Group #18076925 — subgroups

Subgroups — #18076925

Order 4864.

396 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 119 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	#14566821	C₂	1	1	yes	maximal, characteristic
2	#14564605	C₂	1	1	yes	maximal, characteristic
3	#14563769	C₂	1	2	yes	maximal
4	#14561517	C₂	1	2	yes	maximal
5	C₂×C₁₉×#330	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597260	C₂²	1	1	yes	characteristic
7	#11926443	—	2	2	no	—
8	#11926430	—	2	2	no	—
9	#11926430	—	2	2	no	—
10	#11926414	C₂²	1	1	yes	characteristic
11	#11926410	C₂²	1	2	yes	—
12	#11926307	—	2	2	no	—
13	#11926307	—	2	2	no	—
14	#11926306	—	2	2	no	—
15	C₂×C₁₉×#125	C₂²	1	1	yes	characteristic
16	C₂×C₁₉×#97	C₂²	1	2	yes	—
17	C₁₉×#330	C₄	1	4	yes	complemented
18	C₂²×C₈×C₁₉	—	2	2	no	—
19	C₁₉×#125	—	2	2	no	—
20	C₁₉×#125	C₂×C₄	1	2	yes	—
21	C₁₉×#125	Q₈	1	2	yes	—
22	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
23	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
24	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
25	C₂×C₄⋊Dic₁₉	—	2	1	no	—
26	C₂×C₄⋊Dic₁₉	—	2	2	no	—
27	C₂×C₄⋊Dic₁₉	—	2	2	no	—
28	C₂×C₄⋊Dic₁₉	D₄	1	2	yes	—
29	C₂×C₄⋊Dic₁₉	D₄	1	2	yes	—
30	C₂×C₄×Dic₁₉	—	2	2	no	—
31	C₂²×C₁₉⋊C₈	—	2	1	no	—
32	C₁₉×#97	—	2	2	no	—
33	C₁₉×#97	C₂×C₄	1	4	yes	—
34	C₁₉⋊#95	—	2	2	no	—
35	C₁₉⋊#95	—	2	2	no	—
36	C₁₉⋊#95	—	2	2	no	—
37	C₁₉⋊#95	—	2	2	no	—
38	#10597260	—	2	2	no	—
39	C₂²×C₄×C₁₉	—	2	2	no	—
40	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
42	C₂²×Dic₁₉	—	2	2	no	—
43	C₂²×Dic₁₉	—	2	2	no	—
44	C₂²×Dic₁₉	—	4	1	no	—
45	C₂×C₈×C₁₉	—	2	4	no	—
46	C₂×C₈×C₁₉	—	4	2	no	—
47	C₂×C₈×C₁₉	—	4	2	no	—
48	C₄⋊C₄×C₁₉	—	2	4	no	—
49	C₄⋊C₄×C₁₉	—	2	4	no	—
50	C₄⋊C₄×C₁₉	—	2	2	no	—
51	C₄⋊C₄×C₁₉	C₄⋊C₄	1	4	yes	—
52	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
54	C₄²×C₁₉	#41	1	2	yes	—
55	C₄⋊Dic₁₉	—	4	2	no	—
56	C₄⋊Dic₁₉	—	4	2	no	—
57	C₄⋊Dic₁₉	—	4	2	no	—
58	C₄⋊Dic₁₉	—	4	4	no	—
59	C₄⋊Dic₁₉	—	8	2	no	—
60	C₄×Dic₁₉	—	8	2	no	—
61	C₂×C₁₉⋊C₈	—	2	2	no	—
62	C₂×C₁₉⋊C₈	—	4	1	no	—
63	C₂×C₁₉⋊C₈	—	4	1	no	—
64	#8915	—	19	1	no	maximal, Sylow
65	C₂³×C₁₉	#120	1	1	yes	characteristic
66	C₂×C₄×C₁₉	—	2	1	no	—
67	C₂×C₄×C₁₉	—	2	2	no	—
68	C₂×C₄×C₁₉	—	2	2	no	—
69	C₂×C₄×C₁₉	—	2	1	no	—
70	C₂×C₄×C₁₉	—	2	4	no	—
71	C₂×C₄×C₁₉	—	4	2	no	—
72	C₂×C₄×C₁₉	—	4	2	no	—
73	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
75	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
76	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
77	C₂×Dic₁₉	—	4	2	no	—
78	C₂×Dic₁₉	—	4	2	no	—
79	C₂×Dic₁₉	—	4	2	no	—
80	C₂×Dic₁₉	—	4	2	no	—
81	C₂×Dic₁₉	—	4	2	no	—
82	C₂×Dic₁₉	—	4	2	no	—
83	C₂×Dic₁₉	—	8	2	no	—
84	C₂×Dic₁₉	—	8	1	no	—
85	C₈×C₁₉	—	4	4	no	—
86	C₁₉⋊C₈	—	4	2	no	—
87	#2382	—	19	1	no	—
88	#1992	—	19	1	no	—
89	#1836	—	19	2	no	—
90	C₂×#330	D₁₉	1	1	yes	characteristic
91	#1295	—	19	2	no	—
92	C₂²×C₁₉	#385	1	1	yes	characteristic
93	C₂²×C₁₉	#454	1	2	yes	—
94	C₂²×C₁₉	#454	1	1	yes	characteristic
95	C₂²×C₁₉	#455	1	2	yes	—
96	C₂²×C₁₉	#456	1	1	yes	characteristic
97	C₄×C₁₉	—	2	1	no	—
98	C₄×C₁₉	—	2	1	no	—
99	C₄×C₁₉	—	2	2	no	—
100	C₄×C₁₉	—	2	2	no	—
101	C₄×C₁₉	—	4	4	no	—
102	Dic₁₉	—	8	2	no	—
103	Dic₁₉	—	8	2	no	—
104	Dic₁₉	—	16	1	no	—
105	C₂×#125	C₂×D₁₉	1	1	yes	characteristic
106	C₂×#97	—	19	1	no	—
107	C₂×#97	C₂×D₁₉	1	2	yes	—
108	#389	—	38	2	no	—
109	#387	—	38	2	no	—
110	#387	—	38	2	no	—
111	#384	—	19	1	no	—
112	#382	—	19	2	no	—
113	#340	—	38	2	no	—
114	#340	—	38	2	no	—
115	#340	—	38	2	no	—
116	#330	Dic₁₉	1	4	yes	complemented
117	C₂×C₁₉	#1658	1	1	yes	characteristic
118	C₂×C₁₉	#1662	1	1	yes	characteristic
119	C₂×C₁₉	#1664	1	2	yes	—
120	C₂×C₁₉	#1857	1	1	yes	characteristic
121	C₂×C₁₉	#1860	1	2	yes	—
122	C₂²×C₈	—	2	2	no	—
123	C₂²×C₈	—	38	1	no	—
124	#125	—	2	2	no	—
125	#125	Dic₃₈	1	2	yes	—
126	#125	C₂×Dic₁₉	1	2	yes	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	—	19	2	no	—
129	C₂×C₄⋊C₄	—	19	2	no	—
130	C₂×C₄⋊C₄	—	38	1	no	—
131	C₂×C₄⋊C₄	—	38	2	no	—
132	C₂×C₄⋊C₄	—	38	2	no	—
133	C₂×C₄⋊C₄	D₇₆	1	2	yes	—
134	C₂×C₄²	—	38	2	no	—
135	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
136	#97	—	2	2	no	—
137	#97	—	38	2	no	—
138	#97	C₂×Dic₁₉	1	4	yes	—
139	#95	—	38	2	no	—
140	#95	—	38	2	no	—
141	#95	—	38	2	no	—
142	#95	—	38	2	no	—
143	C₁₉	#8915	1	1	yes	characteristic, complemented, Sylow
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	—	38	2	no	—
146	C₂²×C₄	—	38	2	no	—
147	C₂²×C₄	—	76	1	no	—
148	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
149	C₂²×C₄	D₄×D₁₉	1	2	yes	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	38	2	no	—
154	C₂×C₈	—	76	1	no	—
155	C₂×C₈	—	76	1	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	76	2	no	—
160	C₄⋊C₄	—	76	2	no	—
161	C₄⋊C₄	—	76	4	no	—
162	C₄⋊C₄	—	76	2	no	—
163	C₄⋊C₄	—	152	2	no	—
164	C₄⋊C₄	C₄⋊Dic₁₉	1	4	yes	—
165	C₄²	—	152	2	no	—
166	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
167	C₄²	#64543	1	2	yes	—
168	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
169	C₂³	#10597328	1	1	yes	characteristic
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	76	2	no	—
178	C₂×C₄	—	76	2	no	—
179	C₂×C₄	—	76	2	no	—
180	C₂×C₄	—	76	2	no	—
181	C₂×C₄	—	76	2	no	—
182	C₂×C₄	—	76	2	no	—
183	C₂×C₄	—	152	2	no	—
184	C₂×C₄	—	152	1	no	—
185	C₂×C₄	#10597324	1	2	yes	—
186	C₂×C₄	#10597332	1	1	yes	characteristic
187	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
188	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
189	C₈	—	4	4	no	—
190	C₈	—	76	2	no	—
191	C₂²	#11926419	1	1	yes	characteristic
192	C₂²	#11926718	1	2	yes	—
193	C₂²	#11926719	1	1	yes	characteristic
194	C₂²	#11926723	1	2	yes	—
195	C₂²	#11926729	1	1	yes	characteristic
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	2	no	—
199	C₄	—	2	2	no	—
200	C₄	—	4	4	no	—
201	C₄	—	304	1	no	—
202	C₄	—	152	2	no	—
203	C₄	—	152	2	no	—
204	C₂	#14562938	1	1	yes	characteristic
205	C₂	#14562959	1	1	yes	characteristic
206	C₂	#14562971	1	2	yes	—
207	C₂	#14563852	1	1	yes	characteristic
208	C₂	#14563870	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14566821

C₂

yes

maximal, characteristic

#14564605

C₂

yes

maximal, characteristic

#14563769

C₂

yes

maximal

#14561517

C₂

yes

maximal

C₂×C₁₉×#330

C₂

yes

maximal, characteristic

C₂×#10597260

C₂²

yes

—

—

—

C₂²

yes

characteristic

#11926410

C₂²

yes

—

#11926307

—

#11926307

—

#11926306

—

C₂×C₁₉×#125

C₂²

yes

characteristic

C₂×C₁₉×#97

C₂²

yes

—

C₁₉×#330

C₄

yes

complemented

C₂²×C₈×C₁₉

—

C₁₉×#125

—

C₁₉×#125

C₂×C₄

yes

—

C₁₉×#125

Q₈

yes

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

D₄

yes

—

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₉

—

C₂×C₄⋊Dic₁₉

—

C₂×C₄⋊Dic₁₉

—

C₂×C₄⋊Dic₁₉

D₄

yes

—

C₂×C₄⋊Dic₁₉

D₄

yes

—

C₂×C₄×Dic₁₉

—

C₂²×C₁₉⋊C₈

—

C₁₉×#97

—

C₁₉×#97

C₂×C₄

yes

—

C₁₉⋊#95

—

C₁₉⋊#95

—

C₁₉⋊#95

—

C₁₉⋊#95

—

#10597260

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×D₄

yes

—

C₂²×Dic₁₉

—

C₂²×Dic₁₉

—

C₂²×Dic₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

C₄⋊C₄

yes

—

C₄²×C₁₉

C₂²×C₄

yes

characteristic

C₄²×C₁₉

C₂×Q₈

yes

characteristic

C₄²×C₁₉

#41

yes

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄×Dic₁₉

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

—

C₂³×C₁₉

yes

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⋊C₄

yes

characteristic

C₂×C₄×C₁₉

C₄×D₄

yes

—

C₂×C₄×C₁₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₉

#123

yes

characteristic

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₈×C₁₉

—

C₁₉⋊C₈

—

—

—

—

D₁₉

yes

characteristic

#1295

—

C₂²×C₁₉

#385

yes

characteristic

C₂²×C₁₉

#454

yes

—

C₂²×C₁₉

#454

yes

characteristic

C₂²×C₁₉

#455

yes

—

C₂²×C₁₉

#456

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

100

C₄×C₁₉

—

101

C₄×C₁₉

—

102

Dic₁₉

—

103

Dic₁₉

—

104

Dic₁₉

—

105

C₂×#125

C₂×D₁₉

yes

characteristic

106

C₂×#97

—

107

C₂×#97

C₂×D₁₉

yes

—

108

#389

—

109

#387

—

110

#387

—

111

#384

—

112

#382

—

113

#340

—

114

#340

—

115

#340

—

116

#330

Dic₁₉

yes

complemented

117

C₂×C₁₉

#1658

yes

characteristic

118

C₂×C₁₉

#1662

yes

characteristic

119

C₂×C₁₉

#1664

yes

—

120

C₂×C₁₉

#1857

yes

characteristic

121

C₂×C₁₉

#1860

yes

—

122

C₂²×C₈

—

123

C₂²×C₈

—

124

#125

—

125

#125

Dic₃₈

yes

—

126

#125

C₂×Dic₁₉

yes

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

D₇₆

yes

—

134

C₂×C₄²

—

135

C₂×C₄²

C₂²×D₁₉

yes

characteristic

136

#97

—

137

#97

—

138

#97

C₂×Dic₁₉

yes

—

139

#95

—

140

#95

—

141

#95

—

142

#95

—

143

C₁₉

#8915

yes

characteristic, complemented, Sylow

144

C₂²×C₄

—

145

C₂²×C₄

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

C₂×D₇₆

yes

characteristic

149

C₂²×C₄

D₄×D₁₉

yes

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

152

—

164

C₄⋊C₄

C₄⋊Dic₁₉

yes

—

165

C₄²

—

152

—

166

C₄²

C₂×Dic₃₈

yes

characteristic

167

C₄²

#64543

yes

—

168

C₄²

C₂²×Dic₁₉

yes

characteristic

169

C₂³

#10597328

yes

characteristic

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

152

—

184

C₂×C₄

—

152

—

185

C₂×C₄

#10597324

yes

—

186

C₂×C₄

#10597332

yes

characteristic

187

C₂×C₄

C₂×C₄⋊Dic₁₉

yes

characteristic

188

C₂×C₄

D₄×Dic₁₉

yes

—

189

C₈

—

190

C₈

—

191

C₂²

#11926419

yes

characteristic

192

C₂²

#11926718

yes

—

193

C₂²

#11926719

yes

characteristic

194

C₂²

#11926723

yes

—

195

C₂²

#11926729

yes

characteristic

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

—

304

—

202

C₄

—

152

—

203

C₄

—

152

—

204

C₂

#14562938

yes

characteristic

205

C₂

#14562959

yes

characteristic

206

C₂

#14562971

yes

—

207

C₂

#14563852

yes

characteristic

208

C₂

#14563870

yes

—

	subgroup	G/H
center	C₂³	#10597328
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#330	C₂
socle, prime radical	C₂³×C₁₉	#120
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#330	D₁₉
hypocenter	C₁₉	#8915
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#330

subgroup

G/H

center

C₂³

#10597328

commutator

C₄²×C₁₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₂×C₁₉×#330

C₂

socle, prime radical

C₂³×C₁₉

#120

cosocle

C₂×C₄²×C₁₉

C₂³

D₁₉

C₁₉

G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#330

N	H ≅ G/N	type
17. C₁₉×#330	C₄	semidirect
116. #330	Dic₁₉	semidirect
143. C₁₉	#8915	semidirect

H ≅ G/N

type

17. C₁₉×#330

C₄

semidirect

116. #330

Dic₁₉

semidirect

143. C₁₉

#8915

semidirect

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

subgroup	index
#14566821	2
#14564605	2
#14563769	2
#14561517	2
C₂×C₁₉×#330	2
#8915	19

subgroup

index

C₂×C₁₉×#330

Subgroups — #18076925

Conjugacy classes of subgroups

Complements

Maximal subgroups