Group #18018266 — subgroups

Subgroups — C₁₉×#12307 #18018266

Order 4864.

318 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 70 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	C₁₉×#3253	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2238	C₂	1	2	yes	maximal, complemented
3	C₁₉×#2102	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1339	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1327	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#532	—	2	1	no	—
7	C₁₉×#476	—	2	2	no	—
8	C₁₉×#469	—	2	1	no	—
9	C₁₉×#469	C₂²	1	2	yes	—
10	C₁₉×#459	—	2	2	no	—
11	C₁₉×#430	C₂²	1	1	yes	characteristic
12	C₁₉×#423	C₂²	1	1	yes	characteristic
13	C₁₉×#365	—	2	2	no	—
14	C₁₉×#361	—	2	2	no	—
15	C₁₉×#349	C₂²	1	2	yes	—
16	C₁₉×#344	—	2	1	no	—
17	C₁₉×#339	—	2	1	no	—
18	C₁₉×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
19	C₂×#41×C₁₉	—	4	1	no	—
20	C₂×D₈×C₁₉	—	2	1	no	—
21	C₂×D₈×C₁₉	D₄	1	2	yes	—
22	C₂×M₁₆×C₁₉	—	2	1	no	—
23	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
24	C₂×M₁₆×C₁₉	C₂³	1	1	yes	characteristic
25	C₁₉×C₄⋊Q₈	—	2	1	no	—
26	C₁₉×#127	—	2	1	no	—
27	C₁₉×#124	—	4	1	no	—
28	C₁₉×#121	—	2	2	no	—
29	C₁₉×#121	—	4	1	no	—
30	C₄×D₄×C₁₉	—	2	2	no	—
31	C₄×D₄×C₁₉	—	4	1	no	—
32	C₁₉×#117	—	2	1	no	—
33	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
34	C₁₉×M₃₂	—	2	2	no	—
35	C₁₉×#108	—	2	2	no	—
36	C₁₉×#106	—	2	1	no	—
37	C₁₉×#106	D₄	1	2	yes	—
38	C₁₉×C₄⋊C₈	—	2	2	no	—
39	C₁₉×#102	—	2	2	no	—
40	C₁₉×#102	—	2	2	no	—
41	#41×C₁₉	—	8	4	no	—
42	C₂×Q₈×C₁₉	—	4	1	no	—
43	C₂×D₄×C₁₉	—	2	2	no	—
44	C₂×D₄×C₁₉	—	4	1	no	—
45	C₂×D₄×C₁₉	—	4	1	no	—
46	C₂×D₄×C₁₉	—	4	2	no	—
47	C₂²×C₄×C₁₉	—	2	1	no	—
48	C₂²×C₄×C₁₉	—	4	2	no	—
49	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
50	D₈×C₁₉	—	4	2	no	—
51	D₈×C₁₉	—	8	1	no	—
52	M₁₆×C₁₉	—	4	1	no	—
53	M₁₆×C₁₉	—	4	1	no	—
54	M₁₆×C₁₉	—	4	1	no	—
55	M₁₆×C₁₉	—	4	1	no	—
56	C₂×C₈×C₁₉	—	2	1	no	—
57	C₂×C₈×C₁₉	—	4	1	no	—
58	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
59	C₄⋊C₄×C₁₉	—	2	2	no	—
60	C₄⋊C₄×C₁₉	—	4	1	no	—
61	C₄⋊C₄×C₁₉	—	4	1	no	—
62	C₂²⋊C₄×C₁₉	—	4	2	no	—
63	C₂²⋊C₄×C₁₉	—	4	2	no	—
64	C₂²⋊C₄×C₁₉	—	4	1	no	—
65	C₄²×C₁₉	—	2	1	no	—
66	C₄²×C₁₉	—	4	1	no	—
67	C₄²×C₁₉	SD₁₆	1	1	yes	characteristic
68	C₄²×C₁₉	SD₁₆	1	1	yes	characteristic
69	C₁₆×C₁₉	—	4	2	no	—
70	#12307	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂³×C₁₉	—	4	2	no	—
72	C₂³×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
73	Q₈×C₁₉	—	8	2	no	—
74	D₄×C₁₉	—	4	2	no	—
75	D₄×C₁₉	—	4	2	no	—
76	D₄×C₁₉	—	8	2	no	—
77	D₄×C₁₉	—	8	2	no	—
78	D₄×C₁₉	—	8	4	no	—
79	C₂×C₄×C₁₉	—	2	1	no	—
80	C₂×C₄×C₁₉	—	2	1	no	—
81	C₂×C₄×C₁₉	—	2	1	no	—
82	C₂×C₄×C₁₉	—	2	1	no	—
83	C₂×C₄×C₁₉	—	4	2	no	—
84	C₂×C₄×C₁₉	—	4	2	no	—
85	C₂×C₄×C₁₉	—	8	4	no	—
86	C₂×C₄×C₁₉	—	8	1	no	—
87	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
88	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
89	C₈×C₁₉	—	2	2	no	—
90	C₈×C₁₉	—	4	1	no	—
91	C₈×C₁₉	—	8	1	no	—
92	#3253	C₂×C₁₉	1	1	yes	characteristic
93	#2238	C₂×C₁₉	1	2	yes	complemented
94	#2102	C₂×C₁₉	1	1	yes	characteristic, complemented
95	#1339	C₂×C₁₉	1	2	yes	complemented
96	#1327	C₂×C₁₉	1	1	yes	characteristic, complemented
97	C₂²×C₁₉	—	2	1	no	—
98	C₂²×C₁₉	—	4	1	no	—
99	C₂²×C₁₉	—	8	2	no	—
100	C₂²×C₁₉	—	8	2	no	—
101	C₂²×C₁₉	—	8	2	no	—
102	C₂²×C₁₉	#450	1	1	yes	characteristic
103	C₄×C₁₉	—	2	2	no	—
104	C₄×C₁₉	—	2	1	no	—
105	C₄×C₁₉	—	4	1	no	—
106	C₄×C₁₉	—	8	2	no	—
107	C₄×C₁₉	—	16	1	no	—
108	C₄×C₁₉	#454	1	1	yes	characteristic
109	C₄×C₁₉	#458	1	1	yes	characteristic
110	#532	—	2	1	no	—
111	#476	—	2	2	no	—
112	#469	—	2	1	no	—
113	#469	C₂²×C₁₉	1	2	yes	—
114	#459	—	2	2	no	—
115	#430	C₂²×C₁₉	1	1	yes	characteristic
116	#423	C₂²×C₁₉	1	1	yes	characteristic
117	#365	—	2	2	no	—
118	#361	—	2	2	no	—
119	#349	C₂²×C₁₉	1	2	yes	—
120	#344	—	2	1	no	—
121	#339	—	2	1	no	—
122	H(3,ℤ/4)	C₂²×C₁₉	1	1	yes	characteristic
123	C₂×C₁₉	—	2	1	no	—
124	C₂×C₁₉	—	4	1	no	—
125	C₂×C₁₉	—	16	2	no	—
126	C₂×C₁₉	#1616	1	1	yes	characteristic
127	C₂×#41	—	4	1	no	—
128	C₂×D₈	—	2	1	no	—
129	C₂×D₈	D₄×C₁₉	1	2	yes	—
130	C₂×M₁₆	—	2	1	no	—
131	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
132	C₂×M₁₆	C₂³×C₁₉	1	1	yes	characteristic
133	C₄⋊Q₈	—	2	1	no	—
134	#127	—	2	1	no	—
135	#124	—	4	1	no	—
136	#121	—	2	2	no	—
137	#121	—	4	1	no	—
138	C₄×D₄	—	2	2	no	—
139	C₄×D₄	—	4	1	no	—
140	#117	—	2	1	no	—
141	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
142	M₃₂	—	2	2	no	—
143	#108	—	2	2	no	—
144	#106	—	2	1	no	—
145	#106	D₄×C₁₉	1	2	yes	—
146	C₄⋊C₈	—	2	2	no	—
147	#102	—	2	2	no	—
148	#102	—	2	2	no	—
149	C₁₉	#12307	1	1	yes	characteristic, direct product factor, Sylow
150	#41	—	8	4	no	—
151	C₂×Q₈	—	4	1	no	—
152	C₂×D₄	—	2	2	no	—
153	C₂×D₄	—	4	1	no	—
154	C₂×D₄	—	4	1	no	—
155	C₂×D₄	—	4	2	no	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	4	2	no	—
158	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
159	D₈	—	4	2	no	—
160	D₈	—	8	1	no	—
161	M₁₆	—	4	1	no	—
162	M₁₆	—	4	1	no	—
163	M₁₆	—	4	1	no	—
164	M₁₆	—	4	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	4	1	no	—
167	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	4	1	no	—
170	C₄⋊C₄	—	4	1	no	—
171	C₂²⋊C₄	—	4	2	no	—
172	C₂²⋊C₄	—	4	2	no	—
173	C₂²⋊C₄	—	4	1	no	—
174	C₄²	—	2	1	no	—
175	C₄²	—	4	1	no	—
176	C₄²	SD₁₆×C₁₉	1	1	yes	characteristic
177	C₄²	SD₁₆×C₁₉	1	1	yes	characteristic
178	C₁₆	—	4	2	no	—
179	C₂³	—	4	2	no	—
180	C₂³	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
181	Q₈	—	8	2	no	—
182	D₄	—	4	2	no	—
183	D₄	—	4	2	no	—
184	D₄	—	8	2	no	—
185	D₄	—	8	2	no	—
186	D₄	—	8	4	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	8	4	no	—
194	C₂×C₄	—	8	1	no	—
195	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
196	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
197	C₈	—	2	2	no	—
198	C₈	—	4	1	no	—
199	C₈	—	8	1	no	—
200	C₂²	—	2	1	no	—
201	C₂²	—	4	1	no	—
202	C₂²	—	8	2	no	—
203	C₂²	—	8	2	no	—
204	C₂²	—	8	2	no	—
205	C₂²	C₁₉×#450	1	1	yes	characteristic
206	C₄	—	2	2	no	—
207	C₄	—	2	1	no	—
208	C₄	—	4	1	no	—
209	C₄	—	8	2	no	—
210	C₄	—	16	1	no	—
211	C₄	C₁₉×#454	1	1	yes	characteristic
212	C₄	C₁₉×#458	1	1	yes	characteristic
213	C₂	—	2	1	no	—
214	C₂	—	4	1	no	—
215	C₂	—	16	2	no	—
216	C₂	C₁₉×#1616	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#3253

C₂

yes

maximal, characteristic

C₁₉×#2238

C₂

yes

maximal, complemented

C₁₉×#2102

C₂

yes

maximal, characteristic, complemented

C₁₉×#1339

C₂

yes

maximal, complemented

C₁₉×#1327

C₂

yes

maximal, characteristic, complemented

C₁₉×#532

—

C₁₉×#476

—

C₁₉×#469

—

C₁₉×#469

C₂²

yes

—

C₁₉×#459

—

C₁₉×#430

C₂²

yes

characteristic

C₁₉×#423

C₂²

yes

characteristic

C₁₉×#365

—

C₁₉×#361

—

C₁₉×#349

C₂²

yes

—

C₁₉×#344

—

C₁₉×#339

—

C₁₉×H(3,ℤ/4)

C₂²

yes

characteristic

C₂×#41×C₁₉

—

C₂×D₈×C₁₉

—

C₂×D₈×C₁₉

D₄

yes

—

C₂×M₁₆×C₁₉

—

C₂×M₁₆×C₁₉

D₄

yes

characteristic

C₂×M₁₆×C₁₉

C₂³

yes

characteristic

C₁₉×C₄⋊Q₈

—

C₁₉×#127

—

C₁₉×#124

—

C₁₉×#121

—

C₁₉×#121

—

C₄×D₄×C₁₉

—

C₄×D₄×C₁₉

—

C₁₉×#117

—

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₁₉×M₃₂

—

C₁₉×#108

—

C₁₉×#106

—

C₁₉×#106

D₄

yes

—

C₁₉×C₄⋊C₈

—

C₁₉×#102

—

C₁₉×#102

—

#41×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

D₈×C₁₉

—

D₈×C₁₉

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₂×D₄

yes

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

SD₁₆

yes

characteristic

C₄²×C₁₉

SD₁₆

yes

characteristic

C₁₆×C₁₉

—

#12307

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

AGL(1,ℤ/8)

yes

characteristic

Q₈×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

#120

yes

characteristic

C₂×C₄×C₁₉

C₂×SD₁₆

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

#3253

C₂×C₁₉

yes

characteristic

#2238

C₂×C₁₉

yes

complemented

#2102

C₂×C₁₉

yes

characteristic, complemented

#1339

C₂×C₁₉

yes

complemented

#1327

C₂×C₁₉

yes

characteristic, complemented

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

100

C₂²×C₁₉

—

101

C₂²×C₁₉

—

102

C₂²×C₁₉

#450

yes

characteristic

103

C₄×C₁₉

—

104

C₄×C₁₉

—

105

C₄×C₁₉

—

106

C₄×C₁₉

—

107

C₄×C₁₉

—

108

C₄×C₁₉

#454

yes

characteristic

109

C₄×C₁₉

#458

yes

characteristic

110

#532

—

111

#476

—

112

#469

—

113

#469

C₂²×C₁₉

yes

—

114

#459

—

115

#430

C₂²×C₁₉

yes

characteristic

116

#423

C₂²×C₁₉

yes

characteristic

117

#365

—

118

#361

—

119

#349

C₂²×C₁₉

yes

—

120

#344

—

121

#339

—

122

H(3,ℤ/4)

C₂²×C₁₉

yes

characteristic

123

C₂×C₁₉

—

124

C₂×C₁₉

—

125

C₂×C₁₉

—

126

C₂×C₁₉

#1616

yes

characteristic

127

C₂×#41

—

128

C₂×D₈

—

129

C₂×D₈

D₄×C₁₉

yes

—

130

C₂×M₁₆

—

131

C₂×M₁₆

D₄×C₁₉

yes

characteristic

132

C₂×M₁₆

C₂³×C₁₉

yes

characteristic

133

C₄⋊Q₈

—

134

#127

—

135

#124

—

136

#121

—

137

#121

—

138

C₄×D₄

—

139

C₄×D₄

—

140

#117

—

141

C₂×C₄²

D₄×C₁₉

yes

characteristic

142

M₃₂

—

143

#108

—

144

#106

—

145

#106

D₄×C₁₉

yes

—

146

C₄⋊C₈

—

147

#102

—

148

#102

—

149

C₁₉

#12307

yes

characteristic, direct product factor, Sylow

150

#41

—

151

C₂×Q₈

—

152

C₂×D₄

—

153

C₂×D₄

—

154

C₂×D₄

—

155

C₂×D₄

—

156

C₂²×C₄

—

157

C₂²×C₄

—

158

C₂²×C₄

C₂×D₄×C₁₉

yes

characteristic

159

D₈

—

160

D₈

—

161

M₁₆

—

162

M₁₆

—

163

M₁₆

—

164

M₁₆

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

C₂×D₄×C₁₉

yes

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₂²⋊C₄

—

172

C₂²⋊C₄

—

173

C₂²⋊C₄

—

174

C₄²

—

175

C₄²

—

176

C₄²

SD₁₆×C₁₉

yes

characteristic

177

C₄²

SD₁₆×C₁₉

yes

characteristic

178

C₁₆

—

179

C₂³

—

180

C₂³

C₁₉×AGL(1,ℤ/8)

yes

characteristic

181

Q₈

—

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

C₁₉×#120

yes

characteristic

196

C₂×C₄

C₂×SD₁₆×C₁₉

yes

characteristic

197

C₈

—

198

C₈

—

199

C₈

—

200

C₂²

—

201

C₂²

—

202

C₂²

—

203

C₂²

—

204

C₂²

—

205

C₂²

C₁₉×#450

yes

characteristic

206

C₄

—

207

C₄

—

208

C₄

—

209

C₄

—

210

C₄

—

211

C₄

C₁₉×#454

yes

characteristic

212

C₄

C₁₉×#458

yes

characteristic

213

C₂

—

214

C₂

—

215

C₂

—

216

C₂

C₁₉×#1616

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₁₉	#1616
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₁₉
prime radical	C₁₉×#3253	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂×M₁₆×C₁₉ ◁ G

subgroup

G/H

center, socle

C₂×C₁₉

#1616

commutator, Frattini, cosocle

C₂×M₁₆

C₂³×C₁₉

prime radical

C₁₉×#3253

C₂

LCS

G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂×M₁₆×C₁₉ ◁ G

N	H ≅ G/N	type
2. C₁₉×#2238	C₂	semidirect
3. C₁₉×#2102	C₂	semidirect
4. C₁₉×#1339	C₂	semidirect
5. C₁₉×#1327	C₂	semidirect
70. #12307	C₁₉	direct (N × H)
93. #2238	C₂×C₁₉	semidirect
94. #2102	C₂×C₁₉	semidirect
95. #1339	C₂×C₁₉	semidirect
96. #1327	C₂×C₁₉	semidirect
149. C₁₉	#12307	direct (N × H)

H ≅ G/N

type

2. C₁₉×#2238

C₂

semidirect

3. C₁₉×#2102

C₂

semidirect

4. C₁₉×#1339

C₂

semidirect

5. C₁₉×#1327

C₂

semidirect

70. #12307

C₁₉

direct (N × H)

93. #2238

C₂×C₁₉

semidirect

94. #2102

C₂×C₁₉

semidirect

95. #1339

C₂×C₁₉

semidirect

96. #1327

C₂×C₁₉

semidirect

149. C₁₉

#12307

direct (N × H)

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

subgroup	index
C₁₉×#3253	2
C₁₉×#2238	2
C₁₉×#2102	2
C₁₉×#1339	2
C₁₉×#1327	2
#12307	19

subgroup

index

C₁₉×#3253

C₁₉×#2238

C₁₉×#2102

C₁₉×#1339

C₁₉×#1327

#12307

Subgroups — C₁₉×#12307 #18018266

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#12307 #18018266

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12307 #18018266