Group #18069326 — subgroups

Subgroups — #18069326

Order 4864.

222 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 87 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	#14564368	C₂	1	1	yes	maximal, characteristic
2	#14561470	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1374	C₂	1	1	yes	maximal, characteristic
4	C₂×#10597275	C₄	1	1	yes	characteristic
5	C₂×#10597275	C₄	1	1	yes	characteristic
6	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
7	#11926442	—	2	1	no	—
8	#11926307	—	2	1	no	—
9	C₂²×C₁₆×C₁₉	—	2	1	no	—
10	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
13	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₉	—	2	1	no	—
15	C₂×C₄⋊Dic₁₉	—	2	1	no	—
16	C₂×Dic₁₉⋊C₄	—	4	1	no	—
17	C₂²×C₁₉⋊C₈	—	4	1	no	—
18	C₂×C₁₆×C₁₉	—	2	2	no	—
19	C₂×C₁₆×C₁₉	—	4	1	no	—
20	C₂×C₁₆×C₁₉	—	4	1	no	—
21	C₄×C₈×C₁₉	—	4	1	no	—
22	C₁₉⋊#95	—	4	1	no	—
23	#10597275	—	2	1	no	—
24	#10597275	—	4	1	no	—
25	#10597275	C₈	1	2	yes	complemented
26	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
27	C₄⋊(C₁₉⋊C₈)	—	4	1	no	—
28	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
30	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
31	C₂²×Dic₁₉	—	4	1	no	—
32	C₂²×Dic₁₉	—	4	1	no	—
33	C₂×C₈×C₁₉	—	2	1	no	—
34	C₂×C₈×C₁₉	—	2	1	no	—
35	C₂×C₈×C₁₉	—	2	1	no	—
36	C₂×C₈×C₁₉	—	2	1	no	—
37	C₂×C₈×C₁₉	C₂×C₈	1	1	yes	characteristic
38	C₂×C₈×C₁₉	M₁₆	1	1	yes	characteristic
39	C₂×C₈×C₁₉	SD₁₆	1	1	yes	characteristic
40	C₂×C₈×C₁₉	Q₁₆	1	1	yes	characteristic
41	C₄²×C₁₉	—	2	1	no	—
42	C₄²×C₁₉	—	2	1	no	—
43	C₄⋊Dic₁₉	—	2	2	no	—
44	C₄⋊Dic₁₉	—	4	1	no	—
45	C₄⋊Dic₁₉	—	4	1	no	—
46	C₄⋊Dic₁₉	—	4	1	no	—
47	Dic₁₉⋊C₄	—	8	1	no	—
48	Dic₁₉⋊C₄	—	8	1	no	—
49	C₂×C₁₉⋊C₈	—	4	2	no	—
50	C₂×C₁₉⋊C₈	—	8	1	no	—
51	C₂×C₁₉⋊C₈	—	8	1	no	—
52	C₁₆×C₁₉	—	4	2	no	—
53	#7120	—	19	1	no	maximal, Sylow
54	C₂³×C₁₉	#102	1	1	yes	characteristic
55	C₂×C₄×C₁₉	—	2	1	no	—
56	C₂×C₄×C₁₉	—	2	1	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	C₂²⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
62	C₂×C₄×C₁₉	#101	1	1	yes	characteristic
63	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
64	C₂×C₄×C₁₉	#104	1	1	yes	characteristic
65	C₂×C₄×C₁₉	#104	1	1	yes	characteristic
66	C₂×C₄×C₁₉	D₁₆	1	1	yes	characteristic
67	C₂×C₄×C₁₉	SD₃₂	1	1	yes	characteristic
68	C₂×Dic₁₉	—	4	2	no	—
69	C₂×Dic₁₉	—	8	1	no	—
70	C₂×Dic₁₉	—	8	1	no	—
71	C₂×Dic₁₉	—	8	1	no	—
72	C₂×Dic₁₉	—	8	1	no	—
73	C₂×Dic₁₉	—	8	1	no	—
74	C₈×C₁₉	—	2	1	no	—
75	C₈×C₁₉	—	2	1	no	—
76	C₈×C₁₉	—	2	1	no	—
77	C₈×C₁₉	—	2	1	no	—
78	C₁₉⋊C₈	—	8	2	no	—
79	#1942	—	19	1	no	—
80	#1374	D₁₉	1	1	yes	characteristic
81	#1277	—	19	1	no	—
82	C₂²×C₁₉	D₄⋊C₈	1	1	yes	characteristic
83	C₂²×C₁₉	#327	1	1	yes	characteristic
84	C₂²×C₁₉	#332	1	1	yes	characteristic
85	C₂²×C₁₉	#357	1	1	yes	characteristic
86	C₂²×C₁₉	#359	1	1	yes	characteristic
87	C₂²×C₁₉	#360	1	1	yes	characteristic
88	C₂²×C₁₉	#362	1	1	yes	characteristic
89	C₄×C₁₉	—	2	1	no	—
90	C₄×C₁₉	—	2	1	no	—
91	C₄×C₁₉	—	4	1	no	—
92	C₄×C₁₉	#324	1	1	yes	characteristic
93	C₄×C₁₉	Q₈⋊C₈	1	1	yes	characteristic
94	C₄×C₁₉	#328	1	1	yes	characteristic
95	C₄×C₁₉	#330	1	1	yes	characteristic
96	Dic₁₉	—	8	2	no	—
97	Dic₁₉	—	16	1	no	—
98	C₂²×C₁₆	—	2	1	no	—
99	C₂×#107	—	19	1	no	—
100	C₂×#107	—	19	1	no	—
101	C₂×C₄⋊C₈	—	38	1	no	—
102	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
103	#389	—	38	1	no	—
104	#340	—	38	1	no	—
105	C₂×C₁₉	#1271	1	1	yes	characteristic
106	C₂×C₁₉	#1331	1	1	yes	characteristic
107	C₂×C₁₉	#1334	1	1	yes	characteristic
108	C₂×C₁₉	#1347	1	1	yes	characteristic
109	C₂×C₁₉	#1350	1	1	yes	characteristic
110	C₂×C₁₉	#1363	1	1	yes	characteristic
111	C₂×C₁₉	#1364	1	1	yes	characteristic
112	C₂²×C₈	—	76	1	no	—
113	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
114	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
115	C₂×C₄⋊C₄	—	38	1	no	—
116	C₂×C₄⋊C₄	—	38	1	no	—
117	C₂×C₄⋊C₄	—	76	1	no	—
118	C₂×C₄²	D₇₆	1	1	yes	characteristic
119	C₂×C₁₆	—	2	2	no	—
120	C₂×C₁₆	—	4	1	no	—
121	C₂×C₁₆	—	4	1	no	—
122	#107	—	19	2	no	—
123	#107	—	38	1	no	—
124	#107	—	76	1	no	—
125	C₄⋊C₈	—	38	2	no	—
126	C₄⋊C₈	—	76	1	no	—
127	C₄×C₈	—	4	1	no	—
128	#95	—	76	1	no	—
129	C₁₉	#7120	1	1	yes	characteristic, complemented, Sylow
130	C₂²×C₄	—	76	1	no	—
131	C₂²×C₄	—	76	1	no	—
132	C₂²×C₄	#64516	1	1	yes	characteristic
133	C₂²×C₄	D₁₅₂	1	1	yes	characteristic
134	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	76	2	no	—
140	C₂×C₈	—	152	1	no	—
141	C₂×C₈	—	152	1	no	—
142	C₂×C₈	C₈×D₁₉	1	1	yes	characteristic
143	C₂×C₈	#64515	1	1	yes	characteristic
144	C₂×C₈	#64526	1	1	yes	characteristic
145	C₂×C₈	#64528	1	1	yes	characteristic
146	C₄⋊C₄	—	38	2	no	—
147	C₄⋊C₄	—	76	1	no	—
148	C₄⋊C₄	—	76	1	no	—
149	C₄⋊C₄	—	76	1	no	—
150	C₄⋊C₄	—	152	1	no	—
151	C₄⋊C₄	—	152	1	no	—
152	C₄²	—	2	1	no	—
153	C₄²	—	2	1	no	—
154	C₁₆	—	4	2	no	—
155	C₂³	#10597278	1	1	yes	characteristic
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	76	2	no	—
162	C₂×C₄	—	152	1	no	—
163	C₂×C₄	—	152	1	no	—
164	C₂×C₄	—	152	1	no	—
165	C₂×C₄	—	152	1	no	—
166	C₂×C₄	—	152	1	no	—
167	C₂×C₄	D₃₀₄	1	1	yes	characteristic
168	C₂×C₄	#10597257	1	1	yes	characteristic
169	C₂×C₄	#10597262	1	1	yes	characteristic
170	C₂×C₄	#10597263	1	1	yes	characteristic
171	C₂×C₄	#10597267	1	1	yes	characteristic
172	C₂×C₄	(C₂×D₁₉)⋊C₈	1	1	yes	characteristic
173	C₂×C₄	#10597281	1	1	yes	characteristic
174	C₂×C₄	#10597282	1	1	yes	characteristic
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	2	1	no	—
178	C₈	—	2	1	no	—
179	C₈	—	152	2	no	—
180	C₂²	#11926273	1	1	yes	characteristic
181	C₂²	#11926279	1	1	yes	characteristic
182	C₂²	#11926289	1	1	yes	characteristic
183	C₂²	#11926346	1	1	yes	characteristic
184	C₂²	#11926352	1	1	yes	characteristic
185	C₂²	#11926355	1	1	yes	characteristic
186	C₂²	#11926361	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	2	1	no	—
189	C₄	—	4	1	no	—
190	C₄	—	304	1	no	—
191	C₄	—	152	2	no	—
192	C₄	#11926269	1	1	yes	characteristic
193	C₄	#11926274	1	1	yes	characteristic
194	C₄	#11926280	1	1	yes	characteristic
195	C₄	#11926285	1	1	yes	characteristic
196	C₂	#14561457	1	1	yes	characteristic
197	C₂	#14561597	1	1	yes	characteristic
198	C₂	#14561606	1	1	yes	characteristic
199	C₂	#14561645	1	1	yes	characteristic
200	C₂	#14561654	1	1	yes	characteristic
201	C₂	#14561686	1	1	yes	characteristic
202	C₂	#14561689	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14564368

C₂

yes

maximal, characteristic

#14561470

C₂

yes

maximal, characteristic

C₁₉×#1374

C₂

yes

maximal, characteristic

C₂×#10597275

C₄

yes

characteristic

C₂×#10597275

C₄

yes

characteristic

C₂×C₄⋊(C₁₉⋊C₈)

—

#11926442

—

#11926307

—

C₂²×C₁₆×C₁₉

—

C₂×C₄×C₈×C₁₉

C₂²

yes

characteristic

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₂×C₄⋊Dic₁₉

—

C₂×C₄⋊Dic₁₉

—

C₂×Dic₁₉⋊C₄

—

C₂²×C₁₉⋊C₈

—

C₂×C₁₆×C₁₉

—

C₂×C₁₆×C₁₉

—

C₂×C₁₆×C₁₉

—

C₄×C₈×C₁₉

—

C₁₉⋊#95

—

#10597275

—

#10597275

—

#10597275

C₈

yes

complemented

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

D₈

yes

characteristic

C₂²×C₄×C₁₉

SD₁₆

yes

characteristic

C₂²×Dic₁₉

—

C₂²×Dic₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₂×C₈

yes

characteristic

C₂×C₈×C₁₉

M₁₆

yes

characteristic

C₂×C₈×C₁₉

SD₁₆

yes

characteristic

C₂×C₈×C₁₉

Q₁₆

yes

characteristic

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

Dic₁₉⋊C₄

—

Dic₁₉⋊C₄

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₁₆×C₁₉

—

—

C₂³×C₁₉

yes

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂²⋊C₈

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

D₁₆

yes

characteristic

C₂×C₄×C₁₉

SD₃₂

yes

characteristic

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₁₉⋊C₈

—

#1942

—

#1374

D₁₉

yes

characteristic

#1277

—

C₂²×C₁₉

D₄⋊C₈

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#324

yes

characteristic

C₄×C₁₉

Q₈⋊C₈

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

Dic₁₉

—

Dic₁₉

—

C₂²×C₁₆

—

C₂×#107

—

100

C₂×#107

—

101

C₂×C₄⋊C₈

—

102

C₂×C₄×C₈

C₂×D₁₉

yes

characteristic

103

#389

—

104

#340

—

105

C₂×C₁₉

#1271

yes

characteristic

106

C₂×C₁₉

#1331

yes

characteristic

107

C₂×C₁₉

#1334

yes

characteristic

108

C₂×C₁₉

#1347

yes

characteristic

109

C₂×C₁₉

#1350

yes

characteristic

110

C₂×C₁₉

#1363

yes

characteristic

111

C₂×C₁₉

#1364

yes

characteristic

112

C₂²×C₈

—

113

C₂²×C₈

C₄×D₁₉

yes

characteristic

114

C₂²×C₈

C₂²⋊D₁₉

yes

characteristic

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄²

D₇₆

yes

characteristic

119

C₂×C₁₆

—

120

C₂×C₁₆

—

121

C₂×C₁₆

—

122

#107

—

123

#107

—

124

#107

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₄×C₈

—

128

#95

—

129

C₁₉

#7120

yes

characteristic, complemented, Sylow

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

#64516

yes

characteristic

133

C₂²×C₄

D₁₅₂

yes

characteristic

134

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

152

—

141

C₂×C₈

—

152

—

142

C₂×C₈

C₈×D₁₉

yes

characteristic

143

C₂×C₈

#64515

yes

characteristic

144

C₂×C₈

#64526

yes

characteristic

145

C₂×C₈

#64528

yes

characteristic

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

152

—

151

C₄⋊C₄

—

152

—

152

C₄²

—

153

C₄²

—

154

C₁₆

—

155

C₂³

#10597278

yes

characteristic

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

152

—

163

C₂×C₄

—

152

—

164

C₂×C₄

—

152

—

165

C₂×C₄

—

152

—

166

C₂×C₄

—

152

—

167

C₂×C₄

D₃₀₄

yes

characteristic

168

C₂×C₄

#10597257

yes

characteristic

169

C₂×C₄

#10597262

yes

characteristic

170

C₂×C₄

#10597263

yes

characteristic

171

C₂×C₄

#10597267

yes

characteristic

172

C₂×C₄

(C₂×D₁₉)⋊C₈

yes

characteristic

173

C₂×C₄

#10597281

yes

characteristic

174

C₂×C₄

#10597282

yes

characteristic

175

C₈

—

176

C₈

—

177

C₈

—

178

C₈

—

179

C₈

—

152

—

180

C₂²

#11926273

yes

characteristic

181

C₂²

#11926279

yes

characteristic

182

C₂²

#11926289

yes

characteristic

183

C₂²

#11926346

yes

characteristic

184

C₂²

#11926352

yes

characteristic

185

C₂²

#11926355

yes

characteristic

186

C₂²

#11926361

yes

characteristic

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

304

—

191

C₄

—

152

—

192

C₄

#11926269

yes

characteristic

193

C₄

#11926274

yes

characteristic

194

C₄

#11926280

yes

characteristic

195

C₄

#11926285

yes

characteristic

196

C₂

#14561457

yes

characteristic

197

C₂

#14561597

yes

characteristic

198

C₂

#14561606

yes

characteristic

199

C₂

#14561645

yes

characteristic

200

C₂

#14561654

yes

characteristic

201

C₂

#14561686

yes

characteristic

202

C₂

#14561689

yes

characteristic

	subgroup	G/H
center	C₂³	#10597278
commutator	C₂×C₈×C₁₉	C₂×C₈
Frattini	C₂×C₄×C₈	C₂×D₁₉
Fitting	C₁₉×#1374	C₂
socle, prime radical	C₂³×C₁₉	#102
cosocle	C₂×C₄×C₈×C₁₉	C₂²
hypercenter	#1374	D₁₉
hypocenter	C₁₉	#7120
LCS	G ▷ C₂×C₈×C₁₉ ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1374

subgroup

G/H

center

C₂³

#10597278

commutator

C₂×C₈×C₁₉

C₂×C₈

Frattini

C₂×C₄×C₈

C₂×D₁₉

Fitting

C₁₉×#1374

C₂

socle, prime radical

C₂³×C₁₉

#102

cosocle

C₂×C₄×C₈×C₁₉

C₂²

D₁₉

C₁₉

G ▷ C₂×C₈×C₁₉ ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1374

N	H ≅ G/N	type
25. #10597275	C₈	semidirect
129. C₁₉	#7120	semidirect

H ≅ G/N

type

25. #10597275

C₈

semidirect

129. C₁₉

#7120

semidirect

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

subgroup	index
#14564368	2
#14561470	2
C₁₉×#1374	2
#7120	19

subgroup

index

#14564368

#14561470

C₁₉×#1374

#7120

Subgroups — #18069326

Conjugacy classes of subgroups

Complements

Maximal subgroups