Group #16789619 — subgroups

Subgroups — #16789619

Order 3328.

334 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 128 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	#13131048	C₂	1	2	yes	maximal
2	C₄×D₁₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#13130638	C₂	1	1	yes	maximal, characteristic
4	#13129892	C₂	1	2	yes	maximal
5	D₁₃×#364	C₂	1	1	yes	maximal, characteristic
6	D₁₃⋊#108	C₄	1	1	yes	characteristic
7	#11719092	C₄	1	1	yes	characteristic
8	#11719089	C₂²	1	2	yes	—
9	C₂×D₁₃⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×D₁₃⋊C₁₆	C₂²	1	1	yes	characteristic
11	D₁₃×M₃₂	C₂²	1	2	yes	—
12	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
13	#11718257	C₄	1	2	yes	complemented
14	#11718234	C₄	1	2	yes	—
15	C₄×#6092	C₄	1	2	yes	complemented
16	#11718216	—	2	2	no	—
17	C₁₆×#261	—	2	2	no	—
18	C₁₃×#364	C₄	1	1	yes	characteristic, complemented
19	#11718103	C₄	1	2	yes	complemented
20	#11718055	C₄	1	1	yes	characteristic, complemented
21	#88237	C₂×C₄	1	2	yes	—
22	D₁₃⋊M₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
25	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂×#6092	C₂×C₄	1	2	yes	—
27	C₂×#6092	C₂×C₄	1	2	yes	—
28	#88104	C₈	1	2	yes	—
29	#88103	C₈	1	2	yes	—
30	#88102	C₈	1	2	yes	—
31	#88101	C₈	1	2	yes	—
32	#88100	—	2	2	no	—
33	C₈×#261	—	2	2	no	—
34	D₁₃⋊C₁₆	—	2	1	no	—
35	D₁₃⋊C₁₆	—	2	1	no	—
36	D₁₃⋊C₁₆	D₄	1	1	yes	characteristic
37	D₁₃⋊C₁₆	D₄	1	1	yes	characteristic
38	D₁₃⋊C₁₆	Q₈	1	2	yes	—
39	C₁₃×M₃₂	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
42	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
43	#88052	C₂×C₄	1	2	yes	—
44	C₄×#1055	C₂×C₄	1	1	yes	characteristic
45	#88038	—	2	2	no	—
46	C₁₆×D₁₃	—	2	2	no	—
47	#12664	—	13	1	no	maximal, Sylow
48	C₂×C₄×D₁₃	—	2	1	no	—
49	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²⋊#261	—	2	2	no	—
51	#6122	—	2	2	no	—
52	C₄⋊#261	C₂×C₈	1	4	yes	—
53	C₄×#261	—	2	2	no	—
54	#6118	C₂×C₈	1	4	yes	—
55	D₁₃⋊C₈	—	2	2	no	—
56	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
57	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₁₃	C₄²	1	1	yes	characteristic
59	C₄×Dic₁₃	—	2	1	no	—
60	C₄×Dic₁₃	C₄²	1	1	yes	characteristic
61	C₂×#1055	C₄²	1	1	yes	characteristic
62	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
63	C₈×D₁₃	C₂×C₈	1	1	yes	characteristic
64	C₈×D₁₃	C₂×C₈	1	1	yes	characteristic
65	C₈×D₁₃	C₂×C₈	1	1	yes	characteristic
66	C₈×D₁₃	C₂×C₈	1	1	yes	characteristic
67	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
68	C₈×D₁₃	C₂×Q₈	1	1	yes	characteristic
69	C₈×D₁₃	#41	1	1	yes	characteristic
70	C₈×D₁₃	#41	1	1	yes	characteristic
71	#6092	—	2	2	no	—
72	#6092	C₄⋊C₄	1	2	yes	—
73	#6092	C₄⋊C₄	1	2	yes	—
74	C₁₃×C₁₆	—	2	2	no	—
75	#6090	—	2	2	no	—
76	C₂×#364	—	13	1	no	—
77	#2108	—	13	2	no	—
78	C₂×C₄×C₁₆	—	13	1	no	—
79	#1772	—	13	1	no	—
80	#1379	—	13	2	no	—
81	C₂²×D₁₃	#131	1	1	yes	characteristic
82	C₂×#261	—	2	4	no	—
83	C₂×C₄×C₁₃	—	2	1	no	—
84	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
85	C₂×Dic₁₃	—	2	1	no	—
86	C₂×Dic₁₃	#131	1	1	yes	characteristic
87	C₄×D₁₃	—	2	4	no	—
88	C₄×D₁₃	C₄×D₄	1	1	yes	characteristic
89	C₄×D₁₃	C₄×Q₈	1	1	yes	characteristic
90	C₄×D₁₃	C₂²×C₈	1	1	yes	characteristic
91	C₄×D₁₃	C₂²×C₈	1	1	yes	characteristic
92	#1057	—	2	4	no	—
93	C₈×C₁₃	C₄×C₈	1	1	yes	characteristic
94	C₈×C₁₃	C₄×C₈	1	1	yes	characteristic
95	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
96	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
97	#1055	C₄×C₈	1	1	yes	characteristic
98	#1055	C₄×C₈	1	1	yes	characteristic
99	#1055	C₂×C₄⋊C₄	1	1	yes	characteristic
100	#1055	#117	1	1	yes	characteristic
101	C₂×M₃₂	—	13	2	no	—
102	C₂²×C₁₆	—	13	1	no	—
103	C₂²×C₁₆	—	13	1	no	—
104	C₂×#108	—	13	1	no	—
105	#427	—	13	1	no	—
106	#405	—	13	2	no	—
107	C₂×C₄×C₈	—	13	1	no	—
108	#364	—	13	2	no	—
109	#364	—	13	1	no	—
110	#364	#261	1	1	yes	characteristic, complemented
111	#346	—	26	2	no	—
112	C₄×C₁₆	—	13	2	no	—
113	C₄×C₁₆	—	13	2	no	—
114	C₄×C₁₆	—	26	2	no	—
115	#339	—	13	2	no	—
116	C₂²×C₁₃	#405	1	1	yes	characteristic
117	C₂×D₁₃	—	2	2	no	—
118	C₂×D₁₃	C₈×D₄	1	1	yes	characteristic
119	C₂×D₁₃	C₈×Q₈	1	1	yes	characteristic
120	#261	—	4	4	no	—
121	C₄×C₁₃	—	2	2	no	—
122	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
123	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
124	Dic₁₃	—	2	2	no	—
125	Dic₁₃	C₈×D₄	1	1	yes	characteristic
126	Dic₁₃	C₈×Q₈	1	1	yes	characteristic
127	C₂×M₁₆	—	13	2	no	—
128	C₂²×C₈	—	13	1	no	—
129	C₂²×C₈	—	13	1	no	—
130	#117	—	13	2	no	—
131	C₂×C₄²	—	13	1	no	—
132	M₃₂	—	13	2	no	—
133	M₃₂	—	26	2	no	—
134	M₃₂	C₂×#261	1	2	yes	—
135	C₂×C₁₆	—	13	1	no	—
136	C₂×C₁₆	—	13	1	no	—
137	C₂×C₁₆	—	13	2	no	—
138	C₂×C₁₆	—	13	2	no	—
139	C₂×C₁₆	—	13	2	no	—
140	C₂×C₁₆	—	26	2	no	—
141	C₂×C₁₆	—	26	1	no	—
142	C₂×C₁₆	—	26	1	no	—
143	#108	—	13	2	no	—
144	#108	—	13	2	no	—
145	#107	—	13	2	no	—
146	#106	—	13	2	no	—
147	#97	—	26	2	no	—
148	C₄×C₈	—	13	1	no	—
149	C₄×C₈	—	13	1	no	—
150	C₄×C₈	—	13	1	no	—
151	C₄×C₈	—	26	2	no	—
152	C₄×C₈	C₂×#261	1	1	yes	characteristic
153	C₂×C₁₃	—	2	1	no	—
154	C₂×C₁₃	C₈×C₄⋊C₄	1	1	yes	characteristic
155	D₁₃	—	2	1	no	—
156	D₁₃	#2170	1	2	yes	—
157	C₂²×C₄	—	13	1	no	—
158	C₂²×C₄	—	26	1	no	—
159	M₁₆	—	13	4	no	—
160	M₁₆	—	26	2	no	—
161	C₂×C₈	—	13	1	no	—
162	C₂×C₈	—	13	1	no	—
163	C₂×C₈	—	13	1	no	—
164	C₂×C₈	—	13	1	no	—
165	C₂×C₈	—	13	1	no	—
166	C₂×C₈	—	13	1	no	—
167	C₂×C₈	—	13	1	no	—
168	C₂×C₈	—	13	1	no	—
169	C₂×C₈	—	13	1	no	—
170	C₂×C₈	—	13	1	no	—
171	C₂×C₈	—	26	2	no	—
172	C₂×C₈	C₄×#261	1	1	yes	characteristic
173	C₂×C₈	C₂²×#261	1	1	yes	characteristic
174	C₄⋊C₄	—	13	4	no	—
175	C₂²⋊C₄	—	26	2	no	—
176	C₄²	—	13	1	no	—
177	C₄²	—	26	1	no	—
178	C₄²	—	26	2	no	—
179	C₄²	C₄×#261	1	1	yes	characteristic
180	C₁₆	—	2	2	no	—
181	C₁₆	—	13	2	no	—
182	C₁₆	—	13	2	no	—
183	C₁₆	—	26	2	no	—
184	C₁₆	—	26	2	no	—
185	C₁₃	#12664	1	1	yes	characteristic, complemented, Sylow
186	C₂³	—	13	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	13	1	no	—
189	C₂×C₄	—	13	1	no	—
190	C₂×C₄	—	13	1	no	—
191	C₂×C₄	—	13	1	no	—
192	C₂×C₄	—	13	1	no	—
193	C₂×C₄	—	26	4	no	—
194	C₂×C₄	—	26	1	no	—
195	C₂×C₄	—	26	4	no	—
196	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
197	C₈	—	13	1	no	—
198	C₈	—	13	1	no	—
199	C₈	—	13	1	no	—
200	C₈	—	13	1	no	—
201	C₈	—	26	4	no	—
202	C₈	C₈×#261	1	1	yes	characteristic
203	C₈	C₈×#261	1	1	yes	characteristic
204	C₈	D₄×#261	1	1	yes	characteristic
205	C₈	Q₈×#261	1	1	yes	characteristic
206	C₂²	—	13	1	no	—
207	C₂²	—	13	1	no	—
208	C₂²	—	26	2	no	—
209	C₂²	#11719099	1	1	yes	characteristic
210	C₄	—	2	2	no	—
211	C₄	—	13	1	no	—
212	C₄	—	13	1	no	—
213	C₄	—	26	2	no	—
214	C₄	—	52	4	no	—
215	C₄	C₄⋊C₄×#261	1	1	yes	characteristic
216	C₄	C₂×C₈×#261	1	1	yes	characteristic
217	C₂	—	2	1	no	—
218	C₂	—	13	2	no	—
219	C₂	—	26	1	no	—
220	C₂	#13130629	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13131048

C₂

yes

maximal

C₄×D₁₃⋊C₁₆

C₂

yes

maximal, characteristic

#13130638

C₂

yes

maximal, characteristic

#13129892

C₂

yes

maximal

D₁₃×#364

C₂

yes

maximal, characteristic

D₁₃⋊#108

C₄

yes

characteristic

#11719092

C₄

yes

characteristic

#11719089

C₂²

yes

—

C₂×D₁₃⋊C₁₆

C₂²

yes

characteristic

C₂×D₁₃⋊C₁₆

C₂²

yes

characteristic

D₁₃×M₃₂

C₂²

yes

—

C₄×C₈×D₁₃

C₂²

yes

characteristic

#11718257

C₄

yes

complemented

#11718234

C₄

yes

—

C₄×#6092

C₄

yes

complemented

#11718216

—

C₁₆×#261

—

C₁₃×#364

C₄

yes

characteristic, complemented

#11718103

C₄

yes

complemented

#11718055

C₄

yes

characteristic, complemented

#88237

C₂×C₄

yes

—

D₁₃⋊M₁₆

C₂×C₄

yes

—

C₂×C₈×D₁₃

C₂×C₄

yes

characteristic

C₂×C₈×D₁₃

C₂³

yes

characteristic

C₄²×D₁₃

C₂×C₄

yes

characteristic

C₂×#6092

C₂×C₄

yes

—

C₂×#6092

C₂×C₄

yes

—

#88104

C₈

yes

—

#88103

C₈

yes

—

#88102

C₈

yes

—

#88101

C₈

yes

—

#88100

—

C₈×#261

—

D₁₃⋊C₁₆

—

D₁₃⋊C₁₆

—

D₁₃⋊C₁₆

D₄

yes

characteristic

D₁₃⋊C₁₆

D₄

yes

characteristic

D₁₃⋊C₁₆

Q₈

yes

—

C₁₃×M₃₂

C₂×C₄

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₈×Dic₁₃

C₂×C₄

yes

characteristic

C₈×Dic₁₃

C₂×C₄

yes

characteristic

#88052

C₂×C₄

yes

—

C₄×#1055

C₂×C₄

yes

characteristic

#88038

—

C₁₆×D₁₃

—

#12664

—

maximal, Sylow

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

C₂²×C₄

yes

characteristic

C₂²⋊#261

—

#6122

—

C₄⋊#261

C₂×C₈

yes

—

C₄×#261

—

#6118

C₂×C₈

yes

—

D₁₃⋊C₈

—

C₂×C₈×C₁₃

C₄²

yes

characteristic

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₄²

yes

characteristic

C₄×Dic₁₃

—

C₄×Dic₁₃

C₄²

yes

characteristic

C₂×#1055

C₄²

yes

characteristic

C₂×#1055

C₂²×C₄

yes

characteristic

C₈×D₁₃

C₂×C₈

yes

characteristic

C₈×D₁₃

C₂×C₈

yes

characteristic

C₈×D₁₃

C₂×C₈

yes

characteristic

C₈×D₁₃

C₂×C₈

yes

characteristic

C₈×D₁₃

C₂×D₄

yes

characteristic

C₈×D₁₃

C₂×Q₈

yes

C₈×D₁₃

yes

C₈×D₁₃

yes

—

C₄⋊C₄

yes

—

#6092

C₄⋊C₄

yes

—

C₁₃×C₁₆

—

#6090

—

C₂×#364

—

#2108

—

C₂×C₄×C₁₆

—

#1772

—

#1379

—

C₂²×D₁₃

#131

yes

characteristic

C₂×#261

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₄²

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

#131

yes

characteristic

C₄×D₁₃

—

C₄×D₁₃

C₄×D₄

yes

characteristic

C₄×D₁₃

C₄×Q₈

yes

characteristic

C₄×D₁₃

C₂²×C₈

yes

characteristic

C₄×D₁₃

C₂²×C₈

yes

characteristic

#1057

—

C₈×C₁₃

C₄×C₈

yes

characteristic

C₈×C₁₃

C₄×C₈

yes

characteristic

C₈×C₁₃

C₄×D₄

yes

characteristic

C₈×C₁₃

C₄×Q₈

yes

characteristic

#1055

C₄×C₈

yes

characteristic

#1055

C₄×C₈

yes

characteristic

#1055

C₂×C₄⋊C₄

yes

100

yes

101

C₂×M₃₂

—

102

C₂²×C₁₆

—

103

C₂²×C₁₆

—

104

C₂×#108

—

105

#427

—

106

#405

—

107

C₂×C₄×C₈

—

108

#364

—

109

#364

—

110

#364

#261

yes

characteristic, complemented

111

#346

—

112

C₄×C₁₆

—

113

C₄×C₁₆

—

114

C₄×C₁₆

—

115

#339

—

116

C₂²×C₁₃

#405

yes

characteristic

117

C₂×D₁₃

—

118

C₂×D₁₃

C₈×D₄

yes

characteristic

119

C₂×D₁₃

C₈×Q₈

yes

characteristic

120

#261

—

121

C₄×C₁₃

—

122

C₄×C₁₃

C₄×C₄⋊C₄

yes

characteristic

123

C₄×C₁₃

C₂×C₄×C₈

yes

characteristic

124

Dic₁₃

—

125

Dic₁₃

C₈×D₄

yes

characteristic

126

Dic₁₃

C₈×Q₈

yes

characteristic

127

C₂×M₁₆

—

128

C₂²×C₈

—

129

C₂²×C₈

—

130

#117

—

131

C₂×C₄²

—

132

M₃₂

—

133

M₃₂

—

134

M₃₂

C₂×#261

yes

—

135

C₂×C₁₆

—

136

C₂×C₁₆

—

137

C₂×C₁₆

—

138

C₂×C₁₆

—

139

C₂×C₁₆

—

140

C₂×C₁₆

—

141

C₂×C₁₆

—

142

C₂×C₁₆

—

143

#108

—

144

#108

—

145

#107

—

146

#106

—

147

#97

—

148

C₄×C₈

—

149

C₄×C₈

—

150

C₄×C₈

—

151

C₄×C₈

—

152

C₄×C₈

C₂×#261

yes

characteristic

153

C₂×C₁₃

—

154

C₂×C₁₃

C₈×C₄⋊C₄

yes

characteristic

155

D₁₃

—

156

D₁₃

#2170

yes

—

157

C₂²×C₄

—

158

C₂²×C₄

—

159

M₁₆

—

160

M₁₆

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

—

172

C₂×C₈

C₄×#261

yes

characteristic

173

C₂×C₈

C₂²×#261

yes

characteristic

174

C₄⋊C₄

—

175

C₂²⋊C₄

—

176

C₄²

—

177

C₄²

—

178

C₄²

—

179

C₄²

C₄×#261

yes

characteristic

180

C₁₆

—

181

C₁₆

—

182

C₁₆

—

183

C₁₆

—

184

C₁₆

—

185

C₁₃

#12664

yes

characteristic, complemented, Sylow

186

C₂³

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

C₂×C₄×#261

yes

characteristic

197

C₈

—

198

C₈

—

199

C₈

—

200

C₈

—

201

C₈

—

202

C₈

C₈×#261

yes

characteristic

203

C₈

C₈×#261

yes

characteristic

204

C₈

D₄×#261

yes

characteristic

205

C₈

Q₈×#261

yes

characteristic

206

C₂²

—

207

C₂²

—

208

C₂²

—

209

C₂²

#11719099

yes

characteristic

210

C₄

—

211

C₄

—

212

C₄

—

213

C₄

—

214

C₄

—

215

C₄

C₄⋊C₄×#261

yes

characteristic

216

C₄

C₂×C₈×#261

yes

characteristic

217

C₂

—

218

C₂

—

219

C₂

—

220

C₂

#13130629

yes

characteristic

	subgroup	G/H
center	C₈	D₄×#261
commutator	C₄×C₁₃	C₂×C₄×C₈
Frattini	C₂×C₈	C₂²×#261
Fitting	C₁₃×#364	C₄
socle	C₂×C₁₃	C₈×C₄⋊C₄
prime radical	C₂²×D₁₃	#131
cosocle	C₂×C₈×D₁₃	C₂³
hypercenter	#364	#261
hypocenter	C₁₃	#12664
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #364

subgroup

G/H

center

C₈

D₄×#261

commutator

C₄×C₁₃

C₂×C₄×C₈

Frattini

C₂×C₈

C₂²×#261

Fitting

C₁₃×#364

C₄

socle

C₂×C₁₃

C₈×C₄⋊C₄

prime radical

C₂²×D₁₃

#131

cosocle

C₂×C₈×D₁₃

C₂³

C₁₃

G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ #364

N	H ≅ G/N	type
13. #11718257	C₄	semidirect
15. C₄×#6092	C₄	semidirect
18. C₁₃×#364	C₄	semidirect
19. #11718103	C₄	semidirect
20. #11718055	C₄	semidirect
110. #364	#261	semidirect
185. C₁₃	#12664	semidirect

H ≅ G/N

type

13. #11718257

C₄

semidirect

15. C₄×#6092

C₄

semidirect

18. C₁₃×#364

C₄

semidirect

19. #11718103

C₄

semidirect

20. #11718055

C₄

semidirect

110. #364

#261

semidirect

185. C₁₃

#12664

semidirect

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

subgroup	index
#13131048	2
C₄×D₁₃⋊C₁₆	2
#13130638	2
#13129892	2
D₁₃×#364	2
#12664	13

subgroup

index

C₄×D₁₃⋊C₁₆

D₁₃×#364

Subgroups — #16789619

Conjugacy classes of subgroups

Complements

Maximal subgroups