Group #18016475 — subgroups

Subgroups — C₁₉×#10516 #18016475

Order 4864.

364 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 94 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×#3249	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1843	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1552	C₂	1	2	yes	maximal
4	C₁₉×#1506	C₂	1	2	yes	maximal
5	C₁₉×#1290	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#533	C₂²	1	2	yes	—
7	C₁₉×#477	—	2	2	no	—
8	C₁₉×#467	—	2	2	no	—
9	C₁₉×#462	—	2	2	no	—
10	C₁₉×#432	C₂²	1	2	yes	—
11	C₂×C₁₉×C₄⋊C₈	—	2	2	no	—
12	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₁₉×#376	C₂²	1	1	yes	characteristic
15	C₁₉×#332	—	2	2	no	—
16	C₁₉×#332	—	2	2	no	—
17	C₁₉×#328	—	2	4	no	—
18	C₂×Q₁₆×C₁₉	—	2	2	no	—
19	C₂²×C₈×C₁₉	—	2	1	no	—
20	C₂²×C₈×C₁₉	D₄	1	2	yes	—
21	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
22	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
23	C₁₉×#125	—	2	2	no	—
24	C₁₉×C₂²⋊Q₈	—	2	2	no	—
25	C₁₉×C₂²⋊Q₈	—	2	2	no	—
26	C₄×Q₈×C₁₉	—	2	2	no	—
27	C₄×Q₈×C₁₉	—	2	2	no	—
28	C₁₉×#117	—	2	2	no	—
29	C₂×C₄²×C₁₉	—	2	1	no	—
30	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
31	C₁₉×#107	—	2	2	no	—
32	C₁₉×C₄⋊C₈	—	2	2	no	—
33	C₁₉×C₄⋊C₈	—	2	2	no	—
34	C₁₉×C₄⋊C₈	—	2	2	no	—
35	C₁₉×C₄⋊C₈	—	4	4	no	—
36	C₁₉×#103	—	2	4	no	—
37	C₁₉×C₂²⋊C₈	—	2	2	no	—
38	C₁₉×#97	—	2	2	no	—
39	C₁₉×#95	—	2	1	no	—
40	C₁₉×#95	D₄	1	2	yes	—
41	C₂×Q₈×C₁₉	—	2	2	no	—
42	C₂×Q₈×C₁₉	—	2	2	no	—
43	C₂²×C₄×C₁₉	—	2	2	no	—
44	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
47	Q₁₆×C₁₉	—	8	2	no	—
48	C₂×C₈×C₁₉	—	2	2	no	—
49	C₂×C₈×C₁₉	—	4	2	no	—
50	C₂×C₈×C₁₉	—	4	2	no	—
51	C₂×C₈×C₁₉	—	4	1	no	—
52	C₂×C₈×C₁₉	—	4	2	no	—
53	C₄⋊C₄×C₁₉	—	2	2	no	—
54	C₄⋊C₄×C₁₉	—	2	2	no	—
55	C₄⋊C₄×C₁₉	—	2	2	no	—
56	C₄⋊C₄×C₁₉	—	2	2	no	—
57	C₄⋊C₄×C₁₉	—	4	2	no	—
58	C₄⋊C₄×C₁₉	—	4	2	no	—
59	C₂²⋊C₄×C₁₉	—	4	2	no	—
60	C₄²×C₁₉	—	2	2	no	—
61	C₄²×C₁₉	—	2	2	no	—
62	C₄²×C₁₉	—	4	2	no	—
63	C₄²×C₁₉	C₂×D₄	1	2	yes	—
64	C₄²×C₁₉	#41	1	2	yes	—
65	#10516	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂³×C₁₉	#121	1	1	yes	characteristic
67	Q₈×C₁₉	—	4	2	no	—
68	Q₈×C₁₉	—	4	2	no	—
69	Q₈×C₁₉	—	8	2	no	—
70	C₂×C₄×C₁₉	—	2	2	no	—
71	C₂×C₄×C₁₉	—	2	1	no	—
72	C₂×C₄×C₁₉	—	2	1	no	—
73	C₂×C₄×C₁₉	—	2	4	no	—
74	C₂×C₄×C₁₉	—	4	1	no	—
75	C₂×C₄×C₁₉	—	4	2	no	—
76	C₂×C₄×C₁₉	—	4	2	no	—
77	C₂×C₄×C₁₉	—	4	2	no	—
78	C₂×C₄×C₁₉	—	4	2	no	—
79	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
80	C₂×C₄×C₁₉	#121	1	2	yes	—
81	C₂×C₄×C₁₉	#123	1	2	yes	—
82	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
83	C₈×C₁₉	—	4	2	no	—
84	C₈×C₁₉	—	8	2	no	—
85	C₈×C₁₉	—	8	1	no	—
86	#3249	C₂×C₁₉	1	1	yes	characteristic
87	#1843	C₂×C₁₉	1	1	yes	characteristic
88	#1552	C₂×C₁₉	1	2	yes	—
89	#1506	C₂×C₁₉	1	2	yes	—
90	#1290	C₂×C₁₉	1	1	yes	characteristic
91	C₂²×C₁₉	—	2	1	no	—
92	C₂²×C₁₉	—	2	1	no	—
93	C₂²×C₁₉	#394	1	1	yes	characteristic
94	C₂²×C₁₉	#473	1	2	yes	—
95	C₄×C₁₉	—	2	2	no	—
96	C₄×C₁₉	—	2	1	no	—
97	C₄×C₁₉	—	2	2	no	—
98	C₄×C₁₉	—	4	4	no	—
99	C₄×C₁₉	—	8	2	no	—
100	C₄×C₁₉	—	8	2	no	—
101	C₄×C₁₉	#453	1	1	yes	characteristic
102	C₄×C₁₉	#456	1	1	yes	characteristic
103	#533	C₂²×C₁₉	1	2	yes	—
104	#477	—	2	2	no	—
105	#467	—	2	2	no	—
106	#462	—	2	2	no	—
107	#432	C₂²×C₁₉	1	2	yes	—
108	C₂×C₄⋊C₈	—	2	2	no	—
109	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
110	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
111	#376	C₂²×C₁₉	1	1	yes	characteristic
112	#332	—	2	2	no	—
113	#332	—	2	2	no	—
114	#328	—	2	4	no	—
115	C₂×C₁₉	—	2	2	no	—
116	C₂×C₁₉	#2039	1	1	yes	characteristic
117	C₂×C₁₉	#2047	1	1	yes	characteristic
118	C₂×C₁₉	#2053	1	1	yes	characteristic
119	C₂×Q₁₆	—	2	2	no	—
120	C₂²×C₈	—	2	1	no	—
121	C₂²×C₈	D₄×C₁₉	1	2	yes	—
122	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
123	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
124	#125	—	2	2	no	—
125	C₂²⋊Q₈	—	2	2	no	—
126	C₂²⋊Q₈	—	2	2	no	—
127	C₄×Q₈	—	2	2	no	—
128	C₄×Q₈	—	2	2	no	—
129	#117	—	2	2	no	—
130	C₂×C₄²	—	2	1	no	—
131	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
132	#107	—	2	2	no	—
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	—	4	4	no	—
137	#103	—	2	4	no	—
138	C₂²⋊C₈	—	2	2	no	—
139	#97	—	2	2	no	—
140	#95	—	2	1	no	—
141	#95	D₄×C₁₉	1	2	yes	—
142	C₁₉	#10516	1	1	yes	characteristic, direct product factor, Sylow
143	C₂×Q₈	—	2	2	no	—
144	C₂×Q₈	—	2	2	no	—
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
147	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
148	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
149	Q₁₆	—	8	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	4	1	no	—
154	C₂×C₈	—	4	2	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₂²⋊C₄	—	4	2	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	4	2	no	—
165	C₄²	C₂×D₄×C₁₉	1	2	yes	—
166	C₄²	#41×C₁₉	1	2	yes	—
167	C₂³	C₁₉×#121	1	1	yes	characteristic
168	Q₈	—	4	2	no	—
169	Q₈	—	4	2	no	—
170	Q₈	—	8	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
181	C₂×C₄	C₁₉×#121	1	2	yes	—
182	C₂×C₄	C₁₉×#123	1	2	yes	—
183	C₂×C₄	C₁₉×#124	1	1	yes	characteristic
184	C₈	—	4	2	no	—
185	C₈	—	8	2	no	—
186	C₈	—	8	1	no	—
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	C₁₉×#394	1	1	yes	characteristic
190	C₂²	C₁₉×#473	1	2	yes	—
191	C₄	—	2	1	no	—
192	C₄	—	2	2	no	—
193	C₄	—	2	2	no	—
194	C₄	—	4	4	no	—
195	C₄	—	8	2	no	—
196	C₄	—	8	2	no	—
197	C₄	C₁₉×#453	1	1	yes	characteristic
198	C₄	C₁₉×#456	1	1	yes	characteristic
199	C₂	—	2	2	no	—
200	C₂	C₁₉×#2039	1	1	yes	characteristic
201	C₂	C₁₉×#2047	1	1	yes	characteristic
202	C₂	C₁₉×#2053	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#3249

C₂

yes

maximal, characteristic

C₁₉×#1843

C₂

yes

maximal, characteristic

C₁₉×#1552

C₂

yes

maximal

C₁₉×#1506

C₂

yes

maximal

C₁₉×#1290

C₂

yes

maximal, characteristic

C₁₉×#533

C₂²

yes

—

C₁₉×#477

—

C₁₉×#467

—

C₁₉×#462

—

C₁₉×#432

C₂²

yes

—

C₂×C₁₉×C₄⋊C₈

—

C₂×C₁₉×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₁₉×C₄⋊C₈

C₂²

yes

characteristic

C₁₉×#376

C₂²

yes

characteristic

C₁₉×#332

—

C₁₉×#332

—

C₁₉×#328

—

C₂×Q₁₆×C₁₉

—

C₂²×C₈×C₁₉

—

C₂²×C₈×C₁₉

D₄

yes

—

C₁₉×C₄⋊Q₈

D₄

yes

—

C₁₉×C₄⋊Q₈

D₄

yes

—

C₁₉×#125

—

C₁₉×C₂²⋊Q₈

—

C₁₉×C₂²⋊Q₈

—

C₄×Q₈×C₁₉

—

C₄×Q₈×C₁₉

—

C₁₉×#117

—

C₂×C₄²×C₁₉

—

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₁₉×#107

—

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

C₁₉×#103

—

C₁₉×C₂²⋊C₈

—

C₁₉×#97

—

C₁₉×#95

—

C₁₉×#95

D₄

yes

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₉

#41

yes

characteristic

Q₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

C₂×D₄

yes

—

C₄²×C₁₉

#41

yes

—

#10516

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

#121

yes

characteristic

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

—

C₂×C₄×C₁₉

#120

yes

characteristic

C₂×C₄×C₁₉

#121

yes

—

C₂×C₄×C₁₉

#123

yes

—

C₂×C₄×C₁₉

#124

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

#3249

C₂×C₁₉

yes

characteristic

#1843

C₂×C₁₉

yes

characteristic

#1552

C₂×C₁₉

yes

—

#1506

C₂×C₁₉

yes

—

#1290

C₂×C₁₉

yes

characteristic

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#394

yes

characteristic

C₂²×C₁₉

#473

yes

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

100

C₄×C₁₉

—

101

C₄×C₁₉

#453

yes

characteristic

102

C₄×C₁₉

#456

yes

characteristic

103

#533

C₂²×C₁₉

yes

—

104

#477

—

105

#467

—

106

#462

—

107

#432

C₂²×C₁₉

yes

—

108

C₂×C₄⋊C₈

—

109

C₂×C₄⋊C₈

C₂²×C₁₉

yes

characteristic

110

C₂×C₄⋊C₈

C₂²×C₁₉

yes

characteristic

111

#376

C₂²×C₁₉

yes

characteristic

112

#332

—

113

#332

—

114

#328

—

115

C₂×C₁₉

—

116

C₂×C₁₉

#2039

yes

characteristic

117

C₂×C₁₉

#2047

yes

characteristic

118

C₂×C₁₉

#2053

yes

characteristic

119

C₂×Q₁₆

—

120

C₂²×C₈

—

121

C₂²×C₈

D₄×C₁₉

yes

—

122

C₄⋊Q₈

D₄×C₁₉

yes

—

123

C₄⋊Q₈

D₄×C₁₉

yes

—

124

#125

—

125

C₂²⋊Q₈

—

126

C₂²⋊Q₈

—

127

C₄×Q₈

—

128

C₄×Q₈

—

129

#117

—

130

C₂×C₄²

—

131

C₂×C₄²

C₂³×C₁₉

yes

characteristic

132

#107

—

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

C₄⋊C₈

—

136

C₄⋊C₈

—

137

#103

—

138

C₂²⋊C₈

—

139

#97

—

140

#95

—

141

#95

D₄×C₁₉

yes

—

142

C₁₉

#10516

yes

characteristic, direct product factor, Sylow

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂²×C₄

—

146

C₂²×C₄

C₂×D₄×C₁₉

yes

characteristic

147

C₂²×C₄

C₂×D₄×C₁₉

yes

characteristic

148

C₂²×C₄

#41×C₁₉

yes

characteristic

149

Q₁₆

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₂²⋊C₄

—

162

C₄²

—

163

C₄²

—

164

C₄²

—

165

C₄²

C₂×D₄×C₁₉

yes

—

166

C₄²

#41×C₁₉

yes

—

167

C₂³

C₁₉×#121

yes

characteristic

168

Q₈

—

169

Q₈

—

170

Q₈

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

C₁₉×#120

yes

characteristic

181

C₂×C₄

C₁₉×#121

yes

—

182

C₂×C₄

C₁₉×#123

yes

—

183

C₂×C₄

C₁₉×#124

yes

characteristic

184

C₈

—

185

C₈

—

186

C₈

—

187

C₂²

—

188

C₂²

—

189

C₂²

C₁₉×#394

yes

characteristic

190

C₂²

C₁₉×#473

yes

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

C₁₉×#453

yes

characteristic

198

C₄

C₁₉×#456

yes

characteristic

199

C₂

—

200

C₂

C₁₉×#2039

yes

characteristic

201

C₂

C₁₉×#2047

yes

characteristic

202

C₂

C₁₉×#2053

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
prime radical	C₂³×C₁₉	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₉

#394

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₁₉

prime radical

C₂³×C₁₉

#121

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

N	H ≅ G/N	type
65. #10516	C₁₉	direct (N × H)
142. C₁₉	#10516	direct (N × H)

H ≅ G/N

type

65. #10516

C₁₉

direct (N × H)

142. C₁₉

#10516

direct (N × H)

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

subgroup	index
C₁₉×#3249	2
C₁₉×#1843	2
C₁₉×#1552	2
C₁₉×#1506	2
C₁₉×#1290	2
#10516	19

subgroup

index

C₁₉×#3249

C₁₉×#1843

C₁₉×#1552

C₁₉×#1506

C₁₉×#1290

#10516

Subgroups — C₁₉×#10516 #18016475

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#10516 #18016475

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#10516 #18016475