Group #18069283 — subgroups

Subgroups — #18069283

Order 4864.

288 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 95 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	#14561713	C₂	1	2	yes	maximal, complemented
2	C₁₉×#1954	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₉⋊C₁₆	—	2	2	no	—
4	C₂×C₄×D₄×C₁₉	C₄	1	1	yes	characteristic
5	C₂×C₁₉×C₄⋊C₈	C₄	1	1	yes	characteristic
6	C₂×C₁₉×C₂²⋊C₈	—	2	2	no	—
7	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
8	C₂²×D₄×C₁₉	C₈	1	1	yes	characteristic
9	C₂³×C₄×C₁₉	—	2	1	no	—
10	C₂²×C₈×C₁₉	—	2	1	no	—
11	C₂²×C₈×C₁₉	D₄	1	2	yes	—
12	C₄×D₄×C₁₉	—	4	1	no	—
13	C₄×D₄×C₁₉	—	4	1	no	—
14	C₂×C₄⋊C₄×C₁₉	C₈	1	1	yes	characteristic
15	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
16	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
17	C₁₉×C₄⋊C₈	—	4	1	no	—
18	C₁₉×C₂²⋊C₈	—	4	2	no	—
19	C₁₉×C₂²⋊C₈	—	4	2	no	—
20	C₄×C₈×C₁₉	—	4	1	no	—
21	C₂×C₁₉⋊C₁₆	—	2	4	no	—
22	C₂×C₁₉⋊C₁₆	—	4	2	no	—
23	C₂×C₁₉⋊C₁₆	—	4	2	no	—
24	C₂⁴×C₁₉	—	2	1	no	—
25	C₂×D₄×C₁₉	—	2	1	no	—
26	C₂×D₄×C₁₉	—	4	1	no	—
27	C₂×D₄×C₁₉	—	4	1	no	—
28	C₂×D₄×C₁₉	C₁₆	1	2	yes	complemented
29	C₂²×C₄×C₁₉	—	2	1	no	—
30	C₂²×C₄×C₁₉	—	4	1	no	—
31	C₂²×C₄×C₁₉	—	4	1	no	—
32	C₂²×C₄×C₁₉	—	4	2	no	—
33	C₂²×C₄×C₁₉	—	4	2	no	—
34	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₉	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₁₉	M₁₆	1	1	yes	characteristic
37	C₂×C₈×C₁₉	—	2	2	no	—
38	C₂×C₈×C₁₉	—	2	2	no	—
39	C₂×C₈×C₁₉	—	4	1	no	—
40	C₂×C₈×C₁₉	—	4	2	no	—
41	C₂×C₈×C₁₉	D₈	1	2	yes	—
42	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
43	C₄⋊C₄×C₁₉	—	4	1	no	—
44	C₂²⋊C₄×C₁₉	—	4	2	no	—
45	C₄²×C₁₉	—	2	1	no	—
46	C₄²×C₁₉	—	2	1	no	—
47	C₁₉⋊C₁₆	—	4	4	no	—
48	#7102	—	19	1	no	maximal, Sylow
49	C₂³×C₁₉	—	2	2	no	—
50	C₂³×C₁₉	—	4	1	no	—
51	C₂³×C₁₉	—	4	2	no	—
52	C₂³×C₁₉	—	4	1	no	—
53	C₂³×C₁₉	—	4	2	no	—
54	C₂³×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
55	D₄×C₁₉	—	4	2	no	—
56	D₄×C₁₉	—	4	2	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	2	no	—
61	C₂×C₄×C₁₉	—	4	1	no	—
62	C₂×C₄×C₁₉	—	4	2	no	—
63	C₂×C₄×C₁₉	—	8	2	no	—
64	C₂×C₄×C₁₉	—	8	1	no	—
65	C₂×C₄×C₁₉	—	8	1	no	—
66	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
67	C₂×C₄×C₁₉	#100	1	1	yes	characteristic
68	C₂×C₄×C₁₉	#102	1	2	yes	—
69	C₂×C₄×C₁₉	#104	1	2	yes	—
70	C₂×C₄×C₁₉	C₂×C₁₆	1	1	yes	characteristic
71	C₂×C₄×C₁₉	M₃₂	1	1	yes	characteristic
72	C₈×C₁₉	—	2	2	no	—
73	C₈×C₁₉	—	2	2	no	—
74	C₈×C₁₉	—	8	1	no	—
75	#1954	D₁₉	1	1	yes	characteristic
76	#1374	—	19	2	no	—
77	C₂²×C₁₉	—	4	2	no	—
78	C₂²×C₁₉	—	4	4	no	—
79	C₂²×C₁₉	—	8	1	no	—
80	C₂²×C₁₉	—	8	2	no	—
81	C₂²×C₁₉	—	8	1	no	—
82	C₂²×C₁₉	#323	1	1	yes	characteristic
83	C₂²×C₁₉	D₄⋊C₈	1	2	yes	—
84	C₂²×C₁₉	C₂²⋊C₁₆	1	1	yes	characteristic
85	C₂²×C₁₉	#349	1	1	yes	characteristic
86	C₂²×C₁₉	#350	1	2	yes	—
87	C₄×C₁₉	—	2	1	no	—
88	C₄×C₁₉	—	2	1	no	—
89	C₄×C₁₉	—	4	1	no	—
90	C₄×C₁₉	—	8	1	no	—
91	C₄×C₁₉	#327	1	2	yes	—
92	C₄×C₁₉	#329	1	1	yes	characteristic
93	C₄×C₁₉	#331	1	1	yes	characteristic
94	C₂×C₄×D₄	Dic₁₉	1	1	yes	characteristic
95	C₂²×C₁₆	—	38	2	no	—
96	C₂×C₄⋊C₈	Dic₁₉	1	1	yes	characteristic
97	C₂×C₂²⋊C₈	—	2	2	no	—
98	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
99	C₂×C₁₉	—	8	2	no	—
100	C₂×C₁₉	#1270	1	1	yes	characteristic
101	C₂×C₁₉	#1314	1	1	yes	characteristic
102	C₂×C₁₉	#1315	1	1	yes	characteristic
103	C₂×C₁₉	D₄⋊C₁₆	1	2	yes	—
104	C₂×C₁₉	#1330	1	2	yes	—
105	C₂²×D₄	C₁₉⋊C₈	1	1	yes	characteristic
106	C₂³×C₄	—	2	1	no	—
107	C₂²×C₈	—	2	1	no	—
108	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
109	C₄×D₄	—	4	1	no	—
110	C₄×D₄	—	4	1	no	—
111	C₂×C₄⋊C₄	C₁₉⋊C₈	1	1	yes	characteristic
112	C₂×C₂²⋊C₄	—	2	1	no	—
113	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
114	C₂×C₁₆	—	38	4	no	—
115	C₂×C₁₆	—	76	2	no	—
116	C₂×C₁₆	—	76	2	no	—
117	C₄⋊C₈	—	4	1	no	—
118	C₂²⋊C₈	—	4	2	no	—
119	C₂²⋊C₈	—	4	2	no	—
120	C₄×C₈	—	4	1	no	—
121	C₁₉	#7102	1	1	yes	characteristic, complemented, Sylow
122	C₂⁴	—	2	1	no	—
123	C₂×D₄	—	2	1	no	—
124	C₂×D₄	—	4	1	no	—
125	C₂×D₄	—	4	1	no	—
126	C₂×D₄	C₁₉⋊C₁₆	1	2	yes	complemented
127	C₂²×C₄	—	2	1	no	—
128	C₂²×C₄	—	4	2	no	—
129	C₂²×C₄	—	4	1	no	—
130	C₂²×C₄	—	4	1	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	C₂×C₁₉⋊C₈	1	1	yes	characteristic
133	C₂²×C₄	#64520	1	1	yes	characteristic
134	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	4	1	no	—
139	C₂×C₈	#64525	1	2	yes	—
140	C₂×C₈	#64526	1	2	yes	—
141	C₄⋊C₄	—	4	1	no	—
142	C₂²⋊C₄	—	4	2	no	—
143	C₄²	—	2	1	no	—
144	C₄²	—	2	1	no	—
145	C₁₆	—	76	4	no	—
146	C₂³	—	2	2	no	—
147	C₂³	—	4	2	no	—
148	C₂³	—	4	1	no	—
149	C₂³	—	4	2	no	—
150	C₂³	—	4	1	no	—
151	C₂³	C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
152	D₄	—	4	2	no	—
153	D₄	—	4	2	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	8	1	no	—
163	C₂×C₄	C₂×C₁₉⋊C₁₆	1	1	yes	characteristic
164	C₂×C₄	#10597269	1	1	yes	characteristic
165	C₂×C₄	#10597289	1	2	yes	—
166	C₂×C₄	#10597290	1	1	yes	characteristic
167	C₂×C₄	#10597291	1	1	yes	characteristic
168	C₂×C₄	#10597294	1	2	yes	—
169	C₈	—	2	2	no	—
170	C₈	—	2	2	no	—
171	C₈	—	8	1	no	—
172	C₂²	—	4	2	no	—
173	C₂²	—	4	4	no	—
174	C₂²	—	8	1	no	—
175	C₂²	—	8	2	no	—
176	C₂²	—	8	1	no	—
177	C₂²	#11926268	1	1	yes	characteristic
178	C₂²	D₄⋊(C₁₉⋊C₈)	1	2	yes	—
179	C₂²	C₂²⋊(C₁₉⋊C₁₆)	1	1	yes	characteristic
180	C₂²	#11926326	1	1	yes	characteristic
181	C₂²	#11926328	1	2	yes	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	4	1	no	—
185	C₄	—	8	1	no	—
186	C₄	#11926278	1	2	yes	—
187	C₄	#11926284	1	1	yes	characteristic
188	C₄	#11926288	1	1	yes	characteristic
189	C₂	—	8	2	no	—
190	C₂	#14561456	1	1	yes	characteristic
191	C₂	#14561559	1	1	yes	characteristic
192	C₂	#14561561	1	1	yes	characteristic
193	C₂	D₄⋊(C₁₉⋊C₁₆)	1	2	yes	—
194	C₂	#14561593	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14561713

C₂

yes

maximal, complemented

C₁₉×#1954

C₂

yes

maximal, characteristic

C₂²×C₁₉⋊C₁₆

—

C₂×C₄×D₄×C₁₉

C₄

yes

characteristic

C₂×C₁₉×C₄⋊C₈

C₄

yes

characteristic

C₂×C₁₉×C₂²⋊C₈

—

C₂×C₄×C₈×C₁₉

C₂²

yes

characteristic

C₂²×D₄×C₁₉

C₈

yes

characteristic

C₂³×C₄×C₁₉

—

C₂²×C₈×C₁₉

—

C₂²×C₈×C₁₉

D₄

yes

—

C₄×D₄×C₁₉

—

C₄×D₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

C₈

yes

characteristic

C₂×C₂²⋊C₄×C₁₉

—

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₁₉×C₄⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₄×C₈×C₁₉

—

C₂×C₁₉⋊C₁₆

—

C₂×C₁₉⋊C₁₆

—

C₂×C₁₉⋊C₁₆

—

C₂⁴×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

C₁₆

yes

complemented

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₉

M₁₆

yes

characteristic

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

D₈

yes

—

C₂×C₈×C₁₉

SD₁₆

yes

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₁₉⋊C₁₆

—

#7102

—

maximal, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

C₂²⋊C₈

yes

characteristic

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

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

—

C₂×C₄×C₁₉

#104

yes

—

C₂×C₄×C₁₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₁₉

M₃₂

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

#1954

D₁₉

yes

characteristic

#1374

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#323

yes

characteristic

C₂²×C₁₉

D₄⋊C₈

yes

—

C₂²×C₁₉

C₂²⋊C₁₆

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#327

yes

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₂×C₄×D₄

Dic₁₉

yes

characteristic

C₂²×C₁₆

—

C₂×C₄⋊C₈

Dic₁₉

yes

characteristic

C₂×C₂²⋊C₈

—

C₂×C₄×C₈

C₂×D₁₉

yes

characteristic

C₂×C₁₉

—

100

C₂×C₁₉

#1270

yes

characteristic

101

C₂×C₁₉

#1314

yes

characteristic

102

C₂×C₁₉

#1315

yes

characteristic

103

C₂×C₁₉

D₄⋊C₁₆

yes

—

104

C₂×C₁₉

#1330

yes

—

105

C₂²×D₄

C₁₉⋊C₈

yes

characteristic

106

C₂³×C₄

—

107

C₂²×C₈

—

108

C₂²×C₈

C₂²⋊D₁₉

yes

—

109

C₄×D₄

—

110

C₄×D₄

—

111

C₂×C₄⋊C₄

C₁₉⋊C₈

yes

characteristic

112

C₂×C₂²⋊C₄

—

113

C₂×C₄²

C₂×Dic₁₉

yes

characteristic

114

C₂×C₁₆

—

115

C₂×C₁₆

—

116

C₂×C₁₆

—

117

C₄⋊C₈

—

118

C₂²⋊C₈

—

119

C₂²⋊C₈

—

120

C₄×C₈

—

121

C₁₉

#7102

yes

characteristic, complemented, Sylow

122

C₂⁴

—

123

C₂×D₄

—

124

C₂×D₄

—

125

C₂×D₄

—

126

C₂×D₄

C₁₉⋊C₁₆

yes

complemented

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

C₂×C₁₉⋊C₈

yes

characteristic

133

C₂²×C₄

#64520

yes

characteristic

134

C₂²×C₄

C₂²⋊Dic₁₉

yes

characteristic

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

#64525

yes

—

140

C₂×C₈

#64526

yes

—

141

C₄⋊C₄

—

142

C₂²⋊C₄

—

143

C₄²

—

144

C₄²

—

145

C₁₆

—

146

C₂³

—

147

C₂³

—

148

C₂³

—

149

C₂³

—

150

C₂³

—

151

C₂³

C₂²⋊(C₁₉⋊C₈)

yes

characteristic

152

D₄

—

153

D₄

—

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

C₂×C₁₉⋊C₁₆

yes

characteristic

164

C₂×C₄

#10597269

yes

characteristic

165

C₂×C₄

#10597289

yes

—

166

C₂×C₄

#10597290

yes

characteristic

167

C₂×C₄

#10597291

yes

characteristic

168

C₂×C₄

#10597294

yes

—

169

C₈

—

170

C₈

—

171

C₈

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

—

176

C₂²

—

177

C₂²

#11926268

yes

characteristic

178

C₂²

D₄⋊(C₁₉⋊C₈)

yes

—

179

C₂²

C₂²⋊(C₁₉⋊C₁₆)

yes

characteristic

180

C₂²

#11926326

yes

characteristic

181

C₂²

#11926328

yes

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

#11926278

yes

—

187

C₄

#11926284

yes

characteristic

188

C₄

#11926288

yes

characteristic

189

C₂

—

190

C₂

#14561456

yes

characteristic

191

C₂

#14561559

yes

characteristic

192

C₂

#14561561

yes

characteristic

193

C₂

D₄⋊(C₁₉⋊C₁₆)

yes

—

194

C₂

#14561593

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₁₉
commutator	C₂×C₄×C₁₉	C₂×C₁₆
Frattini	C₂×C₄×C₈	C₂×D₁₉
Fitting	C₁₉×#1954	C₂
socle	C₂³×C₁₉	C₂²⋊C₈
prime radical	C₂²×D₄×C₁₉	C₈
cosocle	C₂×C₄×C₈×C₁₉	C₂²
hypercenter	#1954	D₁₉
hypocenter	C₁₉	#7102
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1954

subgroup

G/H

center

C₂²×C₄

C₂²⋊Dic₁₉

commutator

C₂×C₄×C₁₉

C₂×C₁₆

Frattini

C₂×C₄×C₈

C₂×D₁₉

Fitting

C₁₉×#1954

C₂

socle

C₂³×C₁₉

C₂²⋊C₈

prime radical

C₂²×D₄×C₁₉

C₈

cosocle

C₂×C₄×C₈×C₁₉

C₂²

D₁₉

C₁₉

G ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1954

N	H ≅ G/N	type
1. #14561713	C₂	semidirect
28. C₂×D₄×C₁₉	C₁₆	semidirect
121. C₁₉	#7102	semidirect
126. C₂×D₄	C₁₉⋊C₁₆	semidirect

H ≅ G/N

type

1. #14561713

C₂

semidirect

28. C₂×D₄×C₁₉

C₁₆

semidirect

121. C₁₉

#7102

semidirect

126. C₂×D₄

C₁₉⋊C₁₆

semidirect

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

subgroup	index
#14561713	2
C₁₉×#1954	2
#7102	19

subgroup

index

#14561713

C₁₉×#1954

#7102

Subgroups — #18069283

Conjugacy classes of subgroups

Complements

Maximal subgroups