Group #18069331 — subgroups

Subgroups — #18069331

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	#14561713	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1277	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₉⋊C₁₆	—	2	1	no	—
5	C₂×#10597275	C₄	1	1	yes	characteristic
6	C₂×#10597275	C₄	1	1	yes	characteristic
7	#11926442	—	2	1	no	—
8	#11926307	—	2	1	no	—
9	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
10	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₁₉	—	4	1	no	—
12	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
14	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
15	C₂×C₄⋊Dic₁₉	—	2	1	no	—
16	C₂×C₄⋊Dic₁₉	—	2	1	no	—
17	C₂×Dic₁₉⋊C₄	—	4	1	no	—
18	C₁₉×C₄⋊C₈	—	2	2	no	—
19	C₁₉×C₄⋊C₈	—	4	1	no	—
20	C₄×C₈×C₁₉	—	4	1	no	—
21	C₁₉⋊#95	—	4	1	no	—
22	#10597275	—	2	1	no	—
23	#10597275	—	4	1	no	—
24	#10597275	C₈	1	2	yes	complemented
25	C₂×C₁₉⋊C₁₆	—	2	2	no	—
26	C₂×C₁₉⋊C₁₆	—	4	1	no	—
27	C₂×C₁₉⋊C₁₆	—	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	—	4	2	no	—
38	C₂×C₈×C₁₉	—	8	1	no	—
39	C₂×C₈×C₁₉	—	8	1	no	—
40	C₂×C₈×C₁₉	C₂×C₈	1	1	yes	characteristic
41	C₂×C₈×C₁₉	M₁₆	1	1	yes	characteristic
42	C₂×C₈×C₁₉	SD₁₆	1	1	yes	characteristic
43	C₂×C₈×C₁₉	Q₁₆	1	1	yes	characteristic
44	C₄²×C₁₉	—	2	1	no	—
45	C₄²×C₁₉	—	2	1	no	—
46	C₄⋊Dic₁₉	—	2	2	no	—
47	C₄⋊Dic₁₉	—	4	1	no	—
48	C₄⋊Dic₁₉	—	4	1	no	—
49	C₄⋊Dic₁₉	—	4	1	no	—
50	Dic₁₉⋊C₄	—	8	2	no	—
51	C₁₉⋊C₁₆	—	4	2	no	—
52	#7121	—	19	1	no	maximal, Sylow
53	C₂³×C₁₉	#102	1	1	yes	characteristic
54	C₂×C₄×C₁₉	—	2	1	no	—
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	C₂²⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
61	C₂×C₄×C₁₉	#101	1	1	yes	characteristic
62	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
63	C₂×C₄×C₁₉	#104	1	1	yes	characteristic
64	C₂×C₄×C₁₉	#104	1	1	yes	characteristic
65	C₂×C₄×C₁₉	SD₃₂	1	1	yes	characteristic
66	C₂×C₄×C₁₉	Q₃₂	1	1	yes	characteristic
67	C₂×Dic₁₉	—	4	2	no	—
68	C₂×Dic₁₉	—	8	1	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	C₈×C₁₉	—	2	1	no	—
74	C₈×C₁₉	—	2	1	no	—
75	C₈×C₁₉	—	2	1	no	—
76	C₈×C₁₉	—	2	1	no	—
77	C₈×C₁₉	—	8	2	no	—
78	#1942	—	19	1	no	—
79	#1374	—	19	1	no	—
80	#1277	D₁₉	1	1	yes	characteristic
81	C₂²×C₁₉	D₄⋊C₈	1	1	yes	characteristic
82	C₂²×C₁₉	#327	1	1	yes	characteristic
83	C₂²×C₁₉	#332	1	1	yes	characteristic
84	C₂²×C₁₉	#358	1	1	yes	characteristic
85	C₂²×C₁₉	#359	1	1	yes	characteristic
86	C₂²×C₁₉	#360	1	1	yes	characteristic
87	C₂²×C₁₉	#361	1	1	yes	characteristic
88	C₄×C₁₉	—	2	1	no	—
89	C₄×C₁₉	—	2	1	no	—
90	C₄×C₁₉	—	4	1	no	—
91	C₄×C₁₉	#324	1	1	yes	characteristic
92	C₄×C₁₉	Q₈⋊C₈	1	1	yes	characteristic
93	C₄×C₁₉	#328	1	1	yes	characteristic
94	C₄×C₁₉	#330	1	1	yes	characteristic
95	Dic₁₉	—	8	2	no	—
96	Dic₁₉	—	16	1	no	—
97	C₂²×C₁₆	—	38	1	no	—
98	C₂×#107	—	19	1	no	—
99	C₂×#107	—	19	1	no	—
100	C₂×C₄⋊C₈	—	2	1	no	—
101	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
102	#389	—	38	1	no	—
103	#340	—	38	1	no	—
104	C₂×C₁₉	#1271	1	1	yes	characteristic
105	C₂×C₁₉	#1332	1	1	yes	characteristic
106	C₂×C₁₉	#1333	1	1	yes	characteristic
107	C₂×C₁₉	#1348	1	1	yes	characteristic
108	C₂×C₁₉	#1349	1	1	yes	characteristic
109	C₂×C₁₉	#1363	1	1	yes	characteristic
110	C₂×C₁₉	#1365	1	1	yes	characteristic
111	C₂²×C₈	—	4	1	no	—
112	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
113	C₂²×C₈	D₇₆	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	38	1	no	—
115	C₂×C₄⋊C₄	—	38	1	no	—
116	C₂×C₄⋊C₄	—	76	1	no	—
117	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
118	C₂×C₁₆	—	38	2	no	—
119	C₂×C₁₆	—	76	1	no	—
120	C₂×C₁₆	—	76	1	no	—
121	#107	—	19	2	no	—
122	#107	—	38	1	no	—
123	#107	—	76	1	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	4	1	no	—
126	C₄×C₈	—	4	1	no	—
127	#95	—	76	1	no	—
128	C₁₉	#7121	1	1	yes	characteristic, complemented, Sylow
129	C₂²×C₄	—	76	1	no	—
130	C₂²×C₄	—	76	1	no	—
131	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
132	C₂²×C₄	#64525	1	1	yes	characteristic
133	C₂²×C₄	#64527	1	1	yes	characteristic
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	8	1	no	—
140	C₂×C₈	—	8	1	no	—
141	C₂×C₈	C₈×D₁₉	1	1	yes	characteristic
142	C₂×C₈	#64515	1	1	yes	characteristic
143	C₂×C₈	#64516	1	1	yes	characteristic
144	C₂×C₈	Dic₇₆	1	1	yes	characteristic
145	C₄⋊C₄	—	38	2	no	—
146	C₄⋊C₄	—	76	1	no	—
147	C₄⋊C₄	—	76	1	no	—
148	C₄⋊C₄	—	76	1	no	—
149	C₄⋊C₄	—	152	2	no	—
150	C₄²	—	2	1	no	—
151	C₄²	—	2	1	no	—
152	C₁₆	—	76	2	no	—
153	C₂³	#10597266	1	1	yes	characteristic
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	76	2	no	—
160	C₂×C₄	—	152	1	no	—
161	C₂×C₄	—	152	1	no	—
162	C₂×C₄	—	152	1	no	—
163	C₂×C₄	—	152	1	no	—
164	C₂×C₄	—	152	1	no	—
165	C₂×C₄	#10597262	1	1	yes	characteristic
166	C₂×C₄	#10597263	1	1	yes	characteristic
167	C₂×C₄	#10597273	1	1	yes	characteristic
168	C₂×C₄	(C₂×D₁₉)⋊C₈	1	1	yes	characteristic
169	C₂×C₄	#10597281	1	1	yes	characteristic
170	C₂×C₄	#10597282	1	1	yes	characteristic
171	C₂×C₄	#10597284	1	1	yes	characteristic
172	C₂×C₄	#10597286	1	1	yes	characteristic
173	C₈	—	2	1	no	—
174	C₈	—	2	1	no	—
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	8	2	no	—
178	C₂²	#11926271	1	1	yes	characteristic
179	C₂²	#11926277	1	1	yes	characteristic
180	C₂²	#11926291	1	1	yes	characteristic
181	C₂²	#11926347	1	1	yes	characteristic
182	C₂²	#11926350	1	1	yes	characteristic
183	C₂²	#11926353	1	1	yes	characteristic
184	C₂²	#11926356	1	1	yes	characteristic
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	4	1	no	—
188	C₄	—	304	1	no	—
189	C₄	—	152	2	no	—
190	C₄	#11926269	1	1	yes	characteristic
191	C₄	#11926276	1	1	yes	characteristic
192	C₄	#11926282	1	1	yes	characteristic
193	C₄	#11926285	1	1	yes	characteristic
194	C₂	#14561459	1	1	yes	characteristic
195	C₂	#14561598	1	1	yes	characteristic
196	C₂	#14561601	1	1	yes	characteristic
197	C₂	#14561646	1	1	yes	characteristic
198	C₂	#14561649	1	1	yes	characteristic
199	C₂	#14561688	1	1	yes	characteristic
200	C₂	#14561694	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14564368

C₂

yes

maximal, characteristic

#14561713

C₂

yes

maximal, characteristic

C₁₉×#1277

C₂

yes

maximal, characteristic

C₂²×C₁₉⋊C₁₆

—

C₂×#10597275

C₄

yes

characteristic

C₂×#10597275

C₄

yes

characteristic

#11926442

—

#11926307

—

C₂×C₁₉×C₄⋊C₈

—

C₂×C₄×C₈×C₁₉

C₂²

yes

characteristic

C₂²×C₈×C₁₉

—

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

—

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

C₄×C₈×C₁₉

—

C₁₉⋊#95

—

#10597275

—

#10597275

—

#10597275

C₈

yes

complemented

C₂×C₁₉⋊C₁₆

—

C₂×C₁₉⋊C₁₆

—

C₂×C₁₉⋊C₁₆

—

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×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

—

C₁₉⋊C₁₆

—

—

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

SD₃₂

yes

characteristic

C₂×C₄×C₁₉

Q₃₂

yes

characteristic

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

#1942

—

#1374

—

#1277

D₁₉

yes

characteristic

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

—

C₂×#107

—

100

C₂×C₄⋊C₈

—

101

C₂×C₄×C₈

C₂×D₁₉

yes

characteristic

102

#389

—

103

#340

—

104

C₂×C₁₉

#1271

yes

characteristic

105

C₂×C₁₉

#1332

yes

characteristic

106

C₂×C₁₉

#1333

yes

characteristic

107

C₂×C₁₉

#1348

yes

characteristic

108

C₂×C₁₉

#1349

yes

characteristic

109

C₂×C₁₉

#1363

yes

characteristic

110

C₂×C₁₉

#1365

yes

characteristic

111

C₂²×C₈

—

112

C₂²×C₈

C₄×D₁₉

yes

characteristic

113

C₂²×C₈

D₇₆

yes

characteristic

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄²

C₂²⋊D₁₉

yes

characteristic

118

C₂×C₁₆

—

119

C₂×C₁₆

—

120

C₂×C₁₆

—

121

#107

—

122

#107

—

123

#107

—

124

C₄⋊C₈

—

125

C₄⋊C₈

—

126

C₄×C₈

—

127

#95

—

128

C₁₉

#7121

yes

characteristic, complemented, Sylow

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

132

C₂²×C₄

#64525

yes

characteristic

133

C₂²×C₄

#64527

yes

characteristic

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

C₈×D₁₉

yes

characteristic

142

C₂×C₈

#64515

yes

characteristic

143

C₂×C₈

#64516

yes

characteristic

144

C₂×C₈

Dic₇₆

yes

characteristic

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

152

—

150

C₄²

—

151

C₄²

—

152

C₁₆

—

153

C₂³

#10597266

yes

characteristic

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

152

—

161

C₂×C₄

—

152

—

162

C₂×C₄

—

152

—

163

C₂×C₄

—

152

—

164

C₂×C₄

—

152

—

165

C₂×C₄

#10597262

yes

characteristic

166

C₂×C₄

#10597263

yes

characteristic

167

C₂×C₄

#10597273

yes

characteristic

168

C₂×C₄

(C₂×D₁₉)⋊C₈

yes

characteristic

169

C₂×C₄

#10597281

yes

characteristic

170

C₂×C₄

#10597282

yes

characteristic

171

C₂×C₄

#10597284

yes

characteristic

172

C₂×C₄

#10597286

yes

characteristic

173

C₈

—

174

C₈

—

175

C₈

—

176

C₈

—

177

C₈

—

178

C₂²

#11926271

yes

characteristic

179

C₂²

#11926277

yes

characteristic

180

C₂²

#11926291

yes

characteristic

181

C₂²

#11926347

yes

characteristic

182

C₂²

#11926350

yes

characteristic

183

C₂²

#11926353

yes

characteristic

184

C₂²

#11926356

yes

characteristic

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

304

—

189

C₄

—

152

—

190

C₄

#11926269

yes

characteristic

191

C₄

#11926276

yes

characteristic

192

C₄

#11926282

yes

characteristic

193

C₄

#11926285

yes

characteristic

194

C₂

#14561459

yes

characteristic

195

C₂

#14561598

yes

characteristic

196

C₂

#14561601

yes

characteristic

197

C₂

#14561646

yes

characteristic

198

C₂

#14561649

yes

characteristic

199

C₂

#14561688

yes

characteristic

200

C₂

#14561694

yes

characteristic

	subgroup	G/H
center	C₂³	#10597266
commutator	C₂×C₈×C₁₉	C₂×C₈
Frattini	C₂×C₄×C₈	C₂×D₁₉
Fitting	C₁₉×#1277	C₂
socle, prime radical	C₂³×C₁₉	#102
cosocle	C₂×C₄×C₈×C₁₉	C₂²
hypercenter	#1277	D₁₉
hypocenter	C₁₉	#7121
LCS	G ▷ C₂×C₈×C₁₉ ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1277

subgroup

G/H

center

C₂³

#10597266

commutator

C₂×C₈×C₁₉

C₂×C₈

Frattini

C₂×C₄×C₈

C₂×D₁₉

Fitting

C₁₉×#1277

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 ◁ #1277

N	H ≅ G/N	type
24. #10597275	C₈	semidirect
128. C₁₉	#7121	semidirect

H ≅ G/N

type

24. #10597275

C₈

semidirect

128. C₁₉

#7121

semidirect

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

subgroup	index
#14564368	2
#14561713	2
C₁₉×#1277	2
#7121	19

subgroup

index

#14564368

#14561713

C₁₉×#1277

#7121

Subgroups — #18069331

Conjugacy classes of subgroups

Complements

Maximal subgroups