Group #18085215 — subgroups

Subgroups — #18085215

Order 4864.

566 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 113 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	#14574562	C₂	1	2	yes	maximal
2	#14572013	C₂	1	1	yes	maximal, characteristic, complemented
3	#14564463	C₂	1	1	yes	maximal, characteristic
4	#14562315	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1276	C₂	1	1	yes	maximal, characteristic, complemented
6	#11927094	C₂²	1	1	yes	characteristic
7	C₂×#10597315	C₂²	1	2	yes	—
8	#11926917	—	2	2	no	—
9	#11926895	—	2	2	no	—
10	#11926750	—	2	4	no	—
11	#11926685	—	2	4	no	—
12	#11926568	C₂²	1	1	yes	characteristic
13	C₂×#10597262	—	2	2	no	—
14	#11926450	—	2	4	no	—
15	#11926410	—	2	2	no	—
16	#11926292	—	2	4	no	—
17	C₁₉×#431	C₂²	1	2	yes	—
18	C₄³×C₁₉	C₂²	1	1	yes	characteristic
19	C₂×#64541	—	2	1	no	—
20	C₂²×Dic₃₈	D₄	1	4	yes	—
21	C₂×C₄²×C₁₉	—	2	2	no	—
22	C₂×C₄²×C₁₉	D₄	1	2	yes	—
23	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
24	#10597376	—	2	2	no	—
25	C₂×C₄⋊Dic₁₉	—	2	2	no	—
26	C₂×C₄⋊Dic₁₉	—	2	2	no	—
27	C₂×#64520	—	2	1	no	—
28	C₂×#64516	—	2	8	no	—
29	#10597321	—	2	1	no	—
30	#10597320	D₄	1	1	yes	characteristic
31	C₄×D₇₆	—	2	2	no	—
32	#10597315	—	2	2	no	—
33	#10597315	—	2	2	no	—
34	#10597315	—	2	2	no	—
35	#10597315	—	2	2	no	—
36	#10597315	D₄	1	1	yes	characteristic
37	C₁₉×C₄⋊C₈	D₄	1	4	yes	—
38	C₁₉×C₂²⋊C₈	—	2	2	no	—
39	C₄×C₈×C₁₉	—	2	2	no	—
40	C₁₉⋊#95	—	2	8	no	—
41	#10597273	—	2	8	no	—
42	#10597262	—	4	4	no	—
43	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
44	C₂²×C₄×C₁₉	—	2	2	no	—
45	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
47	C₂×C₂²⋊D₁₉	—	4	1	no	—
48	C₂²×Dic₁₉	—	2	4	no	—
49	#64541	—	4	2	no	—
50	#64541	—	4	2	no	—
51	C₂×D₇₆	—	2	2	no	—
52	C₂×D₇₆	—	2	1	no	—
53	C₂×C₄×D₁₉	—	4	1	no	—
54	C₂×Dic₃₈	—	2	4	no	—
55	C₂×Dic₃₈	—	2	1	no	—
56	C₂×Dic₃₈	—	2	4	no	—
57	C₂×Dic₃₈	—	4	8	no	—
58	C₂×C₈×C₁₉	—	2	4	no	—
59	C₄²×C₁₉	—	2	2	no	—
60	C₄²×C₁₉	—	4	4	no	—
61	C₄²×C₁₉	SD₁₆	1	4	yes	complemented
62	C₄²×C₁₉	C₂×D₄	1	2	yes	—
63	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₁₉	#41	1	1	yes	characteristic
65	(C₂×D₁₉)⋊C₄	—	4	2	no	—
66	C₄⋊Dic₁₉	—	2	2	no	—
67	C₄⋊Dic₁₉	—	4	2	no	—
68	C₄⋊Dic₁₉	—	4	2	no	—
69	C₄⋊Dic₁₉	—	4	2	no	—
70	C₄⋊Dic₁₉	—	4	2	no	—
71	#64520	—	4	2	no	—
72	C₂×C₁₉⋊C₈	—	4	1	no	—
73	#64516	—	8	8	no	—
74	#10287	—	19	1	no	maximal, Sylow
75	C₂³×C₁₉	#120	1	1	yes	characteristic
76	C₂²×D₁₉	—	4	1	no	—
77	C₂×C₄×C₁₉	—	2	4	no	—
78	C₂×C₄×C₁₉	—	2	4	no	—
79	C₂×C₄×C₁₉	—	4	4	no	—
80	C₂×C₄×C₁₉	#120	1	2	yes	—
81	C₂×C₄×C₁₉	#121	1	2	yes	—
82	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
83	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
84	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
85	C₂×C₄×C₁₉	#137	1	2	yes	—
86	C₂²⋊D₁₉	—	8	2	no	—
87	C₂×Dic₁₉	—	4	8	no	—
88	C₂×Dic₁₉	—	4	1	no	—
89	C₂×Dic₁₉	—	4	4	no	—
90	D₇₆	—	4	4	no	—
91	D₇₆	—	4	2	no	—
92	C₄×D₁₉	—	8	1	no	—
93	C₄×D₁₉	—	8	1	no	—
94	Dic₃₈	—	4	8	no	—
95	Dic₃₈	—	4	2	no	—
96	Dic₃₈	—	8	4	no	—
97	C₈×C₁₉	—	4	4	no	—
98	C₁₉⋊C₈	—	8	1	no	—
99	#3206	—	19	2	no	—
100	#2862	—	19	1	no	—
101	#1962	—	19	1	no	—
102	#1548	—	19	2	no	—
103	#1276	D₁₉	1	1	yes	characteristic, complemented
104	C₂²×C₁₉	—	2	2	no	—
105	C₂²×C₁₉	#392	1	1	yes	characteristic
106	C₂²×C₁₉	#448	1	2	yes	—
107	C₂×D₁₉	—	8	2	no	—
108	C₂×D₁₉	—	8	1	no	—
109	C₄×C₁₉	—	2	2	no	—
110	C₄×C₁₉	—	4	4	no	—
111	C₄×C₁₉	#455	1	4	yes	—
112	C₄×C₁₉	#461	1	2	yes	—
113	C₄×C₁₉	#492	1	1	yes	characteristic
114	C₄×C₁₉	#497	1	1	yes	characteristic
115	Dic₁₉	—	8	4	no	—
116	Dic₁₉	—	8	1	no	—
117	#532	—	19	1	no	—
118	C₂×C₄⋊Q₈	—	19	2	no	—
119	#492	—	38	2	no	—
120	#487	—	38	2	no	—
121	#461	—	38	4	no	—
122	#448	—	38	4	no	—
123	#431	C₂×D₁₉	1	2	yes	—
124	#423	—	19	1	no	—
125	C₂×#104	—	38	2	no	—
126	#391	—	38	4	no	—
127	#382	—	38	2	no	—
128	C₄³	C₂×D₁₉	1	1	yes	characteristic
129	#333	—	38	4	no	—
130	C₂×C₁₉	—	2	2	no	—
131	C₂×C₁₉	#2004	1	2	yes	—
132	C₂×C₁₉	#2013	1	1	yes	characteristic
133	D₁₉	—	16	1	no	—
134	C₂×#41	—	38	1	no	—
135	C₂²×Q₈	—	19	4	no	—
136	C₂×SD₁₆	—	38	8	no	—
137	C₂×M₁₆	—	38	1	no	—
138	C₄⋊Q₈	—	19	1	no	—
139	C₄⋊Q₈	—	38	2	no	—
140	C₄⋊Q₈	—	38	2	no	—
141	C₄⋊Q₈	—	38	2	no	—
142	C₄⋊Q₈	—	38	2	no	—
143	#127	—	19	1	no	—
144	#124	—	38	1	no	—
145	#121	—	38	2	no	—
146	C₄×D₄	—	38	2	no	—
147	C₂×C₄⋊C₄	—	38	2	no	—
148	C₂×C₄⋊C₄	—	38	2	no	—
149	C₂×C₄²	—	2	2	no	—
150	C₂×C₄²	D₇₆	1	2	yes	—
151	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
152	C₄⋊C₈	—	38	2	no	—
153	C₄⋊C₈	D₇₆	1	4	yes	—
154	#104	—	76	4	no	—
155	#103	—	38	8	no	—
156	C₂²⋊C₈	—	2	2	no	—
157	C₄×C₈	—	2	2	no	—
158	#95	—	38	8	no	—
159	C₁₉	#10287	1	1	yes	characteristic, complemented, Sylow
160	#41	—	76	2	no	—
161	#41	—	76	2	no	—
162	C₂×Q₈	—	38	4	no	—
163	C₂×Q₈	—	38	1	no	—
164	C₂×Q₈	—	38	4	no	—
165	C₂×Q₈	—	76	8	no	—
166	C₂×D₄	—	38	2	no	—
167	C₂×D₄	—	38	1	no	—
168	C₂×D₄	—	76	1	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	38	4	no	—
171	C₂²×C₄	—	76	1	no	—
172	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
173	C₂²×C₄	D₄×D₁₉	1	2	yes	—
174	SD₁₆	—	152	8	no	—
175	M₁₆	—	76	2	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	76	1	no	—
178	C₄⋊C₄	—	38	2	no	—
179	C₄⋊C₄	—	76	2	no	—
180	C₄⋊C₄	—	76	2	no	—
181	C₄⋊C₄	—	76	2	no	—
182	C₄⋊C₄	—	76	2	no	—
183	C₂²⋊C₄	—	76	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	4	4	no	—
186	C₄²	#64516	1	4	yes	complemented
187	C₄²	C₂×D₇₆	1	2	yes	—
188	C₄²	D₄×D₁₉	1	1	yes	characteristic
189	C₄²	#64545	1	1	yes	characteristic
190	C₂³	—	76	1	no	—
191	C₂³	#10597328	1	1	yes	characteristic
192	Q₈	—	76	8	no	—
193	Q₈	—	76	2	no	—
194	Q₈	—	152	4	no	—
195	D₄	—	76	4	no	—
196	D₄	—	76	2	no	—
197	D₄	—	152	2	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	76	8	no	—
202	C₂×C₄	—	76	1	no	—
203	C₂×C₄	—	76	4	no	—
204	C₂×C₄	—	152	1	no	—
205	C₂×C₄	—	152	1	no	—
206	C₂×C₄	#10597320	1	1	yes	characteristic
207	C₂×C₄	#10597328	1	2	yes	—
208	C₂×C₄	#10597341	1	2	yes	—
209	C₂×C₄	#10597341	1	1	yes	characteristic
210	C₂×C₄	C₂×#64516	1	2	yes	—
211	C₂×C₄	#10597355	1	2	yes	—
212	C₈	—	4	4	no	—
213	C₈	—	152	1	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	152	2	no	—
216	C₂²	—	152	1	no	—
217	C₂²	#11926454	1	1	yes	characteristic
218	C₂²	#11926685	1	2	yes	—
219	C₄	—	2	2	no	—
220	C₄	—	4	4	no	—
221	C₄	—	152	4	no	—
222	C₄	—	152	1	no	—
223	C₄	#11926723	1	4	yes	—
224	C₄	#11926750	1	2	yes	—
225	C₄	#11926917	1	1	yes	characteristic
226	C₄	#11926938	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	—	304	1	no	—
229	C₂	#14564667	1	2	yes	—
230	C₂	#14564720	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14574562

C₂

yes

maximal

#14572013

C₂

yes

maximal, characteristic, complemented

#14564463

C₂

yes

maximal, characteristic

#14562315

C₂

yes

maximal, complemented

C₁₉×#1276

C₂

yes

maximal, characteristic, complemented

#11927094

C₂²

yes

characteristic

C₂×#10597315

C₂²

yes

—

—

—

—

—

C₂²

yes

—

—

—

—

C₁₉×#431

C₂²

yes

—

C₄³×C₁₉

C₂²

yes

characteristic

C₂×#64541

—

C₂²×Dic₃₈

D₄

yes

—

C₂×C₄²×C₁₉

—

C₂×C₄²×C₁₉

D₄

yes

—

C₂×C₄²×C₁₉

C₂³

yes

characteristic

#10597376

—

C₂×C₄⋊Dic₁₉

—

C₂×C₄⋊Dic₁₉

—

—

—

—

D₄

yes

characteristic

C₄×D₇₆

—

—

—

—

—

D₄

yes

characteristic

C₁₉×C₄⋊C₈

D₄

yes

—

C₁₉×C₂²⋊C₈

—

C₄×C₈×C₁₉

—

C₁₉⋊#95

—

#10597273

—

#10597262

—

C₄⋊(C₁₉⋊C₈)

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×D₄

yes

—

C₂×C₂²⋊D₁₉

—

C₂²×Dic₁₉

—

#64541

—

#64541

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×C₄×D₁₉

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×C₈×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

SD₁₆

yes

complemented

C₄²×C₁₉

C₂×D₄

yes

—

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄²×C₁₉

#41

yes

characteristic

(C₂×D₁₉)⋊C₄

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

#64520

—

C₂×C₁₉⋊C₈

—

—

—

C₂³×C₁₉

yes

C₂²×D₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#120

yes

—

C₂×C₄×C₁₉

#121

yes

—

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

C₂×SD₁₆

yes

—

C₂×C₄×C₁₉

#137

yes

—

C₂²⋊D₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

D₇₆

—

D₇₆

—

C₄×D₁₉

—

C₄×D₁₉

—

Dic₃₈

—

Dic₃₈

—

Dic₃₈

—

C₈×C₁₉

—

C₁₉⋊C₈

—

#3206

—

100

#2862

—

101

#1962

—

102

#1548

—

103

#1276

D₁₉

yes

characteristic, complemented

104

C₂²×C₁₉

—

105

C₂²×C₁₉

#392

yes

characteristic

106

C₂²×C₁₉

#448

yes

—

107

C₂×D₁₉

—

108

C₂×D₁₉

—

109

C₄×C₁₉

—

110

C₄×C₁₉

—

111

C₄×C₁₉

#455

yes

—

112

C₄×C₁₉

#461

yes

—

113

C₄×C₁₉

#492

yes

characteristic

114

C₄×C₁₉

#497

yes

characteristic

115

Dic₁₉

—

116

Dic₁₉

—

117

#532

—

118

C₂×C₄⋊Q₈

—

119

#492

—

120

#487

—

121

#461

—

122

#448

—

123

#431

C₂×D₁₉

yes

—

124

#423

—

125

C₂×#104

—

126

#391

—

127

#382

—

128

C₄³

C₂×D₁₉

yes

characteristic

129

#333

—

130

C₂×C₁₉

—

131

C₂×C₁₉

#2004

yes

—

132

C₂×C₁₉

#2013

yes

characteristic

133

D₁₉

—

134

C₂×#41

—

135

C₂²×Q₈

—

136

C₂×SD₁₆

—

137

C₂×M₁₆

—

138

C₄⋊Q₈

—

139

C₄⋊Q₈

—

140

C₄⋊Q₈

—

141

C₄⋊Q₈

—

142

C₄⋊Q₈

—

143

#127

—

144

#124

—

145

#121

—

146

C₄×D₄

—

147

C₂×C₄⋊C₄

—

148

C₂×C₄⋊C₄

—

149

C₂×C₄²

—

150

C₂×C₄²

D₇₆

yes

—

151

C₂×C₄²

C₂²×D₁₉

yes

characteristic

152

C₄⋊C₈

—

153

C₄⋊C₈

D₇₆

yes

—

154

#104

—

155

#103

—

156

C₂²⋊C₈

—

157

C₄×C₈

—

158

#95

—

159

C₁₉

#10287

yes

characteristic, complemented, Sylow

160

#41

—

161

#41

—

162

C₂×Q₈

—

163

C₂×Q₈

—

164

C₂×Q₈

—

165

C₂×Q₈

—

166

C₂×D₄

—

167

C₂×D₄

—

168

C₂×D₄

—

169

C₂²×C₄

—

170

C₂²×C₄

—

171

C₂²×C₄

—

172

C₂²×C₄

C₂×D₇₆

yes

characteristic

173

C₂²×C₄

D₄×D₁₉

yes

—

174

SD₁₆

—

152

—

175

M₁₆

—

176

C₂×C₈

—

177

C₂×C₈

—

178

C₄⋊C₄

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₄⋊C₄

—

183

C₂²⋊C₄

—

184

C₄²

—

185

C₄²

—

186

C₄²

#64516

yes

complemented

187

C₄²

C₂×D₇₆

yes

—

188

C₄²

D₄×D₁₉

yes

characteristic

189

C₄²

#64545

yes

characteristic

190

C₂³

—

191

C₂³

#10597328

yes

characteristic

192

Q₈

—

193

Q₈

—

194

Q₈

—

152

—

195

D₄

—

196

D₄

—

197

D₄

—

152

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

152

—

205

C₂×C₄

—

152

—

206

C₂×C₄

#10597320

yes

characteristic

207

C₂×C₄

#10597328

yes

—

208

C₂×C₄

#10597341

yes

—

209

C₂×C₄

#10597341

yes

characteristic

210

C₂×C₄

C₂×#64516

yes

—

211

C₂×C₄

#10597355

yes

—

212

C₈

—

213

C₈

—

152

—

214

C₂²

—

215

C₂²

—

152

—

216

C₂²

—

152

—

217

C₂²

#11926454

yes

characteristic

218

C₂²

#11926685

yes

—

219

C₄

—

220

C₄

—

221

C₄

—

152

—

222

C₄

—

152

—

223

C₄

#11926723

yes

—

224

C₄

#11926750

yes

—

225

C₄

#11926917

yes

characteristic

226

C₄

#11926938

yes

characteristic

227

C₂

—

228

C₂

—

304

—

229

C₂

#14564667

yes

—

230

C₂

#14564720

yes

characteristic

	subgroup	G/H
center	C₂²	#11926454
commutator, cosocle	C₂×C₄²×C₁₉	C₂³
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1276	C₂
socle	C₂²×C₁₉	#392
prime radical	#11927094	C₂²
hypercenter	#1276	D₁₉
hypocenter	C₁₉	#10287
LCS	G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1276

subgroup

G/H

center

C₂²

#11926454

commutator, cosocle

C₂×C₄²×C₁₉

C₂³

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#1276

C₂

C₂²×C₁₉

C₂²

D₁₉

C₁₉

G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1276

N	H ≅ G/N	type
2. #14572013	C₂	semidirect
4. #14562315	C₂	semidirect
5. C₁₉×#1276	C₂	semidirect
61. C₄²×C₁₉	SD₁₆	semidirect
103. #1276	D₁₉	semidirect
159. C₁₉	#10287	semidirect
186. C₄²	#64516	semidirect

H ≅ G/N

type

2. #14572013

C₂

semidirect

4. #14562315

C₂

semidirect

5. C₁₉×#1276

C₂

semidirect

61. C₄²×C₁₉

SD₁₆

semidirect

103. #1276

D₁₉

semidirect

159. C₁₉

#10287

semidirect

186. C₄²

#64516

semidirect

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

subgroup	index
#14574562	2
#14572013	2
#14564463	2
#14562315	2
C₁₉×#1276	2
#10287	19

subgroup

index

C₁₉×#1276

Subgroups — #18085215

Conjugacy classes of subgroups

Complements

Maximal subgroups