Group #18069577 — subgroups

Subgroups — #18069577

Order 4864.

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

normal subgroups only

This group has 226 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	#14565434	C₂	1	1	yes	maximal, characteristic
2	#14563763	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1567	C₂	1	1	yes	maximal, characteristic
4	#11926414	—	2	1	no	—
5	C₄⋊(C₁₉⋊C₁₆)	—	2	1	no	—
6	C₂²⋊(C₁₉⋊C₁₆)	—	2	1	no	—
7	C₄×C₁₉⋊C₁₆	C₄	1	2	yes	complemented
8	#11926307	—	2	1	no	—
9	#11926307	C₄	1	2	yes	—
10	C₁₉×#423	—	2	1	no	—
11	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
12	C₁₉×#335	—	2	1	no	—
13	C₁₉×#335	C₄	1	2	yes	complemented
14	C₂×M₁₆×C₁₉	—	4	1	no	—
15	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
17	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₉	—	2	2	no	—
19	C₂×C₄⋊Dic₁₉	—	4	1	no	—
20	C₂×C₄⋊Dic₁₉	—	4	1	no	—
21	C₁₉×C₄⋊C₈	—	2	2	no	—
22	C₁₉×C₄⋊C₈	—	4	1	no	—
23	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
26	C₄×C₈×C₁₉	Q₈	1	1	yes	characteristic
27	C₂×C₁₉⋊C₁₆	—	2	2	no	—
28	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
30	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
31	C₂²×Dic₁₉	—	4	2	no	—
32	M₁₆×C₁₉	—	8	1	no	—
33	M₁₆×C₁₉	—	8	1	no	—
34	C₂×C₈×C₁₉	—	4	1	no	—
35	C₂×C₈×C₁₉	—	4	1	no	—
36	C₂×C₈×C₁₉	—	4	2	no	—
37	C₂×C₈×C₁₉	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
41	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₄²×C₁₉	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₁₉	SD₁₆	1	1	yes	characteristic
44	C₄²×C₁₉	Q₁₆	1	1	yes	characteristic
45	C₄⋊Dic₁₉	—	4	2	no	—
46	C₄⋊Dic₁₉	—	4	2	no	—
47	C₄⋊Dic₁₉	—	8	2	no	—
48	C₄⋊Dic₁₉	—	8	1	no	—
49	C₄⋊Dic₁₉	—	8	1	no	—
50	C₁₉⋊C₁₆	—	4	2	no	—
51	#7217	—	19	1	no	maximal, Sylow
52	C₂³×C₁₉	#102	1	1	yes	characteristic
53	C₂×C₄×C₁₉	—	2	1	no	—
54	C₂×C₄×C₁₉	—	2	1	no	—
55	C₂×C₄×C₁₉	—	2	2	no	—
56	C₂×C₄×C₁₉	#95	1	1	yes	characteristic
57	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
58	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
59	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
60	C₂×C₄×C₁₉	#106	1	1	yes	characteristic
61	C₂×C₄×C₁₉	#107	1	1	yes	characteristic
62	C₂×C₄×C₁₉	D₁₆	1	1	yes	characteristic
63	C₂×C₄×C₁₉	SD₃₂	1	1	yes	characteristic
64	C₂×C₄×C₁₉	SD₃₂	1	1	yes	characteristic
65	C₂×C₄×C₁₉	Q₃₂	1	1	yes	characteristic
66	C₂×Dic₁₉	—	8	2	no	—
67	C₂×Dic₁₉	—	8	2	no	—
68	C₂×Dic₁₉	—	8	2	no	—
69	C₈×C₁₉	—	2	1	no	—
70	C₈×C₁₉	—	2	1	no	—
71	C₈×C₁₉	—	2	1	no	—
72	C₈×C₁₉	—	2	1	no	—
73	C₈×C₁₉	—	8	2	no	—
74	#2162	—	19	1	no	—
75	#1834	—	19	1	no	—
76	#1567	D₁₉	1	1	yes	characteristic
77	C₂²×C₁₉	—	2	1	no	—
78	C₂²×C₁₉	—	2	1	no	—
79	C₂²×C₁₉	#340	1	1	yes	characteristic
80	C₂²×C₁₉	#357	1	1	yes	characteristic
81	C₂²×C₁₉	#358	1	1	yes	characteristic
82	C₄×C₁₉	—	2	1	no	—
83	C₄×C₁₉	—	2	1	no	—
84	C₄×C₁₉	H(3,ℤ/4)	1	1	yes	characteristic
85	C₄×C₁₉	#344	1	1	yes	characteristic
86	C₄×C₁₉	#357	1	1	yes	characteristic
87	C₄×C₁₉	#358	1	1	yes	characteristic
88	C₄×C₁₉	#365	1	2	yes	—
89	C₄×C₁₉	#366	1	1	yes	characteristic
90	C₄×C₁₉	#367	1	1	yes	characteristic
91	Dic₁₉	—	16	2	no	—
92	#423	—	2	1	no	—
93	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
94	#384	—	38	1	no	—
95	C₄⋊C₁₆	—	38	1	no	—
96	C₂²⋊C₁₆	—	38	1	no	—
97	C₄×C₁₆	—	19	2	no	—
98	#340	—	19	2	no	—
99	#340	—	38	1	no	—
100	#335	—	2	1	no	—
101	#335	Dic₁₉	1	2	yes	complemented
102	C₂×C₁₉	—	2	1	no	—
103	C₂×C₁₉	—	2	1	no	—
104	C₂×C₁₉	#1286	1	1	yes	characteristic
105	C₂×C₁₉	#1380	1	1	yes	characteristic
106	C₂×C₁₉	#1383	1	1	yes	characteristic
107	C₂×M₁₆	—	4	1	no	—
108	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
109	C₂²×C₈	D₇₆	1	1	yes	characteristic
110	C₂×C₄⋊C₄	—	38	2	no	—
111	C₂×C₄⋊C₄	—	76	1	no	—
112	C₂×C₄⋊C₄	—	76	1	no	—
113	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
114	C₂×C₁₆	—	38	2	no	—
115	C₄⋊C₈	—	2	2	no	—
116	C₄⋊C₈	—	4	1	no	—
117	C₄×C₈	Dic₃₈	1	1	yes	characteristic
118	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
119	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
120	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
121	C₁₉	#7217	1	1	yes	characteristic, complemented, Sylow
122	C₂²×C₄	—	76	2	no	—
123	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
124	C₂²×C₄	#64525	1	1	yes	characteristic
125	C₂²×C₄	#64527	1	1	yes	characteristic
126	M₁₆	—	8	1	no	—
127	M₁₆	—	8	1	no	—
128	C₂×C₈	—	4	1	no	—
129	C₂×C₈	—	4	1	no	—
130	C₂×C₈	—	4	2	no	—
131	C₂×C₈	C₄×Dic₁₉	1	1	yes	characteristic
132	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
133	C₂×C₈	C₄⋊Dic₁₉	1	1	yes	characteristic
134	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
135	C₄⋊C₄	—	76	2	no	—
136	C₄⋊C₄	—	76	2	no	—
137	C₄⋊C₄	—	152	2	no	—
138	C₄⋊C₄	—	152	1	no	—
139	C₄⋊C₄	—	152	1	no	—
140	C₄²	Dic₁₉⋊C₄	1	1	yes	characteristic
141	C₄²	#64526	1	1	yes	characteristic
142	C₄²	#64528	1	1	yes	characteristic
143	C₄²	C₂²⋊Dic₁₉	1	1	yes	characteristic
144	C₁₆	—	76	2	no	—
145	C₂³	#10597266	1	1	yes	characteristic
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	152	2	no	—
150	C₂×C₄	—	152	2	no	—
151	C₂×C₄	—	152	2	no	—
152	C₂×C₄	#10597264	1	1	yes	characteristic
153	C₂×C₄	#10597265	1	1	yes	characteristic
154	C₂×C₄	#10597267	1	1	yes	characteristic
155	C₂×C₄	#10597283	1	1	yes	characteristic
156	C₂×C₄	#10597284	1	1	yes	characteristic
157	C₂×C₄	#10597285	1	1	yes	characteristic
158	C₂×C₄	#10597286	1	1	yes	characteristic
159	C₂×C₄	C₁₉⋊#95	1	1	yes	characteristic
160	C₂×C₄	#10597289	1	1	yes	characteristic
161	C₂×C₄	#10597292	1	1	yes	characteristic
162	C₈	—	2	1	no	—
163	C₈	—	2	1	no	—
164	C₈	—	2	1	no	—
165	C₈	—	2	1	no	—
166	C₈	—	8	2	no	—
167	C₂²	—	2	1	no	—
168	C₂²	—	2	1	no	—
169	C₂²	#11926306	1	1	yes	characteristic
170	C₂²	#11926344	1	1	yes	characteristic
171	C₂²	#11926347	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	304	2	no	—
175	C₄	#11926301	1	1	yes	characteristic
176	C₄	#11926316	1	1	yes	characteristic
177	C₄	#11926345	1	1	yes	characteristic
178	C₄	#11926348	1	1	yes	characteristic
179	C₄	#11926368	1	2	yes	—
180	C₄	#11926369	1	1	yes	characteristic
181	C₄	#11926371	1	1	yes	characteristic
182	C₂	—	2	1	no	—
183	C₂	—	2	1	no	—
184	C₂	#14561495	1	1	yes	characteristic
185	C₂	#14561726	1	1	yes	characteristic
186	C₂	#14561735	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14565434

C₂

yes

maximal, characteristic

#14563763

C₂

yes

maximal, characteristic

C₁₉×#1567

C₂

yes

maximal, characteristic

#11926414

—

C₄⋊(C₁₉⋊C₁₆)

—

C₂²⋊(C₁₉⋊C₁₆)

—

C₄×C₁₉⋊C₁₆

C₄

yes

complemented

#11926307

—

#11926307

C₄

yes

—

C₁₉×#423

—

C₂×C₄×C₈×C₁₉

C₂²

yes

characteristic

C₁₉×#335

—

C₁₉×#335

C₄

yes

complemented

C₂×M₁₆×C₁₉

—

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₂×C₄⋊Dic₁₉

—

C₂×C₄⋊Dic₁₉

—

C₂×C₄⋊Dic₁₉

—

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

C₄×C₈×C₁₉

C₂×C₄

yes

characteristic

C₄×C₈×C₁₉

C₂×C₄

yes

characteristic

C₄×C₈×C₁₉

D₄

yes

characteristic

C₄×C₈×C₁₉

Q₈

yes

characteristic

C₂×C₁₉⋊C₁₆

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

D₈

yes

characteristic

C₂²×C₄×C₁₉

SD₁₆

yes

characteristic

C₂²×Dic₁₉

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₄²

yes

characteristic

C₂×C₈×C₁₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₄⋊C₄

yes

characteristic

C₄²×C₁₉

C₂²⋊C₄

yes

characteristic

C₄²×C₁₉

C₄⋊C₄

yes

characteristic

C₄²×C₁₉

SD₁₆

yes

characteristic

C₄²×C₁₉

Q₁₆

yes

characteristic

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₁₉⋊C₁₆

—

—

C₂³×C₁₉

yes

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

D₁₆

yes

characteristic

C₂×C₄×C₁₉

SD₃₂

yes

characteristic

C₂×C₄×C₁₉

SD₃₂

yes

characteristic

C₂×C₄×C₁₉

Q₃₂

yes

characteristic

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

#2162

—

#1834

—

#1567

D₁₉

yes

characteristic

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

—

C₄×C₁₉

yes

C₄×C₁₉

yes

Dic₁₉

—

#423

—

C₂×C₄×C₈

C₂×D₁₉

yes

characteristic

#384

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

#340

—

#340

—

100

#335

—

101

#335

Dic₁₉

yes

complemented

102

C₂×C₁₉

—

103

C₂×C₁₉

—

104

C₂×C₁₉

#1286

yes

characteristic

105

C₂×C₁₉

#1380

yes

characteristic

106

C₂×C₁₉

#1383

yes

characteristic

107

C₂×M₁₆

—

108

C₂²×C₈

C₄×D₁₉

yes

characteristic

109

C₂²×C₈

D₇₆

yes

characteristic

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄²

C₂²⋊D₁₉

yes

characteristic

114

C₂×C₁₆

—

115

C₄⋊C₈

—

116

C₄⋊C₈

—

117

C₄×C₈

Dic₃₈

yes

characteristic

118

C₄×C₈

C₄×D₁₉

yes

characteristic

119

C₄×C₈

C₂×Dic₁₉

yes

characteristic

120

C₄×C₈

C₂²⋊D₁₉

yes

characteristic

121

C₁₉

#7217

yes

characteristic, complemented, Sylow

122

C₂²×C₄

—

123

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

124

C₂²×C₄

#64525

yes

characteristic

125

C₂²×C₄

#64527

yes

characteristic

126

M₁₆

—

127

M₁₆

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

C₄×Dic₁₉

yes

characteristic

132

C₂×C₈

Dic₁₉⋊C₄

yes

characteristic

133

C₂×C₈

C₄⋊Dic₁₉

yes

characteristic

134

C₂×C₈

(C₂×D₁₉)⋊C₄

yes

characteristic

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

152

—

138

C₄⋊C₄

—

152

—

139

C₄⋊C₄

—

152

—

140

C₄²

Dic₁₉⋊C₄

yes

characteristic

141

C₄²

#64526

yes

characteristic

142

C₄²

#64528

yes

characteristic

143

C₄²

C₂²⋊Dic₁₉

yes

characteristic

144

C₁₆

—

145

C₂³

#10597266

yes

characteristic

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

152

—

150

C₂×C₄

—

152

—

151

C₂×C₄

—

152

—

152

C₂×C₄

#10597264

yes

characteristic

153

C₂×C₄

#10597265

yes

characteristic

154

C₂×C₄

#10597267

yes

characteristic

155

C₂×C₄

#10597283

yes

characteristic

156

C₂×C₄

#10597284

yes

characteristic

157

C₂×C₄

#10597285

yes

characteristic

158

C₂×C₄

#10597286

yes

characteristic

159

C₂×C₄

C₁₉⋊#95

yes

characteristic

160

C₂×C₄

#10597289

yes

characteristic

161

C₂×C₄

#10597292

yes

characteristic

162

C₈

—

163

C₈

—

164

C₈

—

165

C₈

—

166

C₈

—

167

C₂²

—

168

C₂²

—

169

C₂²

#11926306

yes

characteristic

170

C₂²

#11926344

yes

characteristic

171

C₂²

#11926347

yes

characteristic

172

C₄

—

173

C₄

—

174

C₄

—

304

—

175

C₄

#11926301

yes

characteristic

176

C₄

#11926316

yes

characteristic

177

C₄

#11926345

yes

characteristic

178

C₄

#11926348

yes

characteristic

179

C₄

#11926368

yes

—

180

C₄

#11926369

yes

characteristic

181

C₄

#11926371

yes

characteristic

182

C₂

—

183

C₂

—

184

C₂

#14561495

yes

characteristic

185

C₂

#14561726

yes

characteristic

186

C₂

#14561735

yes

characteristic

	subgroup	G/H
center	C₂²	#11926306
commutator	C₂×C₈×C₁₉	C₄²
Frattini	C₂×C₄×C₈	C₂×D₁₉
Fitting	C₁₉×#1567	C₂
socle	C₂²×C₁₉	#340
prime radical	C₂³×C₁₉	#102
cosocle	C₂×C₄×C₈×C₁₉	C₂²
hypercenter	#1567	D₁₉
hypocenter	C₁₉	#7217
LCS	G ▷ C₂×C₈×C₁₉ ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1567

subgroup

G/H

center

C₂²

#11926306

commutator

C₂×C₈×C₁₉

C₄²

Frattini

C₂×C₄×C₈

C₂×D₁₉

Fitting

C₁₉×#1567

C₂

C₂²×C₁₉

C₂³×C₁₉

C₂×C₄×C₈×C₁₉

C₂²

D₁₉

C₁₉

G ▷ C₂×C₈×C₁₉ ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1567

N	H ≅ G/N	type
7. C₄×C₁₉⋊C₁₆	C₄	semidirect
13. C₁₉×#335	C₄	semidirect
101. #335	Dic₁₉	semidirect
121. C₁₉	#7217	semidirect

H ≅ G/N

type

7. C₄×C₁₉⋊C₁₆

C₄

semidirect

13. C₁₉×#335

C₄

semidirect

101. #335

Dic₁₉

semidirect

121. C₁₉

#7217

semidirect

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

subgroup	index
#14565434	2
#14563763	2
C₁₉×#1567	2
#7217	19

subgroup

index

#14565434

#14563763

C₁₉×#1567

#7217

Subgroups — #18069577

Conjugacy classes of subgroups

Complements

Maximal subgroups