Group #18018127 — subgroups

Subgroups — C₁₉×#12168 #18018127

Order 4864.

330 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 74 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×#3261	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2188	C₂	1	2	yes	maximal
3	C₁₉×#2047	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1350	C₂	1	2	yes	maximal
5	C₁₉×#1323	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
7	C₁₉×#541	—	2	1	no	—
8	C₂×C₁₉×Q₃₂	—	2	2	no	—
9	C₁₉×#479	—	2	2	no	—
10	C₁₉×#470	—	2	1	no	—
11	C₁₉×#467	C₂²	1	2	yes	—
12	C₁₉×#462	—	2	2	no	—
13	C₁₉×#424	C₂²	1	1	yes	characteristic
14	C₁₉×#419	—	2	1	no	—
15	C₂×C₁₉×#101	—	2	1	no	—
16	C₁₉×#358	—	2	2	no	—
17	C₁₉×C₂²⋊C₁₆	C₂²	1	2	yes	—
18	C₁₉×#341	C₂²	1	1	yes	characteristic
19	C₂²×Q₈×C₁₉	—	2	1	no	—
20	C₂×Q₁₆×C₁₉	—	2	1	no	—
21	C₂×Q₁₆×C₁₉	—	2	2	no	—
22	C₂×Q₁₆×C₁₉	—	4	2	no	—
23	C₂×Q₁₆×C₁₉	D₄	1	2	yes	—
24	C₂×M₁₆×C₁₉	—	2	1	no	—
25	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
27	C₁₉×C₄⋊Q₈	—	2	1	no	—
28	C₁₉×#126	—	4	2	no	—
29	C₁₉×#125	—	2	1	no	—
30	C₁₉×#125	—	4	2	no	—
31	C₁₉×C₂²⋊Q₈	—	2	2	no	—
32	C₄×Q₈×C₁₉	—	2	2	no	—
33	C₁₉×#117	D₄	1	1	yes	characteristic
34	C₁₉×Q₃₂	—	4	4	no	—
35	C₂×C₁₆×C₁₉	—	2	2	no	—
36	C₁₉×#107	D₄	1	2	yes	—
37	C₁₉×#106	—	2	1	no	—
38	C₁₉×C₄⋊C₈	—	2	2	no	—
39	C₁₉×#103	—	2	2	no	—
40	C₁₉×#103	—	2	2	no	—
41	C₁₉×#103	—	4	2	no	—
42	C₁₉×#101	—	4	2	no	—
43	C₂×Q₈×C₁₉	—	2	2	no	—
44	C₂×Q₈×C₁₉	—	2	2	no	—
45	C₂×Q₈×C₁₉	—	4	1	no	—
46	C₂×Q₈×C₁₉	—	4	1	no	—
47	C₂×Q₈×C₁₉	—	4	1	no	—
48	C₂×Q₈×C₁₉	—	4	1	no	—
49	C₂×Q₈×C₁₉	—	4	1	no	—
50	C₂²×C₄×C₁₉	—	4	1	no	—
51	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
52	Q₁₆×C₁₉	—	2	4	no	—
53	Q₁₆×C₁₉	—	4	2	no	—
54	Q₁₆×C₁₉	—	4	2	no	—
55	Q₁₆×C₁₉	—	8	1	no	—
56	M₁₆×C₁₉	—	4	1	no	—
57	M₁₆×C₁₉	—	4	1	no	—
58	M₁₆×C₁₉	—	4	1	no	—
59	C₂×C₈×C₁₉	—	2	1	no	—
60	C₂×C₈×C₁₉	—	4	1	no	—
61	C₂×C₈×C₁₉	—	4	1	no	—
62	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
63	C₄⋊C₄×C₁₉	—	2	2	no	—
64	C₄⋊C₄×C₁₉	—	4	2	no	—
65	C₄⋊C₄×C₁₉	—	4	2	no	—
66	C₄⋊C₄×C₁₉	—	4	2	no	—
67	C₄⋊C₄×C₁₉	—	4	2	no	—
68	C₄⋊C₄×C₁₉	D₈	1	2	yes	—
69	C₂²⋊C₄×C₁₉	—	2	1	no	—
70	C₂²⋊C₄×C₁₉	—	4	2	no	—
71	C₄²×C₁₉	—	2	1	no	—
72	C₄²×C₁₉	—	4	2	no	—
73	C₁₆×C₁₉	—	4	2	no	—
74	#12168	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
76	Q₈×C₁₉	—	4	2	no	—
77	Q₈×C₁₉	—	4	2	no	—
78	Q₈×C₁₉	—	4	2	no	—
79	Q₈×C₁₉	—	4	2	no	—
80	Q₈×C₁₉	—	8	1	no	—
81	Q₈×C₁₉	—	8	1	no	—
82	C₂×C₄×C₁₉	—	2	1	no	—
83	C₂×C₄×C₁₉	—	2	1	no	—
84	C₂×C₄×C₁₉	—	2	2	no	—
85	C₂×C₄×C₁₉	—	4	2	no	—
86	C₂×C₄×C₁₉	—	4	2	no	—
87	C₂×C₄×C₁₉	—	4	2	no	—
88	C₂×C₄×C₁₉	—	8	1	no	—
89	C₂×C₄×C₁₉	—	8	1	no	—
90	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
91	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
92	C₈×C₁₉	—	2	2	no	—
93	C₈×C₁₉	—	4	1	no	—
94	C₈×C₁₉	—	8	1	no	—
95	#3261	C₂×C₁₉	1	1	yes	characteristic
96	#2188	C₂×C₁₉	1	2	yes	—
97	#2047	C₂×C₁₉	1	1	yes	characteristic
98	#1350	C₂×C₁₉	1	2	yes	—
99	#1323	C₂×C₁₉	1	1	yes	characteristic
100	C₂²×C₁₉	—	2	1	no	—
101	C₂²×C₁₉	—	2	2	no	—
102	C₂²×C₁₉	#447	1	1	yes	characteristic
103	C₄×C₁₉	—	2	1	no	—
104	C₄×C₁₉	—	4	2	no	—
105	C₄×C₁₉	—	8	2	no	—
106	C₄×C₁₉	—	8	2	no	—
107	C₄×C₁₉	—	8	2	no	—
108	C₄×C₁₉	#454	1	1	yes	characteristic
109	C₄×C₁₉	B(4,𝔽₂)	1	1	yes	characteristic
110	C₂²×Q₁₆	C₂²×C₁₉	1	1	yes	characteristic
111	#541	—	2	1	no	—
112	C₂×Q₃₂	—	2	2	no	—
113	#479	—	2	2	no	—
114	#470	—	2	1	no	—
115	#467	C₂²×C₁₉	1	2	yes	—
116	#462	—	2	2	no	—
117	#424	C₂²×C₁₉	1	1	yes	characteristic
118	#419	—	2	1	no	—
119	C₂×#101	—	2	1	no	—
120	#358	—	2	2	no	—
121	C₂²⋊C₁₆	C₂²×C₁₉	1	2	yes	—
122	#341	C₂²×C₁₉	1	1	yes	characteristic
123	C₂×C₁₉	—	4	1	no	—
124	C₂×C₁₉	#1598	1	1	yes	characteristic
125	C₂×C₁₉	#2195	1	2	yes	—
126	C₂²×Q₈	—	2	1	no	—
127	C₂×Q₁₆	—	2	1	no	—
128	C₂×Q₁₆	—	2	2	no	—
129	C₂×Q₁₆	—	4	2	no	—
130	C₂×Q₁₆	D₄×C₁₉	1	2	yes	—
131	C₂×M₁₆	—	2	1	no	—
132	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
133	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
134	C₄⋊Q₈	—	2	1	no	—
135	#126	—	4	2	no	—
136	#125	—	2	1	no	—
137	#125	—	4	2	no	—
138	C₂²⋊Q₈	—	2	2	no	—
139	C₄×Q₈	—	2	2	no	—
140	#117	D₄×C₁₉	1	1	yes	characteristic
141	Q₃₂	—	4	4	no	—
142	C₂×C₁₆	—	2	2	no	—
143	#107	D₄×C₁₉	1	2	yes	—
144	#106	—	2	1	no	—
145	C₄⋊C₈	—	2	2	no	—
146	#103	—	2	2	no	—
147	#103	—	2	2	no	—
148	#103	—	4	2	no	—
149	#101	—	4	2	no	—
150	C₁₉	#12168	1	1	yes	characteristic, direct product factor, Sylow
151	C₂×Q₈	—	2	2	no	—
152	C₂×Q₈	—	2	2	no	—
153	C₂×Q₈	—	4	1	no	—
154	C₂×Q₈	—	4	1	no	—
155	C₂×Q₈	—	4	1	no	—
156	C₂×Q₈	—	4	1	no	—
157	C₂×Q₈	—	4	1	no	—
158	C₂²×C₄	—	4	1	no	—
159	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
160	Q₁₆	—	2	4	no	—
161	Q₁₆	—	4	2	no	—
162	Q₁₆	—	4	2	no	—
163	Q₁₆	—	8	1	no	—
164	M₁₆	—	4	1	no	—
165	M₁₆	—	4	1	no	—
166	M₁₆	—	4	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	4	1	no	—
169	C₂×C₈	—	4	1	no	—
170	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	D₈×C₁₉	1	2	yes	—
177	C₂²⋊C₄	—	2	1	no	—
178	C₂²⋊C₄	—	4	2	no	—
179	C₄²	—	2	1	no	—
180	C₄²	—	4	2	no	—
181	C₁₆	—	4	2	no	—
182	C₂³	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
183	Q₈	—	4	2	no	—
184	Q₈	—	4	2	no	—
185	Q₈	—	4	2	no	—
186	Q₈	—	4	2	no	—
187	Q₈	—	8	1	no	—
188	Q₈	—	8	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	8	1	no	—
196	C₂×C₄	—	8	1	no	—
197	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
198	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
199	C₈	—	2	2	no	—
200	C₈	—	4	1	no	—
201	C₈	—	8	1	no	—
202	C₂²	—	2	2	no	—
203	C₂²	—	2	1	no	—
204	C₂²	C₁₉×#447	1	1	yes	characteristic
205	C₄	—	2	1	no	—
206	C₄	—	4	2	no	—
207	C₄	—	8	2	no	—
208	C₄	—	8	2	no	—
209	C₄	—	8	2	no	—
210	C₄	C₁₉×#454	1	1	yes	characteristic
211	C₄	C₁₉×B(4,𝔽₂)	1	1	yes	characteristic
212	C₂	—	4	1	no	—
213	C₂	C₁₉×#1598	1	1	yes	characteristic
214	C₂	C₁₉×#2195	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#3261

C₂

yes

maximal, characteristic

C₁₉×#2188

C₂

yes

maximal

C₁₉×#2047

C₂

yes

maximal, characteristic

C₁₉×#1350

C₂

yes

maximal

C₁₉×#1323

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₁₉

C₂²

yes

characteristic

C₁₉×#541

—

C₂×C₁₉×Q₃₂

—

C₁₉×#479

—

C₁₉×#470

—

C₁₉×#467

C₂²

yes

—

C₁₉×#462

—

C₁₉×#424

C₂²

yes

characteristic

C₁₉×#419

—

C₂×C₁₉×#101

—

C₁₉×#358

—

C₁₉×C₂²⋊C₁₆

C₂²

yes

—

C₁₉×#341

C₂²

yes

characteristic

C₂²×Q₈×C₁₉

—

C₂×Q₁₆×C₁₉

—

C₂×Q₁₆×C₁₉

—

C₂×Q₁₆×C₁₉

—

C₂×Q₁₆×C₁₉

D₄

yes

—

C₂×M₁₆×C₁₉

—

C₂×M₁₆×C₁₉

D₄

yes

characteristic

C₂²×C₈×C₁₉

C₂³

yes

characteristic

C₁₉×C₄⋊Q₈

—

C₁₉×#126

—

C₁₉×#125

—

C₁₉×#125

—

C₁₉×C₂²⋊Q₈

—

C₄×Q₈×C₁₉

—

C₁₉×#117

D₄

yes

characteristic

C₁₉×Q₃₂

—

C₂×C₁₆×C₁₉

—

C₁₉×#107

D₄

yes

—

C₁₉×#106

—

C₁₉×C₄⋊C₈

—

C₁₉×#103

—

C₁₉×#103

—

C₁₉×#103

—

C₁₉×#101

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

Q₁₆×C₁₉

—

Q₁₆×C₁₉

—

Q₁₆×C₁₉

—

Q₁₆×C₁₉

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₂×D₄

yes

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

D₈

yes

—

C₂²⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₁₆×C₁₉

—

#12168

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

AGL(1,ℤ/8)

yes

characteristic

Q₈×C₁₉

—

Q₈×C₁₉

—

Q₈×C₁₉

—

Q₈×C₁₉

—

Q₈×C₁₉

—

Q₈×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×D₈

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

#3261

C₂×C₁₉

yes

characteristic

#2188

C₂×C₁₉

yes

—

#2047

C₂×C₁₉

yes

characteristic

#1350

C₂×C₁₉

yes

—

#1323

C₂×C₁₉

yes

characteristic

100

C₂²×C₁₉

—

101

C₂²×C₁₉

—

102

C₂²×C₁₉

#447

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

B(4,𝔽₂)

yes

characteristic

110

C₂²×Q₁₆

C₂²×C₁₉

yes

characteristic

111

#541

—

112

C₂×Q₃₂

—

113

#479

—

114

#470

—

115

#467

C₂²×C₁₉

yes

—

116

#462

—

117

#424

C₂²×C₁₉

yes

characteristic

118

#419

—

119

C₂×#101

—

120

#358

—

121

C₂²⋊C₁₆

C₂²×C₁₉

yes

—

122

#341

C₂²×C₁₉

yes

characteristic

123

C₂×C₁₉

—

124

C₂×C₁₉

#1598

yes

characteristic

125

C₂×C₁₉

#2195

yes

—

126

C₂²×Q₈

—

127

C₂×Q₁₆

—

128

C₂×Q₁₆

—

129

C₂×Q₁₆

—

130

C₂×Q₁₆

D₄×C₁₉

yes

—

131

C₂×M₁₆

—

132

C₂×M₁₆

D₄×C₁₉

yes

characteristic

133

C₂²×C₈

C₂³×C₁₉

yes

characteristic

134

C₄⋊Q₈

—

135

#126

—

136

#125

—

137

#125

—

138

C₂²⋊Q₈

—

139

C₄×Q₈

—

140

#117

D₄×C₁₉

yes

characteristic

141

Q₃₂

—

142

C₂×C₁₆

—

143

#107

D₄×C₁₉

yes

—

144

#106

—

145

C₄⋊C₈

—

146

#103

—

147

#103

—

148

#103

—

149

#101

—

150

C₁₉

#12168

yes

characteristic, direct product factor, Sylow

151

C₂×Q₈

—

152

C₂×Q₈

—

153

C₂×Q₈

—

154

C₂×Q₈

—

155

C₂×Q₈

—

156

C₂×Q₈

—

157

C₂×Q₈

—

158

C₂²×C₄

—

159

C₂²×C₄

C₂×D₄×C₁₉

yes

characteristic

160

Q₁₆

—

161

Q₁₆

—

162

Q₁₆

—

163

Q₁₆

—

164

M₁₆

—

165

M₁₆

—

166

M₁₆

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

C₂×D₄×C₁₉

yes

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

—

175

C₄⋊C₄

—

176

C₄⋊C₄

D₈×C₁₉

yes

—

177

C₂²⋊C₄

—

178

C₂²⋊C₄

—

179

C₄²

—

180

C₄²

—

181

C₁₆

—

182

C₂³

C₁₉×AGL(1,ℤ/8)

yes

characteristic

183

Q₈

—

184

Q₈

—

185

Q₈

—

186

Q₈

—

187

Q₈

—

188

Q₈

—

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

C₁₉×#120

yes

characteristic

198

C₂×C₄

C₂×D₈×C₁₉

yes

characteristic

199

C₈

—

200

C₈

—

201

C₈

—

202

C₂²

—

203

C₂²

—

204

C₂²

C₁₉×#447

yes

characteristic

205

C₄

—

206

C₄

—

207

C₄

—

208

C₄

—

209

C₄

—

210

C₄

C₁₉×#454

yes

characteristic

211

C₄

C₁₉×B(4,𝔽₂)

yes

characteristic

212

C₂

—

213

C₂

C₁₉×#1598

yes

characteristic

214

C₂

C₁₉×#2195

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₉	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₂³×C₁₉	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#424 ◁ G

subgroup

G/H

center, socle

C₂²×C₁₉

#447

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₁₉

prime radical

C₂³×C₁₉

AGL(1,ℤ/8)

LCS

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#424 ◁ G

N	H ≅ G/N	type
74. #12168	C₁₉	direct (N × H)
150. C₁₉	#12168	direct (N × H)

H ≅ G/N

type

74. #12168

C₁₉

direct (N × H)

150. C₁₉

#12168

direct (N × H)

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

subgroup	index
C₁₉×#3261	2
C₁₉×#2188	2
C₁₉×#2047	2
C₁₉×#1350	2
C₁₉×#1323	2
#12168	19

subgroup

index

C₁₉×#3261

C₁₉×#2188

C₁₉×#2047

C₁₉×#1350

C₁₉×#1323

#12168

Subgroups — C₁₉×#12168 #18018127

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#12168 #18018127

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12168 #18018127