Group #16784426 — subgroups

Subgroups — #16784426

Order 3328.

378 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 120 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	#13130934	C₂	1	2	yes	maximal
2	C₂×#11718253	C₂	1	1	yes	maximal, characteristic
3	C₂×#1057⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#13129672	C₂	1	2	yes	maximal
5	#13114728	C₂	1	1	yes	maximal, characteristic
6	C₂×#88112	—	2	2	no	—
7	C₂×C₄⋊#1057	C₂²	1	2	yes	—
8	C₂×C₄×#1057	C₂²	1	1	yes	characteristic
9	C₂×C₄×#1057	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
11	C₄⋊C₄×Dic₁₃	C₂²	1	2	yes	—
12	C₂×#88047	C₄	1	2	yes	—
13	#11718290	—	2	2	no	—
14	#11718253	—	2	2	no	—
15	#1057⋊C₈	—	2	2	no	—
16	C₁₃×#340	C₄	1	1	yes	characteristic
17	#11718143	C₄	1	1	yes	characteristic
18	C₂²×#1057	—	2	1	no	—
19	C₂²×#1057	—	2	2	no	—
20	C₂²×#1057	D₄	1	2	yes	—
21	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
23	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
24	C₂×C₄×Dic₁₃	—	2	2	no	—
25	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
26	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
27	#88112	—	4	4	no	—
28	C₄⋊#1057	—	2	2	no	—
29	C₄⋊#1057	—	2	2	no	—
30	C₄×#1057	—	2	1	no	—
31	C₄×#1057	—	2	1	no	—
32	C₄×#1057	—	2	1	no	—
33	C₄×#1057	Q₈	1	2	yes	—
34	#88071	—	2	2	no	—
35	C₈×Dic₁₃	D₄	1	2	yes	—
36	C₈×Dic₁₃	Q₈	1	2	yes	—
37	#88047	—	2	4	no	—
38	#9599	—	13	1	no	maximal, Sylow
39	C₂²×C₄×C₁₃	—	2	2	no	—
40	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×Dic₁₃	—	2	2	no	—
42	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₁₃	#41	1	1	yes	characteristic
44	C₂×#1057	—	2	4	no	—
45	C₂×#1057	—	2	2	no	—
46	C₂×#1057	—	2	2	no	—
47	C₂×#1057	—	2	2	no	—
48	C₂×#1057	—	2	2	no	—
49	C₂×#1057	—	4	2	no	—
50	C₂×#1057	—	4	2	no	—
51	C₂×#1057	—	4	2	no	—
52	C₂×C₈×C₁₃	—	2	2	no	—
53	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
54	C₁₃×C₄⋊C₄	—	2	2	no	—
55	C₁₃×C₄⋊C₄	—	2	2	no	—
56	C₄⋊Dic₁₃	—	2	2	no	—
57	C₄⋊Dic₁₃	M₁₆	1	4	yes	—
58	C₄×Dic₁₃	—	4	2	no	—
59	C₄×Dic₁₃	—	4	2	no	—
60	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
61	C₄×Dic₁₃	C₂×Q₈	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	2	yes	—
66	#1939	—	13	2	no	—
67	C₄×#107	—	13	1	no	—
68	C₂×#334	—	13	1	no	—
69	C₂×#322	—	13	1	no	—
70	#1279	—	13	2	no	—
71	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₁₃	—	4	2	no	—
73	C₂×C₄×C₁₃	—	4	2	no	—
74	C₂×C₄×C₁₃	—	4	2	no	—
75	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
78	C₂×C₄×C₁₃	#131	1	2	yes	—
79	C₂×Dic₁₃	—	2	2	no	—
80	C₂×Dic₁₃	—	2	4	no	—
81	C₂×Dic₁₃	—	4	2	no	—
82	C₂×Dic₁₃	—	4	2	no	—
83	C₂×Dic₁₃	#121	1	1	yes	characteristic
84	C₂×Dic₁₃	C₂²⋊Q₈	1	1	yes	characteristic
85	C₂×Dic₁₃	#125	1	1	yes	characteristic
86	C₂×Dic₁₃	C₄⋊Q₈	1	1	yes	characteristic
87	C₂×Dic₁₃	AGL(1,ℤ/8)	1	2	yes	—
88	C₂×Dic₁₃	#137	1	2	yes	—
89	#1057	—	2	4	no	—
90	#1057	—	4	2	no	—
91	#1057	—	8	2	no	—
92	C₈×C₁₃	—	2	2	no	—
93	#1055	—	2	2	no	—
94	C₂×#107	—	13	2	no	—
95	C₂×C₄⋊C₈	—	13	2	no	—
96	C₂×C₄⋊C₈	—	26	2	no	—
97	C₂×C₄×C₈	—	13	1	no	—
98	C₂×C₄×C₈	—	13	1	no	—
99	C₂×C₄×C₈	—	13	1	no	—
100	C₄×C₄⋊C₄	—	13	2	no	—
101	#340	—	13	1	no	—
102	#340	#261	1	1	yes	characteristic
103	#336	—	26	2	no	—
104	#334	—	26	2	no	—
105	#322	—	26	2	no	—
106	C₂²×C₁₃	#389	1	1	yes	characteristic
107	C₂²×C₁₃	#435	1	2	yes	—
108	C₂²×C₁₃	#441	1	1	yes	characteristic
109	C₂²×C₁₃	#442	1	1	yes	characteristic
110	C₂²×C₁₃	#443	1	2	yes	—
111	C₄×C₁₃	—	8	2	no	—
112	C₄×C₁₃	#423	1	1	yes	characteristic
113	C₄×C₁₃	#424	1	1	yes	characteristic
114	C₄×C₁₃	#446	1	2	yes	—
115	Dic₁₃	—	2	1	no	—
116	Dic₁₃	—	2	1	no	—
117	Dic₁₃	—	4	4	no	—
118	Dic₁₃	#468	1	2	yes	—
119	Dic₁₃	#501	1	2	yes	—
120	C₂²×C₈	—	13	1	no	—
121	C₂²×C₈	—	13	2	no	—
122	C₂²×C₈	—	26	1	no	—
123	C₂²×C₈	—	26	2	no	—
124	C₂²×C₈	C₂×#261	1	1	yes	characteristic
125	C₂×C₄⋊C₄	—	13	2	no	—
126	C₂×C₄⋊C₄	C₂×#261	1	2	yes	—
127	C₂×C₄²	—	13	1	no	—
128	C₂×C₄²	—	26	2	no	—
129	#107	—	26	4	no	—
130	C₄⋊C₈	—	26	2	no	—
131	C₄⋊C₈	—	26	2	no	—
132	C₄⋊C₈	—	52	4	no	—
133	C₄×C₈	—	13	2	no	—
134	C₄×C₈	—	13	2	no	—
135	C₄×C₈	—	13	2	no	—
136	C₄×C₈	—	26	1	no	—
137	C₄×C₈	—	26	1	no	—
138	C₄×C₈	—	26	1	no	—
139	#95	—	26	2	no	—
140	C₂×C₁₃	#1586	1	2	yes	—
141	C₂×C₁₃	#1587	1	2	yes	—
142	C₂×C₁₃	#1940	1	1	yes	characteristic
143	C₂×C₁₃	#1943	1	1	yes	characteristic
144	C₂×C₁₃	#1945	1	1	yes	characteristic
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	—	13	1	no	—
147	C₂²×C₄	—	13	1	no	—
148	C₂²×C₄	—	26	2	no	—
149	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	13	2	no	—
152	C₂×C₈	—	26	2	no	—
153	C₂×C₈	—	26	4	no	—
154	C₂×C₈	—	26	2	no	—
155	C₂×C₈	—	26	2	no	—
156	C₂×C₈	—	26	2	no	—
157	C₂×C₈	—	26	2	no	—
158	C₂×C₈	—	52	2	no	—
159	C₂×C₈	—	52	2	no	—
160	C₂×C₈	—	52	2	no	—
161	C₂×C₈	C₄⋊#261	1	2	yes	—
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	—	13	4	no	—
165	C₄⋊C₄	—	26	2	no	—
166	C₄²	—	13	1	no	—
167	C₄²	—	13	1	no	—
168	C₄²	—	13	1	no	—
169	C₄²	—	13	1	no	—
170	C₄²	—	52	2	no	—
171	C₄²	—	52	2	no	—
172	C₁₃	#9599	1	1	yes	characteristic, complemented, Sylow
173	C₂³	D₄×#261	1	1	yes	characteristic
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	13	1	no	—
178	C₂×C₄	—	13	1	no	—
179	C₂×C₄	—	13	2	no	—
180	C₂×C₄	—	13	2	no	—
181	C₂×C₄	—	13	1	no	—
182	C₂×C₄	—	13	1	no	—
183	C₂×C₄	—	26	2	no	—
184	C₂×C₄	—	26	4	no	—
185	C₂×C₄	—	52	2	no	—
186	C₂×C₄	—	52	2	no	—
187	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
188	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
189	C₂×C₄	#88238	1	2	yes	—
190	C₂×C₄	Q₈×#261	1	1	yes	characteristic
191	C₈	—	2	2	no	—
192	C₈	—	26	4	no	—
193	C₈	—	26	2	no	—
194	C₈	—	52	2	no	—
195	C₈	—	104	2	no	—
196	C₂²	#11719065	1	2	yes	—
197	C₂²	#11719083	1	1	yes	characteristic
198	C₂²	#11719102	1	1	yes	characteristic
199	C₂²	#11719107	1	2	yes	—
200	C₂²	#11719110	1	1	yes	characteristic
201	C₄	—	8	2	no	—
202	C₄	—	13	2	no	—
203	C₄	—	13	2	no	—
204	C₄	—	26	1	no	—
205	C₄	—	26	1	no	—
206	C₄	—	52	4	no	—
207	C₄	#11719053	1	1	yes	characteristic
208	C₄	#11719067	1	1	yes	characteristic
209	C₄	#11719078	1	2	yes	—
210	C₂	#13130458	1	2	yes	—
211	C₂	#13130462	1	2	yes	—
212	C₂	#13130937	1	1	yes	characteristic
213	C₂	#13130941	1	1	yes	characteristic
214	C₂	#13130943	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13130934

C₂

yes

maximal

C₂×#11718253

C₂

yes

maximal, characteristic

C₂×#1057⋊C₈

C₂

yes

maximal, characteristic

#13129672

C₂

yes

maximal

#13114728

C₂

yes

maximal, characteristic

C₂×#88112

—

C₂×C₄⋊#1057

C₂²

yes

—

C₂×C₄×#1057

C₂²

yes

characteristic

C₂×C₄×#1057

C₂²

yes

characteristic

C₂×C₈×Dic₁₃

C₂²

yes

characteristic

C₄⋊C₄×Dic₁₃

C₂²

yes

—

C₂×#88047

C₄

yes

—

#11718290

—

#11718253

—

#1057⋊C₈

—

C₁₃×#340

C₄

yes

characteristic

#11718143

C₄

yes

characteristic

C₂²×#1057

—

C₂²×#1057

—

C₂²×#1057

D₄

yes

—

C₂²×C₈×C₁₃

C₂×C₄

yes

characteristic

C₂×C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄⋊Dic₁₃

C₂×C₄

yes

—

C₂×C₄×Dic₁₃

—

C₂×C₄×Dic₁₃

C₂³

yes

characteristic

C₂²×#1055

C₂×C₄

yes

—

—

—

—

—

—

Q₈

yes

—

#88071

—

C₈×Dic₁₃

D₄

yes

—

C₈×Dic₁₃

Q₈

yes

—

#88047

—

#9599

—

maximal, Sylow

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

C₂×D₄

yes

C₂²×Dic₁₃

yes

—

—

—

—

—

—

—

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

M₁₆

yes

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

C₂×D₄

yes

characteristic

C₄×Dic₁₃

C₂×Q₈

yes

characteristic

C₄×Dic₁₃

C₂×Q₈

yes

C₄×Dic₁₃

yes

—

C₄⋊C₄

yes

—

—

—

—

—

—

C₂³×C₁₃

C₄×D₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₄×Q₈

yes

characteristic

C₂×C₄×C₁₃

C₂×M₁₆

yes

—

C₂×C₄×C₁₃

#131

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

#121

yes

characteristic

C₂×Dic₁₃

C₂²⋊Q₈

yes

characteristic

C₂×Dic₁₃

#125

yes

characteristic

C₂×Dic₁₃

C₄⋊Q₈

yes

characteristic

C₂×Dic₁₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₁₃

#137

yes

—

#1057

—

#1057

—

#1057

—

C₈×C₁₃

—

#1055

—

C₂×#107

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

100

C₄×C₄⋊C₄

—

101

#340

—

102

yes

103

—

104

#334

—

105

#322

—

106

C₂²×C₁₃

#389

yes

characteristic

107

C₂²×C₁₃

#435

yes

—

108

C₂²×C₁₃

#441

yes

characteristic

109

C₂²×C₁₃

#442

yes

characteristic

110

C₂²×C₁₃

#443

yes

—

111

C₄×C₁₃

—

112

C₄×C₁₃

#423

yes

characteristic

113

C₄×C₁₃

#424

yes

characteristic

114

C₄×C₁₃

#446

yes

—

115

Dic₁₃

—

116

Dic₁₃

—

117

Dic₁₃

—

118

Dic₁₃

#468

yes

—

119

Dic₁₃

#501

yes

—

120

C₂²×C₈

—

121

C₂²×C₈

—

122

C₂²×C₈

—

123

C₂²×C₈

—

124

C₂²×C₈

C₂×#261

yes

characteristic

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

C₂×#261

yes

—

127

C₂×C₄²

—

128

C₂×C₄²

—

129

#107

—

130

C₄⋊C₈

—

131

C₄⋊C₈

—

132

C₄⋊C₈

—

133

C₄×C₈

—

134

C₄×C₈

—

135

C₄×C₈

—

136

C₄×C₈

—

137

C₄×C₈

—

138

C₄×C₈

—

139

#95

—

140

C₂×C₁₃

#1586

yes

—

141

C₂×C₁₃

#1587

yes

—

142

C₂×C₁₃

#1940

yes

characteristic

143

C₂×C₁₃

#1943

yes

characteristic

144

C₂×C₁₃

#1945

yes

characteristic

145

C₂²×C₄

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

C₂²×#261

yes

characteristic

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

C₄⋊#261

yes

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄²

—

167

C₄²

—

168

C₄²

—

169

C₄²

—

170

C₄²

—

171

C₄²

—

172

C₁₃

#9599

yes

characteristic, complemented, Sylow

173

C₂³

D₄×#261

yes

characteristic

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

D₁₃⋊M₁₆

yes

—

188

C₂×C₄

C₂×C₄⋊#261

yes

characteristic

189

C₂×C₄

#88238

yes

—

190

C₂×C₄

Q₈×#261

yes

characteristic

191

C₈

—

192

C₈

—

193

C₈

—

194

C₈

—

195

C₈

—

104

—

196

C₂²

#11719065

yes

—

197

C₂²

#11719083

yes

characteristic

198

C₂²

#11719102

yes

characteristic

199

C₂²

#11719107

yes

—

200

C₂²

#11719110

yes

characteristic

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

—

206

C₄

—

207

C₄

#11719053

yes

characteristic

208

C₄

#11719067

yes

characteristic

209

C₄

#11719078

yes

—

210

C₂

#13130458

yes

—

211

C₂

#13130462

yes

—

212

C₂

#13130937

yes

characteristic

213

C₂

#13130941

yes

characteristic

214

C₂

#13130943

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×#261
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₁₃×#340	C₄
socle, prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	#340	#261
hypocenter	C₁₃	#9599
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×#261

commutator

C₂²×C₄×C₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×#261

Fitting

C₁₃×#340

C₄

socle, prime radical

C₂³×C₁₃

C₄×D₄

cosocle

C₂×C₄×Dic₁₃

C₂³

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
172. C₁₃	#9599	semidirect

H ≅ G/N

type

172. C₁₃

#9599

semidirect

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

subgroup	index
#13130934	2
C₂×#11718253	2
C₂×#1057⋊C₈	2
#13129672	2
#13114728	2
#9599	13

subgroup

index

C₂×#1057⋊C₈

Subgroups — #16784426

Conjugacy classes of subgroups

Complements

Maximal subgroups