Group #18018124 — subgroups

Subgroups — C₁₉×#12165 #18018124

Order 4864.

350 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 74 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	C₁₉×#3261	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₉×#2187	C₂	1	2	yes	maximal
3	C₁₉×#2045	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1350	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1323	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈×C₁₉	C₂²	1	1	yes	characteristic
7	C₁₉×#541	—	2	1	no	—
8	C₂×C₁₉×SD₃₂	—	2	2	no	—
9	C₁₉×#479	—	2	2	no	—
10	C₁₉×#470	—	2	1	no	—
11	C₁₉×#467	C₂²	1	2	yes	—
12	C₁₉×#462	—	2	2	no	—
13	C₁₉×#424	C₂²	1	1	yes	characteristic
14	C₁₉×#418	—	2	1	no	—
15	C₂×C₁₉×#100	—	2	1	no	—
16	C₁₉×#357	—	2	2	no	—
17	C₁₉×C₂²⋊C₁₆	C₂²	1	2	yes	—
18	C₁₉×#341	C₂²	1	1	yes	characteristic
19	C₂²×D₄×C₁₉	—	2	1	no	—
20	C₂×Q₁₆×C₁₉	—	2	1	no	—
21	C₂×Q₁₆×C₁₉	D₄	1	2	yes	—
22	C₂×D₈×C₁₉	—	2	2	no	—
23	C₂×D₈×C₁₉	—	4	2	no	—
24	C₂×M₁₆×C₁₉	—	2	1	no	—
25	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
27	C₁₉×C₄⋊Q₈	—	2	1	no	—
28	C₁₉×#126	—	4	2	no	—
29	C₁₉×#125	—	2	1	no	—
30	C₁₉×#125	—	4	2	no	—
31	C₁₉×C₂²⋊Q₈	—	2	2	no	—
32	C₄×Q₈×C₁₉	—	2	2	no	—
33	C₁₉×#117	D₄	1	1	yes	characteristic
34	C₁₉×SD₃₂	—	4	4	no	—
35	C₂×C₁₆×C₁₉	—	2	2	no	—
36	C₁₉×#107	D₄	1	2	yes	—
37	C₁₉×#106	—	2	1	no	—
38	C₁₉×C₄⋊C₈	—	2	2	no	—
39	C₁₉×#103	—	2	2	no	—
40	C₁₉×#103	—	2	2	no	—
41	C₁₉×#102	—	4	2	no	—
42	C₁₉×#100	—	4	2	no	—
43	C₂⁴×C₁₉	—	4	1	no	—
44	C₂×Q₈×C₁₉	—	2	2	no	—
45	C₂×D₄×C₁₉	—	2	2	no	—
46	C₂×D₄×C₁₉	—	4	1	no	—
47	C₂×D₄×C₁₉	—	4	1	no	—
48	C₂×D₄×C₁₉	—	4	1	no	—
49	C₂×D₄×C₁₉	—	4	1	no	—
50	C₂×D₄×C₁₉	—	4	1	no	—
51	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
52	Q₁₆×C₁₉	—	4	2	no	—
53	Q₁₆×C₁₉	—	8	1	no	—
54	D₈×C₁₉	—	2	4	no	—
55	D₈×C₁₉	—	4	2	no	—
56	M₁₆×C₁₉	—	4	1	no	—
57	M₁₆×C₁₉	—	4	1	no	—
58	M₁₆×C₁₉	—	4	1	no	—
59	C₂×C₈×C₁₉	—	2	1	no	—
60	C₂×C₈×C₁₉	—	4	1	no	—
61	C₂×C₈×C₁₉	—	4	1	no	—
62	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
63	C₄⋊C₄×C₁₉	—	2	2	no	—
64	C₄⋊C₄×C₁₉	—	4	2	no	—
65	C₄⋊C₄×C₁₉	—	4	2	no	—
66	C₄⋊C₄×C₁₉	—	4	2	no	—
67	C₄⋊C₄×C₁₉	—	4	2	no	—
68	C₄⋊C₄×C₁₉	D₈	1	2	yes	—
69	C₂²⋊C₄×C₁₉	—	2	1	no	—
70	C₂²⋊C₄×C₁₉	—	4	2	no	—
71	C₄²×C₁₉	—	2	1	no	—
72	C₄²×C₁₉	—	4	2	no	—
73	C₁₆×C₁₉	—	4	2	no	—
74	#12165	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×C₁₉	—	4	2	no	—
76	C₂³×C₁₉	—	8	1	no	—
77	C₂³×C₁₉	—	8	1	no	—
78	C₂³×C₁₉	—	16	2	no	—
79	C₂³×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
80	Q₈×C₁₉	—	4	2	no	—
81	Q₈×C₁₉	—	4	2	no	—
82	D₄×C₁₉	—	4	2	no	—
83	D₄×C₁₉	—	4	2	no	—
84	D₄×C₁₉	—	8	1	no	—
85	D₄×C₁₉	—	8	1	no	—
86	C₂×C₄×C₁₉	—	2	1	no	—
87	C₂×C₄×C₁₉	—	2	1	no	—
88	C₂×C₄×C₁₉	—	2	2	no	—
89	C₂×C₄×C₁₉	—	4	2	no	—
90	C₂×C₄×C₁₉	—	4	2	no	—
91	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
92	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
93	C₈×C₁₉	—	2	2	no	—
94	C₈×C₁₉	—	4	1	no	—
95	C₈×C₁₉	—	8	1	no	—
96	#3261	C₂×C₁₉	1	1	yes	characteristic, complemented
97	#2187	C₂×C₁₉	1	2	yes	—
98	#2045	C₂×C₁₉	1	1	yes	characteristic
99	#1350	C₂×C₁₉	1	2	yes	complemented
100	#1323	C₂×C₁₉	1	1	yes	characteristic, complemented
101	C₂²×C₁₉	—	2	1	no	—
102	C₂²×C₁₉	—	2	2	no	—
103	C₂²×C₁₉	—	8	2	no	—
104	C₂²×C₁₉	—	16	2	no	—
105	C₂²×C₁₉	—	16	2	no	—
106	C₂²×C₁₉	—	16	2	no	—
107	C₂²×C₁₉	#447	1	1	yes	characteristic
108	C₄×C₁₉	—	2	1	no	—
109	C₄×C₁₉	—	4	2	no	—
110	C₄×C₁₉	—	8	2	no	—
111	C₄×C₁₉	—	8	2	no	—
112	C₄×C₁₉	#454	1	1	yes	characteristic
113	C₄×C₁₉	B(4,𝔽₂)	1	1	yes	characteristic
114	C₂²×D₈	C₂²×C₁₉	1	1	yes	characteristic
115	#541	—	2	1	no	—
116	C₂×SD₃₂	—	2	2	no	—
117	#479	—	2	2	no	—
118	#470	—	2	1	no	—
119	#467	C₂²×C₁₉	1	2	yes	—
120	#462	—	2	2	no	—
121	#424	C₂²×C₁₉	1	1	yes	characteristic
122	#418	—	2	1	no	—
123	C₂×#100	—	2	1	no	—
124	#357	—	2	2	no	—
125	C₂²⋊C₁₆	C₂²×C₁₉	1	2	yes	—
126	#341	C₂²×C₁₉	1	1	yes	characteristic
127	C₂×C₁₉	—	4	1	no	—
128	C₂×C₁₉	—	16	2	no	—
129	C₂×C₁₉	#1598	1	1	yes	characteristic
130	C₂×C₁₉	#2193	1	2	yes	—
131	C₂²×D₄	—	2	1	no	—
132	C₂×Q₁₆	—	2	1	no	—
133	C₂×Q₁₆	D₄×C₁₉	1	2	yes	—
134	C₂×D₈	—	2	2	no	—
135	C₂×D₈	—	4	2	no	—
136	C₂×M₁₆	—	2	1	no	—
137	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
138	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
139	C₄⋊Q₈	—	2	1	no	—
140	#126	—	4	2	no	—
141	#125	—	2	1	no	—
142	#125	—	4	2	no	—
143	C₂²⋊Q₈	—	2	2	no	—
144	C₄×Q₈	—	2	2	no	—
145	#117	D₄×C₁₉	1	1	yes	characteristic
146	SD₃₂	—	4	4	no	—
147	C₂×C₁₆	—	2	2	no	—
148	#107	D₄×C₁₉	1	2	yes	—
149	#106	—	2	1	no	—
150	C₄⋊C₈	—	2	2	no	—
151	#103	—	2	2	no	—
152	#103	—	2	2	no	—
153	#102	—	4	2	no	—
154	#100	—	4	2	no	—
155	C₁₉	#12165	1	1	yes	characteristic, direct product factor, Sylow
156	C₂⁴	—	4	1	no	—
157	C₂×Q₈	—	2	2	no	—
158	C₂×D₄	—	2	2	no	—
159	C₂×D₄	—	4	1	no	—
160	C₂×D₄	—	4	1	no	—
161	C₂×D₄	—	4	1	no	—
162	C₂×D₄	—	4	1	no	—
163	C₂×D₄	—	4	1	no	—
164	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
165	Q₁₆	—	4	2	no	—
166	Q₁₆	—	8	1	no	—
167	D₈	—	2	4	no	—
168	D₈	—	4	2	no	—
169	M₁₆	—	4	1	no	—
170	M₁₆	—	4	1	no	—
171	M₁₆	—	4	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	4	1	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	D₈×C₁₉	1	2	yes	—
182	C₂²⋊C₄	—	2	1	no	—
183	C₂²⋊C₄	—	4	2	no	—
184	C₄²	—	2	1	no	—
185	C₄²	—	4	2	no	—
186	C₁₆	—	4	2	no	—
187	C₂³	—	4	2	no	—
188	C₂³	—	8	1	no	—
189	C₂³	—	8	1	no	—
190	C₂³	—	16	2	no	—
191	C₂³	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
192	Q₈	—	4	2	no	—
193	Q₈	—	4	2	no	—
194	D₄	—	4	2	no	—
195	D₄	—	4	2	no	—
196	D₄	—	8	1	no	—
197	D₄	—	8	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
204	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	—	4	1	no	—
207	C₈	—	8	1	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	8	2	no	—
211	C₂²	—	16	2	no	—
212	C₂²	—	16	2	no	—
213	C₂²	—	16	2	no	—
214	C₂²	C₁₉×#447	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	4	2	no	—
217	C₄	—	8	2	no	—
218	C₄	—	8	2	no	—
219	C₄	C₁₉×#454	1	1	yes	characteristic
220	C₄	C₁₉×B(4,𝔽₂)	1	1	yes	characteristic
221	C₂	—	4	1	no	—
222	C₂	—	16	2	no	—
223	C₂	C₁₉×#1598	1	1	yes	characteristic
224	C₂	C₁₉×#2193	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#3261

C₂

yes

maximal, characteristic, complemented

C₁₉×#2187

C₂

yes

maximal

C₁₉×#2045

C₂

yes

maximal, characteristic

C₁₉×#1350

C₂

yes

maximal, complemented

C₁₉×#1323

C₂

yes

maximal, characteristic, complemented

C₂²×D₈×C₁₉

C₂²

yes

characteristic

C₁₉×#541

—

C₂×C₁₉×SD₃₂

—

C₁₉×#479

—

C₁₉×#470

—

C₁₉×#467

C₂²

yes

—

C₁₉×#462

—

C₁₉×#424

C₂²

yes

characteristic

C₁₉×#418

—

C₂×C₁₉×#100

—

C₁₉×#357

—

C₁₉×C₂²⋊C₁₆

C₂²

yes

—

C₁₉×#341

C₂²

yes

characteristic

C₂²×D₄×C₁₉

—

C₂×Q₁₆×C₁₉

—

C₂×Q₁₆×C₁₉

D₄

yes

—

C₂×D₈×C₁₉

—

C₂×D₈×C₁₉

—

C₂×M₁₆×C₁₉

—

C₂×M₁₆×C₁₉

D₄

yes

characteristic

C₂²×C₈×C₁₉

C₂³

yes

characteristic

C₁₉×C₄⋊Q₈

—

C₁₉×#126

—

C₁₉×#125

—

C₁₉×#125

—

C₁₉×C₂²⋊Q₈

—

C₄×Q₈×C₁₉

—

C₁₉×#117

D₄

yes

characteristic

C₁₉×SD₃₂

—

C₂×C₁₆×C₁₉

—

C₁₉×#107

D₄

yes

—

C₁₉×#106

—

C₁₉×C₄⋊C₈

—

C₁₉×#103

—

C₁₉×#103

—

C₁₉×#102

—

C₁₉×#100

—

C₂⁴×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

Q₁₆×C₁₉

—

Q₁₆×C₁₉

—

D₈×C₁₉

—

D₈×C₁₉

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₂×D₄

yes

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

D₈

yes

—

C₂²⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₁₆×C₁₉

—

#12165

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

AGL(1,ℤ/8)

yes

characteristic

Q₈×C₁₉

—

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

#120

yes

characteristic

C₂×C₄×C₁₉

C₂×D₈

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

#3261

C₂×C₁₉

yes

characteristic, complemented

#2187

C₂×C₁₉

yes

—

#2045

C₂×C₁₉

yes

characteristic

#1350

C₂×C₁₉

yes

complemented

100

#1323

C₂×C₁₉

yes

characteristic, complemented

101

C₂²×C₁₉

—

102

C₂²×C₁₉

—

103

C₂²×C₁₉

—

104

C₂²×C₁₉

—

105

C₂²×C₁₉

—

106

C₂²×C₁₉

—

107

C₂²×C₁₉

#447

yes

characteristic

108

C₄×C₁₉

—

109

C₄×C₁₉

—

110

C₄×C₁₉

—

111

C₄×C₁₉

—

112

C₄×C₁₉

#454

yes

characteristic

113

C₄×C₁₉

B(4,𝔽₂)

yes

characteristic

114

C₂²×D₈

C₂²×C₁₉

yes

characteristic

115

#541

—

116

C₂×SD₃₂

—

117

#479

—

118

#470

—

119

#467

C₂²×C₁₉

yes

—

120

#462

—

121

#424

C₂²×C₁₉

yes

characteristic

122

#418

—

123

C₂×#100

—

124

#357

—

125

C₂²⋊C₁₆

C₂²×C₁₉

yes

—

126

#341

C₂²×C₁₉

yes

characteristic

127

C₂×C₁₉

—

128

C₂×C₁₉

—

129

C₂×C₁₉

#1598

yes

characteristic

130

C₂×C₁₉

#2193

yes

—

131

C₂²×D₄

—

132

C₂×Q₁₆

—

133

C₂×Q₁₆

D₄×C₁₉

yes

—

134

C₂×D₈

—

135

C₂×D₈

—

136

C₂×M₁₆

—

137

C₂×M₁₆

D₄×C₁₉

yes

characteristic

138

C₂²×C₈

C₂³×C₁₉

yes

characteristic

139

C₄⋊Q₈

—

140

#126

—

141

#125

—

142

#125

—

143

C₂²⋊Q₈

—

144

C₄×Q₈

—

145

#117

D₄×C₁₉

yes

characteristic

146

SD₃₂

—

147

C₂×C₁₆

—

148

#107

D₄×C₁₉

yes

—

149

#106

—

150

C₄⋊C₈

—

151

#103

—

152

#103

—

153

#102

—

154

#100

—

155

C₁₉

#12165

yes

characteristic, direct product factor, Sylow

156

C₂⁴

—

157

C₂×Q₈

—

158

C₂×D₄

—

159

C₂×D₄

—

160

C₂×D₄

—

161

C₂×D₄

—

162

C₂×D₄

—

163

C₂×D₄

—

164

C₂²×C₄

C₂×D₄×C₁₉

yes

characteristic

165

Q₁₆

—

166

Q₁₆

—

167

D₈

—

168

D₈

—

169

M₁₆

—

170

M₁₆

—

171

M₁₆

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

C₂×D₄×C₁₉

yes

—

176

C₄⋊C₄

—

177

C₄⋊C₄

—

178

C₄⋊C₄

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₄⋊C₄

D₈×C₁₉

yes

—

182

C₂²⋊C₄

—

183

C₂²⋊C₄

—

184

C₄²

—

185

C₄²

—

186

C₁₆

—

187

C₂³

—

188

C₂³

—

189

C₂³

—

190

C₂³

—

191

C₂³

C₁₉×AGL(1,ℤ/8)

yes

characteristic

192

Q₈

—

193

Q₈

—

194

D₄

—

195

D₄

—

196

D₄

—

197

D₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

C₁₉×#120

yes

characteristic

204

C₂×C₄

C₂×D₈×C₁₉

yes

characteristic

205

C₈

—

206

C₈

—

207

C₈

—

208

C₂²

—

209

C₂²

—

210

C₂²

—

211

C₂²

—

212

C₂²

—

213

C₂²

—

214

C₂²

C₁₉×#447

yes

characteristic

215

C₄

—

216

C₄

—

217

C₄

—

218

C₄

—

219

C₄

C₁₉×#454

yes

characteristic

220

C₄

C₁₉×B(4,𝔽₂)

yes

characteristic

221

C₂

—

222

C₂

—

223

C₂

C₁₉×#1598

yes

characteristic

224

C₂

C₁₉×#2193

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₉	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₂²×D₈×C₁₉	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#424 ◁ G

subgroup

G/H

center, socle

C₂²×C₁₉

#447

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₁₉

prime radical

C₂²×D₈×C₁₉

C₂²

LCS

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#424 ◁ G

N	H ≅ G/N	type
1. C₁₉×#3261	C₂	semidirect
4. C₁₉×#1350	C₂	semidirect
5. C₁₉×#1323	C₂	semidirect
74. #12165	C₁₉	direct (N × H)
96. #3261	C₂×C₁₉	semidirect
99. #1350	C₂×C₁₉	semidirect
100. #1323	C₂×C₁₉	semidirect
155. C₁₉	#12165	direct (N × H)

H ≅ G/N

type

1. C₁₉×#3261

C₂

semidirect

4. C₁₉×#1350

C₂

semidirect

5. C₁₉×#1323

C₂

semidirect

74. #12165

C₁₉

direct (N × H)

96. #3261

C₂×C₁₉

semidirect

99. #1350

C₂×C₁₉

semidirect

100. #1323

C₂×C₁₉

semidirect

155. C₁₉

#12165

direct (N × H)

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

subgroup	index
C₁₉×#3261	2
C₁₉×#2187	2
C₁₉×#2045	2
C₁₉×#1350	2
C₁₉×#1323	2
#12165	19

subgroup

index

C₁₉×#3261

C₁₉×#2187

C₁₉×#2045

C₁₉×#1350

C₁₉×#1323

#12165

Subgroups — C₁₉×#12165 #18018124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#12165 #18018124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12165 #18018124