Group #18081922 — subgroups

Subgroups — #18081922

Order 4864.

400 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 115 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	#14573958	C₂	1	1	yes	maximal, characteristic
2	#14563766	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562200	C₂	1	2	yes	maximal, complemented
4	#14562196	C₂	1	2	yes	maximal
5	C₁₉×#1450	C₂	1	1	yes	maximal, characteristic, complemented
6	#11927101	C₂²	1	1	yes	characteristic
7	#11927094	C₂²	1	1	yes	characteristic
8	#11926899	—	2	1	no	—
9	#11926899	—	2	1	no	—
10	#11926895	C₄	1	2	yes	complemented
11	#11926891	C₄	1	2	yes	complemented
12	#11926541	—	2	2	no	—
13	#11926410	C₂²	1	2	yes	—
14	#11926307	—	2	2	no	—
15	#11926282	—	2	4	no	—
16	#11926279	—	2	4	no	—
17	C₁₉×#431	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
19	C₁₉×#322	—	2	1	no	—
20	C₈²×C₁₉	—	2	1	no	—
21	C₂×#64541	—	2	1	no	—
22	C₂²×C₈×C₁₉	D₄	1	2	yes	—
23	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
24	#10597376	—	2	2	no	—
25	#10597372	—	2	1	no	—
26	C₂×C₄⋊Dic₁₉	—	2	2	no	—
27	#10597370	—	2	2	no	—
28	#10597321	—	2	1	no	—
29	#10597320	C₂×C₄	1	1	yes	characteristic
30	C₄×D₇₆	—	2	2	no	—
31	#10597316	—	2	1	no	—
32	#10597315	C₂×C₄	1	1	yes	characteristic
33	#10597315	C₂×C₄	1	2	yes	—
34	C₄×Dic₃₈	—	2	2	no	—
35	C₁₉×C₄⋊C₈	—	2	2	no	—
36	C₁₉×C₂²⋊C₈	D₄	1	4	yes	—
37	C₄×C₈×C₁₉	—	2	2	no	—
38	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
40	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
41	C₁₉⋊#95	—	2	2	no	—
42	C₁₉⋊#95	—	2	2	no	—
43	#10597278	—	2	2	no	—
44	#10597278	—	2	2	no	—
45	#10597273	—	2	2	no	—
46	#10597273	—	2	2	no	—
47	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₂²⋊D₁₉	—	4	1	no	—
50	C₂²×Dic₁₉	—	4	2	no	—
51	#64541	—	4	4	no	—
52	C₂×D₇₆	—	2	1	no	—
53	C₂×D₇₆	—	2	2	no	—
54	C₂×C₄×D₁₉	—	4	1	no	—
55	C₂×Dic₃₈	—	2	1	no	—
56	C₂×Dic₃₈	—	2	2	no	—
57	C₂×C₈×C₁₉	—	2	4	no	—
58	C₂×C₈×C₁₉	—	2	4	no	—
59	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
61	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₁₉	#41	1	2	yes	—
64	C₂²⋊Dic₁₉	—	4	1	no	—
65	(C₂×D₁₉)⋊C₄	—	4	2	no	—
66	C₄⋊Dic₁₉	—	2	2	no	—
67	C₄⋊Dic₁₉	—	2	2	no	—
68	C₄⋊Dic₁₉	—	2	2	no	—
69	C₄⋊Dic₁₉	—	4	4	no	—
70	Dic₁₉⋊C₄	—	4	2	no	—
71	C₄×Dic₁₉	—	4	1	no	—
72	#9757	—	19	1	no	maximal, Sylow
73	C₂³×C₁₉	#127	1	1	yes	characteristic
74	C₂²×D₁₉	—	4	1	no	—
75	C₂×C₄×C₁₉	—	4	2	no	—
76	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
78	C₂×C₄×C₁₉	#121	1	2	yes	—
79	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
80	C₂×C₄×C₁₉	#135	1	4	yes	—
81	C₂²⋊D₁₉	—	8	2	no	—
82	C₂×Dic₁₉	—	4	2	no	—
83	C₂×Dic₁₉	—	4	1	no	—
84	C₂×Dic₁₉	—	8	4	no	—
85	C₂×Dic₁₉	—	8	2	no	—
86	D₇₆	—	4	2	no	—
87	D₇₆	—	8	2	no	—
88	C₄×D₁₉	—	8	2	no	—
89	Dic₃₈	—	4	2	no	—
90	Dic₃₈	—	8	2	no	—
91	C₈×C₁₉	—	2	2	no	—
92	C₈×C₁₉	—	2	2	no	—
93	C₈×C₁₉	—	4	4	no	—
94	#3131	—	19	1	no	—
95	#1835	—	19	1	no	—
96	#1518	—	19	2	no	—
97	#1517	—	19	2	no	—
98	#1450	D₁₉	1	1	yes	characteristic, complemented
99	C₂²×C₁₉	—	2	2	no	—
100	C₂²×C₁₉	#390	1	1	yes	characteristic
101	C₂²×C₁₉	#495	1	2	yes	—
102	C₂×D₁₉	—	8	1	no	—
103	C₂×D₁₉	—	8	2	no	—
104	C₄×C₁₉	—	2	2	no	—
105	C₄×C₁₉	—	2	2	no	—
106	C₄×C₁₉	#416	1	2	yes	—
107	C₄×C₁₉	#488	1	2	yes	—
108	Dic₁₉	—	8	1	no	—
109	Dic₁₉	—	8	2	no	—
110	Dic₁₉	—	16	2	no	—
111	#533	—	19	1	no	—
112	#532	—	19	1	no	—
113	#488	—	38	1	no	—
114	#488	—	38	1	no	—
115	#487	—	19	2	no	—
116	#486	—	19	2	no	—
117	#431	C₂×D₁₉	1	2	yes	—
118	#416	—	38	2	no	—
119	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
120	#382	—	19	2	no	—
121	#340	—	38	2	no	—
122	#328	—	38	4	no	—
123	#327	—	38	4	no	—
124	#322	—	2	1	no	—
125	C₈²	—	2	1	no	—
126	C₂×C₁₉	—	2	2	no	—
127	C₂×C₁₉	#1958	1	1	yes	characteristic
128	C₂×C₁₉	#1976	1	2	yes	—
129	D₁₉	—	16	1	no	—
130	C₂×#41	—	38	1	no	—
131	C₂²×C₈	D₇₆	1	2	yes	—
132	C₄⋊Q₈	—	19	1	no	—
133	C₄⋊Q₈	—	19	2	no	—
134	#127	—	19	1	no	—
135	#125	—	38	1	no	—
136	#124	—	38	1	no	—
137	C₂²⋊Q₈	—	38	2	no	—
138	#121	—	38	2	no	—
139	C₄×Q₈	—	38	2	no	—
140	C₄×D₄	—	38	2	no	—
141	#117	—	38	1	no	—
142	C₂×C₄⋊C₄	—	38	2	no	—
143	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
144	C₄⋊C₈	—	2	2	no	—
145	#103	—	38	2	no	—
146	#103	—	38	2	no	—
147	#102	—	38	2	no	—
148	#102	—	38	2	no	—
149	C₂²⋊C₈	D₇₆	1	4	yes	—
150	C₄×C₈	—	2	2	no	—
151	C₄×C₈	C₄×D₁₉	1	2	yes	—
152	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
153	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
154	#95	—	38	2	no	—
155	#95	—	38	2	no	—
156	C₁₉	#9757	1	1	yes	characteristic, complemented, Sylow
157	#41	—	76	4	no	—
158	C₂×Q₈	—	38	2	no	—
159	C₂×Q₈	—	38	1	no	—
160	C₂×D₄	—	38	2	no	—
161	C₂×D₄	—	38	1	no	—
162	C₂×D₄	—	76	1	no	—
163	C₂²×C₄	—	76	1	no	—
164	C₂²×C₄	—	76	2	no	—
165	C₂²×C₄	C₂×D₇₆	1	2	yes	—
166	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
170	C₂×C₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
171	C₄⋊C₄	—	38	2	no	—
172	C₄⋊C₄	—	38	2	no	—
173	C₄⋊C₄	—	38	2	no	—
174	C₄⋊C₄	—	76	2	no	—
175	C₄⋊C₄	—	76	4	no	—
176	C₂²⋊C₄	—	76	1	no	—
177	C₂²⋊C₄	—	76	2	no	—
178	C₄²	—	76	1	no	—
179	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
180	C₄²	#64541	1	2	yes	—
181	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
182	C₂³	—	76	1	no	—
183	C₂³	#10597320	1	1	yes	characteristic
184	Q₈	—	76	2	no	—
185	Q₈	—	152	2	no	—
186	D₄	—	76	2	no	—
187	D₄	—	152	2	no	—
188	D₄	—	152	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	76	2	no	—
191	C₂×C₄	—	76	1	no	—
192	C₂×C₄	—	152	4	no	—
193	C₂×C₄	—	152	2	no	—
194	C₂×C₄	—	152	2	no	—
195	C₂×C₄	C₄×D₇₆	1	2	yes	—
196	C₂×C₄	#10597321	1	1	yes	characteristic
197	C₂×C₄	#10597350	1	4	yes	—
198	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
199	C₂×C₄	#10597376	1	2	yes	—
200	C₈	—	2	2	no	—
201	C₈	—	2	2	no	—
202	C₈	—	4	4	no	—
203	C₂²	—	2	2	no	—
204	C₂²	—	152	1	no	—
205	C₂²	—	152	2	no	—
206	C₂²	#11926446	1	1	yes	characteristic
207	C₂²	#11926928	1	2	yes	—
208	C₄	—	2	2	no	—
209	C₄	—	2	2	no	—
210	C₄	—	304	2	no	—
211	C₄	—	152	1	no	—
212	C₄	—	152	2	no	—
213	C₄	#11926541	1	2	yes	—
214	C₄	#11926899	1	2	yes	—
215	C₂	—	2	2	no	—
216	C₂	—	304	1	no	—
217	C₂	#14564446	1	1	yes	characteristic
218	C₂	#14564533	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14573958

C₂

yes

maximal, characteristic

#14563766

C₂

yes

maximal, characteristic, complemented

#14562200

C₂

yes

maximal, complemented

#14562196

C₂

yes

maximal

C₁₉×#1450

C₂

yes

maximal, characteristic, complemented

#11927101

C₂²

yes

characteristic

#11927094

C₂²

yes

—

—

C₄

yes

complemented

#11926891

C₄

yes

complemented

#11926541

—

#11926410

C₂²

yes

—

#11926307

—

#11926282

—

#11926279

—

C₁₉×#431

C₂²

yes

—

C₂×C₄×C₈×C₁₉

C₂²

yes

characteristic

C₁₉×#322

—

C₈²×C₁₉

—

C₂×#64541

—

C₂²×C₈×C₁₉

D₄

yes

—

C₂×C₄²×C₁₉

C₂³

yes

characteristic

#10597376

—

#10597372

—

C₂×C₄⋊Dic₁₉

—

#10597370

—

#10597321

—

#10597320

C₂×C₄

yes

characteristic

C₄×D₇₆

—

#10597316

—

#10597315

C₂×C₄

yes

characteristic

#10597315

C₂×C₄

yes

—

C₄×Dic₃₈

—

C₁₉×C₄⋊C₈

—

C₁₉×C₂²⋊C₈

D₄

yes

—

C₄×C₈×C₁₉

—

C₄×C₈×C₁₉

C₂×C₄

yes

—

C₄×C₈×C₁₉

D₄

yes

characteristic

C₄×C₈×C₁₉

D₄

yes

characteristic

C₁₉⋊#95

—

C₁₉⋊#95

—

—

—

—

—

C₂²×C₄×C₁₉

C₂×D₄

yes

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂×C₂²⋊D₁₉

—

C₂²×Dic₁₉

—

#64541

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×C₄×D₁₉

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₂²⋊C₄

yes

—

C₂×C₈×C₁₉

C₂²⋊C₄

yes

—

C₄²×C₁₉

C₂²×C₄

yes

characteristic

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄²×C₁₉

#41

yes

—

C₂²⋊Dic₁₉

—

(C₂×D₁₉)⋊C₄

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

Dic₁₉⋊C₄

—

C₄×Dic₁₉

—

—

C₂³×C₁₉

yes

C₂²×D₁₉

—

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

#135

yes

—

C₂²⋊D₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

D₇₆

—

D₇₆

—

C₄×D₁₉

—

Dic₃₈

—

Dic₃₈

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

—

—

—

—

D₁₉

yes

characteristic, complemented

C₂²×C₁₉

—

100

C₂²×C₁₉

#390

yes

characteristic

101

C₂²×C₁₉

#495

yes

—

102

C₂×D₁₉

—

103

C₂×D₁₉

—

104

C₄×C₁₉

—

105

C₄×C₁₉

—

106

C₄×C₁₉

#416

yes

—

107

C₄×C₁₉

#488

yes

—

108

Dic₁₉

—

109

Dic₁₉

—

110

Dic₁₉

—

111

#533

—

112

#532

—

113

#488

—

114

#488

—

115

#487

—

116

#486

—

117

#431

C₂×D₁₉

yes

—

118

#416

—

119

C₂×C₄×C₈

C₂×D₁₉

yes

characteristic

120

#382

—

121

#340

—

122

#328

—

123

#327

—

124

#322

—

125

C₈²

—

126

C₂×C₁₉

—

127

C₂×C₁₉

#1958

yes

characteristic

128

C₂×C₁₉

#1976

yes

—

129

D₁₉

—

130

C₂×#41

—

131

C₂²×C₈

D₇₆

yes

—

132

C₄⋊Q₈

—

133

C₄⋊Q₈

—

134

#127

—

135

#125

—

136

#124

—

137

C₂²⋊Q₈

—

138

#121

—

139

C₄×Q₈

—

140

C₄×D₄

—

141

#117

—

142

C₂×C₄⋊C₄

—

143

C₂×C₄²

C₂²×D₁₉

yes

characteristic

144

C₄⋊C₈

—

145

#103

—

146

#103

—

147

#102

—

148

#102

—

149

C₂²⋊C₈

D₇₆

yes

—

150

C₄×C₈

—

151

C₄×C₈

C₄×D₁₉

yes

—

152

C₄×C₈

C₂²⋊D₁₉

yes

characteristic

153

C₄×C₈

C₂²⋊D₁₉

yes

characteristic

154

#95

—

155

#95

—

156

C₁₉

#9757

yes

characteristic, complemented, Sylow

157

#41

—

158

C₂×Q₈

—

159

C₂×Q₈

—

160

C₂×D₄

—

161

C₂×D₄

—

162

C₂×D₄

—

163

C₂²×C₄

—

164

C₂²×C₄

—

165

C₂²×C₄

C₂×D₇₆

yes

—

166

C₂²×C₄

C₂×D₇₆

yes

characteristic

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

(C₂×D₁₉)⋊C₄

yes

—

170

C₂×C₈

(C₂×D₁₉)⋊C₄

yes

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

—

175

C₄⋊C₄

—

176

C₂²⋊C₄

—

177

C₂²⋊C₄

—

178

C₄²

—

179

C₄²

C₂×C₄×D₁₉

yes

characteristic

180

C₄²

#64541

yes

—

181

C₄²

C₂×C₂²⋊D₁₉

yes

characteristic

182

C₂³

—

183

C₂³

#10597320

yes

characteristic

184

Q₈

—

185

Q₈

—

152

—

186

D₄

—

187

D₄

—

152

—

188

D₄

—

152

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

152

—

193

C₂×C₄

—

152

—

194

C₂×C₄

—

152

—

195

C₂×C₄

C₄×D₇₆

yes

—

196

C₂×C₄

#10597321

yes

characteristic

197

C₂×C₄

#10597350

yes

—

198

C₂×C₄

C₂×(C₂×D₁₉)⋊C₄

yes

characteristic

199

C₂×C₄

#10597376

yes

—

200

C₈

—

201

C₈

—

202

C₈

—

203

C₂²

—

204

C₂²

—

152

—

205

C₂²

—

152

—

206

C₂²

#11926446

yes

characteristic

207

C₂²

#11926928

yes

—

208

C₄

—

209

C₄

—

210

C₄

—

304

—

211

C₄

—

152

—

212

C₄

—

152

—

213

C₄

#11926541

yes

—

214

C₄

#11926899

yes

—

215

C₂

—

216

C₂

—

304

—

217

C₂

#14564446

yes

characteristic

218

C₂

#14564533

yes

—

	subgroup	G/H
center	C₂²	#11926446
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1450	C₂
socle	C₂²×C₁₉	#390
prime radical	#11927094	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1450	D₁₉
hypocenter	C₁₉	#9757
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

subgroup

G/H

center

C₂²

#11926446

commutator

C₄²×C₁₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#1450

C₂

C₂²×C₁₉

C₂²

C₂×C₄²×C₁₉

C₂³

D₁₉

C₁₉

G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

N	H ≅ G/N	type
2. #14563766	C₂	semidirect
3. #14562200	C₂	semidirect
5. C₁₉×#1450	C₂	semidirect
10. #11926895	C₄	semidirect
11. #11926891	C₄	semidirect
98. #1450	D₁₉	semidirect
156. C₁₉	#9757	semidirect

H ≅ G/N

type

2. #14563766

C₂

semidirect

3. #14562200

C₂

semidirect

5. C₁₉×#1450

C₂

semidirect

10. #11926895

C₄

semidirect

11. #11926891

C₄

semidirect

98. #1450

D₁₉

semidirect

156. C₁₉

#9757

semidirect

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

subgroup	index
#14573958	2
#14563766	2
#14562200	2
#14562196	2
C₁₉×#1450	2
#9757	19

subgroup

index

C₁₉×#1450

Subgroups — #18081922

Conjugacy classes of subgroups

Complements

Maximal subgroups