Group #18069621 — subgroups

Subgroups — #18069621

Order 4864.

264 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 83 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	#14565246	C₂	1	1	yes	maximal, characteristic
2	#14563673	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1810	C₂	1	1	yes	maximal, characteristic
4	C₂×#10597269	C₄	1	2	yes	—
5	C₂×#10597276	—	2	1	no	—
6	#11926508	—	2	1	no	—
7	C₂²⋊(C₁₉⋊C₁₆)	—	2	1	no	—
8	C₂²⋊(C₁₉⋊C₁₆)	—	2	1	no	—
9	#11926309	C₄	1	2	yes	—
10	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×C₁₉×#108	—	2	1	no	—
12	C₁₉×#407	—	2	1	no	—
13	C₁₉×#341	C₄	1	2	yes	—
14	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
15	C₂×M₁₆×C₁₉	—	2	2	no	—
16	C₂²×C₈×C₁₉	—	4	2	no	—
17	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₉	Q₈	1	1	yes	characteristic
23	C₂×#64520	—	2	2	no	—
24	C₁₉×#108	—	4	2	no	—
25	C₁₉×C₂²⋊C₈	—	4	1	no	—
26	C₁₉×C₂²⋊C₈	—	4	1	no	—
27	C₂²⋊(C₁₉⋊C₈)	—	4	1	no	—
28	C₂²⋊(C₁₉⋊C₈)	—	4	1	no	—
29	#10597276	—	4	2	no	—
30	#10597269	—	4	2	no	—
31	C₂×C₁₉⋊C₁₆	—	2	2	no	—
32	C₂⁴×C₁₉	Q₁₆	1	1	yes	characteristic
33	C₂²×C₄×C₁₉	—	4	1	no	—
34	C₂²×C₄×C₁₉	—	4	1	no	—
35	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
39	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
40	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
41	M₁₆×C₁₉	—	4	2	no	—
42	M₁₆×C₁₉	—	4	2	no	—
43	C₂×C₈×C₁₉	—	2	2	no	—
44	C₂×C₈×C₁₉	—	2	2	no	—
45	C₂×C₈×C₁₉	—	2	2	no	—
46	C₂×C₈×C₁₉	—	4	1	no	—
47	C₂×C₈×C₁₉	—	4	1	no	—
48	C₂×C₈×C₁₉	—	4	1	no	—
49	C₂×C₈×C₁₉	—	4	2	no	—
50	C₂×C₈×C₁₉	C₄²	1	1	yes	characteristic
51	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
54	#64520	—	4	2	no	—
55	#64520	—	4	2	no	—
56	C₂×C₁₉⋊C₈	—	4	2	no	—
57	C₁₉⋊C₁₆	—	4	2	no	—
58	#7234	—	19	1	no	maximal, Sylow
59	C₂³×C₁₉	—	4	1	no	—
60	C₂³×C₁₉	—	4	1	no	—
61	C₂³×C₁₉	—	4	1	no	—
62	C₂³×C₁₉	#103	1	1	yes	characteristic
63	C₂³×C₁₉	#103	1	1	yes	characteristic
64	C₂³×C₁₉	#107	1	1	yes	characteristic
65	C₂×C₄×C₁₉	—	2	1	no	—
66	C₂×C₄×C₁₉	—	2	1	no	—
67	C₂×C₄×C₁₉	—	2	1	no	—
68	C₂×C₄×C₁₉	—	2	1	no	—
69	C₂×C₄×C₁₉	—	2	1	no	—
70	C₂×C₄×C₁₉	—	2	1	no	—
71	C₂×C₄×C₁₉	—	4	1	no	—
72	C₂×C₄×C₁₉	—	4	1	no	—
73	C₂×C₄×C₁₉	—	4	1	no	—
74	C₂×C₄×C₁₉	#95	1	1	yes	characteristic
75	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
76	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
77	C₂×C₄×C₁₉	#106	1	1	yes	characteristic
78	C₈×C₁₉	—	2	1	no	—
79	C₈×C₁₉	—	2	1	no	—
80	C₈×C₁₉	—	2	1	no	—
81	C₈×C₁₉	—	8	2	no	—
82	C₈×C₁₉	#99	1	2	yes	—
83	C₁₉⋊C₈	—	8	2	no	—
84	#2112	—	19	1	no	—
85	#1810	—	19	1	no	—
86	#1810	D₁₉	1	1	yes	characteristic
87	C₂²×C₁₉	—	2	1	no	—
88	C₂²×C₁₉	—	2	1	no	—
89	C₂²×C₁₉	—	2	1	no	—
90	C₂²×C₁₉	—	2	1	no	—
91	C₂²×C₁₉	—	2	1	no	—
92	C₂²×C₁₉	—	2	1	no	—
93	C₂²×C₁₉	—	2	1	no	—
94	C₂²×C₁₉	—	2	1	no	—
95	C₂²×C₁₉	—	2	1	no	—
96	C₂²×C₁₉	—	4	4	no	—
97	C₂²×C₁₉	#340	1	1	yes	characteristic
98	C₄×C₁₉	—	2	1	no	—
99	C₄×C₁₉	—	2	1	no	—
100	C₄×C₁₉	—	2	1	no	—
101	C₄×C₁₉	H(3,ℤ/4)	1	1	yes	characteristic
102	C₄×C₁₉	#342	1	1	yes	characteristic
103	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
104	C₂×M₃₂	—	19	2	no	—
105	C₂×#108	—	2	1	no	—
106	C₂×#108	—	38	1	no	—
107	#407	—	2	1	no	—
108	#407	—	38	1	no	—
109	C₂²⋊C₁₆	—	38	1	no	—
110	C₂²⋊C₁₆	—	38	1	no	—
111	#341	—	19	2	no	—
112	#341	Dic₁₉	1	2	yes	—
113	C₂×C₁₉	—	4	1	no	—
114	C₂×C₁₉	—	4	1	no	—
115	C₂×C₁₉	—	4	1	no	—
116	C₂×C₁₉	#1289	1	1	yes	characteristic
117	C₂×C₁₉	#1388	1	1	yes	characteristic
118	C₂×C₁₉	#1388	1	1	yes	characteristic
119	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
120	C₂×M₁₆	—	2	2	no	—
121	C₂×M₁₆	—	38	2	no	—
122	C₂²×C₈	—	4	2	no	—
123	C₂²×C₈	Dic₃₈	1	1	yes	characteristic
124	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
125	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
126	C₂²×C₈	D₇₆	1	1	yes	characteristic
127	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
128	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
129	M₃₂	—	76	2	no	—
130	C₂×C₁₆	—	38	2	no	—
131	#108	—	4	2	no	—
132	#108	—	76	2	no	—
133	C₂²⋊C₈	—	4	1	no	—
134	C₂²⋊C₈	—	4	1	no	—
135	C₂²⋊C₈	—	76	1	no	—
136	C₂²⋊C₈	—	76	1	no	—
137	C₁₉	#7234	1	1	yes	characteristic, complemented, Sylow
138	C₂⁴	#64528	1	1	yes	characteristic
139	C₂²×C₄	—	4	1	no	—
140	C₂²×C₄	—	4	1	no	—
141	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
142	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
143	C₂²×C₄	#64525	1	1	yes	characteristic
144	C₂²×C₄	#64526	1	1	yes	characteristic
145	C₂²×C₄	#64527	1	1	yes	characteristic
146	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
147	M₁₆	—	4	2	no	—
148	M₁₆	—	4	2	no	—
149	M₁₆	—	76	2	no	—
150	M₁₆	—	76	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	4	1	no	—
155	C₂×C₈	—	4	1	no	—
156	C₂×C₈	—	4	1	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	76	2	no	—
159	C₂×C₈	C₄×Dic₁₉	1	1	yes	characteristic
160	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
161	C₂×C₈	C₄⋊Dic₁₉	1	1	yes	characteristic
162	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
163	C₁₆	—	76	2	no	—
164	C₂³	—	4	1	no	—
165	C₂³	—	4	1	no	—
166	C₂³	—	4	1	no	—
167	C₂³	#10597264	1	1	yes	characteristic
168	C₂³	#10597267	1	1	yes	characteristic
169	C₂³	#10597292	1	1	yes	characteristic
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	4	1	no	—
177	C₂×C₄	—	4	1	no	—
178	C₂×C₄	—	4	1	no	—
179	C₂×C₄	#10597265	1	1	yes	characteristic
180	C₂×C₄	#10597266	1	1	yes	characteristic
181	C₂×C₄	C₁₉⋊#95	1	1	yes	characteristic
182	C₂×C₄	#10597289	1	1	yes	characteristic
183	C₈	—	2	1	no	—
184	C₈	—	2	1	no	—
185	C₈	—	2	1	no	—
186	C₈	—	8	2	no	—
187	C₈	—	152	2	no	—
188	C₈	#10597263	1	2	yes	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	2	1	no	—
198	C₂²	—	4	4	no	—
199	C₂²	#11926306	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	#11926300	1	1	yes	characteristic
204	C₄	#11926310	1	1	yes	characteristic
205	C₂	—	4	1	no	—
206	C₂	—	4	1	no	—
207	C₂	—	4	1	no	—
208	C₂	#14561502	1	1	yes	characteristic
209	C₂	#14561746	1	1	yes	characteristic
210	C₂	#14561748	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14565246

C₂

yes

maximal, characteristic

#14563673

C₂

yes

maximal, characteristic

C₁₉×#1810

C₂

yes

maximal, characteristic

C₂×#10597269

C₄

yes

—

C₂×#10597276

—

#11926508

—

C₂²⋊(C₁₉⋊C₁₆)

—

C₂²⋊(C₁₉⋊C₁₆)

—

#11926309

C₄

yes

—

C₂³×C₈×C₁₉

C₂²

yes

characteristic

C₂×C₁₉×#108

—

C₁₉×#407

—

C₁₉×#341

C₄

yes

—

C₂³×C₄×C₁₉

D₄

yes

characteristic

C₂×M₁₆×C₁₉

—

C₂²×C₈×C₁₉

—

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂²×C₈×C₁₉

Q₈

yes

characteristic

C₂×#64520

—

C₁₉×#108

—

C₁₉×C₂²⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₂²⋊(C₁₉⋊C₈)

—

C₂²⋊(C₁₉⋊C₈)

—

#10597276

—

#10597269

—

C₂×C₁₉⋊C₁₆

—

C₂⁴×C₁₉

Q₁₆

yes

characteristic

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

D₈

yes

characteristic

C₂²×C₄×C₁₉

SD₁₆

yes

characteristic

C₂²×C₄×C₁₉

SD₁₆

yes

characteristic

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×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

#64520

—

#64520

—

C₂×C₁₉⋊C₈

—

C₁₉⋊C₁₆

—

#7234

—

maximal, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

yes

C₂³×C₁₉

yes

C₂³×C₁₉

yes

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×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

#99

yes

—

C₁₉⋊C₈

—

#2112

—

#1810

—

#1810

D₁₉

yes

characteristic

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#340

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

100

C₄×C₁₉

—

101

C₄×C₁₉

H(3,ℤ/4)

yes

characteristic

102

C₄×C₁₉

#342

yes

characteristic

103

C₂³×C₈

C₂×D₁₉

yes

characteristic

104

C₂×M₃₂

—

105

C₂×#108

—

106

C₂×#108

—

107

#407

—

108

#407

—

109

C₂²⋊C₁₆

—

110

C₂²⋊C₁₆

—

111

#341

—

112

#341

Dic₁₉

yes

—

113

C₂×C₁₉

—

114

C₂×C₁₉

—

115

C₂×C₁₉

—

116

C₂×C₁₉

#1289

yes

characteristic

117

C₂×C₁₉

#1388

yes

characteristic

118

C₂×C₁₉

#1388

yes

characteristic

119

C₂³×C₄

C₂²⋊D₁₉

yes

characteristic

120

C₂×M₁₆

—

121

C₂×M₁₆

—

122

C₂²×C₈

—

123

C₂²×C₈

Dic₃₈

yes

characteristic

124

C₂²×C₈

C₄×D₁₉

yes

characteristic

125

C₂²×C₈

C₄×D₁₉

yes

characteristic

126

C₂²×C₈

D₇₆

yes

characteristic

127

C₂²×C₈

C₂×Dic₁₉

yes

characteristic

128

C₂²×C₈

C₂²⋊D₁₉

yes

characteristic

129

M₃₂

—

130

C₂×C₁₆

—

131

#108

—

132

#108

—

133

C₂²⋊C₈

—

134

C₂²⋊C₈

—

135

C₂²⋊C₈

—

136

C₂²⋊C₈

—

137

C₁₉

#7234

yes

characteristic, complemented, Sylow

138

C₂⁴

#64528

yes

characteristic

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

Dic₁₉⋊C₄

yes

characteristic

142

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

143

C₂²×C₄

#64525

yes

characteristic

144

C₂²×C₄

#64526

yes

characteristic

145

C₂²×C₄

#64527

yes

characteristic

146

C₂²×C₄

C₂²⋊Dic₁₉

yes

characteristic

147

M₁₆

—

148

M₁₆

—

149

M₁₆

—

150

M₁₆

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

C₄×Dic₁₉

yes

characteristic

160

C₂×C₈

Dic₁₉⋊C₄

yes

characteristic

161

C₂×C₈

C₄⋊Dic₁₉

yes

characteristic

162

C₂×C₈

(C₂×D₁₉)⋊C₄

yes

characteristic

163

C₁₆

—

164

C₂³

—

165

C₂³

—

166

C₂³

—

167

C₂³

#10597264

yes

characteristic

168

C₂³

#10597267

yes

characteristic

169

C₂³

#10597292

yes

characteristic

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

#10597265

yes

characteristic

180

C₂×C₄

#10597266

yes

characteristic

181

C₂×C₄

C₁₉⋊#95

yes

characteristic

182

C₂×C₄

#10597289

yes

characteristic

183

C₈

—

184

C₈

—

185

C₈

—

186

C₈

—

187

C₈

—

152

—

188

C₈

#10597263

yes

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

194

C₂²

—

195

C₂²

—

196

C₂²

—

197

C₂²

—

198

C₂²

—

199

C₂²

#11926306

yes

characteristic

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

#11926300

yes

characteristic

204

C₄

#11926310

yes

characteristic

205

C₂

—

206

C₂

—

207

C₂

—

208

C₂

#14561502

yes

characteristic

209

C₂

#14561746

yes

characteristic

210

C₂

#14561748

yes

characteristic

	subgroup	G/H
center	C₂²	#11926306
commutator	C₂×C₈×C₁₉	C₄²
Frattini	C₂³×C₈	C₂×D₁₉
Fitting	C₁₉×#1810	C₂
socle	C₂²×C₁₉	#340
prime radical	C₂⁴×C₁₉	Q₁₆
cosocle	C₂³×C₈×C₁₉	C₂²
hypercenter	#1810	D₁₉
hypocenter	C₁₉	#7234
LCS	G ▷ C₂×C₈×C₁₉ ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1810

subgroup

G/H

center

C₂²

#11926306

commutator

C₂×C₈×C₁₉

C₄²

Frattini

C₂³×C₈

C₂×D₁₉

Fitting

C₁₉×#1810

C₂

socle

C₂²×C₁₉

#340

prime radical

C₂⁴×C₁₉

Q₁₆

cosocle

C₂³×C₈×C₁₉

C₂²

D₁₉

C₁₉

G ▷ C₂×C₈×C₁₉ ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1810

N	H ≅ G/N	type
137. C₁₉	#7234	semidirect

H ≅ G/N

type

137. C₁₉

#7234

semidirect

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

subgroup	index
#14565246	2
#14563673	2
C₁₉×#1810	2
#7234	19

subgroup

index

#14565246

#14563673

C₁₉×#1810

#7234

Subgroups — #18069621

Conjugacy classes of subgroups

Complements

Maximal subgroups