Group #16795096 — subgroups

Subgroups — #16795096

Order 3328.

702 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 270 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	#13131482	C₂	1	2	yes	maximal
2	#13130406	C₂	1	6	yes	maximal, complemented
3	#13130395	C₂	1	3	yes	maximal, complemented
4	D₁₃⋊#335	C₂	1	3	yes	maximal, complemented
5	C₈×#6131	C₂	1	1	yes	maximal, characteristic
6	C₂×#88240	—	2	2	no	—
7	C₄×#6131	C₂²	1	1	yes	characteristic
8	#11719079	—	2	6	no	—
9	#11719052	C₂²	1	6	yes	—
10	D₁₃⋊(C₄⋊C₈)	C₂²	1	6	yes	—
11	C₈×Q₈×C₁₃	C₄	1	1	yes	characteristic
12	Q₈×#1055	C₄	1	1	yes	characteristic
13	#11718495	C₄	1	3	yes	—
14	#11718494	C₂²	1	3	yes	—
15	C₈×D₅₂	C₄	1	3	yes	—
16	C₄×C₈×D₁₃	C₂²	1	3	yes	—
17	#11718256	C₂²	1	1	yes	characteristic
18	#11718256	C₂²	1	3	yes	—
19	#11718256	C₂²	1	6	yes	—
20	#11718254	C₂²	1	6	yes	—
21	C₂×#6131	D₄	1	1	yes	characteristic
22	#88240	—	4	8	no	—
23	C₂×#6118	—	2	6	no	—
24	C₂×D₁₃⋊C₈	—	2	6	no	—
25	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	Q₈×Dic₁₃	Q₈	1	1	yes	characteristic
27	C₂×C₈×D₁₃	C₂×C₄	1	3	yes	—
28	C₂×C₈×D₁₃	C₂×C₄	1	3	yes	—
29	#88147	Q₈	1	3	yes	—
30	#88146	D₄	1	3	yes	—
31	C₄×D₅₂	C₂×C₄	1	3	yes	—
32	C₄²×D₁₃	C₂³	1	3	yes	—
33	#88112	—	2	6	no	—
34	C₄⋊#1057	—	2	6	no	—
35	C₄⋊#1057	C₂³	1	2	yes	—
36	C₄⋊#1057	C₂³	1	6	yes	—
37	C₁₃×C₄⋊C₈	C₂×C₄	1	3	yes	—
38	C₄×C₈×C₁₃	C₂×C₄	1	3	yes	—
39	#88060	C₂×C₄	1	6	yes	—
40	C₈×Dic₁₃	C₂³	1	3	yes	—
41	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
42	C₄⋊#1055	C₂×C₄	1	3	yes	—
43	C₄×#1055	C₂×C₄	1	3	yes	—
44	#17828	—	13	1	no	maximal, Sylow
45	C₂×Q₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
46	#6131	D₈	1	4	yes	—
47	#6131	Q₁₆	1	4	yes	—
48	C₂×D₅₂	C₄⋊C₄	1	3	yes	—
49	C₂×C₄×D₁₃	C₂²×C₄	1	3	yes	—
50	C₂×C₄×D₁₃	C₂×D₄	1	3	yes	—
51	C₂×C₄×D₁₃	C₂×Q₈	1	3	yes	—
52	C₂×#1057	—	2	6	no	—
53	C₂×#1057	—	2	2	no	—
54	#6118	—	4	12	no	—
55	D₁₃⋊C₈	—	4	12	no	—
56	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
58	C₁₃×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
59	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
60	#6103	C₄⋊C₄	1	6	yes	—
61	C₄⋊Dic₁₃	C₄⋊C₄	1	3	yes	—
62	C₄×Dic₁₃	C₂×D₄	1	3	yes	—
63	C₄×Dic₁₃	C₂×Q₈	1	3	yes	—
64	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
65	C₂×#1055	C₂²×C₄	1	3	yes	—
66	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
67	C₈×D₁₃	—	2	6	no	—
68	C₈×D₁₃	—	4	3	no	—
69	C₈×#41	—	13	1	no	—
70	#2906	—	13	2	no	—
71	#1571	—	13	6	no	—
72	#1567	—	13	3	no	—
73	C₂×#335	—	13	3	no	—
74	C₂²×D₁₃	C₂×C₄⋊C₄	1	3	yes	—
75	Q₈×C₁₃	#107	1	4	yes	—
76	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
77	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
78	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
79	C₂×Dic₁₃	C₂×C₄⋊C₄	1	3	yes	—
80	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
81	C₂×Dic₁₃	C₂²×Q₈	1	1	yes	characteristic
82	D₅₂	#107	1	12	yes	—
83	C₄×D₁₃	—	2	3	no	—
84	C₄×D₁₃	—	2	3	no	—
85	C₄×D₁₃	—	4	3	no	—
86	C₄×D₁₃	C₂×D₈	1	6	yes	—
87	C₄×D₁₃	C₂×Q₁₆	1	6	yes	—
88	#1057	—	4	2	no	—
89	#1057	—	4	6	no	—
90	C₈×C₁₃	—	2	3	no	—
91	C₈×C₁₃	—	2	1	no	—
92	#1055	—	2	3	no	—
93	#1055	#131	1	2	yes	complemented
94	C₂×#131	—	26	2	no	—
95	C₄×#41	—	13	1	no	—
96	C₈×Q₈	—	13	1	no	—
97	C₈×Q₈	#261	1	1	yes	characteristic
98	C₈×D₄	—	13	3	no	—
99	C₈×D₄	—	13	3	no	—
100	#431	—	13	3	no	—
101	#424	—	26	6	no	—
102	#423	—	13	6	no	—
103	C₂×C₄⋊C₈	—	13	6	no	—
104	C₂×C₄×C₈	—	13	3	no	—
105	#335	—	13	1	no	—
106	#335	—	13	3	no	—
107	#335	—	13	6	no	—
108	#335	—	13	6	no	—
109	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
110	C₂×D₁₃	—	2	3	no	—
111	C₂×D₁₃	—	2	3	no	—
112	C₂×D₁₃	C₂×#107	1	6	yes	—
113	C₄×C₁₃	—	2	3	no	—
114	C₄×C₁₃	C₂×#107	1	6	yes	—
115	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
116	C₄×C₁₃	#568	1	1	yes	characteristic
117	Dic₁₃	—	2	3	no	—
118	Dic₁₃	—	2	1	no	—
119	Dic₁₃	C₂²×D₈	1	1	yes	characteristic
120	Dic₁₃	C₂²×Q₁₆	1	1	yes	characteristic
121	C₂×#41	—	13	1	no	—
122	#131	—	52	8	no	—
123	C₂×M₁₆	—	26	6	no	—
124	C₂²×C₈	—	13	3	no	—
125	C₂²×C₈	—	13	3	no	—
126	C₂²×C₈	—	26	6	no	—
127	C₄×Q₈	—	13	1	no	—
128	C₄×Q₈	C₂×#261	1	1	yes	characteristic
129	C₄×D₄	—	13	3	no	—
130	C₄×D₄	—	13	3	no	—
131	#117	—	13	3	no	—
132	C₂×C₄²	—	13	3	no	—
133	C₄⋊C₈	—	13	2	no	—
134	C₄⋊C₈	—	13	6	no	—
135	C₄⋊C₈	—	13	3	no	—
136	C₄⋊C₈	—	26	6	no	—
137	C₄⋊C₈	—	26	6	no	—
138	C₄⋊C₈	C₂×#261	1	3	yes	—
139	C₂²⋊C₈	—	13	6	no	—
140	C₄×C₈	—	13	1	no	—
141	C₄×C₈	—	13	3	no	—
142	C₄×C₈	—	13	3	no	—
143	C₄×C₈	C₂×#261	1	3	yes	—
144	C₂×C₁₃	#2906	1	1	yes	characteristic
145	C₂×C₁₃	C₂²×#107	1	1	yes	characteristic
146	C₂×C₁₃	#2916	1	1	yes	characteristic
147	D₁₃	—	2	6	no	—
148	#41	—	13	4	no	—
149	#41	—	13	4	no	—
150	C₂×Q₈	C₄⋊#261	1	1	yes	characteristic
151	C₂×D₄	—	13	3	no	—
152	C₂²×C₄	—	13	3	no	—
153	C₂²×C₄	—	13	3	no	—
154	C₂²×C₄	—	13	3	no	—
155	M₁₆	—	52	12	no	—
156	C₂×C₈	—	13	1	no	—
157	C₂×C₈	—	13	3	no	—
158	C₂×C₈	—	26	6	no	—
159	C₂×C₈	—	26	2	no	—
160	C₂×C₈	—	26	6	no	—
161	C₂×C₈	—	52	3	no	—
162	C₂×C₈	—	52	12	no	—
163	C₂×C₈	C₂²×#261	1	1	yes	characteristic
164	C₂×C₈	C₂²×#261	1	3	yes	—
165	C₄⋊C₄	—	13	3	no	—
166	C₄⋊C₄	C₄⋊#261	1	3	yes	—
167	C₂²⋊C₄	—	13	6	no	—
168	C₄²	—	13	3	no	—
169	C₄²	—	13	3	no	—
170	C₄²	—	13	1	no	—
171	C₄²	C₂²×#261	1	3	yes	—
172	C₁₃	#17828	1	1	yes	characteristic, complemented, Sylow
173	C₂³	—	13	3	no	—
174	Q₈	#88102	1	4	yes	—
175	D₄	—	13	12	no	—
176	C₂×C₄	—	13	1	no	—
177	C₂×C₄	—	13	6	no	—
178	C₂×C₄	—	13	6	no	—
179	C₂×C₄	—	13	1	no	—
180	C₂×C₄	—	13	3	no	—
181	C₂×C₄	—	26	3	no	—
182	C₂×C₄	—	26	3	no	—
183	C₂×C₄	—	52	3	no	—
184	C₂×C₄	C₂×C₄⋊#261	1	3	yes	—
185	C₂×C₄	C₂×C₄⋊#261	1	3	yes	—
186	C₂×C₄	C₂³×#261	1	1	yes	characteristic
187	C₈	—	2	1	no	—
188	C₈	—	2	3	no	—
189	C₈	—	13	2	no	—
190	C₈	—	26	3	no	—
191	C₈	—	52	2	no	—
192	C₈	—	52	6	no	—
193	C₂²	—	13	6	no	—
194	C₂²	—	26	3	no	—
195	C₂²	—	26	3	no	—
196	C₂²	C₂²×C₄⋊#261	1	1	yes	characteristic
197	C₄	—	2	3	no	—
198	C₄	—	13	1	no	—
199	C₄	—	13	1	no	—
200	C₄	—	26	3	no	—
201	C₄	—	26	1	no	—
202	C₄	C₂×#88102	1	6	yes	—
203	C₄	#11719625	1	1	yes	characteristic
204	C₄	D₁₃⋊#131	1	1	yes	characteristic
205	C₂	—	26	6	no	—
206	C₂	#13131484	1	1	yes	characteristic
207	C₂	C₂²×#88102	1	1	yes	characteristic
208	C₂	#13131497	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13131482

C₂

yes

maximal

#13130406

C₂

yes

maximal, complemented

#13130395

C₂

yes

maximal, complemented

D₁₃⋊#335

C₂

yes

maximal, complemented

C₈×#6131

C₂

yes

maximal, characteristic

C₂×#88240

—

C₄×#6131

C₂²

yes

characteristic

#11719079

—

#11719052

C₂²

yes

—

D₁₃⋊(C₄⋊C₈)

C₂²

yes

—

C₈×Q₈×C₁₃

C₄

yes

characteristic

Q₈×#1055

C₄

yes

characteristic

#11718495

C₄

yes

—

#11718494

C₂²

yes

—

C₈×D₅₂

C₄

yes

—

C₄×C₈×D₁₃

C₂²

yes

—

#11718256

C₂²

yes

characteristic

#11718256

C₂²

yes

—

#11718256

C₂²

yes

—

#11718254

C₂²

yes

—

C₂×#6131

D₄

yes

characteristic

#88240

—

C₂×#6118

—

C₂×D₁₃⋊C₈

—

C₄×Q₈×C₁₃

C₂×C₄

yes

characteristic

Q₈×Dic₁₃

Q₈

yes

characteristic

C₂×C₈×D₁₃

C₂×C₄

yes

—

C₂×C₈×D₁₃

C₂×C₄

yes

—

#88147

Q₈

yes

—

#88146

D₄

yes

—

C₄×D₅₂

C₂×C₄

yes

—

C₄²×D₁₃

C₂³

yes

—

#88112

—

C₄⋊#1057

—

C₄⋊#1057

C₂³

yes

—

C₄⋊#1057

C₂³

yes

—

C₁₃×C₄⋊C₈

C₂×C₄

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

#88060

C₂×C₄

yes

—

C₈×Dic₁₃

C₂³

yes

—

C₈×Dic₁₃

C₂³

yes

characteristic

C₄⋊#1055

C₂×C₄

yes

—

C₄×#1055

C₂×C₄

yes

—

#17828

—

maximal, Sylow

C₂×Q₈×C₁₃

C₄⋊C₄

yes

characteristic

#6131

D₈

yes

—

#6131

Q₁₆

yes

—

C₂×D₅₂

C₄⋊C₄

yes

—

C₂×C₄×D₁₃

C₂²×C₄

yes

—

C₂×C₄×D₁₃

C₂×D₄

yes

—

C₂×C₄×D₁₃

C₂×Q₈

yes

—

C₂×#1057

—

C₂×#1057

—

#6118

—

D₁₃⋊C₈

—

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₃

C₂²×C₄

yes

—

C₁₃×C₄⋊C₄

C₄⋊C₄

yes

—

C₄²×C₁₃

C₂²×C₄

yes

—

#6103

C₄⋊C₄

yes

—

C₄⋊Dic₁₃

C₄⋊C₄

yes

—

C₄×Dic₁₃

C₂×D₄

yes

—

C₄×Dic₁₃

C₂×Q₈

yes

—

C₄×Dic₁₃

C₂⁴

yes

characteristic

C₂×#1055

C₂²×C₄

yes

—

C₂×#1055

C₂²×C₄

yes

characteristic

C₈×D₁₃

—

C₈×D₁₃

—

C₈×#41

—

—

—

—

—

C₂²×D₁₃

C₂×C₄⋊C₄

yes

—

Q₈×C₁₃

#107

yes

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₁₃

C₂³×C₄

yes

characteristic

C₂×Dic₁₃

C₂×C₄⋊C₄

yes

—

C₂×Dic₁₃

C₂²×D₄

yes

characteristic

C₂×Dic₁₃

C₂²×Q₈

yes

characteristic

D₅₂

#107

yes

—

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

C₂×D₈

yes

—

C₄×D₁₃

C₂×Q₁₆

yes

—

#1057

—

#1057

—

C₈×C₁₃

—

C₈×C₁₃

—

—

yes

—

C₄×#41

—

C₈×Q₈

—

C₈×Q₈

#261

yes

characteristic

C₈×D₄

—

C₈×D₄

—

100

#431

—

101

#424

—

102

#423

—

103

C₂×C₄⋊C₈

—

104

C₂×C₄×C₈

—

105

#335

—

106

#335

—

107

#335

—

108

#335

—

109

C₂²×C₁₃

C₂²×C₄⋊C₄

yes

characteristic

110

C₂×D₁₃

—

111

C₂×D₁₃

—

112

C₂×D₁₃

C₂×#107

yes

—

113

C₄×C₁₃

—

114

C₄×C₁₃

C₂×#107

yes

—

115

C₄×C₁₃

C₂×#131

yes

characteristic

116

C₄×C₁₃

#568

yes

characteristic

117

Dic₁₃

—

118

Dic₁₃

—

119

Dic₁₃

C₂²×D₈

yes

characteristic

120

Dic₁₃

C₂²×Q₁₆

yes

characteristic

121

C₂×#41

—

122

#131

—

123

C₂×M₁₆

—

124

C₂²×C₈

—

125

C₂²×C₈

—

126

C₂²×C₈

—

127

C₄×Q₈

—

128

C₄×Q₈

C₂×#261

yes

characteristic

129

C₄×D₄

—

130

C₄×D₄

—

131

#117

—

132

C₂×C₄²

—

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

C₄⋊C₈

—

136

C₄⋊C₈

—

137

C₄⋊C₈

—

138

C₄⋊C₈

C₂×#261

yes

—

139

C₂²⋊C₈

—

140

C₄×C₈

—

141

C₄×C₈

—

142

C₄×C₈

—

143

C₄×C₈

C₂×#261

yes

—

144

C₂×C₁₃

#2906

yes

characteristic

145

C₂×C₁₃

C₂²×#107

yes

characteristic

146

C₂×C₁₃

#2916

yes

characteristic

147

D₁₃

—

148

#41

—

149

#41

—

150

C₂×Q₈

C₄⋊#261

yes

characteristic

151

C₂×D₄

—

152

C₂²×C₄

—

153

C₂²×C₄

—

154

C₂²×C₄

—

155

M₁₆

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

C₂²×#261

yes

characteristic

164

C₂×C₈

C₂²×#261

yes

—

165

C₄⋊C₄

—

166

C₄⋊C₄

C₄⋊#261

yes

—

167

C₂²⋊C₄

—

168

C₄²

—

169

C₄²

—

170

C₄²

—

171

C₄²

C₂²×#261

yes

—

172

C₁₃

#17828

yes

characteristic, complemented, Sylow

173

C₂³

—

174

Q₈

#88102

yes

—

175

D₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

C₂×C₄⋊#261

yes

—

185

C₂×C₄

C₂×C₄⋊#261

yes

—

186

C₂×C₄

C₂³×#261

yes

characteristic

187

C₈

—

188

C₈

—

189

C₈

—

190

C₈

—

191

C₈

—

192

C₈

—

193

C₂²

—

194

C₂²

—

195

C₂²

—

196

C₂²

C₂²×C₄⋊#261

yes

characteristic

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

C₂×#88102

yes

—

203

C₄

#11719625

yes

characteristic

204

C₄

D₁₃⋊#131

yes

characteristic

205

C₂

—

206

C₂

#13131484

yes

characteristic

207

C₂

C₂²×#88102

yes

characteristic

208

C₂

#13131497

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#261
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₈×Q₈×C₁₃	C₄
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂×#6131	D₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	C₈×Q₈	#261
hypocenter	C₁₃	#17828
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×C₄⋊#261

commutator

C₂×C₄×C₁₃

C₂³×C₄

Frattini

C₂×C₄

C₂³×#261

Fitting

C₈×Q₈×C₁₃

C₄

socle

C₂²×C₁₃

C₂²×C₄⋊C₄

prime radical

C₂×#6131

D₄

cosocle

C₄×Dic₁₃

C₂⁴

C₈×Q₈

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
2. #13130406	C₂	semidirect
3. #13130395	C₂	semidirect
4. D₁₃⋊#335	C₂	semidirect
93. #1055	#131	semidirect
172. C₁₃	#17828	semidirect

H ≅ G/N

type

2. #13130406

C₂

semidirect

3. #13130395

C₂

semidirect

4. D₁₃⋊#335

C₂

semidirect

93. #1055

#131

semidirect

172. C₁₃

#17828

semidirect

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

subgroup	index
#13131482	2
#13130406	2
#13130395	2
D₁₃⋊#335	2
C₈×#6131	2
#17828	13

subgroup

index

D₁₃⋊#335

Subgroups — #16795096

Conjugacy classes of subgroups

Complements

Maximal subgroups