Group #18086713 — subgroups

Subgroups — #18086713

Order 4864.

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

normal subgroups only

This group has 386 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	#14563772	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	#11926568	C₂²	1	2	yes	—
7	#11926568	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
9	#11926308	—	2	2	no	—
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	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
36	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
37	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
38	C₄×C₁₉⋊C₈	—	2	1	no	—
39	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
40	#41×C₁₉	—	8	2	no	—
41	#41×C₁₉	—	8	2	no	—
42	C₂×Q₈×C₁₉	—	2	2	no	—
43	C₂×D₄×C₁₉	—	2	2	no	—
44	C₂×D₄×C₁₉	—	2	2	no	—
45	C₂×D₄×C₁₉	—	2	2	no	—
46	C₂×D₄×C₁₉	—	4	2	no	—
47	C₂²×C₄×C₁₉	—	4	2	no	—
48	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	#41	1	2	yes	—
50	D₈×C₁₉	—	8	2	no	—
51	M₁₆×C₁₉	—	4	2	no	—
52	C₂×C₈×C₁₉	—	2	2	no	—
53	C₄⋊C₄×C₁₉	—	2	2	no	—
54	C₄⋊C₄×C₁₉	—	2	2	no	—
55	C₂²⋊C₄×C₁₉	—	4	2	no	—
56	C₂²⋊C₄×C₁₉	—	4	2	no	—
57	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₁₉	C₂×D₄	1	2	yes	—
59	C₄²×C₁₉	#41	1	1	yes	characteristic
60	#64520	—	4	2	no	—
61	#64520	—	8	2	no	—
62	C₂×C₁₉⋊C₈	—	2	2	no	—
63	C₂×C₁₉⋊C₈	—	4	1	no	—
64	C₂×C₁₉⋊C₈	—	4	1	no	—
65	C₂×C₁₉⋊C₈	—	4	1	no	—
66	C₂×C₁₉⋊C₈	—	4	2	no	—
67	#10531	—	19	1	no	maximal, Sylow
68	C₂³×C₁₉	—	4	2	no	—
69	C₂³×C₁₉	#124	1	1	yes	characteristic
70	Q₈×C₁₉	—	8	2	no	—
71	D₄×C₁₉	—	4	2	no	—
72	D₄×C₁₉	—	4	2	no	—
73	D₄×C₁₉	—	4	2	no	—
74	D₄×C₁₉	—	4	2	no	—
75	D₄×C₁₉	—	8	2	no	—
76	D₄×C₁₉	—	8	4	no	—
77	C₂×C₄×C₁₉	—	2	1	no	—
78	C₂×C₄×C₁₉	—	2	1	no	—
79	C₂×C₄×C₁₉	—	2	2	no	—
80	C₂×C₄×C₁₉	—	2	2	no	—
81	C₂×C₄×C₁₉	—	4	2	no	—
82	C₂×C₄×C₁₉	—	8	2	no	—
83	C₂×C₄×C₁₉	—	8	2	no	—
84	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
85	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
86	C₂×C₄×C₁₉	#121	1	2	yes	—
87	C₂×C₄×C₁₉	#123	1	2	yes	—
88	C₈×C₁₉	—	4	2	no	—
89	C₁₉⋊C₈	—	4	2	no	—
90	C₁₉⋊C₈	—	8	1	no	—
91	C₁₉⋊C₈	—	8	2	no	—
92	#3253	D₁₉	1	1	yes	characteristic
93	#1837	—	19	1	no	—
94	#1560	—	19	2	no	—
95	#1492	—	19	2	no	—
96	#1309	—	19	1	no	—
97	C₂²×C₁₉	—	2	2	no	—
98	C₂²×C₁₉	—	2	1	no	—
99	C₂²×C₁₉	—	8	2	no	—
100	C₂²×C₁₉	—	8	2	no	—
101	C₂²×C₁₉	—	8	2	no	—
102	C₂²×C₁₉	#394	1	1	yes	characteristic
103	C₄×C₁₉	—	2	2	no	—
104	C₄×C₁₉	—	2	1	no	—
105	C₄×C₁₉	—	8	2	no	—
106	C₄×C₁₉	#454	1	1	yes	characteristic
107	C₄×C₁₉	#455	1	1	yes	characteristic
108	C₄×C₁₉	#471	1	2	yes	—
109	C₄×C₁₉	#472	1	2	yes	—
110	#532	C₂×D₁₉	1	2	yes	—
111	#476	—	2	2	no	—
112	#469	—	2	2	no	—
113	#459	—	2	2	no	—
114	#423	—	19	1	no	—
115	#423	—	19	2	no	—
116	#423	C₂×D₁₉	1	1	yes	characteristic
117	C₂×C₄×C₈	—	19	1	no	—
118	#341	—	38	2	no	—
119	#333	—	38	2	no	—
120	#331	—	38	2	no	—
121	D₄⋊C₈	—	38	4	no	—
122	C₂×C₁₉	—	4	1	no	—
123	C₂×C₁₉	—	16	2	no	—
124	C₂×C₁₉	#2040	1	1	yes	characteristic
125	C₂×C₁₉	#2051	1	2	yes	—
126	C₂×#41	—	2	2	no	—
127	C₂×D₈	—	2	2	no	—
128	C₂×M₁₆	—	38	1	no	—
129	C₂×M₁₆	—	38	2	no	—
130	C₂×M₁₆	C₂²⋊D₁₉	1	2	yes	—
131	C₂²×C₈	—	38	1	no	—
132	C₄⋊Q₈	C₂²⋊D₁₉	1	2	yes	—
133	#127	C₂²⋊D₁₉	1	2	yes	—
134	#124	—	2	2	no	—
135	#121	—	2	2	no	—
136	#121	—	2	2	no	—
137	C₄×D₄	—	2	2	no	—
138	C₄×D₄	—	2	2	no	—
139	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
140	#106	—	2	2	no	—
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	—	38	2	no	—
143	C₄⋊C₈	—	38	2	no	—
144	C₄⋊C₈	—	38	2	no	—
145	#102	—	2	4	no	—
146	C₄×C₈	—	19	2	no	—
147	C₄×C₈	—	38	1	no	—
148	C₁₉	#10531	1	1	yes	characteristic, complemented, Sylow
149	#41	—	8	2	no	—
150	#41	—	8	2	no	—
151	C₂×Q₈	—	2	2	no	—
152	C₂×D₄	—	2	2	no	—
153	C₂×D₄	—	2	2	no	—
154	C₂×D₄	—	2	2	no	—
155	C₂×D₄	—	4	2	no	—
156	C₂²×C₄	—	4	2	no	—
157	C₂²×C₄	#64543	1	2	yes	—
158	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
159	D₈	—	8	2	no	—
160	M₁₆	—	4	2	no	—
161	M₁₆	—	76	2	no	—
162	M₁₆	—	152	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	38	2	no	—
165	C₂×C₈	—	76	1	no	—
166	C₂×C₈	—	76	1	no	—
167	C₂×C₈	—	76	2	no	—
168	C₂×C₈	—	76	1	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₂²⋊C₄	—	4	2	no	—
172	C₂²⋊C₄	—	4	2	no	—
173	C₄²	D₄×D₁₉	1	1	yes	characteristic
174	C₄²	#64543	1	1	yes	characteristic
175	C₄²	C₂×C₂²⋊D₁₉	1	2	yes	—
176	C₂³	—	4	2	no	—
177	C₂³	#10597382	1	1	yes	characteristic
178	Q₈	—	8	2	no	—
179	D₄	—	4	2	no	—
180	D₄	—	4	2	no	—
181	D₄	—	4	2	no	—
182	D₄	—	4	2	no	—
183	D₄	—	8	2	no	—
184	D₄	—	8	4	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	#10597381	1	2	yes	—
193	C₂×C₄	#10597384	1	1	yes	characteristic
194	C₂×C₄	#10597385	1	2	yes	—
195	C₂×C₄	#10597399	1	1	yes	characteristic
196	C₈	—	4	2	no	—
197	C₈	—	76	2	no	—
198	C₈	—	152	2	no	—
199	C₈	—	152	1	no	—
200	C₂²	—	2	2	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	8	2	no	—
203	C₂²	—	8	2	no	—
204	C₂²	—	8	2	no	—
205	C₂²	#11926458	1	1	yes	characteristic
206	C₄	—	2	2	no	—
207	C₄	—	2	1	no	—
208	C₄	—	8	2	no	—
209	C₄	#11926717	1	1	yes	characteristic
210	C₄	#11926722	1	1	yes	characteristic
211	C₄	#11926805	1	2	yes	—
212	C₄	#11926812	1	2	yes	—
213	C₂	—	4	1	no	—
214	C₂	—	16	2	no	—
215	C₂	#14564861	1	1	yes	characteristic
216	C₂	#14564924	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14563772

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

#11926568

C₂²

yes

—

#11926568

C₂²

yes

characteristic

C₂×C₄×C₁₉⋊C₈

C₂²

yes

—

—

—

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

—

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

—

#10531

—

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

—

C₂²×C₁₉

—

100

C₂²×C₁₉

—

101

C₂²×C₁₉

—

102

C₂²×C₁₉

#394

yes

characteristic

103

C₄×C₁₉

—

104

C₄×C₁₉

—

105

C₄×C₁₉

—

106

C₄×C₁₉

#454

yes

characteristic

107

C₄×C₁₉

#455

yes

characteristic

108

C₄×C₁₉

#471

yes

—

109

C₄×C₁₉

#472

yes

—

110

#532

C₂×D₁₉

yes

—

111

#476

—

112

#469

—

113

#459

—

114

#423

—

115

#423

—

116

#423

C₂×D₁₉

yes

characteristic

117

C₂×C₄×C₈

—

118

#341

—

119

#333

—

120

#331

—

121

D₄⋊C₈

—

122

C₂×C₁₉

—

123

C₂×C₁₉

—

124

C₂×C₁₉

#2040

yes

characteristic

125

C₂×C₁₉

#2051

yes

—

126

C₂×#41

—

127

C₂×D₈

—

128

C₂×M₁₆

—

129

C₂×M₁₆

—

130

C₂×M₁₆

C₂²⋊D₁₉

yes

—

131

C₂²×C₈

—

132

C₄⋊Q₈

C₂²⋊D₁₉

yes

—

133

#127

C₂²⋊D₁₉

yes

—

134

#124

—

135

#121

—

136

#121

—

137

C₄×D₄

—

138

C₄×D₄

—

139

C₂×C₄²

C₂²×D₁₉

yes

characteristic

140

#106

—

141

C₄⋊C₈

—

142

C₄⋊C₈

—

143

C₄⋊C₈

—

144

C₄⋊C₈

—

145

#102

—

146

C₄×C₈

—

147

C₄×C₈

—

148

C₁₉

#10531

yes

characteristic, complemented, Sylow

149

#41

—

150

#41

—

151

C₂×Q₈

—

152

C₂×D₄

—

153

C₂×D₄

—

154

C₂×D₄

—

155

C₂×D₄

—

156

C₂²×C₄

—

157

C₂²×C₄

#64543

yes

—

158

C₂²×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

159

D₈

—

160

M₁₆

—

161

M₁₆

—

162

M₁₆

—

152

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₂²⋊C₄

—

172

C₂²⋊C₄

—

173

C₄²

D₄×D₁₉

yes

characteristic

174

C₄²

#64543

yes

characteristic

175

C₄²

C₂×C₂²⋊D₁₉

yes

—

176

C₂³

—

177

C₂³

#10597382

yes

characteristic

178

Q₈

—

179

D₄

—

180

D₄

—

181

D₄

—

182

D₄

—

183

D₄

—

184

D₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

#10597381

yes

—

193

C₂×C₄

#10597384

yes

characteristic

194

C₂×C₄

#10597385

yes

—

195

C₂×C₄

#10597399

yes

characteristic

196

C₈

—

197

C₈

—

198

C₈

—

152

—

199

C₈

—

152

—

200

C₂²

—

201

C₂²

—

202

C₂²

—

203

C₂²

—

204

C₂²

—

205

C₂²

#11926458

yes

characteristic

206

C₄

—

207

C₄

—

208

C₄

—

209

C₄

#11926717

yes

characteristic

210

C₄

#11926722

yes

characteristic

211

C₄

#11926805

yes

—

212

C₄

#11926812

yes

—

213

C₂

—

214

C₂

—

215

C₂

#14564861

yes

characteristic

216

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	#10531
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. #14563772	C₂	semidirect
2. #14562370	C₂	semidirect
3. #14562086	C₂	semidirect
4. C₁₉⋊#1309	C₂	semidirect
148. C₁₉	#10531	semidirect

H ≅ G/N

type

1. #14563772

C₂

semidirect

2. #14562370

C₂

semidirect

3. #14562086

C₂

semidirect

4. C₁₉⋊#1309

C₂

semidirect

148. C₁₉

#10531

semidirect

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

subgroup	index
#14563772	2
#14562370	2
#14562086	2
C₁₉⋊#1309	2
C₁₉×#3253	2
#10531	19

subgroup

index

#14563772

#14562370

#14562086

C₁₉⋊#1309

C₁₉×#3253

#10531

Subgroups — #18086713

Conjugacy classes of subgroups

Complements

Maximal subgroups