Group #16790862 — subgroups

Subgroups — #16790862

Order 3328.

450 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 124 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	#13131168	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11718270	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×#6092	C₂	1	1	yes	maximal, characteristic, complemented
4	#13123547	C₂	1	1	yes	maximal, characteristic
5	C₂×#88161	C₂²	1	2	yes	—
6	#11719182	C₄	1	1	yes	characteristic
7	C₂×#88106	C₂²	1	2	yes	—
8	C₂²×#6092	C₂²	1	2	yes	—
9	C₁₃×#423	C₄	1	1	yes	characteristic
10	#11718790	C₄	1	2	yes	—
11	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
12	#11718499	—	2	2	no	—
13	#11718495	—	2	2	no	—
14	#11718270	—	2	8	no	—
15	C₄⋊#6092	—	2	2	no	—
16	C₄×#6092	—	2	2	no	—
17	C₂×#6127	C₂×C₄	1	2	yes	—
18	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
20	#88184	—	2	2	no	—
21	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
22	C₂²×#1055	C₂³	1	1	yes	characteristic
23	#88161	—	2	4	no	—
24	#88161	—	2	4	no	—
25	C₂×#6094	—	2	2	no	—
26	C₂×C₈×D₁₃	—	2	2	no	—
27	#88129	C₈	1	2	yes	—
28	#88128	C₈	1	1	yes	characteristic
29	C₄×D₅₂	—	2	2	no	—
30	#88123	C₈	1	1	yes	characteristic
31	#88106	—	2	4	no	—
32	C₂×#6092	—	2	2	no	—
33	C₂×#6092	—	2	2	no	—
34	C₂×#6092	—	2	2	no	—
35	C₂×#6092	D₄	1	4	yes	complemented
36	C₁₃×C₄⋊C₈	—	2	2	no	—
37	#88060	—	2	4	no	—
38	C₄×#1055	D₄	1	2	yes	—
39	C₄×#1055	D₄	1	2	yes	—
40	#13191	—	13	1	no	maximal, Sylow
41	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
42	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₁₃	—	2	2	no	—
44	#6127	—	2	4	no	—
45	#6127	—	2	4	no	—
46	C₂×D₅₂	—	2	2	no	—
47	C₂×D₅₂	C₂×C₈	1	2	yes	—
48	C₂×C₄×D₁₃	—	2	2	no	—
49	C₂×Dic₂₆	C₂×C₈	1	2	yes	—
50	C₁₃×M₁₆	—	2	4	no	—
51	C₂×C₈×C₁₃	—	2	2	no	—
52	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
53	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
54	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
55	#6103	—	2	4	no	—
56	C₄⋊Dic₁₃	—	2	2	no	—
57	C₂×#1055	—	2	2	no	—
58	C₂×#1055	C₂²⋊C₄	1	2	yes	—
59	C₂×#1055	C₂×D₄	1	1	yes	characteristic
60	C₂×#1055	C₂×D₄	1	2	yes	—
61	C₂×#1055	C₂×D₄	1	1	yes	characteristic
62	C₂×#1055	#41	1	2	yes	—
63	#6094	—	4	4	no	—
64	C₈×D₁₃	—	4	2	no	—
65	C₈×D₁₃	—	4	2	no	—
66	#6092	—	2	4	no	—
67	#6092	—	4	2	no	—
68	#2931	—	13	1	no	—
69	#2150	—	13	1	no	—
70	C₂×#350	—	13	4	no	—
71	C₂×C₄×C₁₆	—	13	1	no	—
72	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
73	C₂²×D₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	4	no	—
75	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
76	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
77	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
79	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₁₃	#131	1	2	yes	—
81	C₂²⋊D₁₃	—	4	4	no	—
82	C₂×Dic₁₃	—	2	2	no	—
83	D₅₂	—	2	4	no	—
84	D₅₂	—	4	4	no	—
85	C₄×D₁₃	—	4	2	no	—
86	C₄×D₁₃	—	4	2	no	—
87	Dic₂₆	—	2	4	no	—
88	C₈×C₁₃	—	4	2	no	—
89	#1055	—	2	2	no	—
90	#1055	#121	1	2	yes	—
91	#1055	#124	1	1	yes	characteristic
92	#1055	#127	1	1	yes	characteristic
93	C₂×#131	—	13	2	no	—
94	#532	—	13	1	no	—
95	C₂×M₃₂	—	13	2	no	—
96	C₂²×C₁₆	—	13	2	no	—
97	#436	—	26	2	no	—
98	C₈×D₄	—	26	2	no	—
99	#423	#261	1	1	yes	characteristic
100	#408	—	13	2	no	—
101	C₂×C₄×C₈	—	13	1	no	—
102	C₄⋊C₁₆	—	26	2	no	—
103	#350	—	26	8	no	—
104	C₄×C₁₆	—	26	2	no	—
105	C₂²×C₁₃	—	2	2	no	—
106	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
107	C₂²×C₁₃	#436	1	2	yes	—
108	C₂×D₁₃	—	4	4	no	—
109	C₂×D₁₃	—	4	2	no	—
110	C₄×C₁₃	—	2	2	no	—
111	C₄×C₁₃	#350	1	4	yes	complemented
112	C₄×C₁₃	#390	1	1	yes	characteristic
113	C₄×C₁₃	#408	1	1	yes	characteristic
114	C₄×C₁₃	C₈×D₄	1	2	yes	—
115	Dic₁₃	—	4	2	no	—
116	C₂×#41	—	13	2	no	—
117	#131	—	26	4	no	—
118	#131	—	26	4	no	—
119	C₂×M₁₆	—	26	2	no	—
120	C₂×M₁₆	C₂×#261	1	2	yes	—
121	C₂²×C₈	—	13	1	no	—
122	C₂²×C₈	—	13	1	no	—
123	C₂²×C₈	—	26	2	no	—
124	C₄⋊Q₈	—	13	1	no	—
125	#127	—	13	1	no	—
126	#124	—	13	2	no	—
127	#121	—	26	2	no	—
128	C₄×D₄	—	26	2	no	—
129	C₂×C₄²	C₂×#261	1	1	yes	characteristic
130	M₃₂	—	26	4	no	—
131	C₂×C₁₆	—	13	4	no	—
132	C₂×C₁₆	—	26	2	no	—
133	C₂×C₁₆	—	26	2	no	—
134	C₂×C₁₆	—	26	2	no	—
135	C₄⋊C₈	—	2	2	no	—
136	C₂²⋊C₈	—	26	4	no	—
137	C₄×C₈	—	13	2	no	—
138	C₄×C₈	—	13	2	no	—
139	C₂×C₁₃	—	2	2	no	—
140	C₂×C₁₃	#1954	1	1	yes	characteristic
141	C₂×C₁₃	C₂×#350	1	2	yes	—
142	D₁₃	—	8	2	no	—
143	#41	—	26	4	no	—
144	#41	—	26	4	no	—
145	C₂×Q₈	—	13	2	no	—
146	C₂×D₄	—	13	2	no	—
147	C₂×D₄	—	26	2	no	—
148	C₂×D₄	—	26	2	no	—
149	C₂²×C₄	—	26	2	no	—
150	C₂²×C₄	#6118	1	2	yes	—
151	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
152	M₁₆	—	2	4	no	—
153	M₁₆	—	52	4	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	13	2	no	—
156	C₂×C₈	—	13	2	no	—
157	C₂×C₈	—	13	1	no	—
158	C₂×C₈	—	13	1	no	—
159	C₂×C₈	—	13	2	no	—
160	C₂×C₈	—	26	2	no	—
161	C₂×C₈	—	52	2	no	—
162	C₂×C₈	—	52	2	no	—
163	C₄⋊C₄	—	26	2	no	—
164	C₂²⋊C₄	—	26	4	no	—
165	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
166	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
167	C₄²	C₂²⋊#261	1	2	yes	—
168	C₁₆	—	26	4	no	—
169	C₁₆	—	52	2	no	—
170	C₁₃	#13191	1	1	yes	characteristic, complemented, Sylow
171	C₂³	—	26	2	no	—
172	C₂³	C₂×#6118	1	1	yes	characteristic
173	Q₈	—	26	4	no	—
174	D₄	—	26	4	no	—
175	D₄	—	52	4	no	—
176	D₄	—	52	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	26	2	no	—
179	C₂×C₄	—	52	2	no	—
180	C₂×C₄	—	52	2	no	—
181	C₂×C₄	#88111	1	2	yes	—
182	C₂×C₄	#88111	1	2	yes	—
183	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
184	C₂×C₄	D₄×#261	1	2	yes	—
185	C₂×C₄	#88240	1	2	yes	—
186	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
187	C₈	—	4	2	no	—
188	C₈	—	13	2	no	—
189	C₈	—	13	1	no	—
190	C₈	—	13	1	no	—
191	C₈	—	26	2	no	—
192	C₂²	—	2	2	no	—
193	C₂²	—	52	4	no	—
194	C₂²	—	52	2	no	—
195	C₂²	#11719076	1	2	yes	—
196	C₂²	C₂×#88111	1	1	yes	characteristic
197	C₄	—	2	2	no	—
198	C₄	—	52	2	no	—
199	C₄	#11718270	1	4	yes	complemented
200	C₄	#11719073	1	2	yes	—
201	C₄	#11719151	1	1	yes	characteristic
202	C₄	#11719159	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	—	104	2	no	—
205	C₂	#13130958	1	1	yes	characteristic
206	C₂	C₂×#11718270	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13131168

C₂

yes

maximal, characteristic, complemented

C₂×#11718270

C₂

yes

maximal, complemented

C₂×C₄×#6092

C₂

yes

maximal, characteristic, complemented

#13123547

C₂

yes

maximal, characteristic

C₂×#88161

C₂²

yes

—

#11719182

C₄

yes

characteristic

C₂×#88106

C₂²

yes

—

C₂²×#6092

C₂²

yes

—

C₁₃×#423

C₄

yes

characteristic

#11718790

C₄

yes

—

C₂×C₄×#1055

C₂²

yes

—

—

—

—

—

C₂×C₄

yes

—

C₂×C₁₃×M₁₆

C₂×C₄

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

#88184

—

C₂²×#1055

C₂×C₄

yes

characteristic

C₂²×#1055

C₂³

yes

—

—

—

C₂×C₈×D₁₃

—

#88129

C₈

yes

—

#88128

C₈

yes

characteristic

C₄×D₅₂

—

#88123

C₈

yes

—

—

—

—

D₄

yes

complemented

C₁₃×C₄⋊C₈

—

#88060

—

C₄×#1055

D₄

yes

—

C₄×#1055

D₄

yes

—

#13191

—

maximal, Sylow

C₂²×C₄×C₁₃

M₁₆

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₂²⋊D₁₃

—

#6127

—

#6127

—

C₂×D₅₂

—

C₂×D₅₂

C₂×C₈

yes

—

C₂×C₄×D₁₃

—

C₂×Dic₂₆

C₂×C₈

yes

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

C₂²⋊C₄

yes

—

C₄²×C₁₃

C₂×C₈

yes

characteristic

C₄²×C₁₃

C₂×C₈

yes

characteristic

#6103

—

C₄⋊Dic₁₃

—

C₂×#1055

—

C₂×#1055

C₂²⋊C₄

yes

—

C₂×#1055

C₂×D₄

yes

characteristic

C₂×#1055

C₂×D₄

yes

—

C₂×#1055

C₂×D₄

yes

characteristic

C₂×#1055

#41

yes

—

#6094

—

C₈×D₁₃

—

C₈×D₁₃

—

—

—

—

—

—

C₂×C₄×C₁₆

—

C₂³×C₁₃

C₂×M₁₆

yes

characteristic

C₂²×D₁₃

—

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

C₄×D₄

yes

—

C₂×C₄×C₁₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₃

#131

yes

—

C₂²⋊D₁₃

—

C₂×Dic₁₃

—

D₅₂

—

D₅₂

—

C₄×D₁₃

—

C₄×D₁₃

—

Dic₂₆

—

C₈×C₁₃

—

#1055

—

#1055

#121

yes

—

yes

yes

—

—

C₂×M₃₂

—

C₂²×C₁₆

—

#436

—

C₈×D₄

—

yes

100

—

101

C₂×C₄×C₈

—

102

C₄⋊C₁₆

—

103

#350

—

104

C₄×C₁₆

—

105

C₂²×C₁₃

—

106

C₂²×C₁₃

C₂×C₂²⋊C₈

yes

characteristic

107

C₂²×C₁₃

#436

yes

—

108

C₂×D₁₃

—

109

C₂×D₁₃

—

110

C₄×C₁₃

—

111

C₄×C₁₃

#350

yes

complemented

112

C₄×C₁₃

#390

yes

characteristic

113

C₄×C₁₃

#408

yes

characteristic

114

C₄×C₁₃

C₈×D₄

yes

—

115

Dic₁₃

—

116

C₂×#41

—

117

#131

—

118

#131

—

119

C₂×M₁₆

—

120

C₂×M₁₆

C₂×#261

yes

—

121

C₂²×C₈

—

122

C₂²×C₈

—

123

C₂²×C₈

—

124

C₄⋊Q₈

—

125

#127

—

126

#124

—

127

#121

—

128

C₄×D₄

—

129

C₂×C₄²

C₂×#261

yes

characteristic

130

M₃₂

—

131

C₂×C₁₆

—

132

C₂×C₁₆

—

133

C₂×C₁₆

—

134

C₂×C₁₆

—

135

C₄⋊C₈

—

136

C₂²⋊C₈

—

137

C₄×C₈

—

138

C₄×C₈

—

139

C₂×C₁₃

—

140

C₂×C₁₃

#1954

yes

characteristic

141

C₂×C₁₃

C₂×#350

yes

—

142

D₁₃

—

143

#41

—

144

#41

—

145

C₂×Q₈

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂²×C₄

—

150

C₂²×C₄

#6118

yes

—

151

C₂²×C₄

C₂²×#261

yes

characteristic

152

M₁₆

—

153

M₁₆

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₄⋊C₄

—

164

C₂²⋊C₄

—

165

C₄²

D₁₃⋊C₈

yes

characteristic

166

C₄²

D₁₃⋊C₈

yes

characteristic

167

C₄²

C₂²⋊#261

yes

—

168

C₁₆

—

169

C₁₆

—

170

C₁₃

#13191

yes

characteristic, complemented, Sylow

171

C₂³

—

172

C₂³

C₂×#6118

yes

characteristic

173

Q₈

—

174

D₄

—

175

D₄

—

176

D₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

#88111

yes

—

182

C₂×C₄

#88111

yes

—

183

C₂×C₄

C₂×D₁₃⋊C₈

yes

characteristic

184

C₂×C₄

D₄×#261

yes

—

185

C₂×C₄

#88240

yes

—

186

C₂×C₄

C₂×C₂²⋊#261

yes

characteristic

187

C₈

—

188

C₈

—

189

C₈

—

190

C₈

—

191

C₈

—

192

C₂²

—

193

C₂²

—

194

C₂²

—

195

C₂²

#11719076

yes

—

196

C₂²

C₂×#88111

yes

characteristic

197

C₄

—

198

C₄

—

199

C₄

#11718270

yes

complemented

200

C₄

#11719073

yes

—

201

C₄

#11719151

yes

characteristic

202

C₄

#11719159

yes

characteristic

203

C₂

—

204

C₂

—

104

—

205

C₂

#13130958

yes

characteristic

206

C₂

C₂×#11718270

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊#261
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₁₃×#423	C₄
socle	C₂²×C₁₃	C₂×C₂²⋊C₈
prime radical	#11719182	C₄
cosocle	C₂²×#1055	C₂³
hypercenter	#423	#261
hypocenter	C₁₃	#13191
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊#261

commutator

C₂×C₄×C₁₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#261

Fitting

C₁₃×#423

C₄

socle

C₂²×C₁₃

C₂×C₂²⋊C₈

prime radical

#11719182

C₄

cosocle

C₂²×#1055

C₂³

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
1. #13131168	C₂	semidirect
2. C₂×#11718270	C₂	semidirect
3. C₂×C₄×#6092	C₂	semidirect
35. C₂×#6092	D₄	semidirect
111. C₄×C₁₃	#350	semidirect
170. C₁₃	#13191	semidirect
199. C₄	#11718270	semidirect

H ≅ G/N

type

1. #13131168

C₂

semidirect

2. C₂×#11718270

C₂

semidirect

3. C₂×C₄×#6092

C₂

semidirect

35. C₂×#6092

D₄

semidirect

111. C₄×C₁₃

#350

semidirect

170. C₁₃

#13191

semidirect

199. C₄

#11718270

semidirect

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

subgroup	index
#13131168	2
C₂×#11718270	2
C₂×C₄×#6092	2
#13123547	2
#13191	13

subgroup

index

C₂×C₄×#6092

Subgroups — #16790862

Conjugacy classes of subgroups

Complements

Maximal subgroups