Group #18017948 — subgroups

Subgroups — C₁₉×#11989 #18017948

Order 4864.

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

normal subgroups only

This group has 292 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	C₁₉×#3176	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2166	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#2153	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1366	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1357	C₂	1	2	yes	maximal, complemented
6	C₁₉×#540	C₄	1	2	yes	—
7	C₂×C₁₉×M₃₂	—	2	1	no	—
8	C₁₉×#499	C₂²	1	1	yes	characteristic
9	C₁₉×#493	C₂²	1	1	yes	characteristic
10	C₁₉×#490	C₄	1	2	yes	—
11	C₁₉×#469	—	2	1	no	—
12	C₁₉×#466	—	2	1	no	—
13	C₁₉×#442	—	2	1	no	—
14	C₄×D₈×C₁₉	—	2	1	no	—
15	C₁₉×#405	C₂²	1	1	yes	characteristic
16	C₁₉×#364	—	2	1	no	—
17	C₁₉×#364	C₂²	1	2	yes	—
18	C₁₉×#364	C₂²	1	2	yes	—
19	C₁₉×#349	—	2	1	no	—
20	C₁₉×#347	—	2	1	no	—
21	C₂×D₈×C₁₉	—	2	1	no	—
22	C₂×D₈×C₁₉	—	2	1	no	—
23	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
25	C₁₉×#127	C₂×C₄	1	2	yes	—
26	C₁₉×#125	C₂×C₄	1	2	yes	—
27	C₁₉×#123	—	4	2	no	—
28	C₁₉×#121	—	2	2	no	—
29	C₁₉×#121	—	4	2	no	—
30	C₄×D₄×C₁₉	—	2	2	no	—
31	C₁₉×#117	C₂×C₄	1	1	yes	characteristic
32	C₁₉×M₃₂	—	2	2	no	—
33	C₁₉×M₃₂	—	2	2	no	—
34	C₁₉×M₃₂	—	4	1	no	—
35	C₂×C₁₆×C₁₉	—	4	1	no	—
36	C₁₉×#107	—	2	1	no	—
37	C₁₉×#106	—	2	1	no	—
38	C₁₉×#102	—	2	2	no	—
39	C₁₉×#102	—	2	2	no	—
40	C₁₉×#97	C₂×C₄	1	1	yes	characteristic
41	C₁₉×#97	D₄	1	1	yes	characteristic
42	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
43	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
44	C₂×D₄×C₁₉	—	2	2	no	—
45	C₂×D₄×C₁₉	—	4	2	no	—
46	C₂×D₄×C₁₉	—	4	2	no	—
47	C₂²×C₄×C₁₉	—	4	2	no	—
48	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
49	D₈×C₁₉	—	4	2	no	—
50	D₈×C₁₉	—	4	1	no	—
51	D₈×C₁₉	—	4	1	no	—
52	M₁₆×C₁₉	—	2	2	no	—
53	C₂×C₈×C₁₉	—	2	1	no	—
54	C₂×C₈×C₁₉	—	2	1	no	—
55	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
59	C₄⋊C₄×C₁₉	—	2	1	no	—
60	C₄⋊C₄×C₁₉	—	2	2	no	—
61	C₄⋊C₄×C₁₉	—	4	2	no	—
62	C₂²⋊C₄×C₁₉	—	2	1	no	—
63	C₂²⋊C₄×C₁₉	—	4	2	no	—
64	C₂²⋊C₄×C₁₉	—	4	2	no	—
65	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
67	C₁₆×C₁₉	—	4	2	no	—
68	C₁₆×C₁₉	—	4	2	no	—
69	#11989	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂³×C₁₉	—	4	2	no	—
71	C₂³×C₁₉	#100	1	1	yes	characteristic
72	D₄×C₁₉	—	4	2	no	—
73	D₄×C₁₉	—	4	2	no	—
74	D₄×C₁₉	—	16	2	no	—
75	D₄×C₁₉	—	16	2	no	—
76	C₂×C₄×C₁₉	—	2	1	no	—
77	C₂×C₄×C₁₉	—	2	1	no	—
78	C₂×C₄×C₁₉	—	2	1	no	—
79	C₂×C₄×C₁₉	—	2	1	no	—
80	C₂×C₄×C₁₉	—	4	2	no	—
81	C₂×C₄×C₁₉	—	16	2	no	—
82	C₂×C₄×C₁₉	#100	1	1	yes	characteristic
83	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₈×C₁₉	—	2	1	no	—
85	C₈×C₁₉	—	2	1	no	—
86	C₈×C₁₉	#135	1	2	yes	—
87	C₈×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
88	C₈×C₁₉	#137	1	1	yes	characteristic
89	#3176	C₂×C₁₉	1	1	yes	characteristic
90	#2166	C₂×C₁₉	1	1	yes	characteristic, complemented
91	#2153	C₂×C₁₉	1	1	yes	characteristic, complemented
92	#1366	C₂×C₁₉	1	2	yes	complemented
93	#1357	C₂×C₁₉	1	2	yes	complemented
94	C₂²×C₁₉	—	2	1	no	—
95	C₂²×C₁₉	—	4	1	no	—
96	C₂²×C₁₉	—	8	2	no	—
97	C₂²×C₁₉	—	8	2	no	—
98	C₂²×C₁₉	—	8	2	no	—
99	C₂²×C₁₉	C₂×#100	1	1	yes	characteristic
100	C₄×C₁₉	—	2	1	no	—
101	C₄×C₁₉	—	4	1	no	—
102	C₄×C₁₉	—	4	1	no	—
103	C₄×C₁₉	—	8	2	no	—
104	C₄×C₁₉	#416	1	1	yes	characteristic
105	C₄×C₁₉	#417	1	1	yes	characteristic
106	#540	C₄×C₁₉	1	2	yes	—
107	C₂×M₃₂	—	2	1	no	—
108	#499	C₂²×C₁₉	1	1	yes	characteristic
109	#493	C₂²×C₁₉	1	1	yes	characteristic
110	#490	C₄×C₁₉	1	2	yes	—
111	#469	—	2	1	no	—
112	#466	—	2	1	no	—
113	#442	—	2	1	no	—
114	C₄×D₈	—	2	1	no	—
115	#405	C₂²×C₁₉	1	1	yes	characteristic
116	#364	—	2	1	no	—
117	#364	C₂²×C₁₉	1	2	yes	—
118	#364	C₂²×C₁₉	1	2	yes	—
119	#349	—	2	1	no	—
120	#347	—	2	1	no	—
121	C₂×C₁₉	—	2	1	no	—
122	C₂×C₁₉	—	4	1	no	—
123	C₂×C₁₉	—	16	2	no	—
124	C₂×C₁₉	#1551	1	1	yes	characteristic
125	C₂×D₈	—	2	1	no	—
126	C₂×D₈	—	2	1	no	—
127	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
128	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
129	#127	C₂×C₄×C₁₉	1	2	yes	—
130	#125	C₂×C₄×C₁₉	1	2	yes	—
131	#123	—	4	2	no	—
132	#121	—	2	2	no	—
133	#121	—	4	2	no	—
134	C₄×D₄	—	2	2	no	—
135	#117	C₂×C₄×C₁₉	1	1	yes	characteristic
136	M₃₂	—	2	2	no	—
137	M₃₂	—	2	2	no	—
138	M₃₂	—	4	1	no	—
139	C₂×C₁₆	—	4	1	no	—
140	#107	—	2	1	no	—
141	#106	—	2	1	no	—
142	#102	—	2	2	no	—
143	#102	—	2	2	no	—
144	#97	C₂×C₄×C₁₉	1	1	yes	characteristic
145	#97	D₄×C₁₉	1	1	yes	characteristic
146	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
147	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
148	C₁₉	#11989	1	1	yes	characteristic, direct product factor, Sylow
149	C₂×D₄	—	2	2	no	—
150	C₂×D₄	—	4	2	no	—
151	C₂×D₄	—	4	2	no	—
152	C₂²×C₄	—	4	2	no	—
153	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
154	D₈	—	4	2	no	—
155	D₈	—	4	1	no	—
156	D₈	—	4	1	no	—
157	M₁₆	—	2	2	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
161	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
162	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
163	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₂²⋊C₄	—	2	1	no	—
168	C₂²⋊C₄	—	4	2	no	—
169	C₂²⋊C₄	—	4	2	no	—
170	C₄²	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
171	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
172	C₁₆	—	4	2	no	—
173	C₁₆	—	4	2	no	—
174	C₂³	—	4	2	no	—
175	C₂³	C₁₉×#100	1	1	yes	characteristic
176	D₄	—	4	2	no	—
177	D₄	—	4	2	no	—
178	D₄	—	16	2	no	—
179	D₄	—	16	2	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	—	16	2	no	—
186	C₂×C₄	C₁₉×#100	1	1	yes	characteristic
187	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	C₁₉×#135	1	2	yes	—
191	C₈	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
192	C₈	C₁₉×#137	1	1	yes	characteristic
193	C₂²	—	2	1	no	—
194	C₂²	—	4	1	no	—
195	C₂²	—	8	2	no	—
196	C₂²	—	8	2	no	—
197	C₂²	—	8	2	no	—
198	C₂²	C₂×C₁₉×#100	1	1	yes	characteristic
199	C₄	—	2	1	no	—
200	C₄	—	4	1	no	—
201	C₄	—	4	1	no	—
202	C₄	—	8	2	no	—
203	C₄	C₁₉×#416	1	1	yes	characteristic
204	C₄	C₁₉×#417	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	4	1	no	—
207	C₂	—	16	2	no	—
208	C₂	C₁₉×#1551	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#3176

C₂

yes

maximal, characteristic

C₁₉×#2166

C₂

yes

maximal, characteristic, complemented

C₁₉×#2153

C₂

yes

maximal, characteristic, complemented

C₁₉×#1366

C₂

yes

maximal, complemented

C₁₉×#1357

C₂

yes

maximal, complemented

C₁₉×#540

C₄

yes

—

C₂×C₁₉×M₃₂

—

C₁₉×#499

C₂²

yes

characteristic

C₁₉×#493

C₂²

yes

characteristic

C₁₉×#490

C₄

yes

—

C₁₉×#469

—

C₁₉×#466

—

C₁₉×#442

—

C₄×D₈×C₁₉

—

C₁₉×#405

C₂²

yes

characteristic

C₁₉×#364

—

C₁₉×#364

C₂²

yes

—

C₁₉×#364

C₂²

yes

—

C₁₉×#349

—

C₁₉×#347

—

C₂×D₈×C₁₉

—

C₂×D₈×C₁₉

—

C₂×M₁₆×C₁₉

D₄

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₁₉×#127

C₂×C₄

yes

—

C₁₉×#125

C₂×C₄

yes

—

C₁₉×#123

—

C₁₉×#121

—

C₁₉×#121

—

C₄×D₄×C₁₉

—

C₁₉×#117

C₂×C₄

yes

characteristic

C₁₉×M₃₂

—

C₁₉×M₃₂

—

C₁₉×M₃₂

—

C₂×C₁₆×C₁₉

—

C₁₉×#107

—

C₁₉×#106

—

C₁₉×#102

—

C₁₉×#102

—

C₁₉×#97

C₂×C₄

yes

characteristic

C₁₉×#97

D₄

yes

characteristic

C₄×C₈×C₁₉

D₄

yes

characteristic

C₄×C₈×C₁₉

C₂³

yes

characteristic

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

D₈×C₁₉

—

D₈×C₁₉

—

D₈×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₂×D₄

yes

characteristic

C₂×C₈×C₁₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

C₂²⋊C₄

yes

characteristic

C₄²×C₁₉

C₂²×C₄

yes

characteristic

C₁₆×C₁₉

—

C₁₆×C₁₉

—

#11989

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

#100

yes

characteristic

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

#100

yes

characteristic

C₂×C₄×C₁₉

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

#135

yes

—

C₈×C₁₉

yes

C₈×C₁₉

yes

C₂×C₁₉

yes

characteristic

#2166

C₂×C₁₉

yes

characteristic, complemented

#2153

C₂×C₁₉

yes

characteristic, complemented

#1366

C₂×C₁₉

yes

complemented

#1357

C₂×C₁₉

yes

complemented

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

C₂×#100

yes

characteristic

100

C₄×C₁₉

—

101

C₄×C₁₉

—

102

C₄×C₁₉

—

103

C₄×C₁₉

—

104

C₄×C₁₉

#416

yes

characteristic

105

C₄×C₁₉

#417

yes

characteristic

106

#540

C₄×C₁₉

yes

—

107

C₂×M₃₂

—

108

#499

C₂²×C₁₉

yes

characteristic

109

#493

C₂²×C₁₉

yes

characteristic

110

#490

C₄×C₁₉

yes

—

111

#469

—

112

#466

—

113

#442

—

114

C₄×D₈

—

115

#405

C₂²×C₁₉

yes

characteristic

116

#364

—

117

#364

C₂²×C₁₉

yes

—

118

#364

C₂²×C₁₉

yes

—

119

#349

—

120

#347

—

121

C₂×C₁₉

—

122

C₂×C₁₉

—

123

C₂×C₁₉

—

124

C₂×C₁₉

#1551

yes

characteristic

125

C₂×D₈

—

126

C₂×D₈

—

127

C₂×M₁₆

D₄×C₁₉

yes

characteristic

128

C₂²×C₈

D₄×C₁₉

yes

characteristic

129

#127

C₂×C₄×C₁₉

yes

—

130

#125

C₂×C₄×C₁₉

yes

—

131

#123

—

132

#121

—

133

#121

—

134

C₄×D₄

—

135

#117

C₂×C₄×C₁₉

yes

characteristic

136

M₃₂

—

137

M₃₂

—

138

M₃₂

—

139

C₂×C₁₆

—

140

#107

—

141

#106

—

142

#102

—

143

#102

—

144

#97

C₂×C₄×C₁₉

yes

characteristic

145

#97

D₄×C₁₉

yes

characteristic

146

C₄×C₈

D₄×C₁₉

yes

characteristic

147

C₄×C₈

C₂³×C₁₉

yes

characteristic

148

C₁₉

#11989

yes

characteristic, direct product factor, Sylow

149

C₂×D₄

—

150

C₂×D₄

—

151

C₂×D₄

—

152

C₂²×C₄

—

153

C₂²×C₄

C₂²⋊C₄×C₁₉

yes

characteristic

154

D₈

—

155

D₈

—

156

D₈

—

157

M₁₆

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

C₂²⋊C₄×C₁₉

yes

characteristic

161

C₂×C₈

C₂²⋊C₄×C₁₉

yes

characteristic

162

C₂×C₈

C₂×D₄×C₁₉

yes

characteristic

163

C₂×C₈

C₂×D₄×C₁₉

yes

characteristic

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₂²⋊C₄

—

168

C₂²⋊C₄

—

169

C₂²⋊C₄

—

170

C₄²

C₂²⋊C₄×C₁₉

yes

characteristic

171

C₄²

C₂²×C₄×C₁₉

yes

characteristic

172

C₁₆

—

173

C₁₆

—

174

C₂³

—

175

C₂³

C₁₉×#100

yes

characteristic

176

D₄

—

177

D₄

—

178

D₄

—

179

D₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

C₁₉×#100

yes

characteristic

187

C₂×C₄

C₂×C₂²⋊C₄×C₁₉

yes

characteristic

188

C₈

—

189

C₈

—

190

C₈

C₁₉×#135

yes

—

191

C₈

C₁₉×AGL(1,ℤ/8)

yes

characteristic

192

C₈

C₁₉×#137

yes

characteristic

193

C₂²

—

194

C₂²

—

195

C₂²

—

196

C₂²

—

197

C₂²

—

198

C₂²

C₂×C₁₉×#100

yes

characteristic

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

C₁₉×#416

yes

characteristic

204

C₄

C₁₉×#417

yes

characteristic

205

C₂

—

206

C₂

—

207

C₂

—

208

C₂

C₁₉×#1551

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₁₉	#1551
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	C₄×C₈	C₂³×C₁₉
prime radical	C₁₉×#3176	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₁₉×#405 ◁ G

subgroup

G/H

center, socle

C₂×C₁₉

#1551

commutator

C₄²

C₂²×C₄×C₁₉

Frattini, cosocle

C₄×C₈

C₂³×C₁₉

prime radical

C₁₉×#3176

C₂

LCS

G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₁₉×#405 ◁ G

N	H ≅ G/N	type
2. C₁₉×#2166	C₂	semidirect
3. C₁₉×#2153	C₂	semidirect
4. C₁₉×#1366	C₂	semidirect
5. C₁₉×#1357	C₂	semidirect
69. #11989	C₁₉	direct (N × H)
90. #2166	C₂×C₁₉	semidirect
91. #2153	C₂×C₁₉	semidirect
92. #1366	C₂×C₁₉	semidirect
93. #1357	C₂×C₁₉	semidirect
148. C₁₉	#11989	direct (N × H)

H ≅ G/N

type

2. C₁₉×#2166

C₂

semidirect

3. C₁₉×#2153

C₂

semidirect

4. C₁₉×#1366

C₂

semidirect

5. C₁₉×#1357

C₂

semidirect

69. #11989

C₁₉

direct (N × H)

90. #2166

C₂×C₁₉

semidirect

91. #2153

C₂×C₁₉

semidirect

92. #1366

C₂×C₁₉

semidirect

93. #1357

C₂×C₁₉

semidirect

148. C₁₉

#11989

direct (N × H)

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

subgroup	index
C₁₉×#3176	2
C₁₉×#2166	2
C₁₉×#2153	2
C₁₉×#1366	2
C₁₉×#1357	2
#11989	19

subgroup

index

C₁₉×#3176

C₁₉×#2166

C₁₉×#2153

C₁₉×#1366

C₁₉×#1357

#11989

Subgroups — C₁₉×#11989 #18017948

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#11989 #18017948

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#11989 #18017948