Group #18087328 — subgroups

Subgroups — #18087328

Order 4864.

362 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 93 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	#14564459	C₂	1	1	yes	maximal, characteristic
2	#14562304	C₂	1	2	yes	maximal
3	#14561495	C₂	1	2	yes	maximal, complemented
4	#14561480	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1849	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10597320	C₂²	1	1	yes	characteristic
7	#11926597	C₂²	1	2	yes	—
8	C₂×#10597278	—	2	1	no	—
9	C₂×#10597278	—	2	1	no	—
10	#11926410	C₂²	1	1	yes	characteristic
11	#11926287	—	2	4	no	—
12	C₂×C₁₉×#108	—	2	1	no	—
13	C₂×C₁₉×#108	—	2	1	no	—
14	C₁₉×#423	C₂²	1	2	yes	—
15	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
16	C₂²×D₇₆	—	2	1	no	—
17	C₂²×D₇₆	—	2	2	no	—
18	C₂×M₁₆×C₁₉	—	2	2	no	—
19	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₉	—	2	1	no	—
23	#10597320	—	2	2	no	—
24	#10597320	—	2	1	no	—
25	#10597320	D₄	1	2	yes	—
26	C₁₉×#108	—	4	2	no	—
27	C₁₉×#108	—	4	2	no	—
28	C₁₉×C₄⋊C₈	—	2	2	no	—
29	C₁₉×C₄⋊C₈	—	2	2	no	—
30	C₁₉×#97	—	2	2	no	—
31	C₁₉⋊#95	—	2	2	no	—
32	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
33	#10597278	—	4	2	no	—
34	#10597278	—	4	2	no	—
35	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
36	C₄×C₁₉⋊C₈	—	2	2	no	—
37	C₂³×D₁₉	—	4	1	no	—
38	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₉	#41	1	2	yes	—
40	C₂²×Dic₁₉	—	4	1	no	—
41	C₂×D₇₆	—	2	2	no	—
42	C₂×D₇₆	—	2	4	no	—
43	C₂×D₇₆	—	4	1	no	—
44	C₂×D₇₆	—	4	1	no	—
45	C₂×D₇₆	—	4	1	no	—
46	C₂×D₇₆	—	4	1	no	—
47	C₂×D₇₆	—	4	1	no	—
48	C₂×D₇₆	—	4	2	no	—
49	C₂×D₇₆	—	8	4	no	—
50	M₁₆×C₁₉	—	4	2	no	—
51	M₁₆×C₁₉	—	4	2	no	—
52	C₂×C₈×C₁₉	—	2	1	no	—
53	C₂×C₈×C₁₉	—	2	1	no	—
54	C₂×C₈×C₁₉	—	2	1	no	—
55	C₂×C₈×C₁₉	—	2	1	no	—
56	C₂×C₈×C₁₉	—	4	2	no	—
57	C₂×C₈×C₁₉	D₈	1	2	yes	—
58	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
59	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₁₉	#41	1	2	yes	—
61	C₄²×C₁₉	#41	1	1	yes	characteristic
62	C₄⋊Dic₁₉	—	4	1	no	—
63	C₄⋊Dic₁₉	—	4	1	no	—
64	C₂×C₁₉⋊C₈	—	4	2	no	—
65	#10631	—	19	1	no	maximal, Sylow
66	C₂³×C₁₉	#124	1	1	yes	characteristic
67	C₂²×D₁₉	—	4	2	no	—
68	C₂²×D₁₉	—	8	4	no	—
69	C₂²×D₁₉	—	8	1	no	—
70	C₂²×D₁₉	—	8	1	no	—
71	C₂×C₄×C₁₉	—	4	2	no	—
72	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
73	C₂×C₄×C₁₉	#123	1	2	yes	—
74	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
75	C₂×C₄×C₁₉	#126	1	2	yes	—
76	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
80	C₂×Dic₁₉	—	8	2	no	—
81	C₂×Dic₁₉	—	8	1	no	—
82	D₇₆	—	4	2	no	—
83	D₇₆	—	4	2	no	—
84	D₇₆	—	8	4	no	—
85	D₇₆	—	8	4	no	—
86	D₇₆	—	8	1	no	—
87	D₇₆	—	8	1	no	—
88	C₈×C₁₉	—	2	2	no	—
89	C₈×C₁₉	—	2	2	no	—
90	C₈×C₁₉	—	8	2	no	—
91	C₁₉⋊C₈	—	8	2	no	—
92	#1961	—	19	1	no	—
93	#1849	D₁₉	1	1	yes	characteristic, complemented
94	#1545	—	19	2	no	—
95	#1286	—	19	2	no	—
96	#1281	—	19	1	no	—
97	C₂²×C₁₉	—	2	2	no	—
98	C₂²×C₁₉	#397	1	1	yes	characteristic
99	C₂²×C₁₉	#486	1	1	yes	characteristic
100	C₂²×C₁₉	#489	1	1	yes	characteristic
101	C₂×D₁₉	—	8	2	no	—
102	C₂×D₁₉	—	8	4	no	—
103	C₂×D₁₉	—	16	2	no	—
104	C₂×D₁₉	—	16	2	no	—
105	C₄×C₁₉	—	2	2	no	—
106	C₄×C₁₉	—	2	2	no	—
107	C₄×C₁₉	#459	1	1	yes	characteristic
108	C₄×C₁₉	#461	1	1	yes	characteristic
109	C₄×C₁₉	#480	1	1	yes	characteristic
110	C₄×C₁₉	#481	1	1	yes	characteristic
111	Dic₁₉	—	16	1	no	—
112	C₂×#127	—	19	1	no	—
113	#431	—	19	2	no	—
114	C₂×#108	—	2	1	no	—
115	C₂×#108	—	2	1	no	—
116	#423	C₂×D₁₉	1	2	yes	—
117	C₂×#102	—	38	1	no	—
118	C₂×#102	—	38	1	no	—
119	C₂×#97	C₂×D₁₉	1	1	yes	characteristic
120	#382	—	19	1	no	—
121	#331	—	38	4	no	—
122	C₂×C₁₉	—	2	1	no	—
123	C₂×C₁₉	—	2	1	no	—
124	C₂×C₁₉	#2071	1	1	yes	characteristic
125	C₂×C₁₉	#2081	1	2	yes	—
126	D₁₉	—	16	2	no	—
127	C₂²×D₄	—	38	2	no	—
128	C₂²×D₄	—	38	1	no	—
129	C₂×M₁₆	—	2	2	no	—
130	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
131	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
132	#127	—	19	2	no	—
133	#127	—	38	2	no	—
134	#127	—	38	1	no	—
135	C₂×C₄⋊C₄	—	38	1	no	—
136	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
137	#108	—	4	2	no	—
138	#108	—	4	2	no	—
139	C₄⋊C₈	—	2	2	no	—
140	C₄⋊C₈	—	2	2	no	—
141	C₄⋊C₈	—	38	2	no	—
142	#102	—	76	2	no	—
143	#102	—	76	2	no	—
144	C₂²⋊C₈	—	38	2	no	—
145	#97	—	2	2	no	—
146	C₄×C₈	—	38	2	no	—
147	#95	—	38	2	no	—
148	C₁₉	#10631	1	1	yes	characteristic, complemented, Sylow
149	C₂⁴	—	76	1	no	—
150	C₂×D₄	—	38	2	no	—
151	C₂×D₄	—	38	4	no	—
152	C₂×D₄	—	76	1	no	—
153	C₂×D₄	—	76	1	no	—
154	C₂×D₄	—	76	1	no	—
155	C₂×D₄	—	76	1	no	—
156	C₂×D₄	—	76	2	no	—
157	C₂×D₄	—	76	1	no	—
158	C₂×D₄	—	152	4	no	—
159	C₂²×C₄	—	76	1	no	—
160	C₂²×C₄	#64545	1	2	yes	—
161	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
162	M₁₆	—	4	2	no	—
163	M₁₆	—	4	2	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	76	2	no	—
170	C₂×C₈	#64525	1	2	yes	—
171	C₂×C₈	#64527	1	2	yes	—
172	C₄⋊C₄	—	76	1	no	—
173	C₄⋊C₄	—	76	1	no	—
174	C₄²	#64541	1	2	yes	—
175	C₄²	D₄×D₁₉	1	1	yes	characteristic
176	C₄²	#64543	1	1	yes	characteristic
177	C₂³	—	76	2	no	—
178	C₂³	—	152	4	no	—
179	C₂³	—	152	1	no	—
180	C₂³	—	152	1	no	—
181	C₂³	#10597393	1	1	yes	characteristic
182	D₄	—	76	2	no	—
183	D₄	—	76	2	no	—
184	D₄	—	152	4	no	—
185	D₄	—	152	4	no	—
186	D₄	—	152	1	no	—
187	D₄	—	152	1	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	152	2	no	—
190	C₂×C₄	—	152	1	no	—
191	C₂×C₄	#10597340	1	2	yes	—
192	C₂×C₄	#10597344	1	2	yes	—
193	C₂×C₄	#10597375	1	1	yes	characteristic
194	C₂×C₄	C₂×#64525	1	1	yes	characteristic
195	C₂×C₄	#10597378	1	1	yes	characteristic
196	C₂×C₄	#10597384	1	1	yes	characteristic
197	C₂×C₄	C₂×#64527	1	1	yes	characteristic
198	C₂×C₄	#10597388	1	1	yes	characteristic
199	C₈	—	2	2	no	—
200	C₈	—	2	2	no	—
201	C₈	—	8	2	no	—
202	C₈	—	152	2	no	—
203	C₂²	—	2	2	no	—
204	C₂²	—	304	2	no	—
205	C₂²	—	304	2	no	—
206	C₂²	—	152	2	no	—
207	C₂²	—	152	4	no	—
208	C₂²	#11926470	1	1	yes	characteristic
209	C₂²	#11926892	1	1	yes	characteristic
210	C₂²	#11926906	1	1	yes	characteristic
211	C₄	—	2	2	no	—
212	C₄	—	2	2	no	—
213	C₄	—	304	1	no	—
214	C₄	#11926736	1	1	yes	characteristic
215	C₄	#11926746	1	1	yes	characteristic
216	C₄	#11926857	1	1	yes	characteristic
217	C₄	#11926862	1	1	yes	characteristic
218	C₂	—	2	1	no	—
219	C₂	—	2	1	no	—
220	C₂	—	304	2	no	—
221	C₂	#14565039	1	1	yes	characteristic
222	C₂	#14565093	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14564459

C₂

yes

maximal, characteristic

#14562304

C₂

yes

maximal

#14561495

C₂

yes

maximal, complemented

#14561480

C₂

yes

maximal, characteristic, complemented

C₁₉×#1849

C₂

yes

maximal, characteristic, complemented

C₂×#10597320

C₂²

yes

characteristic

#11926597

C₂²

yes

—

C₂×#10597278

—

C₂×#10597278

—

#11926410

C₂²

yes

characteristic

#11926287

—

C₂×C₁₉×#108

—

C₂×C₁₉×#108

—

C₁₉×#423

C₂²

yes

—

C₂×C₁₉×#97

C₂²

yes

characteristic

C₂²×D₇₆

—

C₂²×D₇₆

—

C₂×M₁₆×C₁₉

—

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₉

—

#10597320

—

#10597320

—

#10597320

D₄

yes

—

C₁₉×#108

—

C₁₉×#108

—

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

C₁₉×#97

—

C₁₉⋊#95

—

C₂²⋊(C₁₉⋊C₈)

—

#10597278

—

#10597278

—

C₄⋊(C₁₉⋊C₈)

—

C₄×C₁₉⋊C₈

—

C₂³×D₁₉

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₉

#41

yes

—

C₂²×Dic₁₉

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

D₈

yes

—

C₂×C₈×C₁₉

SD₁₆

yes

—

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄²×C₁₉

#41

yes

—

C₄²×C₁₉

#41

yes

characteristic

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₂×C₁₉⋊C₈

—

—

C₂³×C₁₉

yes

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#121

yes

characteristic

C₂×C₄×C₁₉

#123

yes

—

C₂×C₄×C₁₉

#123

yes

characteristic

C₂×C₄×C₁₉

#126

yes

—

C₂×C₄×C₁₉

C₂×D₈

yes

characteristic

C₂×C₄×C₁₉

C₂×SD₁₆

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×Dic₁₉

—

C₂×Dic₁₉

—

D₇₆

—

D₇₆

—

D₇₆

—

D₇₆

—

D₇₆

—

D₇₆

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₁₉⋊C₈

—

#1961

—

#1849

D₁₉

yes

characteristic, complemented

#1545

—

#1286

—

#1281

—

C₂²×C₁₉

—

C₂²×C₁₉

yes

C₂²×C₁₉

yes

100

C₂²×C₁₉

#489

yes

characteristic

101

C₂×D₁₉

—

102

C₂×D₁₉

—

103

C₂×D₁₉

—

104

C₂×D₁₉

—

105

C₄×C₁₉

—

106

C₄×C₁₉

—

107

C₄×C₁₉

#459

yes

characteristic

108

C₄×C₁₉

#461

yes

characteristic

109

C₄×C₁₉

#480

yes

characteristic

110

C₄×C₁₉

#481

yes

characteristic

111

Dic₁₉

—

112

C₂×#127

—

113

#431

—

114

C₂×#108

—

115

C₂×#108

—

116

#423

C₂×D₁₉

yes

—

117

C₂×#102

—

118

C₂×#102

—

119

C₂×#97

C₂×D₁₉

yes

characteristic

120

#382

—

121

#331

—

122

C₂×C₁₉

—

123

C₂×C₁₉

—

124

C₂×C₁₉

#2071

yes

characteristic

125

C₂×C₁₉

#2081

yes

—

126

D₁₉

—

127

C₂²×D₄

—

128

C₂²×D₄

—

129

C₂×M₁₆

—

130

C₂²×C₈

C₂²⋊D₁₉

yes

characteristic

131

C₂²×C₈

C₂²⋊D₁₉

yes

characteristic

132

#127

—

133

#127

—

134

#127

—

135

C₂×C₄⋊C₄

—

136

C₂×C₄²

C₂²×D₁₉

yes

characteristic

137

#108

—

138

#108

—

139

C₄⋊C₈

—

140

C₄⋊C₈

—

141

C₄⋊C₈

—

142

#102

—

143

#102

—

144

C₂²⋊C₈

—

145

#97

—

146

C₄×C₈

—

147

#95

—

148

C₁₉

#10631

yes

characteristic, complemented, Sylow

149

C₂⁴

—

150

C₂×D₄

—

151

C₂×D₄

—

152

C₂×D₄

—

153

C₂×D₄

—

154

C₂×D₄

—

155

C₂×D₄

—

156

C₂×D₄

—

157

C₂×D₄

—

158

C₂×D₄

—

152

—

159

C₂²×C₄

—

160

C₂²×C₄

#64545

yes

—

161

C₂²×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

162

M₁₆

—

163

M₁₆

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

#64525

yes

—

171

C₂×C₈

#64527

yes

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄²

#64541

yes

—

175

C₄²

D₄×D₁₉

yes

characteristic

176

C₄²

#64543

yes

characteristic

177

C₂³

—

178

C₂³

—

152

—

179

C₂³

—

152

—

180

C₂³

—

152

—

181

C₂³

#10597393

yes

characteristic

182

D₄

—

183

D₄

—

184

D₄

—

152

—

185

D₄

—

152

—

186

D₄

—

152

—

187

D₄

—

152

—

188

C₂×C₄

—

189

C₂×C₄

—

152

—

190

C₂×C₄

—

152

—

191

C₂×C₄

#10597340

yes

—

192

C₂×C₄

#10597344

yes

—

193

C₂×C₄

#10597375

yes

characteristic

194

C₂×C₄

C₂×#64525

yes

characteristic

195

C₂×C₄

#10597378

yes

characteristic

196

C₂×C₄

#10597384

yes

characteristic

197

C₂×C₄

C₂×#64527

yes

characteristic

198

C₂×C₄

#10597388

yes

characteristic

199

C₈

—

200

C₈

—

201

C₈

—

202

C₈

—

152

—

203

C₂²

—

204

C₂²

—

304

—

205

C₂²

—

304

—

206

C₂²

—

152

—

207

C₂²

—

152

—

208

C₂²

#11926470

yes

characteristic

209

C₂²

#11926892

yes

characteristic

210

C₂²

#11926906

yes

characteristic

211

C₄

—

212

C₄

—

213

C₄

—

304

—

214

C₄

#11926736

yes

characteristic

215

C₄

#11926746

yes

characteristic

216

C₄

#11926857

yes

characteristic

217

C₄

#11926862

yes

characteristic

218

C₂

—

219

C₂

—

220

C₂

—

304

—

221

C₂

#14565039

yes

characteristic

222

C₂

#14565093

yes

—

	subgroup	G/H
center	C₂²	#11926470
commutator, cosocle	C₂×C₄²×C₁₉	C₂³
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1849	C₂
socle	C₂²×C₁₉	#397
prime radical	C₂×#10597320	C₂²
hypercenter	#1849	D₁₉
hypocenter	C₁₉	#10631
LCS	G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1849

subgroup

G/H

center

C₂²

#11926470

commutator, cosocle

C₂×C₄²×C₁₉

C₂³

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#1849

C₂

C₂²×C₁₉

C₂²

D₁₉

C₁₉

G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1849

N	H ≅ G/N	type
3. #14561495	C₂	semidirect
4. #14561480	C₂	semidirect
5. C₁₉×#1849	C₂	semidirect
93. #1849	D₁₉	semidirect
148. C₁₉	#10631	semidirect

H ≅ G/N

type

3. #14561495

C₂

semidirect

4. #14561480

C₂

semidirect

5. C₁₉×#1849

C₂

semidirect

93. #1849

D₁₉

semidirect

148. C₁₉

#10631

semidirect

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

subgroup	index
#14564459	2
#14562304	2
#14561495	2
#14561480	2
C₁₉×#1849	2
#10631	19

subgroup

index

C₁₉×#1849

Subgroups — #18087328

Conjugacy classes of subgroups

Complements

Maximal subgroups