Group #18018126 — subgroups

Subgroups — C₁₉×#12167 #18018126

Order 4864.

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

normal subgroups only

This group has 354 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×#3259	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2185	C₂	1	2	yes	maximal, complemented
3	C₁₉×#2047	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1349	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1323	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
7	C₁₉×#540	—	2	1	no	—
8	C₂×C₁₉×SD₃₂	—	2	2	no	—
9	C₁₉×#478	—	2	2	no	—
10	C₁₉×#469	—	2	1	no	—
11	C₁₉×#466	C₂²	1	2	yes	—
12	C₁₉×#459	—	2	2	no	—
13	C₁₉×#424	C₂²	1	1	yes	characteristic
14	C₁₉×#419	—	2	1	no	—
15	C₂×C₁₉×#101	—	2	1	no	—
16	C₁₉×#358	—	2	2	no	—
17	C₁₉×C₂²⋊C₁₆	C₂²	1	2	yes	—
18	C₁₉×#341	C₂²	1	1	yes	characteristic
19	C₂²×Q₈×C₁₉	—	2	1	no	—
20	C₂×Q₁₆×C₁₉	—	2	2	no	—
21	C₂×Q₁₆×C₁₉	—	4	2	no	—
22	C₂×D₈×C₁₉	—	2	1	no	—
23	C₂×D₈×C₁₉	D₄	1	2	yes	—
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×#127	—	2	1	no	—
28	C₁₉×#125	—	2	1	no	—
29	C₁₉×#123	—	4	2	no	—
30	C₁₉×#121	—	2	2	no	—
31	C₁₉×#121	—	4	2	no	—
32	C₄×D₄×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	—	4	2	no	—
40	C₁₉×#102	—	2	2	no	—
41	C₁₉×#102	—	2	2	no	—
42	C₁₉×#101	—	4	2	no	—
43	C₂×Q₈×C₁₉	—	2	2	no	—
44	C₂×Q₈×C₁₉	—	4	1	no	—
45	C₂×Q₈×C₁₉	—	4	1	no	—
46	C₂×Q₈×C₁₉	—	4	1	no	—
47	C₂×Q₈×C₁₉	—	4	1	no	—
48	C₂×Q₈×C₁₉	—	4	1	no	—
49	C₂×D₄×C₁₉	—	2	2	no	—
50	C₂×D₄×C₁₉	—	4	2	no	—
51	C₂×D₄×C₁₉	—	4	2	no	—
52	C₂²×C₄×C₁₉	—	4	2	no	—
53	C₂²×C₄×C₁₉	—	4	1	no	—
54	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
55	Q₁₆×C₁₉	—	2	4	no	—
56	Q₁₆×C₁₉	—	4	2	no	—
57	D₈×C₁₉	—	4	2	no	—
58	D₈×C₁₉	—	8	1	no	—
59	M₁₆×C₁₉	—	4	1	no	—
60	M₁₆×C₁₉	—	4	1	no	—
61	M₁₆×C₁₉	—	4	1	no	—
62	C₂×C₈×C₁₉	—	2	1	no	—
63	C₂×C₈×C₁₉	—	4	1	no	—
64	C₂×C₈×C₁₉	—	4	1	no	—
65	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
66	C₄⋊C₄×C₁₉	—	2	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	C₂²⋊C₄×C₁₉	—	4	2	no	—
72	C₄²×C₁₉	—	2	1	no	—
73	C₁₆×C₁₉	—	4	2	no	—
74	#12167	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×C₁₉	—	4	2	no	—
76	C₂³×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
77	Q₈×C₁₉	—	4	2	no	—
78	Q₈×C₁₉	—	4	2	no	—
79	Q₈×C₁₉	—	8	1	no	—
80	Q₈×C₁₉	—	8	1	no	—
81	D₄×C₁₉	—	4	2	no	—
82	D₄×C₁₉	—	4	2	no	—
83	D₄×C₁₉	—	16	2	no	—
84	D₄×C₁₉	—	16	2	no	—
85	C₂×C₄×C₁₉	—	2	1	no	—
86	C₂×C₄×C₁₉	—	2	1	no	—
87	C₂×C₄×C₁₉	—	2	2	no	—
88	C₂×C₄×C₁₉	—	4	2	no	—
89	C₂×C₄×C₁₉	—	4	2	no	—
90	C₂×C₄×C₁₉	—	8	1	no	—
91	C₂×C₄×C₁₉	—	8	1	no	—
92	C₂×C₄×C₁₉	—	16	2	no	—
93	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
94	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
95	C₈×C₁₉	—	2	2	no	—
96	C₈×C₁₉	—	4	1	no	—
97	C₈×C₁₉	—	8	1	no	—
98	#3259	C₂×C₁₉	1	1	yes	characteristic
99	#2185	C₂×C₁₉	1	2	yes	complemented
100	#2047	C₂×C₁₉	1	1	yes	characteristic, complemented
101	#1349	C₂×C₁₉	1	2	yes	complemented
102	#1323	C₂×C₁₉	1	1	yes	characteristic, complemented
103	C₂²×C₁₉	—	2	1	no	—
104	C₂²×C₁₉	—	2	2	no	—
105	C₂²×C₁₉	—	8	2	no	—
106	C₂²×C₁₉	—	8	2	no	—
107	C₂²×C₁₉	—	8	2	no	—
108	C₂²×C₁₉	#447	1	1	yes	characteristic
109	C₄×C₁₉	—	2	1	no	—
110	C₄×C₁₉	—	4	2	no	—
111	C₄×C₁₉	—	8	2	no	—
112	C₄×C₁₉	—	8	2	no	—
113	C₄×C₁₉	#454	1	1	yes	characteristic
114	C₄×C₁₉	B(4,𝔽₂)	1	1	yes	characteristic
115	C₂²×Q₁₆	C₂²×C₁₉	1	1	yes	characteristic
116	#540	—	2	1	no	—
117	C₂×SD₃₂	—	2	2	no	—
118	#478	—	2	2	no	—
119	#469	—	2	1	no	—
120	#466	C₂²×C₁₉	1	2	yes	—
121	#459	—	2	2	no	—
122	#424	C₂²×C₁₉	1	1	yes	characteristic
123	#419	—	2	1	no	—
124	C₂×#101	—	2	1	no	—
125	#358	—	2	2	no	—
126	C₂²⋊C₁₆	C₂²×C₁₉	1	2	yes	—
127	#341	C₂²×C₁₉	1	1	yes	characteristic
128	C₂×C₁₉	—	4	1	no	—
129	C₂×C₁₉	—	16	2	no	—
130	C₂×C₁₉	#1598	1	1	yes	characteristic
131	C₂×C₁₉	#2194	1	2	yes	—
132	C₂²×Q₈	—	2	1	no	—
133	C₂×Q₁₆	—	2	2	no	—
134	C₂×Q₁₆	—	4	2	no	—
135	C₂×D₈	—	2	1	no	—
136	C₂×D₈	D₄×C₁₉	1	2	yes	—
137	C₂×M₁₆	—	2	1	no	—
138	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
139	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
140	#127	—	2	1	no	—
141	#125	—	2	1	no	—
142	#123	—	4	2	no	—
143	#121	—	2	2	no	—
144	#121	—	4	2	no	—
145	C₄×D₄	—	2	2	no	—
146	#117	D₄×C₁₉	1	1	yes	characteristic
147	SD₃₂	—	4	4	no	—
148	C₂×C₁₆	—	2	2	no	—
149	#107	D₄×C₁₉	1	2	yes	—
150	#106	—	2	1	no	—
151	C₄⋊C₈	—	2	2	no	—
152	#103	—	4	2	no	—
153	#102	—	2	2	no	—
154	#102	—	2	2	no	—
155	#101	—	4	2	no	—
156	C₁₉	#12167	1	1	yes	characteristic, direct product factor, Sylow
157	C₂×Q₈	—	2	2	no	—
158	C₂×Q₈	—	4	1	no	—
159	C₂×Q₈	—	4	1	no	—
160	C₂×Q₈	—	4	1	no	—
161	C₂×Q₈	—	4	1	no	—
162	C₂×Q₈	—	4	1	no	—
163	C₂×D₄	—	2	2	no	—
164	C₂×D₄	—	4	2	no	—
165	C₂×D₄	—	4	2	no	—
166	C₂²×C₄	—	4	2	no	—
167	C₂²×C₄	—	4	1	no	—
168	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
169	Q₁₆	—	2	4	no	—
170	Q₁₆	—	4	2	no	—
171	D₈	—	4	2	no	—
172	D₈	—	8	1	no	—
173	M₁₆	—	4	1	no	—
174	M₁₆	—	4	1	no	—
175	M₁₆	—	4	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	4	1	no	—
178	C₂×C₈	—	4	1	no	—
179	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	D₈×C₁₉	1	2	yes	—
183	C₂²⋊C₄	—	2	1	no	—
184	C₂²⋊C₄	—	4	2	no	—
185	C₂²⋊C₄	—	4	2	no	—
186	C₄²	—	2	1	no	—
187	C₁₆	—	4	2	no	—
188	C₂³	—	4	2	no	—
189	C₂³	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
190	Q₈	—	4	2	no	—
191	Q₈	—	4	2	no	—
192	Q₈	—	8	1	no	—
193	Q₈	—	8	1	no	—
194	D₄	—	4	2	no	—
195	D₄	—	4	2	no	—
196	D₄	—	16	2	no	—
197	D₄	—	16	2	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	—	8	1	no	—
204	C₂×C₄	—	8	1	no	—
205	C₂×C₄	—	16	2	no	—
206	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
207	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
208	C₈	—	2	2	no	—
209	C₈	—	4	1	no	—
210	C₈	—	8	1	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	1	no	—
213	C₂²	—	8	2	no	—
214	C₂²	—	8	2	no	—
215	C₂²	—	8	2	no	—
216	C₂²	C₁₉×#447	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	4	2	no	—
219	C₄	—	8	2	no	—
220	C₄	—	8	2	no	—
221	C₄	C₁₉×#454	1	1	yes	characteristic
222	C₄	C₁₉×B(4,𝔽₂)	1	1	yes	characteristic
223	C₂	—	4	1	no	—
224	C₂	—	16	2	no	—
225	C₂	C₁₉×#1598	1	1	yes	characteristic
226	C₂	C₁₉×#2194	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#3259

C₂

yes

maximal, characteristic

C₁₉×#2185

C₂

yes

maximal, complemented

C₁₉×#2047

C₂

yes

maximal, characteristic, complemented

C₁₉×#1349

C₂

yes

maximal, complemented

C₁₉×#1323

C₂

yes

maximal, characteristic, complemented

C₂²×Q₁₆×C₁₉

C₂²

yes

characteristic

C₁₉×#540

—

C₂×C₁₉×SD₃₂

—

C₁₉×#478

—

C₁₉×#469

—

C₁₉×#466

C₂²

yes

—

C₁₉×#459

—

C₁₉×#424

C₂²

yes

characteristic

C₁₉×#419

—

C₂×C₁₉×#101

—

C₁₉×#358

—

C₁₉×C₂²⋊C₁₆

C₂²

yes

—

C₁₉×#341

C₂²

yes

characteristic

C₂²×Q₈×C₁₉

—

C₂×Q₁₆×C₁₉

—

C₂×Q₁₆×C₁₉

—

C₂×D₈×C₁₉

—

C₂×D₈×C₁₉

D₄

yes

—

C₂×M₁₆×C₁₉

—

C₂×M₁₆×C₁₉

D₄

yes

characteristic

C₂²×C₈×C₁₉

C₂³

yes

characteristic

C₁₉×#127

—

C₁₉×#125

—

C₁₉×#123

—

C₁₉×#121

—

C₁₉×#121

—

C₄×D₄×C₁₉

—

C₁₉×#117

D₄

yes

characteristic

C₁₉×SD₃₂

—

C₂×C₁₆×C₁₉

—

C₁₉×#107

D₄

yes

—

C₁₉×#106

—

C₁₉×C₄⋊C₈

—

C₁₉×#103

—

C₁₉×#102

—

C₁₉×#102

—

C₁₉×#101

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×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

D₈

yes

—

C₂²⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₁₆×C₁₉

—

#12167

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

AGL(1,ℤ/8)

yes

characteristic

Q₈×C₁₉

—

Q₈×C₁₉

—

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

—

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

—

#3259

C₂×C₁₉

yes

characteristic

#2185

C₂×C₁₉

yes

complemented

100

#2047

C₂×C₁₉

yes

characteristic, complemented

101

#1349

C₂×C₁₉

yes

complemented

102

#1323

C₂×C₁₉

yes

characteristic, complemented

103

C₂²×C₁₉

—

104

C₂²×C₁₉

—

105

C₂²×C₁₉

—

106

C₂²×C₁₉

—

107

C₂²×C₁₉

—

108

C₂²×C₁₉

#447

yes

characteristic

109

C₄×C₁₉

—

110

C₄×C₁₉

—

111

C₄×C₁₉

—

112

C₄×C₁₉

—

113

C₄×C₁₉

#454

yes

characteristic

114

C₄×C₁₉

B(4,𝔽₂)

yes

characteristic

115

C₂²×Q₁₆

C₂²×C₁₉

yes

characteristic

116

#540

—

117

C₂×SD₃₂

—

118

#478

—

119

#469

—

120

#466

C₂²×C₁₉

yes

—

121

#459

—

122

#424

C₂²×C₁₉

yes

characteristic

123

#419

—

124

C₂×#101

—

125

#358

—

126

C₂²⋊C₁₆

C₂²×C₁₉

yes

—

127

#341

C₂²×C₁₉

yes

characteristic

128

C₂×C₁₉

—

129

C₂×C₁₉

—

130

C₂×C₁₉

#1598

yes

characteristic

131

C₂×C₁₉

#2194

yes

—

132

C₂²×Q₈

—

133

C₂×Q₁₆

—

134

C₂×Q₁₆

—

135

C₂×D₈

—

136

C₂×D₈

D₄×C₁₉

yes

—

137

C₂×M₁₆

—

138

C₂×M₁₆

D₄×C₁₉

yes

characteristic

139

C₂²×C₈

C₂³×C₁₉

yes

characteristic

140

#127

—

141

#125

—

142

#123

—

143

#121

—

144

#121

—

145

C₄×D₄

—

146

#117

D₄×C₁₉

yes

characteristic

147

SD₃₂

—

148

C₂×C₁₆

—

149

#107

D₄×C₁₉

yes

—

150

#106

—

151

C₄⋊C₈

—

152

#103

—

153

#102

—

154

#102

—

155

#101

—

156

C₁₉

#12167

yes

characteristic, direct product factor, Sylow

157

C₂×Q₈

—

158

C₂×Q₈

—

159

C₂×Q₈

—

160

C₂×Q₈

—

161

C₂×Q₈

—

162

C₂×Q₈

—

163

C₂×D₄

—

164

C₂×D₄

—

165

C₂×D₄

—

166

C₂²×C₄

—

167

C₂²×C₄

—

168

C₂²×C₄

C₂×D₄×C₁₉

yes

characteristic

169

Q₁₆

—

170

Q₁₆

—

171

D₈

—

172

D₈

—

173

M₁₆

—

174

M₁₆

—

175

M₁₆

—

176

C₂×C₈

—

177

C₂×C₈

—

178

C₂×C₈

—

179

C₂×C₈

C₂×D₄×C₁₉

yes

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₄⋊C₄

D₈×C₁₉

yes

—

183

C₂²⋊C₄

—

184

C₂²⋊C₄

—

185

C₂²⋊C₄

—

186

C₄²

—

187

C₁₆

—

188

C₂³

—

189

C₂³

C₁₉×AGL(1,ℤ/8)

yes

characteristic

190

Q₈

—

191

Q₈

—

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

—

204

C₂×C₄

—

205

C₂×C₄

—

206

C₂×C₄

C₁₉×#120

yes

characteristic

207

C₂×C₄

C₂×D₈×C₁₉

yes

characteristic

208

C₈

—

209

C₈

—

210

C₈

—

211

C₂²

—

212

C₂²

—

213

C₂²

—

214

C₂²

—

215

C₂²

—

216

C₂²

C₁₉×#447

yes

characteristic

217

C₄

—

218

C₄

—

219

C₄

—

220

C₄

—

221

C₄

C₁₉×#454

yes

characteristic

222

C₄

C₁₉×B(4,𝔽₂)

yes

characteristic

223

C₂

—

224

C₂

—

225

C₂

C₁₉×#1598

yes

characteristic

226

C₂

C₁₉×#2194

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₉	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₁₉×#3259	C₂
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

C₁₉×#3259

C₂

LCS

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

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

N	H ≅ G/N	type
2. C₁₉×#2185	C₂	semidirect
3. C₁₉×#2047	C₂	semidirect
4. C₁₉×#1349	C₂	semidirect
5. C₁₉×#1323	C₂	semidirect
74. #12167	C₁₉	direct (N × H)
99. #2185	C₂×C₁₉	semidirect
100. #2047	C₂×C₁₉	semidirect
101. #1349	C₂×C₁₉	semidirect
102. #1323	C₂×C₁₉	semidirect
156. C₁₉	#12167	direct (N × H)

H ≅ G/N

type

2. C₁₉×#2185

C₂

semidirect

3. C₁₉×#2047

C₂

semidirect

4. C₁₉×#1349

C₂

semidirect

5. C₁₉×#1323

C₂

semidirect

74. #12167

C₁₉

direct (N × H)

99. #2185

C₂×C₁₉

semidirect

100. #2047

C₂×C₁₉

semidirect

101. #1349

C₂×C₁₉

semidirect

102. #1323

C₂×C₁₉

semidirect

156. C₁₉

#12167

direct (N × H)

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

subgroup	index
C₁₉×#3259	2
C₁₉×#2185	2
C₁₉×#2047	2
C₁₉×#1349	2
C₁₉×#1323	2
#12167	19

subgroup

index

C₁₉×#3259

C₁₉×#2185

C₁₉×#2047

C₁₉×#1349

C₁₉×#1323

#12167

Subgroups — C₁₉×#12167 #18018126

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#12167 #18018126

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12167 #18018126