Group #16786947 — subgroups

Subgroups — #16786947

Order 3328.

418 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 98 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	#13131197	C₂	1	2	yes	maximal, complemented
2	#13129757	C₂	1	4	yes	maximal, complemented
3	C₂×#11718495	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₈×D₁₃	C₄	1	2	yes	—
5	C₂×C₄×D₅₂	C₄	1	1	yes	characteristic
6	C₂×C₁₃×C₄⋊C₈	C₄	1	1	yes	characteristic
7	C₂×#88060	C₂²	1	2	yes	—
8	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
9	#11718495	—	2	2	no	—
10	#11718495	—	2	2	no	—
11	C₂²⋊#6092	C₂²	1	4	yes	—
12	#11718231	—	2	2	no	—
13	C₄⋊#6092	—	2	2	no	—
14	C₂²×D₅₂	C₈	1	1	yes	characteristic
15	C₂²×C₄×D₁₃	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂×#6103	C₈	1	2	yes	—
19	C₂×C₄⋊Dic₁₃	C₈	1	1	yes	characteristic
20	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
21	C₂²×#1055	C₂³	1	1	yes	characteristic
22	C₂×C₈×D₁₃	—	2	2	no	—
23	C₂×C₈×D₁₃	—	2	2	no	—
24	C₂×C₈×D₁₃	—	4	4	no	—
25	C₄×D₅₂	—	2	2	no	—
26	C₄×D₅₂	—	2	2	no	—
27	C₂×#6092	—	2	4	no	—
28	C₁₃×C₄⋊C₈	—	2	2	no	—
29	#88060	—	2	4	no	—
30	C₄×#1055	D₄	1	2	yes	—
31	C₄×#1055	D₄	1	2	yes	—
32	#11488	—	13	1	no	maximal, Sylow
33	C₂³×D₁₃	C₂×C₈	1	2	yes	—
34	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
38	C₂×D₅₂	—	2	4	no	—
39	C₂×D₅₂	—	2	2	no	—
40	C₂×C₄×D₁₃	—	2	2	no	—
41	C₂×C₄×D₁₃	—	2	2	no	—
42	C₂×C₄×D₁₃	—	4	2	no	—
43	C₂×C₄×D₁₃	—	4	2	no	—
44	C₂×C₈×C₁₃	—	2	2	no	—
45	C₂×C₈×C₁₃	—	4	2	no	—
46	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
47	C₄²×C₁₃	M₁₆	1	2	yes	—
48	#6103	—	2	4	no	—
49	C₄⋊Dic₁₃	—	2	2	no	—
50	C₂×#1055	—	2	2	no	—
51	C₂×#1055	—	4	1	no	—
52	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×#1055	C₂×D₄	1	2	yes	—
55	C₈×D₁₃	—	4	4	no	—
56	C₈×D₁₃	—	4	4	no	—
57	#6092	—	4	4	no	—
58	C₂×C₈×D₄	—	13	1	no	—
59	#2163	—	13	2	no	—
60	#1314	—	13	4	no	—
61	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
62	C₂²×D₁₃	—	2	2	no	—
63	C₂²×D₁₃	—	2	2	no	—
64	C₂²×D₁₃	—	4	2	no	—
65	C₂²×D₁₃	—	4	2	no	—
66	C₂²×D₁₃	M₃₂	1	4	yes	—
67	C₂×C₄×C₁₃	—	2	2	no	—
68	C₂×C₄×C₁₃	—	2	1	no	—
69	C₂×C₄×C₁₃	—	2	1	no	—
70	C₂×C₄×C₁₃	—	4	1	no	—
71	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
72	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
73	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
75	C₂×Dic₁₃	—	2	2	no	—
76	C₂×Dic₁₃	—	4	2	no	—
77	D₅₂	—	2	8	no	—
78	C₄×D₁₃	—	4	4	no	—
79	C₄×D₁₃	—	4	2	no	—
80	C₄×D₁₃	—	4	2	no	—
81	C₈×C₁₃	—	4	2	no	—
82	#1055	—	2	2	no	—
83	#1055	—	2	1	no	—
84	#1055	#100	1	2	yes	—
85	C₂³×C₈	—	13	2	no	—
86	C₂×C₄×D₄	—	13	1	no	—
87	C₈×D₄	—	26	2	no	—
88	C₈×D₄	—	26	2	no	—
89	C₂×C₄⋊C₈	#261	1	1	yes	characteristic
90	C₂×C₂²⋊C₈	—	13	2	no	—
91	C₂×C₄×C₈	—	13	1	no	—
92	C₄⋊C₁₆	—	26	2	no	—
93	C₂²⋊C₁₆	—	13	4	no	—
94	#346	—	26	2	no	—
95	C₂²×C₁₃	—	2	1	no	—
96	C₂²×C₁₃	—	2	1	no	—
97	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
98	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
99	C₂×D₁₃	—	2	4	no	—
100	C₂×D₁₃	—	4	4	no	—
101	C₂×D₁₃	—	4	2	no	—
102	C₂×D₁₃	—	4	2	no	—
103	C₂×D₁₃	—	4	4	no	—
104	C₄×C₁₃	—	2	2	no	—
105	C₄×C₁₃	—	2	2	no	—
106	C₄×C₁₃	—	2	1	no	—
107	C₄×C₁₃	#410	1	1	yes	characteristic
108	C₄×C₁₃	C₂×#100	1	1	yes	characteristic
109	Dic₁₃	—	4	2	no	—
110	C₂²×D₄	—	13	1	no	—
111	C₂³×C₄	—	13	2	no	—
112	C₂²×C₈	—	13	1	no	—
113	C₂²×C₈	—	13	1	no	—
114	C₂²×C₈	—	26	2	no	—
115	C₂²×C₈	—	26	2	no	—
116	C₂²×C₈	—	52	4	no	—
117	C₂²×C₈	C₂×#261	1	2	yes	—
118	C₄×D₄	—	26	2	no	—
119	C₄×D₄	—	26	2	no	—
120	C₂×C₄⋊C₄	—	13	1	no	—
121	C₂×C₂²⋊C₄	—	13	2	no	—
122	C₂×C₄²	C₂×#261	1	1	yes	characteristic
123	C₂×C₁₆	—	26	4	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₂²⋊C₈	—	26	4	no	—
126	C₄×C₈	—	13	2	no	—
127	C₄×C₈	—	13	2	no	—
128	C₂×C₁₃	—	2	2	no	—
129	C₂×C₁₃	#1460	1	1	yes	characteristic
130	C₂×C₁₃	#2112	1	1	yes	characteristic
131	C₂×C₁₃	#2115	1	1	yes	characteristic
132	D₁₃	—	4	4	no	—
133	C₂⁴	—	13	2	no	—
134	C₂×D₄	—	26	4	no	—
135	C₂×D₄	—	26	2	no	—
136	C₂²×C₄	—	13	2	no	—
137	C₂²×C₄	—	26	2	no	—
138	C₂²×C₄	—	26	2	no	—
139	C₂²×C₄	—	52	2	no	—
140	C₂²×C₄	—	52	2	no	—
141	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
142	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
143	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	4	2	no	—
146	C₂×C₈	—	13	1	no	—
147	C₂×C₈	—	13	2	no	—
148	C₂×C₈	—	13	1	no	—
149	C₂×C₈	—	26	2	no	—
150	C₂×C₈	—	52	4	no	—
151	C₂×C₈	—	52	4	no	—
152	C₂×C₈	—	52	1	no	—
153	C₄⋊C₄	—	26	2	no	—
154	C₂²⋊C₄	—	26	4	no	—
155	C₄²	#6118	1	2	yes	—
156	C₄²	C₂²⋊#261	1	2	yes	—
157	C₁₆	—	52	4	no	—
158	C₁₃	#11488	1	1	yes	characteristic, complemented, Sylow
159	C₂³	—	13	4	no	—
160	C₂³	—	26	2	no	—
161	C₂³	—	26	2	no	—
162	C₂³	—	52	2	no	—
163	C₂³	—	52	2	no	—
164	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
165	D₄	—	26	8	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	—	26	2	no	—
171	C₂×C₄	—	52	4	no	—
172	C₂×C₄	—	52	2	no	—
173	C₂×C₄	—	52	2	no	—
174	C₂×C₄	—	52	2	no	—
175	C₂×C₄	#88111	1	2	yes	—
176	C₂×C₄	#88111	1	2	yes	—
177	C₂×C₄	C₂×#6118	1	1	yes	characteristic
178	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
179	C₈	—	4	2	no	—
180	C₈	—	13	2	no	—
181	C₈	—	26	1	no	—
182	C₈	—	26	2	no	—
183	C₂²	—	2	1	no	—
184	C₂²	—	2	1	no	—
185	C₂²	—	26	4	no	—
186	C₂²	—	52	4	no	—
187	C₂²	—	52	4	no	—
188	C₂²	—	52	2	no	—
189	C₂²	—	52	2	no	—
190	C₂²	D₁₃⋊M₃₂	1	2	yes	—
191	C₂²	C₂×#88111	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	2	2	no	—
194	C₄	—	2	1	no	—
195	C₄	—	52	2	no	—
196	C₄	D₁₃⋊#100	1	1	yes	characteristic
197	C₄	#11719156	1	1	yes	characteristic
198	C₂	—	2	2	no	—
199	C₂	—	52	4	no	—
200	C₂	#13130077	1	1	yes	characteristic
201	C₂	#13131065	1	1	yes	characteristic
202	C₂	#13131074	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13131197

C₂

yes

maximal, complemented

#13129757

C₂

yes

maximal, complemented

C₂×#11718495

C₂

yes

maximal, characteristic

C₂²×C₈×D₁₃

C₄

yes

—

C₂×C₄×D₅₂

C₄

yes

characteristic

C₂×C₁₃×C₄⋊C₈

C₄

yes

characteristic

C₂×#88060

C₂²

yes

—

C₂×C₄×#1055

C₂²

yes

characteristic

#11718495

—

#11718495

—

C₂²⋊#6092

C₂²

yes

—

#11718231

—

C₄⋊#6092

—

C₂²×D₅₂

C₈

yes

characteristic

C₂²×C₄×D₁₃

C₂×C₄

yes

—

C₂²×C₈×C₁₃

C₂×C₄

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×#6103

C₈

yes

—

C₂×C₄⋊Dic₁₃

C₈

yes

characteristic

C₂²×#1055

C₂×C₄

yes

characteristic

C₂²×#1055

C₂³

yes

characteristic

C₂×C₈×D₁₃

—

C₂×C₈×D₁₃

—

C₂×C₈×D₁₃

—

C₄×D₅₂

—

C₄×D₅₂

—

C₂×#6092

—

C₁₃×C₄⋊C₈

—

#88060

—

C₄×#1055

D₄

yes

—

C₄×#1055

D₄

yes

—

#11488

—

maximal, Sylow

C₂³×D₁₃

C₂×C₈

yes

—

C₂²×C₄×C₁₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²×Dic₁₃

C₂×C₈

yes

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

C₂²⋊C₄

yes

—

C₄²×C₁₃

M₁₆

yes

—

#6103

—

C₄⋊Dic₁₃

—

C₂×#1055

—

C₂×#1055

—

C₂×#1055

C₂²⋊C₄

yes

characteristic

C₂×#1055

C₂²⋊C₄

yes

characteristic

C₂×#1055

C₂×D₄

yes

—

C₈×D₁₃

—

C₈×D₁₃

—

#6092

—

C₂×C₈×D₄

—

#2163

—

#1314

—

C₂³×C₁₃

C₂²×C₈

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

M₃₂

yes

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₈

yes

—

C₂×C₄×C₁₃

C₂²⋊C₈

yes

—

C₂×C₄×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×M₁₆

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

D₅₂

—

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

—

C₈×C₁₃

—

—

—

yes

—

C₂³×C₈

—

C₂×C₄×D₄

—

C₈×D₄

—

C₈×D₄

—

C₂×C₄⋊C₈

#261

yes

characteristic

C₂×C₂²⋊C₈

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₁₃

C₂×M₃₂

yes

—

C₂×D₁₃

—

100

C₂×D₁₃

—

101

C₂×D₁₃

—

102

C₂×D₁₃

—

103

C₂×D₁₃

—

104

C₄×C₁₃

—

105

C₄×C₁₃

—

106

C₄×C₁₃

—

107

C₄×C₁₃

#410

yes

characteristic

108

C₄×C₁₃

C₂×#100

yes

characteristic

109

Dic₁₃

—

110

C₂²×D₄

—

111

C₂³×C₄

—

112

C₂²×C₈

—

113

C₂²×C₈

—

114

C₂²×C₈

—

115

C₂²×C₈

—

116

C₂²×C₈

—

117

C₂²×C₈

C₂×#261

yes

—

118

C₄×D₄

—

119

C₄×D₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₂²⋊C₄

—

122

C₂×C₄²

C₂×#261

yes

characteristic

123

C₂×C₁₆

—

124

C₄⋊C₈

—

125

C₂²⋊C₈

—

126

C₄×C₈

—

127

C₄×C₈

—

128

C₂×C₁₃

—

129

C₂×C₁₃

#1460

yes

characteristic

130

C₂×C₁₃

#2112

yes

characteristic

131

C₂×C₁₃

#2115

yes

characteristic

132

D₁₃

—

133

C₂⁴

—

134

C₂×D₄

—

135

C₂×D₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

D₁₃⋊C₈

yes

characteristic

142

C₂²×C₄

D₁₃⋊C₈

yes

characteristic

143

C₂²×C₄

C₂²×#261

yes

characteristic

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₄⋊C₄

—

154

C₂²⋊C₄

—

155

C₄²

#6118

yes

—

156

C₄²

C₂²⋊#261

yes

—

157

C₁₆

—

158

C₁₃

#11488

yes

characteristic, complemented, Sylow

159

C₂³

—

160

C₂³

—

161

C₂³

—

162

C₂³

—

163

C₂³

—

164

C₂³

C₂×D₁₃⋊C₈

yes

characteristic

165

D₄

—

166

C₂×C₄

—

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

#88111

yes

—

176

C₂×C₄

#88111

yes

—

177

C₂×C₄

C₂×#6118

yes

characteristic

178

C₂×C₄

C₂×C₂²⋊#261

yes

characteristic

179

C₈

—

180

C₈

—

181

C₈

—

182

C₈

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

190

C₂²

D₁₃⋊M₃₂

yes

—

191

C₂²

C₂×#88111

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

D₁₃⋊#100

yes

characteristic

197

C₄

#11719156

yes

characteristic

198

C₂

—

199

C₂

—

200

C₂

#13130077

yes

characteristic

201

C₂

#13131065

yes

characteristic

202

C₂

#13131074

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊#261
commutator	C₂³×C₁₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₁₃×C₄⋊C₈	C₄
socle	C₂²×C₁₃	C₂×C₂²⋊C₈
prime radical	C₂²×D₅₂	C₈
cosocle	C₂²×#1055	C₂³
hypercenter	C₂×C₄⋊C₈	#261
hypocenter	C₁₃	#11488
LCS	G ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊#261

commutator

C₂³×C₁₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#261

Fitting

C₂×C₁₃×C₄⋊C₈

C₄

socle

C₂²×C₁₃

C₂×C₂²⋊C₈

prime radical

C₂²×D₅₂

C₈

cosocle

C₂²×#1055

C₂³

C₂×C₄⋊C₈

C₁₃

G ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
1. #13131197	C₂	semidirect
2. #13129757	C₂	semidirect
158. C₁₃	#11488	semidirect

H ≅ G/N

type

1. #13131197

C₂

semidirect

2. #13129757

C₂

semidirect

158. C₁₃

#11488

semidirect

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

subgroup	index
#13131197	2
#13129757	2
C₂×#11718495	2
#11488	13

subgroup

index

Subgroups — #16786947

Conjugacy classes of subgroups

Complements

Maximal subgroups