Group #18014591 — subgroups

Subgroups — C₁₉×#8632 #18014591

Order 4864.

388 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 130 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×#2931	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1568	C₂	1	2	yes	maximal, complemented
3	C₁₉×#1536	C₂	1	2	yes	maximal
4	C₁₉×#1307	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₉×#131	C₄	1	2	yes	—
6	C₁₉×#532	C₂²	1	1	yes	characteristic
7	C₁₉×#436	—	2	2	no	—
8	C₈×D₄×C₁₉	—	2	2	no	—
9	C₁₉×#432	C₂²	1	4	yes	—
10	C₁₉×#423	C₂²	1	1	yes	characteristic
11	C₁₉×#408	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
13	C₁₉×#335	—	2	2	no	—
14	C₁₉×#334	—	2	2	no	—
15	C₁₉×#329	—	2	4	no	—
16	C₂×#41×C₁₉	C₂×C₄	1	2	yes	—
17	C₁₉×#131	—	4	4	no	—
18	C₂×M₁₆×C₁₉	—	2	2	no	—
19	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₁₉	—	2	2	no	—
21	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
23	C₁₉×C₄⋊Q₈	Q₈	1	1	yes	characteristic
24	C₁₉×#127	Q₈	1	1	yes	characteristic
25	C₁₉×#124	D₄	1	2	yes	—
26	C₁₉×#121	—	2	2	no	—
27	C₄×D₄×C₁₉	—	2	2	no	—
28	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
29	C₁₉×C₄⋊C₈	—	2	2	no	—
30	C₁₉×C₄⋊C₈	—	2	4	no	—
31	C₁₉×C₂²⋊C₈	—	2	4	no	—
32	C₁₉×C₂²⋊C₈	—	2	4	no	—
33	C₁₉×#97	—	2	4	no	—
34	C₄×C₈×C₁₉	D₄	1	2	yes	—
35	C₄×C₈×C₁₉	D₄	1	2	yes	—
36	#41×C₁₉	—	4	2	no	—
37	#41×C₁₉	—	4	2	no	—
38	C₂×Q₈×C₁₉	C₄⋊C₄	1	2	yes	—
39	C₂×D₄×C₁₉	—	2	2	no	—
40	C₂×D₄×C₁₉	—	2	2	no	—
41	C₂×D₄×C₁₉	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₁₉	—	2	2	no	—
43	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	#41	1	2	yes	—
45	M₁₆×C₁₉	—	4	4	no	—
46	M₁₆×C₁₉	—	4	2	no	—
47	C₂×C₈×C₁₉	—	2	2	no	—
48	C₂×C₈×C₁₉	—	2	2	no	—
49	C₂×C₈×C₁₉	—	4	1	no	—
50	C₂×C₈×C₁₉	—	4	4	no	—
51	C₂×C₈×C₁₉	—	4	4	no	—
52	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₄⋊C₄×C₁₉	—	2	2	no	—
56	C₂²⋊C₄×C₁₉	—	2	4	no	—
57	C₄²×C₁₉	C₂×D₄	1	2	yes	—
58	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
60	#8632	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂³×C₁₉	—	2	2	no	—
62	C₂³×C₁₉	#117	1	1	yes	characteristic
63	Q₈×C₁₉	—	4	2	no	—
64	D₄×C₁₉	—	4	2	no	—
65	D₄×C₁₉	—	4	4	no	—
66	D₄×C₁₉	—	4	4	no	—
67	C₂×C₄×C₁₉	—	2	1	no	—
68	C₂×C₄×C₁₉	—	2	1	no	—
69	C₂×C₄×C₁₉	—	2	2	no	—
70	C₂×C₄×C₁₉	—	4	2	no	—
71	C₂×C₄×C₁₉	—	4	2	no	—
72	C₂×C₄×C₁₉	—	4	2	no	—
73	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₉	#121	1	2	yes	—
76	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
77	C₈×C₁₉	—	2	1	no	—
78	C₈×C₁₉	—	2	2	no	—
79	C₈×C₁₉	—	4	2	no	—
80	C₈×C₁₉	—	8	4	no	—
81	C₈×C₁₉	#100	1	2	yes	—
82	#2931	C₂×C₁₉	1	1	yes	characteristic
83	#1568	C₂×C₁₉	1	2	yes	complemented
84	#1536	C₂×C₁₉	1	2	yes	—
85	#1307	C₂×C₁₉	1	2	yes	complemented
86	C₂²×C₁₉	—	2	1	no	—
87	C₂²×C₁₉	—	2	2	no	—
88	C₂²×C₁₉	—	4	2	no	—
89	C₂²×C₁₉	—	4	4	no	—
90	C₂²×C₁₉	#380	1	1	yes	characteristic
91	C₄×C₁₉	—	2	1	no	—
92	C₄×C₁₉	—	2	2	no	—
93	C₄×C₁₉	—	2	2	no	—
94	C₄×C₁₉	—	4	2	no	—
95	C₄×C₁₉	C₂×#100	1	1	yes	characteristic
96	C₄×C₁₉	#413	1	1	yes	characteristic
97	C₂×#131	C₄×C₁₉	1	2	yes	—
98	#532	C₂²×C₁₉	1	1	yes	characteristic
99	#436	—	2	2	no	—
100	C₈×D₄	—	2	2	no	—
101	#432	C₂²×C₁₉	1	4	yes	—
102	#423	C₂²×C₁₉	1	1	yes	characteristic
103	#408	C₄×C₁₉	1	2	yes	—
104	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
105	#335	—	2	2	no	—
106	#334	—	2	2	no	—
107	#329	—	2	4	no	—
108	C₂×C₁₉	—	4	1	no	—
109	C₂×C₁₉	—	8	2	no	—
110	C₂×C₁₉	#1800	1	1	yes	characteristic
111	C₂×C₁₉	#1814	1	2	yes	—
112	C₂×#41	C₂×C₄×C₁₉	1	2	yes	—
113	#131	—	4	4	no	—
114	C₂×M₁₆	—	2	2	no	—
115	C₂×M₁₆	C₂×C₄×C₁₉	1	2	yes	—
116	C₂²×C₈	—	2	2	no	—
117	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
118	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
119	C₄⋊Q₈	Q₈×C₁₉	1	1	yes	characteristic
120	#127	Q₈×C₁₉	1	1	yes	characteristic
121	#124	D₄×C₁₉	1	2	yes	—
122	#121	—	2	2	no	—
123	C₄×D₄	—	2	2	no	—
124	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	2	4	no	—
127	C₂²⋊C₈	—	2	4	no	—
128	C₂²⋊C₈	—	2	4	no	—
129	#97	—	2	4	no	—
130	C₄×C₈	D₄×C₁₉	1	2	yes	—
131	C₄×C₈	D₄×C₁₉	1	2	yes	—
132	C₁₉	#8632	1	1	yes	characteristic, direct product factor, Sylow
133	#41	—	4	2	no	—
134	#41	—	4	2	no	—
135	C₂×Q₈	C₄⋊C₄×C₁₉	1	2	yes	—
136	C₂×D₄	—	2	2	no	—
137	C₂×D₄	—	2	2	no	—
138	C₂×D₄	C₄⋊C₄×C₁₉	1	2	yes	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
141	C₂²×C₄	#41×C₁₉	1	2	yes	—
142	M₁₆	—	4	4	no	—
143	M₁₆	—	4	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	4	1	no	—
147	C₂×C₈	—	4	4	no	—
148	C₂×C₈	—	4	4	no	—
149	C₂×C₈	C₂²⋊C₄×C₁₉	1	2	yes	—
150	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
151	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
152	C₄⋊C₄	—	2	2	no	—
153	C₂²⋊C₄	—	2	4	no	—
154	C₄²	C₂×D₄×C₁₉	1	2	yes	—
155	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
156	C₄²	C₂×Q₈×C₁₉	1	1	yes	characteristic
157	C₂³	—	2	2	no	—
158	C₂³	C₁₉×#117	1	1	yes	characteristic
159	Q₈	—	4	2	no	—
160	D₄	—	4	2	no	—
161	D₄	—	4	4	no	—
162	D₄	—	4	4	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
170	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
171	C₂×C₄	C₁₉×#121	1	2	yes	—
172	C₂×C₄	C₁₉×C₂²⋊Q₈	1	2	yes	—
173	C₈	—	2	1	no	—
174	C₈	—	2	2	no	—
175	C₈	—	4	2	no	—
176	C₈	—	8	4	no	—
177	C₈	C₁₉×#100	1	2	yes	—
178	C₂²	—	2	1	no	—
179	C₂²	—	2	2	no	—
180	C₂²	—	4	2	no	—
181	C₂²	—	4	4	no	—
182	C₂²	C₁₉×#380	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	2	2	no	—
185	C₄	—	2	2	no	—
186	C₄	—	4	2	no	—
187	C₄	C₂×C₁₉×#100	1	1	yes	characteristic
188	C₄	C₁₉×#413	1	1	yes	characteristic
189	C₂	—	4	1	no	—
190	C₂	—	8	2	no	—
191	C₂	C₁₉×#1800	1	1	yes	characteristic
192	C₂	C₁₉×#1814	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#2931

C₂

yes

maximal, characteristic

C₁₉×#1568

C₂

yes

maximal, complemented

C₁₉×#1536

C₂

yes

maximal

C₁₉×#1307

C₂

yes

maximal, complemented

C₂×C₁₉×#131

C₄

yes

—

C₁₉×#532

C₂²

yes

characteristic

C₁₉×#436

—

C₈×D₄×C₁₉

—

C₁₉×#432

C₂²

yes

—

C₁₉×#423

C₂²

yes

characteristic

C₁₉×#408

C₄

yes

—

C₂×C₄×C₈×C₁₉

C₂²

yes

characteristic

C₁₉×#335

—

C₁₉×#334

—

C₁₉×#329

—

C₂×#41×C₁₉

C₂×C₄

yes

—

C₁₉×#131

—

C₂×M₁₆×C₁₉

—

C₂×M₁₆×C₁₉

C₂×C₄

yes

—

C₂²×C₈×C₁₉

—

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₁₉×C₄⋊Q₈

Q₈

yes

characteristic

C₁₉×#127

Q₈

yes

characteristic

C₁₉×#124

D₄

yes

—

C₁₉×#121

—

C₄×D₄×C₁₉

—

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₁₉×#97

—

C₄×C₈×C₁₉

D₄

yes

—

C₄×C₈×C₁₉

D₄

yes

—

#41×C₁₉

—

#41×C₁₉

—

C₂×Q₈×C₁₉

C₄⋊C₄

yes

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

C₄⋊C₄

yes

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₉

#41

yes

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₂²⋊C₄

yes

—

C₂×C₈×C₁₉

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

C₂×D₄

yes

—

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄²×C₁₉

C₂×Q₈

yes

characteristic

#8632

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

#117

yes

characteristic

Q₈×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×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉

#121

yes

—

C₂×C₄×C₁₉

C₂²⋊Q₈

yes

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

#100

yes

—

#2931

C₂×C₁₉

yes

characteristic

#1568

C₂×C₁₉

yes

complemented

#1536

C₂×C₁₉

yes

—

#1307

C₂×C₁₉

yes

complemented

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#380

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

—

#532

C₂²×C₁₉

yes

characteristic

#436

—

100

C₈×D₄

—

101

#432

C₂²×C₁₉

yes

—

102

#423

C₂²×C₁₉

yes

characteristic

103

#408

C₄×C₁₉

yes

—

104

C₂×C₄×C₈

C₂²×C₁₉

yes

characteristic

105

#335

—

106

#334

—

107

#329

—

108

C₂×C₁₉

—

109

C₂×C₁₉

—

110

C₂×C₁₉

#1800

yes

characteristic

111

C₂×C₁₉

#1814

yes

—

112

C₂×#41

C₂×C₄×C₁₉

yes

—

113

#131

—

114

C₂×M₁₆

—

115

C₂×M₁₆

C₂×C₄×C₁₉

yes

—

116

C₂²×C₈

—

117

C₂²×C₈

C₂×C₄×C₁₉

yes

characteristic

118

C₂²×C₈

C₂×C₄×C₁₉

yes

characteristic

119

C₄⋊Q₈

Q₈×C₁₉

yes

characteristic

120

#127

Q₈×C₁₉

yes

characteristic

121

#124

D₄×C₁₉

yes

—

122

#121

—

123

C₄×D₄

—

124

C₂×C₄²

C₂³×C₁₉

yes

characteristic

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₂²⋊C₈

—

128

C₂²⋊C₈

—

129

#97

—

130

C₄×C₈

D₄×C₁₉

yes

—

131

C₄×C₈

D₄×C₁₉

yes

—

132

C₁₉

#8632

yes

characteristic, direct product factor, Sylow

133

#41

—

134

#41

—

135

C₂×Q₈

C₄⋊C₄×C₁₉

yes

—

136

C₂×D₄

—

137

C₂×D₄

—

138

C₂×D₄

C₄⋊C₄×C₁₉

yes

—

139

C₂²×C₄

—

140

C₂²×C₄

C₂²×C₄×C₁₉

yes

characteristic

141

C₂²×C₄

#41×C₁₉

yes

—

142

M₁₆

—

143

M₁₆

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

C₂²⋊C₄×C₁₉

yes

—

150

C₂×C₈

C₂²⋊C₄×C₁₉

yes

characteristic

151

C₂×C₈

C₂²⋊C₄×C₁₉

yes

characteristic

152

C₄⋊C₄

—

153

C₂²⋊C₄

—

154

C₄²

C₂×D₄×C₁₉

yes

—

155

C₄²

C₂×D₄×C₁₉

yes

characteristic

156

C₄²

C₂×Q₈×C₁₉

yes

characteristic

157

C₂³

—

158

C₂³

C₁₉×#117

yes

characteristic

159

Q₈

—

160

D₄

—

161

D₄

—

162

D₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

C₂×C₂²⋊C₄×C₁₉

yes

characteristic

170

C₂×C₄

C₂×C₄⋊C₄×C₁₉

yes

characteristic

171

C₂×C₄

C₁₉×#121

yes

—

172

C₂×C₄

C₁₉×C₂²⋊Q₈

yes

—

173

C₈

—

174

C₈

—

175

C₈

—

176

C₈

—

177

C₈

C₁₉×#100

yes

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

—

182

C₂²

C₁₉×#380

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

C₂×C₁₉×#100

yes

characteristic

188

C₄

C₁₉×#413

yes

characteristic

189

C₂

—

190

C₂

—

191

C₂

C₁₉×#1800

yes

characteristic

192

C₂

C₁₉×#1814

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₉	#380
commutator	C₂²×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
prime radical	C₁₉×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₉

#380

commutator

C₂²×C₄

C₂²×C₄×C₁₉

Frattini, cosocle

C₂×C₄²

C₂³×C₁₉

prime radical

C₁₉×#532

C₂²

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

N	H ≅ G/N	type
2. C₁₉×#1568	C₂	semidirect
4. C₁₉×#1307	C₂	semidirect
60. #8632	C₁₉	direct (N × H)
83. #1568	C₂×C₁₉	semidirect
85. #1307	C₂×C₁₉	semidirect
132. C₁₉	#8632	direct (N × H)

H ≅ G/N

type

2. C₁₉×#1568

C₂

semidirect

4. C₁₉×#1307

C₂

semidirect

60. #8632

C₁₉

direct (N × H)

83. #1568

C₂×C₁₉

semidirect

85. #1307

C₂×C₁₉

semidirect

132. C₁₉

#8632

direct (N × H)

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

subgroup	index
C₁₉×#2931	2
C₁₉×#1568	2
C₁₉×#1536	2
C₁₉×#1307	2
#8632	19

subgroup

index

C₁₉×#2931

C₁₉×#1568

C₁₉×#1536

C₁₉×#1307

#8632

Subgroups — C₁₉×#8632 #18014591

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#8632 #18014591

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#8632 #18014591