Group #18082681 — subgroups

Subgroups — #18082681

Order 4864.

438 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 159 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	#14571998	C₂	1	1	yes	maximal, characteristic
2	#14563488	C₂	1	1	yes	maximal, characteristic
3	#14561519	C₂	1	4	yes	maximal
4	C₁₉×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×#10597316	—	2	1	no	—
6	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×#10597260	—	2	1	no	—
8	#11926442	—	2	2	no	—
9	#11926442	C₄	1	4	yes	—
10	#11926414	C₂²	1	2	yes	—
11	#11926410	—	2	1	no	—
12	#11926299	—	2	1	no	—
13	C₂×C₁₉×C₄⋊C₈	—	2	2	no	—
14	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
15	C₄³×C₁₉	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₁₉	—	2	2	no	—
17	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₉	D₄	1	2	yes	—
20	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₉	—	2	2	no	—
23	C₂×C₄⋊Dic₁₉	—	2	2	no	—
24	C₂×C₄⋊Dic₁₉	C₂×C₄	1	4	yes	—
25	C₂×Dic₁₉⋊C₄	—	2	4	no	—
26	C₂²×C₁₉⋊C₈	—	2	2	no	—
27	#10597316	—	4	4	no	—
28	C₁₉×C₄⋊C₈	—	2	4	no	—
29	C₁₉×C₄⋊C₈	—	4	4	no	—
30	C₁₉⋊#95	—	2	4	no	—
31	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
32	C₄⋊(C₁₉⋊C₈)	Q₈	1	4	yes	—
33	#10597260	—	4	2	no	—
34	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
36	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₁₉	—	2	4	no	—
40	C₂×C₈×C₁₉	—	4	2	no	—
41	C₂×C₈×C₁₉	—	4	4	no	—
42	C₄²×C₁₉	—	2	2	no	—
43	C₄²×C₁₉	—	2	2	no	—
44	C₄²×C₁₉	—	2	2	no	—
45	C₄²×C₁₉	—	4	2	no	—
46	C₄²×C₁₉	SD₁₆	1	2	yes	—
47	C₄²×C₁₉	Q₁₆	1	2	yes	—
48	C₄²×C₁₉	C₂×Q₈	1	2	yes	—
49	C₄²×C₁₉	#41	1	2	yes	—
50	C₄⋊Dic₁₉	—	4	4	no	—
51	C₄⋊Dic₁₉	—	4	2	no	—
52	C₄⋊Dic₁₉	—	4	2	no	—
53	C₄⋊Dic₁₉	—	4	4	no	—
54	Dic₁₉⋊C₄	—	4	8	no	—
55	C₂×C₁₉⋊C₈	—	2	4	no	—
56	C₂×C₁₉⋊C₈	—	4	2	no	—
57	C₂×C₁₉⋊C₈	—	4	2	no	—
58	#9902	—	19	1	no	maximal, Sylow
59	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₉	—	2	2	no	—
61	C₂×C₄×C₁₉	—	2	2	no	—
62	C₂×C₄×C₁₉	—	2	2	no	—
63	C₂×C₄×C₁₉	—	2	2	no	—
64	C₂×C₄×C₁₉	—	2	2	no	—
65	C₂×C₄×C₁₉	—	2	2	no	—
66	C₂×C₄×C₁₉	#103	1	4	yes	—
67	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
68	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
69	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
70	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
73	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
74	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
75	C₂×C₄×C₁₉	#137	1	2	yes	—
76	C₂×Dic₁₉	—	4	4	no	—
77	C₂×Dic₁₉	—	4	4	no	—
78	C₂×Dic₁₉	—	4	4	no	—
79	C₈×C₁₉	—	8	2	no	—
80	C₁₉⋊C₈	—	4	4	no	—
81	#2859	—	19	1	no	—
82	#1842	D₁₉	1	1	yes	characteristic
83	#1767	—	19	1	no	—
84	#1296	—	19	4	no	—
85	C₂²×C₁₉	#391	1	1	yes	characteristic
86	C₂²×C₁₉	#413	1	2	yes	—
87	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
88	C₂²×C₁₉	#417	1	2	yes	—
89	C₂²×C₁₉	#418	1	1	yes	characteristic
90	C₄×C₁₉	—	2	4	no	—
91	C₄×C₁₉	—	2	2	no	—
92	C₄×C₁₉	—	4	2	no	—
93	C₄×C₁₉	#476	1	2	yes	—
94	C₄×C₁₉	#477	1	2	yes	—
95	C₄×C₁₉	#490	1	1	yes	characteristic
96	C₄×C₁₉	#491	1	1	yes	characteristic
97	C₄×C₁₉	#501	1	2	yes	—
98	Dic₁₉	—	8	4	no	—
99	C₂×#125	—	38	1	no	—
100	C₂×C₄⋊C₈	—	2	2	no	—
101	C₂×C₄⋊C₈	—	19	2	no	—
102	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
103	C₂×#97	—	38	1	no	—
104	#389	—	19	4	no	—
105	#389	—	38	2	no	—
106	#384	—	19	2	no	—
107	#382	—	38	1	no	—
108	C₄³	C₂×D₁₉	1	1	yes	characteristic
109	#336	—	38	1	no	—
110	C₂×C₁₉	#1550	1	2	yes	—
111	C₂×C₁₉	#1557	1	2	yes	—
112	C₂×C₁₉	#1987	1	1	yes	characteristic
113	C₂×C₁₉	#1992	1	1	yes	characteristic
114	C₂×C₁₉	#1998	1	1	yes	characteristic
115	C₂²×C₈	—	2	2	no	—
116	C₂²×C₈	—	38	2	no	—
117	#125	—	76	4	no	—
118	C₂×C₄⋊C₄	—	19	4	no	—
119	C₂×C₄⋊C₄	—	38	4	no	—
120	C₂×C₄⋊C₄	—	38	2	no	—
121	C₂×C₄⋊C₄	—	38	2	no	—
122	C₂×C₄²	C₄×D₁₉	1	2	yes	—
123	C₂×C₄²	D₇₆	1	1	yes	characteristic
124	C₂×C₄²	D₇₆	1	1	yes	characteristic
125	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
126	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
127	C₄⋊C₈	—	2	4	no	—
128	C₄⋊C₈	—	4	4	no	—
129	C₄⋊C₈	—	19	4	no	—
130	C₄⋊C₈	—	38	2	no	—
131	#97	—	76	2	no	—
132	#95	—	38	4	no	—
133	C₁₉	#9902	1	1	yes	characteristic, complemented, Sylow
134	C₂²×C₄	—	38	4	no	—
135	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
136	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
137	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
138	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
140	C₂×C₈	—	4	4	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	38	4	no	—
143	C₂×C₈	—	76	2	no	—
144	C₂×C₈	—	76	2	no	—
145	C₄⋊C₄	—	76	4	no	—
146	C₄⋊C₄	—	76	8	no	—
147	C₄⋊C₄	—	76	2	no	—
148	C₄⋊C₄	—	76	2	no	—
149	C₄⋊C₄	—	76	4	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	2	2	no	—
153	C₄²	—	4	2	no	—
154	C₄²	#64516	1	2	yes	—
155	C₄²	Dic₇₆	1	2	yes	—
156	C₄²	#64543	1	2	yes	—
157	C₄²	Q₈×D₁₉	1	2	yes	—
158	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	76	4	no	—
166	C₂×C₄	—	76	4	no	—
167	C₂×C₄	—	76	4	no	—
168	C₂×C₄	#10597273	1	4	yes	—
169	C₂×C₄	#10597333	1	2	yes	—
170	C₂×C₄	#10597343	1	2	yes	—
171	C₂×C₄	C₂×#64516	1	1	yes	characteristic
172	C₂×C₄	C₂×Dic₇₆	1	1	yes	characteristic
173	C₂×C₄	#10597354	1	2	yes	—
174	C₂×C₄	#10597378	1	2	yes	—
175	C₂×C₄	#10597382	1	1	yes	characteristic
176	C₂×C₄	#10597388	1	2	yes	—
177	C₂×C₄	#10597390	1	1	yes	characteristic
178	C₈	—	8	2	no	—
179	C₈	—	76	4	no	—
180	C₂²	#11926449	1	1	yes	characteristic
181	C₂²	#11926526	1	2	yes	—
182	C₂²	C₂×#10597273	1	1	yes	characteristic
183	C₂²	#11926543	1	2	yes	—
184	C₂²	#11926550	1	1	yes	characteristic
185	C₄	—	2	4	no	—
186	C₄	—	2	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	152	4	no	—
189	C₄	#11926835	1	2	yes	—
190	C₄	#11926840	1	2	yes	—
191	C₄	#11926908	1	1	yes	characteristic
192	C₄	#11926912	1	1	yes	characteristic
193	C₄	#11926950	1	2	yes	—
194	C₂	#14562323	1	2	yes	—
195	C₂	#14562355	1	2	yes	—
196	C₂	#14564582	1	1	yes	characteristic
197	C₂	#14564603	1	1	yes	characteristic
198	C₂	#14564627	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14571998

C₂

yes

maximal, characteristic

#14563488

C₂

yes

maximal, characteristic

#14561519

C₂

yes

maximal

C₁₉×#1842

C₂

yes

maximal, characteristic

C₂×#10597316

—

C₂×C₄⋊(C₁₉⋊C₈)

C₂²

yes

—

C₂×#10597260

—

#11926442

—

#11926442

C₄

yes

—

#11926414

C₂²

yes

—

#11926410

—

#11926299

—

C₂×C₁₉×C₄⋊C₈

—

C₂×C₁₉×C₄⋊C₈

C₂²

yes

—

C₄³×C₁₉

C₂²

yes

characteristic

C₂²×C₈×C₁₉

—

C₂×C₄²×C₁₉

C₂×C₄

yes

—

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

D₄

yes

—

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₉

—

C₂×C₄⋊Dic₁₉

—

C₂×C₄⋊Dic₁₉

C₂×C₄

yes

—

C₂×Dic₁₉⋊C₄

—

C₂²×C₁₉⋊C₈

—

#10597316

—

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

C₁₉⋊#95

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

Q₈

yes

—

#10597260

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂²×Dic₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

SD₁₆

yes

—

C₄²×C₁₉

Q₁₆

yes

—

C₄²×C₁₉

C₂×Q₈

yes

—

C₄²×C₁₉

#41

yes

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

Dic₁₉⋊C₄

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

#9902

—

maximal, Sylow

C₂³×C₁₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#103

yes

—

C₂×C₄×C₁₉

C₄×Q₈

yes

—

C₂×C₄×C₁₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₉

#124

yes

characteristic

C₂×C₄×C₁₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₁₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₁₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₁₉

#137

yes

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₈×C₁₉

—

C₁₉⋊C₈

—

#2859

—

#1842

D₁₉

yes

characteristic

#1767

—

#1296

—

C₂²×C₁₉

#391

yes

characteristic

C₂²×C₁₉

#413

yes

—

C₂²×C₁₉

C₂×#103

yes

characteristic

C₂²×C₁₉

#417

yes

—

C₂²×C₁₉

#418

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#476

yes

—

C₄×C₁₉

#477

yes

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

—

Dic₁₉

—

C₂×#125

—

100

C₂×C₄⋊C₈

—

101

C₂×C₄⋊C₈

—

102

C₂×C₄⋊C₈

C₂×D₁₉

yes

—

103

C₂×#97

—

104

#389

—

105

#389

—

106

#384

—

107

#382

—

108

C₄³

C₂×D₁₉

yes

characteristic

109

#336

—

110

C₂×C₁₉

#1550

yes

—

111

C₂×C₁₉

#1557

yes

—

112

C₂×C₁₉

#1987

yes

characteristic

113

C₂×C₁₉

#1992

yes

characteristic

114

C₂×C₁₉

#1998

yes

characteristic

115

C₂²×C₈

—

116

C₂²×C₈

—

117

#125

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄²

C₄×D₁₉

yes

—

123

C₂×C₄²

D₇₆

yes

characteristic

124

C₂×C₄²

D₇₆

yes

characteristic

125

C₂×C₄²

C₂²⋊D₁₉

yes

—

126

C₂×C₄²

C₂²×D₁₉

yes

characteristic

127

C₄⋊C₈

—

128

C₄⋊C₈

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

131

#97

—

132

#95

—

133

C₁₉

#9902

yes

characteristic, complemented, Sylow

134

C₂²×C₄

—

135

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

—

136

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

—

137

C₂²×C₄

C₂×C₄×D₁₉

yes

characteristic

138

C₂²×C₄

C₂×D₇₆

yes

characteristic

139

C₂²×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄²

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

#64516

yes

—

155

C₄²

Dic₇₆

yes

—

156

C₄²

#64543

yes

—

157

C₄²

Q₈×D₁₉

yes

—

158

C₂³

C₂×(C₂×D₁₉)⋊C₄

yes

characteristic

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

#10597273

yes

—

169

C₂×C₄

#10597333

yes

—

170

C₂×C₄

#10597343

yes

—

171

C₂×C₄

C₂×#64516

yes

characteristic

172

C₂×C₄

C₂×Dic₇₆

yes

characteristic

173

C₂×C₄

#10597354

yes

—

174

C₂×C₄

#10597378

yes

—

175

C₂×C₄

#10597382

yes

characteristic

176

C₂×C₄

#10597388

yes

—

177

C₂×C₄

#10597390

yes

characteristic

178

C₈

—

179

C₈

—

180

C₂²

#11926449

yes

characteristic

181

C₂²

#11926526

yes

—

182

C₂²

C₂×#10597273

yes

characteristic

183

C₂²

#11926543

yes

—

184

C₂²

#11926550

yes

characteristic

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

152

—

189

C₄

#11926835

yes

—

190

C₄

#11926840

yes

—

191

C₄

#11926908

yes

characteristic

192

C₄

#11926912

yes

characteristic

193

C₄

#11926950

yes

—

194

C₂

#14562323

yes

—

195

C₂

#14562355

yes

—

196

C₂

#14564582

yes

characteristic

197

C₂

#14564603

yes

characteristic

198

C₂

#14564627

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₁₉)⋊C₄
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1842	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1842	D₁₉
hypocenter	C₁₉	#9902
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

subgroup

G/H

center

C₂³

C₂×(C₂×D₁₉)⋊C₄

commutator

C₂²×C₄×C₁₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#1842

C₂

socle, prime radical

C₂³×C₁₉

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₁₉

C₂³

D₁₉

C₁₉

G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

N	H ≅ G/N	type
133. C₁₉	#9902	semidirect

H ≅ G/N

type

133. C₁₉

#9902

semidirect

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

subgroup	index
#14571998	2
#14563488	2
#14561519	2
C₁₉×#1842	2
#9902	19

subgroup

index

C₁₉×#1842

Subgroups — #18082681

Conjugacy classes of subgroups

Complements

Maximal subgroups