Group #18075529 — subgroups

Subgroups — #18075529

Order 4864.

388 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 101 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	#14572347	C₂	1	1	yes	maximal, characteristic
2	#14562396	C₂	1	2	yes	maximal, complemented
3	#14562270	C₂	1	1	yes	maximal, characteristic
4	#14561545	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1536	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597394	C₄	1	2	yes	—
7	#11926622	—	2	2	no	—
8	D₄×C₁₉⋊C₈	—	2	2	no	—
9	#11926601	C₂²	1	2	yes	—
10	#11926568	C₂²	1	1	yes	characteristic
11	#11926511	C₄	1	2	yes	—
12	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
13	#11926296	—	2	2	no	—
14	#11926294	—	2	2	no	—
15	#11926284	—	2	2	no	—
16	C₁₉×#532	C₂²	1	1	yes	characteristic
17	C₁₉×#432	C₂²	1	2	yes	—
18	C₁₉×#329	—	2	2	no	—
19	C₂×#41×C₁₉	C₂×C₄	1	2	yes	—
20	C₁₉×C₄⋊Q₈	Q₈	1	1	yes	characteristic
21	C₁₉×#127	Q₈	1	1	yes	characteristic
22	C₁₉×#124	D₄	1	2	yes	—
23	C₁₉×#121	—	2	2	no	—
24	C₄×D₄×C₁₉	—	2	2	no	—
25	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
26	#10597394	—	4	4	no	—
27	C₂×#64520	—	2	2	no	—
28	C₂×#64520	C₂×C₄	1	2	yes	—
29	C₂²×C₁₉⋊C₈	—	2	2	no	—
30	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
32	C₁₉×C₄⋊C₈	—	2	2	no	—
33	C₁₉×C₂²⋊C₈	—	2	2	no	—
34	C₁₉×#97	—	2	2	no	—
35	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
36	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
37	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
38	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
39	#10597260	—	2	2	no	—
40	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
41	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
42	#41×C₁₉	—	4	2	no	—
43	#41×C₁₉	—	4	2	no	—
44	C₂×Q₈×C₁₉	C₄⋊C₄	1	2	yes	—
45	C₂×D₄×C₁₉	—	2	2	no	—
46	C₂×D₄×C₁₉	—	2	2	no	—
47	C₂×D₄×C₁₉	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₁₉	—	2	2	no	—
49	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	#41	1	2	yes	—
51	C₂×C₈×C₁₉	—	4	2	no	—
52	C₄⋊C₄×C₁₉	—	2	2	no	—
53	C₂²⋊C₄×C₁₉	—	2	4	no	—
54	C₄²×C₁₉	C₂×D₄	1	2	yes	—
55	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
57	#64520	—	4	4	no	—
58	#64520	—	4	2	no	—
59	C₂×C₁₉⋊C₈	—	2	2	no	—
60	C₂×C₁₉⋊C₈	—	2	2	no	—
61	C₂×C₁₉⋊C₈	—	4	1	no	—
62	C₂×C₁₉⋊C₈	—	4	4	no	—
63	C₂×C₁₉⋊C₈	—	4	2	no	—
64	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	2	yes	—
65	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
67	#8630	—	19	1	no	maximal, Sylow
68	C₂³×C₁₉	—	2	2	no	—
69	C₂³×C₁₉	#117	1	1	yes	characteristic
70	Q₈×C₁₉	—	4	2	no	—
71	D₄×C₁₉	—	4	2	no	—
72	D₄×C₁₉	—	4	4	no	—
73	D₄×C₁₉	—	4	4	no	—
74	C₂×C₄×C₁₉	—	2	1	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	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₉	#121	1	2	yes	—
83	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
84	C₈×C₁₉	—	8	2	no	—
85	C₁₉⋊C₈	—	2	1	no	—
86	C₁₉⋊C₈	—	2	2	no	—
87	C₁₉⋊C₈	—	4	2	no	—
88	C₁₉⋊C₈	—	8	2	no	—
89	C₁₉⋊C₈	#100	1	2	yes	—
90	#2931	—	19	1	no	—
91	#1568	—	19	2	no	—
92	#1536	—	19	1	no	—
93	#1536	D₁₉	1	1	yes	characteristic
94	#1307	—	19	2	no	—
95	C₂²×C₁₉	—	2	1	no	—
96	C₂²×C₁₉	—	2	2	no	—
97	C₂²×C₁₉	—	4	2	no	—
98	C₂²×C₁₉	—	4	4	no	—
99	C₂²×C₁₉	#380	1	1	yes	characteristic
100	C₄×C₁₉	—	2	1	no	—
101	C₄×C₁₉	—	2	2	no	—
102	C₄×C₁₉	—	2	2	no	—
103	C₄×C₁₉	—	4	2	no	—
104	C₄×C₁₉	C₂×#100	1	1	yes	characteristic
105	C₄×C₁₉	#413	1	1	yes	characteristic
106	C₂×#131	—	19	2	no	—
107	#532	C₂×D₁₉	1	1	yes	characteristic
108	#436	—	38	2	no	—
109	C₈×D₄	—	38	2	no	—
110	#432	—	19	2	no	—
111	#432	C₂×D₁₉	1	2	yes	—
112	#423	—	19	1	no	—
113	#408	—	19	2	no	—
114	C₂×C₄×C₈	—	19	1	no	—
115	#335	—	38	2	no	—
116	#334	—	38	2	no	—
117	#329	—	2	2	no	—
118	#329	—	38	2	no	—
119	C₂×C₁₉	—	4	1	no	—
120	C₂×C₁₉	—	8	2	no	—
121	C₂×C₁₉	#1800	1	1	yes	characteristic
122	C₂×C₁₉	#1814	1	2	yes	—
123	C₂×#41	C₄×D₁₉	1	2	yes	—
124	#131	—	76	4	no	—
125	C₂×M₁₆	—	19	2	no	—
126	C₂×M₁₆	—	38	2	no	—
127	C₂²×C₈	—	19	1	no	—
128	C₂²×C₈	—	19	1	no	—
129	C₂²×C₈	—	38	2	no	—
130	C₄⋊Q₈	Dic₃₈	1	1	yes	characteristic
131	#127	Dic₃₈	1	1	yes	characteristic
132	#124	C₂²⋊D₁₉	1	2	yes	—
133	#121	—	2	2	no	—
134	C₄×D₄	—	2	2	no	—
135	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	—	38	2	no	—
138	C₄⋊C₈	—	38	2	no	—
139	C₂²⋊C₈	—	2	2	no	—
140	C₂²⋊C₈	—	38	4	no	—
141	C₂²⋊C₈	—	38	2	no	—
142	#97	—	2	2	no	—
143	#97	—	38	2	no	—
144	C₄×C₈	—	19	2	no	—
145	C₄×C₈	—	19	2	no	—
146	C₁₉	#8630	1	1	yes	characteristic, complemented, Sylow
147	#41	—	4	2	no	—
148	#41	—	4	2	no	—
149	C₂×Q₈	Dic₁₉⋊C₄	1	2	yes	—
150	C₂×D₄	—	2	2	no	—
151	C₂×D₄	—	2	2	no	—
152	C₂×D₄	Dic₁₉⋊C₄	1	2	yes	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
155	C₂²×C₄	#64543	1	2	yes	—
156	M₁₆	—	76	4	no	—
157	M₁₆	—	76	2	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	—	19	1	no	—
160	C₂×C₈	—	19	2	no	—
161	C₂×C₈	—	19	1	no	—
162	C₂×C₈	—	38	2	no	—
163	C₂×C₈	—	38	2	no	—
164	C₂×C₈	—	76	4	no	—
165	C₂×C₈	—	76	1	no	—
166	C₂×C₈	—	76	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₂²⋊C₄	—	2	4	no	—
169	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
170	C₄²	D₄×D₁₉	1	2	yes	—
171	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
172	C₂³	—	2	2	no	—
173	C₂³	#10597323	1	1	yes	characteristic
174	Q₈	—	4	2	no	—
175	D₄	—	4	2	no	—
176	D₄	—	4	4	no	—
177	D₄	—	4	4	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	#10597324	1	2	yes	—
185	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
186	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
187	C₂×C₄	#10597385	1	2	yes	—
188	C₈	—	8	2	no	—
189	C₈	—	19	2	no	—
190	C₈	—	38	1	no	—
191	C₈	—	38	2	no	—
192	C₈	—	76	2	no	—
193	C₈	—	152	2	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	2	no	—
196	C₂²	—	4	2	no	—
197	C₂²	—	4	4	no	—
198	C₂²	#11926402	1	1	yes	characteristic
199	C₄	—	2	1	no	—
200	C₄	—	2	2	no	—
201	C₄	—	2	2	no	—
202	C₄	—	4	2	no	—
203	C₄	#100×D₁₉	1	1	yes	characteristic
204	C₄	#11926525	1	1	yes	characteristic
205	C₂	—	4	1	no	—
206	C₂	—	8	2	no	—
207	C₂	#14563625	1	1	yes	characteristic
208	C₂	#14563687	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14572347

C₂

yes

maximal, characteristic

#14562396

C₂

yes

maximal, complemented

#14562270

C₂

yes

maximal, characteristic

#14561545

C₂

yes

maximal, complemented

C₁₉×#1536

C₂

yes

maximal, characteristic

C₂×#10597394

C₄

yes

—

#11926622

—

D₄×C₁₉⋊C₈

—

#11926601

C₂²

yes

—

#11926568

C₂²

yes

characteristic

#11926511

C₄

yes

—

C₂×C₄×C₁₉⋊C₈

C₂²

yes

—

—

—

C₁₉×#532

C₂²

yes

characteristic

C₁₉×#432

C₂²

yes

—

C₁₉×#329

—

C₂×#41×C₁₉

C₂×C₄

yes

—

C₁₉×C₄⋊Q₈

Q₈

yes

characteristic

C₁₉×#127

Q₈

yes

characteristic

C₁₉×#124

D₄

yes

—

C₁₉×#121

—

C₄×D₄×C₁₉

—

C₂×C₄²×C₁₉

C₂³

yes

—

—

C₂×C₄

yes

—

C₂²×C₁₉⋊C₈

—

C₂²×C₁₉⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₁₉⋊C₈

C₂×C₄

yes

characteristic

C₁₉×C₄⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₁₉×#97

—

C₂²⋊(C₁₉⋊C₈)

—

C₂²⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

#10597260

—

C₄×C₁₉⋊C₈

D₄

yes

—

C₄×C₁₉⋊C₈

D₄

yes

—

#41×C₁₉

—

#41×C₁₉

—

C₂×Q₈×C₁₉

C₄⋊C₄

yes

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

C₄⋊C₄

yes

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₉

#41

yes

—

C₂×C₈×C₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

C₂×D₄

yes

—

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄²×C₁₉

C₂×Q₈

yes

characteristic

#64520

—

#64520

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

C₂²⋊C₄

yes

—

C₂×C₁₉⋊C₈

C₂²⋊C₄

yes

characteristic

C₂×C₁₉⋊C₈

C₂²⋊C₄

yes

characteristic

#8630

—

maximal, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

#117

yes

characteristic

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×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉

#121

yes

—

C₂×C₄×C₁₉

C₂²⋊Q₈

yes

—

C₈×C₁₉

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

C₁₉⋊C₈

#100

yes

—

—

—

—

D₁₉

yes

characteristic

#1307

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#380

yes

characteristic

100

C₄×C₁₉

—

101

C₄×C₁₉

—

102

C₄×C₁₉

—

103

C₄×C₁₉

—

104

C₄×C₁₉

C₂×#100

yes

characteristic

105

C₄×C₁₉

#413

yes

characteristic

106

C₂×#131

—

107

#532

C₂×D₁₉

yes

characteristic

108

#436

—

109

C₈×D₄

—

110

#432

—

111

#432

C₂×D₁₉

yes

—

112

#423

—

113

#408

—

114

C₂×C₄×C₈

—

115

#335

—

116

#334

—

117

#329

—

118

#329

—

119

C₂×C₁₉

—

120

C₂×C₁₉

—

121

C₂×C₁₉

#1800

yes

characteristic

122

C₂×C₁₉

#1814

yes

—

123

C₂×#41

C₄×D₁₉

yes

—

124

#131

—

125

C₂×M₁₆

—

126

C₂×M₁₆

—

127

C₂²×C₈

—

128

C₂²×C₈

—

129

C₂²×C₈

—

130

C₄⋊Q₈

Dic₃₈

yes

characteristic

131

#127

Dic₃₈

yes

characteristic

132

#124

C₂²⋊D₁₉

yes

—

133

#121

—

134

C₄×D₄

—

135

C₂×C₄²

C₂²×D₁₉

yes

characteristic

136

C₄⋊C₈

—

137

C₄⋊C₈

—

138

C₄⋊C₈

—

139

C₂²⋊C₈

—

140

C₂²⋊C₈

—

141

C₂²⋊C₈

—

142

#97

—

143

#97

—

144

C₄×C₈

—

145

C₄×C₈

—

146

C₁₉

#8630

yes

characteristic, complemented, Sylow

147

#41

—

148

#41

—

149

C₂×Q₈

Dic₁₉⋊C₄

yes

—

150

C₂×D₄

—

151

C₂×D₄

—

152

C₂×D₄

Dic₁₉⋊C₄

yes

—

153

C₂²×C₄

—

154

C₂²×C₄

C₂×C₄×D₁₉

yes

characteristic

155

C₂²×C₄

#64543

yes

—

156

M₁₆

—

157

M₁₆

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₄⋊C₄

—

168

C₂²⋊C₄

—

169

C₄²

C₂×Dic₃₈

yes

characteristic

170

C₄²

D₄×D₁₉

yes

—

171

C₄²

C₂×C₂²⋊D₁₉

yes

characteristic

172

C₂³

—

173

C₂³

#10597323

yes

characteristic

174

Q₈

—

175

D₄

—

176

D₄

—

177

D₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

#10597324

yes

—

185

C₂×C₄

C₂²⋊C₄×D₁₉

yes

characteristic

186

C₂×C₄

C₂×Dic₁₉⋊C₄

yes

characteristic

187

C₂×C₄

#10597385

yes

—

188

C₈

—

189

C₈

—

190

C₈

—

191

C₈

—

192

C₈

—

193

C₈

—

152

—

194

C₂²

—

195

C₂²

—

196

C₂²

—

197

C₂²

—

198

C₂²

#11926402

yes

characteristic

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

#100×D₁₉

yes

characteristic

204

C₄

#11926525

yes

characteristic

205

C₂

—

206

C₂

—

207

C₂

#14563625

yes

characteristic

208

C₂

#14563687

yes

—

	subgroup	G/H
center	C₂²	#11926402
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1536	C₂
socle	C₂²×C₁₉	#380
prime radical	C₁₉×#532	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1536	D₁₉
hypocenter	C₁₉	#8630
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1536

subgroup

G/H

center

C₂²

#11926402

commutator

C₂²×C₄×C₁₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#1536

C₂

socle

C₂²×C₁₉

#380

prime radical

C₁₉×#532

C₂²

cosocle

C₂×C₄²×C₁₉

C₂³

D₁₉

C₁₉

G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1536

N	H ≅ G/N	type
2. #14562396	C₂	semidirect
4. #14561545	C₂	semidirect
146. C₁₉	#8630	semidirect

H ≅ G/N

type

2. #14562396

C₂

semidirect

4. #14561545

C₂

semidirect

146. C₁₉

#8630

semidirect

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

subgroup	index
#14572347	2
#14562396	2
#14562270	2
#14561545	2
C₁₉×#1536	2
#8630	19

subgroup

index

C₁₉×#1536

Subgroups — #18075529

Conjugacy classes of subgroups

Complements

Maximal subgroups