Group #18013208 — subgroups

Subgroups — C₁₉×#7249 #18013208

Order 4864.

312 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 78 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×#1954	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1465	C₂	1	2	yes	maximal
3	C₂×C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
4	C₂×C₁₉×C₄⋊C₈	C₄	1	1	yes	characteristic
5	C₁₉×#407	—	2	2	no	—
6	C₂×C₁₉×C₂²⋊C₈	—	2	2	no	—
7	C₂×C₄×C₈×C₁₉	C₄	1	1	yes	characteristic
8	C₁₉×#324	—	2	2	no	—
9	C₁₉×#323	C₄	1	4	yes	—
10	C₂²×D₄×C₁₉	Q₈	1	1	yes	characteristic
11	C₂³×C₄×C₁₉	D₄	1	2	yes	—
12	C₂×M₁₆×C₁₉	—	4	2	no	—
13	C₂²×C₈×C₁₉	—	2	1	no	—
14	C₂²×C₈×C₁₉	—	2	1	no	—
15	C₄×D₄×C₁₉	—	4	1	no	—
16	C₄×D₄×C₁₉	—	4	1	no	—
17	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
18	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
20	C₁₉×C₄⋊C₈	—	4	1	no	—
21	C₁₉×C₂²⋊C₈	—	2	4	no	—
22	C₁₉×C₂²⋊C₈	—	4	4	no	—
23	C₁₉×C₂²⋊C₈	—	4	2	no	—
24	C₄×C₈×C₁₉	—	4	1	no	—
25	C₂⁴×C₁₉	C₄⋊C₄	1	2	yes	—
26	C₂×D₄×C₁₉	—	2	2	no	—
27	C₂×D₄×C₁₉	—	4	1	no	—
28	C₂×D₄×C₁₉	—	4	1	no	—
29	C₂²×C₄×C₁₉	—	2	2	no	—
30	C₂²×C₄×C₁₉	—	2	2	no	—
31	C₂²×C₄×C₁₉	—	4	2	no	—
32	C₂²×C₄×C₁₉	—	4	2	no	—
33	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
34	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
37	M₁₆×C₁₉	—	8	2	no	—
38	M₁₆×C₁₉	—	8	2	no	—
39	C₂×C₈×C₁₉	—	2	2	no	—
40	C₂×C₈×C₁₉	—	4	1	no	—
41	C₂×C₈×C₁₉	—	4	1	no	—
42	C₂×C₈×C₁₉	—	4	2	no	—
43	C₂×C₈×C₁₉	—	4	4	no	—
44	C₂×C₈×C₁₉	—	4	1	no	—
45	C₄⋊C₄×C₁₉	—	4	1	no	—
46	C₂²⋊C₄×C₁₉	—	4	2	no	—
47	C₄²×C₁₉	—	2	1	no	—
48	C₄²×C₁₉	—	2	1	no	—
49	#7249	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂³×C₁₉	—	2	2	no	—
51	C₂³×C₁₉	—	2	2	no	—
52	C₂³×C₁₉	—	2	2	no	—
53	C₂³×C₁₉	—	4	2	no	—
54	C₂³×C₁₉	—	4	2	no	—
55	C₂³×C₁₉	#95	1	1	yes	characteristic
56	D₄×C₁₉	—	4	4	no	—
57	C₂×C₄×C₁₉	—	2	1	no	—
58	C₂×C₄×C₁₉	—	2	1	no	—
59	C₂×C₄×C₁₉	—	2	1	no	—
60	C₂×C₄×C₁₉	—	2	1	no	—
61	C₂×C₄×C₁₉	—	2	1	no	—
62	C₂×C₄×C₁₉	—	2	2	no	—
63	C₂×C₄×C₁₉	—	4	2	no	—
64	C₂×C₄×C₁₉	—	4	2	no	—
65	C₂×C₄×C₁₉	—	4	2	no	—
66	C₂×C₄×C₁₉	—	4	1	no	—
67	C₂×C₄×C₁₉	—	8	2	no	—
68	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
69	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
70	C₂×C₄×C₁₉	#108	1	2	yes	—
71	C₈×C₁₉	—	4	2	no	—
72	C₈×C₁₉	—	8	4	no	—
73	C₈×C₁₉	—	8	1	no	—
74	#1954	C₂×C₁₉	1	1	yes	characteristic
75	#1465	C₂×C₁₉	1	2	yes	—
76	C₂²×C₁₉	—	2	1	no	—
77	C₂²×C₁₉	—	2	1	no	—
78	C₂²×C₁₉	—	4	4	no	—
79	C₂²×C₁₉	—	4	2	no	—
80	C₂²×C₁₉	—	4	2	no	—
81	C₂²×C₁₉	—	4	2	no	—
82	C₂²×C₁₉	—	8	2	no	—
83	C₂²×C₁₉	#338	1	1	yes	characteristic
84	C₂²×C₁₉	#341	1	1	yes	characteristic
85	C₂²×C₁₉	#342	1	1	yes	characteristic
86	C₄×C₁₉	—	2	2	no	—
87	C₄×C₁₉	—	2	1	no	—
88	C₄×C₁₉	—	4	1	no	—
89	C₄×C₁₉	—	4	2	no	—
90	C₄×C₁₉	#353	1	1	yes	characteristic
91	C₄×C₁₉	#354	1	1	yes	characteristic
92	C₂×C₄×D₄	C₂²×C₁₉	1	1	yes	characteristic
93	C₂×C₄⋊C₈	C₄×C₁₉	1	1	yes	characteristic
94	#407	—	2	2	no	—
95	C₂×C₂²⋊C₈	—	2	2	no	—
96	C₂×C₄×C₈	C₄×C₁₉	1	1	yes	characteristic
97	#324	—	2	2	no	—
98	#323	C₄×C₁₉	1	4	yes	—
99	C₂×C₁₉	—	2	1	no	—
100	C₂×C₁₉	—	2	1	no	—
101	C₂×C₁₉	—	8	2	no	—
102	C₂×C₁₉	#1293	1	1	yes	characteristic
103	C₂×C₁₉	#1392	1	1	yes	characteristic
104	C₂×C₁₉	#1393	1	1	yes	characteristic
105	C₂²×D₄	Q₈×C₁₉	1	1	yes	characteristic
106	C₂³×C₄	D₄×C₁₉	1	2	yes	—
107	C₂×M₁₆	—	4	2	no	—
108	C₂²×C₈	—	2	1	no	—
109	C₂²×C₈	—	2	1	no	—
110	C₄×D₄	—	4	1	no	—
111	C₄×D₄	—	4	1	no	—
112	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
113	C₂×C₂²⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
114	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
115	C₄⋊C₈	—	4	1	no	—
116	C₂²⋊C₈	—	2	4	no	—
117	C₂²⋊C₈	—	4	4	no	—
118	C₂²⋊C₈	—	4	2	no	—
119	C₄×C₈	—	4	1	no	—
120	C₁₉	#7249	1	1	yes	characteristic, direct product factor, Sylow
121	C₂⁴	C₄⋊C₄×C₁₉	1	2	yes	—
122	C₂×D₄	—	2	2	no	—
123	C₂×D₄	—	4	1	no	—
124	C₂×D₄	—	4	1	no	—
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	—	2	2	no	—
127	C₂²×C₄	—	4	2	no	—
128	C₂²×C₄	—	4	2	no	—
129	C₂²×C₄	C₄²×C₁₉	1	1	yes	characteristic
130	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
131	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
132	C₂²×C₄	C₄⋊C₄×C₁₉	1	1	yes	characteristic
133	M₁₆	—	8	2	no	—
134	M₁₆	—	8	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	4	1	no	—
137	C₂×C₈	—	4	1	no	—
138	C₂×C₈	—	4	4	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	4	1	no	—
141	C₄⋊C₄	—	4	1	no	—
142	C₂²⋊C₄	—	4	2	no	—
143	C₄²	—	2	1	no	—
144	C₄²	—	2	1	no	—
145	C₂³	—	2	2	no	—
146	C₂³	—	2	2	no	—
147	C₂³	—	2	2	no	—
148	C₂³	—	4	2	no	—
149	C₂³	—	4	2	no	—
150	C₂³	C₁₉×#95	1	1	yes	characteristic
151	D₄	—	4	4	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	4	1	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	C₁₉×#99	1	1	yes	characteristic
164	C₂×C₄	C₁₉×#99	1	1	yes	characteristic
165	C₂×C₄	C₁₉×#108	1	2	yes	—
166	C₈	—	4	2	no	—
167	C₈	—	8	4	no	—
168	C₈	—	8	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	—	4	4	no	—
172	C₂²	—	4	2	no	—
173	C₂²	—	4	2	no	—
174	C₂²	—	4	2	no	—
175	C₂²	—	8	2	no	—
176	C₂²	C₁₉×#338	1	1	yes	characteristic
177	C₂²	C₁₉×#341	1	1	yes	characteristic
178	C₂²	C₁₉×#342	1	1	yes	characteristic
179	C₄	—	2	1	no	—
180	C₄	—	2	2	no	—
181	C₄	—	4	1	no	—
182	C₄	—	4	2	no	—
183	C₄	C₁₉×#353	1	1	yes	characteristic
184	C₄	C₁₉×#354	1	1	yes	characteristic
185	C₂	—	2	1	no	—
186	C₂	—	2	1	no	—
187	C₂	—	8	2	no	—
188	C₂	C₁₉×#1293	1	1	yes	characteristic
189	C₂	C₁₉×#1392	1	1	yes	characteristic
190	C₂	C₁₉×#1393	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#1954

C₂

yes

maximal, characteristic

C₁₉×#1465

C₂

yes

maximal

C₂×C₄×D₄×C₁₉

C₂²

yes

characteristic

C₂×C₁₉×C₄⋊C₈

C₄

yes

characteristic

C₁₉×#407

—

C₂×C₁₉×C₂²⋊C₈

—

C₂×C₄×C₈×C₁₉

C₄

yes

characteristic

C₁₉×#324

—

C₁₉×#323

C₄

yes

—

C₂²×D₄×C₁₉

Q₈

yes

characteristic

C₂³×C₄×C₁₉

D₄

yes

—

C₂×M₁₆×C₁₉

—

C₂²×C₈×C₁₉

—

C₂²×C₈×C₁₉

—

C₄×D₄×C₁₉

—

C₄×D₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

D₄

yes

characteristic

C₂×C₂²⋊C₄×C₁₉

C₂×C₄

yes

—

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₁₉×C₄⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₄×C₈×C₁₉

—

C₂⁴×C₁₉

C₄⋊C₄

yes

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₄²

yes

characteristic

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₄⋊C₄

yes

characteristic

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

#7249

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

#95

yes

characteristic

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

—

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

#1954

C₂×C₁₉

yes

characteristic

#1465

C₂×C₁₉

yes

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

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

C₂×C₄×D₄

C₂²×C₁₉

yes

characteristic

C₂×C₄⋊C₈

C₄×C₁₉

yes

characteristic

#407

—

C₂×C₂²⋊C₈

—

C₂×C₄×C₈

C₄×C₁₉

yes

characteristic

#324

—

#323

C₄×C₁₉

yes

—

C₂×C₁₉

—

100

C₂×C₁₉

—

101

C₂×C₁₉

—

102

C₂×C₁₉

#1293

yes

characteristic

103

C₂×C₁₉

#1392

yes

characteristic

104

C₂×C₁₉

#1393

yes

characteristic

105

C₂²×D₄

Q₈×C₁₉

yes

characteristic

106

C₂³×C₄

D₄×C₁₉

yes

—

107

C₂×M₁₆

—

108

C₂²×C₈

—

109

C₂²×C₈

—

110

C₄×D₄

—

111

C₄×D₄

—

112

C₂×C₄⋊C₄

D₄×C₁₉

yes

characteristic

113

C₂×C₂²⋊C₄

C₂×C₄×C₁₉

yes

—

114

C₂×C₄²

C₂×C₄×C₁₉

yes

characteristic

115

C₄⋊C₈

—

116

C₂²⋊C₈

—

117

C₂²⋊C₈

—

118

C₂²⋊C₈

—

119

C₄×C₈

—

120

C₁₉

#7249

yes

characteristic, direct product factor, Sylow

121

C₂⁴

C₄⋊C₄×C₁₉

yes

—

122

C₂×D₄

—

123

C₂×D₄

—

124

C₂×D₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

C₄²×C₁₉

yes

characteristic

130

C₂²×C₄

C₂²⋊C₄×C₁₉

yes

characteristic

131

C₂²×C₄

C₂²⋊C₄×C₁₉

yes

—

132

C₂²×C₄

C₄⋊C₄×C₁₉

yes

characteristic

133

M₁₆

—

134

M₁₆

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₄⋊C₄

—

142

C₂²⋊C₄

—

143

C₄²

—

144

C₄²

—

145

C₂³

—

146

C₂³

—

147

C₂³

—

148

C₂³

—

149

C₂³

—

150

C₂³

C₁₉×#95

yes

characteristic

151

D₄

—

152

C₂×C₄

—

153

C₂×C₄

—

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

—

163

C₂×C₄

C₁₉×#99

yes

characteristic

164

C₂×C₄

C₁₉×#99

yes

characteristic

165

C₂×C₄

C₁₉×#108

yes

—

166

C₈

—

167

C₈

—

168

C₈

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

—

176

C₂²

C₁₉×#338

yes

characteristic

177

C₂²

C₁₉×#341

yes

characteristic

178

C₂²

C₁₉×#342

yes

characteristic

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

C₁₉×#353

yes

characteristic

184

C₄

C₁₉×#354

yes

characteristic

185

C₂

—

186

C₂

—

187

C₂

—

188

C₂

C₁₉×#1293

yes

characteristic

189

C₂

C₁₉×#1392

yes

characteristic

190

C₂

C₁₉×#1393

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#342
commutator	C₂²×C₄	C₄²×C₁₉
Frattini, cosocle	C₂×C₄×D₄	C₂²×C₁₉
prime radical	C₂²×D₄×C₁₉	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₄×D₄×C₁₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₉

#342

commutator

C₂²×C₄

C₄²×C₁₉

Frattini, cosocle

C₂×C₄×D₄

C₂²×C₁₉

prime radical

C₂²×D₄×C₁₉

Q₈

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₄×D₄×C₁₉ ◁ G

N	H ≅ G/N	type
49. #7249	C₁₉	direct (N × H)
120. C₁₉	#7249	direct (N × H)

H ≅ G/N

type

49. #7249

C₁₉

direct (N × H)

120. C₁₉

#7249

direct (N × H)

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

subgroup	index
C₁₉×#1954	2
C₁₉×#1465	2
#7249	19

subgroup

index

C₁₉×#1954

C₁₉×#1465

#7249

Subgroups — C₁₉×#7249 #18013208

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#7249 #18013208

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#7249 #18013208