Group #18019551 — subgroups

Subgroups — C₁₉×#13592 #18019551

Order 4864.

398 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 90 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×#3152	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2132	C₂	1	2	yes	maximal
3	C₁₉×#1411	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₉×#137	—	2	2	no	—
5	C₁₉×#532	C₂²	1	1	yes	characteristic
6	C₂×C₁₉×C₄⋊Q₈	C₄	1	1	yes	characteristic
7	C₁₉×#497	—	2	2	no	—
8	C₁₉×#494	C₂²	1	2	yes	—
9	C₁₉×#492	C₄	1	2	yes	—
10	C₁₉×#413	—	2	2	no	—
11	C₄×M₁₆×C₁₉	C₄	1	1	yes	characteristic
12	C₁₉×#356	C₂²	1	4	yes	—
13	C₁₉×#355	—	2	2	no	—
14	C₂×#41×C₁₉	D₄	1	2	yes	—
15	C₂²×Q₈×C₁₉	—	2	1	no	—
16	C₁₉×#137	—	4	4	no	—
17	C₁₉×#137	—	4	4	no	—
18	C₂×Q₁₆×C₁₉	—	2	4	no	—
19	C₂×SD₁₆×C₁₉	—	2	4	no	—
20	C₂×M₁₆×C₁₉	—	2	1	no	—
21	C₂×M₁₆×C₁₉	—	4	2	no	—
22	C₁₉×C₄⋊Q₈	—	2	1	no	—
23	C₁₉×C₄⋊Q₈	—	4	1	no	—
24	C₁₉×C₄⋊Q₈	C₂×C₄	1	2	yes	—
25	C₁₉×C₄⋊Q₈	C₂³	1	1	yes	characteristic
26	C₁₉×#127	C₂×C₄	1	1	yes	characteristic
27	C₁₉×#124	D₄	1	2	yes	—
28	C₁₉×#121	—	4	1	no	—
29	C₄×D₄×C₁₉	—	4	1	no	—
30	C₂×C₄⋊C₄×C₁₉	—	4	1	no	—
31	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
32	C₁₉×#101	—	2	4	no	—
33	C₁₉×#100	—	4	2	no	—
34	C₁₉×#97	—	2	1	no	—
35	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
36	#41×C₁₉	—	4	2	no	—
37	#41×C₁₉	—	4	2	no	—
38	C₂×Q₈×C₁₉	—	2	2	no	—
39	C₂×Q₈×C₁₉	—	4	2	no	—
40	C₂×Q₈×C₁₉	—	4	2	no	—
41	C₂×Q₈×C₁₉	—	4	1	no	—
42	C₂×Q₈×C₁₉	C₂×D₄	1	2	yes	—
43	C₂×D₄×C₁₉	—	2	2	no	—
44	C₂×D₄×C₁₉	—	4	1	no	—
45	C₂×D₄×C₁₉	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₁₉	—	2	1	no	—
47	C₂²×C₄×C₁₉	—	2	2	no	—
48	C₂²×C₄×C₁₉	—	4	1	no	—
49	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
50	Q₁₆×C₁₉	—	4	8	no	—
51	SD₁₆×C₁₉	—	4	8	no	—
52	M₁₆×C₁₉	—	2	4	no	—
53	M₁₆×C₁₉	—	4	4	no	—
54	M₁₆×C₁₉	—	8	2	no	—
55	C₂×C₈×C₁₉	—	2	2	no	—
56	C₂×C₈×C₁₉	—	8	2	no	—
57	C₄⋊C₄×C₁₉	—	4	1	no	—
58	C₄⋊C₄×C₁₉	—	4	2	no	—
59	C₄⋊C₄×C₁₉	—	4	2	no	—
60	C₂²⋊C₄×C₁₉	—	4	2	no	—
61	C₄²×C₁₉	—	2	1	no	—
62	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
64	#13592	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂³×C₁₉	—	2	2	no	—
66	C₂³×C₁₉	#99	1	1	yes	characteristic
67	Q₈×C₁₉	—	4	2	no	—
68	Q₈×C₁₉	—	4	4	no	—
69	Q₈×C₁₉	—	8	2	no	—
70	D₄×C₁₉	—	4	2	no	—
71	D₄×C₁₉	—	8	2	no	—
72	D₄×C₁₉	—	8	2	no	—
73	C₂×C₄×C₁₉	—	2	1	no	—
74	C₂×C₄×C₁₉	—	2	2	no	—
75	C₂×C₄×C₁₉	—	2	1	no	—
76	C₂×C₄×C₁₉	—	2	2	no	—
77	C₂×C₄×C₁₉	—	4	2	no	—
78	C₂×C₄×C₁₉	—	4	2	no	—
79	C₂×C₄×C₁₉	—	4	2	no	—
80	C₂×C₄×C₁₉	—	4	2	no	—
81	C₂×C₄×C₁₉	—	8	1	no	—
82	C₂×C₄×C₁₉	—	8	1	no	—
83	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
84	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₈×C₁₉	—	2	4	no	—
86	C₈×C₁₉	—	8	4	no	—
87	#3152	C₂×C₁₉	1	1	yes	characteristic
88	#2132	C₂×C₁₉	1	2	yes	—
89	#1411	C₂×C₁₉	1	4	yes	complemented
90	C₂²×C₁₉	—	2	1	no	—
91	C₂²×C₁₉	—	4	1	no	—
92	C₂²×C₁₉	—	4	2	no	—
93	C₂²×C₁₉	—	8	2	no	—
94	C₂²×C₁₉	C₂×#99	1	1	yes	characteristic
95	C₄×C₁₉	—	2	2	no	—
96	C₄×C₁₉	—	2	1	no	—
97	C₄×C₁₉	—	4	1	no	—
98	C₄×C₁₉	—	4	2	no	—
99	C₄×C₁₉	—	8	2	no	—
100	C₄×C₁₉	#353	1	2	yes	—
101	C₂×#137	—	2	2	no	—
102	#532	C₂²×C₁₉	1	1	yes	characteristic
103	C₂×C₄⋊Q₈	C₄×C₁₉	1	1	yes	characteristic
104	#497	—	2	2	no	—
105	#494	C₂²×C₁₉	1	2	yes	—
106	#492	C₄×C₁₉	1	2	yes	—
107	#413	—	2	2	no	—
108	C₄×M₁₆	C₄×C₁₉	1	1	yes	characteristic
109	#356	C₂²×C₁₉	1	4	yes	—
110	#355	—	2	2	no	—
111	C₂×C₁₉	—	2	1	no	—
112	C₂×C₁₉	—	4	1	no	—
113	C₂×C₁₉	—	8	2	no	—
114	C₂×C₁₉	C₂×#353	1	1	yes	characteristic
115	C₂×#41	D₄×C₁₉	1	2	yes	—
116	C₂²×Q₈	—	2	1	no	—
117	#137	—	4	4	no	—
118	#137	—	4	4	no	—
119	C₂×Q₁₆	—	2	4	no	—
120	C₂×SD₁₆	—	2	4	no	—
121	C₂×M₁₆	—	2	1	no	—
122	C₂×M₁₆	—	4	2	no	—
123	C₄⋊Q₈	—	2	1	no	—
124	C₄⋊Q₈	—	4	1	no	—
125	C₄⋊Q₈	C₂×C₄×C₁₉	1	2	yes	—
126	C₄⋊Q₈	C₂³×C₁₉	1	1	yes	characteristic
127	#127	C₂×C₄×C₁₉	1	1	yes	characteristic
128	#124	D₄×C₁₉	1	2	yes	—
129	#121	—	4	1	no	—
130	C₄×D₄	—	4	1	no	—
131	C₂×C₄⋊C₄	—	4	1	no	—
132	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
133	#101	—	2	4	no	—
134	#100	—	4	2	no	—
135	#97	—	2	1	no	—
136	C₄×C₈	C₂×C₄×C₁₉	1	2	yes	—
137	C₁₉	#13592	1	1	yes	characteristic, direct product factor, Sylow
138	#41	—	4	2	no	—
139	#41	—	4	2	no	—
140	C₂×Q₈	—	2	2	no	—
141	C₂×Q₈	—	4	2	no	—
142	C₂×Q₈	—	4	2	no	—
143	C₂×Q₈	—	4	1	no	—
144	C₂×Q₈	C₂×D₄×C₁₉	1	2	yes	—
145	C₂×D₄	—	2	2	no	—
146	C₂×D₄	—	4	1	no	—
147	C₂×D₄	C₂²⋊C₄×C₁₉	1	2	yes	—
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	—	4	1	no	—
151	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
152	Q₁₆	—	4	8	no	—
153	SD₁₆	—	4	8	no	—
154	M₁₆	—	2	4	no	—
155	M₁₆	—	4	4	no	—
156	M₁₆	—	8	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	8	2	no	—
159	C₄⋊C₄	—	4	1	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₂²⋊C₄	—	4	2	no	—
163	C₄²	—	2	1	no	—
164	C₄²	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
165	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
166	C₂³	—	2	2	no	—
167	C₂³	C₁₉×#99	1	1	yes	characteristic
168	Q₈	—	4	2	no	—
169	Q₈	—	4	4	no	—
170	Q₈	—	8	2	no	—
171	D₄	—	4	2	no	—
172	D₄	—	8	2	no	—
173	D₄	—	8	2	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	8	1	no	—
183	C₂×C₄	—	8	1	no	—
184	C₂×C₄	C₁₉×#99	1	1	yes	characteristic
185	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
186	C₈	—	2	4	no	—
187	C₈	—	8	4	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	4	1	no	—
190	C₂²	—	4	2	no	—
191	C₂²	—	8	2	no	—
192	C₂²	C₂×C₁₉×#99	1	1	yes	characteristic
193	C₄	—	2	2	no	—
194	C₄	—	2	1	no	—
195	C₄	—	4	1	no	—
196	C₄	—	4	2	no	—
197	C₄	—	8	2	no	—
198	C₄	C₁₉×#353	1	2	yes	—
199	C₂	—	2	1	no	—
200	C₂	—	4	1	no	—
201	C₂	—	8	2	no	—
202	C₂	C₂×C₁₉×#353	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#3152

C₂

yes

maximal, characteristic

C₁₉×#2132

C₂

yes

maximal

C₁₉×#1411

C₂

yes

maximal, complemented

C₂×C₁₉×#137

—

C₁₉×#532

C₂²

yes

characteristic

C₂×C₁₉×C₄⋊Q₈

C₄

yes

characteristic

C₁₉×#497

—

C₁₉×#494

C₂²

yes

—

C₁₉×#492

C₄

yes

—

C₁₉×#413

—

C₄×M₁₆×C₁₉

C₄

yes

characteristic

C₁₉×#356

C₂²

yes

—

C₁₉×#355

—

C₂×#41×C₁₉

D₄

yes

—

C₂²×Q₈×C₁₉

—

C₁₉×#137

—

C₁₉×#137

—

C₂×Q₁₆×C₁₉

—

C₂×SD₁₆×C₁₉

—

C₂×M₁₆×C₁₉

—

C₂×M₁₆×C₁₉

—

C₁₉×C₄⋊Q₈

—

C₁₉×C₄⋊Q₈

—

C₁₉×C₄⋊Q₈

C₂×C₄

yes

—

C₁₉×C₄⋊Q₈

C₂³

yes

characteristic

C₁₉×#127

C₂×C₄

yes

characteristic

C₁₉×#124

D₄

yes

—

C₁₉×#121

—

C₄×D₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₁₉×#101

—

C₁₉×#100

—

C₁₉×#97

—

C₄×C₈×C₁₉

C₂×C₄

yes

—

#41×C₁₉

—

#41×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

C₂×D₄

yes

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

Q₁₆×C₁₉

—

SD₁₆×C₁₉

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

C₂²⋊C₄

yes

characteristic

C₄²×C₁₉

C₂²×C₄

yes

characteristic

#13592

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

#99

yes

characteristic

Q₈×C₁₉

—

Q₈×C₁₉

—

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×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#99

yes

characteristic

C₂×C₄×C₁₉

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

—

#3152

C₂×C₁₉

yes

characteristic

#2132

C₂×C₁₉

yes

—

#1411

C₂×C₁₉

yes

complemented

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

C₂×#99

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

100

C₄×C₁₉

#353

yes

—

101

C₂×#137

—

102

#532

C₂²×C₁₉

yes

characteristic

103

C₂×C₄⋊Q₈

C₄×C₁₉

yes

characteristic

104

#497

—

105

#494

C₂²×C₁₉

yes

—

106

#492

C₄×C₁₉

yes

—

107

#413

—

108

C₄×M₁₆

C₄×C₁₉

yes

characteristic

109

#356

C₂²×C₁₉

yes

—

110

#355

—

111

C₂×C₁₉

—

112

C₂×C₁₉

—

113

C₂×C₁₉

—

114

C₂×C₁₉

C₂×#353

yes

characteristic

115

C₂×#41

D₄×C₁₉

yes

—

116

C₂²×Q₈

—

117

#137

—

118

#137

—

119

C₂×Q₁₆

—

120

C₂×SD₁₆

—

121

C₂×M₁₆

—

122

C₂×M₁₆

—

123

C₄⋊Q₈

—

124

C₄⋊Q₈

—

125

C₄⋊Q₈

C₂×C₄×C₁₉

yes

—

126

C₄⋊Q₈

C₂³×C₁₉

yes

characteristic

127

#127

C₂×C₄×C₁₉

yes

characteristic

128

#124

D₄×C₁₉

yes

—

129

#121

—

130

C₄×D₄

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄²

C₂×C₄×C₁₉

yes

characteristic

133

#101

—

134

#100

—

135

#97

—

136

C₄×C₈

C₂×C₄×C₁₉

yes

—

137

C₁₉

#13592

yes

characteristic, direct product factor, Sylow

138

#41

—

139

#41

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂×Q₈

C₂×D₄×C₁₉

yes

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

C₂²⋊C₄×C₁₉

yes

—

148

C₂²×C₄

—

149

C₂²×C₄

—

150

C₂²×C₄

—

151

C₂²×C₄

C₂²⋊C₄×C₁₉

yes

characteristic

152

Q₁₆

—

153

SD₁₆

—

154

M₁₆

—

155

M₁₆

—

156

M₁₆

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₂²⋊C₄

—

163

C₄²

—

164

C₄²

C₂²⋊C₄×C₁₉

yes

characteristic

165

C₄²

C₂²×C₄×C₁₉

yes

characteristic

166

C₂³

—

167

C₂³

C₁₉×#99

yes

characteristic

168

Q₈

—

169

Q₈

—

170

Q₈

—

171

D₄

—

172

D₄

—

173

D₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

C₁₉×#99

yes

characteristic

185

C₂×C₄

C₂×C₂²⋊C₄×C₁₉

yes

characteristic

186

C₈

—

187

C₈

—

188

C₂²

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

192

C₂²

C₂×C₁₉×#99

yes

characteristic

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

C₁₉×#353

yes

—

199

C₂

—

200

C₂

—

201

C₂

—

202

C₂

C₂×C₁₉×#353

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₁₉	C₂×#353
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	C₄⋊Q₈	C₂³×C₁₉
prime radical	C₁₉×#532	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#532 ◁ G

subgroup

G/H

center, socle

C₂×C₁₉

C₂×#353

commutator

C₄²

C₂²×C₄×C₁₉

Frattini, cosocle

C₄⋊Q₈

C₂³×C₁₉

prime radical

C₁₉×#532

C₂²

LCS

G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#532 ◁ G

N	H ≅ G/N	type
3. C₁₉×#1411	C₂	semidirect
64. #13592	C₁₉	direct (N × H)
89. #1411	C₂×C₁₉	semidirect
137. C₁₉	#13592	direct (N × H)

H ≅ G/N

type

3. C₁₉×#1411

C₂

semidirect

64. #13592

C₁₉

direct (N × H)

89. #1411

C₂×C₁₉

semidirect

137. C₁₉

#13592

direct (N × H)

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

subgroup	index
C₁₉×#3152	2
C₁₉×#2132	2
C₁₉×#1411	2
#13592	19

subgroup

index

C₁₉×#3152

C₁₉×#2132

C₁₉×#1411

#13592

Subgroups — C₁₉×#13592 #18019551

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#13592 #18019551

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13592 #18019551