

108

C₄×C₁₃

C₈×Q₈

yes

—

109

C₄×C₁₃

#446

yes

—

110

Dic₁₃

—

111

Dic₁₃

—

112

Dic₁₃

—

113

Dic₁₃

#488

yes

—

114

Dic₁₃

#499

yes

—

115

C₂²×C₈

—

116

C₂²×C₈

—

117

#125

—

118

#125

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

C₂×#261

yes

—

123

C₂×C₄²

—

124

C₂×C₄²

—

125

C₂×C₄²

—

126

C₂×C₄²

—

127

C₂×C₄²

—

128

C₂×C₄²

C₂×#261

yes

characteristic

129

C₄⋊C₈

—

130

C₄×C₈

—

131

C₄×C₈

—

132

C₄×C₈

—

133

#95

—

134

#95

—

135

#95

—

136

C₂×C₁₃

#1476

yes

—

137

C₂×C₁₃

#1494

yes

—

138

C₂×C₁₃

#1984

yes

characteristic

139

C₂×C₁₃

#1991

yes

characteristic

140

C₂×C₁₃

#1994

yes

characteristic

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

C₂²×C₄

C₂²⋊#261

yes

—

147

C₂²×C₄

C₂²×#261

yes

characteristic

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄²

—

158

C₄²

—

159

C₄²

—

160

C₄²

—

161

C₄²

—

162

C₄²

—

163

C₄²

D₁₃⋊C₈

yes

—

164

C₄²

#6118

yes

—

165

C₁₃

#9871

yes

characteristic, complemented, Sylow

166

C₂³

C₂×C₂²⋊#261

yes

characteristic

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

#88111

yes

—

184

C₂×C₄

C₂×D₁₃⋊C₈

yes

characteristic

185

C₂×C₄

C₂×#6118

yes

characteristic

186

C₂×C₄

#88238

yes

—

187

C₂×C₄

Q₈×#261

yes

—

188

C₈

—

189

C₈

—

104

—

190

C₂²

C₂×#88111

yes

characteristic

191

C₂²

#11719147

yes

characteristic

192

C₂²

(C₂×#261)⋊Q₈

yes

characteristic

193

C₂²

D₁₃⋊#104

yes

—

194

C₂²

#11719165

yes

—

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

#11719074

yes

—

203

C₄

#11719080

yes

—

204

C₂

#13130129

yes

—

205

C₂

#13130188

yes

—

206

C₂

#13131006

yes

characteristic

207

C₂

#13131018

yes

characteristic

208

C₂

#13131022

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊#261
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₁₃×#382	C₄
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	#382	#261
hypocenter	C₁₃	#9871
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊#261

commutator

C₂×C₄×C₁₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#261

Fitting

C₁₃×#382

C₄

socle, prime radical

C₂³×C₁₃

C₂×C₂²⋊C₄

cosocle

C₂×C₄×Dic₁₃

C₂³

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
29. #88124	C₈	semidirect
165. C₁₃	#9871	semidirect

H ≅ G/N

type

29. #88124

C₈

semidirect

165. C₁₃

#9871

semidirect

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

subgroup	index
#13130751	2
C₄²×#1057	2
#13129672	2
#13117391	2
#9871	13

subgroup

index

C₄²×#1057

Subgroups — #16785051

Conjugacy classes of subgroups

Complements

Maximal subgroups