Group #18087521 — subgroups

Subgroups — #18087521

Order 4864.

372 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 101 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	#14564433	C₂	1	1	yes	maximal, characteristic
2	#14563810	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562406	C₂	1	2	yes	maximal, complemented
4	#14562061	C₂	1	2	yes	maximal
5	C₁₉×#1275	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₇₆	C₂²	1	1	yes	characteristic
7	C₂×#10597276	—	2	2	no	—
8	#11926568	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×(C₂×D₁₉)⋊C₈	—	2	2	no	—
11	#11926295	—	2	4	no	—
12	#11926273	—	2	4	no	—
13	C₁₉×#431	C₂²	1	2	yes	—
14	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
15	C₂²×D₇₆	Q₈	1	1	yes	characteristic
16	C₂²×C₄×D₁₉	—	2	1	no	—
17	C₂²×C₈×C₁₉	D₄	1	2	yes	—
18	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
19	C₂×(C₂×D₁₉)⋊C₄	—	2	1	no	—
20	C₂×C₄⋊Dic₁₉	Q₈	1	1	yes	characteristic
21	C₂×#64520	—	2	2	no	—
22	C₂²×C₁₉⋊C₈	—	2	1	no	—
23	C₄×D₇₆	—	2	2	no	—
24	C₄×D₇₆	—	2	2	no	—
25	C₁₉×C₄⋊C₈	—	2	2	no	—
26	C₁₉×C₂²⋊C₈	—	2	2	no	—
27	C₁₉×#97	—	2	2	no	—
28	C₄×C₈×C₁₉	D₄	1	4	yes	—
29	(C₂×D₁₉)⋊C₈	—	4	4	no	—
30	#10597276	—	4	4	no	—
31	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
32	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
33	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
34	C₂³×D₁₉	—	2	1	no	—
35	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
37	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
38	C₂²×Dic₁₉	—	2	1	no	—
39	C₂×D₇₆	—	2	2	no	—
40	C₂×D₇₆	—	4	2	no	—
41	C₂×C₄×D₁₉	—	4	1	no	—
42	C₂×C₄×D₁₉	—	4	1	no	—
43	C₂×C₄×D₁₉	—	4	2	no	—
44	C₂×C₄×D₁₉	—	4	2	no	—
45	C₂×C₈×C₁₉	—	2	2	no	—
46	C₂×C₈×C₁₉	—	2	2	no	—
47	C₂×C₈×C₁₉	—	2	4	no	—
48	C₂×C₈×C₁₉	SD₁₆	1	4	yes	—
49	C₄²×C₁₉	C₂×D₄	1	2	yes	—
50	C₄²×C₁₉	#41	1	2	yes	—
51	(C₂×D₁₉)⋊C₄	—	4	2	no	—
52	C₄⋊Dic₁₉	—	2	2	no	—
53	#64520	—	4	4	no	—
54	C₂×C₁₉⋊C₈	—	4	1	no	—
55	C₂×C₁₉⋊C₈	—	4	2	no	—
56	C₂×C₁₉⋊C₈	—	4	2	no	—
57	#10661	—	19	1	no	maximal, Sylow
58	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
59	C₂²×D₁₉	—	2	2	no	—
60	C₂²×D₁₉	—	4	1	no	—
61	C₂²×D₁₉	—	4	2	no	—
62	C₂²×D₁₉	—	4	1	no	—
63	C₂²×D₁₉	—	4	2	no	—
64	C₂×C₄×C₁₉	—	4	1	no	—
65	C₂×C₄×C₁₉	—	4	1	no	—
66	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
67	C₂×C₄×C₁₉	#123	1	2	yes	—
68	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
69	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
70	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
71	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
72	C₂×Dic₁₉	—	4	1	no	—
73	C₂×Dic₁₉	—	4	2	no	—
74	D₇₆	—	2	4	no	—
75	D₇₆	—	4	2	no	—
76	C₄×D₁₉	—	8	2	no	—
77	C₄×D₁₉	—	8	1	no	—
78	C₄×D₁₉	—	8	1	no	—
79	C₈×C₁₉	—	2	4	no	—
80	C₈×C₁₉	—	4	4	no	—
81	C₁₉⋊C₈	—	8	1	no	—
82	C₁₉⋊C₈	—	8	2	no	—
83	#1955	—	19	1	no	—
84	#1849	—	19	1	no	—
85	#1570	—	19	2	no	—
86	#1486	—	19	2	no	—
87	#1275	D₁₉	1	1	yes	characteristic, complemented
88	C₂²×C₁₉	—	2	1	no	—
89	C₂²×C₁₉	—	2	1	no	—
90	C₂²×C₁₉	#399	1	1	yes	characteristic
91	C₂²×C₁₉	#476	1	2	yes	—
92	C₂×D₁₉	—	4	4	no	—
93	C₂×D₁₉	—	4	2	no	—
94	C₂×D₁₉	—	8	2	no	—
95	C₂×D₁₉	—	8	1	no	—
96	C₂×D₁₉	—	8	1	no	—
97	C₄×C₁₉	—	2	2	no	—
98	C₄×C₁₉	—	2	2	no	—
99	C₄×C₁₉	#481	1	2	yes	—
100	C₄×C₁₉	#492	1	1	yes	characteristic
101	C₄×C₁₉	#496	1	1	yes	characteristic
102	Dic₁₉	—	8	1	no	—
103	C₂×C₄×D₄	—	19	1	no	—
104	#431	C₂×D₁₉	1	2	yes	—
105	C₂×#108	—	38	2	no	—
106	#423	—	19	2	no	—
107	C₂×C₄⋊C₈	—	19	1	no	—
108	C₂×C₂²⋊C₈	—	38	2	no	—
109	C₂×#97	C₂×D₁₉	1	1	yes	characteristic
110	#334	—	38	4	no	—
111	D₄⋊C₈	—	38	4	no	—
112	C₂×C₁₉	—	2	2	no	—
113	C₂×C₁₉	#2091	1	1	yes	characteristic
114	C₂×C₁₉	#2094	1	1	yes	characteristic
115	C₂×C₁₉	#2098	1	1	yes	characteristic
116	D₁₉	—	8	2	no	—
117	C₂²×D₄	—	19	1	no	—
118	C₂³×C₄	—	38	1	no	—
119	C₂×M₁₆	—	38	2	no	—
120	C₂²×C₈	—	38	1	no	—
121	C₂²×C₈	D₇₆	1	2	yes	—
122	C₄×D₄	—	38	2	no	—
123	C₄×D₄	—	38	2	no	—
124	C₂×C₄⋊C₄	—	19	1	no	—
125	C₂×C₂²⋊C₄	—	38	1	no	—
126	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
127	#108	—	76	4	no	—
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	—	38	2	no	—
130	C₄⋊C₈	—	38	2	no	—
131	C₄⋊C₈	—	38	2	no	—
132	C₂²⋊C₈	—	2	2	no	—
133	C₂²⋊C₈	—	76	4	no	—
134	#97	—	2	2	no	—
135	C₄×C₈	D₇₆	1	4	yes	—
136	C₁₉	#10661	1	1	yes	characteristic, complemented, Sylow
137	C₂⁴	—	38	1	no	—
138	C₂×D₄	—	38	2	no	—
139	C₂×D₄	—	76	2	no	—
140	C₂²×C₄	—	38	1	no	—
141	C₂²×C₄	—	76	1	no	—
142	C₂²×C₄	—	76	1	no	—
143	C₂²×C₄	—	76	2	no	—
144	C₂²×C₄	—	76	2	no	—
145	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
146	C₂²×C₄	#64543	1	1	yes	characteristic
147	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
148	M₁₆	—	76	4	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	76	1	no	—
153	C₂×C₈	—	76	2	no	—
154	C₂×C₈	—	76	2	no	—
155	C₂×C₈	#64516	1	4	yes	—
156	C₄⋊C₄	—	38	2	no	—
157	C₂²⋊C₄	—	76	2	no	—
158	C₄²	C₂×D₇₆	1	2	yes	—
159	C₄²	#64543	1	2	yes	—
160	C₂³	—	38	2	no	—
161	C₂³	—	76	1	no	—
162	C₂³	—	76	2	no	—
163	C₂³	—	76	2	no	—
164	C₂³	—	76	1	no	—
165	C₂³	#10597343	1	1	yes	characteristic
166	D₄	—	38	4	no	—
167	D₄	—	76	2	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	—	76	2	no	—
171	C₂×C₄	—	76	1	no	—
172	C₂×C₄	—	152	2	no	—
173	C₂×C₄	—	152	1	no	—
174	C₂×C₄	—	152	1	no	—
175	C₂×C₄	#10597320	1	1	yes	characteristic
176	C₂×C₄	#10597332	1	2	yes	—
177	C₂×C₄	#10597332	1	1	yes	characteristic
178	C₂×C₄	#10597343	1	2	yes	—
179	C₂×C₄	C₂×#64516	1	2	yes	—
180	C₂×C₄	#10597354	1	2	yes	—
181	C₈	—	2	4	no	—
182	C₈	—	4	4	no	—
183	C₈	—	152	1	no	—
184	C₈	—	152	2	no	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	1	no	—
187	C₂²	—	76	4	no	—
188	C₂²	—	76	2	no	—
189	C₂²	—	152	2	no	—
190	C₂²	—	152	1	no	—
191	C₂²	—	152	1	no	—
192	C₂²	#11926481	1	1	yes	characteristic
193	C₂²	#11926835	1	2	yes	—
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	—	152	1	no	—
197	C₄	#11926863	1	2	yes	—
198	C₄	#11926917	1	1	yes	characteristic
199	C₄	#11926933	1	1	yes	characteristic
200	C₂	—	2	2	no	—
201	C₂	—	152	2	no	—
202	C₂	#14565152	1	1	yes	characteristic
203	C₂	#14565167	1	1	yes	characteristic
204	C₂	#14565189	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14564433

C₂

yes

maximal, characteristic

#14563810

C₂

yes

maximal, characteristic, complemented

#14562406

C₂

yes

maximal, complemented

#14562061

C₂

yes

maximal

C₁₉×#1275

C₂

yes

maximal, characteristic, complemented

C₂×C₄×D₇₆

C₂²

yes

characteristic

C₂×#10597276

—

#11926568

C₂²

yes

—

C₂×C₄⋊(C₁₉⋊C₈)

C₂²

yes

characteristic

C₂×(C₂×D₁₉)⋊C₈

—

#11926295

—

#11926273

—

C₁₉×#431

C₂²

yes

—

C₂×C₁₉×#97

C₂²

yes

characteristic

C₂²×D₇₆

Q₈

yes

characteristic

C₂²×C₄×D₁₉

—

C₂²×C₈×C₁₉

D₄

yes

—

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₂×(C₂×D₁₉)⋊C₄

—

C₂×C₄⋊Dic₁₉

Q₈

yes

characteristic

C₂×#64520

—

C₂²×C₁₉⋊C₈

—

C₄×D₇₆

—

C₄×D₇₆

—

C₁₉×C₄⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₁₉×#97

—

C₄×C₈×C₁₉

D₄

yes

—

(C₂×D₁₉)⋊C₈

—

#10597276

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₂³×D₁₉

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×Q₈

yes

characteristic

C₂²×C₄×C₁₉

#41

yes

characteristic

C₂²×Dic₁₉

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×C₄×D₁₉

—

C₂×C₄×D₁₉

—

C₂×C₄×D₁₉

—

C₂×C₄×D₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

SD₁₆

yes

—

C₄²×C₁₉

C₂×D₄

yes

—

C₄²×C₁₉

#41

yes

—

(C₂×D₁₉)⋊C₄

—

C₄⋊Dic₁₉

—

#64520

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

#10661

—

maximal, Sylow

C₂³×C₁₉

C₂²⋊Q₈

yes

characteristic

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₉

#123

yes

—

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

C₂×SD₁₆

yes

—

C₂×C₄×C₁₉

AGL(1,ℤ/8)

yes

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

D₇₆

—

D₇₆

—

C₄×D₁₉

—

C₄×D₁₉

—

C₄×D₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

—

—

—

—

D₁₉

yes

characteristic, complemented

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#399

yes

characteristic

C₂²×C₁₉

#476

yes

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#481

yes

—

100

C₄×C₁₉

#492

yes

characteristic

101

C₄×C₁₉

#496

yes

characteristic

102

Dic₁₉

—

103

C₂×C₄×D₄

—

104

#431

C₂×D₁₉

yes

—

105

C₂×#108

—

106

#423

—

107

C₂×C₄⋊C₈

—

108

C₂×C₂²⋊C₈

—

109

C₂×#97

C₂×D₁₉

yes

characteristic

110

#334

—

111

D₄⋊C₈

—

112

C₂×C₁₉

—

113

C₂×C₁₉

#2091

yes

characteristic

114

C₂×C₁₉

#2094

yes

characteristic

115

C₂×C₁₉

#2098

yes

characteristic

116

D₁₉

—

117

C₂²×D₄

—

118

C₂³×C₄

—

119

C₂×M₁₆

—

120

C₂²×C₈

—

121

C₂²×C₈

D₇₆

yes

—

122

C₄×D₄

—

123

C₄×D₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₂²⋊C₄

—

126

C₂×C₄²

C₂²×D₁₉

yes

characteristic

127

#108

—

128

C₄⋊C₈

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

131

C₄⋊C₈

—

132

C₂²⋊C₈

—

133

C₂²⋊C₈

—

134

#97

—

135

C₄×C₈

D₇₆

yes

—

136

C₁₉

#10661

yes

characteristic, complemented, Sylow

137

C₂⁴

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

C₂×D₇₆

yes

characteristic

146

C₂²×C₄

#64543

yes

characteristic

147

C₂²×C₄

Q₈×D₁₉

yes

characteristic

148

M₁₆

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

#64516

yes

—

156

C₄⋊C₄

—

157

C₂²⋊C₄

—

158

C₄²

C₂×D₇₆

yes

—

159

C₄²

#64543

yes

—

160

C₂³

—

161

C₂³

—

162

C₂³

—

163

C₂³

—

164

C₂³

—

165

C₂³

#10597343

yes

characteristic

166

D₄

—

167

D₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

152

—

173

C₂×C₄

—

152

—

174

C₂×C₄

—

152

—

175

C₂×C₄

#10597320

yes

characteristic

176

C₂×C₄

#10597332

yes

—

177

C₂×C₄

#10597332

yes

characteristic

178

C₂×C₄

#10597343

yes

—

179

C₂×C₄

C₂×#64516

yes

—

180

C₂×C₄

#10597354

yes

—

181

C₈

—

182

C₈

—

183

C₈

—

152

—

184

C₈

—

152

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

152

—

190

C₂²

—

152

—

191

C₂²

—

152

—

192

C₂²

#11926481

yes

characteristic

193

C₂²

#11926835

yes

—

194

C₄

—

195

C₄

—

196

C₄

—

152

—

197

C₄

#11926863

yes

—

198

C₄

#11926917

yes

characteristic

199

C₄

#11926933

yes

characteristic

200

C₂

—

201

C₂

—

152

—

202

C₂

#14565152

yes

characteristic

203

C₂

#14565167

yes

characteristic

204

C₂

#14565189

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10597320
commutator, cosocle	C₂×C₄²×C₁₉	C₂³
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1275	C₂
socle	C₂²×C₁₉	#399
prime radical	C₂²×D₇₆	Q₈
hypercenter	#1275	D₁₉
hypocenter	C₁₉	#10661
LCS	G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1275

subgroup

G/H

center

C₂×C₄

#10597320

commutator, cosocle

C₂×C₄²×C₁₉

C₂³

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#1275

C₂

socle

C₂²×C₁₉

#399

prime radical

C₂²×D₇₆

Q₈

D₁₉

C₁₉

G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1275

N	H ≅ G/N	type
2. #14563810	C₂	semidirect
3. #14562406	C₂	semidirect
5. C₁₉×#1275	C₂	semidirect
87. #1275	D₁₉	semidirect
136. C₁₉	#10661	semidirect

H ≅ G/N

type

2. #14563810

C₂

semidirect

3. #14562406

C₂

semidirect

5. C₁₉×#1275

C₂

semidirect

87. #1275

D₁₉

semidirect

136. C₁₉

#10661

semidirect

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

subgroup	index
#14564433	2
#14563810	2
#14562406	2
#14562061	2
C₁₉×#1275	2
#10661	19

subgroup

index

C₁₉×#1275

Subgroups — #18087521

Conjugacy classes of subgroups

Complements

Maximal subgroups