Group #18070052 — subgroups

Subgroups — #18070052

Order 4864.

304 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 43 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	#14561762	C₂	1	1	yes	maximal, characteristic, complemented
2	#14561569	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#2014	C₂	1	1	yes	maximal, characteristic
4	C₂×#10597293	—	2	1	no	—
5	#11926270	—	2	1	no	—
6	C₂²×SD₁₆×C₁₉	—	2	1	no	—
7	C₂×C₁₉×#127	C₄	1	1	yes	characteristic
8	C₂×C₁₉×#102	—	2	1	no	—
9	C₁₉×#391	C₂²	1	1	yes	characteristic
10	C₁₉×#336	C₄	1	1	yes	characteristic
11	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
12	C₂²×D₄×C₁₉	—	2	1	no	—
13	C₂²×D₄×C₁₉	—	4	1	no	—
14	C₂×SD₁₆×C₁₉	—	4	2	no	—
15	C₂×SD₁₆×C₁₉	—	4	2	no	—
16	C₂×SD₁₆×C₁₉	—	4	1	no	—
17	C₂×SD₁₆×C₁₉	—	4	1	no	—
18	C₂²×C₈×C₁₉	—	2	1	no	—
19	C₁₉×#127	—	4	2	no	—
20	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂×#64520	—	4	1	no	—
23	C₁₉×#102	—	4	2	no	—
24	C₁₉×#95	—	4	1	no	—
25	#10597293	—	2	2	no	—
26	#10597293	—	4	1	no	—
27	C₂²⋊(C₁₉⋊C₈)	—	4	1	no	—
28	C₂⁴×C₁₉	—	4	1	no	—
29	C₂×Q₈×C₁₉	—	2	1	no	—
30	C₂×Q₈×C₁₉	—	4	1	no	—
31	C₂×Q₈×C₁₉	—	4	1	no	—
32	C₂×Q₈×C₁₉	D₈	1	1	yes	characteristic
33	C₂×Q₈×C₁₉	SD₁₆	1	1	yes	characteristic
34	C₂×D₄×C₁₉	—	2	2	no	—
35	C₂×D₄×C₁₉	—	4	1	no	—
36	C₂×D₄×C₁₉	—	4	1	no	—
37	C₂×D₄×C₁₉	—	4	1	no	—
38	C₂×D₄×C₁₉	—	4	2	no	—
39	C₂×D₄×C₁₉	—	4	2	no	—
40	C₂×D₄×C₁₉	—	4	1	no	—
41	C₂×D₄×C₁₉	—	4	1	no	—
42	C₂×D₄×C₁₉	—	8	2	no	—
43	C₂×D₄×C₁₉	—	8	2	no	—
44	C₂²×C₄×C₁₉	—	2	1	no	—
45	C₂²×C₄×C₁₉	—	2	1	no	—
46	C₂²×C₄×C₁₉	—	2	1	no	—
47	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
48	SD₁₆×C₁₉	—	4	4	no	—
49	SD₁₆×C₁₉	—	4	4	no	—
50	C₂×C₈×C₁₉	—	2	2	no	—
51	C₂×C₈×C₁₉	—	4	1	no	—
52	C₂×C₈×C₁₉	—	4	1	no	—
53	C₄⋊C₄×C₁₉	—	4	1	no	—
54	C₄²×C₁₉	—	4	1	no	—
55	#64520	—	4	2	no	—
56	#64520	—	8	1	no	—
57	C₂×C₁₉⋊C₈	—	8	1	no	—
58	C₂×C₁₉⋊C₈	—	8	1	no	—
59	#7404	—	19	1	no	maximal, Sylow
60	C₂³×C₁₉	—	4	2	no	—
61	C₂³×C₁₉	—	8	2	no	—
62	C₂³×C₁₉	—	8	2	no	—
63	C₂³×C₁₉	—	8	1	no	—
64	C₂³×C₁₉	—	8	1	no	—
65	C₂³×C₁₉	#99	1	1	yes	characteristic
66	Q₈×C₁₉	—	4	1	no	—
67	Q₈×C₁₉	—	4	1	no	—
68	Q₈×C₁₉	—	8	1	no	—
69	D₄×C₁₉	—	4	2	no	—
70	D₄×C₁₉	—	4	2	no	—
71	D₄×C₁₉	—	8	1	no	—
72	D₄×C₁₉	—	8	1	no	—
73	D₄×C₁₉	—	8	4	no	—
74	D₄×C₁₉	—	8	4	no	—
75	C₂×C₄×C₁₉	—	2	1	no	—
76	C₂×C₄×C₁₉	—	2	1	no	—
77	C₂×C₄×C₁₉	—	2	2	no	—
78	C₂×C₄×C₁₉	—	2	1	no	—
79	C₂×C₄×C₁₉	—	2	1	no	—
80	C₂×C₄×C₁₉	—	4	1	no	—
81	C₂×C₄×C₁₉	—	4	1	no	—
82	C₂×C₄×C₁₉	—	4	1	no	—
83	C₂×C₄×C₁₉	—	4	1	no	—
84	C₂×C₄×C₁₉	—	4	1	no	—
85	C₂×C₄×C₁₉	—	8	1	no	—
86	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
87	C₂×C₄×C₁₉	#104	1	1	yes	characteristic
88	C₈×C₁₉	—	4	2	no	—
89	C₁₉⋊C₈	—	8	2	no	—
90	C₁₉⋊C₈	—	16	1	no	—
91	#2014	D₁₉	1	1	yes	characteristic
92	#1394	—	19	1	no	—
93	#1319	—	19	1	no	—
94	C₂²×C₁₉	—	2	1	no	—
95	C₂²×C₁₉	—	2	1	no	—
96	C₂²×C₁₉	—	8	4	no	—
97	C₂²×C₁₉	—	8	2	no	—
98	C₂²×C₁₉	—	16	2	no	—
99	C₂²×C₁₉	—	16	1	no	—
100	C₂²×C₁₉	—	16	1	no	—
101	C₂²×C₁₉	#327	1	1	yes	characteristic
102	C₂²×C₁₉	AΣL(1,𝔽₁₆)	1	1	yes	characteristic
103	C₂²×C₁₉	#353	1	1	yes	characteristic
104	C₄×C₁₉	—	2	1	no	—
105	C₄×C₁₉	—	2	1	no	—
106	C₄×C₁₉	—	4	2	no	—
107	C₄×C₁₉	—	4	1	no	—
108	C₄×C₁₉	—	4	1	no	—
109	C₄×C₁₉	—	8	1	no	—
110	C₂²×SD₁₆	—	2	1	no	—
111	C₂×#127	Dic₁₉	1	1	yes	characteristic
112	C₂×#102	—	2	1	no	—
113	C₂×#101	—	38	1	no	—
114	#391	C₂×D₁₉	1	1	yes	characteristic
115	#336	Dic₁₉	1	1	yes	characteristic
116	#324	—	38	1	no	—
117	C₂×C₁₉	—	2	1	no	—
118	C₂×C₁₉	—	2	1	no	—
119	C₂×C₁₉	—	16	2	no	—
120	C₂×C₁₉	#1343	1	1	yes	characteristic
121	C₂×C₁₉	#1406	1	1	yes	characteristic
122	C₂×C₁₉	#1410	1	1	yes	characteristic
123	C₂²×Q₈	C₂²⋊D₁₉	1	1	yes	characteristic
124	C₂²×D₄	—	2	1	no	—
125	C₂²×D₄	—	4	1	no	—
126	C₂×SD₁₆	—	4	2	no	—
127	C₂×SD₁₆	—	4	2	no	—
128	C₂×SD₁₆	—	4	1	no	—
129	C₂×SD₁₆	—	4	1	no	—
130	C₂×M₁₆	—	76	1	no	—
131	C₂²×C₈	—	2	1	no	—
132	#127	—	4	2	no	—
133	C₂×C₄⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
134	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
135	#102	—	4	2	no	—
136	#101	—	38	2	no	—
137	#101	—	76	1	no	—
138	C₂²⋊C₈	—	76	1	no	—
139	#95	—	4	1	no	—
140	C₁₉	#7404	1	1	yes	characteristic, complemented, Sylow
141	C₂⁴	—	4	1	no	—
142	C₂×Q₈	—	2	1	no	—
143	C₂×Q₈	—	4	1	no	—
144	C₂×Q₈	—	4	1	no	—
145	C₂×Q₈	#64525	1	1	yes	characteristic
146	C₂×Q₈	#64526	1	1	yes	characteristic
147	C₂×D₄	—	2	2	no	—
148	C₂×D₄	—	4	1	no	—
149	C₂×D₄	—	4	1	no	—
150	C₂×D₄	—	4	1	no	—
151	C₂×D₄	—	4	2	no	—
152	C₂×D₄	—	4	2	no	—
153	C₂×D₄	—	4	1	no	—
154	C₂×D₄	—	4	1	no	—
155	C₂×D₄	—	8	2	no	—
156	C₂×D₄	—	8	2	no	—
157	C₂²×C₄	—	2	1	no	—
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
161	SD₁₆	—	4	4	no	—
162	SD₁₆	—	4	4	no	—
163	M₁₆	—	76	2	no	—
164	M₁₆	—	152	1	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	4	1	no	—
167	C₂×C₈	—	4	1	no	—
168	C₂×C₈	—	152	1	no	—
169	C₂×C₈	—	152	1	no	—
170	C₄⋊C₄	—	4	1	no	—
171	C₄²	—	4	1	no	—
172	C₂³	—	4	2	no	—
173	C₂³	—	8	2	no	—
174	C₂³	—	8	2	no	—
175	C₂³	—	8	1	no	—
176	C₂³	—	8	1	no	—
177	C₂³	#10597291	1	1	yes	characteristic
178	Q₈	—	4	1	no	—
179	Q₈	—	4	1	no	—
180	Q₈	—	8	1	no	—
181	D₄	—	4	2	no	—
182	D₄	—	4	2	no	—
183	D₄	—	8	1	no	—
184	D₄	—	8	1	no	—
185	D₄	—	8	4	no	—
186	D₄	—	8	4	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	8	1	no	—
198	C₂×C₄	#10597289	1	1	yes	characteristic
199	C₂×C₄	#10597294	1	1	yes	characteristic
200	C₈	—	4	2	no	—
201	C₈	—	304	1	no	—
202	C₈	—	152	2	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	8	4	no	—
206	C₂²	—	8	2	no	—
207	C₂²	—	16	2	no	—
208	C₂²	—	16	1	no	—
209	C₂²	—	16	1	no	—
210	C₂²	#11926278	1	1	yes	characteristic
211	C₂²	#11926331	1	1	yes	characteristic
212	C₂²	#11926337	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	4	2	no	—
216	C₄	—	4	1	no	—
217	C₄	—	4	1	no	—
218	C₄	—	8	1	no	—
219	C₂	—	2	1	no	—
220	C₂	—	2	1	no	—
221	C₂	—	16	2	no	—
222	C₂	#14561632	1	1	yes	characteristic
223	C₂	#14561792	1	1	yes	characteristic
224	C₂	#14561802	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14561762

C₂

yes

maximal, characteristic, complemented

#14561569

C₂

yes

maximal, characteristic, complemented

C₁₉×#2014

C₂

yes

maximal, characteristic

C₂×#10597293

—

#11926270

—

C₂²×SD₁₆×C₁₉

—

C₂×C₁₉×#127

C₄

yes

characteristic

C₂×C₁₉×#102

—

C₁₉×#391

C₂²

yes

characteristic

C₁₉×#336

C₄

yes

characteristic

C₂²×Q₈×C₁₉

D₄

yes

characteristic

C₂²×D₄×C₁₉

—

C₂²×D₄×C₁₉

—

C₂×SD₁₆×C₁₉

—

C₂×SD₁₆×C₁₉

—

C₂×SD₁₆×C₁₉

—

C₂×SD₁₆×C₁₉

—

C₂²×C₈×C₁₉

—

C₁₉×#127

—

C₂×C₄⋊C₄×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₂×#64520

—

C₁₉×#102

—

C₁₉×#95

—

#10597293

—

#10597293

—

C₂²⋊(C₁₉⋊C₈)

—

C₂⁴×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

D₈

yes

characteristic

C₂×Q₈×C₁₉

SD₁₆

yes

characteristic

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

SD₁₆×C₁₉

—

SD₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄²×C₁₉

—

#64520

—

#64520

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

#7404

—

maximal, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

#99

yes

characteristic

Q₈×C₁₉

—

Q₈×C₁₉

—

Q₈×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₈×C₁₉

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

#2014

D₁₉

yes

characteristic

#1394

—

#1319

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

100

C₂²×C₁₉

—

101

C₂²×C₁₉

#327

yes

characteristic

102

C₂²×C₁₉

AΣL(1,𝔽₁₆)

yes

characteristic

103

C₂²×C₁₉

#353

yes

characteristic

104

C₄×C₁₉

—

105

C₄×C₁₉

—

106

C₄×C₁₉

—

107

C₄×C₁₉

—

108

C₄×C₁₉

—

109

C₄×C₁₉

—

110

C₂²×SD₁₆

—

111

C₂×#127

Dic₁₉

yes

characteristic

112

C₂×#102

—

113

C₂×#101

—

114

#391

C₂×D₁₉

yes

characteristic

115

#336

Dic₁₉

yes

characteristic

116

#324

—

117

C₂×C₁₉

—

118

C₂×C₁₉

—

119

C₂×C₁₉

—

120

C₂×C₁₉

#1343

yes

characteristic

121

C₂×C₁₉

#1406

yes

characteristic

122

C₂×C₁₉

#1410

yes

characteristic

123

C₂²×Q₈

C₂²⋊D₁₉

yes

characteristic

124

C₂²×D₄

—

125

C₂²×D₄

—

126

C₂×SD₁₆

—

127

C₂×SD₁₆

—

128

C₂×SD₁₆

—

129

C₂×SD₁₆

—

130

C₂×M₁₆

—

131

C₂²×C₈

—

132

#127

—

133

C₂×C₄⋊C₄

C₂²⋊D₁₉

yes

characteristic

134

C₂×C₄²

C₂×Dic₁₉

yes

characteristic

135

#102

—

136

#101

—

137

#101

—

138

C₂²⋊C₈

—

139

#95

—

140

C₁₉

#7404

yes

characteristic, complemented, Sylow

141

C₂⁴

—

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂×Q₈

#64525

yes

characteristic

146

C₂×Q₈

#64526

yes

characteristic

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

—

150

C₂×D₄

—

151

C₂×D₄

—

152

C₂×D₄

—

153

C₂×D₄

—

154

C₂×D₄

—

155

C₂×D₄

—

156

C₂×D₄

—

157

C₂²×C₄

—

158

C₂²×C₄

—

159

C₂²×C₄

—

160

C₂²×C₄

C₂²⋊Dic₁₉

yes

characteristic

161

SD₁₆

—

162

SD₁₆

—

163

M₁₆

—

164

M₁₆

—

152

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

152

—

169

C₂×C₈

—

152

—

170

C₄⋊C₄

—

171

C₄²

—

172

C₂³

—

173

C₂³

—

174

C₂³

—

175

C₂³

—

176

C₂³

—

177

C₂³

#10597291

yes

characteristic

178

Q₈

—

179

Q₈

—

180

Q₈

—

181

D₄

—

182

D₄

—

183

D₄

—

184

D₄

—

185

D₄

—

186

D₄

—

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

—

198

C₂×C₄

#10597289

yes

characteristic

199

C₂×C₄

#10597294

yes

characteristic

200

C₈

—

201

C₈

—

304

—

202

C₈

—

152

—

203

C₂²

—

204

C₂²

—

205

C₂²

—

206

C₂²

—

207

C₂²

—

208

C₂²

—

209

C₂²

—

210

C₂²

#11926278

yes

characteristic

211

C₂²

#11926331

yes

characteristic

212

C₂²

#11926337

yes

characteristic

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

—

217

C₄

—

218

C₄

—

219

C₂

—

220

C₂

—

221

C₂

—

222

C₂

#14561632

yes

characteristic

223

C₂

#14561792

yes

characteristic

224

C₂

#14561802

yes

characteristic

	subgroup	G/H
center	C₂²	#11926337
commutator	C₂×C₄²×C₁₉	C₂×C₄
Frattini	#391	C₂×D₁₉
Fitting	C₁₉×#2014	C₂
socle	C₂²×C₁₉	#353
prime radical	C₂×C₁₉×#127	C₄
cosocle	C₁₉×#391	C₂²
hypercenter	#2014	D₁₉
hypocenter	C₁₉	#7404
LCS	G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #391 ◁ #2014

subgroup

G/H

center

C₂²

#11926337

commutator

C₂×C₄²×C₁₉

C₂×C₄

Frattini

#391

C₂×D₁₉

Fitting

C₁₉×#2014

C₂

socle

C₂²×C₁₉

#353

prime radical

C₂×C₁₉×#127

C₄

cosocle

C₁₉×#391

C₂²

D₁₉

C₁₉

G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #391 ◁ #2014

N	H ≅ G/N	type
1. #14561762	C₂	semidirect
2. #14561569	C₂	semidirect
140. C₁₉	#7404	semidirect

H ≅ G/N

type

1. #14561762

C₂

semidirect

2. #14561569

C₂

semidirect

140. C₁₉

#7404

semidirect

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

subgroup	index
#14561762	2
#14561569	2
C₁₉×#2014	2
#7404	19

subgroup

index

#14561762

#14561569

C₁₉×#2014

#7404

Subgroups — #18070052

Conjugacy classes of subgroups

Complements

Maximal subgroups