Group #18069608 — subgroups

Subgroups — #18069608

Order 4864.

252 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 79 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	#14565249	C₂	1	1	yes	maximal, characteristic
2	#14563681	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1812	C₂	1	1	yes	maximal, characteristic
4	C₂×#10597269	C₄	1	1	yes	characteristic, complemented
5	C₂²×C₁₉⋊C₁₆	C₄	1	1	yes	characteristic, complemented
6	#11926582	—	2	1	no	—
7	#11926518	—	2	1	no	—
8	C₂²⋊(C₁₉⋊C₁₆)	—	2	1	no	—
9	C₂²⋊(C₁₉⋊C₁₆)	—	2	1	no	—
10	#11926301	C₄	1	2	yes	complemented
11	C₂×C₁₉×#131	C₂²	1	1	yes	characteristic
12	C₁₉×#427	—	2	1	no	—
13	C₁₉×#410	—	2	1	no	—
14	C₁₉×H(3,ℤ/4)	C₄	1	2	yes	complemented
15	C₂×#41×C₁₉	D₄	1	1	yes	characteristic
16	C₁₉×#131	—	4	1	no	—
17	C₁₉×#131	—	4	1	no	—
18	C₂×M₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
19	C₂×M₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
20	C₂×M₁₆×C₁₉	Q₈	1	1	yes	characteristic
21	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
24	C₁₉×#117	—	2	2	no	—
25	#10597372	—	2	2	no	—
26	C₁₉×#107	—	4	1	no	—
27	C₁₉×#106	—	4	1	no	—
28	C₁₉×#99	—	4	1	no	—
29	C₁₉×#99	—	4	1	no	—
30	#10597291	—	4	1	no	—
31	#10597291	—	4	1	no	—
32	#10597275	—	4	1	no	—
33	#10597274	—	4	1	no	—
34	#10597269	—	4	1	no	—
35	C₂×C₁₉⋊C₁₆	—	2	1	no	—
36	C₂×C₁₉⋊C₁₆	—	2	1	no	—
37	C₂×C₁₉⋊C₁₆	—	4	1	no	—
38	#41×C₁₉	—	4	1	no	—
39	#41×C₁₉	—	4	1	no	—
40	C₂×Q₈×C₁₉	Q₁₆	1	1	yes	characteristic
41	C₂×D₄×C₁₉	D₈	1	1	yes	characteristic
42	C₂×D₄×C₁₉	SD₁₆	1	1	yes	characteristic
43	C₂×D₄×C₁₉	SD₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
47	M₁₆×C₁₉	—	4	1	no	—
48	M₁₆×C₁₉	—	4	1	no	—
49	M₁₆×C₁₉	—	4	1	no	—
50	C₂×C₈×C₁₉	—	2	1	no	—
51	C₂×C₈×C₁₉	—	2	1	no	—
52	C₂×C₈×C₁₉	—	2	1	no	—
53	C₂×C₈×C₁₉	—	2	1	no	—
54	C₂×C₈×C₁₉	—	2	1	no	—
55	C₂×C₈×C₁₉	—	2	1	no	—
56	C₂×C₈×C₁₉	C₄²	1	1	yes	characteristic
57	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
60	C₄⋊C₄×C₁₉	—	4	2	no	—
61	C₂²⋊C₄×C₁₉	—	4	2	no	—
62	C₄²×C₁₉	—	4	2	no	—
63	C₂²⋊Dic₁₉	—	4	2	no	—
64	C₄⋊Dic₁₉	—	4	2	no	—
65	C₄×Dic₁₉	—	4	2	no	—
66	C₁₉⋊C₁₆	—	4	1	no	—
67	C₁₉⋊C₁₆	—	4	1	no	—
68	#7228	—	19	1	no	maximal, Sylow
69	C₂³×C₁₉	#102	1	1	yes	characteristic
70	C₂³×C₁₉	#102	1	1	yes	characteristic
71	C₂³×C₁₉	#106	1	1	yes	characteristic
72	Q₈×C₁₉	—	4	1	no	—
73	D₄×C₁₉	—	4	1	no	—
74	D₄×C₁₉	—	4	1	no	—
75	D₄×C₁₉	—	4	1	no	—
76	C₂×C₄×C₁₉	—	2	1	no	—
77	C₂×C₄×C₁₉	—	2	1	no	—
78	C₂×C₄×C₁₉	—	2	1	no	—
79	C₂×C₄×C₁₉	—	2	1	no	—
80	C₂×C₄×C₁₉	—	2	1	no	—
81	C₂×C₄×C₁₉	—	2	1	no	—
82	C₂×C₄×C₁₉	—	8	2	no	—
83	C₂×C₄×C₁₉	#95	1	1	yes	characteristic
84	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
85	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
86	C₂×C₄×C₁₉	#107	1	1	yes	characteristic
87	C₂×Dic₁₉	—	8	2	no	—
88	C₈×C₁₉	—	2	1	no	—
89	C₈×C₁₉	—	2	1	no	—
90	C₈×C₁₉	—	2	1	no	—
91	C₈×C₁₉	#99	1	1	yes	characteristic
92	C₈×C₁₉	#99	1	1	yes	characteristic
93	#2113	—	19	1	no	—
94	#1812	—	19	1	no	—
95	#1812	D₁₉	1	1	yes	characteristic
96	C₂²×C₁₉	—	2	1	no	—
97	C₂²×C₁₉	—	2	1	no	—
98	C₂²×C₁₉	—	2	1	no	—
99	C₂²×C₁₉	—	4	1	no	—
100	C₂²×C₁₉	—	4	1	no	—
101	C₂²×C₁₉	—	4	1	no	—
102	C₂²×C₁₉	#340	1	1	yes	characteristic
103	C₄×C₁₉	—	2	1	no	—
104	C₄×C₁₉	—	2	1	no	—
105	C₄×C₁₉	—	2	1	no	—
106	C₄×C₁₉	—	16	2	no	—
107	C₄×C₁₉	H(3,ℤ/4)	1	1	yes	characteristic
108	C₄×C₁₉	#342	1	1	yes	characteristic
109	Dic₁₉	—	16	2	no	—
110	C₂×#131	C₂×D₁₉	1	1	yes	characteristic
111	C₂×M₃₂	—	19	1	no	—
112	C₂²×C₁₆	—	19	1	no	—
113	#427	—	2	1	no	—
114	#427	—	38	1	no	—
115	#410	—	2	1	no	—
116	#410	—	38	1	no	—
117	C₂²⋊C₁₆	—	38	1	no	—
118	C₂²⋊C₁₆	—	38	1	no	—
119	H(3,ℤ/4)	—	19	2	no	—
120	H(3,ℤ/4)	Dic₁₉	1	2	yes	complemented
121	C₂×C₁₉	—	2	1	no	—
122	C₂×C₁₉	—	4	1	no	—
123	C₂×C₁₉	—	4	1	no	—
124	C₂×C₁₉	—	4	1	no	—
125	C₂×C₁₉	#1289	1	1	yes	characteristic
126	C₂×#41	C₂²⋊D₁₉	1	1	yes	characteristic
127	#131	—	4	1	no	—
128	#131	—	4	1	no	—
129	C₂×M₁₆	Dic₃₈	1	1	yes	characteristic
130	C₂×M₁₆	C₄×D₁₉	1	1	yes	characteristic
131	C₂×M₁₆	C₂×Dic₁₉	1	1	yes	characteristic
132	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
133	C₂²×C₈	D₇₆	1	1	yes	characteristic
134	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
135	#117	—	2	2	no	—
136	#117	—	38	2	no	—
137	M₃₂	—	76	1	no	—
138	C₂×C₁₆	—	38	1	no	—
139	C₂×C₁₆	—	38	1	no	—
140	C₂×C₁₆	—	76	1	no	—
141	#107	—	4	1	no	—
142	#107	—	76	1	no	—
143	#106	—	4	1	no	—
144	#106	—	76	1	no	—
145	#99	—	4	1	no	—
146	#99	—	4	1	no	—
147	#99	—	76	1	no	—
148	#99	—	76	1	no	—
149	C₁₉	#7228	1	1	yes	characteristic, complemented, Sylow
150	#41	—	4	1	no	—
151	#41	—	4	1	no	—
152	C₂×Q₈	#64528	1	1	yes	characteristic
153	C₂×D₄	#64525	1	1	yes	characteristic
154	C₂×D₄	#64526	1	1	yes	characteristic
155	C₂×D₄	#64527	1	1	yes	characteristic
156	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
157	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
158	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
159	M₁₆	—	4	1	no	—
160	M₁₆	—	4	1	no	—
161	M₁₆	—	4	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	C₄×Dic₁₉	1	1	yes	characteristic
169	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
170	C₂×C₈	C₄⋊Dic₁₉	1	1	yes	characteristic
171	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	76	2	no	—
174	C₂²⋊C₄	—	4	2	no	—
175	C₂²⋊C₄	—	76	2	no	—
176	C₄²	—	4	2	no	—
177	C₄²	—	76	2	no	—
178	C₁₆	—	76	1	no	—
179	C₁₆	—	76	1	no	—
180	C₂³	#10597265	1	1	yes	characteristic
181	C₂³	#10597266	1	1	yes	characteristic
182	C₂³	#10597289	1	1	yes	characteristic
183	Q₈	—	4	1	no	—
184	D₄	—	4	1	no	—
185	D₄	—	4	1	no	—
186	D₄	—	4	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	8	2	no	—
194	C₂×C₄	—	152	2	no	—
195	C₂×C₄	#10597264	1	1	yes	characteristic
196	C₂×C₄	#10597267	1	1	yes	characteristic
197	C₂×C₄	C₁₉⋊#95	1	1	yes	characteristic
198	C₂×C₄	#10597292	1	1	yes	characteristic
199	C₈	—	2	1	no	—
200	C₈	—	2	1	no	—
201	C₈	—	2	1	no	—
202	C₈	#10597263	1	1	yes	characteristic
203	C₈	#10597263	1	1	yes	characteristic
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	4	1	no	—
208	C₂²	—	4	1	no	—
209	C₂²	—	4	1	no	—
210	C₂²	#11926306	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	16	2	no	—
215	C₄	—	304	2	no	—
216	C₄	#11926300	1	1	yes	characteristic
217	C₄	#11926310	1	1	yes	characteristic
218	C₂	—	2	1	no	—
219	C₂	—	4	1	no	—
220	C₂	—	4	1	no	—
221	C₂	—	4	1	no	—
222	C₂	#14561502	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14565249

C₂

yes

maximal, characteristic

#14563681

C₂

yes

maximal, characteristic

C₁₉×#1812

C₂

yes

maximal, characteristic

C₂×#10597269

C₄

yes

characteristic, complemented

C₂²×C₁₉⋊C₁₆

C₄

yes

characteristic, complemented

#11926582

—

#11926518

—

C₂²⋊(C₁₉⋊C₁₆)

—

C₂²⋊(C₁₉⋊C₁₆)

—

#11926301

C₄

yes

complemented

C₂×C₁₉×#131

C₂²

yes

characteristic

C₁₉×#427

—

C₁₉×#410

—

C₁₉×H(3,ℤ/4)

C₄

yes

complemented

C₂×#41×C₁₉

D₄

yes

characteristic

C₁₉×#131

—

C₁₉×#131

—

C₂×M₁₆×C₁₉

C₂×C₄

yes

characteristic

C₂×M₁₆×C₁₉

C₂×C₄

yes

characteristic

C₂×M₁₆×C₁₉

Q₈

yes

characteristic

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₁₉×#117

—

#10597372

—

C₁₉×#107

—

C₁₉×#106

—

C₁₉×#99

—

C₁₉×#99

—

—

—

—

—

—

C₂×C₁₉⋊C₁₆

—

C₂×C₁₉⋊C₁₆

—

C₂×C₁₉⋊C₁₆

—

#41×C₁₉

—

#41×C₁₉

—

C₂×Q₈×C₁₉

Q₁₆

yes

characteristic

C₂×D₄×C₁₉

D₈

yes

characteristic

C₂×D₄×C₁₉

SD₁₆

yes

characteristic

C₂×D₄×C₁₉

SD₁₆

yes

characteristic

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₄⋊C₄

yes

characteristic

M₁₆×C₁₉

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₄²

yes

characteristic

C₂×C₈×C₁₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₄⋊C₄

yes

characteristic

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₂²⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄×Dic₁₉

—

C₁₉⋊C₁₆

—

C₁₉⋊C₁₆

—

—

C₂³×C₁₉

yes

C₂³×C₁₉

yes

C₂³×C₁₉

yes

Q₈×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

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×Dic₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

yes

C₈×C₁₉

yes

—

—

D₁₉

yes

characteristic

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

100

C₂²×C₁₉

—

101

C₂²×C₁₉

—

102

C₂²×C₁₉

#340

yes

characteristic

103

C₄×C₁₉

—

104

C₄×C₁₉

—

105

C₄×C₁₉

—

106

C₄×C₁₉

—

107

C₄×C₁₉

H(3,ℤ/4)

yes

characteristic

108

C₄×C₁₉

#342

yes

characteristic

109

Dic₁₉

—

110

C₂×#131

C₂×D₁₉

yes

characteristic

111

C₂×M₃₂

—

112

C₂²×C₁₆

—

113

#427

—

114

#427

—

115

#410

—

116

#410

—

117

C₂²⋊C₁₆

—

118

C₂²⋊C₁₆

—

119

H(3,ℤ/4)

—

120

H(3,ℤ/4)

Dic₁₉

yes

complemented

121

C₂×C₁₉

—

122

C₂×C₁₉

—

123

C₂×C₁₉

—

124

C₂×C₁₉

—

125

C₂×C₁₉

#1289

yes

characteristic

126

C₂×#41

C₂²⋊D₁₉

yes

characteristic

127

#131

—

128

#131

—

129

C₂×M₁₆

Dic₃₈

yes

characteristic

130

C₂×M₁₆

C₄×D₁₉

yes

characteristic

131

C₂×M₁₆

C₂×Dic₁₉

yes

characteristic

132

C₂²×C₈

C₄×D₁₉

yes

characteristic

133

C₂²×C₈

D₇₆

yes

characteristic

134

C₂²×C₈

C₂²⋊D₁₉

yes

characteristic

135

#117

—

136

#117

—

137

M₃₂

—

138

C₂×C₁₆

—

139

C₂×C₁₆

—

140

C₂×C₁₆

—

141

#107

—

142

#107

—

143

#106

—

144

#106

—

145

#99

—

146

#99

—

147

#99

—

148

#99

—

149

C₁₉

#7228

yes

characteristic, complemented, Sylow

150

#41

—

151

#41

—

152

C₂×Q₈

#64528

yes

characteristic

153

C₂×D₄

#64525

yes

characteristic

154

C₂×D₄

#64526

yes

characteristic

155

C₂×D₄

#64527

yes

characteristic

156

C₂²×C₄

Dic₁₉⋊C₄

yes

characteristic

157

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

158

C₂²×C₄

C₂²⋊Dic₁₉

yes

characteristic

159

M₁₆

—

160

M₁₆

—

161

M₁₆

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

C₄×Dic₁₉

yes

characteristic

169

C₂×C₈

Dic₁₉⋊C₄

yes

characteristic

170

C₂×C₈

C₄⋊Dic₁₉

yes

characteristic

171

C₂×C₈

(C₂×D₁₉)⋊C₄

yes

characteristic

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₂²⋊C₄

—

175

C₂²⋊C₄

—

176

C₄²

—

177

C₄²

—

178

C₁₆

—

179

C₁₆

—

180

C₂³

#10597265

yes

characteristic

181

C₂³

#10597266

yes

characteristic

182

C₂³

#10597289

yes

characteristic

183

Q₈

—

184

D₄

—

185

D₄

—

186

D₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

152

—

195

C₂×C₄

#10597264

yes

characteristic

196

C₂×C₄

#10597267

yes

characteristic

197

C₂×C₄

C₁₉⋊#95

yes

characteristic

198

C₂×C₄

#10597292

yes

characteristic

199

C₈

—

200

C₈

—

201

C₈

—

202

C₈

#10597263

yes

characteristic

203

C₈

#10597263

yes

characteristic

204

C₂²

—

205

C₂²

—

206

C₂²

—

207

C₂²

—

208

C₂²

—

209

C₂²

—

210

C₂²

#11926306

yes

characteristic

211

C₄

—

212

C₄

—

213

C₄

—

214

C₄

—

215

C₄

—

304

—

216

C₄

#11926300

yes

characteristic

217

C₄

#11926310

yes

characteristic

218

C₂

—

219

C₂

—

220

C₂

—

221

C₂

—

222

C₂

#14561502

yes

characteristic

	subgroup	G/H
center	C₄	#11926300
commutator	C₂×C₈×C₁₉	C₄²
Frattini	C₂×#131	C₂×D₁₉
Fitting	C₁₉×#1812	C₂
socle	C₂×C₁₉	#1289
prime radical	C₂×#41×C₁₉	D₄
cosocle	C₂×C₁₉×#131	C₂²
hypercenter	#1812	D₁₉
hypocenter	C₁₉	#7228
LCS	G ▷ C₂×C₈×C₁₉ ▷ C₄×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #1812

subgroup

G/H

center

C₄

#11926300

commutator

C₂×C₈×C₁₉

C₄²

Frattini

C₂×#131

C₂×D₁₉

Fitting

C₁₉×#1812

C₂

socle

C₂×C₁₉

#1289

prime radical

C₂×#41×C₁₉

D₄

cosocle

C₂×C₁₉×#131

C₂²

D₁₉

C₁₉

G ▷ C₂×C₈×C₁₉ ▷ C₄×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ #1812

N	H ≅ G/N	type
4. C₂×#10597269	C₄	semidirect
5. C₂²×C₁₉⋊C₁₆	C₄	semidirect
10. #11926301	C₄	semidirect
14. C₁₉×H(3,ℤ/4)	C₄	semidirect
120. H(3,ℤ/4)	Dic₁₉	semidirect
149. C₁₉	#7228	semidirect

H ≅ G/N

type

4. C₂×#10597269

C₄

semidirect

5. C₂²×C₁₉⋊C₁₆

C₄

semidirect

10. #11926301

C₄

semidirect

14. C₁₉×H(3,ℤ/4)

C₄

semidirect

120. H(3,ℤ/4)

Dic₁₉

semidirect

149. C₁₉

#7228

semidirect

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

subgroup	index
#14565249	2
#14563681	2
C₁₉×#1812	2
#7228	19

subgroup

index

#14565249

#14563681

C₁₉×#1812

#7228

Subgroups — #18069608

Conjugacy classes of subgroups

Complements

Maximal subgroups