Group #16807351 — subgroups

Subgroups — #16807351

Order 3328.

572 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 218 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	#13131222	C₂	1	4	yes	maximal, complemented
2	#13131215	C₂	1	4	yes	maximal, complemented
3	#13131193	C₂	1	2	yes	maximal, complemented
4	#13131067	C₂	1	4	yes	maximal, complemented
5	#13123547	C₂	1	1	yes	maximal, characteristic
6	C₂×#88161	C₂²	1	2	yes	—
7	#11719182	C₄	1	1	yes	characteristic
8	C₂×#88098	C₂²	1	4	yes	—
9	C₂×D₁₃⋊C₁₆	C₂²	1	4	yes	—
10	C₁₃×#423	C₄	1	1	yes	characteristic
11	#11718790	C₄	1	2	yes	—
12	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
13	#11718499	C₄	1	4	yes	—
14	#11718495	C₂²	1	4	yes	—
15	C₂²⋊#6092	C₂²	1	4	yes	—
16	#11718259	C₂²	1	8	yes	—
17	C₄⋊#6092	C₂²	1	2	yes	—
18	C₄⋊#6092	C₂²	1	2	yes	—
19	C₄×#6092	C₂²	1	4	yes	—
20	C₂×#6127	C₂×C₄	1	2	yes	—
21	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
23	#88184	C₈	1	4	yes	—
24	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
25	C₂²×#1055	C₂³	1	1	yes	characteristic
26	#88161	—	4	2	no	—
27	#88161	—	4	2	no	—
28	C₂×#6094	C₂×C₄	1	4	yes	—
29	C₂×C₈×D₁₃	C₂³	1	4	yes	—
30	#88129	C₈	1	2	yes	—
31	#88128	C₈	1	1	yes	characteristic
32	C₄×D₅₂	C₂×C₄	1	4	yes	—
33	#88123	C₈	1	1	yes	characteristic
34	C₂×#6092	C₂³	1	4	yes	—
35	C₂×#6092	C₂³	1	4	yes	—
36	#88098	—	4	4	no	—
37	D₁₃⋊C₁₆	—	4	4	no	—
38	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
39	#88060	C₂×C₄	1	8	yes	—
40	C₄×#1055	C₂×C₄	1	2	yes	—
41	C₄×#1055	C₂³	1	2	yes	—
42	#33493	—	13	1	no	maximal, Sylow
43	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
44	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₂²⋊D₁₃	C₂×C₈	1	4	yes	—
46	#6127	—	4	2	no	—
47	#6127	—	4	2	no	—
48	C₂×D₅₂	C₂×C₈	1	4	yes	—
49	C₂×D₅₂	C₂×C₈	1	2	yes	—
50	C₂×C₄×D₁₃	C₂²×C₄	1	4	yes	—
51	C₂×Dic₂₆	C₂×C₈	1	2	yes	—
52	C₁₃×M₁₆	—	4	2	no	—
53	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
54	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
55	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
56	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
57	#6103	C₂×C₈	1	8	yes	—
58	C₄⋊Dic₁₃	C₂×C₈	1	4	yes	—
59	C₂×#1055	—	2	1	no	—
60	C₂×#1055	—	2	1	no	—
61	C₂×#1055	—	4	1	no	—
62	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
63	C₂×#1055	C₂²×C₄	1	2	yes	—
64	C₂×#1055	C₂⁴	1	1	yes	characteristic
65	#6094	—	4	4	no	—
66	C₈×D₁₃	—	2	4	no	—
67	C₈×D₁₃	—	2	4	no	—
68	#6092	—	2	4	no	—
69	#6092	—	2	4	no	—
70	#2931	—	13	1	no	—
71	#2169	—	13	4	no	—
72	D₄×C₁₆	—	13	4	no	—
73	#2162	—	13	2	no	—
74	#2113	—	13	4	no	—
75	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
76	C₂²×D₁₃	C₂²×C₈	1	4	yes	—
77	C₂×C₄×C₁₃	—	2	1	no	—
78	C₂×C₄×C₁₃	—	2	1	no	—
79	C₂×C₄×C₁₃	—	2	4	no	—
80	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
81	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
83	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
84	C₂²⋊D₁₃	—	4	4	no	—
85	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
86	D₅₂	—	2	8	no	—
87	D₅₂	—	4	2	no	—
88	C₄×D₁₃	—	2	4	no	—
89	C₄×D₁₃	—	2	4	no	—
90	Dic₂₆	—	4	2	no	—
91	C₈×C₁₃	—	2	4	no	—
92	#1055	—	2	1	no	—
93	#1055	—	2	2	no	—
94	#1055	H(5,𝔽₂)	1	1	yes	characteristic
95	#1055	H(5,𝔽₂)	1	1	yes	characteristic
96	C₂×#131	—	13	2	no	—
97	#532	—	13	1	no	—
98	C₂×M₃₂	—	13	4	no	—
99	C₂²×C₁₆	—	13	4	no	—
100	#436	—	13	4	no	—
101	C₈×D₄	—	13	4	no	—
102	#423	#261	1	1	yes	characteristic
103	#408	—	13	2	no	—
104	C₂×C₄×C₈	—	13	1	no	—
105	C₄⋊C₁₆	—	13	2	no	—
106	C₄⋊C₁₆	—	13	2	no	—
107	C₂²⋊C₁₆	—	13	4	no	—
108	C₂²⋊C₁₆	—	13	8	no	—
109	C₄×C₁₆	—	13	4	no	—
110	C₂²×C₁₃	—	2	2	no	—
111	C₂²×C₁₃	—	2	1	no	—
112	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
113	C₂×D₁₃	—	2	8	no	—
114	C₂×D₁₃	—	2	4	no	—
115	C₄×C₁₃	—	2	2	no	—
116	C₄×C₁₃	—	2	1	no	—
117	C₄×C₁₃	#504	1	4	yes	—
118	C₄×C₁₃	#518	1	1	yes	characteristic
119	C₄×C₁₃	#568	1	1	yes	characteristic
120	Dic₁₃	—	2	4	no	—
121	C₂×#41	—	13	2	no	—
122	#131	—	52	2	no	—
123	#131	—	52	2	no	—
124	C₂×M₁₆	—	13	4	no	—
125	C₂×M₁₆	C₂×#261	1	2	yes	—
126	C₂²×C₈	—	13	1	no	—
127	C₂²×C₈	—	13	1	no	—
128	C₂²×C₈	—	13	4	no	—
129	C₄⋊Q₈	—	13	1	no	—
130	#127	—	13	1	no	—
131	#124	—	13	2	no	—
132	#121	—	13	4	no	—
133	C₄×D₄	—	13	4	no	—
134	C₂×C₄²	C₂×#261	1	1	yes	characteristic
135	M₃₂	—	52	4	no	—
136	C₂×C₁₆	—	13	4	no	—
137	C₂×C₁₆	—	13	4	no	—
138	C₂×C₁₆	—	52	4	no	—
139	C₄⋊C₈	C₂×#261	1	4	yes	—
140	C₂²⋊C₈	—	13	8	no	—
141	C₄×C₈	—	13	2	no	—
142	C₄×C₈	—	13	2	no	—
143	C₂×C₁₃	—	4	1	no	—
144	C₂×C₁₃	#2972	1	1	yes	characteristic
145	C₂×C₁₃	C₂×#504	1	2	yes	—
146	D₁₃	—	4	4	no	—
147	#41	—	52	2	no	—
148	#41	—	52	2	no	—
149	C₂×Q₈	—	13	2	no	—
150	C₂×D₄	—	13	4	no	—
151	C₂×D₄	—	13	4	no	—
152	C₂×D₄	—	13	2	no	—
153	C₂²×C₄	—	13	4	no	—
154	C₂²×C₄	D₁₃⋊C₈	1	2	yes	—
155	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
156	M₁₆	—	4	2	no	—
157	M₁₆	—	52	4	no	—
158	C₂×C₈	—	13	1	no	—
159	C₂×C₈	—	13	2	no	—
160	C₂×C₈	—	13	1	no	—
161	C₂×C₈	—	26	4	no	—
162	C₂×C₈	—	26	4	no	—
163	C₂×C₈	—	26	1	no	—
164	C₂×C₈	—	26	1	no	—
165	C₂×C₈	—	52	1	no	—
166	C₂×C₈	C₂²×#261	1	4	yes	—
167	C₄⋊C₄	—	13	4	no	—
168	C₂²⋊C₄	—	13	8	no	—
169	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
170	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
171	C₄²	C₂²×#261	1	2	yes	—
172	C₁₆	—	26	4	no	—
173	C₁₆	—	26	4	no	—
174	C₁₃	#33493	1	1	yes	characteristic, complemented, Sylow
175	C₂³	—	13	4	no	—
176	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
177	Q₈	—	52	2	no	—
178	D₄	—	26	8	no	—
179	D₄	—	52	2	no	—
180	D₄	—	52	4	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	13	4	no	—
185	C₂×C₄	—	26	4	no	—
186	C₂×C₄	—	26	4	no	—
187	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
188	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
189	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
190	C₂×C₄	C₂³×#261	1	1	yes	characteristic
191	C₈	—	2	4	no	—
192	C₈	—	13	1	no	—
193	C₈	—	13	1	no	—
194	C₈	—	26	1	no	—
195	C₈	—	26	2	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	1	no	—
198	C₂²	—	26	8	no	—
199	C₂²	—	26	4	no	—
200	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
201	C₄	—	2	2	no	—
202	C₄	—	2	1	no	—
203	C₄	—	26	4	no	—
204	C₄	#11719096	1	4	yes	—
205	C₄	#11719623	1	1	yes	characteristic
206	C₄	#11719625	1	1	yes	characteristic
207	C₂	—	4	1	no	—
208	C₂	—	52	4	no	—
209	C₂	#13131686	1	1	yes	characteristic
210	C₂	C₂×#11719096	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13131222

C₂

yes

maximal, complemented

#13131215

C₂

yes

maximal, complemented

#13131193

C₂

yes

maximal, complemented

#13131067

C₂

yes

maximal, complemented

#13123547

C₂

yes

maximal, characteristic

C₂×#88161

C₂²

yes

—

#11719182

C₄

yes

characteristic

C₂×#88098

C₂²

yes

—

C₂×D₁₃⋊C₁₆

C₂²

yes

—

C₁₃×#423

C₄

yes

characteristic

#11718790

C₄

yes

—

C₂×C₄×#1055

C₂²

yes

characteristic

#11718499

C₄

yes

—

#11718495

C₂²

yes

—

C₂²⋊#6092

C₂²

yes

—

#11718259

C₂²

yes

—

C₄⋊#6092

C₂²

yes

—

C₄⋊#6092

C₂²

yes

—

C₄×#6092

C₂²

yes

—

C₂×#6127

C₂×C₄

yes

—

C₂×C₁₃×M₁₆

C₂×C₄

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

#88184

C₈

yes

—

C₂²×#1055

C₂×C₄

yes

characteristic

C₂²×#1055

C₂³

yes

—

—

C₂×C₄

yes

—

C₂×C₈×D₁₃

C₂³

yes

—

#88129

C₈

yes

—

#88128

C₈

yes

characteristic

C₄×D₅₂

C₂×C₄

yes

—

#88123

C₈

yes

characteristic

C₂×#6092

C₂³

yes

—

C₂×#6092

C₂³

yes

—

#88098

—

D₁₃⋊C₁₆

—

C₁₃×C₄⋊C₈

C₂×C₄

yes

—

#88060

C₂×C₄

yes

—

C₄×#1055

C₂×C₄

yes

—

C₄×#1055

C₂³

yes

—

#33493

—

maximal, Sylow

C₂²×C₄×C₁₃

C₂×C₈

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₂²⋊D₁₃

C₂×C₈

yes

—

#6127

—

#6127

—

C₂×D₅₂

C₂×C₈

yes

—

C₂×D₅₂

C₂×C₈

yes

—

C₂×C₄×D₁₃

C₂²×C₄

yes

—

C₂×Dic₂₆

C₂×C₈

yes

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

C₂²×C₄

yes

—

C₄²×C₁₃

C₂×C₈

yes

characteristic

C₄²×C₁₃

C₂×C₈

yes

characteristic

C₄²×C₁₃

C₂²×C₄

yes

—

#6103

C₂×C₈

yes

—

C₄⋊Dic₁₃

C₂×C₈

yes

—

—

—

—

C₂²×C₄

yes

characteristic

C₂×#1055

C₂²×C₄

yes

—

C₂×#1055

C₂⁴

yes

characteristic

#6094

—

C₈×D₁₃

—

C₈×D₁₃

—

—

—

—

—

D₄×C₁₆

—

#2162

—

#2113

—

C₂³×C₁₃

C₂²×C₈

yes

characteristic

C₂²×D₁₃

C₂²×C₈

yes

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²×C₈

yes

—

C₂×C₄×C₁₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₈

yes

—

C₂×C₄×C₁₃

C₂³×C₄

yes

characteristic

C₂²⋊D₁₃

—

C₂×Dic₁₃

C₂²×C₈

yes

—

D₅₂

—

D₅₂

—

C₄×D₁₃

—

C₄×D₁₃

—

Dic₂₆

—

C₈×C₁₃

—

—

—

yes

yes

—

—

C₂×M₃₂

—

C₂²×C₁₆

—

100

#436

—

101

C₈×D₄

—

102

yes

103

—

104

C₂×C₄×C₈

—

105

C₄⋊C₁₆

—

106

C₄⋊C₁₆

—

107

C₂²⋊C₁₆

—

108

C₂²⋊C₁₆

—

109

C₄×C₁₆

—

110

C₂²×C₁₃

—

111

C₂²×C₁₃

—

112

C₂²×C₁₃

C₂³×C₈

yes

characteristic

113

C₂×D₁₃

—

114

C₂×D₁₃

—

115

C₄×C₁₃

—

116

C₄×C₁₃

—

117

C₄×C₁₃

#504

yes

—

118

C₄×C₁₃

#518

yes

characteristic

119

C₄×C₁₃

#568

yes

characteristic

120

Dic₁₃

—

121

C₂×#41

—

122

#131

—

123

#131

—

124

C₂×M₁₆

—

125

C₂×M₁₆

C₂×#261

yes

—

126

C₂²×C₈

—

127

C₂²×C₈

—

128

C₂²×C₈

—

129

C₄⋊Q₈

—

130

#127

—

131

#124

—

132

#121

—

133

C₄×D₄

—

134

C₂×C₄²

C₂×#261

yes

characteristic

135

M₃₂

—

136

C₂×C₁₆

—

137

C₂×C₁₆

—

138

C₂×C₁₆

—

139

C₄⋊C₈

C₂×#261

yes

—

140

C₂²⋊C₈

—

141

C₄×C₈

—

142

C₄×C₈

—

143

C₂×C₁₃

—

144

C₂×C₁₃

#2972

yes

characteristic

145

C₂×C₁₃

C₂×#504

yes

—

146

D₁₃

—

147

#41

—

148

#41

—

149

C₂×Q₈

—

150

C₂×D₄

—

151

C₂×D₄

—

152

C₂×D₄

—

153

C₂²×C₄

—

154

C₂²×C₄

D₁₃⋊C₈

yes

—

155

C₂²×C₄

C₂²×#261

yes

characteristic

156

M₁₆

—

157

M₁₆

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

C₂²×#261

yes

—

167

C₄⋊C₄

—

168

C₂²⋊C₄

—

169

C₄²

D₁₃⋊C₈

yes

characteristic

170

C₄²

D₁₃⋊C₈

yes

characteristic

171

C₄²

C₂²×#261

yes

—

172

C₁₆

—

173

C₁₆

—

174

C₁₃

#33493

yes

characteristic, complemented, Sylow

175

C₂³

—

176

C₂³

C₂×D₁₃⋊C₈

yes

characteristic

177

Q₈

—

178

D₄

—

179

D₄

—

180

D₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

C₂×D₁₃⋊C₈

yes

characteristic

188

C₂×C₄

C₂×D₁₃⋊C₈

yes

—

189

C₂×C₄

C₂×D₁₃⋊C₈

yes

—

190

C₂×C₄

C₂³×#261

yes

characteristic

191

C₈

—

192

C₈

—

193

C₈

—

194

C₈

—

195

C₈

—

196

C₂²

—

197

C₂²

—

198

C₂²

—

199

C₂²

—

200

C₂²

C₂²×D₁₃⋊C₈

yes

characteristic

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

#11719096

yes

—

205

C₄

#11719623

yes

characteristic

206

C₄

#11719625

yes

characteristic

207

C₂

—

208

C₂

—

209

C₂

#13131686

yes

characteristic

210

C₂

C₂×#11719096

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#423	C₄
prime radical	#11719182	C₄
cosocle	C₂×#1055	C₂⁴
hypercenter	#423	#261
hypocenter	C₁₃	#33493
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×#261

commutator, socle

C₂²×C₁₃

C₂³×C₈

Fitting

C₁₃×#423

C₄

C₄

C₂⁴

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
1. #13131222	C₂	semidirect
2. #13131215	C₂	semidirect
3. #13131193	C₂	semidirect
4. #13131067	C₂	semidirect
174. C₁₃	#33493	semidirect

H ≅ G/N

type

1. #13131222

C₂

semidirect

2. #13131215

C₂

semidirect

3. #13131193

C₂

semidirect

4. #13131067

C₂

semidirect

174. C₁₃

#33493

semidirect

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

subgroup	index
#13131222	2
#13131215	2
#13131193	2
#13131067	2
#13123547	2
#33493	13

subgroup

index

Subgroups — #16807351

Conjugacy classes of subgroups

Complements

Maximal subgroups