Group #18080151 — subgroups

Subgroups — #18080151

Order 4864.

452 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 103 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	#14574011	C₂	1	1	yes	maximal, characteristic
2	#14572737	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562304	C₂	1	2	yes	maximal
4	#14561524	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1561	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10597320	C₂²	1	1	yes	characteristic
7	#11926910	C₄	1	4	yes	—
8	#11926892	—	2	2	no	—
9	#11926675	—	2	2	no	—
10	Q₈×C₁₉⋊C₈	—	2	2	no	—
11	#11926605	—	2	2	no	—
12	#11926597	C₂²	1	2	yes	—
13	#11926548	—	2	2	no	—
14	C₄×#64520	C₂²	1	1	yes	characteristic
15	#11926287	—	2	2	no	—
16	#11926287	—	2	2	no	—
17	C₁₉×#533	C₂²	1	1	yes	characteristic
18	C₁₉×#423	C₂²	1	2	yes	—
19	C₁₉×#332	—	2	2	no	—
20	C₂²×D₇₆	—	2	1	no	—
21	C₂²×D₇₆	—	2	2	no	—
22	C₂×M₁₆×C₁₉	—	2	2	no	—
23	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
24	C₁₉×#125	C₂×C₄	1	2	yes	—
25	C₁₉×C₂²⋊Q₈	—	2	2	no	—
26	C₄×Q₈×C₁₉	—	2	2	no	—
27	C₁₉×#117	D₄	1	2	yes	—
28	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
29	C₂×#64520	—	2	1	no	—
30	#10597320	—	2	2	no	—
31	#10597320	—	2	1	no	—
32	#10597320	C₂×C₄	1	2	yes	—
33	C₁₉×C₄⋊C₈	—	2	2	no	—
34	C₁₉×C₄⋊C₈	—	2	2	no	—
35	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
36	#10597266	—	2	8	no	—
37	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
38	C₄⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
39	C₄⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
40	#10597260	—	2	2	no	—
41	#10597260	C₂×C₄	1	2	yes	—
42	C₄×C₁₉⋊C₈	—	2	2	no	—
43	C₄×C₁₉⋊C₈	—	2	2	no	—
44	C₄×C₁₉⋊C₈	—	2	1	no	—
45	C₂³×D₁₉	—	4	1	no	—
46	C₂×Q₈×C₁₉	—	2	2	no	—
47	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₉	#41	1	2	yes	—
49	C₂×D₇₆	—	2	2	no	—
50	C₂×D₇₆	—	2	4	no	—
51	C₂×D₇₆	—	4	4	no	—
52	C₂×D₇₆	—	4	1	no	—
53	C₂×D₇₆	—	4	2	no	—
54	C₂×D₇₆	—	8	2	no	—
55	C₂×D₇₆	—	8	2	no	—
56	M₁₆×C₁₉	—	8	2	no	—
57	C₂×C₈×C₁₉	—	4	2	no	—
58	C₄⋊C₄×C₁₉	—	2	4	no	—
59	C₄⋊C₄×C₁₉	—	2	2	no	—
60	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
61	C₂²⋊C₄×C₁₉	—	2	2	no	—
62	C₄²×C₁₉	—	2	2	no	—
63	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₁₉	C₂×D₄	1	2	yes	—
65	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
66	#64520	—	4	2	no	—
67	C₂×C₁₉⋊C₈	—	2	2	no	—
68	C₂×C₁₉⋊C₈	—	2	2	no	—
69	C₂×C₁₉⋊C₈	—	2	2	no	—
70	#9438	—	19	1	no	maximal, Sylow
71	C₂³×C₁₉	#123	1	1	yes	characteristic
72	C₂²×D₁₉	—	4	2	no	—
73	C₂²×D₁₉	—	8	2	no	—
74	C₂²×D₁₉	—	8	4	no	—
75	Q₈×C₁₉	—	4	2	no	—
76	Q₈×C₁₉	—	4	2	no	—
77	C₂×C₄×C₁₉	—	2	2	no	—
78	C₂×C₄×C₁₉	—	2	2	no	—
79	C₂×C₄×C₁₉	—	2	2	no	—
80	C₂×C₄×C₁₉	—	2	4	no	—
81	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
83	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
84	C₂×C₄×C₁₉	#121	1	2	yes	—
85	D₇₆	—	4	4	no	—
86	D₇₆	—	8	4	no	—
87	D₇₆	—	8	2	no	—
88	D₇₆	—	8	2	no	—
89	D₇₆	—	8	2	no	—
90	C₈×C₁₉	—	8	2	no	—
91	C₁₉⋊C₈	—	4	2	no	—
92	C₁₉⋊C₈	—	4	2	no	—
93	C₁₉⋊C₈	—	4	2	no	—
94	#3138	—	19	1	no	—
95	#2986	—	19	1	no	—
96	#1561	D₁₉	1	1	yes	characteristic, complemented
97	#1545	—	19	2	no	—
98	#1299	—	19	2	no	—
99	C₂²×C₁₉	—	2	1	no	—
100	C₂²×C₁₉	—	2	2	no	—
101	C₂²×C₁₉	#386	1	1	yes	characteristic
102	C₂×D₁₉	—	8	2	no	—
103	C₂×D₁₉	—	8	4	no	—
104	C₂×D₁₉	—	16	4	no	—
105	C₄×C₁₉	—	2	1	no	—
106	C₄×C₁₉	—	2	2	no	—
107	C₄×C₁₉	—	4	4	no	—
108	C₄×C₁₉	#454	1	2	yes	—
109	C₄×C₁₉	#471	1	2	yes	—
110	C₄×C₁₉	#472	1	2	yes	—
111	#533	C₂×D₁₉	1	1	yes	characteristic
112	C₂×#127	—	19	1	no	—
113	#490	—	19	4	no	—
114	#486	—	38	2	no	—
115	#446	—	38	2	no	—
116	C₈×Q₈	—	38	2	no	—
117	#433	—	38	2	no	—
118	#431	—	19	2	no	—
119	#423	C₂×D₁₉	1	2	yes	—
120	#418	—	38	2	no	—
121	C₄×M₁₆	—	19	1	no	—
122	#332	—	2	2	no	—
123	#331	—	38	2	no	—
124	#331	—	38	2	no	—
125	C₂×C₁₉	—	4	1	no	—
126	C₂×C₁₉	#1888	1	1	yes	characteristic
127	C₂×C₁₉	#1907	1	2	yes	—
128	D₁₉	—	16	2	no	—
129	C₂²×D₄	—	38	1	no	—
130	C₂²×D₄	—	38	2	no	—
131	C₂×M₁₆	—	2	2	no	—
132	C₂×M₁₆	—	38	1	no	—
133	C₄⋊Q₈	C₂²⋊D₁₉	1	2	yes	—
134	#127	—	19	2	no	—
135	#127	—	38	2	no	—
136	#127	—	38	1	no	—
137	#125	C₄×D₁₉	1	2	yes	—
138	C₂²⋊Q₈	—	2	2	no	—
139	C₄×Q₈	—	2	2	no	—
140	#117	D₇₆	1	2	yes	—
141	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
142	C₄⋊C₈	—	2	2	no	—
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	—	19	2	no	—
145	C₄⋊C₈	—	19	2	no	—
146	C₄⋊C₈	—	38	4	no	—
147	#102	—	38	8	no	—
148	C₂²⋊C₈	—	38	2	no	—
149	#97	—	19	2	no	—
150	#97	—	38	2	no	—
151	C₄×C₈	—	38	2	no	—
152	C₄×C₈	—	38	2	no	—
153	C₄×C₈	—	38	1	no	—
154	C₁₉	#9438	1	1	yes	characteristic, complemented, Sylow
155	C₂⁴	—	76	1	no	—
156	C₂×Q₈	—	2	2	no	—
157	C₂×D₄	—	38	4	no	—
158	C₂×D₄	—	38	2	no	—
159	C₂×D₄	—	76	4	no	—
160	C₂×D₄	—	76	2	no	—
161	C₂×D₄	—	76	1	no	—
162	C₂×D₄	—	152	2	no	—
163	C₂×D₄	—	152	2	no	—
164	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
165	C₂²×C₄	#64543	1	2	yes	—
166	M₁₆	—	8	2	no	—
167	M₁₆	—	76	2	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	38	2	no	—
170	C₂×C₈	—	38	2	no	—
171	C₂×C₈	—	38	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	4	no	—
174	C₄⋊C₄	(C₂×D₁₉)⋊C₄	1	4	yes	—
175	C₂²⋊C₄	—	2	2	no	—
176	C₄²	—	2	2	no	—
177	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
178	C₄²	D₄×D₁₉	1	2	yes	—
179	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
180	C₂³	—	76	2	no	—
181	C₂³	—	152	2	no	—
182	C₂³	—	152	4	no	—
183	C₂³	#10597332	1	1	yes	characteristic
184	Q₈	—	4	2	no	—
185	Q₈	—	4	2	no	—
186	D₄	—	76	4	no	—
187	D₄	—	152	4	no	—
188	D₄	—	152	2	no	—
189	D₄	—	152	2	no	—
190	D₄	—	152	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
196	C₂×C₄	#10597328	1	1	yes	characteristic
197	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
198	C₂×C₄	#10597385	1	2	yes	—
199	C₈	—	8	2	no	—
200	C₈	—	76	2	no	—
201	C₈	—	76	2	no	—
202	C₈	—	76	2	no	—
203	C₂²	—	2	2	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	304	4	no	—
206	C₂²	—	152	4	no	—
207	C₂²	—	152	2	no	—
208	C₂²	#11926421	1	1	yes	characteristic
209	C₄	—	2	2	no	—
210	C₄	—	2	1	no	—
211	C₄	—	4	4	no	—
212	C₄	#11926719	1	2	yes	—
213	C₄	#11926809	1	2	yes	—
214	C₄	#11926812	1	2	yes	—
215	C₂	—	4	1	no	—
216	C₂	—	304	2	no	—
217	C₂	#14564036	1	1	yes	characteristic
218	C₂	#14564141	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14574011

C₂

yes

maximal, characteristic

#14572737

C₂

yes

maximal, characteristic, complemented

#14562304

C₂

yes

maximal

#14561524

C₂

yes

maximal, complemented

C₁₉×#1561

C₂

yes

maximal, characteristic, complemented

C₂×#10597320

C₂²

yes

characteristic

#11926910

C₄

yes

—

#11926892

—

#11926675

—

Q₈×C₁₉⋊C₈

—

#11926605

—

#11926597

C₂²

yes

—

#11926548

—

C₄×#64520

C₂²

yes

characteristic

#11926287

—

#11926287

—

C₁₉×#533

C₂²

yes

characteristic

C₁₉×#423

C₂²

yes

—

C₁₉×#332

—

C₂²×D₇₆

—

C₂²×D₇₆

—

C₂×M₁₆×C₁₉

—

C₁₉×C₄⋊Q₈

D₄

yes

—

C₁₉×#125

C₂×C₄

yes

—

C₁₉×C₂²⋊Q₈

—

C₄×Q₈×C₁₉

—

C₁₉×#117

D₄

yes

—

C₂×C₄²×C₁₉

C₂³

yes

—

—

—

C₂×C₄

yes

—

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

C₂²⋊(C₁₉⋊C₈)

—

#10597266

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

D₄

yes

—

C₄⋊(C₁₉⋊C₈)

D₄

yes

—

#10597260

—

#10597260

C₂×C₄

yes

—

C₄×C₁₉⋊C₈

—

C₄×C₁₉⋊C₈

—

C₄×C₁₉⋊C₈

—

C₂³×D₁₉

—

C₂×Q₈×C₁₉

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₉

#41

yes

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

C₂²⋊C₄

yes

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

C₂²×C₄

yes

characteristic

C₄²×C₁₉

C₂×D₄

yes

—

C₄²×C₁₉

C₂×D₄

yes

characteristic

#64520

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

—

C₂³×C₁₉

yes

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

Q₈×C₁₉

—

Q₈×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉

C₄×D₄

yes

—

C₂×C₄×C₁₉

#120

yes

characteristic

C₂×C₄×C₁₉

#121

yes

—

D₇₆

—

D₇₆

—

D₇₆

—

D₇₆

—

D₇₆

—

C₈×C₁₉

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

#3138

—

#2986

—

#1561

D₁₉

yes

characteristic, complemented

#1545

—

#1299

—

C₂²×C₁₉

—

100

C₂²×C₁₉

—

101

C₂²×C₁₉

#386

yes

characteristic

102

C₂×D₁₉

—

103

C₂×D₁₉

—

104

C₂×D₁₉

—

105

C₄×C₁₉

—

106

C₄×C₁₉

—

107

C₄×C₁₉

—

108

C₄×C₁₉

#454

yes

—

109

C₄×C₁₉

#471

yes

—

110

C₄×C₁₉

#472

yes

—

111

#533

C₂×D₁₉

yes

characteristic

112

C₂×#127

—

113

#490

—

114

#486

—

115

#446

—

116

C₈×Q₈

—

117

#433

—

118

#431

—

119

#423

C₂×D₁₉

yes

—

120

#418

—

121

C₄×M₁₆

—

122

#332

—

123

#331

—

124

#331

—

125

C₂×C₁₉

—

126

C₂×C₁₉

#1888

yes

characteristic

127

C₂×C₁₉

#1907

yes

—

128

D₁₉

—

129

C₂²×D₄

—

130

C₂²×D₄

—

131

C₂×M₁₆

—

132

C₂×M₁₆

—

133

C₄⋊Q₈

C₂²⋊D₁₉

yes

—

134

#127

—

135

#127

—

136

#127

—

137

#125

C₄×D₁₉

yes

—

138

C₂²⋊Q₈

—

139

C₄×Q₈

—

140

#117

D₇₆

yes

—

141

C₂×C₄²

C₂²×D₁₉

yes

characteristic

142

C₄⋊C₈

—

143

C₄⋊C₈

—

144

C₄⋊C₈

—

145

C₄⋊C₈

—

146

C₄⋊C₈

—

147

#102

—

148

C₂²⋊C₈

—

149

#97

—

150

#97

—

151

C₄×C₈

—

152

C₄×C₈

—

153

C₄×C₈

—

154

C₁₉

#9438

yes

characteristic, complemented, Sylow

155

C₂⁴

—

156

C₂×Q₈

—

157

C₂×D₄

—

158

C₂×D₄

—

159

C₂×D₄

—

160

C₂×D₄

—

161

C₂×D₄

—

162

C₂×D₄

—

152

—

163

C₂×D₄

—

152

—

164

C₂²×C₄

C₂×D₇₆

yes

characteristic

165

C₂²×C₄

#64543

yes

—

166

M₁₆

—

167

M₁₆

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

(C₂×D₁₉)⋊C₄

yes

—

175

C₂²⋊C₄

—

176

C₄²

—

177

C₄²

C₂×C₄×D₁₉

yes

characteristic

178

C₄²

D₄×D₁₉

yes

—

179

C₄²

C₂×C₂²⋊D₁₉

yes

characteristic

180

C₂³

—

181

C₂³

—

152

—

182

C₂³

—

152

—

183

C₂³

#10597332

yes

characteristic

184

Q₈

—

185

Q₈

—

186

D₄

—

187

D₄

—

152

—

188

D₄

—

152

—

189

D₄

—

152

—

190

D₄

—

152

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

(C₂²⋊D₁₉)⋊C₄

yes

—

196

C₂×C₄

#10597328

yes

characteristic

197

C₂×C₄

C₂×(C₂×D₁₉)⋊C₄

yes

characteristic

198

C₂×C₄

#10597385

yes

—

199

C₈

—

200

C₈

—

201

C₈

—

202

C₈

—

203

C₂²

—

204

C₂²

—

205

C₂²

—

304

—

206

C₂²

—

152

—

207

C₂²

—

152

—

208

C₂²

#11926421

yes

characteristic

209

C₄

—

210

C₄

—

211

C₄

—

212

C₄

#11926719

yes

—

213

C₄

#11926809

yes

—

214

C₄

#11926812

yes

—

215

C₂

—

216

C₂

—

304

—

217

C₂

#14564036

yes

characteristic

218

C₂

#14564141

yes

—

	subgroup	G/H
center	C₂²	#11926421
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1561	C₂
socle	C₂²×C₁₉	#386
prime radical	C₂×#10597320	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1561	D₁₉
hypocenter	C₁₉	#9438
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1561

subgroup

G/H

center

C₂²

#11926421

commutator

C₄²×C₁₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#1561

C₂

C₂²×C₁₉

C₂²

C₂×C₄²×C₁₉

C₂³

D₁₉

C₁₉

G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1561

N	H ≅ G/N	type
2. #14572737	C₂	semidirect
4. #14561524	C₂	semidirect
5. C₁₉×#1561	C₂	semidirect
96. #1561	D₁₉	semidirect
154. C₁₉	#9438	semidirect

H ≅ G/N

type

2. #14572737

C₂

semidirect

4. #14561524

C₂

semidirect

5. C₁₉×#1561

C₂

semidirect

96. #1561

D₁₉

semidirect

154. C₁₉

#9438

semidirect

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

subgroup	index
#14574011	2
#14572737	2
#14562304	2
#14561524	2
C₁₉×#1561	2
#9438	19

subgroup

index

C₁₉×#1561

Subgroups — #18080151

Conjugacy classes of subgroups

Complements

Maximal subgroups