Group #16784487 — subgroups

Subgroups — #16784487

Order 3328.

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

normal subgroups only

This group has 322 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	#13130775	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊#261	C₂	1	1	yes	maximal, characteristic
3	#13130406	C₂	1	2	yes	maximal
4	#13129733	C₂	1	2	yes	maximal
5	#13113581	C₂	1	1	yes	maximal, characteristic
6	C₂×#88103	—	2	1	no	—
7	C₂×#88103	—	2	1	no	—
8	C₂×C₈×#261	—	2	1	no	—
9	C₄×C₄⋊#261	C₂²	1	1	yes	characteristic
10	#11719052	C₂²	1	2	yes	—
11	D₁₃⋊(C₄⋊C₈)	C₂²	1	1	yes	characteristic
12	#11718494	C₂²	1	2	yes	—
13	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
14	(C₂×#261)⋊C₈	—	2	1	no	—
15	#11718256	—	2	2	no	—
16	#11718254	—	2	2	no	—
17	C₁₃×#334	C₄	1	1	yes	characteristic, complemented
18	#11718071	C₄	1	2	yes	complemented
19	#11718049	C₄	1	1	yes	characteristic, complemented
20	C₂×C₄⋊#261	—	2	1	no	—
21	C₂×C₄×#261	—	2	1	no	—
22	C₂×#6118	—	2	2	no	—
23	C₂×D₁₃⋊C₈	—	2	1	no	—
24	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
25	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
26	C₄²×D₁₃	C₂³	1	1	yes	characteristic
27	(C₂×#261)⋊C₄	Q₈	1	2	yes	—
28	#88112	—	2	1	no	—
29	C₄⋊#1057	—	2	2	no	—
30	C₄⋊#1057	—	2	2	no	—
31	C₄⋊#1057	D₄	1	2	yes	—
32	#88103	—	4	2	no	—
33	#88103	—	4	2	no	—
34	C₈×#261	—	4	2	no	—
35	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
37	#88060	—	2	2	no	—
38	C₈×Dic₁₃	—	2	2	no	—
39	C₈×Dic₁₃	Q₈	1	1	yes	characteristic
40	C₈×Dic₁₃	Q₈	1	1	yes	characteristic
41	C₄⋊#1055	C₂×C₄	1	2	yes	—
42	C₄×#1055	C₂×C₄	1	1	yes	characteristic
43	#9627	—	13	1	no	maximal, Sylow
44	C₂²×#261	—	2	2	no	—
45	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
48	C₂×#1057	—	2	2	no	—
49	C₂×#1057	—	4	2	no	—
50	C₄⋊#261	—	2	4	no	—
51	C₄×#261	—	4	1	no	—
52	C₄×#261	—	4	1	no	—
53	#6118	—	4	2	no	—
54	#6118	—	4	2	no	—
55	D₁₃⋊C₈	—	4	1	no	—
56	D₁₃⋊C₈	—	4	1	no	—
57	C₂×C₈×C₁₃	—	2	2	no	—
58	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
60	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
62	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
63	C₄×Dic₁₃	#41	1	1	yes	characteristic
64	C₂×#1055	—	2	2	no	—
65	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
66	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
67	C₈×D₁₃	—	2	1	no	—
68	C₈×D₁₃	—	2	1	no	—
69	C₈×D₁₃	D₈	1	2	yes	—
70	C₈×D₁₃	Q₁₆	1	2	yes	—
71	#1848	—	13	1	no	—
72	C₈×C₄⋊C₄	—	13	1	no	—
73	#1571	—	13	2	no	—
74	#1564	—	13	1	no	—
75	#1303	—	13	2	no	—
76	C₂²×D₁₃	C₂²⋊Q₈	1	1	yes	characteristic
77	C₂×#261	—	2	4	no	—
78	C₂×#261	—	4	2	no	—
79	C₂×#261	—	4	2	no	—
80	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
83	C₂×Dic₁₃	#121	1	1	yes	characteristic
84	C₂×Dic₁₃	#125	1	1	yes	characteristic
85	C₂×Dic₁₃	C₄⋊Q₈	1	1	yes	characteristic
86	C₄×D₁₃	—	2	2	no	—
87	C₄×D₁₃	—	4	1	no	—
88	C₄×D₁₃	C₂×D₈	1	1	yes	characteristic
89	C₄×D₁₃	C₂×Q₁₆	1	1	yes	characteristic
90	C₄×D₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
91	C₄×D₁₃	#137	1	1	yes	characteristic
92	#1057	—	4	2	no	—
93	#1057	—	8	2	no	—
94	C₈×C₁₃	—	2	1	no	—
95	C₈×C₁₃	—	4	2	no	—
96	C₈×C₁₃	#107	1	2	yes	—
97	#1055	—	2	1	no	—
98	#1055	—	4	2	no	—
99	#1055	#107	1	2	yes	—
100	#431	—	13	2	no	—
101	C₂×#108	—	26	1	no	—
102	C₂×#108	—	26	1	no	—
103	#423	—	13	2	no	—
104	C₂×C₄⋊C₈	—	13	1	no	—
105	C₂×C₄×C₈	—	13	1	no	—
106	C₂×C₄×C₈	—	26	1	no	—
107	C₄×C₄⋊C₄	—	13	1	no	—
108	#336	—	26	1	no	—
109	#335	—	13	2	no	—
110	#335	—	26	2	no	—
111	#335	—	26	2	no	—
112	#334	—	13	1	no	—
113	#334	#261	1	1	yes	characteristic, complemented
114	C₂²×C₁₃	#389	1	1	yes	characteristic
115	C₂×D₁₃	—	2	1	no	—
116	C₂×D₁₃	—	2	1	no	—
117	C₂×D₁₃	#474	1	1	yes	characteristic
118	C₂×D₁₃	#477	1	1	yes	characteristic
119	#261	—	4	4	no	—
120	C₄×C₁₃	—	2	1	no	—
121	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
122	C₄×C₁₃	#428	1	1	yes	characteristic
123	C₄×C₁₃	#443	1	2	yes	—
124	Dic₁₃	—	2	1	no	—
125	Dic₁₃	—	2	1	no	—
126	Dic₁₃	#459	1	1	yes	characteristic
127	Dic₁₃	#462	1	1	yes	characteristic
128	C₂×M₁₆	—	26	2	no	—
129	C₂²×C₈	—	13	1	no	—
130	C₂²×C₈	—	13	1	no	—
131	C₂²×C₈	—	26	1	no	—
132	C₂×C₄⋊C₄	—	26	1	no	—
133	C₂×C₄²	—	13	1	no	—
134	C₂×C₄²	—	26	1	no	—
135	#108	—	52	2	no	—
136	#108	—	52	2	no	—
137	C₄⋊C₈	—	13	2	no	—
138	C₄⋊C₈	—	13	2	no	—
139	C₄⋊C₈	—	26	1	no	—
140	C₄⋊C₈	—	26	2	no	—
141	C₄⋊C₈	—	26	2	no	—
142	C₄⋊C₈	C₂×#261	1	2	yes	—
143	C₂²⋊C₈	—	26	2	no	—
144	C₄×C₈	—	13	1	no	—
145	C₄×C₈	—	13	1	no	—
146	C₄×C₈	—	13	1	no	—
147	C₄×C₈	—	26	2	no	—
148	C₄×C₈	—	52	2	no	—
149	C₄×C₈	C₂×#261	1	1	yes	characteristic
150	#95	—	13	2	no	—
151	C₂×C₁₃	#1945	1	1	yes	characteristic
152	C₂×C₁₃	#1947	1	1	yes	characteristic
153	C₂×C₁₃	#1953	1	1	yes	characteristic
154	D₁₃	—	2	1	no	—
155	D₁₃	—	2	1	no	—
156	C₂²×C₄	—	13	1	no	—
157	C₂²×C₄	—	13	1	no	—
158	C₂²×C₄	—	13	1	no	—
159	C₂²×C₄	—	26	2	no	—
160	M₁₆	—	52	2	no	—
161	M₁₆	—	52	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	13	1	no	—
164	C₂×C₈	—	13	2	no	—
165	C₂×C₈	—	13	2	no	—
166	C₂×C₈	—	13	1	no	—
167	C₂×C₈	—	26	1	no	—
168	C₂×C₈	—	26	1	no	—
169	C₂×C₈	—	26	2	no	—
170	C₂×C₈	—	26	2	no	—
171	C₂×C₈	—	52	1	no	—
172	C₂×C₈	—	52	1	no	—
173	C₂×C₈	—	52	2	no	—
174	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
175	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
176	C₄⋊C₄	—	26	4	no	—
177	C₄²	—	13	1	no	—
178	C₄²	—	13	1	no	—
179	C₄²	—	13	1	no	—
180	C₄²	—	52	1	no	—
181	C₄²	—	52	1	no	—
182	C₄²	C₂²×#261	1	1	yes	characteristic
183	C₁₃	#9627	1	1	yes	characteristic, complemented, Sylow
184	C₂³	—	13	1	no	—
185	C₂×C₄	—	13	1	no	—
186	C₂×C₄	—	13	1	no	—
187	C₂×C₄	—	13	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	—	26	2	no	—
193	C₂×C₄	—	26	4	no	—
194	C₂×C₄	—	52	1	no	—
195	C₂×C₄	—	52	2	no	—
196	C₂×C₄	—	52	2	no	—
197	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
198	C₂×C₄	D₄×#261	1	1	yes	characteristic
199	C₂×C₄	Q₈×#261	1	1	yes	characteristic
200	C₈	—	2	1	no	—
201	C₈	—	4	2	no	—
202	C₈	—	13	2	no	—
203	C₈	—	26	1	no	—
204	C₈	—	52	2	no	—
205	C₈	—	52	2	no	—
206	C₈	—	104	2	no	—
207	C₈	#88102	1	2	yes	—
208	C₂²	—	13	1	no	—
209	C₂²	—	13	1	no	—
210	C₂²	—	26	1	no	—
211	C₂²	—	26	1	no	—
212	C₂²	#11719083	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	13	1	no	—
215	C₄	—	13	1	no	—
216	C₄	—	26	1	no	—
217	C₄	—	26	1	no	—
218	C₄	—	52	4	no	—
219	C₄	C₂×#88102	1	1	yes	characteristic
220	C₄	#11719098	1	1	yes	characteristic
221	C₄	#11719107	1	2	yes	—
222	C₂	—	26	1	no	—
223	C₂	—	26	1	no	—
224	C₂	#13130943	1	1	yes	characteristic
225	C₂	#13130947	1	1	yes	characteristic
226	C₂	#13130955	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13130775

C₂

yes

maximal, characteristic

C₈×C₄⋊#261

C₂

yes

maximal, characteristic

#13130406

C₂

yes

maximal

#13129733

C₂

yes

maximal

#13113581

C₂

yes

maximal, characteristic

C₂×#88103

—

C₂×#88103

—

C₂×C₈×#261

—

C₄×C₄⋊#261

C₂²

yes

characteristic

#11719052

C₂²

yes

—

D₁₃⋊(C₄⋊C₈)

C₂²

yes

characteristic

#11718494

C₂²

yes

—

C₄×C₈×D₁₃

C₂²

yes

characteristic

(C₂×#261)⋊C₈

—

#11718256

—

#11718254

—

C₁₃×#334

C₄

yes

characteristic, complemented

#11718071

C₄

yes

complemented

#11718049

C₄

yes

characteristic, complemented

C₂×C₄⋊#261

—

C₂×C₄×#261

—

C₂×#6118

—

C₂×D₁₃⋊C₈

—

C₂×C₈×D₁₃

D₄

yes

characteristic

C₂×C₈×D₁₃

D₄

yes

characteristic

C₄²×D₁₃

C₂³

yes

characteristic

(C₂×#261)⋊C₄

Q₈

yes

—

—

—

—

D₄

yes

—

#88103

—

#88103

—

C₈×#261

—

C₁₃×C₄⋊C₈

C₂×C₄

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

#88060

—

C₈×Dic₁₃

—

C₈×Dic₁₃

Q₈

yes

characteristic

C₈×Dic₁₃

Q₈

yes

characteristic

C₄⋊#1055

C₂×C₄

yes

—

C₄×#1055

C₂×C₄

yes

characteristic

#9627

—

maximal, Sylow

C₂²×#261

—

C₂×C₄×D₁₃

C₂×D₄

yes

characteristic

C₂×C₄×D₁₃

C₂×Q₈

yes

C₂×C₄×D₁₃

yes

—

—

—

—

—

—

—

D₁₃⋊C₈

—

D₁₃⋊C₈

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₃

C₄⋊C₄

yes

characteristic

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄×Dic₁₃

C₂×D₄

yes

characteristic

C₄×Dic₁₃

C₂×Q₈

yes

C₄×Dic₁₃

yes

—

C₄⋊C₄

yes

characteristic

C₂×#1055

C₄⋊C₄

yes

characteristic

C₈×D₁₃

—

C₈×D₁₃

—

C₈×D₁₃

D₈

yes

—

C₈×D₁₃

Q₁₆

yes

—

#1848

—

C₈×C₄⋊C₄

—

#1571

—

#1564

—

#1303

—

C₂²×D₁₃

C₂²⋊Q₈

yes

—

—

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₄×D₄

yes

characteristic

C₂×C₄×C₁₃

C₄×Q₈

yes

C₂×Dic₁₃

yes

C₂×Dic₁₃

yes

C₂×Dic₁₃

C₄⋊Q₈

yes

characteristic

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

C₂×D₈

yes

characteristic

C₄×D₁₃

C₂×Q₁₆

yes

C₄×D₁₃

yes

C₄×D₁₃

yes

—

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

#107

yes

—

—

—

yes

—

100

#431

—

101

C₂×#108

—

102

C₂×#108

—

103

#423

—

104

C₂×C₄⋊C₈

—

105

C₂×C₄×C₈

—

106

C₂×C₄×C₈

—

107

C₄×C₄⋊C₄

—

108

#336

—

109

#335

—

110

#335

—

111

#335

—

112

#334

—

113

#334

#261

yes

characteristic, complemented

114

C₂²×C₁₃

#389

yes

characteristic

115

C₂×D₁₃

—

116

C₂×D₁₃

—

117

C₂×D₁₃

#474

yes

characteristic

118

C₂×D₁₃

#477

yes

characteristic

119

#261

—

120

C₄×C₁₃

—

121

C₄×C₁₃

C₂×#107

yes

characteristic

122

C₄×C₁₃

#428

yes

characteristic

123

C₄×C₁₃

#443

yes

—

124

Dic₁₃

—

125

Dic₁₃

—

126

Dic₁₃

#459

yes

characteristic

127

Dic₁₃

#462

yes

characteristic

128

C₂×M₁₆

—

129

C₂²×C₈

—

130

C₂²×C₈

—

131

C₂²×C₈

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄²

—

134

C₂×C₄²

—

135

#108

—

136

#108

—

137

C₄⋊C₈

—

138

C₄⋊C₈

—

139

C₄⋊C₈

—

140

C₄⋊C₈

—

141

C₄⋊C₈

—

142

C₄⋊C₈

C₂×#261

yes

—

143

C₂²⋊C₈

—

144

C₄×C₈

—

145

C₄×C₈

—

146

C₄×C₈

—

147

C₄×C₈

—

148

C₄×C₈

—

149

C₄×C₈

C₂×#261

yes

characteristic

150

#95

—

151

C₂×C₁₃

#1945

yes

characteristic

152

C₂×C₁₃

#1947

yes

characteristic

153

C₂×C₁₃

#1953

yes

characteristic

154

D₁₃

—

155

D₁₃

—

156

C₂²×C₄

—

157

C₂²×C₄

—

158

C₂²×C₄

—

159

C₂²×C₄

—

160

M₁₆

—

161

M₁₆

—

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

—

173

C₂×C₈

—

174

C₂×C₈

C₄⋊#261

yes

characteristic

175

C₂×C₈

C₄⋊#261

yes

characteristic

176

C₄⋊C₄

—

177

C₄²

—

178

C₄²

—

179

C₄²

—

180

C₄²

—

181

C₄²

—

182

C₄²

C₂²×#261

yes

characteristic

183

C₁₃

#9627

yes

characteristic, complemented, Sylow

184

C₂³

—

185

C₂×C₄

—

186

C₂×C₄

—

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

—

197

C₂×C₄

C₂×C₄⋊#261

yes

characteristic

198

C₂×C₄

D₄×#261

yes

characteristic

199

C₂×C₄

Q₈×#261

yes

characteristic

200

C₈

—

201

C₈

—

202

C₈

—

203

C₈

—

204

C₈

—

205

C₈

—

206

C₈

—

104

—

207

C₈

#88102

yes

—

208

C₂²

—

209

C₂²

—

210

C₂²

—

211

C₂²

—

212

C₂²

#11719083

yes

characteristic

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

—

217

C₄

—

218

C₄

—

219

C₄

C₂×#88102

yes

characteristic

220

C₄

#11719098

yes

characteristic

221

C₄

#11719107

yes

—

222

C₂

—

223

C₂

—

224

C₂

#13130943

yes

characteristic

225

C₂

#13130947

yes

characteristic

226

C₂

#13130955

yes

characteristic

	subgroup	G/H
center	C₂²	#11719083
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₄²	C₂²×#261
Fitting	C₁₃×#334	C₄
socle	C₂²×C₁₃	#389
prime radical	C₂²×D₁₃	C₂²⋊Q₈
cosocle	C₄²×D₁₃	C₂³
hypercenter	#334	#261
hypocenter	C₁₃	#9627
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#11719083

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₄²

C₂²×#261

Fitting

C₁₃×#334

C₄

socle

C₂²×C₁₃

#389

prime radical

C₂²×D₁₃

C₂²⋊Q₈

cosocle

C₄²×D₁₃

C₂³

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #334

N	H ≅ G/N	type
17. C₁₃×#334	C₄	semidirect
18. #11718071	C₄	semidirect
19. #11718049	C₄	semidirect
113. #334	#261	semidirect
183. C₁₃	#9627	semidirect

H ≅ G/N

type

17. C₁₃×#334

C₄

semidirect

18. #11718071

C₄

semidirect

19. #11718049

C₄

semidirect

113. #334

#261

semidirect

183. C₁₃

#9627

semidirect

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

subgroup	index
#13130775	2
C₈×C₄⋊#261	2
#13130406	2
#13129733	2
#13113581	2
#9627	13

subgroup

index

C₈×C₄⋊#261

Subgroups — #16784487

Conjugacy classes of subgroups

Complements

Maximal subgroups