Group #16018343 — subgroups

Subgroups — #16018343

Order 3328.

826 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 219 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×#11718565	C₂	1	4	yes	maximal, complemented
2	#13125264	C₂	1	2	yes	maximal, complemented
3	#13117107	C₂	1	4	yes	maximal, complemented
4	#13117102	C₂	1	4	yes	maximal, complemented
5	C₂×C₁₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#88158	—	2	2	no	—
7	C₂×#88158	C₂²	1	4	yes	—
8	#11719182	—	2	2	no	—
9	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₀₄	C₂²	1	4	yes	—
12	#11718565	—	2	16	no	—
13	C₂×#88040	C₂²	1	8	yes	—
14	C₂×D₂₀₈	C₂²	1	4	yes	—
15	#11718360	C₂²	1	2	yes	—
16	#11718357	—	2	2	no	—
17	#11718357	C₂²	1	4	yes	—
18	#11718356	C₂²	1	2	yes	—
19	#11718354	—	2	2	no	—
20	C₁₃×#346	C₂²	1	4	yes	—
21	C₂×#6127	—	2	4	no	—
22	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
25	#88184	—	4	4	no	—
26	C₂×Dic₅₂	—	2	2	no	—
27	C₂×Dic₅₂	C₂³	1	4	yes	—
28	#88158	—	2	16	no	—
29	#88158	—	4	8	no	—
30	C₂×D₁₀₄	—	2	2	no	—
31	C₂×D₁₀₄	C₂³	1	4	yes	—
32	C₂×#6095	—	2	4	no	—
33	C₂×#6095	—	2	4	no	—
34	#88129	—	2	4	no	—
35	#88128	—	2	2	no	—
36	C₄×D₅₂	—	4	4	no	—
37	#88123	—	2	2	no	—
38	C₂×C₁₃×C₁₆	D₄	1	8	yes	complemented
39	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
40	C₄×C₈×C₁₃	D₄	1	2	yes	—
41	C₄×C₈×C₁₃	C₂³	1	2	yes	—
42	Dic₁₀₄	—	2	8	no	—
43	#88040	—	2	16	no	—
44	D₂₀₈	—	2	8	no	—
45	#33716	—	13	1	no	maximal, Sylow
46	C₂²×C₄×C₁₃	D₈	1	2	yes	—
47	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₂²⋊D₁₃	—	4	4	no	—
49	#6127	—	4	8	no	—
50	#6127	—	4	8	no	—
51	C₂×D₅₂	—	2	4	no	—
52	C₂×D₅₂	—	4	4	no	—
53	C₂×C₄×D₁₃	—	4	4	no	—
54	C₂×Dic₂₆	—	2	4	no	—
55	C₂×C₈×C₁₃	D₈	1	4	yes	—
56	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
58	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
60	C₄²×C₁₃	D₈	1	2	yes	—
61	C₄²×C₁₃	C₂×D₄	1	2	yes	—
62	#6103	—	4	8	no	—
63	C₄⋊Dic₁₃	—	4	4	no	—
64	Dic₅₂	—	2	8	no	—
65	Dic₅₂	—	4	4	no	—
66	D₁₀₄	—	2	8	no	—
67	D₁₀₄	—	4	4	no	—
68	#6095	—	4	8	no	—
69	#6095	—	4	8	no	—
70	C₁₃×C₁₆	C₂×D₄	1	8	yes	—
71	C₂×#508	—	13	4	no	—
72	#3147	—	13	2	no	—
73	#2251	—	13	4	no	—
74	#2250	—	13	4	no	—
75	C₂×#346	D₁₃	1	1	yes	characteristic, complemented
76	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
77	C₂²×D₁₃	—	4	4	no	—
78	C₂×C₄×C₁₃	—	4	2	no	—
79	C₂×C₄×C₁₃	#127	1	2	yes	—
80	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
81	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
82	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
84	C₂²⋊D₁₃	—	8	8	no	—
85	C₂×Dic₁₃	—	4	4	no	—
86	D₅₂	—	4	4	no	—
87	D₅₂	—	4	4	no	—
88	D₅₂	—	8	8	no	—
89	C₄×D₁₃	—	8	8	no	—
90	Dic₂₆	—	4	4	no	—
91	Dic₂₆	—	4	4	no	—
92	C₈×C₁₃	#127	1	2	yes	—
93	C₈×C₁₃	C₂×D₈	1	4	yes	—
94	C₈×C₁₃	C₂²×D₄	1	2	yes	—
95	C₂×#135	—	13	4	no	—
96	C₂×#135	—	26	2	no	—
97	#532	—	26	2	no	—
98	#508	—	26	16	no	—
99	C₂×Q₃₂	—	13	4	no	—
100	C₂×SD₃₂	—	13	8	no	—
101	C₂×D₁₆	—	13	4	no	—
102	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
103	#495	—	13	4	no	—
104	#495	—	26	2	no	—
105	#494	—	13	2	no	—
106	#493	—	13	2	no	—
107	#492	—	26	2	no	—
108	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
109	#346	C₂×D₁₃	1	4	yes	—
110	C₂²×C₁₃	—	2	2	no	—
111	C₂²×C₁₃	#493	1	2	yes	—
112	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
113	C₂×D₁₃	—	8	4	no	—
114	C₂×D₁₃	—	8	8	no	—
115	C₄×C₁₃	—	2	4	no	—
116	C₄×C₁₃	#493	1	2	yes	—
117	C₄×C₁₃	C₂×#127	1	1	yes	characteristic
118	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
119	Dic₁₃	—	8	4	no	—
120	C₂×#41	—	26	4	no	—
121	#135	—	26	16	no	—
122	#135	—	52	8	no	—
123	C₂×Q₁₆	—	13	4	no	—
124	C₂×Q₁₆	—	26	2	no	—
125	C₂×SD₁₆	—	26	4	no	—
126	C₂×SD₁₆	—	26	4	no	—
127	C₂×D₈	—	13	4	no	—
128	C₂×D₈	—	26	2	no	—
129	C₂²×C₈	D₅₂	1	1	yes	characteristic
130	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
131	C₄⋊Q₈	—	26	2	no	—
132	#127	—	26	2	no	—
133	#124	—	26	4	no	—
134	#121	—	52	4	no	—
135	C₄×D₄	—	52	4	no	—
136	C₂×C₄²	D₅₂	1	1	yes	characteristic
137	Q₃₂	—	26	8	no	—
138	SD₃₂	—	26	16	no	—
139	D₁₆	—	26	8	no	—
140	C₂×C₁₆	D₅₂	1	8	yes	complemented
141	C₂×C₁₆	C₂²×D₁₃	1	4	yes	—
142	C₄×C₈	D₅₂	1	2	yes	—
143	C₄×C₈	C₂²×D₁₃	1	2	yes	—
144	C₂×C₁₃	—	2	2	no	—
145	C₂×C₁₃	C₂×#493	1	1	yes	characteristic
146	C₂×C₁₃	#3414	1	2	yes	—
147	D₁₃	—	16	4	no	—
148	#41	—	52	8	no	—
149	#41	—	52	8	no	—
150	C₂×Q₈	—	26	4	no	—
151	C₂×D₄	—	26	4	no	—
152	C₂×D₄	—	52	4	no	—
153	C₂×D₄	—	52	4	no	—
154	C₂²×C₄	—	52	4	no	—
155	C₂²×C₄	D₁₀₄	1	2	yes	—
156	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
157	Q₁₆	—	26	8	no	—
158	Q₁₆	—	52	4	no	—
159	SD₁₆	—	52	8	no	—
160	SD₁₆	—	52	8	no	—
161	D₈	—	26	8	no	—
162	D₈	—	52	4	no	—
163	C₂×C₈	D₁₀₄	1	4	yes	—
164	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
165	C₂×C₈	C₂×D₅₂	1	4	yes	—
166	C₂×C₈	C₂×D₅₂	1	2	yes	—
167	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
168	C₄⋊C₄	—	52	4	no	—
169	C₂²⋊C₄	—	52	8	no	—
170	C₄²	D₁₀₄	1	2	yes	—
171	C₄²	C₂×D₅₂	1	2	yes	—
172	C₁₆	C₂×D₅₂	1	8	yes	—
173	C₁₃	#33716	1	1	yes	characteristic, complemented, Sylow
174	C₂³	—	52	4	no	—
175	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
176	Q₈	—	52	4	no	—
177	Q₈	—	52	4	no	—
178	D₄	—	52	4	no	—
179	D₄	—	52	4	no	—
180	D₄	—	104	8	no	—
181	D₄	—	104	8	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	52	4	no	—
184	C₂×C₄	—	104	8	no	—
185	C₂×C₄	#88128	1	2	yes	—
186	C₂×C₄	C₂×D₁₀₄	1	4	yes	—
187	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
188	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
189	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
190	C₈	#88128	1	2	yes	—
191	C₈	C₂×D₁₀₄	1	4	yes	—
192	C₈	C₂²×D₅₂	1	2	yes	—
193	C₂²	—	2	2	no	—
194	C₂²	—	104	8	no	—
195	C₂²	—	104	4	no	—
196	C₂²	#11718356	1	2	yes	—
197	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
198	C₄	—	2	4	no	—
199	C₄	—	104	4	no	—
200	C₄	#11718356	1	2	yes	—
201	C₄	C₂×#88128	1	1	yes	characteristic
202	C₄	C₂²×D₁₀₄	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	—	208	4	no	—
205	C₂	C₂×#11718356	1	1	yes	characteristic
206	C₂	#13127972	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718565

C₂

yes

maximal, complemented

#13125264

C₂

yes

maximal, complemented

#13117107

C₂

yes

maximal, complemented

#13117102

C₂

yes

maximal, complemented

C₂×C₁₃×#346

C₂

yes

maximal, characteristic, complemented

C₂×#88158

—

C₂×#88158

C₂²

yes

—

#11719182

—

C₂²×C₁₃×C₁₆

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×Dic₁₀₄

C₂²

yes

—

#11718565

—

C₂×#88040

C₂²

yes

—

C₂×D₂₀₈

C₂²

yes

—

#11718360

C₂²

yes

—

#11718357

—

#11718357

C₂²

yes

—

#11718356

C₂²

yes

—

#11718354

—

C₁₃×#346

C₂²

yes

—

C₂×#6127

—

C₂²×C₈×C₁₃

D₄

yes

characteristic

C₂²×C₈×C₁₃

C₂³

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

characteristic

#88184

—

C₂×Dic₅₂

—

C₂×Dic₅₂

C₂³

yes

—

#88158

—

#88158

—

C₂×D₁₀₄

—

C₂×D₁₀₄

C₂³

yes

—

—

—

—

—

C₄×D₅₂

—

#88123

—

C₂×C₁₃×C₁₆

D₄

yes

complemented

C₂×C₁₃×C₁₆

C₂³

yes

—

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

C₂³

yes

—

Dic₁₀₄

—

#88040

—

D₂₀₈

—

#33716

—

maximal, Sylow

C₂²×C₄×C₁₃

D₈

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₂²⋊D₁₃

—

#6127

—

#6127

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×C₄×D₁₃

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₂×C₈×C₁₃

C₂⁴

yes

characteristic

C₄²×C₁₃

D₈

yes

—

C₄²×C₁₃

C₂×D₄

yes

—

#6103

—

C₄⋊Dic₁₃

—

Dic₅₂

—

Dic₅₂

—

D₁₀₄

—

D₁₀₄

—

#6095

—

#6095

—

C₁₃×C₁₆

C₂×D₄

yes

—

—

—

—

—

D₁₃

yes

characteristic, complemented

C₂³×C₁₃

C₂×D₈

yes

characteristic

C₂²×D₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#127

yes

—

C₂×C₄×C₁₃

C₂×D₈

yes

—

C₂×C₄×C₁₃

C₂×D₈

yes

—

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

C₂²×D₄

yes

characteristic

C₂²⋊D₁₃

—

C₂×Dic₁₃

—

D₅₂

—

D₅₂

—

D₅₂

—

C₄×D₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

#127

yes

—

C₈×C₁₃

C₂×D₈

yes

—

C₈×C₁₃

C₂²×D₄

yes

—

—

—

—

—

C₂×Q₃₂

—

100

C₂×SD₃₂

—

101

C₂×D₁₆

—

102

C₂²×C₁₆

C₂×D₁₃

yes

—

103

#495

—

104

#495

—

105

#494

—

106

#493

—

107

#492

—

108

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

109

#346

C₂×D₁₃

yes

—

110

C₂²×C₁₃

—

111

C₂²×C₁₃

#493

yes

—

112

C₂²×C₁₃

C₂²×D₈

yes

characteristic

113

C₂×D₁₃

—

114

C₂×D₁₃

—

115

C₄×C₁₃

—

116

C₄×C₁₃

#493

yes

—

117

C₄×C₁₃

C₂×#127

yes

characteristic

118

C₄×C₁₃

C₂²×D₈

yes

characteristic

119

Dic₁₃

—

120

C₂×#41

—

121

#135

—

122

#135

—

123

C₂×Q₁₆

—

124

C₂×Q₁₆

—

125

C₂×SD₁₆

—

126

C₂×SD₁₆

—

127

C₂×D₈

—

128

C₂×D₈

—

129

C₂²×C₈

D₅₂

yes

characteristic

130

C₂²×C₈

C₂²×D₁₃

yes

characteristic

131

C₄⋊Q₈

—

132

#127

—

133

#124

—

134

#121

—

135

C₄×D₄

—

136

C₂×C₄²

D₅₂

yes

characteristic

137

Q₃₂

—

138

SD₃₂

—

139

D₁₆

—

140

C₂×C₁₆

D₅₂

yes

complemented

141

C₂×C₁₆

C₂²×D₁₃

yes

—

142

C₄×C₈

D₅₂

yes

—

143

C₄×C₈

C₂²×D₁₃

yes

—

144

C₂×C₁₃

—

145

C₂×C₁₃

C₂×#493

yes

characteristic

146

C₂×C₁₃

#3414

yes

—

147

D₁₃

—

148

#41

—

149

#41

—

150

C₂×Q₈

—

151

C₂×D₄

—

152

C₂×D₄

—

153

C₂×D₄

—

154

C₂²×C₄

—

155

C₂²×C₄

D₁₀₄

yes

—

156

C₂²×C₄

C₂×D₅₂

yes

characteristic

157

Q₁₆

—

158

Q₁₆

—

159

SD₁₆

—

160

SD₁₆

—

161

D₈

—

162

D₈

—

163

C₂×C₈

D₁₀₄

yes

—

164

C₂×C₈

C₂×D₅₂

yes

characteristic

165

C₂×C₈

C₂×D₅₂

yes

—

166

C₂×C₈

C₂×D₅₂

yes

—

167

C₂×C₈

C₂³×D₁₃

yes

characteristic

168

C₄⋊C₄

—

169

C₂²⋊C₄

—

170

C₄²

D₁₀₄

yes

—

171

C₄²

C₂×D₅₂

yes

—

172

C₁₆

C₂×D₅₂

yes

—

173

C₁₃

#33716

yes

characteristic, complemented, Sylow

174

C₂³

—

175

C₂³

C₂×D₁₀₄

yes

characteristic

176

Q₈

—

177

Q₈

—

178

D₄

—

179

D₄

—

180

D₄

—

104

—

181

D₄

—

104

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

104

—

185

C₂×C₄

#88128

yes

—

186

C₂×C₄

C₂×D₁₀₄

yes

—

187

C₂×C₄

C₂×D₁₀₄

yes

—

188

C₂×C₄

C₂×D₁₀₄

yes

characteristic

189

C₂×C₄

C₂²×D₅₂

yes

characteristic

190

C₈

#88128

yes

—

191

C₈

C₂×D₁₀₄

yes

—

192

C₈

C₂²×D₅₂

yes

—

193

C₂²

—

194

C₂²

—

104

—

195

C₂²

—

104

—

196

C₂²

#11718356

yes

—

197

C₂²

C₂²×D₁₀₄

yes

characteristic

198

C₄

—

199

C₄

—

104

—

200

C₄

#11718356

yes

—

201

C₄

C₂×#88128

yes

characteristic

202

C₄

C₂²×D₁₀₄

yes

characteristic

203

C₂

—

204

C₂

—

208

—

205

C₂

C₂×#11718356

yes

characteristic

206

C₂

#13127972

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#346	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	C₂×#346	D₁₃
hypocenter	C₁₃	#33716
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

C₂×D₁₀₄

commutator, cosocle

C₂×C₈×C₁₃

C₂⁴

Frattini

C₂×C₈

C₂³×D₁₃

Fitting

C₂×C₁₃×#346

C₂

socle

C₂²×C₁₃

C₂²×D₈

D₁₃

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
1. C₂×#11718565	C₂	semidirect
2. #13125264	C₂	semidirect
3. #13117107	C₂	semidirect
4. #13117102	C₂	semidirect
5. C₂×C₁₃×#346	C₂	semidirect
38. C₂×C₁₃×C₁₆	D₄	semidirect
75. C₂×#346	D₁₃	semidirect
140. C₂×C₁₆	D₅₂	semidirect
173. C₁₃	#33716	semidirect

H ≅ G/N

type

1. C₂×#11718565

C₂

semidirect

2. #13125264

C₂

semidirect

3. #13117107

C₂

semidirect

4. #13117102

C₂

semidirect

5. C₂×C₁₃×#346

C₂

semidirect

38. C₂×C₁₃×C₁₆

D₄

semidirect

75. C₂×#346

D₁₃

semidirect

140. C₂×C₁₆

D₅₂

semidirect

173. C₁₃

#33716

semidirect

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

subgroup	index
C₂×#11718565	2
#13125264	2
#13117107	2
#13117102	2
C₂×C₁₃×#346	2
#33716	13

subgroup

index

C₂×C₁₃×#346

Subgroups — #16018343

Conjugacy classes of subgroups

Complements

Maximal subgroups