Group #18016256 — subgroups

Subgroups — C₁₉×#10297 #18016256

Order 4864.

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

normal subgroups only

This group has 528 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×#3209	C₂	1	2	yes	maximal
2	C₁₉×#2862	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1994	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1548	C₂	1	2	yes	maximal
5	C₁₉×#1276	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#533	C₂²	1	1	yes	characteristic
7	C₂×C₁₉×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₁₉×#494	—	2	2	no	—
9	C₁₉×#487	—	2	2	no	—
10	C₁₉×#462	—	2	4	no	—
11	C₁₉×#451	—	2	4	no	—
12	C₁₉×#431	C₂²	1	2	yes	—
13	C₁₉×#423	C₂²	1	1	yes	characteristic
14	C₁₉×#391	—	2	4	no	—
15	C₁₉×#382	—	2	2	no	—
16	C₄³×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#339	—	2	2	no	—
18	C₁₉×#333	—	2	4	no	—
19	C₂²×Q₈×C₁₉	D₄	1	4	yes	—
20	C₂×Q₁₆×C₁₉	—	2	8	no	—
21	C₂×M₁₆×C₁₉	—	2	1	no	—
22	C₁₉×C₄⋊Q₈	—	2	2	no	—
23	C₁₉×C₄⋊Q₈	—	2	2	no	—
24	C₁₉×C₄⋊Q₈	—	2	2	no	—
25	C₁₉×C₄⋊Q₈	—	2	2	no	—
26	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
27	C₁₉×#125	—	2	1	no	—
28	C₁₉×C₂²⋊Q₈	—	2	2	no	—
29	C₄×Q₈×C₁₉	—	2	2	no	—
30	C₁₉×#117	—	2	1	no	—
31	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
32	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
33	C₂×C₄²×C₁₉	—	2	2	no	—
34	C₂×C₄²×C₁₉	D₄	1	2	yes	—
35	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
36	C₁₉×C₄⋊C₈	—	2	2	no	—
37	C₁₉×C₄⋊C₈	D₄	1	4	yes	—
38	C₁₉×#103	—	2	8	no	—
39	C₁₉×C₂²⋊C₈	—	2	2	no	—
40	C₄×C₈×C₁₉	—	2	2	no	—
41	C₁₉×#95	—	2	8	no	—
42	C₂×Q₈×C₁₉	—	2	2	no	—
43	C₂×Q₈×C₁₉	—	2	4	no	—
44	C₂×Q₈×C₁₉	—	2	4	no	—
45	C₂×Q₈×C₁₉	—	4	8	no	—
46	C₂²×C₄×C₁₉	—	2	2	no	—
47	C₂²×C₄×C₁₉	—	2	4	no	—
48	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
50	Q₁₆×C₁₉	—	8	8	no	—
51	M₁₆×C₁₉	—	4	2	no	—
52	C₂×C₈×C₁₉	—	2	4	no	—
53	C₂×C₈×C₁₉	—	4	1	no	—
54	C₄⋊C₄×C₁₉	—	2	2	no	—
55	C₄⋊C₄×C₁₉	—	2	2	no	—
56	C₄⋊C₄×C₁₉	—	4	2	no	—
57	C₄⋊C₄×C₁₉	—	4	2	no	—
58	C₄⋊C₄×C₁₉	—	4	2	no	—
59	C₄⋊C₄×C₁₉	—	4	2	no	—
60	C₄⋊C₄×C₁₉	—	4	2	no	—
61	C₂²⋊C₄×C₁₉	—	4	1	no	—
62	C₄²×C₁₉	—	2	2	no	—
63	C₄²×C₁₉	—	4	4	no	—
64	C₄²×C₁₉	—	4	1	no	—
65	C₄²×C₁₉	Q₁₆	1	4	yes	complemented
66	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₁₉	C₂×D₄	1	2	yes	—
68	C₄²×C₁₉	#41	1	1	yes	characteristic
69	#10297	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂³×C₁₉	#120	1	1	yes	characteristic
71	Q₈×C₁₉	—	4	4	no	—
72	Q₈×C₁₉	—	4	8	no	—
73	Q₈×C₁₉	—	8	4	no	—
74	C₂×C₄×C₁₉	—	2	4	no	—
75	C₂×C₄×C₁₉	—	2	4	no	—
76	C₂×C₄×C₁₉	—	4	4	no	—
77	C₂×C₄×C₁₉	—	4	8	no	—
78	C₂×C₄×C₁₉	—	4	2	no	—
79	C₂×C₄×C₁₉	—	4	4	no	—
80	C₂×C₄×C₁₉	#120	1	2	yes	—
81	C₂×C₄×C₁₉	#121	1	2	yes	—
82	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
83	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
84	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
85	C₂×C₄×C₁₉	#137	1	2	yes	—
86	C₈×C₁₉	—	4	4	no	—
87	C₈×C₁₉	—	8	1	no	—
88	#3209	C₂×C₁₉	1	2	yes	—
89	#2862	C₂×C₁₉	1	1	yes	characteristic
90	#1994	C₂×C₁₉	1	1	yes	characteristic
91	#1548	C₂×C₁₉	1	2	yes	—
92	#1276	C₂×C₁₉	1	1	yes	characteristic
93	C₂²×C₁₉	—	2	2	no	—
94	C₂²×C₁₉	#392	1	1	yes	characteristic
95	C₂²×C₁₉	#451	1	2	yes	—
96	C₄×C₁₉	—	2	2	no	—
97	C₄×C₁₉	—	4	4	no	—
98	C₄×C₁₉	—	8	4	no	—
99	C₄×C₁₉	—	8	2	no	—
100	C₄×C₁₉	#455	1	4	yes	—
101	C₄×C₁₉	#462	1	2	yes	—
102	C₄×C₁₉	#494	1	1	yes	characteristic
103	C₄×C₁₉	#497	1	1	yes	characteristic
104	#533	C₂²×C₁₉	1	1	yes	characteristic
105	C₂×C₄⋊Q₈	C₂²×C₁₉	1	2	yes	—
106	#494	—	2	2	no	—
107	#487	—	2	2	no	—
108	#462	—	2	4	no	—
109	#451	—	2	4	no	—
110	#431	C₂²×C₁₉	1	2	yes	—
111	#423	C₂²×C₁₉	1	1	yes	characteristic
112	#391	—	2	4	no	—
113	#382	—	2	2	no	—
114	C₄³	C₂²×C₁₉	1	1	yes	characteristic
115	#339	—	2	2	no	—
116	#333	—	2	4	no	—
117	C₂×C₁₉	—	2	2	no	—
118	C₂×C₁₉	#2004	1	2	yes	—
119	C₂×C₁₉	#2016	1	1	yes	characteristic
120	C₂²×Q₈	D₄×C₁₉	1	4	yes	—
121	C₂×Q₁₆	—	2	8	no	—
122	C₂×M₁₆	—	2	1	no	—
123	C₄⋊Q₈	—	2	2	no	—
124	C₄⋊Q₈	—	2	2	no	—
125	C₄⋊Q₈	—	2	2	no	—
126	C₄⋊Q₈	—	2	2	no	—
127	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
128	#125	—	2	1	no	—
129	C₂²⋊Q₈	—	2	2	no	—
130	C₄×Q₈	—	2	2	no	—
131	#117	—	2	1	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄²	—	2	2	no	—
135	C₂×C₄²	D₄×C₁₉	1	2	yes	—
136	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	D₄×C₁₉	1	4	yes	—
139	#103	—	2	8	no	—
140	C₂²⋊C₈	—	2	2	no	—
141	C₄×C₈	—	2	2	no	—
142	#95	—	2	8	no	—
143	C₁₉	#10297	1	1	yes	characteristic, direct product factor, Sylow
144	C₂×Q₈	—	2	4	no	—
145	C₂×Q₈	—	2	2	no	—
146	C₂×Q₈	—	2	4	no	—
147	C₂×Q₈	—	4	8	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	2	4	no	—
150	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
151	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
152	Q₁₆	—	8	8	no	—
153	M₁₆	—	4	2	no	—
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	—	4	1	no	—
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₂²⋊C₄	—	4	1	no	—
164	C₄²	—	2	2	no	—
165	C₄²	—	4	4	no	—
166	C₄²	—	4	1	no	—
167	C₄²	Q₁₆×C₁₉	1	4	yes	complemented
168	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
169	C₄²	C₂×D₄×C₁₉	1	2	yes	—
170	C₄²	#41×C₁₉	1	1	yes	characteristic
171	C₂³	C₁₉×#120	1	1	yes	characteristic
172	Q₈	—	4	4	no	—
173	Q₈	—	4	8	no	—
174	Q₈	—	8	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	4	8	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	C₁₉×#120	1	2	yes	—
182	C₂×C₄	C₁₉×#121	1	2	yes	—
183	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
184	C₂×C₄	C₁₉×#127	1	1	yes	characteristic
185	C₂×C₄	C₂×Q₁₆×C₁₉	1	2	yes	—
186	C₂×C₄	C₁₉×#137	1	2	yes	—
187	C₈	—	4	4	no	—
188	C₈	—	8	1	no	—
189	C₂²	—	2	2	no	—
190	C₂²	C₁₉×#392	1	1	yes	characteristic
191	C₂²	C₁₉×#451	1	2	yes	—
192	C₄	—	2	2	no	—
193	C₄	—	4	4	no	—
194	C₄	—	8	4	no	—
195	C₄	—	8	2	no	—
196	C₄	C₁₉×#455	1	4	yes	—
197	C₄	C₁₉×#462	1	2	yes	—
198	C₄	C₁₉×#494	1	1	yes	characteristic
199	C₄	C₁₉×#497	1	1	yes	characteristic
200	C₂	—	2	2	no	—
201	C₂	C₁₉×#2004	1	2	yes	—
202	C₂	C₁₉×#2016	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#3209

C₂

yes

maximal

C₁₉×#2862

C₂

yes

maximal, characteristic

C₁₉×#1994

C₂

yes

maximal, characteristic

C₁₉×#1548

C₂

yes

maximal

C₁₉×#1276

C₂

yes

maximal, characteristic

C₁₉×#533

C₂²

yes

characteristic

C₂×C₁₉×C₄⋊Q₈

C₂²

yes

—

C₁₉×#494

—

C₁₉×#487

—

C₁₉×#462

—

C₁₉×#451

—

C₁₉×#431

C₂²

yes

—

C₁₉×#423

C₂²

yes

characteristic

C₁₉×#391

—

C₁₉×#382

—

C₄³×C₁₉

C₂²

yes

characteristic

C₁₉×#339

—

C₁₉×#333

—

C₂²×Q₈×C₁₉

D₄

yes

—

C₂×Q₁₆×C₁₉

—

C₂×M₁₆×C₁₉

—

C₁₉×C₄⋊Q₈

—

C₁₉×C₄⋊Q₈

—

C₁₉×C₄⋊Q₈

—

C₁₉×C₄⋊Q₈

—

C₁₉×C₄⋊Q₈

D₄

yes

—

C₁₉×#125

—

C₁₉×C₂²⋊Q₈

—

C₄×Q₈×C₁₉

—

C₁₉×#117

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄²×C₁₉

—

C₂×C₄²×C₁₉

D₄

yes

—

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

D₄

yes

—

C₁₉×#103

—

C₁₉×C₂²⋊C₈

—

C₄×C₈×C₁₉

—

C₁₉×#95

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂×D₄

yes

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

Q₁₆×C₁₉

—

M₁₆×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

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

Q₁₆

yes

complemented

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄²×C₁₉

C₂×D₄

yes

—

C₄²×C₁₉

#41

yes

characteristic

#10297

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

#120

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

#120

yes

—

C₂×C₄×C₁₉

#121

yes

—

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

C₂×Q₁₆

yes

—

C₂×C₄×C₁₉

#137

yes

—

C₈×C₁₉

—

C₈×C₁₉

—

#3209

C₂×C₁₉

yes

—

#2862

C₂×C₁₉

yes

characteristic

#1994

C₂×C₁₉

yes

characteristic

#1548

C₂×C₁₉

yes

—

#1276

C₂×C₁₉

yes

characteristic

C₂²×C₁₉

—

C₂²×C₁₉

#392

yes

characteristic

C₂²×C₁₉

#451

yes

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

100

C₄×C₁₉

#455

yes

—

101

C₄×C₁₉

#462

yes

—

102

C₄×C₁₉

#494

yes

characteristic

103

C₄×C₁₉

#497

yes

characteristic

104

#533

C₂²×C₁₉

yes

characteristic

105

C₂×C₄⋊Q₈

C₂²×C₁₉

yes

—

106

#494

—

107

#487

—

108

#462

—

109

#451

—

110

#431

C₂²×C₁₉

yes

—

111

#423

C₂²×C₁₉

yes

characteristic

112

#391

—

113

#382

—

114

C₄³

C₂²×C₁₉

yes

characteristic

115

#339

—

116

#333

—

117

C₂×C₁₉

—

118

C₂×C₁₉

#2004

yes

—

119

C₂×C₁₉

#2016

yes

characteristic

120

C₂²×Q₈

D₄×C₁₉

yes

—

121

C₂×Q₁₆

—

122

C₂×M₁₆

—

123

C₄⋊Q₈

—

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

D₄×C₁₉

yes

—

128

#125

—

129

C₂²⋊Q₈

—

130

C₄×Q₈

—

131

#117

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

134

C₂×C₄²

—

135

C₂×C₄²

D₄×C₁₉

yes

—

136

C₂×C₄²

C₂³×C₁₉

yes

characteristic

137

C₄⋊C₈

—

138

C₄⋊C₈

D₄×C₁₉

yes

—

139

#103

—

140

C₂²⋊C₈

—

141

C₄×C₈

—

142

#95

—

143

C₁₉

#10297

yes

characteristic, direct product factor, Sylow

144

C₂×Q₈

—

145

C₂×Q₈

—

146

C₂×Q₈

—

147

C₂×Q₈

—

148

C₂²×C₄

—

149

C₂²×C₄

—

150

C₂²×C₄

C₂×D₄×C₁₉

yes

—

151

C₂²×C₄

C₂×D₄×C₁₉

yes

characteristic

152

Q₁₆

—

153

M₁₆

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₂²⋊C₄

—

164

C₄²

—

165

C₄²

—

166

C₄²

—

167

C₄²

Q₁₆×C₁₉

yes

complemented

168

C₄²

C₂×D₄×C₁₉

yes

characteristic

169

C₄²

C₂×D₄×C₁₉

yes

—

170

C₄²

#41×C₁₉

yes

characteristic

171

C₂³

C₁₉×#120

yes

characteristic

172

Q₈

—

173

Q₈

—

174

Q₈

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

C₁₉×#120

yes

—

182

C₂×C₄

C₁₉×#121

yes

—

183

C₂×C₄

C₁₉×#121

yes

characteristic

184

C₂×C₄

C₁₉×#127

yes

characteristic

185

C₂×C₄

C₂×Q₁₆×C₁₉

yes

—

186

C₂×C₄

C₁₉×#137

yes

—

187

C₈

—

188

C₈

—

189

C₂²

—

190

C₂²

C₁₉×#392

yes

characteristic

191

C₂²

C₁₉×#451

yes

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

C₁₉×#455

yes

—

197

C₄

C₁₉×#462

yes

—

198

C₄

C₁₉×#494

yes

characteristic

199

C₄

C₁₉×#497

yes

characteristic

200

C₂

—

201

C₂

C₁₉×#2004

yes

—

202

C₂

C₁₉×#2016

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
prime radical	C₂³×C₁₉	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₉

#392

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₁₉

prime radical

C₂³×C₁₉

#120

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

N	H ≅ G/N	type
65. C₄²×C₁₉	Q₁₆	semidirect
69. #10297	C₁₉	direct (N × H)
143. C₁₉	#10297	direct (N × H)
167. C₄²	Q₁₆×C₁₉	semidirect

H ≅ G/N

type

65. C₄²×C₁₉

Q₁₆

semidirect

69. #10297

C₁₉

direct (N × H)

143. C₁₉

#10297

direct (N × H)

167. C₄²

Q₁₆×C₁₉

semidirect

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

subgroup	index
C₁₉×#3209	2
C₁₉×#2862	2
C₁₉×#1994	2
C₁₉×#1548	2
C₁₉×#1276	2
#10297	19

subgroup

index

C₁₉×#3209

C₁₉×#2862

C₁₉×#1994

C₁₉×#1548

C₁₉×#1276

#10297

Subgroups — C₁₉×#10297 #18016256

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#10297 #18016256

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#10297 #18016256