Group #18075565 — subgroups

Subgroups — #18075565

Order 4864.

404 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 127 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	#14562394	C₂	1	2	yes	maximal, complemented
2	#14562370	C₂	1	2	yes	maximal
3	C₁₉⋊#1309	C₂	1	2	yes	maximal, complemented
4	C₁₉×#2931	C₂	1	1	yes	maximal, characteristic
5	#11926568	C₂²	1	4	yes	—
6	#11926297	—	2	2	no	—
7	#11926297	—	2	2	no	—
8	#11926293	—	2	2	no	—
9	#11926288	—	2	2	no	—
10	C₂×C₁₉×#131	C₄	1	2	yes	—
11	C₁₉×#532	C₂²	1	1	yes	characteristic
12	C₁₉×#436	—	2	2	no	—
13	C₈×D₄×C₁₉	—	2	2	no	—
14	C₁₉×#423	C₂²	1	1	yes	characteristic
15	C₁₉×#408	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×#41×C₁₉	C₂×C₄	1	2	yes	—
18	C₁₉×#131	—	4	4	no	—
19	C₂×M₁₆×C₁₉	—	2	2	no	—
20	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₁₉	—	2	2	no	—
22	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
25	C₁₉×#127	D₄	1	1	yes	characteristic
26	C₁₉×#124	Q₈	1	2	yes	—
27	C₁₉×#121	—	2	2	no	—
28	C₄×D₄×C₁₉	—	2	2	no	—
29	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
30	C₂×#64520	—	2	4	no	—
31	C₁₉×C₄⋊C₈	—	2	2	no	—
32	C₁₉×C₂²⋊C₈	—	2	4	no	—
33	C₄×C₈×C₁₉	D₄	1	2	yes	—
34	C₄×C₈×C₁₉	D₄	1	2	yes	—
35	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
36	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
37	#41×C₁₉	—	4	2	no	—
38	#41×C₁₉	—	4	2	no	—
39	C₂×Q₈×C₁₉	C₄⋊C₄	1	2	yes	—
40	C₂×D₄×C₁₉	—	2	2	no	—
41	C₂×D₄×C₁₉	—	2	2	no	—
42	C₂×D₄×C₁₉	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₁₉	—	2	2	no	—
44	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₉	#41	1	2	yes	—
46	M₁₆×C₁₉	—	4	4	no	—
47	M₁₆×C₁₉	—	4	2	no	—
48	C₂×C₈×C₁₉	—	2	2	no	—
49	C₂×C₈×C₁₉	—	2	2	no	—
50	C₂×C₈×C₁₉	—	4	1	no	—
51	C₂×C₈×C₁₉	—	4	4	no	—
52	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₄⋊C₄×C₁₉	—	2	2	no	—
56	C₂²⋊C₄×C₁₉	—	2	4	no	—
57	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₁₉	C₂×D₄	1	2	yes	—
59	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
60	#64520	—	8	4	no	—
61	C₂×C₁₉⋊C₈	—	4	4	no	—
62	#8638	—	19	1	no	maximal, Sylow
63	C₂³×C₁₉	—	2	2	no	—
64	C₂³×C₁₉	#117	1	1	yes	characteristic
65	Q₈×C₁₉	—	4	2	no	—
66	D₄×C₁₉	—	4	2	no	—
67	D₄×C₁₉	—	4	4	no	—
68	D₄×C₁₉	—	4	2	no	—
69	D₄×C₁₉	—	4	2	no	—
70	C₂×C₄×C₁₉	—	2	1	no	—
71	C₂×C₄×C₁₉	—	2	1	no	—
72	C₂×C₄×C₁₉	—	2	2	no	—
73	C₂×C₄×C₁₉	—	2	2	no	—
74	C₂×C₄×C₁₉	—	2	2	no	—
75	C₂×C₄×C₁₉	—	4	2	no	—
76	C₂×C₄×C₁₉	—	4	2	no	—
77	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₉	#121	1	2	yes	—
80	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
81	C₈×C₁₉	—	2	1	no	—
82	C₈×C₁₉	—	2	2	no	—
83	C₈×C₁₉	—	4	2	no	—
84	C₈×C₁₉	#100	1	2	yes	—
85	C₁₉⋊C₈	—	8	4	no	—
86	#2931	D₁₉	1	1	yes	characteristic
87	#1567	—	19	2	no	—
88	#1560	—	19	2	no	—
89	#1309	—	19	2	no	—
90	C₂²×C₁₉	—	2	1	no	—
91	C₂²×C₁₉	—	2	2	no	—
92	C₂²×C₁₉	—	4	2	no	—
93	C₂²×C₁₉	—	4	4	no	—
94	C₂²×C₁₉	#380	1	1	yes	characteristic
95	C₄×C₁₉	—	2	1	no	—
96	C₄×C₁₉	—	2	2	no	—
97	C₄×C₁₉	—	4	2	no	—
98	C₄×C₁₉	C₂×#100	1	1	yes	characteristic
99	C₄×C₁₉	#413	1	1	yes	characteristic
100	C₄×C₁₉	#472	1	2	yes	—
101	C₄×C₁₉	#473	1	2	yes	—
102	C₂×#131	Dic₁₉	1	2	yes	—
103	#532	C₂×D₁₉	1	1	yes	characteristic
104	#436	—	2	2	no	—
105	C₈×D₄	—	2	2	no	—
106	#423	—	19	4	no	—
107	#423	C₂×D₁₉	1	1	yes	characteristic
108	#408	Dic₁₉	1	2	yes	—
109	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
110	#335	—	38	2	no	—
111	#335	—	38	2	no	—
112	#333	—	38	2	no	—
113	#331	—	38	2	no	—
114	C₂×C₁₉	—	4	1	no	—
115	C₂×C₁₉	—	8	2	no	—
116	C₂×C₁₉	#1800	1	1	yes	characteristic
117	C₂×C₁₉	#1816	1	2	yes	—
118	C₂×#41	C₂×Dic₁₉	1	2	yes	—
119	#131	—	4	4	no	—
120	C₂×M₁₆	—	2	2	no	—
121	C₂×M₁₆	—	38	4	no	—
122	C₂×M₁₆	C₂×Dic₁₉	1	2	yes	—
123	C₂²×C₈	—	2	2	no	—
124	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
125	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
126	C₄⋊Q₈	D₇₆	1	1	yes	characteristic
127	#127	D₇₆	1	1	yes	characteristic
128	#124	Dic₃₈	1	2	yes	—
129	#121	—	2	2	no	—
130	C₄×D₄	—	2	2	no	—
131	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
132	C₄⋊C₈	—	2	2	no	—
133	C₄⋊C₈	—	38	4	no	—
134	C₄⋊C₈	—	38	4	no	—
135	C₂²⋊C₈	—	2	4	no	—
136	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
137	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
138	C₁₉	#8638	1	1	yes	characteristic, complemented, Sylow
139	#41	—	4	2	no	—
140	#41	—	4	2	no	—
141	C₂×Q₈	C₄⋊Dic₁₉	1	2	yes	—
142	C₂×D₄	—	2	2	no	—
143	C₂×D₄	—	2	2	no	—
144	C₂×D₄	C₄⋊Dic₁₉	1	2	yes	—
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	#64541	1	2	yes	—
147	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
148	M₁₆	—	4	4	no	—
149	M₁₆	—	4	2	no	—
150	M₁₆	—	152	4	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	4	1	no	—
154	C₂×C₈	—	4	4	no	—
155	C₂×C₈	—	76	4	no	—
156	C₂×C₈	C₂²⋊Dic₁₉	1	2	yes	—
157	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
158	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
159	C₄⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	2	4	no	—
161	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
162	C₄²	C₂×D₇₆	1	1	yes	characteristic
163	C₄²	C₂×C₂²⋊D₁₉	1	2	yes	—
164	C₂³	—	2	2	no	—
165	C₂³	#10597372	1	1	yes	characteristic
166	Q₈	—	4	2	no	—
167	D₄	—	4	2	no	—
168	D₄	—	4	4	no	—
169	D₄	—	4	2	no	—
170	D₄	—	4	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	#10597370	1	2	yes	—
179	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
180	C₂×C₄	#10597376	1	2	yes	—
181	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
182	C₈	—	2	2	no	—
183	C₈	—	2	1	no	—
184	C₈	—	4	2	no	—
185	C₈	—	152	4	no	—
186	C₈	#10597290	1	2	yes	—
187	C₂²	—	2	1	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	4	2	no	—
190	C₂²	—	4	4	no	—
191	C₂²	#11926404	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	2	no	—
194	C₄	—	4	2	no	—
195	C₄	C₂×#10597290	1	1	yes	characteristic
196	C₄	#11926528	1	1	yes	characteristic
197	C₄	#11926814	1	2	yes	—
198	C₄	#11926819	1	2	yes	—
199	C₂	—	4	1	no	—
200	C₂	—	8	2	no	—
201	C₂	#14563627	1	1	yes	characteristic
202	C₂	#14563699	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14562394

C₂

yes

maximal, complemented

#14562370

C₂

yes

maximal

C₁₉⋊#1309

C₂

yes

maximal, complemented

C₁₉×#2931

C₂

yes

maximal, characteristic

#11926568

C₂²

yes

—

—

—

—

—

C₂×C₁₉×#131

C₄

yes

—

C₁₉×#532

C₂²

yes

characteristic

C₁₉×#436

—

C₈×D₄×C₁₉

—

C₁₉×#423

C₂²

yes

characteristic

C₁₉×#408

C₄

yes

—

C₂×C₄×C₈×C₁₉

C₂²

yes

characteristic

C₂×#41×C₁₉

C₂×C₄

yes

—

C₁₉×#131

—

C₂×M₁₆×C₁₉

—

C₂×M₁₆×C₁₉

C₂×C₄

yes

—

C₂²×C₈×C₁₉

—

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₁₉×C₄⋊Q₈

D₄

yes

characteristic

C₁₉×#127

D₄

yes

characteristic

C₁₉×#124

Q₈

yes

—

C₁₉×#121

—

C₄×D₄×C₁₉

—

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₂×#64520

—

C₁₉×C₄⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₄×C₈×C₁₉

D₄

yes

—

C₄×C₈×C₁₉

D₄

yes

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

#41×C₁₉

—

#41×C₁₉

—

C₂×Q₈×C₁₉

C₄⋊C₄

yes

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

C₄⋊C₄

yes

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₉

#41

yes

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₂²⋊C₄

yes

—

C₂×C₈×C₁₉

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄²×C₁₉

C₂×D₄

yes

—

C₄²×C₁₉

C₂×Q₈

yes

characteristic

#64520

—

C₂×C₁₉⋊C₈

—

#8638

—

maximal, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

#117

yes

characteristic

Q₈×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

D₄×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×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉

#121

yes

—

C₂×C₄×C₁₉

C₂²⋊Q₈

yes

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

#100

yes

—

C₁₉⋊C₈

—

#2931

D₁₉

yes

—

—

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#380

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

100

C₄×C₁₉

#472

yes

—

101

C₄×C₁₉

#473

yes

—

102

C₂×#131

Dic₁₉

yes

—

103

#532

C₂×D₁₉

yes

characteristic

104

#436

—

105

C₈×D₄

—

106

#423

—

107

#423

C₂×D₁₉

yes

characteristic

108

#408

Dic₁₉

yes

—

109

C₂×C₄×C₈

C₂×D₁₉

yes

characteristic

110

#335

—

111

#335

—

112

#333

—

113

#331

—

114

C₂×C₁₉

—

115

C₂×C₁₉

—

116

C₂×C₁₉

#1800

yes

characteristic

117

C₂×C₁₉

#1816

yes

—

118

C₂×#41

C₂×Dic₁₉

yes

—

119

#131

—

120

C₂×M₁₆

—

121

C₂×M₁₆

—

122

C₂×M₁₆

C₂×Dic₁₉

yes

—

123

C₂²×C₈

—

124

C₂²×C₈

C₂×Dic₁₉

yes

characteristic

125

C₂²×C₈

C₂×Dic₁₉

yes

characteristic

126

C₄⋊Q₈

D₇₆

yes

characteristic

127

#127

D₇₆

yes

characteristic

128

#124

Dic₃₈

yes

—

129

#121

—

130

C₄×D₄

—

131

C₂×C₄²

C₂²×D₁₉

yes

characteristic

132

C₄⋊C₈

—

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

C₂²⋊C₈

—

136

C₄×C₈

C₂²⋊D₁₉

yes

—

137

C₄×C₈

C₂²⋊D₁₉

yes

—

138

C₁₉

#8638

yes

characteristic, complemented, Sylow

139

#41

—

140

#41

—

141

C₂×Q₈

C₄⋊Dic₁₉

yes

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

C₄⋊Dic₁₉

yes

—

145

C₂²×C₄

—

146

C₂²×C₄

#64541

yes

—

147

C₂²×C₄

C₂²×Dic₁₉

yes

characteristic

148

M₁₆

—

149

M₁₆

—

150

M₁₆

—

152

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

C₂²⋊Dic₁₉

yes

—

157

C₂×C₈

C₂²⋊Dic₁₉

yes

characteristic

158

C₂×C₈

C₂²⋊Dic₁₉

yes

characteristic

159

C₄⋊C₄

—

160

C₂²⋊C₄

—

161

C₄²

C₂×Dic₃₈

yes

characteristic

162

C₄²

C₂×D₇₆

yes

characteristic

163

C₄²

C₂×C₂²⋊D₁₉

yes

—

164

C₂³

—

165

C₂³

#10597372

yes

characteristic

166

Q₈

—

167

D₄

—

168

D₄

—

169

D₄

—

170

D₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

#10597370

yes

—

179

C₂×C₄

C₂×C₄⋊Dic₁₉

yes

characteristic

180

C₂×C₄

#10597376

yes

—

181

C₂×C₄

C₂×C₂²⋊Dic₁₉

yes

characteristic

182

C₈

—

183

C₈

—

184

C₈

—

185

C₈

—

152

—

186

C₈

#10597290

yes

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

190

C₂²

—

191

C₂²

#11926404

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

C₂×#10597290

yes

characteristic

196

C₄

#11926528

yes

characteristic

197

C₄

#11926814

yes

—

198

C₄

#11926819

yes

—

199

C₂

—

200

C₂

—

201

C₂

#14563627

yes

characteristic

202

C₂

#14563699

yes

—

	subgroup	G/H
center	C₂²	#11926404
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2931	C₂
socle	C₂²×C₁₉	#380
prime radical	C₁₉×#532	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2931	D₁₉
hypocenter	C₁₉	#8638
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931

subgroup

G/H

center

C₂²

#11926404

commutator

C₂²×C₄×C₁₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#2931

C₂

socle

C₂²×C₁₉

#380

prime radical

C₁₉×#532

C₂²

cosocle

C₂×C₄²×C₁₉

C₂³

D₁₉

C₁₉

G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931

N	H ≅ G/N	type
1. #14562394	C₂	semidirect
3. C₁₉⋊#1309	C₂	semidirect
138. C₁₉	#8638	semidirect

H ≅ G/N

type

1. #14562394

C₂

semidirect

3. C₁₉⋊#1309

C₂

semidirect

138. C₁₉

#8638

semidirect

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

subgroup	index
#14562394	2
#14562370	2
C₁₉⋊#1309	2
C₁₉×#2931	2
#8638	19

subgroup

index

#14562394

#14562370

C₁₉⋊#1309

C₁₉×#2931

#8638

Subgroups — #18075565

Conjugacy classes of subgroups

Complements

Maximal subgroups