Group #18082301 — subgroups

Subgroups — #18082301

Order 4864.

726 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 199 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	C₂×#11926921	C₂	1	1	yes	maximal, characteristic
2	#14564440	C₂	1	4	yes	maximal, complemented
3	C₂×C₈×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1845	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₁₅₂	—	2	2	no	—
6	C₂×#10597320	C₂²	1	2	yes	—
7	#11926921	—	2	2	no	—
8	#11926921	C₄	1	4	yes	complemented
9	C₂×#10597266	—	2	8	no	—
10	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
11	C₈×C₁₉⋊C₈	—	2	2	no	—
12	C₂×C₁₉×#107	—	2	2	no	—
13	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
14	C₁₉×#384	C₂²	1	2	yes	—
15	C₂²×D₇₆	—	2	4	no	—
16	C₂²×D₇₆	—	2	2	no	—
17	C₂²×C₈×C₁₉	D₄	1	2	yes	—
18	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
19	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
20	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
21	C₂²×C₁₉⋊C₈	D₄	1	4	yes	—
22	C₂×D₁₅₂	—	2	8	no	—
23	C₂×D₁₅₂	—	4	4	no	—
24	C₂×D₁₅₂	—	4	4	no	—
25	#10597320	—	2	2	no	—
26	#10597320	—	2	4	no	—
27	#10597320	C₂×C₄	1	4	yes	—
28	C₁₉×#107	—	4	4	no	—
29	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₁₉	D₄	1	2	yes	—
31	#10597266	—	4	16	no	—
32	C₄×C₁₉⋊C₈	—	2	2	no	—
33	C₄×C₁₉⋊C₈	—	2	2	no	—
34	C₂³×D₁₉	—	4	2	no	—
35	C₂²×C₄×C₁₉	—	4	2	no	—
36	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
37	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
38	C₂×D₇₆	—	2	4	no	—
39	C₂×D₇₆	—	2	8	no	—
40	C₂×D₇₆	—	4	8	no	—
41	C₂×D₇₆	—	4	4	no	—
42	C₂×D₇₆	—	4	4	no	—
43	C₂×D₇₆	—	4	8	no	—
44	C₂×D₇₆	—	4	4	no	—
45	C₂×D₇₆	—	4	2	no	—
46	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
47	C₂×C₈×C₁₉	D₈	1	4	yes	—
48	C₂×C₈×C₁₉	SD₁₆	1	4	yes	—
49	C₄⋊C₄×C₁₉	—	4	8	no	—
50	C₄⋊C₄×C₁₉	—	4	4	no	—
51	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₁₉	#41	1	2	yes	—
54	C₂×C₁₉⋊C₈	—	2	4	no	—
55	C₂×C₁₉⋊C₈	—	2	4	no	—
56	C₂×C₁₉⋊C₈	D₈	1	8	yes	complemented
57	D₁₅₂	—	4	16	no	—
58	#9824	—	19	1	no	maximal, Sylow
59	C₂³×C₁₉	#127	1	1	yes	characteristic
60	C₂²×D₁₉	—	4	4	no	—
61	C₂²×D₁₉	—	8	2	no	—
62	C₂²×D₁₉	—	8	4	no	—
63	C₂²×D₁₉	—	8	2	no	—
64	C₂²×D₁₉	—	8	4	no	—
65	C₂×C₄×C₁₉	—	8	4	no	—
66	C₂×C₄×C₁₉	—	8	2	no	—
67	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
69	C₂×C₄×C₁₉	#121	1	2	yes	—
70	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
71	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
72	C₂×C₄×C₁₉	C₂×D₈	1	4	yes	—
73	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
74	C₂×C₄×C₁₉	#135	1	4	yes	—
75	D₇₆	—	4	8	no	—
76	D₇₆	—	4	16	no	—
77	D₇₆	—	8	8	no	—
78	D₇₆	—	8	4	no	—
79	C₈×C₁₉	#102	1	8	yes	complemented
80	C₁₉⋊C₈	—	2	8	no	—
81	C₂×#493	—	19	1	no	—
82	#1957	—	19	4	no	—
83	#1845	D₁₉	1	1	yes	characteristic, complemented
84	C₂×C₈²	—	19	1	no	—
85	C₂²×C₁₉	#390	1	1	yes	characteristic
86	C₂²×C₁₉	#492	1	1	yes	characteristic
87	C₂²×C₁₉	#493	1	2	yes	—
88	C₂²×C₁₉	#493	1	1	yes	characteristic
89	C₂²×C₁₉	#495	1	2	yes	—
90	C₂×D₁₉	—	8	4	no	—
91	C₂×D₁₉	—	8	8	no	—
92	C₂×D₁₉	—	16	4	no	—
93	C₂×D₁₉	—	16	2	no	—
94	C₂×D₁₉	—	16	2	no	—
95	C₄×C₁₉	—	16	2	no	—
96	C₄×C₁₉	C₂×#102	1	2	yes	—
97	C₄×C₁₉	C₄×D₈	1	4	yes	—
98	C₄×C₁₉	#466	1	4	yes	—
99	C₄×C₁₉	#486	1	2	yes	—
100	C₂²×D₈	—	38	2	no	—
101	C₂×#127	—	19	2	no	—
102	#493	—	19	4	no	—
103	#493	—	38	2	no	—
104	C₂×#107	—	2	2	no	—
105	C₂×#102	—	38	8	no	—
106	C₂×C₄×C₈	—	19	2	no	—
107	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
108	#384	C₂×D₁₉	1	2	yes	—
109	C₈²	—	38	2	no	—
110	C₂×C₁₉	#1706	1	2	yes	—
111	C₂×C₁₉	#1709	1	1	yes	characteristic
112	C₂×C₁₉	#1712	1	1	yes	characteristic
113	C₂×C₁₉	#1957	1	1	yes	characteristic
114	C₂×C₁₉	#1970	1	2	yes	—
115	D₁₉	—	16	4	no	—
116	C₂²×D₄	—	38	4	no	—
117	C₂²×D₄	—	38	2	no	—
118	C₂×D₈	—	38	8	no	—
119	C₂×D₈	—	76	4	no	—
120	C₂×D₈	—	76	4	no	—
121	C₂²×C₈	—	19	4	no	—
122	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
123	#127	—	19	4	no	—
124	#127	—	38	2	no	—
125	#127	—	38	4	no	—
126	C₂×C₄⋊C₄	—	2	4	no	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
129	#107	—	4	4	no	—
130	#102	—	76	16	no	—
131	C₄×C₈	—	38	2	no	—
132	C₄×C₈	—	38	2	no	—
133	C₄×C₈	C₄×D₁₉	1	2	yes	—
134	C₄×C₈	D₇₆	1	2	yes	—
135	C₁₉	#9824	1	1	yes	characteristic, complemented, Sylow
136	C₂⁴	—	76	2	no	—
137	C₂×D₄	—	38	8	no	—
138	C₂×D₄	—	38	4	no	—
139	C₂×D₄	—	76	4	no	—
140	C₂×D₄	—	76	4	no	—
141	C₂×D₄	—	76	2	no	—
142	C₂×D₄	—	76	4	no	—
143	C₂×D₄	—	76	8	no	—
144	C₂×D₄	—	76	8	no	—
145	C₂²×C₄	—	4	2	no	—
146	C₂²×C₄	D₄×D₁₉	1	2	yes	—
147	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
148	D₈	—	76	16	no	—
149	C₂×C₈	—	19	8	no	—
150	C₂×C₈	—	38	4	no	—
151	C₂×C₈	—	38	4	no	—
152	C₂×C₈	(C₂×D₁₉)⋊C₄	1	4	yes	—
153	C₂×C₈	#64525	1	4	yes	—
154	C₂×C₈	#64527	1	4	yes	—
155	C₄⋊C₄	—	4	4	no	—
156	C₄⋊C₄	—	4	8	no	—
157	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
158	C₄²	C₂×D₇₆	1	1	yes	characteristic
159	C₄²	#64545	1	2	yes	—
160	C₂³	—	76	4	no	—
161	C₂³	—	152	2	no	—
162	C₂³	—	152	4	no	—
163	C₂³	—	152	2	no	—
164	C₂³	—	152	4	no	—
165	C₂³	#10597386	1	1	yes	characteristic
166	D₄	—	76	8	no	—
167	D₄	—	76	16	no	—
168	D₄	—	152	4	no	—
169	D₄	—	152	8	no	—
170	C₂×C₄	—	8	4	no	—
171	C₂×C₄	—	8	2	no	—
172	C₂×C₄	#10597339	1	2	yes	—
173	C₂×C₄	#10597341	1	2	yes	—
174	C₂×C₄	D₈×D₁₉	1	4	yes	—
175	C₂×C₄	#10597365	1	4	yes	—
176	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
177	C₂×C₄	C₂×#64525	1	2	yes	—
178	C₂×C₄	C₂×#64527	1	2	yes	—
179	C₂×C₄	#10597393	1	1	yes	characteristic
180	C₈	—	38	8	no	—
181	C₈	#10597266	1	8	yes	complemented
182	C₂²	—	304	4	no	—
183	C₂²	—	304	2	no	—
184	C₂²	—	304	2	no	—
185	C₂²	—	152	4	no	—
186	C₂²	—	152	8	no	—
187	C₂²	#11926445	1	1	yes	characteristic
188	C₂²	#11926918	1	1	yes	characteristic
189	C₂²	#11926919	1	2	yes	—
190	C₂²	#11926920	1	1	yes	characteristic
191	C₂²	#11926929	1	2	yes	—
192	C₄	—	16	2	no	—
193	C₄	C₂×#10597266	1	2	yes	—
194	C₄	#11926626	1	4	yes	—
195	C₄	#11926781	1	4	yes	—
196	C₄	#11926892	1	2	yes	—
197	C₂	—	304	4	no	—
198	C₂	#14563228	1	2	yes	—
199	C₂	#14563242	1	1	yes	characteristic
200	C₂	#14563249	1	1	yes	characteristic
201	C₂	#14564440	1	1	yes	characteristic
202	C₂	#14564499	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11926921

C₂

yes

maximal, characteristic

#14564440

C₂

yes

maximal, complemented

C₂×C₈×C₁₉⋊C₈

C₂

yes

maximal, characteristic, complemented

C₁₉×#1845

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₅₂

—

C₂×#10597320

C₂²

yes

—

#11926921

—

#11926921

C₄

yes

complemented

C₂×#10597266

—

C₂×C₄×C₁₉⋊C₈

C₂²

yes

—

C₈×C₁₉⋊C₈

—

C₂×C₁₉×#107

—

C₂×C₄×C₈×C₁₉

C₂²

yes

characteristic

C₁₉×#384

C₂²

yes

—

C₂²×D₇₆

—

C₂²×D₇₆

—

C₂²×C₈×C₁₉

D₄

yes

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₂²×C₁₉⋊C₈

D₄

yes

—

C₂×D₁₅₂

—

C₂×D₁₅₂

—

C₂×D₁₅₂

—

#10597320

—

#10597320

—

#10597320

C₂×C₄

yes

—

C₁₉×#107

—

C₄×C₈×C₁₉

C₂×C₄

yes

—

C₄×C₈×C₁₉

D₄

yes

—

#10597266

—

C₄×C₁₉⋊C₈

—

C₄×C₁₉⋊C₈

—

C₂³×D₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂×D₄

yes

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×C₈×C₁₉

C₂²⋊C₄

yes

—

C₂×C₈×C₁₉

D₈

yes

—

C₂×C₈×C₁₉

SD₁₆

yes

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄²×C₁₉

C₂²×C₄

yes

characteristic

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄²×C₁₉

#41

yes

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

D₈

yes

complemented

D₁₅₂

—

—

C₂³×C₁₉

yes

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉

C₄×D₄

yes

—

C₂×C₄×C₁₉

#121

yes

—

C₂×C₄×C₁₉

#124

yes

characteristic

C₂×C₄×C₁₉

C₂×D₈

yes

—

C₂×C₄×C₁₉

C₂×D₈

yes

—

C₂×C₄×C₁₉

C₂×SD₁₆

yes

—

C₂×C₄×C₁₉

#135

yes

—

D₇₆

—

D₇₆

—

D₇₆

—

D₇₆

—

C₈×C₁₉

#102

yes

complemented

C₁₉⋊C₈

—

C₂×#493

—

#1957

—

#1845

D₁₉

yes

characteristic, complemented

C₂×C₈²

—

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

—

C₂²×C₁₉

#493

yes

characteristic

C₂²×C₁₉

#495

yes

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₄×C₁₉

—

C₄×C₁₉

C₂×#102

yes

—

C₄×C₁₉

C₄×D₈

yes

—

C₄×C₁₉

#466

yes

—

C₄×C₁₉

#486

yes

—

100

C₂²×D₈

—

101

C₂×#127

—

102

#493

—

103

#493

—

104

C₂×#107

—

105

C₂×#102

—

106

C₂×C₄×C₈

—

107

C₂×C₄×C₈

C₂×D₁₉

yes

characteristic

108

#384

C₂×D₁₉

yes

—

109

C₈²

—

110

C₂×C₁₉

#1706

yes

—

111

C₂×C₁₉

#1709

yes

characteristic

112

C₂×C₁₉

#1712

yes

characteristic

113

C₂×C₁₉

#1957

yes

characteristic

114

C₂×C₁₉

#1970

yes

—

115

D₁₉

—

116

C₂²×D₄

—

117

C₂²×D₄

—

118

C₂×D₈

—

119

C₂×D₈

—

120

C₂×D₈

—

121

C₂²×C₈

—

122

C₂²×C₈

C₂²⋊D₁₉

yes

—

123

#127

—

124

#127

—

125

#127

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄²

C₂²×D₁₉

yes

characteristic

129

#107

—

130

#102

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

C₄×D₁₉

yes

—

134

C₄×C₈

D₇₆

yes

—

135

C₁₉

#9824

yes

characteristic, complemented, Sylow

136

C₂⁴

—

137

C₂×D₄

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂×D₄

—

141

C₂×D₄

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂²×C₄

—

146

C₂²×C₄

D₄×D₁₉

yes

—

147

C₂²×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

148

D₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

(C₂×D₁₉)⋊C₄

yes

—

153

C₂×C₈

#64525

yes

—

154

C₂×C₈

#64527

yes

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄²

C₂×C₄×D₁₉

yes

characteristic

158

C₄²

C₂×D₇₆

yes

characteristic

159

C₄²

#64545

yes

—

160

C₂³

—

161

C₂³

—

152

—

162

C₂³

—

152

—

163

C₂³

—

152

—

164

C₂³

—

152

—

165

C₂³

#10597386

yes

characteristic

166

D₄

—

167

D₄

—

168

D₄

—

152

—

169

D₄

—

152

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

#10597339

yes

—

173

C₂×C₄

#10597341

yes

—

174

C₂×C₄

D₈×D₁₉

yes

—

175

C₂×C₄

#10597365

yes

—

176

C₂×C₄

C₂×(C₂×D₁₉)⋊C₄

yes

characteristic

177

C₂×C₄

C₂×#64525

yes

—

178

C₂×C₄

C₂×#64527

yes

—

179

C₂×C₄

#10597393

yes

characteristic

180

C₈

—

181

C₈

#10597266

yes

complemented

182

C₂²

—

304

—

183

C₂²

—

304

—

184

C₂²

—

304

—

185

C₂²

—

152

—

186

C₂²

—

152

—

187

C₂²

#11926445

yes

characteristic

188

C₂²

#11926918

yes

characteristic

189

C₂²

#11926919

yes

—

190

C₂²

#11926920

yes

characteristic

191

C₂²

#11926929

yes

—

192

C₄

—

193

C₄

C₂×#10597266

yes

—

194

C₄

#11926626

yes

—

195

C₄

#11926781

yes

—

196

C₄

#11926892

yes

—

197

C₂

—

304

—

198

C₂

#14563228

yes

—

199

C₂

#14563242

yes

characteristic

200

C₂

#14563249

yes

characteristic

201

C₂

#14564440

yes

characteristic

202

C₂

#14564499

yes

—

	subgroup	G/H
center	C₂³	#10597386
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1845	C₂
socle	C₂³×C₁₉	#127
prime radical	C₂×#11926921	C₂
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1845	D₁₉
hypocenter	C₁₉	#9824
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

subgroup

G/H

center

C₂³

#10597386

commutator

C₄²×C₁₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#1845

C₂

C₂³×C₁₉

C₂

C₂×C₄²×C₁₉

C₂³

D₁₉

C₁₉

G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

N	H ≅ G/N	type
2. #14564440	C₂	semidirect
3. C₂×C₈×C₁₉⋊C₈	C₂	semidirect
4. C₁₉×#1845	C₂	semidirect
8. #11926921	C₄	semidirect
56. C₂×C₁₉⋊C₈	D₈	semidirect
79. C₈×C₁₉	#102	semidirect
83. #1845	D₁₉	semidirect
135. C₁₉	#9824	semidirect
181. C₈	#10597266	semidirect

H ≅ G/N

type

2. #14564440

C₂

semidirect

3. C₂×C₈×C₁₉⋊C₈

C₂

semidirect

4. C₁₉×#1845

C₂

semidirect

8. #11926921

C₄

semidirect

56. C₂×C₁₉⋊C₈

D₈

semidirect

79. C₈×C₁₉

#102

semidirect

83. #1845

D₁₉

semidirect

135. C₁₉

#9824

semidirect

181. C₈

#10597266

semidirect

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

subgroup	index
C₂×#11926921	2
#14564440	2
C₂×C₈×C₁₉⋊C₈	2
C₁₉×#1845	2
#9824	19

subgroup

index

C₂×#11926921

#14564440

C₂×C₈×C₁₉⋊C₈

C₁₉×#1845

#9824

Subgroups — #18082301

Conjugacy classes of subgroups

Complements

Maximal subgroups