Group #18069249 — subgroups

Subgroups — #18069249

Order 4864.

248 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 49 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	#14564946	C₂	1	1	yes	maximal, characteristic
2	#14561510	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#1322	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₁₅₂	C₄	1	1	yes	characteristic
5	#11926568	—	2	1	no	—
6	C₂×#10597278	—	2	1	no	—
7	#11926446	—	2	1	no	—
8	#11926307	C₄	1	1	yes	characteristic
9	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
10	C₁₉×C₂²⋊C₁₆	—	2	1	no	—
11	C₂²×D₇₆	—	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×(C₂×D₁₉)⋊C₄	—	4	1	no	—
16	C₂×(C₂×D₁₉)⋊C₄	—	4	1	no	—
17	C₂×C₄⋊Dic₁₉	—	2	1	no	—
18	C₂×#64520	—	4	1	no	—
19	C₂×D₁₅₂	—	4	1	no	—
20	C₂×D₁₅₂	—	4	2	no	—
21	C₂×C₁₆×C₁₉	—	4	1	no	—
22	C₁₉×C₄⋊C₈	—	4	1	no	—
23	#10597278	—	4	2	no	—
24	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
25	C₄⋊(C₁₉⋊C₈)	—	4	1	no	—
26	C₂³×D₁₉	—	4	1	no	—
27	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
29	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
30	C₂²×Dic₁₉	—	4	1	no	—
31	C₂×D₇₆	—	2	2	no	—
32	C₂×D₇₆	—	4	1	no	—
33	C₂×D₇₆	—	4	1	no	—
34	C₂×D₇₆	—	4	1	no	—
35	C₂×D₇₆	—	4	1	no	—
36	C₂×D₇₆	—	4	1	no	—
37	C₂×C₈×C₁₉	—	2	1	no	—
38	C₂×C₈×C₁₉	—	2	1	no	—
39	C₂×C₈×C₁₉	—	4	1	no	—
40	C₂×C₈×C₁₉	—	4	1	no	—
41	C₄²×C₁₉	—	2	1	no	—
42	C₄²×C₁₉	—	2	1	no	—
43	(C₂×D₁₉)⋊C₄	—	8	2	no	—
44	(C₂×D₁₉)⋊C₄	—	8	1	no	—
45	(C₂×D₁₉)⋊C₄	—	8	1	no	—
46	C₄⋊Dic₁₉	—	4	1	no	—
47	C₄⋊Dic₁₉	—	4	1	no	—
48	#64520	—	8	1	no	—
49	#64520	—	8	1	no	—
50	C₂×C₁₉⋊C₈	—	4	2	no	—
51	D₁₅₂	—	4	2	no	—
52	D₁₅₂	—	4	2	no	—
53	C₁₆×C₁₉	—	8	1	no	—
54	#7088	—	19	1	no	maximal, Sylow
55	C₂³×C₁₉	#102	1	1	yes	characteristic
56	C₂²×D₁₉	—	4	2	no	—
57	C₂²×D₁₉	—	8	1	no	—
58	C₂²×D₁₉	—	8	1	no	—
59	C₂²×D₁₉	—	8	1	no	—
60	C₂²×D₁₉	—	8	1	no	—
61	C₂²×D₁₉	—	8	1	no	—
62	C₂²×D₁₉	—	8	1	no	—
63	C₂×C₄×C₁₉	—	2	1	no	—
64	C₂×C₄×C₁₉	—	2	1	no	—
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	—	4	1	no	—
69	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
70	C₂×C₄×C₁₉	#104	1	1	yes	characteristic
71	C₂×C₄×C₁₉	D₁₆	1	1	yes	characteristic
72	C₂×C₄×C₁₉	SD₃₂	1	1	yes	characteristic
73	C₂×Dic₁₉	—	8	1	no	—
74	C₂×Dic₁₉	—	8	1	no	—
75	C₂×Dic₁₉	—	8	1	no	—
76	D₇₆	—	4	2	no	—
77	D₇₆	—	4	2	no	—
78	D₇₆	—	8	1	no	—
79	D₇₆	—	8	1	no	—
80	C₈×C₁₉	—	4	1	no	—
81	C₈×C₁₉	—	4	1	no	—
82	C₁₉⋊C₈	—	8	2	no	—
83	#2054	—	19	1	no	—
84	#1322	D₁₉	1	1	yes	characteristic, complemented
85	#1292	—	19	1	no	—
86	C₂²×C₁₉	—	2	1	no	—
87	C₂²×C₁₉	—	2	1	no	—
88	C₂²×C₁₉	#327	1	1	yes	characteristic
89	C₂²×C₁₉	#357	1	1	yes	characteristic
90	C₂²×C₁₉	#360	1	1	yes	characteristic
91	C₂×D₁₉	—	8	2	no	—
92	C₂×D₁₉	—	8	2	no	—
93	C₂×D₁₉	—	8	2	no	—
94	C₂×D₁₉	—	16	1	no	—
95	C₂×D₁₉	—	16	1	no	—
96	C₂×D₁₉	—	16	1	no	—
97	C₂×D₁₉	—	16	1	no	—
98	C₄×C₁₉	—	2	1	no	—
99	C₄×C₁₉	—	2	1	no	—
100	C₄×C₁₉	—	2	1	no	—
101	C₄×C₁₉	—	4	1	no	—
102	C₄×C₁₉	AΣL(1,𝔽₁₆)	1	1	yes	characteristic
103	C₄×C₁₉	#355	1	1	yes	characteristic
104	Dic₁₉	—	16	1	no	—
105	C₂²×D₈	—	19	1	no	—
106	#423	—	38	1	no	—
107	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
108	C₂×#102	—	38	1	no	—
109	#390	—	38	1	no	—
110	C₂²⋊C₁₆	—	2	1	no	—
111	#340	—	19	1	no	—
112	C₂×C₁₉	—	2	1	no	—
113	C₂×C₁₉	—	2	1	no	—
114	C₂×C₁₉	#1341	1	1	yes	characteristic
115	C₂×C₁₉	#1345	1	1	yes	characteristic
116	C₂×C₁₉	#1347	1	1	yes	characteristic
117	D₁₉	—	16	2	no	—
118	C₂²×D₄	—	38	1	no	—
119	C₂×D₈	—	76	1	no	—
120	C₂×D₈	—	76	2	no	—
121	C₂×M₁₆	—	76	1	no	—
122	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
123	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
124	C₂×C₄⋊C₄	—	38	1	no	—
125	C₂×C₂²⋊C₄	—	76	1	no	—
126	C₂×C₂²⋊C₄	—	76	1	no	—
127	C₂×C₄²	D₇₆	1	1	yes	characteristic
128	C₂×C₁₆	—	4	1	no	—
129	C₄⋊C₈	—	4	1	no	—
130	C₄⋊C₈	—	38	2	no	—
131	C₄⋊C₈	—	76	1	no	—
132	#102	—	76	2	no	—
133	C₁₉	#7088	1	1	yes	characteristic, complemented, Sylow
134	C₂⁴	—	76	1	no	—
135	C₂×D₄	—	38	2	no	—
136	C₂×D₄	—	76	1	no	—
137	C₂×D₄	—	76	1	no	—
138	C₂×D₄	—	76	1	no	—
139	C₂×D₄	—	76	1	no	—
140	C₂×D₄	—	76	1	no	—
141	C₂²×C₄	—	76	1	no	—
142	C₂²×C₄	#64516	1	1	yes	characteristic
143	C₂²×C₄	D₁₅₂	1	1	yes	characteristic
144	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
145	D₈	—	76	2	no	—
146	D₈	—	76	2	no	—
147	M₁₆	—	152	1	no	—
148	M₁₆	—	152	1	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	4	1	no	—
152	C₂×C₈	—	4	1	no	—
153	C₂×C₈	—	76	2	no	—
154	C₄⋊C₄	—	76	1	no	—
155	C₄⋊C₄	—	76	1	no	—
156	C₂²⋊C₄	—	152	2	no	—
157	C₂²⋊C₄	—	152	1	no	—
158	C₂²⋊C₄	—	152	1	no	—
159	C₄²	—	2	1	no	—
160	C₄²	—	2	1	no	—
161	C₁₆	—	8	1	no	—
162	C₂³	—	76	2	no	—
163	C₂³	—	152	1	no	—
164	C₂³	—	152	1	no	—
165	C₂³	—	152	1	no	—
166	C₂³	—	152	1	no	—
167	C₂³	—	152	1	no	—
168	C₂³	—	152	1	no	—
169	C₂³	#10597278	1	1	yes	characteristic
170	D₄	—	76	2	no	—
171	D₄	—	76	2	no	—
172	D₄	—	152	1	no	—
173	D₄	—	152	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	4	1	no	—
180	C₂×C₄	—	152	1	no	—
181	C₂×C₄	—	152	1	no	—
182	C₂×C₄	—	152	1	no	—
183	C₂×C₄	D₃₀₄	1	1	yes	characteristic
184	C₂×C₄	#10597257	1	1	yes	characteristic
185	C₂×C₄	#10597263	1	1	yes	characteristic
186	C₂×C₄	#10597282	1	1	yes	characteristic
187	C₈	—	4	1	no	—
188	C₈	—	4	1	no	—
189	C₈	—	152	2	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	2	1	no	—
192	C₂²	—	304	1	no	—
193	C₂²	—	304	1	no	—
194	C₂²	—	304	1	no	—
195	C₂²	—	304	1	no	—
196	C₂²	—	152	2	no	—
197	C₂²	—	152	2	no	—
198	C₂²	—	152	2	no	—
199	C₂²	#11926279	1	1	yes	characteristic
200	C₂²	#11926346	1	1	yes	characteristic
201	C₂²	#11926355	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	4	1	no	—
206	C₄	—	304	1	no	—
207	C₄	#11926330	1	1	yes	characteristic
208	C₄	#11926340	1	1	yes	characteristic
209	C₂	—	2	1	no	—
210	C₂	—	2	1	no	—
211	C₂	—	304	2	no	—
212	C₂	#14561627	1	1	yes	characteristic
213	C₂	#14561639	1	1	yes	characteristic
214	C₂	#14561645	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14564946

C₂

yes

maximal, characteristic

#14561510

C₂

yes

maximal, characteristic, complemented

C₁₉×#1322

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₅₂

C₄

yes

—

—

—

C₄

yes

characteristic

C₂×C₁₉×C₄⋊C₈

C₂²

yes

characteristic

C₁₉×C₂²⋊C₁₆

—

C₂²×D₇₆

—

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₂×(C₂×D₁₉)⋊C₄

—

C₂×(C₂×D₁₉)⋊C₄

—

C₂×C₄⋊Dic₁₉

—

C₂×#64520

—

C₂×D₁₅₂

—

C₂×D₁₅₂

—

C₂×C₁₆×C₁₉

—

C₁₉×C₄⋊C₈

—

#10597278

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₂³×D₁₉

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

D₈

yes

characteristic

C₂²×C₄×C₁₉

SD₁₆

yes

characteristic

C₂²×Dic₁₉

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

(C₂×D₁₉)⋊C₄

—

(C₂×D₁₉)⋊C₄

—

(C₂×D₁₉)⋊C₄

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

#64520

—

#64520

—

C₂×C₁₉⋊C₈

—

D₁₅₂

—

D₁₅₂

—

C₁₆×C₁₉

—

—

C₂³×C₁₉

yes

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

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

D₁₆

yes

characteristic

C₂×C₄×C₁₉

SD₃₂

yes

characteristic

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

D₇₆

—

D₇₆

—

D₇₆

—

D₇₆

—

C₈×C₁₉

—

C₈×C₁₉

—

C₁₉⋊C₈

—

#2054

—

#1322

D₁₉

yes

characteristic, complemented

#1292

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

100

C₄×C₁₉

—

101

C₄×C₁₉

—

102

C₄×C₁₉

AΣL(1,𝔽₁₆)

yes

characteristic

103

C₄×C₁₉

#355

yes

characteristic

104

Dic₁₉

—

105

C₂²×D₈

—

106

#423

—

107

C₂×C₄⋊C₈

C₂×D₁₉

yes

characteristic

108

C₂×#102

—

109

#390

—

110

C₂²⋊C₁₆

—

111

#340

—

112

C₂×C₁₉

—

113

C₂×C₁₉

—

114

C₂×C₁₉

#1341

yes

characteristic

115

C₂×C₁₉

#1345

yes

characteristic

116

C₂×C₁₉

#1347

yes

characteristic

117

D₁₉

—

118

C₂²×D₄

—

119

C₂×D₈

—

120

C₂×D₈

—

121

C₂×M₁₆

—

122

C₂²×C₈

C₄×D₁₉

yes

characteristic

123

C₂²×C₈

C₂²⋊D₁₉

yes

characteristic

124

C₂×C₄⋊C₄

—

125

C₂×C₂²⋊C₄

—

126

C₂×C₂²⋊C₄

—

127

C₂×C₄²

D₇₆

yes

characteristic

128

C₂×C₁₆

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

131

C₄⋊C₈

—

132

#102

—

133

C₁₉

#7088

yes

characteristic, complemented, Sylow

134

C₂⁴

—

135

C₂×D₄

—

136

C₂×D₄

—

137

C₂×D₄

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂×D₄

—

141

C₂²×C₄

—

142

C₂²×C₄

#64516

yes

characteristic

143

C₂²×C₄

D₁₅₂

yes

characteristic

144

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

145

D₈

—

146

D₈

—

147

M₁₆

—

152

—

148

M₁₆

—

152

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₂²⋊C₄

—

152

—

157

C₂²⋊C₄

—

152

—

158

C₂²⋊C₄

—

152

—

159

C₄²

—

160

C₄²

—

161

C₁₆

—

162

C₂³

—

163

C₂³

—

152

—

164

C₂³

—

152

—

165

C₂³

—

152

—

166

C₂³

—

152

—

167

C₂³

—

152

—

168

C₂³

—

152

—

169

C₂³

#10597278

yes

characteristic

170

D₄

—

171

D₄

—

172

D₄

—

152

—

173

D₄

—

152

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

152

—

181

C₂×C₄

—

152

—

182

C₂×C₄

—

152

—

183

C₂×C₄

D₃₀₄

yes

characteristic

184

C₂×C₄

#10597257

yes

characteristic

185

C₂×C₄

#10597263

yes

characteristic

186

C₂×C₄

#10597282

yes

characteristic

187

C₈

—

188

C₈

—

189

C₈

—

152

—

190

C₂²

—

191

C₂²

—

192

C₂²

—

304

—

193

C₂²

—

304

—

194

C₂²

—

304

—

195

C₂²

—

304

—

196

C₂²

—

152

—

197

C₂²

—

152

—

198

C₂²

—

152

—

199

C₂²

#11926279

yes

characteristic

200

C₂²

#11926346

yes

characteristic

201

C₂²

#11926355

yes

characteristic

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

—

206

C₄

—

304

—

207

C₄

#11926330

yes

characteristic

208

C₄

#11926340

yes

characteristic

209

C₂

—

210

C₂

—

211

C₂

—

304

—

212

C₂

#14561627

yes

characteristic

213

C₂

#14561639

yes

characteristic

214

C₂

#14561645

yes

characteristic

	subgroup	G/H
center	C₂²	#11926279
commutator	C₂²×C₈×C₁₉	C₂×C₄
Frattini	C₂×C₄⋊C₈	C₂×D₁₉
Fitting	C₁₉×#1322	C₂
socle	C₂²×C₁₉	#327
prime radical	C₂²×D₁₅₂	C₄
cosocle	C₂×C₁₉×C₄⋊C₈	C₂²
hypercenter	#1322	D₁₉
hypocenter	C₁₉	#7088
LCS	G ▷ C₂²×C₈×C₁₉ ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ #1322

subgroup

G/H

center

C₂²

#11926279

commutator

C₂²×C₈×C₁₉

C₂×C₄

Frattini

C₂×C₄⋊C₈

C₂×D₁₉

Fitting

C₁₉×#1322

C₂

socle

C₂²×C₁₉

#327

prime radical

C₂²×D₁₅₂

C₄

cosocle

C₂×C₁₉×C₄⋊C₈

C₂²

D₁₉

C₁₉

G ▷ C₂²×C₈×C₁₉ ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ #1322

N	H ≅ G/N	type
2. #14561510	C₂	semidirect
3. C₁₉×#1322	C₂	semidirect
84. #1322	D₁₉	semidirect
133. C₁₉	#7088	semidirect

H ≅ G/N

type

2. #14561510

C₂

semidirect

3. C₁₉×#1322

C₂

semidirect

84. #1322

D₁₉

semidirect

133. C₁₉

#7088

semidirect

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

subgroup	index
#14564946	2
#14561510	2
C₁₉×#1322	2
#7088	19

subgroup

index

#14564946

#14561510

C₁₉×#1322

#7088

Subgroups — #18069249

Conjugacy classes of subgroups

Complements

Maximal subgroups