Group #18069224 — subgroups

Subgroups — #18069224

Order 4864.

240 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 47 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	#14561575	C₂	1	1	yes	maximal, characteristic, complemented
2	#14561505	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#2054	C₂	1	1	yes	maximal, characteristic
4	#11926601	—	2	1	no	—
5	C₂²⋊(C₁₉⋊C₁₆)	—	2	1	no	—
6	C₂²×D₈×C₁₉	C₄	1	1	yes	characteristic
7	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₁₉×#102	—	2	1	no	—
9	C₁₉×#390	—	2	1	no	—
10	C₁₉×#340	C₄	1	1	yes	characteristic
11	C₂²×D₄×C₁₉	—	2	1	no	—
12	C₂×D₈×C₁₉	—	4	1	no	—
13	C₂×D₈×C₁₉	—	4	2	no	—
14	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
17	C₂×C₂²⋊C₄×C₁₉	—	4	1	no	—
18	C₂×C₂²⋊C₄×C₁₉	—	4	1	no	—
19	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
20	C₁₉×C₄⋊C₈	—	4	1	no	—
21	C₁₉×#102	—	4	2	no	—
22	C₂²⋊(C₁₉⋊C₈)	—	4	1	no	—
23	C₂×C₁₉⋊C₁₆	—	4	1	no	—
24	C₄⋊(C₁₉⋊C₈)	—	4	1	no	—
25	#10597260	—	2	2	no	—
26	C₂⁴×C₁₉	—	4	1	no	—
27	C₂×D₄×C₁₉	—	2	2	no	—
28	C₂×D₄×C₁₉	—	4	1	no	—
29	C₂×D₄×C₁₉	—	4	1	no	—
30	C₂×D₄×C₁₉	—	4	1	no	—
31	C₂×D₄×C₁₉	—	4	1	no	—
32	C₂×D₄×C₁₉	—	4	1	no	—
33	C₂²×C₄×C₁₉	—	4	1	no	—
34	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
36	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
37	D₈×C₁₉	—	4	2	no	—
38	D₈×C₁₉	—	4	2	no	—
39	C₂×C₈×C₁₉	—	2	1	no	—
40	C₂×C₈×C₁₉	—	2	1	no	—
41	C₂×C₈×C₁₉	—	4	1	no	—
42	C₂×C₈×C₁₉	—	4	1	no	—
43	C₄⋊C₄×C₁₉	—	4	1	no	—
44	C₄⋊C₄×C₁₉	—	4	1	no	—
45	C₂²⋊C₄×C₁₉	—	8	2	no	—
46	C₂²⋊C₄×C₁₉	—	8	2	no	—
47	C₄²×C₁₉	—	2	1	no	—
48	C₄²×C₁₉	—	2	1	no	—
49	C₂×C₁₉⋊C₈	—	4	2	no	—
50	C₁₉⋊C₁₆	—	8	1	no	—
51	#7080	—	19	1	no	maximal, Sylow
52	C₂³×C₁₉	—	4	2	no	—
53	C₂³×C₁₉	—	8	1	no	—
54	C₂³×C₁₉	—	8	1	no	—
55	C₂³×C₁₉	—	8	1	no	—
56	C₂³×C₁₉	—	8	1	no	—
57	C₂³×C₁₉	—	8	1	no	—
58	C₂³×C₁₉	—	8	1	no	—
59	C₂³×C₁₉	#102	1	1	yes	characteristic
60	D₄×C₁₉	—	4	2	no	—
61	D₄×C₁₉	—	4	2	no	—
62	D₄×C₁₉	—	8	1	no	—
63	D₄×C₁₉	—	8	1	no	—
64	C₂×C₄×C₁₉	—	2	1	no	—
65	C₂×C₄×C₁₉	—	2	1	no	—
66	C₂×C₄×C₁₉	—	2	1	no	—
67	C₂×C₄×C₁₉	—	2	1	no	—
68	C₂×C₄×C₁₉	—	2	1	no	—
69	C₂×C₄×C₁₉	—	2	1	no	—
70	C₂×C₄×C₁₉	—	4	1	no	—
71	C₂×C₄×C₁₉	—	8	1	no	—
72	C₂×C₄×C₁₉	—	8	1	no	—
73	C₂×C₄×C₁₉	—	8	1	no	—
74	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
75	C₂×C₄×C₁₉	#104	1	1	yes	characteristic
76	C₈×C₁₉	—	4	1	no	—
77	C₈×C₁₉	—	4	1	no	—
78	C₁₉⋊C₈	—	8	2	no	—
79	#2054	D₁₉	1	1	yes	characteristic
80	#1322	—	19	1	no	—
81	#1290	—	19	1	no	—
82	C₂²×C₁₉	—	2	1	no	—
83	C₂²×C₁₉	—	2	1	no	—
84	C₂²×C₁₉	—	8	2	no	—
85	C₂²×C₁₉	—	8	2	no	—
86	C₂²×C₁₉	—	8	2	no	—
87	C₂²×C₁₉	—	16	1	no	—
88	C₂²×C₁₉	—	16	1	no	—
89	C₂²×C₁₉	—	16	1	no	—
90	C₂²×C₁₉	—	16	1	no	—
91	C₂²×C₁₉	#327	1	1	yes	characteristic
92	C₂²×C₁₉	#359	1	1	yes	characteristic
93	C₂²×C₁₉	#361	1	1	yes	characteristic
94	C₄×C₁₉	—	2	1	no	—
95	C₄×C₁₉	—	4	1	no	—
96	C₄×C₁₉	—	4	1	no	—
97	C₄×C₁₉	—	16	1	no	—
98	C₄×C₁₉	AΣL(1,𝔽₁₆)	1	1	yes	characteristic
99	C₄×C₁₉	#355	1	1	yes	characteristic
100	C₂²×D₈	Dic₁₉	1	1	yes	characteristic
101	#432	—	38	1	no	—
102	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
103	C₂×#102	—	2	1	no	—
104	#390	—	2	1	no	—
105	C₂²⋊C₁₆	—	38	1	no	—
106	#340	Dic₁₉	1	1	yes	characteristic
107	C₂×C₁₉	—	2	1	no	—
108	C₂×C₁₉	—	2	1	no	—
109	C₂×C₁₉	—	16	2	no	—
110	C₂×C₁₉	#1339	1	1	yes	characteristic
111	C₂×C₁₉	#1345	1	1	yes	characteristic
112	C₂×C₁₉	#1349	1	1	yes	characteristic
113	C₂²×D₄	—	2	1	no	—
114	C₂×D₈	—	4	1	no	—
115	C₂×D₈	—	4	2	no	—
116	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
117	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
118	C₂×C₄⋊C₄	—	2	1	no	—
119	C₂×C₂²⋊C₄	—	4	1	no	—
120	C₂×C₂²⋊C₄	—	4	1	no	—
121	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
122	C₂×C₁₆	—	76	1	no	—
123	C₄⋊C₈	—	4	1	no	—
124	C₄⋊C₈	—	76	1	no	—
125	#102	—	4	2	no	—
126	C₂²⋊C₈	—	76	1	no	—
127	#97	—	38	2	no	—
128	C₁₉	#7080	1	1	yes	characteristic, complemented, Sylow
129	C₂⁴	—	4	1	no	—
130	C₂×D₄	—	2	2	no	—
131	C₂×D₄	—	4	1	no	—
132	C₂×D₄	—	4	1	no	—
133	C₂×D₄	—	4	1	no	—
134	C₂×D₄	—	4	1	no	—
135	C₂×D₄	—	4	1	no	—
136	C₂²×C₄	—	4	1	no	—
137	C₂²×C₄	#64525	1	1	yes	characteristic
138	C₂²×C₄	#64526	1	1	yes	characteristic
139	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
140	D₈	—	4	2	no	—
141	D₈	—	4	2	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	4	1	no	—
145	C₂×C₈	—	4	1	no	—
146	C₂×C₈	—	76	2	no	—
147	C₄⋊C₄	—	4	1	no	—
148	C₄⋊C₄	—	4	1	no	—
149	C₂²⋊C₄	—	8	2	no	—
150	C₂²⋊C₄	—	8	2	no	—
151	C₄²	—	2	1	no	—
152	C₄²	—	2	1	no	—
153	C₁₆	—	152	1	no	—
154	C₂³	—	4	2	no	—
155	C₂³	—	8	1	no	—
156	C₂³	—	8	1	no	—
157	C₂³	—	8	1	no	—
158	C₂³	—	8	1	no	—
159	C₂³	—	8	1	no	—
160	C₂³	—	8	1	no	—
161	C₂³	#10597289	1	1	yes	characteristic
162	D₄	—	4	2	no	—
163	D₄	—	4	2	no	—
164	D₄	—	8	1	no	—
165	D₄	—	8	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	8	1	no	—
174	C₂×C₄	—	8	1	no	—
175	C₂×C₄	—	8	1	no	—
176	C₂×C₄	#10597291	1	1	yes	characteristic
177	C₂×C₄	#10597294	1	1	yes	characteristic
178	C₈	—	4	1	no	—
179	C₈	—	4	1	no	—
180	C₈	—	152	2	no	—
181	C₂²	—	2	1	no	—
182	C₂²	—	2	1	no	—
183	C₂²	—	8	2	no	—
184	C₂²	—	8	2	no	—
185	C₂²	—	8	2	no	—
186	C₂²	—	16	1	no	—
187	C₂²	—	16	1	no	—
188	C₂²	—	16	1	no	—
189	C₂²	—	16	1	no	—
190	C₂²	#11926278	1	1	yes	characteristic
191	C₂²	#11926351	1	1	yes	characteristic
192	C₂²	#11926357	1	1	yes	characteristic
193	C₄	—	2	1	no	—
194	C₄	—	4	1	no	—
195	C₄	—	4	1	no	—
196	C₄	—	16	1	no	—
197	C₄	#11926331	1	1	yes	characteristic
198	C₄	#11926341	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	—	16	2	no	—
202	C₂	#14561620	1	1	yes	characteristic
203	C₂	#14561638	1	1	yes	characteristic
204	C₂	#14561650	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14561575

C₂

yes

maximal, characteristic, complemented

#14561505

C₂

yes

maximal, characteristic, complemented

C₁₉×#2054

C₂

yes

maximal, characteristic

#11926601

—

C₂²⋊(C₁₉⋊C₁₆)

—

C₂²×D₈×C₁₉

C₄

yes

characteristic

C₂×C₁₉×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₁₉×#102

—

C₁₉×#390

—

C₁₉×#340

C₄

yes

characteristic

C₂²×D₄×C₁₉

—

C₂×D₈×C₁₉

—

C₂×D₈×C₁₉

—

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂×C₄⋊C₄×C₁₉

—

C₂×C₂²⋊C₄×C₁₉

—

C₂×C₂²⋊C₄×C₁₉

—

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₁₉×C₄⋊C₈

—

C₁₉×#102

—

C₂²⋊(C₁₉⋊C₈)

—

C₂×C₁₉⋊C₁₆

—

C₄⋊(C₁₉⋊C₈)

—

#10597260

—

C₂⁴×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

D₈

yes

characteristic

C₂²×C₄×C₁₉

SD₁₆

yes

characteristic

D₈×C₁₉

—

D₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₂×C₁₉⋊C₈

—

C₁₉⋊C₁₆

—

#7080

—

maximal, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

#102

yes

characteristic

D₄×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

yes

C₂×C₄×C₁₉

yes

C₈×C₁₉

—

C₈×C₁₉

—

C₁₉⋊C₈

—

#2054

D₁₉

yes

characteristic

#1322

—

#1290

—

C₂²×C₁₉

—

C₂²×C₁₉

—

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

AΣL(1,𝔽₁₆)

yes

characteristic

C₄×C₁₉

#355

yes

characteristic

100

C₂²×D₈

Dic₁₉

yes

characteristic

101

#432

—

102

C₂×C₄⋊C₈

C₂×D₁₉

yes

characteristic

103

C₂×#102

—

104

#390

—

105

C₂²⋊C₁₆

—

106

#340

Dic₁₉

yes

characteristic

107

C₂×C₁₉

—

108

C₂×C₁₉

—

109

C₂×C₁₉

—

110

C₂×C₁₉

#1339

yes

characteristic

111

C₂×C₁₉

#1345

yes

characteristic

112

C₂×C₁₉

#1349

yes

characteristic

113

C₂²×D₄

—

114

C₂×D₈

—

115

C₂×D₈

—

116

C₂²×C₈

C₂×Dic₁₉

yes

characteristic

117

C₂²×C₈

C₂²⋊D₁₉

yes

characteristic

118

C₂×C₄⋊C₄

—

119

C₂×C₂²⋊C₄

—

120

C₂×C₂²⋊C₄

—

121

C₂×C₄²

C₂²⋊D₁₉

yes

characteristic

122

C₂×C₁₆

—

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

#102

—

126

C₂²⋊C₈

—

127

#97

—

128

C₁₉

#7080

yes

characteristic, complemented, Sylow

129

C₂⁴

—

130

C₂×D₄

—

131

C₂×D₄

—

132

C₂×D₄

—

133

C₂×D₄

—

134

C₂×D₄

—

135

C₂×D₄

—

136

C₂²×C₄

—

137

C₂²×C₄

#64525

yes

characteristic

138

C₂²×C₄

#64526

yes

characteristic

139

C₂²×C₄

C₂²⋊Dic₁₉

yes

characteristic

140

D₈

—

141

D₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₂²⋊C₄

—

150

C₂²⋊C₄

—

151

C₄²

—

152

C₄²

—

153

C₁₆

—

152

—

154

C₂³

—

155

C₂³

—

156

C₂³

—

157

C₂³

—

158

C₂³

—

159

C₂³

—

160

C₂³

—

161

C₂³

#10597289

yes

characteristic

162

D₄

—

163

D₄

—

164

D₄

—

165

D₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

#10597291

yes

characteristic

177

C₂×C₄

#10597294

yes

characteristic

178

C₈

—

179

C₈

—

180

C₈

—

152

—

181

C₂²

—

182

C₂²

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

190

C₂²

#11926278

yes

characteristic

191

C₂²

#11926351

yes

characteristic

192

C₂²

#11926357

yes

characteristic

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

#11926331

yes

characteristic

198

C₄

#11926341

yes

characteristic

199

C₂

—

200

C₂

—

201

C₂

—

202

C₂

#14561620

yes

characteristic

203

C₂

#14561638

yes

characteristic

204

C₂

#14561650

yes

characteristic

	subgroup	G/H
center	C₂²	#11926278
commutator	C₂²×C₈×C₁₉	C₂×C₄
Frattini	C₂×C₄⋊C₈	C₂×D₁₉
Fitting	C₁₉×#2054	C₂
socle	C₂²×C₁₉	#327
prime radical	C₂²×D₈×C₁₉	C₄
cosocle	C₂×C₁₉×C₄⋊C₈	C₂²
hypercenter	#2054	D₁₉
hypocenter	C₁₉	#7080
LCS	G ▷ C₂²×C₈×C₁₉ ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ #2054

subgroup

G/H

center

C₂²

#11926278

commutator

C₂²×C₈×C₁₉

C₂×C₄

Frattini

C₂×C₄⋊C₈

C₂×D₁₉

Fitting

C₁₉×#2054

C₂

socle

C₂²×C₁₉

#327

prime radical

C₂²×D₈×C₁₉

C₄

cosocle

C₂×C₁₉×C₄⋊C₈

C₂²

D₁₉

C₁₉

G ▷ C₂²×C₈×C₁₉ ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ #2054

N	H ≅ G/N	type
1. #14561575	C₂	semidirect
2. #14561505	C₂	semidirect
128. C₁₉	#7080	semidirect

H ≅ G/N

type

1. #14561575

C₂

semidirect

2. #14561505

C₂

semidirect

128. C₁₉

#7080

semidirect

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

subgroup	index
#14561575	2
#14561505	2
C₁₉×#2054	2
#7080	19

subgroup

index

#14561575

#14561505

C₁₉×#2054

#7080

Subgroups — #18069224

Conjugacy classes of subgroups

Complements

Maximal subgroups