Group #18069178 — subgroups

Subgroups — #18069178

Order 4864.

224 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 59 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	#14561559	C₂	1	1	yes	maximal, characteristic
2	#14561559	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1850	C₂	1	1	yes	maximal, characteristic
4	C₂²⋊(C₁₉⋊C₁₆)	—	2	1	no	—
5	C₂²⋊(C₁₉⋊C₁₆)	—	2	1	no	—
6	C₂³×C₈×C₁₉	C₄	1	1	yes	characteristic
7	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
8	C₂×C₁₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
9	C₄×C₂²⋊C₄×C₁₉	C₄	1	1	yes	characteristic
10	C₁₉×#336	—	2	1	no	—
11	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₈×C₁₉	—	2	1	no	—
13	C₂²×C₈×C₁₉	—	4	2	no	—
14	C₂²×C₈×C₁₉	—	4	1	no	—
15	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
17	C₂×C₂²⋊C₄×C₁₉	C₈	1	1	yes	characteristic
18	C₂×C₂²⋊C₄×C₁₉	C₈	1	1	yes	characteristic
19	C₂×C₄²×C₁₉	—	2	1	no	—
20	C₁₉×C₄⋊C₈	—	4	2	no	—
21	C₁₉×C₂²⋊C₈	—	4	1	no	—
22	C₁₉×#95	—	2	1	no	—
23	C₂×C₁₉⋊C₁₆	—	4	1	no	—
24	C₂×C₁₉⋊C₁₆	—	4	1	no	—
25	C₂⁴×C₁₉	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₄×C₁₉	—	2	1	no	—
27	C₂²×C₄×C₁₉	—	2	1	no	—
28	C₂²×C₄×C₁₉	—	2	1	no	—
29	C₂²×C₄×C₁₉	—	2	1	no	—
30	C₂²×C₄×C₁₉	—	4	1	no	—
31	C₂²×C₄×C₁₉	—	4	1	no	—
32	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₉	M₁₆	1	1	yes	characteristic
34	C₂×C₈×C₁₉	—	2	2	no	—
35	C₂×C₈×C₁₉	—	2	1	no	—
36	C₂×C₈×C₁₉	—	2	1	no	—
37	C₂×C₈×C₁₉	—	4	1	no	—
38	C₂×C₈×C₁₉	—	4	2	no	—
39	C₂×C₈×C₁₉	—	4	1	no	—
40	C₂×C₈×C₁₉	—	4	1	no	—
41	C₂×C₈×C₁₉	—	4	1	no	—
42	C₂×C₈×C₁₉	—	4	2	no	—
43	C₂²⋊C₄×C₁₉	—	2	1	no	—
44	C₂²⋊C₄×C₁₉	—	4	1	no	—
45	C₂²⋊C₄×C₁₉	C₁₆	1	2	yes	complemented
46	C₄²×C₁₉	—	4	1	no	—
47	C₄²×C₁₉	—	4	1	no	—
48	C₁₉⋊C₁₆	—	8	1	no	—
49	C₁₉⋊C₁₆	—	8	1	no	—
50	#7064	—	19	1	no	maximal, Sylow
51	C₂³×C₁₉	—	2	1	no	—
52	C₂³×C₁₉	—	2	1	no	—
53	C₂³×C₁₉	—	4	1	no	—
54	C₂³×C₁₉	—	4	1	no	—
55	C₂³×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
56	C₂³×C₁₉	C₂×C₁₆	1	1	yes	characteristic
57	C₂³×C₁₉	M₃₂	1	1	yes	characteristic
58	C₂×C₄×C₁₉	—	2	1	no	—
59	C₂×C₄×C₁₉	—	2	1	no	—
60	C₂×C₄×C₁₉	—	2	1	no	—
61	C₂×C₄×C₁₉	—	2	2	no	—
62	C₂×C₄×C₁₉	—	4	2	no	—
63	C₂×C₄×C₁₉	—	4	1	no	—
64	C₂×C₄×C₁₉	—	4	1	no	—
65	C₂×C₄×C₁₉	—	4	1	no	—
66	C₂×C₄×C₁₉	—	4	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	#99	1	1	yes	characteristic
72	C₂×C₄×C₁₉	#101	1	1	yes	characteristic
73	C₈×C₁₉	—	4	1	no	—
74	C₈×C₁₉	—	4	2	no	—
75	C₈×C₁₉	—	4	1	no	—
76	#1850	D₁₉	1	1	yes	characteristic
77	#1314	—	19	1	no	—
78	#1314	—	19	1	no	—
79	C₂²×C₁₉	—	2	1	no	—
80	C₂²×C₁₉	—	2	1	no	—
81	C₂²×C₁₉	—	2	1	no	—
82	C₂²×C₁₉	—	2	1	no	—
83	C₂²×C₁₉	—	4	1	no	—
84	C₂²×C₁₉	—	4	1	no	—
85	C₂²×C₁₉	—	4	2	no	—
86	C₂²×C₁₉	—	4	1	no	—
87	C₂²×C₁₉	—	4	1	no	—
88	C₂²×C₁₉	#324	1	1	yes	characteristic
89	C₂²×C₁₉	C₂²⋊C₁₆	1	1	yes	characteristic
90	C₂²×C₁₉	#349	1	1	yes	characteristic
91	C₄×C₁₉	—	2	1	no	—
92	C₄×C₁₉	—	4	2	no	—
93	C₄×C₁₉	—	4	1	no	—
94	C₄×C₁₉	—	8	1	no	—
95	C₄×C₁₉	AΣL(1,𝔽₁₆)	1	1	yes	characteristic
96	C₄×C₁₉	#352	1	1	yes	characteristic
97	C₂³×C₈	Dic₁₉	1	1	yes	characteristic
98	C₂×C₄⋊C₈	—	2	1	no	—
99	C₂×C₂²⋊C₈	C₂×D₁₉	1	1	yes	characteristic
100	C₄×C₂²⋊C₄	Dic₁₉	1	1	yes	characteristic
101	C₂²⋊C₁₆	—	38	1	no	—
102	C₂²⋊C₁₆	—	38	1	no	—
103	#336	—	2	1	no	—
104	C₂×C₁₉	—	2	1	no	—
105	C₂×C₁₉	—	2	1	no	—
106	C₂×C₁₉	—	4	1	no	—
107	C₂×C₁₉	—	4	1	no	—
108	C₂×C₁₉	#1322	1	1	yes	characteristic
109	C₂×C₁₉	#1326	1	1	yes	characteristic
110	C₂×C₁₉	#1328	1	1	yes	characteristic
111	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
112	C₂²×C₈	—	2	1	no	—
113	C₂²×C₈	—	4	2	no	—
114	C₂²×C₈	—	4	1	no	—
115	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
116	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
117	C₂×C₂²⋊C₄	C₁₉⋊C₈	1	1	yes	characteristic
118	C₂×C₂²⋊C₄	C₁₉⋊C₈	1	1	yes	characteristic
119	C₂×C₄²	—	2	1	no	—
120	C₂×C₁₆	—	76	1	no	—
121	C₂×C₁₆	—	76	1	no	—
122	C₄⋊C₈	—	4	2	no	—
123	C₂²⋊C₈	—	4	1	no	—
124	#95	—	2	1	no	—
125	C₁₉	#7064	1	1	yes	characteristic, complemented, Sylow
126	C₂⁴	C₂×C₁₉⋊C₈	1	1	yes	characteristic
127	C₂²×C₄	—	2	1	no	—
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	—	2	1	no	—
130	C₂²×C₄	—	2	1	no	—
131	C₂²×C₄	—	4	1	no	—
132	C₂²×C₄	—	4	1	no	—
133	C₂²×C₄	#64520	1	1	yes	characteristic
134	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	4	1	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	4	1	no	—
141	C₂×C₈	—	4	1	no	—
142	C₂×C₈	—	4	1	no	—
143	C₂×C₈	—	4	2	no	—
144	C₂²⋊C₄	—	2	1	no	—
145	C₂²⋊C₄	—	4	1	no	—
146	C₂²⋊C₄	C₁₉⋊C₁₆	1	2	yes	complemented
147	C₄²	—	4	1	no	—
148	C₄²	—	4	1	no	—
149	C₁₆	—	152	1	no	—
150	C₁₆	—	152	1	no	—
151	C₂³	—	2	1	no	—
152	C₂³	—	2	1	no	—
153	C₂³	—	4	1	no	—
154	C₂³	—	4	1	no	—
155	C₂³	C₂×C₁₉⋊C₁₆	1	1	yes	characteristic
156	C₂³	#10597269	1	1	yes	characteristic
157	C₂³	C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	4	1	no	—
165	C₂×C₄	—	4	1	no	—
166	C₂×C₄	—	4	1	no	—
167	C₂×C₄	—	4	1	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	#10597291	1	1	yes	characteristic
172	C₂×C₄	#10597293	1	1	yes	characteristic
173	C₈	—	4	1	no	—
174	C₈	—	4	2	no	—
175	C₈	—	4	1	no	—
176	C₂²	—	2	1	no	—
177	C₂²	—	2	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	—	2	1	no	—
180	C₂²	—	4	1	no	—
181	C₂²	—	4	1	no	—
182	C₂²	—	4	2	no	—
183	C₂²	—	4	1	no	—
184	C₂²	—	4	1	no	—
185	C₂²	#11926270	1	1	yes	characteristic
186	C₂²	C₂²⋊(C₁₉⋊C₁₆)	1	1	yes	characteristic
187	C₂²	#11926326	1	1	yes	characteristic
188	C₄	—	2	1	no	—
189	C₄	—	4	2	no	—
190	C₄	—	4	1	no	—
191	C₄	—	8	1	no	—
192	C₄	#11926331	1	1	yes	characteristic
193	C₄	#11926334	1	1	yes	characteristic
194	C₂	—	2	1	no	—
195	C₂	—	2	1	no	—
196	C₂	—	4	1	no	—
197	C₂	—	4	1	no	—
198	C₂	#14561575	1	1	yes	characteristic
199	C₂	#14561583	1	1	yes	characteristic
200	C₂	#14561588	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14561559

C₂

yes

maximal, characteristic

#14561559

C₂

yes

maximal, characteristic

C₁₉×#1850

C₂

yes

maximal, characteristic

C₂²⋊(C₁₉⋊C₁₆)

—

C₂²⋊(C₁₉⋊C₁₆)

—

C₂³×C₈×C₁₉

C₄

yes

characteristic

C₂×C₁₉×C₄⋊C₈

—

C₂×C₁₉×C₂²⋊C₈

C₂²

yes

characteristic

C₄×C₂²⋊C₄×C₁₉

C₄

yes

characteristic

C₁₉×#336

—

C₂³×C₄×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₉

—

C₂²×C₈×C₁₉

—

C₂²×C₈×C₁₉

—

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂×C₂²⋊C₄×C₁₉

C₈

yes

characteristic

C₂×C₂²⋊C₄×C₁₉

C₈

yes

characteristic

C₂×C₄²×C₁₉

—

C₁₉×C₄⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₁₉×#95

—

C₂×C₁₉⋊C₁₆

—

C₂×C₁₉⋊C₁₆

—

C₂⁴×C₁₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

M₁₆

yes

characteristic

C₂×C₈×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

—

C₂²⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

C₁₆

yes

complemented

C₄²×C₁₉

—

C₄²×C₁₉

—

C₁₉⋊C₁₆

—

C₁₉⋊C₁₆

—

#7064

—

maximal, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

C₂²⋊C₈

yes

characteristic

C₂³×C₁₉

C₂×C₁₆

yes

characteristic

C₂³×C₁₉

M₃₂

yes

characteristic

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

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

#1850

D₁₉

yes

characteristic

#1314

—

#1314

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#324

yes

characteristic

C₂²×C₁₉

C₂²⋊C₁₆

yes

characteristic

C₂²×C₁₉

#349

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

AΣL(1,𝔽₁₆)

yes

characteristic

C₄×C₁₉

#352

yes

characteristic

C₂³×C₈

Dic₁₉

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

C₂×D₁₉

yes

characteristic

100

C₄×C₂²⋊C₄

Dic₁₉

yes

characteristic

101

C₂²⋊C₁₆

—

102

C₂²⋊C₁₆

—

103

#336

—

104

C₂×C₁₉

—

105

C₂×C₁₉

—

106

C₂×C₁₉

—

107

C₂×C₁₉

—

108

C₂×C₁₉

#1322

yes

characteristic

109

C₂×C₁₉

#1326

yes

characteristic

110

C₂×C₁₉

#1328

yes

characteristic

111

C₂³×C₄

C₂×Dic₁₉

yes

characteristic

112

C₂²×C₈

—

113

C₂²×C₈

—

114

C₂²×C₈

—

115

C₂²×C₈

C₂²⋊D₁₉

yes

characteristic

116

C₂²×C₈

C₂²⋊D₁₉

yes

characteristic

117

C₂×C₂²⋊C₄

C₁₉⋊C₈

yes

characteristic

118

C₂×C₂²⋊C₄

C₁₉⋊C₈

yes

characteristic

119

C₂×C₄²

—

120

C₂×C₁₆

—

121

C₂×C₁₆

—

122

C₄⋊C₈

—

123

C₂²⋊C₈

—

124

#95

—

125

C₁₉

#7064

yes

characteristic, complemented, Sylow

126

C₂⁴

C₂×C₁₉⋊C₈

yes

characteristic

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

#64520

yes

characteristic

134

C₂²×C₄

C₂²⋊Dic₁₉

yes

characteristic

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂²⋊C₄

—

145

C₂²⋊C₄

—

146

C₂²⋊C₄

C₁₉⋊C₁₆

yes

complemented

147

C₄²

—

148

C₄²

—

149

C₁₆

—

152

—

150

C₁₆

—

152

—

151

C₂³

—

152

C₂³

—

153

C₂³

—

154

C₂³

—

155

C₂³

C₂×C₁₉⋊C₁₆

yes

characteristic

156

C₂³

#10597269

yes

characteristic

157

C₂³

C₂²⋊(C₁₉⋊C₈)

yes

characteristic

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

#10597291

yes

characteristic

172

C₂×C₄

#10597293

yes

characteristic

173

C₈

—

174

C₈

—

175

C₈

—

176

C₂²

—

177

C₂²

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

—

182

C₂²

—

183

C₂²

—

184

C₂²

—

185

C₂²

#11926270

yes

characteristic

186

C₂²

C₂²⋊(C₁₉⋊C₁₆)

yes

characteristic

187

C₂²

#11926326

yes

characteristic

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

#11926331

yes

characteristic

193

C₄

#11926334

yes

characteristic

194

C₂

—

195

C₂

—

196

C₂

—

197

C₂

—

198

C₂

#14561575

yes

characteristic

199

C₂

#14561583

yes

characteristic

200

C₂

#14561588

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10597291
commutator	C₂³×C₁₉	C₂×C₁₆
Frattini	C₂×C₂²⋊C₈	C₂×D₁₉
Fitting	C₁₉×#1850	C₂
socle	C₂²×C₁₉	#324
prime radical	C₂⁴×C₁₉	C₂×C₈
cosocle	C₂×C₁₉×C₂²⋊C₈	C₂²
hypercenter	#1850	D₁₉
hypocenter	C₁₉	#7064
LCS	G ▷ C₂³×C₁₉ ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈ ◁ #1850

subgroup

G/H

center

C₂×C₄

#10597291

commutator

C₂³×C₁₉

C₂×C₁₆

Frattini

C₂×C₂²⋊C₈

C₂×D₁₉

Fitting

C₁₉×#1850

C₂

socle

C₂²×C₁₉

#324

prime radical

C₂⁴×C₁₉

C₂×C₈

cosocle

C₂×C₁₉×C₂²⋊C₈

C₂²

D₁₉

C₁₉

G ▷ C₂³×C₁₉ ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈ ◁ #1850

N	H ≅ G/N	type
45. C₂²⋊C₄×C₁₉	C₁₆	semidirect
125. C₁₉	#7064	semidirect
146. C₂²⋊C₄	C₁₉⋊C₁₆	semidirect

H ≅ G/N

type

45. C₂²⋊C₄×C₁₉

C₁₆

semidirect

125. C₁₉

#7064

semidirect

146. C₂²⋊C₄

C₁₉⋊C₁₆

semidirect

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

subgroup	index
#14561559	2
C₁₉×#1850	2
#7064	19

subgroup

index

#14561559

C₁₉×#1850

#7064

Subgroups — #18069178

Conjugacy classes of subgroups

Complements

Maximal subgroups