Group #18016647 — subgroups

Subgroups — C₁₉×#10688 #18016647

Order 4864.

264 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 106 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×#1940	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1849	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1303	C₂	1	2	yes	maximal
4	C₁₉×#1301	C₂	1	2	yes	maximal
5	C₁₉×#1275	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#431	C₂²	1	2	yes	—
7	C₂×C₁₉×#108	—	2	1	no	—
8	C₂×C₁₉×#108	—	2	1	no	—
9	C₁₉×#423	C₂²	1	2	yes	—
10	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
11	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
14	C₁₉×#336	—	2	1	no	—
15	C₂×M₁₆×C₁₉	—	2	2	no	—
16	C₂²×C₈×C₁₉	—	2	1	no	—
17	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
20	C₂×C₄²×C₁₉	—	2	1	no	—
21	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
22	C₁₉×#108	—	4	2	no	—
23	C₁₉×#108	—	4	2	no	—
24	C₁₉×C₄⋊C₈	—	2	2	no	—
25	C₁₉×C₄⋊C₈	—	2	4	no	—
26	C₁₉×C₄⋊C₈	—	2	2	no	—
27	C₁₉×C₄⋊C₈	—	4	2	no	—
28	C₁₉×C₂²⋊C₈	—	2	2	no	—
29	C₁₉×#97	—	2	2	no	—
30	C₄×C₈×C₁₉	—	2	2	no	—
31	C₁₉×#95	Q₈	1	1	yes	characteristic
32	C₁₉×#95	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₁₉	—	2	1	no	—
34	C₂²×C₄×C₁₉	—	2	1	no	—
35	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
37	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
38	M₁₆×C₁₉	—	4	2	no	—
39	M₁₆×C₁₉	—	4	2	no	—
40	C₂×C₈×C₁₉	—	2	1	no	—
41	C₂×C₈×C₁₉	—	2	1	no	—
42	C₂×C₈×C₁₉	—	2	1	no	—
43	C₂×C₈×C₁₉	—	2	1	no	—
44	C₂×C₈×C₁₉	—	4	1	no	—
45	C₂×C₈×C₁₉	—	4	2	no	—
46	C₂×C₈×C₁₉	—	4	2	no	—
47	C₂×C₈×C₁₉	—	4	1	no	—
48	C₂×C₈×C₁₉	—	4	1	no	—
49	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
50	C₂×C₈×C₁₉	Q₁₆	1	2	yes	—
51	C₄⋊C₄×C₁₉	—	4	2	no	—
52	C₄²×C₁₉	—	4	1	no	—
53	C₄²×C₁₉	—	4	1	no	—
54	C₄²×C₁₉	#41	1	2	yes	—
55	C₄²×C₁₉	#41	1	2	yes	—
56	#10688	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₁₉	—	2	2	no	—
59	C₂×C₄×C₁₉	—	4	1	no	—
60	C₂×C₄×C₁₉	—	4	1	no	—
61	C₂×C₄×C₁₉	—	4	1	no	—
62	C₂×C₄×C₁₉	—	4	1	no	—
63	C₂×C₄×C₁₉	—	4	1	no	—
64	C₂×C₄×C₁₉	—	4	1	no	—
65	C₂×C₄×C₁₉	—	4	1	no	—
66	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
67	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
68	C₂×C₄×C₁₉	#125	1	2	yes	—
69	C₂×C₄×C₁₉	#126	1	2	yes	—
70	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
74	C₈×C₁₉	—	2	2	no	—
75	C₈×C₁₉	—	2	2	no	—
76	C₈×C₁₉	—	8	2	no	—
77	C₈×C₁₉	—	8	1	no	—
78	C₈×C₁₉	—	8	2	no	—
79	#1940	C₂×C₁₉	1	1	yes	characteristic
80	#1849	C₂×C₁₉	1	1	yes	characteristic
81	#1303	C₂×C₁₉	1	2	yes	—
82	#1301	C₂×C₁₉	1	2	yes	—
83	#1275	C₂×C₁₉	1	1	yes	characteristic
84	C₂²×C₁₉	—	2	1	no	—
85	C₂²×C₁₉	—	2	1	no	—
86	C₂²×C₁₉	#400	1	1	yes	characteristic
87	C₂²×C₁₉	#476	1	1	yes	characteristic
88	C₂²×C₁₉	#477	1	1	yes	characteristic
89	C₄×C₁₉	—	2	2	no	—
90	C₄×C₁₉	—	2	2	no	—
91	C₄×C₁₉	—	4	2	no	—
92	C₄×C₁₉	—	8	1	no	—
93	C₄×C₁₉	#481	1	1	yes	characteristic
94	C₄×C₁₉	#484	1	1	yes	characteristic
95	C₄×C₁₉	#487	1	1	yes	characteristic
96	C₄×C₁₉	#490	1	1	yes	characteristic
97	#431	C₂²×C₁₉	1	2	yes	—
98	C₂×#108	—	2	1	no	—
99	C₂×#108	—	2	1	no	—
100	#423	C₂²×C₁₉	1	2	yes	—
101	C₂×C₄⋊C₈	—	2	1	no	—
102	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
103	C₂×#97	C₂²×C₁₉	1	1	yes	characteristic
104	C₄×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
105	#336	—	2	1	no	—
106	C₂×C₁₉	—	2	1	no	—
107	C₂×C₁₉	—	2	1	no	—
108	C₂×C₁₉	#2100	1	1	yes	characteristic
109	C₂×C₁₉	#2102	1	1	yes	characteristic
110	C₂×C₁₉	#2103	1	1	yes	characteristic
111	C₂×M₁₆	—	2	2	no	—
112	C₂²×C₈	—	2	1	no	—
113	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
114	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
115	C₂×C₄⋊C₄	—	2	1	no	—
116	C₂×C₄²	—	2	1	no	—
117	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
118	#108	—	4	2	no	—
119	#108	—	4	2	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	2	4	no	—
122	C₄⋊C₈	—	2	2	no	—
123	C₄⋊C₈	—	4	2	no	—
124	C₂²⋊C₈	—	2	2	no	—
125	#97	—	2	2	no	—
126	C₄×C₈	—	2	2	no	—
127	#95	Q₈×C₁₉	1	1	yes	characteristic
128	#95	Q₈×C₁₉	1	1	yes	characteristic
129	C₁₉	#10688	1	1	yes	characteristic, direct product factor, Sylow
130	C₂²×C₄	—	2	1	no	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
133	C₂²×C₄	C₂×Q₈×C₁₉	1	1	yes	characteristic
134	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
135	M₁₆	—	4	2	no	—
136	M₁₆	—	4	2	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	4	1	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	4	2	no	—
144	C₂×C₈	—	4	1	no	—
145	C₂×C₈	—	4	1	no	—
146	C₂×C₈	SD₁₆×C₁₉	1	2	yes	—
147	C₂×C₈	Q₁₆×C₁₉	1	2	yes	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄²	—	4	1	no	—
150	C₄²	—	4	1	no	—
151	C₄²	#41×C₁₉	1	2	yes	—
152	C₄²	#41×C₁₉	1	2	yes	—
153	C₂³	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	—	4	1	no	—
162	C₂×C₄	C₁₉×#123	1	1	yes	characteristic
163	C₂×C₄	C₁₉×#124	1	1	yes	characteristic
164	C₂×C₄	C₁₉×#125	1	2	yes	—
165	C₂×C₄	C₁₉×#126	1	2	yes	—
166	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
167	C₂×C₄	C₂×Q₁₆×C₁₉	1	1	yes	characteristic
168	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
169	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	2	2	no	—
172	C₈	—	8	2	no	—
173	C₈	—	8	1	no	—
174	C₈	—	8	2	no	—
175	C₂²	—	2	1	no	—
176	C₂²	—	2	1	no	—
177	C₂²	C₁₉×#400	1	1	yes	characteristic
178	C₂²	C₁₉×#476	1	1	yes	characteristic
179	C₂²	C₁₉×#477	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	8	1	no	—
184	C₄	C₁₉×#481	1	1	yes	characteristic
185	C₄	C₁₉×#484	1	1	yes	characteristic
186	C₄	C₁₉×#487	1	1	yes	characteristic
187	C₄	C₁₉×#490	1	1	yes	characteristic
188	C₂	—	2	1	no	—
189	C₂	—	2	1	no	—
190	C₂	C₁₉×#2100	1	1	yes	characteristic
191	C₂	C₁₉×#2102	1	1	yes	characteristic
192	C₂	C₁₉×#2103	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#1940

C₂

yes

maximal, characteristic

C₁₉×#1849

C₂

yes

maximal, characteristic

C₁₉×#1303

C₂

yes

maximal

C₁₉×#1301

C₂

yes

maximal

C₁₉×#1275

C₂

yes

maximal, characteristic

C₁₉×#431

C₂²

yes

—

C₂×C₁₉×#108

—

C₂×C₁₉×#108

—

C₁₉×#423

C₂²

yes

—

C₂×C₁₉×C₄⋊C₈

—

C₂×C₁₉×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₁₉×#97

C₂²

yes

characteristic

C₄×C₄⋊C₄×C₁₉

C₂²

yes

characteristic

C₁₉×#336

—

C₂×M₁₆×C₁₉

—

C₂²×C₈×C₁₉

—

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄²×C₁₉

—

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₁₉×#108

—

C₁₉×#108

—

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₁₉×#97

—

C₄×C₈×C₁₉

—

C₁₉×#95

Q₈

yes

characteristic

C₁₉×#95

Q₈

yes

characteristic

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×Q₈

yes

characteristic

C₂²×C₄×C₁₉

#41

yes

characteristic

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

SD₁₆

yes

—

C₂×C₈×C₁₉

Q₁₆

yes

—

C₄⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

#41

yes

—

C₄²×C₁₉

#41

yes

—

#10688

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

C₂²⋊Q₈

yes

characteristic

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

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

—

C₂×C₄×C₁₉

#126

yes

—

C₂×C₄×C₁₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₉

C₂×Q₁₆

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

#1940

C₂×C₁₉

yes

characteristic

#1849

C₂×C₁₉

yes

characteristic

#1303

C₂×C₁₉

yes

—

#1301

C₂×C₁₉

yes

—

#1275

C₂×C₁₉

yes

characteristic

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

C₂²×C₁₉

yes

—

C₂×#108

—

C₂×#108

—

100

#423

C₂²×C₁₉

yes

—

101

C₂×C₄⋊C₈

—

102

C₂×C₄⋊C₈

C₂²×C₁₉

yes

characteristic

103

C₂×#97

C₂²×C₁₉

yes

characteristic

104

C₄×C₄⋊C₄

C₂²×C₁₉

yes

characteristic

105

#336

—

106

C₂×C₁₉

—

107

C₂×C₁₉

—

108

C₂×C₁₉

#2100

yes

characteristic

109

C₂×C₁₉

#2102

yes

characteristic

110

C₂×C₁₉

#2103

yes

characteristic

111

C₂×M₁₆

—

112

C₂²×C₈

—

113

C₂²×C₈

D₄×C₁₉

yes

characteristic

114

C₂²×C₈

D₄×C₁₉

yes

characteristic

115

C₂×C₄⋊C₄

—

116

C₂×C₄²

—

117

C₂×C₄²

C₂³×C₁₉

yes

characteristic

118

#108

—

119

#108

—

120

C₄⋊C₈

—

121

C₄⋊C₈

—

122

C₄⋊C₈

—

123

C₄⋊C₈

—

124

C₂²⋊C₈

—

125

#97

—

126

C₄×C₈

—

127

#95

Q₈×C₁₉

yes

characteristic

128

#95

Q₈×C₁₉

yes

characteristic

129

C₁₉

#10688

yes

characteristic, direct product factor, Sylow

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

C₂×D₄×C₁₉

yes

characteristic

133

C₂²×C₄

C₂×Q₈×C₁₉

yes

characteristic

134

C₂²×C₄

#41×C₁₉

yes

characteristic

135

M₁₆

—

136

M₁₆

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

SD₁₆×C₁₉

yes

—

147

C₂×C₈

Q₁₆×C₁₉

yes

—

148

C₄⋊C₄

—

149

C₄²

—

150

C₄²

—

151

C₄²

#41×C₁₉

yes

—

152

C₄²

#41×C₁₉

yes

—

153

C₂³

C₁₉×C₂²⋊Q₈

yes

characteristic

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

C₁₉×#123

yes

characteristic

163

C₂×C₄

C₁₉×#124

yes

characteristic

164

C₂×C₄

C₁₉×#125

yes

—

165

C₂×C₄

C₁₉×#126

yes

—

166

C₂×C₄

C₂×SD₁₆×C₁₉

yes

characteristic

167

C₂×C₄

C₂×Q₁₆×C₁₉

yes

characteristic

168

C₂×C₄

C₁₉×AGL(1,ℤ/8)

yes

characteristic

169

C₂×C₄

C₁₉×AGL(1,ℤ/8)

yes

characteristic

170

C₈

—

171

C₈

—

172

C₈

—

173

C₈

—

174

C₈

—

175

C₂²

—

176

C₂²

—

177

C₂²

C₁₉×#400

yes

characteristic

178

C₂²

C₁₉×#476

yes

characteristic

179

C₂²

C₁₉×#477

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

C₁₉×#481

yes

characteristic

185

C₄

C₁₉×#484

yes

characteristic

186

C₄

C₁₉×#487

yes

characteristic

187

C₄

C₁₉×#490

yes

characteristic

188

C₂

—

189

C₂

—

190

C₂

C₁₉×#2100

yes

characteristic

191

C₂

C₁₉×#2102

yes

characteristic

192

C₂

C₁₉×#2103

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₁₉	#124
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
socle	C₂²×C₁₉	#400
prime radical	C₂³×C₁₉	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

subgroup

G/H

center

C₂×C₄×C₁₉

#124

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₁₉

socle

C₂²×C₁₉

#400

prime radical

C₂³×C₁₉

C₂²⋊Q₈

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

N	H ≅ G/N	type
56. #10688	C₁₉	direct (N × H)
129. C₁₉	#10688	direct (N × H)

H ≅ G/N

type

56. #10688

C₁₉

direct (N × H)

129. C₁₉

#10688

direct (N × H)

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

subgroup	index
C₁₉×#1940	2
C₁₉×#1849	2
C₁₉×#1303	2
C₁₉×#1301	2
C₁₉×#1275	2
#10688	19

subgroup

index

C₁₉×#1940

C₁₉×#1849

C₁₉×#1303

C₁₉×#1301

C₁₉×#1275

#10688

Subgroups — C₁₉×#10688 #18016647

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#10688 #18016647

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#10688 #18016647