Group #18070056 — subgroups

Subgroups — #18070056

Order 4864.

252 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 41 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	#14564734	C₂	1	1	yes	maximal, characteristic
2	#14561762	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1319	C₂	1	1	yes	maximal, characteristic
4	C₂²×#64528	—	2	1	no	—
5	C₂×#10597315	C₄	1	1	yes	characteristic
6	C₂×#10597267	—	2	1	no	—
7	C₂×#10597293	—	2	1	no	—
8	#11926299	C₄	1	1	yes	characteristic
9	C₁₉×#391	C₂²	1	1	yes	characteristic
10	C₁₉×#324	—	2	1	no	—
11	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
12	C₂²×Dic₃₈	—	2	1	no	—
13	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
14	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
15	C₂×#64528	—	4	2	no	—
16	C₂×#64528	—	4	2	no	—
17	C₂×#64528	—	4	1	no	—
18	C₂×#64528	—	4	1	no	—
19	C₂×C₄⋊Dic₁₉	—	4	1	no	—
20	C₂×#64520	—	4	1	no	—
21	C₂²×C₁₉⋊C₈	—	2	1	no	—
22	#10597315	—	4	2	no	—
23	C₁₉×C₂²⋊C₈	—	4	1	no	—
24	C₁₉×#95	—	4	1	no	—
25	#10597293	—	2	2	no	—
26	#10597293	—	4	1	no	—
27	#10597267	—	4	2	no	—
28	C₂×Q₈×C₁₉	—	2	1	no	—
29	C₂×Q₈×C₁₉	—	4	1	no	—
30	C₂×Q₈×C₁₉	—	4	1	no	—
31	C₂×Q₈×C₁₉	D₈	1	1	yes	characteristic
32	C₂×Q₈×C₁₉	SD₁₆	1	1	yes	characteristic
33	C₂²×C₄×C₁₉	—	2	1	no	—
34	C₂²×C₄×C₁₉	—	2	1	no	—
35	C₂²×C₄×C₁₉	—	2	1	no	—
36	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×Dic₁₉	—	4	1	no	—
38	C₂×Dic₃₈	—	2	2	no	—
39	C₂×Dic₃₈	—	4	1	no	—
40	C₂×Dic₃₈	—	4	1	no	—
41	C₂×Dic₃₈	—	4	1	no	—
42	C₂×Dic₃₈	—	4	1	no	—
43	C₂×Dic₃₈	—	4	1	no	—
44	C₂×C₈×C₁₉	—	8	1	no	—
45	C₄⋊C₄×C₁₉	—	4	1	no	—
46	C₄²×C₁₉	—	4	1	no	—
47	#64528	—	4	4	no	—
48	#64528	—	4	4	no	—
49	C₄⋊Dic₁₉	—	4	2	no	—
50	C₄⋊Dic₁₉	—	4	2	no	—
51	#64520	—	4	2	no	—
52	#64520	—	8	1	no	—
53	C₂×C₁₉⋊C₈	—	2	2	no	—
54	C₂×C₁₉⋊C₈	—	4	1	no	—
55	C₂×C₁₉⋊C₈	—	4	1	no	—
56	C₂×C₁₉⋊C₈	—	8	1	no	—
57	#7405	—	19	1	no	maximal, Sylow
58	C₂³×C₁₉	#99	1	1	yes	characteristic
59	Q₈×C₁₉	—	4	1	no	—
60	Q₈×C₁₉	—	4	1	no	—
61	Q₈×C₁₉	—	8	1	no	—
62	C₂×C₄×C₁₉	—	2	1	no	—
63	C₂×C₄×C₁₉	—	2	1	no	—
64	C₂×C₄×C₁₉	—	2	2	no	—
65	C₂×C₄×C₁₉	—	2	1	no	—
66	C₂×C₄×C₁₉	—	2	1	no	—
67	C₂×C₄×C₁₉	—	4	1	no	—
68	C₂×C₄×C₁₉	—	4	1	no	—
69	C₂×C₄×C₁₉	—	4	1	no	—
70	C₂×C₄×C₁₉	—	4	1	no	—
71	C₂×C₄×C₁₉	—	4	1	no	—
72	C₂×C₄×C₁₉	—	8	1	no	—
73	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
74	C₂×C₄×C₁₉	#104	1	1	yes	characteristic
75	C₂×Dic₁₉	—	4	2	no	—
76	C₂×Dic₁₉	—	8	1	no	—
77	C₂×Dic₁₉	—	8	1	no	—
78	Dic₃₈	—	4	2	no	—
79	Dic₃₈	—	4	2	no	—
80	Dic₃₈	—	8	1	no	—
81	Dic₃₈	—	8	1	no	—
82	C₈×C₁₉	—	16	1	no	—
83	C₁₉⋊C₈	—	4	2	no	—
84	C₁₉⋊C₈	—	8	2	no	—
85	#2016	—	19	1	no	—
86	#1394	—	19	1	no	—
87	#1319	D₁₉	1	1	yes	characteristic
88	C₂²×C₁₉	—	2	1	no	—
89	C₂²×C₁₉	—	2	1	no	—
90	C₂²×C₁₉	#327	1	1	yes	characteristic
91	C₂²×C₁₉	AΣL(1,𝔽₁₆)	1	1	yes	characteristic
92	C₂²×C₁₉	#353	1	1	yes	characteristic
93	C₄×C₁₉	—	2	1	no	—
94	C₄×C₁₉	—	2	1	no	—
95	C₄×C₁₉	—	4	2	no	—
96	C₄×C₁₉	—	4	1	no	—
97	C₄×C₁₉	—	4	1	no	—
98	C₄×C₁₉	—	8	1	no	—
99	Dic₁₉	—	8	2	no	—
100	C₂²×Q₁₆	—	38	1	no	—
101	C₂×C₄⋊Q₈	—	19	1	no	—
102	C₂×#103	—	38	1	no	—
103	C₂×#101	—	38	1	no	—
104	#391	C₂×D₁₉	1	1	yes	characteristic
105	#336	—	19	1	no	—
106	#324	—	2	1	no	—
107	C₂×C₁₉	—	2	1	no	—
108	C₂×C₁₉	—	2	1	no	—
109	C₂×C₁₉	#1343	1	1	yes	characteristic
110	C₂×C₁₉	#1407	1	1	yes	characteristic
111	C₂×C₁₉	#1411	1	1	yes	characteristic
112	C₂²×Q₈	—	38	1	no	—
113	C₂²×Q₈	D₇₆	1	1	yes	characteristic
114	C₂×Q₁₆	—	76	2	no	—
115	C₂×Q₁₆	—	76	2	no	—
116	C₂×Q₁₆	—	76	1	no	—
117	C₂×Q₁₆	—	76	1	no	—
118	C₂×M₁₆	—	76	1	no	—
119	C₂²×C₈	—	38	1	no	—
120	C₄⋊Q₈	—	76	2	no	—
121	C₂×C₄⋊C₄	—	76	1	no	—
122	C₂×C₄⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
123	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
124	#103	—	76	2	no	—
125	#101	—	38	2	no	—
126	#101	—	76	1	no	—
127	C₂²⋊C₈	—	4	1	no	—
128	#95	—	4	1	no	—
129	C₁₉	#7405	1	1	yes	characteristic, complemented, Sylow
130	C₂×Q₈	—	2	1	no	—
131	C₂×Q₈	—	4	1	no	—
132	C₂×Q₈	—	4	1	no	—
133	C₂×Q₈	—	38	2	no	—
134	C₂×Q₈	—	76	1	no	—
135	C₂×Q₈	—	76	1	no	—
136	C₂×Q₈	—	76	1	no	—
137	C₂×Q₈	—	76	1	no	—
138	C₂×Q₈	—	76	1	no	—
139	C₂×Q₈	#64516	1	1	yes	characteristic
140	C₂×Q₈	D₁₅₂	1	1	yes	characteristic
141	C₂²×C₄	—	2	1	no	—
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	—	76	1	no	—
145	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
146	Q₁₆	—	76	4	no	—
147	Q₁₆	—	76	4	no	—
148	M₁₆	—	76	2	no	—
149	M₁₆	—	152	1	no	—
150	C₂×C₈	—	8	1	no	—
151	C₂×C₈	—	38	2	no	—
152	C₂×C₈	—	76	1	no	—
153	C₂×C₈	—	76	1	no	—
154	C₂×C₈	—	152	1	no	—
155	C₄⋊C₄	—	4	1	no	—
156	C₄⋊C₄	—	76	2	no	—
157	C₄⋊C₄	—	76	2	no	—
158	C₄²	—	4	1	no	—
159	C₂³	#10597263	1	1	yes	characteristic
160	Q₈	—	4	1	no	—
161	Q₈	—	4	1	no	—
162	Q₈	—	8	1	no	—
163	Q₈	—	76	2	no	—
164	Q₈	—	76	2	no	—
165	Q₈	—	152	1	no	—
166	Q₈	—	152	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	—	4	1	no	—
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	—	4	1	no	—
177	C₂×C₄	—	8	1	no	—
178	C₂×C₄	—	76	2	no	—
179	C₂×C₄	—	152	1	no	—
180	C₂×C₄	—	152	1	no	—
181	C₂×C₄	#10597278	1	1	yes	characteristic
182	C₂×C₄	#10597282	1	1	yes	characteristic
183	C₈	—	16	1	no	—
184	C₈	—	76	2	no	—
185	C₈	—	152	2	no	—
186	C₂²	—	2	1	no	—
187	C₂²	—	2	1	no	—
188	C₂²	#11926279	1	1	yes	characteristic
189	C₂²	#11926332	1	1	yes	characteristic
190	C₂²	#11926336	1	1	yes	characteristic
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	—	4	2	no	—
194	C₄	—	4	1	no	—
195	C₄	—	4	1	no	—
196	C₄	—	8	1	no	—
197	C₄	—	152	2	no	—
198	C₂	—	2	1	no	—
199	C₂	—	2	1	no	—
200	C₂	#14561633	1	1	yes	characteristic
201	C₂	#14561794	1	1	yes	characteristic
202	C₂	#14561803	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14564734

C₂

yes

maximal, characteristic

#14561762

C₂

yes

maximal, characteristic

C₁₉×#1319

C₂

yes

maximal, characteristic

C₂²×#64528

—

C₂×#10597315

C₄

yes

—

—

C₄

yes

characteristic

C₁₉×#391

C₂²

yes

characteristic

C₁₉×#324

—

C₂²×Q₈×C₁₉

D₄

yes

characteristic

C₂²×Dic₃₈

—

C₂×C₄⋊C₄×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

C₂×C₄

yes

—

—

—

—

C₂×C₄⋊Dic₁₉

—

C₂×#64520

—

C₂²×C₁₉⋊C₈

—

#10597315

—

C₁₉×C₂²⋊C₈

—

C₁₉×#95

—

#10597293

—

#10597293

—

#10597267

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

D₈

yes

characteristic

C₂×Q₈×C₁₉

SD₁₆

yes

characteristic

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×Dic₁₉

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×C₈×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄²×C₁₉

—

#64528

—

#64528

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

#64520

—

#64520

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

—

C₂³×C₁₉

yes

Q₈×C₁₉

—

Q₈×C₁₉

—

Q₈×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

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

Dic₃₈

—

Dic₃₈

—

Dic₃₈

—

Dic₃₈

—

C₈×C₁₉

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

#2016

—

#1394

—

#1319

D₁₉

yes

characteristic

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#327

yes

characteristic

C₂²×C₁₉

AΣL(1,𝔽₁₆)

yes

characteristic

C₂²×C₁₉

#353

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

Dic₁₉

—

100

C₂²×Q₁₆

—

101

C₂×C₄⋊Q₈

—

102

C₂×#103

—

103

C₂×#101

—

104

#391

C₂×D₁₉

yes

characteristic

105

#336

—

106

#324

—

107

C₂×C₁₉

—

108

C₂×C₁₉

—

109

C₂×C₁₉

#1343

yes

characteristic

110

C₂×C₁₉

#1407

yes

characteristic

111

C₂×C₁₉

#1411

yes

characteristic

112

C₂²×Q₈

—

113

C₂²×Q₈

D₇₆

yes

characteristic

114

C₂×Q₁₆

—

115

C₂×Q₁₆

—

116

C₂×Q₁₆

—

117

C₂×Q₁₆

—

118

C₂×M₁₆

—

119

C₂²×C₈

—

120

C₄⋊Q₈

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

C₂²⋊D₁₉

yes

characteristic

123

C₂×C₄²

C₄×D₁₉

yes

characteristic

124

#103

—

125

#101

—

126

#101

—

127

C₂²⋊C₈

—

128

#95

—

129

C₁₉

#7405

yes

characteristic, complemented, Sylow

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂×Q₈

#64516

yes

characteristic

140

C₂×Q₈

D₁₅₂

yes

characteristic

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

146

Q₁₆

—

147

Q₁₆

—

148

M₁₆

—

149

M₁₆

—

152

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

152

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄²

—

159

C₂³

#10597263

yes

characteristic

160

Q₈

—

161

Q₈

—

162

Q₈

—

163

Q₈

—

164

Q₈

—

165

Q₈

—

152

—

166

Q₈

—

152

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

152

—

180

C₂×C₄

—

152

—

181

C₂×C₄

#10597278

yes

characteristic

182

C₂×C₄

#10597282

yes

characteristic

183

C₈

—

184

C₈

—

185

C₈

—

152

—

186

C₂²

—

187

C₂²

—

188

C₂²

#11926279

yes

characteristic

189

C₂²

#11926332

yes

characteristic

190

C₂²

#11926336

yes

characteristic

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

—

152

—

198

C₂

—

199

C₂

—

200

C₂

#14561633

yes

characteristic

201

C₂

#14561794

yes

characteristic

202

C₂

#14561803

yes

characteristic

	subgroup	G/H
center	C₂²	#11926336
commutator	C₂×C₄²×C₁₉	C₂×C₄
Frattini	#391	C₂×D₁₉
Fitting	C₁₉×#1319	C₂
socle	C₂²×C₁₉	#353
prime radical	C₂³×C₁₉	#99
cosocle	C₁₉×#391	C₂²
hypercenter	#1319	D₁₉
hypocenter	C₁₉	#7405
LCS	G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #391 ◁ #1319

subgroup

G/H

center

C₂²

#11926336

commutator

C₂×C₄²×C₁₉

C₂×C₄

Frattini

#391

C₂×D₁₉

Fitting

C₁₉×#1319

C₂

C₂²×C₁₉

C₂³×C₁₉

C₁₉×#391

C₂²

D₁₉

C₁₉

G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #391 ◁ #1319

N	H ≅ G/N	type
129. C₁₉	#7405	semidirect

H ≅ G/N

type

129. C₁₉

#7405

semidirect

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

subgroup	index
#14564734	2
#14561762	2
C₁₉×#1319	2
#7405	19

subgroup

index

#14564734

#14561762

C₁₉×#1319

#7405

Subgroups — #18070056

Conjugacy classes of subgroups

Complements

Maximal subgroups