Group #18013270 — subgroups

Subgroups — C₁₉×#7311 #18013270

Order 4864.

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

normal subgroups only

This group has 590 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×#1783	C₂	1	2	yes	maximal
2	C₁₉×#1781	C₂	1	1	yes	maximal, characteristic
3	C₂³×D₄×C₁₉	C₂²	1	1	yes	characteristic
4	C₂×C₁₉×#100	—	2	2	no	—
5	C₂×C₁₉×#100	C₄	1	4	yes	complemented
6	C₂×C₁₉×#99	—	2	2	no	—
7	C₁₉×#407	—	2	2	no	—
8	C₁₉×#379	C₄	1	2	yes	—
9	C₂⁵×C₁₉	C₂×C₄	1	1	yes	characteristic
10	C₂⁵×C₁₉	Q₈	1	1	yes	characteristic
11	C₂²×D₄×C₁₉	—	4	4	no	—
12	C₂²×D₄×C₁₉	—	4	2	no	—
13	C₂²×D₄×C₁₉	C₂×C₄	1	2	yes	—
14	C₂²×D₄×C₁₉	D₄	1	2	yes	—
15	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
16	C₂×M₁₆×C₁₉	—	2	4	no	—
17	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
18	C₂×C₂²⋊C₄×C₁₉	—	4	2	no	—
19	C₁₉×#100	—	4	4	no	—
20	C₁₉×#100	—	4	4	no	—
21	C₁₉×#99	—	4	4	no	—
22	C₁₉×C₂²⋊C₈	—	4	4	no	—
23	C₂⁴×C₁₉	—	2	2	no	—
24	C₂⁴×C₁₉	—	4	2	no	—
25	C₂⁴×C₁₉	—	4	2	no	—
26	C₂⁴×C₁₉	—	4	1	no	—
27	C₂⁴×C₁₉	—	8	2	no	—
28	C₂⁴×C₁₉	—	8	2	no	—
29	C₂⁴×C₁₉	C₄²	1	1	yes	characteristic
30	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
31	C₂⁴×C₁₉	C₄⋊C₄	1	1	yes	characteristic
32	C₂⁴×C₁₉	C₄⋊C₄	1	2	yes	—
33	C₂×D₄×C₁₉	—	2	4	no	—
34	C₂×D₄×C₁₉	—	2	4	no	—
35	C₂×D₄×C₁₉	—	2	4	no	—
36	C₂×D₄×C₁₉	—	4	2	no	—
37	C₂×D₄×C₁₉	—	4	2	no	—
38	C₂×D₄×C₁₉	—	4	2	no	—
39	C₂×D₄×C₁₉	—	4	16	no	—
40	C₂²×C₄×C₁₉	—	2	4	no	—
41	C₂²×C₄×C₁₉	—	4	1	no	—
42	C₂²×C₄×C₁₉	—	4	2	no	—
43	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
44	M₁₆×C₁₉	—	4	4	no	—
45	M₁₆×C₁₉	—	4	4	no	—
46	C₂×C₈×C₁₉	—	4	4	no	—
47	C₂²⋊C₄×C₁₉	—	4	4	no	—
48	C₂²⋊C₄×C₁₉	—	8	4	no	—
49	#7311	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂³×C₁₉	—	2	2	no	—
51	C₂³×C₁₉	—	2	1	no	—
52	C₂³×C₁₉	—	2	2	no	—
53	C₂³×C₁₉	—	2	1	no	—
54	C₂³×C₁₉	—	2	2	no	—
55	C₂³×C₁₉	—	2	2	no	—
56	C₂³×C₁₉	—	2	2	no	—
57	C₂³×C₁₉	—	2	2	no	—
58	C₂³×C₁₉	—	4	2	no	—
59	C₂³×C₁₉	—	4	4	no	—
60	C₂³×C₁₉	—	4	4	no	—
61	C₂³×C₁₉	—	4	4	no	—
62	C₂³×C₁₉	—	8	2	no	—
63	C₂³×C₁₉	—	8	4	no	—
64	C₂³×C₁₉	—	8	4	no	—
65	C₂³×C₁₉	—	8	1	no	—
66	C₂³×C₁₉	—	8	1	no	—
67	C₂³×C₁₉	—	8	2	no	—
68	C₂³×C₁₉	—	8	2	no	—
69	C₂³×C₁₉	—	8	4	no	—
70	C₂³×C₁₉	—	8	2	no	—
71	C₂³×C₁₉	—	8	4	no	—
72	C₂³×C₁₉	#95	1	1	yes	characteristic
73	C₂³×C₁₉	#99	1	2	yes	—
74	D₄×C₁₉	—	4	8	no	—
75	D₄×C₁₉	—	4	8	no	—
76	C₂×C₄×C₁₉	—	2	2	no	—
77	C₂×C₄×C₁₉	—	2	2	no	—
78	C₂×C₄×C₁₉	—	4	4	no	—
79	C₂×C₄×C₁₉	—	8	2	no	—
80	C₂×C₄×C₁₉	—	8	4	no	—
81	C₂×C₄×C₁₉	#99	1	4	yes	—
82	C₈×C₁₉	—	8	4	no	—
83	#1783	C₂×C₁₉	1	2	yes	—
84	#1781	C₂×C₁₉	1	1	yes	characteristic
85	C₂²×C₁₉	—	2	2	no	—
86	C₂²×C₁₉	—	2	4	no	—
87	C₂²×C₁₉	—	2	2	no	—
88	C₂²×C₁₉	—	4	1	no	—
89	C₂²×C₁₉	—	4	2	no	—
90	C₂²×C₁₉	—	4	1	no	—
91	C₂²×C₁₉	—	4	2	no	—
92	C₂²×C₁₉	—	4	2	no	—
93	C₂²×C₁₉	—	4	1	no	—
94	C₂²×C₁₉	—	4	1	no	—
95	C₂²×C₁₉	—	4	2	no	—
96	C₂²×C₁₉	—	8	2	no	—
97	C₂²×C₁₉	—	8	2	no	—
98	C₂²×C₁₉	—	8	2	no	—
99	C₂²×C₁₉	—	8	2	no	—
100	C₂²×C₁₉	—	8	2	no	—
101	C₂²×C₁₉	—	8	8	no	—
102	C₂²×C₁₉	—	8	8	no	—
103	C₂²×C₁₉	#342	1	1	yes	characteristic
104	C₂²×C₁₉	#342	1	2	yes	—
105	C₄×C₁₉	—	2	4	no	—
106	C₄×C₁₉	—	16	2	no	—
107	C₂³×D₄	C₂²×C₁₉	1	1	yes	characteristic
108	C₂×#100	—	2	2	no	—
109	C₂×#100	C₄×C₁₉	1	4	yes	complemented
110	C₂×#99	—	2	2	no	—
111	#407	—	2	2	no	—
112	#379	C₄×C₁₉	1	2	yes	—
113	C₂×C₁₉	—	2	1	no	—
114	C₂×C₁₉	—	2	2	no	—
115	C₂×C₁₉	—	4	2	no	—
116	C₂×C₁₉	—	4	2	no	—
117	C₂×C₁₉	—	8	1	no	—
118	C₂×C₁₉	—	8	2	no	—
119	C₂×C₁₉	#1304	1	1	yes	characteristic
120	C₂⁵	C₂×C₄×C₁₉	1	1	yes	characteristic
121	C₂⁵	Q₈×C₁₉	1	1	yes	characteristic
122	C₂²×D₄	—	4	4	no	—
123	C₂²×D₄	—	4	2	no	—
124	C₂²×D₄	C₂×C₄×C₁₉	1	2	yes	—
125	C₂²×D₄	D₄×C₁₉	1	2	yes	—
126	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
127	C₂×M₁₆	—	2	4	no	—
128	C₂×C₂²⋊C₄	—	2	2	no	—
129	C₂×C₂²⋊C₄	—	4	2	no	—
130	#100	—	4	4	no	—
131	#100	—	4	4	no	—
132	#99	—	4	4	no	—
133	C₂²⋊C₈	—	4	4	no	—
134	C₁₉	#7311	1	1	yes	characteristic, direct product factor, Sylow
135	C₂⁴	—	2	2	no	—
136	C₂⁴	—	4	2	no	—
137	C₂⁴	—	4	2	no	—
138	C₂⁴	—	4	1	no	—
139	C₂⁴	—	8	2	no	—
140	C₂⁴	—	8	2	no	—
141	C₂⁴	C₄²×C₁₉	1	1	yes	characteristic
142	C₂⁴	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
143	C₂⁴	C₄⋊C₄×C₁₉	1	1	yes	characteristic
144	C₂⁴	C₄⋊C₄×C₁₉	1	2	yes	—
145	C₂×D₄	—	2	4	no	—
146	C₂×D₄	—	2	4	no	—
147	C₂×D₄	—	2	4	no	—
148	C₂×D₄	—	4	2	no	—
149	C₂×D₄	—	4	2	no	—
150	C₂×D₄	—	4	2	no	—
151	C₂×D₄	—	4	16	no	—
152	C₂²×C₄	—	2	4	no	—
153	C₂²×C₄	—	4	1	no	—
154	C₂²×C₄	—	4	2	no	—
155	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
156	M₁₆	—	4	4	no	—
157	M₁₆	—	4	4	no	—
158	C₂×C₈	—	4	4	no	—
159	C₂²⋊C₄	—	4	4	no	—
160	C₂²⋊C₄	—	8	4	no	—
161	C₂³	—	2	2	no	—
162	C₂³	—	2	1	no	—
163	C₂³	—	2	2	no	—
164	C₂³	—	2	1	no	—
165	C₂³	—	2	2	no	—
166	C₂³	—	2	2	no	—
167	C₂³	—	2	2	no	—
168	C₂³	—	2	2	no	—
169	C₂³	—	4	2	no	—
170	C₂³	—	4	4	no	—
171	C₂³	—	4	4	no	—
172	C₂³	—	4	4	no	—
173	C₂³	—	8	2	no	—
174	C₂³	—	8	4	no	—
175	C₂³	—	8	4	no	—
176	C₂³	—	8	1	no	—
177	C₂³	—	8	1	no	—
178	C₂³	—	8	2	no	—
179	C₂³	—	8	2	no	—
180	C₂³	—	8	4	no	—
181	C₂³	—	8	2	no	—
182	C₂³	—	8	4	no	—
183	C₂³	C₁₉×#95	1	1	yes	characteristic
184	C₂³	C₁₉×#99	1	2	yes	—
185	D₄	—	4	8	no	—
186	D₄	—	4	8	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	4	4	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	8	4	no	—
192	C₂×C₄	C₁₉×#99	1	4	yes	—
193	C₈	—	8	4	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	2	4	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	4	1	no	—
198	C₂²	—	4	2	no	—
199	C₂²	—	4	1	no	—
200	C₂²	—	4	2	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	4	1	no	—
203	C₂²	—	4	1	no	—
204	C₂²	—	4	2	no	—
205	C₂²	—	8	2	no	—
206	C₂²	—	8	2	no	—
207	C₂²	—	8	2	no	—
208	C₂²	—	8	2	no	—
209	C₂²	—	8	2	no	—
210	C₂²	—	8	8	no	—
211	C₂²	—	8	8	no	—
212	C₂²	C₁₉×#342	1	1	yes	characteristic
213	C₂²	C₁₉×#342	1	2	yes	—
214	C₄	—	2	4	no	—
215	C₄	—	16	2	no	—
216	C₂	—	2	1	no	—
217	C₂	—	2	2	no	—
218	C₂	—	4	2	no	—
219	C₂	—	4	2	no	—
220	C₂	—	8	1	no	—
221	C₂	—	8	2	no	—
222	C₂	C₁₉×#1304	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#1783

C₂

yes

maximal

C₁₉×#1781

C₂

yes

maximal, characteristic

C₂³×D₄×C₁₉

C₂²

yes

characteristic

C₂×C₁₉×#100

—

C₂×C₁₉×#100

C₄

yes

complemented

C₂×C₁₉×#99

—

C₁₉×#407

—

C₁₉×#379

C₄

yes

—

C₂⁵×C₁₉

C₂×C₄

yes

characteristic

C₂⁵×C₁₉

Q₈

yes

characteristic

C₂²×D₄×C₁₉

—

C₂²×D₄×C₁₉

—

C₂²×D₄×C₁₉

C₂×C₄

yes

—

C₂²×D₄×C₁₉

D₄

yes

—

C₂³×C₄×C₁₉

D₄

yes

characteristic

C₂×M₁₆×C₁₉

—

C₂×C₂²⋊C₄×C₁₉

—

C₂×C₂²⋊C₄×C₁₉

—

C₁₉×#100

—

C₁₉×#100

—

C₁₉×#99

—

C₁₉×C₂²⋊C₈

—

C₂⁴×C₁₉

—

C₂⁴×C₁₉

—

C₂⁴×C₁₉

—

C₂⁴×C₁₉

—

C₂⁴×C₁₉

—

C₂⁴×C₁₉

—

C₂⁴×C₁₉

C₄²

yes

characteristic

C₂⁴×C₁₉

C₂²⋊C₄

yes

characteristic

C₂⁴×C₁₉

C₄⋊C₄

yes

characteristic

C₂⁴×C₁₉

C₄⋊C₄

yes

—

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

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

#7311

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

#95

yes

characteristic

C₂³×C₁₉

#99

yes

—

D₄×C₁₉

—

D₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#99

yes

—

C₈×C₁₉

—

#1783

C₂×C₁₉

yes

—

#1781

C₂×C₁₉

yes

characteristic

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

100

C₂²×C₁₉

—

101

C₂²×C₁₉

—

102

C₂²×C₁₉

—

103

C₂²×C₁₉

#342

yes

characteristic

104

C₂²×C₁₉

#342

yes

—

105

C₄×C₁₉

—

106

C₄×C₁₉

—

107

C₂³×D₄

C₂²×C₁₉

yes

characteristic

108

C₂×#100

—

109

C₂×#100

C₄×C₁₉

yes

complemented

110

C₂×#99

—

111

#407

—

112

#379

C₄×C₁₉

yes

—

113

C₂×C₁₉

—

114

C₂×C₁₉

—

115

C₂×C₁₉

—

116

C₂×C₁₉

—

117

C₂×C₁₉

—

118

C₂×C₁₉

—

119

C₂×C₁₉

#1304

yes

characteristic

120

C₂⁵

C₂×C₄×C₁₉

yes

characteristic

121

C₂⁵

Q₈×C₁₉

yes

characteristic

122

C₂²×D₄

—

123

C₂²×D₄

—

124

C₂²×D₄

C₂×C₄×C₁₉

yes

—

125

C₂²×D₄

D₄×C₁₉

yes

—

126

C₂³×C₄

D₄×C₁₉

yes

characteristic

127

C₂×M₁₆

—

128

C₂×C₂²⋊C₄

—

129

C₂×C₂²⋊C₄

—

130

#100

—

131

#100

—

132

#99

—

133

C₂²⋊C₈

—

134

C₁₉

#7311

yes

characteristic, direct product factor, Sylow

135

C₂⁴

—

136

C₂⁴

—

137

C₂⁴

—

138

C₂⁴

—

139

C₂⁴

—

140

C₂⁴

—

141

C₂⁴

C₄²×C₁₉

yes

characteristic

142

C₂⁴

C₂²⋊C₄×C₁₉

yes

characteristic

143

C₂⁴

C₄⋊C₄×C₁₉

yes

characteristic

144

C₂⁴

C₄⋊C₄×C₁₉

yes

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

—

150

C₂×D₄

—

151

C₂×D₄

—

152

C₂²×C₄

—

153

C₂²×C₄

—

154

C₂²×C₄

—

155

C₂²×C₄

C₂²⋊C₄×C₁₉

yes

—

156

M₁₆

—

157

M₁₆

—

158

C₂×C₈

—

159

C₂²⋊C₄

—

160

C₂²⋊C₄

—

161

C₂³

—

162

C₂³

—

163

C₂³

—

164

C₂³

—

165

C₂³

—

166

C₂³

—

167

C₂³

—

168

C₂³

—

169

C₂³

—

170

C₂³

—

171

C₂³

—

172

C₂³

—

173

C₂³

—

174

C₂³

—

175

C₂³

—

176

C₂³

—

177

C₂³

—

178

C₂³

—

179

C₂³

—

180

C₂³

—

181

C₂³

—

182

C₂³

—

183

C₂³

C₁₉×#95

yes

characteristic

184

C₂³

C₁₉×#99

yes

—

185

D₄

—

186

D₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

C₁₉×#99

yes

—

193

C₈

—

194

C₂²

—

195

C₂²

—

196

C₂²

—

197

C₂²

—

198

C₂²

—

199

C₂²

—

200

C₂²

—

201

C₂²

—

202

C₂²

—

203

C₂²

—

204

C₂²

—

205

C₂²

—

206

C₂²

—

207

C₂²

—

208

C₂²

—

209

C₂²

—

210

C₂²

—

211

C₂²

—

212

C₂²

C₁₉×#342

yes

characteristic

213

C₂²

C₁₉×#342

yes

—

214

C₄

—

215

C₄

—

216

C₂

—

217

C₂

—

218

C₂

—

219

C₂

—

220

C₂

—

221

C₂

—

222

C₂

C₁₉×#1304

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₁₉	#1304
commutator	C₂⁴	C₄²×C₁₉
Frattini, cosocle	C₂³×D₄	C₂²×C₁₉
prime radical	C₂³×D₄×C₁₉	C₂²
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂³×C₁₉ ◁ C₂³×D₄×C₁₉ ◁ G

subgroup

G/H

center, socle

C₂×C₁₉

#1304

commutator

C₂⁴

C₄²×C₁₉

Frattini, cosocle

C₂³×D₄

C₂²×C₁₉

prime radical

C₂³×D₄×C₁₉

C₂²

LCS

G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₁₉ ◁ C₂³×C₁₉ ◁ C₂³×D₄×C₁₉ ◁ G

N	H ≅ G/N	type
5. C₂×C₁₉×#100	C₄	semidirect
49. #7311	C₁₉	direct (N × H)
109. C₂×#100	C₄×C₁₉	semidirect
134. C₁₉	#7311	direct (N × H)

H ≅ G/N

type

5. C₂×C₁₉×#100

C₄

semidirect

49. #7311

C₁₉

direct (N × H)

109. C₂×#100

C₄×C₁₉

semidirect

134. C₁₉

#7311

direct (N × H)

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

subgroup	index
C₁₉×#1783	2
C₁₉×#1781	2
#7311	19

subgroup

index

C₁₉×#1783

C₁₉×#1781

#7311

Subgroups — C₁₉×#7311 #18013270

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#7311 #18013270

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#7311 #18013270