Group #18082661 — subgroups

Subgroups — #18082661

Order 4864.

430 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 167 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	#14563752	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#14561518	C₂	1	4	yes	maximal
4	C₁₉×#2859	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
6	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
8	#11926299	—	2	1	no	—
9	#11926299	—	2	2	no	—
10	C₂×C₁₉×#125	—	2	1	no	—
11	C₁₉×#389	—	2	2	no	—
12	C₁₉×#389	C₄	1	4	yes	—
13	C₁₉×#384	C₂²	1	2	yes	—
14	C₁₉×#382	—	2	1	no	—
15	C₄³×C₁₉	C₂²	1	1	yes	characteristic
16	C₁₉×#125	—	4	4	no	—
17	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
18	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
19	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
20	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
21	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₁₉	D₄	1	2	yes	—
23	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
26	C₂²×C₁₉⋊C₈	—	2	2	no	—
27	C₂²×C₁₉⋊C₈	—	2	2	no	—
28	C₁₉×#95	—	2	4	no	—
29	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
30	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
31	C₄⋊(C₁₉⋊C₈)	Q₈	1	4	yes	—
32	C₄×C₁₉⋊C₈	—	4	2	no	—
33	C₂²×C₄×C₁₉	—	2	4	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	C₄⋊C₄×C₁₉	—	4	4	no	—
40	C₄⋊C₄×C₁₉	—	4	8	no	—
41	C₄⋊C₄×C₁₉	—	4	4	no	—
42	C₄⋊C₄×C₁₉	—	4	4	no	—
43	C₄²×C₁₉	—	2	4	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	C₂×C₁₉⋊C₈	—	2	4	no	—
51	C₂×C₁₉⋊C₈	—	4	2	no	—
52	C₂×C₁₉⋊C₈	—	4	2	no	—
53	C₂×C₁₉⋊C₈	—	4	2	no	—
54	C₂×C₁₉⋊C₈	—	4	4	no	—
55	#9897	—	19	1	no	maximal, Sylow
56	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₉	—	2	2	no	—
58	C₂×C₄×C₁₉	—	2	2	no	—
59	C₂×C₄×C₁₉	—	2	2	no	—
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	—	4	4	no	—
64	C₂×C₄×C₁₉	—	4	4	no	—
65	C₂×C₄×C₁₉	—	4	4	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	#135	1	4	yes	—
74	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
75	C₁₉⋊C₈	—	4	4	no	—
76	C₁₉⋊C₈	—	8	2	no	—
77	#2859	D₁₉	1	1	yes	characteristic
78	#1831	—	19	1	no	—
79	C₄×C₄⋊C₈	—	19	1	no	—
80	#1296	—	19	4	no	—
81	C₂²×C₁₉	#391	1	1	yes	characteristic
82	C₂²×C₁₉	#413	1	2	yes	—
83	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
84	C₂²×C₁₉	#416	1	2	yes	—
85	C₂²×C₁₉	#418	1	1	yes	characteristic
86	C₄×C₁₉	—	2	4	no	—
87	C₄×C₁₉	—	2	2	no	—
88	C₄×C₁₉	—	4	2	no	—
89	C₄×C₁₉	—	8	4	no	—
90	C₄×C₁₉	#476	1	2	yes	—
91	C₄×C₁₉	#477	1	2	yes	—
92	C₄×C₁₉	#488	1	2	yes	—
93	C₄×C₁₉	#499	1	2	yes	—
94	C₂×#125	—	2	1	no	—
95	C₂×C₄⋊C₈	—	19	2	no	—
96	C₂×C₄⋊C₈	—	19	2	no	—
97	C₂×C₄×C₈	—	38	1	no	—
98	#389	—	2	2	no	—
99	#389	Dic₁₉	1	4	yes	—
100	#384	C₂×D₁₉	1	2	yes	—
101	#382	—	2	1	no	—
102	C₄³	C₂×D₁₉	1	1	yes	characteristic
103	#336	—	38	2	no	—
104	#336	—	38	1	no	—
105	C₂×C₁₉	#1542	1	2	yes	—
106	C₂×C₁₉	#1553	1	2	yes	—
107	C₂×C₁₉	#1987	1	1	yes	characteristic
108	C₂×C₁₉	#1991	1	1	yes	characteristic
109	C₂×C₁₉	#1998	1	1	yes	characteristic
110	C₂²×C₈	—	38	2	no	—
111	C₂²×C₈	—	38	2	no	—
112	#125	—	4	4	no	—
113	C₂×C₄⋊C₄	—	2	4	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	C₂×Dic₁₉	1	4	yes	—
117	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
118	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
119	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
120	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
121	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
122	C₄⋊C₈	—	19	4	no	—
123	C₄⋊C₈	—	38	2	no	—
124	C₄⋊C₈	—	38	4	no	—
125	C₄×C₈	—	76	2	no	—
126	#95	—	2	4	no	—
127	C₁₉	#9897	1	1	yes	characteristic, complemented, Sylow
128	C₂²×C₄	—	2	4	no	—
129	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
130	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
131	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
133	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
134	C₂×C₈	—	38	4	no	—
135	C₂×C₈	—	76	2	no	—
136	C₂×C₈	—	76	2	no	—
137	C₂×C₈	—	76	2	no	—
138	C₂×C₈	—	76	4	no	—
139	C₄⋊C₄	—	4	4	no	—
140	C₄⋊C₄	—	4	8	no	—
141	C₄⋊C₄	—	4	4	no	—
142	C₄⋊C₄	—	4	4	no	—
143	C₄²	—	2	2	no	—
144	C₄²	—	2	4	no	—
145	C₄²	—	4	2	no	—
146	C₄²	#64527	1	2	yes	—
147	C₄²	#64528	1	2	yes	—
148	C₄²	Q₈×D₁₉	1	2	yes	—
149	C₄²	#64545	1	2	yes	—
150	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	#10597292	1	4	yes	—
161	C₂×C₄	#10597378	1	2	yes	—
162	C₂×C₄	C₂×#64527	1	1	yes	characteristic
163	C₂×C₄	C₂×#64528	1	1	yes	characteristic
164	C₂×C₄	#10597390	1	1	yes	characteristic
165	C₂×C₄	Q₈×Dic₁₉	1	2	yes	—
166	C₂×C₄	#10597392	1	2	yes	—
167	C₂×C₄	#10597393	1	1	yes	characteristic
168	C₂×C₄	#10597396	1	4	yes	—
169	C₈	—	76	4	no	—
170	C₈	—	152	2	no	—
171	C₂²	#11926451	1	1	yes	characteristic
172	C₂²	#11926528	1	2	yes	—
173	C₂²	C₂×#10597292	1	1	yes	characteristic
174	C₂²	#11926540	1	2	yes	—
175	C₂²	#11926549	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	2	4	no	—
178	C₄	—	4	2	no	—
179	C₄	—	8	4	no	—
180	C₄	#11926831	1	2	yes	—
181	C₄	#11926836	1	2	yes	—
182	C₄	#11926901	1	2	yes	—
183	C₄	#11926943	1	2	yes	—
184	C₂	#14562292	1	2	yes	—
185	C₂	#14562340	1	2	yes	—
186	C₂	#14564584	1	1	yes	characteristic
187	C₂	#14564601	1	1	yes	characteristic
188	C₂	#14564629	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14563752

C₂

yes

maximal, characteristic

C₄×C₄⋊(C₁₉⋊C₈)

C₂

yes

maximal, characteristic

#14561518

C₂

yes

maximal

C₁₉×#2859

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₁₉⋊C₈)

C₂²

yes

—

C₂×C₄⋊(C₁₉⋊C₈)

C₂²

yes

—

C₂×C₄×C₁₉⋊C₈

—

#11926299

—

#11926299

—

C₂×C₁₉×#125

—

C₁₉×#389

—

C₁₉×#389

C₄

yes

—

C₁₉×#384

C₂²

yes

—

C₁₉×#382

—

C₄³×C₁₉

C₂²

yes

characteristic

C₁₉×#125

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

C₂×C₄

yes

—

C₂×C₄²×C₁₉

C₂×C₄

yes

—

C₂×C₄²×C₁₉

D₄

yes

—

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₂²×C₁₉⋊C₈

—

C₂²×C₁₉⋊C₈

—

C₁₉×#95

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

Q₈

yes

—

C₄×C₁₉⋊C₈

—

C₂²×C₄×C₁₉

—

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

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

SD₁₆

yes

—

C₄²×C₁₉

Q₁₆

yes

—

C₄²×C₁₉

C₂×Q₈

yes

—

C₄²×C₁₉

#41

yes

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

#9897

—

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

—

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

#135

yes

—

C₂×C₄×C₁₉

AGL(1,ℤ/8)

yes

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

#2859

D₁₉

yes

characteristic

#1831

—

C₄×C₄⋊C₈

—

#1296

—

C₂²×C₁₉

#391

yes

characteristic

C₂²×C₁₉

#413

yes

—

C₂²×C₁₉

C₂×#103

yes

characteristic

C₂²×C₁₉

#416

yes

—

C₂²×C₁₉

#418

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#476

yes

—

C₄×C₁₉

#477

yes

—

C₄×C₁₉

#488

yes

—

C₄×C₁₉

#499

yes

—

C₂×#125

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

#389

—

#389

Dic₁₉

yes

—

100

#384

C₂×D₁₉

yes

—

101

#382

—

102

C₄³

C₂×D₁₉

yes

characteristic

103

#336

—

104

#336

—

105

C₂×C₁₉

#1542

yes

—

106

C₂×C₁₉

#1553

yes

—

107

C₂×C₁₉

#1987

yes

characteristic

108

C₂×C₁₉

#1991

yes

characteristic

109

C₂×C₁₉

#1998

yes

characteristic

110

C₂²×C₈

—

111

C₂²×C₈

—

112

#125

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

C₂×Dic₁₉

yes

—

117

C₂×C₄²

C₂×Dic₁₉

yes

—

118

C₂×C₄²

C₂²⋊D₁₉

yes

characteristic

119

C₂×C₄²

C₂²⋊D₁₉

yes

—

120

C₂×C₄²

C₂²⋊D₁₉

yes

characteristic

121

C₂×C₄²

C₂²×D₁₉

yes

characteristic

122

C₄⋊C₈

—

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

C₄×C₈

—

126

#95

—

127

C₁₉

#9897

yes

characteristic, complemented, Sylow

128

C₂²×C₄

—

129

C₂²×C₄

C₂²⋊Dic₁₉

yes

—

130

C₂²×C₄

C₂²⋊Dic₁₉

yes

—

131

C₂²×C₄

C₂²×Dic₁₉

yes

characteristic

132

C₂²×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

133

C₂²×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₄²

#64527

yes

—

147

C₄²

#64528

yes

—

148

C₄²

Q₈×D₁₉

yes

—

149

C₄²

#64545

yes

—

150

C₂³

C₂×C₂²⋊Dic₁₉

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

#10597292

yes

—

161

C₂×C₄

#10597378

yes

—

162

C₂×C₄

C₂×#64527

yes

characteristic

163

C₂×C₄

C₂×#64528

yes

characteristic

164

C₂×C₄

#10597390

yes

characteristic

165

C₂×C₄

Q₈×Dic₁₉

yes

—

166

C₂×C₄

#10597392

yes

—

167

C₂×C₄

#10597393

yes

characteristic

168

C₂×C₄

#10597396

yes

—

169

C₈

—

170

C₈

—

152

—

171

C₂²

#11926451

yes

characteristic

172

C₂²

#11926528

yes

—

173

C₂²

C₂×#10597292

yes

characteristic

174

C₂²

#11926540

yes

—

175

C₂²

#11926549

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

#11926831

yes

—

181

C₄

#11926836

yes

—

182

C₄

#11926901

yes

—

183

C₄

#11926943

yes

—

184

C₂

#14562292

yes

—

185

C₂

#14562340

yes

—

186

C₂

#14564584

yes

characteristic

187

C₂

#14564601

yes

characteristic

188

C₂

#14564629

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2859	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2859	D₁₉
hypocenter	C₁₉	#9897
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₁₉

commutator

C₂²×C₄×C₁₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#2859

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 ◁ #2859

N	H ≅ G/N	type
127. C₁₉	#9897	semidirect

H ≅ G/N

type

127. C₁₉

#9897

semidirect

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

subgroup	index
#14563752	2
C₄×C₄⋊(C₁₉⋊C₈)	2
#14561518	2
C₁₉×#2859	2
#9897	19

subgroup

index

#14563752

C₄×C₄⋊(C₁₉⋊C₈)

#14561518

C₁₉×#2859

#9897

Subgroups — #18082661

Conjugacy classes of subgroups

Complements

Maximal subgroups