Group #18086708 — subgroups

Subgroups — #18086708

Order 4864.

402 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 85 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	#14563806	C₂	1	1	yes	maximal, characteristic, complemented
2	#14562370	C₂	1	2	yes	maximal, complemented
3	#14562086	C₂	1	2	yes	maximal, complemented
4	C₁₉⋊#1309	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#3253	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597279	—	2	2	no	—
7	#11926568	C₂²	1	2	yes	—
8	#11926568	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
10	#11926293	—	2	2	no	—
11	#11926288	—	2	2	no	—
12	D₄⋊(C₁₉⋊C₈)	—	2	4	no	—
13	C₁₉×#532	C₂²	1	2	yes	—
14	C₁₉×#476	—	2	2	no	—
15	C₁₉×#469	—	2	2	no	—
16	C₁₉×#459	—	2	2	no	—
17	C₁₉×#423	C₂²	1	1	yes	characteristic
18	C₂×#41×C₁₉	—	2	2	no	—
19	C₂×D₈×C₁₉	—	2	2	no	—
20	C₂×M₁₆×C₁₉	D₄	1	2	yes	—
21	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
22	C₁₉×#127	D₄	1	2	yes	—
23	C₁₉×#124	—	2	2	no	—
24	C₁₉×#121	—	2	2	no	—
25	C₁₉×#121	—	2	2	no	—
26	C₄×D₄×C₁₉	—	2	2	no	—
27	C₄×D₄×C₁₉	—	2	2	no	—
28	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
29	C₂×#64520	—	2	1	no	—
30	C₂×#64520	—	2	2	no	—
31	C₂²×C₁₉⋊C₈	—	2	1	no	—
32	C₁₉×#106	—	2	2	no	—
33	C₁₉×C₄⋊C₈	—	2	2	no	—
34	C₁₉×#102	—	2	4	no	—
35	#10597279	—	4	4	no	—
36	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
37	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
38	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
39	C₄×C₁₉⋊C₈	—	2	1	no	—
40	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
41	#41×C₁₉	—	8	2	no	—
42	#41×C₁₉	—	8	2	no	—
43	C₂×Q₈×C₁₉	—	2	2	no	—
44	C₂×D₄×C₁₉	—	2	2	no	—
45	C₂×D₄×C₁₉	—	2	2	no	—
46	C₂×D₄×C₁₉	—	2	2	no	—
47	C₂×D₄×C₁₉	—	4	2	no	—
48	C₂²×C₄×C₁₉	—	4	2	no	—
49	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	#41	1	2	yes	—
51	D₈×C₁₉	—	8	2	no	—
52	M₁₆×C₁₉	—	4	2	no	—
53	C₂×C₈×C₁₉	—	2	2	no	—
54	C₄⋊C₄×C₁₉	—	2	2	no	—
55	C₄⋊C₄×C₁₉	—	2	2	no	—
56	C₂²⋊C₄×C₁₉	—	4	2	no	—
57	C₂²⋊C₄×C₁₉	—	4	2	no	—
58	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₁₉	C₂×D₄	1	2	yes	—
60	C₄²×C₁₉	#41	1	1	yes	characteristic
61	#64520	—	4	2	no	—
62	#64520	—	8	2	no	—
63	C₂×C₁₉⋊C₈	—	2	2	no	—
64	C₂×C₁₉⋊C₈	—	2	2	no	—
65	C₂×C₁₉⋊C₈	—	2	2	no	—
66	C₂×C₁₉⋊C₈	—	4	1	no	—
67	C₂×C₁₉⋊C₈	—	4	2	no	—
68	#10530	—	19	1	no	maximal, Sylow
69	C₂³×C₁₉	—	4	2	no	—
70	C₂³×C₁₉	#124	1	1	yes	characteristic
71	Q₈×C₁₉	—	8	2	no	—
72	D₄×C₁₉	—	4	2	no	—
73	D₄×C₁₉	—	4	2	no	—
74	D₄×C₁₉	—	4	2	no	—
75	D₄×C₁₉	—	4	2	no	—
76	D₄×C₁₉	—	8	2	no	—
77	D₄×C₁₉	—	8	4	no	—
78	C₂×C₄×C₁₉	—	2	1	no	—
79	C₂×C₄×C₁₉	—	2	1	no	—
80	C₂×C₄×C₁₉	—	2	2	no	—
81	C₂×C₄×C₁₉	—	2	2	no	—
82	C₂×C₄×C₁₉	—	4	2	no	—
83	C₂×C₄×C₁₉	—	8	2	no	—
84	C₂×C₄×C₁₉	—	8	2	no	—
85	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
86	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
87	C₂×C₄×C₁₉	#121	1	2	yes	—
88	C₂×C₄×C₁₉	#123	1	2	yes	—
89	C₈×C₁₉	—	4	2	no	—
90	C₁₉⋊C₈	—	2	4	no	—
91	C₁₉⋊C₈	—	8	1	no	—
92	C₁₉⋊C₈	—	8	2	no	—
93	#3253	D₁₉	1	1	yes	characteristic
94	#1848	—	19	1	no	—
95	#1560	—	19	2	no	—
96	#1492	—	19	2	no	—
97	#1309	—	19	1	no	—
98	C₂²×C₁₉	—	2	2	no	—
99	C₂²×C₁₉	—	2	1	no	—
100	C₂²×C₁₉	—	8	2	no	—
101	C₂²×C₁₉	—	8	2	no	—
102	C₂²×C₁₉	—	8	2	no	—
103	C₂²×C₁₉	#394	1	1	yes	characteristic
104	C₄×C₁₉	—	2	2	no	—
105	C₄×C₁₉	—	2	1	no	—
106	C₄×C₁₉	—	8	2	no	—
107	C₄×C₁₉	#453	1	1	yes	characteristic
108	C₄×C₁₉	#456	1	1	yes	characteristic
109	C₄×C₁₉	#471	1	2	yes	—
110	C₄×C₁₉	#472	1	2	yes	—
111	#532	C₂×D₁₉	1	2	yes	—
112	#476	—	2	2	no	—
113	#469	—	2	2	no	—
114	#459	—	2	2	no	—
115	C₂×#108	—	38	2	no	—
116	#423	—	19	1	no	—
117	#423	—	19	2	no	—
118	#423	C₂×D₁₉	1	1	yes	characteristic
119	C₂×C₄×C₈	—	19	1	no	—
120	#333	—	38	2	no	—
121	#331	—	38	2	no	—
122	D₄⋊C₈	—	38	4	no	—
123	C₂×C₁₉	—	4	1	no	—
124	C₂×C₁₉	—	16	2	no	—
125	C₂×C₁₉	#2039	1	1	yes	characteristic
126	C₂×C₁₉	#2051	1	2	yes	—
127	C₂×#41	—	2	2	no	—
128	C₂×D₈	—	2	2	no	—
129	C₂×M₁₆	—	38	1	no	—
130	C₂×M₁₆	—	38	2	no	—
131	C₂×M₁₆	C₂²⋊D₁₉	1	2	yes	—
132	C₂²×C₈	—	38	1	no	—
133	C₄⋊Q₈	C₂²⋊D₁₉	1	2	yes	—
134	#127	C₂²⋊D₁₉	1	2	yes	—
135	#124	—	2	2	no	—
136	#121	—	2	2	no	—
137	#121	—	2	2	no	—
138	C₄×D₄	—	2	2	no	—
139	C₄×D₄	—	2	2	no	—
140	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
141	#108	—	76	4	no	—
142	#106	—	2	2	no	—
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	—	38	2	no	—
145	C₄⋊C₈	—	38	2	no	—
146	C₄⋊C₈	—	38	2	no	—
147	#102	—	2	4	no	—
148	C₄×C₈	—	19	2	no	—
149	C₄×C₈	—	38	1	no	—
150	C₁₉	#10530	1	1	yes	characteristic, complemented, Sylow
151	#41	—	8	2	no	—
152	#41	—	8	2	no	—
153	C₂×Q₈	—	2	2	no	—
154	C₂×D₄	—	2	2	no	—
155	C₂×D₄	—	2	2	no	—
156	C₂×D₄	—	2	2	no	—
157	C₂×D₄	—	4	2	no	—
158	C₂²×C₄	—	4	2	no	—
159	C₂²×C₄	#64543	1	2	yes	—
160	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
161	D₈	—	8	2	no	—
162	M₁₆	—	4	2	no	—
163	M₁₆	—	76	2	no	—
164	M₁₆	—	152	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	38	2	no	—
167	C₂×C₈	—	38	2	no	—
168	C₂×C₈	—	38	2	no	—
169	C₂×C₈	—	76	2	no	—
170	C₂×C₈	—	76	1	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₂²⋊C₄	—	4	2	no	—
174	C₂²⋊C₄	—	4	2	no	—
175	C₄²	D₄×D₁₉	1	1	yes	characteristic
176	C₄²	#64543	1	1	yes	characteristic
177	C₄²	C₂×C₂²⋊D₁₉	1	2	yes	—
178	C₂³	—	4	2	no	—
179	C₂³	#10597382	1	1	yes	characteristic
180	Q₈	—	8	2	no	—
181	D₄	—	4	2	no	—
182	D₄	—	4	2	no	—
183	D₄	—	4	2	no	—
184	D₄	—	4	2	no	—
185	D₄	—	8	2	no	—
186	D₄	—	8	4	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	8	2	no	—
193	C₂×C₄	—	8	2	no	—
194	C₂×C₄	#10597381	1	2	yes	—
195	C₂×C₄	#10597384	1	1	yes	characteristic
196	C₂×C₄	#10597385	1	2	yes	—
197	C₂×C₄	#10597399	1	1	yes	characteristic
198	C₈	—	4	2	no	—
199	C₈	—	38	4	no	—
200	C₈	—	152	2	no	—
201	C₈	—	152	1	no	—
202	C₂²	—	2	2	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	8	2	no	—
205	C₂²	—	8	2	no	—
206	C₂²	—	8	2	no	—
207	C₂²	#11926458	1	1	yes	characteristic
208	C₄	—	2	2	no	—
209	C₄	—	2	1	no	—
210	C₄	—	8	2	no	—
211	C₄	#11926712	1	1	yes	characteristic
212	C₄	#11926727	1	1	yes	characteristic
213	C₄	#11926805	1	2	yes	—
214	C₄	#11926812	1	2	yes	—
215	C₂	—	4	1	no	—
216	C₂	—	16	2	no	—
217	C₂	#14564856	1	1	yes	characteristic
218	C₂	#14564924	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14563806

C₂

yes

maximal, characteristic, complemented

#14562370

C₂

yes

maximal, complemented

#14562086

C₂

yes

maximal, complemented

C₁₉⋊#1309

C₂

yes

maximal, characteristic, complemented

C₁₉×#3253

C₂

yes

maximal, characteristic

C₂×#10597279

—

#11926568

C₂²

yes

—

#11926568

C₂²

yes

characteristic

C₂×C₄×C₁₉⋊C₈

C₂²

yes

characteristic

#11926293

—

#11926288

—

D₄⋊(C₁₉⋊C₈)

—

C₁₉×#532

C₂²

yes

—

C₁₉×#476

—

C₁₉×#469

—

C₁₉×#459

—

C₁₉×#423

C₂²

yes

characteristic

C₂×#41×C₁₉

—

C₂×D₈×C₁₉

—

C₂×M₁₆×C₁₉

D₄

yes

—

C₁₉×C₄⋊Q₈

D₄

yes

—

C₁₉×#127

D₄

yes

—

C₁₉×#124

—

C₁₉×#121

—

C₁₉×#121

—

C₄×D₄×C₁₉

—

C₄×D₄×C₁₉

—

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₂×#64520

—

C₂×#64520

—

C₂²×C₁₉⋊C₈

—

C₁₉×#106

—

C₁₉×C₄⋊C₈

—

C₁₉×#102

—

#10597279

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₄×C₁₉⋊C₈

—

C₄×C₁₉⋊C₈

D₄

yes

—

#41×C₁₉

—

#41×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₉

#41

yes

—

D₈×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄²×C₁₉

C₂×D₄

yes

—

C₄²×C₁₉

yes

—

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

#10530

—

maximal, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

#124

yes

characteristic

Q₈×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

D₄×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

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

—

C₂×C₄×C₁₉

#123

yes

—

C₈×C₁₉

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

#3253

D₁₉

yes

—

—

—

—

C₂²×C₁₉

—

C₂²×C₁₉

—

100

C₂²×C₁₉

—

101

C₂²×C₁₉

—

102

C₂²×C₁₉

—

103

C₂²×C₁₉

#394

yes

characteristic

104

C₄×C₁₉

—

105

C₄×C₁₉

—

106

C₄×C₁₉

—

107

C₄×C₁₉

#453

yes

characteristic

108

C₄×C₁₉

#456

yes

characteristic

109

C₄×C₁₉

#471

yes

—

110

C₄×C₁₉

#472

yes

—

111

#532

C₂×D₁₉

yes

—

112

#476

—

113

#469

—

114

#459

—

115

C₂×#108

—

116

#423

—

117

#423

—

118

#423

C₂×D₁₉

yes

characteristic

119

C₂×C₄×C₈

—

120

#333

—

121

#331

—

122

D₄⋊C₈

—

123

C₂×C₁₉

—

124

C₂×C₁₉

—

125

C₂×C₁₉

#2039

yes

characteristic

126

C₂×C₁₉

#2051

yes

—

127

C₂×#41

—

128

C₂×D₈

—

129

C₂×M₁₆

—

130

C₂×M₁₆

—

131

C₂×M₁₆

C₂²⋊D₁₉

yes

—

132

C₂²×C₈

—

133

C₄⋊Q₈

C₂²⋊D₁₉

yes

—

134

#127

C₂²⋊D₁₉

yes

—

135

#124

—

136

#121

—

137

#121

—

138

C₄×D₄

—

139

C₄×D₄

—

140

C₂×C₄²

C₂²×D₁₉

yes

characteristic

141

#108

—

142

#106

—

143

C₄⋊C₈

—

144

C₄⋊C₈

—

145

C₄⋊C₈

—

146

C₄⋊C₈

—

147

#102

—

148

C₄×C₈

—

149

C₄×C₈

—

150

C₁₉

#10530

yes

characteristic, complemented, Sylow

151

#41

—

152

#41

—

153

C₂×Q₈

—

154

C₂×D₄

—

155

C₂×D₄

—

156

C₂×D₄

—

157

C₂×D₄

—

158

C₂²×C₄

—

159

C₂²×C₄

#64543

yes

—

160

C₂²×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

161

D₈

—

162

M₁₆

—

163

M₁₆

—

164

M₁₆

—

152

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₂²⋊C₄

—

174

C₂²⋊C₄

—

175

C₄²

D₄×D₁₉

yes

characteristic

176

C₄²

#64543

yes

characteristic

177

C₄²

C₂×C₂²⋊D₁₉

yes

—

178

C₂³

—

179

C₂³

#10597382

yes

characteristic

180

Q₈

—

181

D₄

—

182

D₄

—

183

D₄

—

184

D₄

—

185

D₄

—

186

D₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

#10597381

yes

—

195

C₂×C₄

#10597384

yes

characteristic

196

C₂×C₄

#10597385

yes

—

197

C₂×C₄

#10597399

yes

characteristic

198

C₈

—

199

C₈

—

200

C₈

—

152

—

201

C₈

—

152

—

202

C₂²

—

203

C₂²

—

204

C₂²

—

205

C₂²

—

206

C₂²

—

207

C₂²

#11926458

yes

characteristic

208

C₄

—

209

C₄

—

210

C₄

—

211

C₄

#11926712

yes

characteristic

212

C₄

#11926727

yes

characteristic

213

C₄

#11926805

yes

—

214

C₄

#11926812

yes

—

215

C₂

—

216

C₂

—

217

C₂

#14564856

yes

characteristic

218

C₂

#14564924

yes

—

	subgroup	G/H
center	C₂²	#11926458
commutator, cosocle	C₂×C₄²×C₁₉	C₂³
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting, prime radical	C₁₉×#3253	C₂
socle	C₂²×C₁₉	#394
hypercenter	#3253	D₁₉
hypocenter	C₁₉	#10530
LCS	G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3253

subgroup

G/H

center

C₂²

#11926458

commutator, cosocle

C₂×C₄²×C₁₉

C₂³

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting, prime radical

C₁₉×#3253

C₂

C₂²×C₁₉

D₁₉

C₁₉

G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3253

N	H ≅ G/N	type
1. #14563806	C₂	semidirect
2. #14562370	C₂	semidirect
3. #14562086	C₂	semidirect
4. C₁₉⋊#1309	C₂	semidirect
150. C₁₉	#10530	semidirect

H ≅ G/N

type

1. #14563806

C₂

semidirect

2. #14562370

C₂

semidirect

3. #14562086

C₂

semidirect

4. C₁₉⋊#1309

C₂

semidirect

150. C₁₉

#10530

semidirect

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

subgroup	index
#14563806	2
#14562370	2
#14562086	2
C₁₉⋊#1309	2
C₁₉×#3253	2
#10530	19

subgroup

index

#14563806

#14562370

#14562086

C₁₉⋊#1309

C₁₉×#3253

#10530

Subgroups — #18086708

Conjugacy classes of subgroups

Complements

Maximal subgroups