Group #18069302 — subgroups

Subgroups — #18069302

Order 4864.

234 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 67 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	#14564506	C₂	1	1	yes	maximal, characteristic
2	#14561466	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1377	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₇₆	C₄	1	1	yes	characteristic
5	#11926597	—	2	1	no	—
6	C₂×#10597274	C₄	1	1	yes	characteristic
7	C₂×#10597273	—	2	1	no	—
8	#11926450	—	2	1	no	—
9	C₂×C₁₉×M₃₂	—	2	1	no	—
10	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
11	C₂²×Dic₃₈	—	2	1	no	—
12	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
14	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
15	C₂×C₄⋊Dic₁₉	—	2	1	no	—
16	C₂×Dic₇₆	—	2	1	no	—
17	C₂×Dic₇₆	—	4	1	no	—
18	C₂×Dic₇₆	—	4	1	no	—
19	C₂×Dic₇₆	C₈	1	2	yes	complemented
20	C₁₉×M₃₂	—	2	2	no	—
21	C₁₉×M₃₂	—	4	1	no	—
22	C₂×C₁₆×C₁₉	—	4	1	no	—
23	C₁₉×#97	—	4	1	no	—
24	C₁₉⋊#95	—	4	1	no	—
25	C₁₉⋊#95	—	4	1	no	—
26	C₂²⋊(C₁₉⋊C₈)	—	4	1	no	—
27	#10597274	—	4	1	no	—
28	#10597273	—	4	1	no	—
29	#10597273	—	4	1	no	—
30	C₄⋊(C₁₉⋊C₈)	—	4	1	no	—
31	C₄×C₁₉⋊C₈	—	2	2	no	—
32	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
34	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
35	C₂²×Dic₁₉	—	4	1	no	—
36	C₂²×Dic₁₉	—	4	1	no	—
37	C₂×Dic₃₈	—	2	2	no	—
38	C₂×Dic₃₈	—	4	1	no	—
39	C₂×Dic₃₈	—	4	1	no	—
40	C₂×Dic₃₈	—	4	1	no	—
41	C₂×Dic₃₈	—	4	1	no	—
42	C₂×Dic₃₈	—	4	1	no	—
43	C₂×C₈×C₁₉	—	2	1	no	—
44	C₂×C₈×C₁₉	—	2	1	no	—
45	C₂×C₈×C₁₉	—	2	1	no	—
46	C₂×C₈×C₁₉	—	2	1	no	—
47	C₂×C₈×C₁₉	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈×C₁₉	M₁₆	1	1	yes	characteristic
49	C₂×C₈×C₁₉	D₈	1	1	yes	characteristic
50	C₂×C₈×C₁₉	SD₁₆	1	1	yes	characteristic
51	C₄²×C₁₉	—	2	1	no	—
52	C₄²×C₁₉	—	2	1	no	—
53	C₄⋊Dic₁₉	—	4	1	no	—
54	C₄⋊Dic₁₉	—	4	1	no	—
55	C₂×C₁₉⋊C₈	—	4	2	no	—
56	Dic₇₆	—	4	2	no	—
57	Dic₇₆	—	4	2	no	—
58	C₁₆×C₁₉	—	4	2	no	—
59	#7111	—	19	1	no	maximal, Sylow
60	C₂³×C₁₉	#102	1	1	yes	characteristic
61	C₂×C₄×C₁₉	—	2	1	no	—
62	C₂×C₄×C₁₉	—	2	1	no	—
63	C₂×C₄×C₁₉	—	4	1	no	—
64	C₂×C₄×C₁₉	—	4	1	no	—
65	C₂×C₄×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
66	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
67	C₂×C₄×C₁₉	#100	1	1	yes	characteristic
68	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
69	C₂×C₄×C₁₉	#104	1	1	yes	characteristic
70	C₂×C₄×C₁₉	#104	1	1	yes	characteristic
71	C₂×Dic₁₉	—	4	2	no	—
72	C₂×Dic₁₉	—	8	1	no	—
73	C₂×Dic₁₉	—	8	1	no	—
74	C₂×Dic₁₉	—	8	1	no	—
75	C₂×Dic₁₉	—	8	1	no	—
76	C₂×Dic₁₉	—	8	1	no	—
77	Dic₃₈	—	4	2	no	—
78	Dic₃₈	—	4	2	no	—
79	Dic₃₈	—	8	1	no	—
80	Dic₃₈	—	8	1	no	—
81	C₈×C₁₉	—	2	1	no	—
82	C₈×C₁₉	—	2	1	no	—
83	C₈×C₁₉	—	2	1	no	—
84	C₈×C₁₉	—	2	1	no	—
85	C₁₉⋊C₈	—	8	2	no	—
86	#1971	—	19	1	no	—
87	#1377	D₁₉	1	1	yes	characteristic
88	#1275	—	19	1	no	—
89	C₂²×C₁₉	—	2	1	no	—
90	C₂²×C₁₉	—	2	1	no	—
91	C₂²×C₁₉	D₄⋊C₈	1	1	yes	characteristic
92	C₂²×C₁₉	#327	1	1	yes	characteristic
93	C₂²×C₁₉	#331	1	1	yes	characteristic
94	C₄×C₁₉	—	4	1	no	—
95	C₄×C₁₉	—	4	1	no	—
96	C₄×C₁₉	#323	1	1	yes	characteristic
97	C₄×C₁₉	D₄⋊C₈	1	1	yes	characteristic
98	C₄×C₁₉	#327	1	1	yes	characteristic
99	C₄×C₁₉	#329	1	1	yes	characteristic
100	Dic₁₉	—	8	2	no	—
101	Dic₁₉	—	16	1	no	—
102	C₂²×Q₁₆	—	19	1	no	—
103	C₂×M₃₂	—	2	1	no	—
104	#431	—	38	1	no	—
105	C₂×#106	—	19	1	no	—
106	C₂×#103	—	38	1	no	—
107	C₂×#97	C₂×D₁₉	1	1	yes	characteristic
108	#391	—	38	1	no	—
109	C₂×C₁₉	—	2	1	no	—
110	C₂×C₁₉	—	2	1	no	—
111	C₂×C₁₉	#1270	1	1	yes	characteristic
112	C₂×C₁₉	#1340	1	1	yes	characteristic
113	C₂×C₁₉	#1342	1	1	yes	characteristic
114	C₂²×Q₈	—	38	1	no	—
115	C₂×Q₁₆	—	19	2	no	—
116	C₂×Q₁₆	—	38	1	no	—
117	C₂×Q₁₆	—	76	1	no	—
118	C₂×Q₁₆	—	76	1	no	—
119	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
120	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
121	C₂×C₄⋊C₄	—	38	1	no	—
122	C₂×C₄²	D₇₆	1	1	yes	characteristic
123	M₃₂	—	2	2	no	—
124	M₃₂	—	4	1	no	—
125	C₂×C₁₆	—	4	1	no	—
126	#106	—	76	1	no	—
127	C₄⋊C₈	—	76	1	no	—
128	#103	—	76	1	no	—
129	#103	—	76	1	no	—
130	C₂²⋊C₈	—	76	1	no	—
131	#97	—	4	1	no	—
132	C₄×C₈	—	38	2	no	—
133	#95	—	76	1	no	—
134	#95	—	76	1	no	—
135	C₁₉	#7111	1	1	yes	characteristic, complemented, Sylow
136	C₂×Q₈	—	38	2	no	—
137	C₂×Q₈	—	76	1	no	—
138	C₂×Q₈	—	76	1	no	—
139	C₂×Q₈	—	76	1	no	—
140	C₂×Q₈	—	76	1	no	—
141	C₂×Q₈	—	76	1	no	—
142	C₂²×C₄	—	76	1	no	—
143	C₂²×C₄	—	76	1	no	—
144	C₂²×C₄	#64516	1	1	yes	characteristic
145	C₂²×C₄	D₁₅₂	1	1	yes	characteristic
146	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
147	Q₁₆	—	76	2	no	—
148	Q₁₆	—	76	2	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	76	2	no	—
154	C₂×C₈	C₈×D₁₉	1	1	yes	characteristic
155	C₂×C₈	#64515	1	1	yes	characteristic
156	C₂×C₈	#64525	1	1	yes	characteristic
157	C₂×C₈	#64527	1	1	yes	characteristic
158	C₄⋊C₄	—	76	1	no	—
159	C₄⋊C₄	—	76	1	no	—
160	C₄²	—	2	1	no	—
161	C₄²	—	2	1	no	—
162	C₁₆	—	4	2	no	—
163	C₂³	#10597278	1	1	yes	characteristic
164	Q₈	—	76	2	no	—
165	Q₈	—	76	2	no	—
166	Q₈	—	152	1	no	—
167	Q₈	—	152	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	76	2	no	—
173	C₂×C₄	—	152	1	no	—
174	C₂×C₄	—	152	1	no	—
175	C₂×C₄	—	152	1	no	—
176	C₂×C₄	—	152	1	no	—
177	C₂×C₄	—	152	1	no	—
178	C₂×C₄	#10597262	1	1	yes	characteristic
179	C₂×C₄	#10597263	1	1	yes	characteristic
180	C₂×C₄	#10597266	1	1	yes	characteristic
181	C₂×C₄	(C₂×D₁₉)⋊C₈	1	1	yes	characteristic
182	C₂×C₄	#10597280	1	1	yes	characteristic
183	C₂×C₄	#10597282	1	1	yes	characteristic
184	C₈	—	2	1	no	—
185	C₈	—	2	1	no	—
186	C₈	—	2	1	no	—
187	C₈	—	2	1	no	—
188	C₈	—	152	2	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	1	no	—
191	C₂²	#11926273	1	1	yes	characteristic
192	C₂²	#11926279	1	1	yes	characteristic
193	C₂²	#11926287	1	1	yes	characteristic
194	C₄	—	4	1	no	—
195	C₄	—	4	1	no	—
196	C₄	—	304	1	no	—
197	C₄	—	152	2	no	—
198	C₄	#11926267	1	1	yes	characteristic
199	C₄	#11926271	1	1	yes	characteristic
200	C₄	#11926277	1	1	yes	characteristic
201	C₄	#11926283	1	1	yes	characteristic
202	C₂	—	2	1	no	—
203	C₂	—	2	1	no	—
204	C₂	#14561455	1	1	yes	characteristic
205	C₂	#14561624	1	1	yes	characteristic
206	C₂	#14561630	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14564506

C₂

yes

maximal, characteristic

#14561466

C₂

yes

maximal, characteristic

C₁₉×#1377

C₂

yes

maximal, characteristic

C₂²×Dic₇₆

C₄

yes

characteristic

#11926597

—

C₂×#10597274

C₄

yes

characteristic

C₂×#10597273

—

#11926450

—

C₂×C₁₉×M₃₂

—

C₂×C₁₉×#97

C₂²

yes

characteristic

C₂²×Dic₃₈

—

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₂×C₄⋊Dic₁₉

—

C₂×Dic₇₆

—

C₂×Dic₇₆

—

C₂×Dic₇₆

—

C₂×Dic₇₆

C₈

yes

complemented

C₁₉×M₃₂

—

C₁₉×M₃₂

—

C₂×C₁₆×C₁₉

—

C₁₉×#97

—

C₁₉⋊#95

—

C₁₉⋊#95

—

C₂²⋊(C₁₉⋊C₈)

—

#10597274

—

#10597273

—

#10597273

—

C₄⋊(C₁₉⋊C₈)

—

C₄×C₁₉⋊C₈

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

D₈

yes

characteristic

C₂²×C₄×C₁₉

SD₁₆

yes

characteristic

C₂²×Dic₁₉

—

C₂²×Dic₁₉

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₂×C₈

yes

characteristic

C₂×C₈×C₁₉

M₁₆

yes

characteristic

C₂×C₈×C₁₉

D₈

yes

characteristic

C₂×C₈×C₁₉

SD₁₆

yes

characteristic

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₂×C₁₉⋊C₈

—

Dic₇₆

—

Dic₇₆

—

C₁₆×C₁₉

—

—

C₂³×C₁₉

yes

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂²⋊C₈

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

Dic₃₈

—

Dic₃₈

—

Dic₃₈

—

Dic₃₈

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₁₉⋊C₈

—

#1971

—

#1377

D₁₉

yes

characteristic

#1275

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

D₄⋊C₈

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#323

yes

characteristic

C₄×C₁₉

D₄⋊C₈

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

100

Dic₁₉

—

101

Dic₁₉

—

102

C₂²×Q₁₆

—

103

C₂×M₃₂

—

104

#431

—

105

C₂×#106

—

106

C₂×#103

—

107

C₂×#97

C₂×D₁₉

yes

characteristic

108

#391

—

109

C₂×C₁₉

—

110

C₂×C₁₉

—

111

C₂×C₁₉

#1270

yes

characteristic

112

C₂×C₁₉

#1340

yes

characteristic

113

C₂×C₁₉

#1342

yes

characteristic

114

C₂²×Q₈

—

115

C₂×Q₁₆

—

116

C₂×Q₁₆

—

117

C₂×Q₁₆

—

118

C₂×Q₁₆

—

119

C₂²×C₈

C₄×D₁₉

yes

characteristic

120

C₂²×C₈

C₂²⋊D₁₉

yes

characteristic

121

C₂×C₄⋊C₄

—

122

C₂×C₄²

D₇₆

yes

characteristic

123

M₃₂

—

124

M₃₂

—

125

C₂×C₁₆

—

126

#106

—

127

C₄⋊C₈

—

128

#103

—

129

#103

—

130

C₂²⋊C₈

—

131

#97

—

132

C₄×C₈

—

133

#95

—

134

#95

—

135

C₁₉

#7111

yes

characteristic, complemented, Sylow

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

#64516

yes

characteristic

145

C₂²×C₄

D₁₅₂

yes

characteristic

146

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

147

Q₁₆

—

148

Q₁₆

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

C₈×D₁₉

yes

characteristic

155

C₂×C₈

#64515

yes

characteristic

156

C₂×C₈

#64525

yes

characteristic

157

C₂×C₈

#64527

yes

characteristic

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄²

—

161

C₄²

—

162

C₁₆

—

163

C₂³

#10597278

yes

characteristic

164

Q₈

—

165

Q₈

—

166

Q₈

—

152

—

167

Q₈

—

152

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

152

—

174

C₂×C₄

—

152

—

175

C₂×C₄

—

152

—

176

C₂×C₄

—

152

—

177

C₂×C₄

—

152

—

178

C₂×C₄

#10597262

yes

characteristic

179

C₂×C₄

#10597263

yes

characteristic

180

C₂×C₄

#10597266

yes

characteristic

181

C₂×C₄

(C₂×D₁₉)⋊C₈

yes

characteristic

182

C₂×C₄

#10597280

yes

characteristic

183

C₂×C₄

#10597282

yes

characteristic

184

C₈

—

185

C₈

—

186

C₈

—

187

C₈

—

188

C₈

—

152

—

189

C₂²

—

190

C₂²

—

191

C₂²

#11926273

yes

characteristic

192

C₂²

#11926279

yes

characteristic

193

C₂²

#11926287

yes

characteristic

194

C₄

—

195

C₄

—

196

C₄

—

304

—

197

C₄

—

152

—

198

C₄

#11926267

yes

characteristic

199

C₄

#11926271

yes

characteristic

200

C₄

#11926277

yes

characteristic

201

C₄

#11926283

yes

characteristic

202

C₂

—

203

C₂

—

204

C₂

#14561455

yes

characteristic

205

C₂

#14561624

yes

characteristic

206

C₂

#14561630

yes

characteristic

	subgroup	G/H
center	C₂²	#11926279
commutator	C₂×C₈×C₁₉	C₂×C₈
Frattini	C₂×#97	C₂×D₁₉
Fitting	C₁₉×#1377	C₂
socle	C₂²×C₁₉	#327
prime radical	C₂³×C₁₉	#102
cosocle	C₂×C₁₉×#97	C₂²
hypercenter	#1377	D₁₉
hypocenter	C₁₉	#7111
LCS	G ▷ C₂×C₈×C₁₉ ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1377

subgroup

G/H

center

C₂²

#11926279

commutator

C₂×C₈×C₁₉

C₂×C₈

Frattini

C₂×#97

C₂×D₁₉

Fitting

C₁₉×#1377

C₂

C₂²×C₁₉

C₂³×C₁₉

C₂×C₁₉×#97

C₂²

D₁₉

C₁₉

G ▷ C₂×C₈×C₁₉ ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1377

N	H ≅ G/N	type
19. C₂×Dic₇₆	C₈	semidirect
135. C₁₉	#7111	semidirect

H ≅ G/N

type

19. C₂×Dic₇₆

C₈

semidirect

135. C₁₉

#7111

semidirect

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

subgroup	index
#14564506	2
#14561466	2
C₁₉×#1377	2
#7111	19

subgroup

index

#14564506

#14561466

C₁₉×#1377

#7111

Subgroups — #18069302

Conjugacy classes of subgroups

Complements

Maximal subgroups