Group #18018263 — subgroups

Subgroups — C₁₉×#12304 #18018263

Order 4864.

286 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 70 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×#2238	C₂	1	2	yes	maximal, complemented
3	C₁₉×#2103	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1341	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1328	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#540	—	2	1	no	—
7	C₁₉×#478	—	2	2	no	—
8	C₁₉×#469	C₂²	1	2	yes	—
9	C₁₉×#466	—	2	1	no	—
10	C₁₉×#459	—	2	2	no	—
11	C₁₉×#430	C₂²	1	1	yes	characteristic
12	C₁₉×#424	C₂²	1	1	yes	characteristic
13	C₁₉×#365	—	2	2	no	—
14	C₁₉×#361	—	2	2	no	—
15	C₁₉×#349	C₂²	1	2	yes	—
16	C₁₉×#344	—	2	1	no	—
17	C₁₉×#341	—	2	1	no	—
18	C₁₉×#338	C₂²	1	1	yes	characteristic
19	C₂×D₈×C₁₉	—	2	1	no	—
20	C₂×D₈×C₁₉	D₄	1	2	yes	—
21	C₂×M₁₆×C₁₉	—	2	1	no	—
22	C₂×M₁₆×C₁₉	—	2	1	no	—
23	C₂×M₁₆×C₁₉	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
25	C₁₉×#127	—	2	1	no	—
26	C₁₉×#125	—	2	1	no	—
27	C₁₉×#123	—	4	2	no	—
28	C₁₉×#121	—	2	2	no	—
29	C₁₉×#121	—	4	2	no	—
30	C₄×D₄×C₁₉	—	2	2	no	—
31	C₁₉×#117	D₄	1	1	yes	characteristic
32	C₁₉×M₃₂	—	2	2	no	—
33	C₁₉×#108	—	2	2	no	—
34	C₁₉×#107	—	2	1	no	—
35	C₁₉×#106	D₄	1	2	yes	—
36	C₁₉×C₄⋊C₈	—	2	2	no	—
37	C₁₉×#102	—	2	2	no	—
38	C₁₉×#102	—	2	2	no	—
39	C₂×D₄×C₁₉	—	2	2	no	—
40	C₂×D₄×C₁₉	—	4	2	no	—
41	C₂×D₄×C₁₉	—	4	2	no	—
42	C₂²×C₄×C₁₉	—	4	2	no	—
43	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
44	D₈×C₁₉	—	4	2	no	—
45	D₈×C₁₉	—	8	1	no	—
46	M₁₆×C₁₉	—	4	1	no	—
47	M₁₆×C₁₉	—	4	1	no	—
48	M₁₆×C₁₉	—	4	1	no	—
49	M₁₆×C₁₉	—	4	1	no	—
50	M₁₆×C₁₉	—	4	1	no	—
51	C₂×C₈×C₁₉	—	2	1	no	—
52	C₂×C₈×C₁₉	—	4	1	no	—
53	C₂×C₈×C₁₉	—	4	1	no	—
54	C₂×C₈×C₁₉	—	4	1	no	—
55	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
56	C₄⋊C₄×C₁₉	—	2	1	no	—
57	C₄⋊C₄×C₁₉	—	2	2	no	—
58	C₄⋊C₄×C₁₉	—	4	2	no	—
59	C₂²⋊C₄×C₁₉	—	4	2	no	—
60	C₂²⋊C₄×C₁₉	—	4	2	no	—
61	C₂²⋊C₄×C₁₉	SD₁₆	1	2	yes	—
62	C₄²×C₁₉	—	2	1	no	—
63	C₁₆×C₁₉	—	4	2	no	—
64	#12304	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂³×C₁₉	—	4	2	no	—
66	C₂³×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
67	D₄×C₁₉	—	4	2	no	—
68	D₄×C₁₉	—	4	2	no	—
69	D₄×C₁₉	—	16	2	no	—
70	D₄×C₁₉	—	16	2	no	—
71	C₂×C₄×C₁₉	—	2	1	no	—
72	C₂×C₄×C₁₉	—	2	1	no	—
73	C₂×C₄×C₁₉	—	2	2	no	—
74	C₂×C₄×C₁₉	—	4	2	no	—
75	C₂×C₄×C₁₉	—	16	2	no	—
76	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
77	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₈×C₁₉	—	2	2	no	—
79	C₈×C₁₉	—	4	1	no	—
80	C₈×C₁₉	—	8	1	no	—
81	C₈×C₁₉	—	8	1	no	—
82	#3259	C₂×C₁₉	1	1	yes	characteristic
83	#2238	C₂×C₁₉	1	2	yes	complemented
84	#2103	C₂×C₁₉	1	1	yes	characteristic, complemented
85	#1341	C₂×C₁₉	1	2	yes	complemented
86	#1328	C₂×C₁₉	1	1	yes	characteristic, complemented
87	C₂²×C₁₉	—	2	1	no	—
88	C₂²×C₁₉	—	4	1	no	—
89	C₂²×C₁₉	—	8	2	no	—
90	C₂²×C₁₉	—	8	2	no	—
91	C₂²×C₁₉	—	8	2	no	—
92	C₂²×C₁₉	#450	1	1	yes	characteristic
93	C₄×C₁₉	—	2	1	no	—
94	C₄×C₁₉	—	4	2	no	—
95	C₄×C₁₉	—	8	2	no	—
96	C₄×C₁₉	#454	1	1	yes	characteristic
97	C₄×C₁₉	#458	1	1	yes	characteristic
98	#540	—	2	1	no	—
99	#478	—	2	2	no	—
100	#469	C₂²×C₁₉	1	2	yes	—
101	#466	—	2	1	no	—
102	#459	—	2	2	no	—
103	#430	C₂²×C₁₉	1	1	yes	characteristic
104	#424	C₂²×C₁₉	1	1	yes	characteristic
105	#365	—	2	2	no	—
106	#361	—	2	2	no	—
107	#349	C₂²×C₁₉	1	2	yes	—
108	#344	—	2	1	no	—
109	#341	—	2	1	no	—
110	#338	C₂²×C₁₉	1	1	yes	characteristic
111	C₂×C₁₉	—	2	1	no	—
112	C₂×C₁₉	—	4	1	no	—
113	C₂×C₁₉	—	16	2	no	—
114	C₂×C₁₉	#1616	1	1	yes	characteristic
115	C₂×D₈	—	2	1	no	—
116	C₂×D₈	D₄×C₁₉	1	2	yes	—
117	C₂×M₁₆	—	2	1	no	—
118	C₂×M₁₆	—	2	1	no	—
119	C₂×M₁₆	C₂³×C₁₉	1	1	yes	characteristic
120	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
121	#127	—	2	1	no	—
122	#125	—	2	1	no	—
123	#123	—	4	2	no	—
124	#121	—	2	2	no	—
125	#121	—	4	2	no	—
126	C₄×D₄	—	2	2	no	—
127	#117	D₄×C₁₉	1	1	yes	characteristic
128	M₃₂	—	2	2	no	—
129	#108	—	2	2	no	—
130	#107	—	2	1	no	—
131	#106	D₄×C₁₉	1	2	yes	—
132	C₄⋊C₈	—	2	2	no	—
133	#102	—	2	2	no	—
134	#102	—	2	2	no	—
135	C₁₉	#12304	1	1	yes	characteristic, direct product factor, Sylow
136	C₂×D₄	—	2	2	no	—
137	C₂×D₄	—	4	2	no	—
138	C₂×D₄	—	4	2	no	—
139	C₂²×C₄	—	4	2	no	—
140	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
141	D₈	—	4	2	no	—
142	D₈	—	8	1	no	—
143	M₁₆	—	4	1	no	—
144	M₁₆	—	4	1	no	—
145	M₁₆	—	4	1	no	—
146	M₁₆	—	4	1	no	—
147	M₁₆	—	4	1	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	4	1	no	—
150	C₂×C₈	—	4	1	no	—
151	C₂×C₈	—	4	1	no	—
152	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
153	C₄⋊C₄	—	2	1	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₂²⋊C₄	—	4	2	no	—
157	C₂²⋊C₄	—	4	2	no	—
158	C₂²⋊C₄	SD₁₆×C₁₉	1	2	yes	—
159	C₄²	—	2	1	no	—
160	C₁₆	—	4	2	no	—
161	C₂³	—	4	2	no	—
162	C₂³	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
163	D₄	—	4	2	no	—
164	D₄	—	4	2	no	—
165	D₄	—	16	2	no	—
166	D₄	—	16	2	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	16	2	no	—
172	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
173	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
174	C₈	—	2	2	no	—
175	C₈	—	4	1	no	—
176	C₈	—	8	1	no	—
177	C₈	—	8	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	—	4	1	no	—
180	C₂²	—	8	2	no	—
181	C₂²	—	8	2	no	—
182	C₂²	—	8	2	no	—
183	C₂²	C₁₉×#450	1	1	yes	characteristic
184	C₄	—	2	1	no	—
185	C₄	—	4	2	no	—
186	C₄	—	8	2	no	—
187	C₄	C₁₉×#454	1	1	yes	characteristic
188	C₄	C₁₉×#458	1	1	yes	characteristic
189	C₂	—	2	1	no	—
190	C₂	—	4	1	no	—
191	C₂	—	16	2	no	—
192	C₂	C₁₉×#1616	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#3259

C₂

yes

maximal, characteristic

C₁₉×#2238

C₂

yes

maximal, complemented

C₁₉×#2103

C₂

yes

maximal, characteristic, complemented

C₁₉×#1341

C₂

yes

maximal, complemented

C₁₉×#1328

C₂

yes

maximal, characteristic, complemented

C₁₉×#540

—

C₁₉×#478

—

C₁₉×#469

C₂²

yes

—

C₁₉×#466

—

C₁₉×#459

—

C₁₉×#430

C₂²

yes

characteristic

C₁₉×#424

C₂²

yes

characteristic

C₁₉×#365

—

C₁₉×#361

—

C₁₉×#349

C₂²

yes

—

C₁₉×#344

—

C₁₉×#341

—

C₁₉×#338

C₂²

yes

characteristic

C₂×D₈×C₁₉

—

C₂×D₈×C₁₉

D₄

yes

—

C₂×M₁₆×C₁₉

—

C₂×M₁₆×C₁₉

—

C₂×M₁₆×C₁₉

C₂³

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₁₉×#127

—

C₁₉×#125

—

C₁₉×#123

—

C₁₉×#121

—

C₁₉×#121

—

C₄×D₄×C₁₉

—

C₁₉×#117

D₄

yes

characteristic

C₁₉×M₃₂

—

C₁₉×#108

—

C₁₉×#107

—

C₁₉×#106

D₄

yes

—

C₁₉×C₄⋊C₈

—

C₁₉×#102

—

C₁₉×#102

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

D₈×C₁₉

—

D₈×C₁₉

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×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

—

C₂²⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

SD₁₆

yes

—

C₄²×C₁₉

—

C₁₆×C₁₉

—

#12304

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

C₂×SD₁₆

yes

characteristic

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

yes

C₂×C₄×C₁₉

yes

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

#3259

C₂×C₁₉

yes

characteristic

#2238

C₂×C₁₉

yes

complemented

#2103

C₂×C₁₉

yes

characteristic, complemented

#1341

C₂×C₁₉

yes

complemented

#1328

C₂×C₁₉

yes

characteristic, complemented

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#450

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

—

—

100

#469

C₂²×C₁₉

yes

—

101

#466

—

102

#459

—

103

#430

C₂²×C₁₉

yes

characteristic

104

#424

C₂²×C₁₉

yes

characteristic

105

#365

—

106

#361

—

107

#349

C₂²×C₁₉

yes

—

108

#344

—

109

#341

—

110

#338

C₂²×C₁₉

yes

characteristic

111

C₂×C₁₉

—

112

C₂×C₁₉

—

113

C₂×C₁₉

—

114

C₂×C₁₉

#1616

yes

characteristic

115

C₂×D₈

—

116

C₂×D₈

D₄×C₁₉

yes

—

117

C₂×M₁₆

—

118

C₂×M₁₆

—

119

C₂×M₁₆

C₂³×C₁₉

yes

characteristic

120

C₂²×C₈

D₄×C₁₉

yes

characteristic

121

#127

—

122

#125

—

123

#123

—

124

#121

—

125

#121

—

126

C₄×D₄

—

127

#117

D₄×C₁₉

yes

characteristic

128

M₃₂

—

129

#108

—

130

#107

—

131

#106

D₄×C₁₉

yes

—

132

C₄⋊C₈

—

133

#102

—

134

#102

—

135

C₁₉

#12304

yes

characteristic, direct product factor, Sylow

136

C₂×D₄

—

137

C₂×D₄

—

138

C₂×D₄

—

139

C₂²×C₄

—

140

C₂²×C₄

C₂×D₄×C₁₉

yes

characteristic

141

D₈

—

142

D₈

—

143

M₁₆

—

144

M₁₆

—

145

M₁₆

—

146

M₁₆

—

147

M₁₆

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

C₂×D₄×C₁₉

yes

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₂²⋊C₄

—

157

C₂²⋊C₄

—

158

C₂²⋊C₄

SD₁₆×C₁₉

yes

—

159

C₄²

—

160

C₁₆

—

161

C₂³

—

162

C₂³

C₂×SD₁₆×C₁₉

yes

characteristic

163

D₄

—

164

D₄

—

165

D₄

—

166

D₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

C₁₉×#120

yes

characteristic

173

C₂×C₄

C₁₉×AGL(1,ℤ/8)

yes

characteristic

174

C₈

—

175

C₈

—

176

C₈

—

177

C₈

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

—

182

C₂²

—

183

C₂²

C₁₉×#450

yes

characteristic

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

C₁₉×#454

yes

characteristic

188

C₄

C₁₉×#458

yes

characteristic

189

C₂

—

190

C₂

—

191

C₂

—

192

C₂

C₁₉×#1616

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₁₉	#1616
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₁₉
prime radical	C₁₉×#3259	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂×M₁₆×C₁₉ ◁ G

subgroup

G/H

center, socle

C₂×C₁₉

#1616

commutator, Frattini, cosocle

C₂×M₁₆

C₂³×C₁₉

prime radical

C₁₉×#3259

C₂

LCS

G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂×M₁₆×C₁₉ ◁ G

N	H ≅ G/N	type
2. C₁₉×#2238	C₂	semidirect
3. C₁₉×#2103	C₂	semidirect
4. C₁₉×#1341	C₂	semidirect
5. C₁₉×#1328	C₂	semidirect
64. #12304	C₁₉	direct (N × H)
83. #2238	C₂×C₁₉	semidirect
84. #2103	C₂×C₁₉	semidirect
85. #1341	C₂×C₁₉	semidirect
86. #1328	C₂×C₁₉	semidirect
135. C₁₉	#12304	direct (N × H)

H ≅ G/N

type

2. C₁₉×#2238

C₂

semidirect

3. C₁₉×#2103

C₂

semidirect

4. C₁₉×#1341

C₂

semidirect

5. C₁₉×#1328

C₂

semidirect

64. #12304

C₁₉

direct (N × H)

83. #2238

C₂×C₁₉

semidirect

84. #2103

C₂×C₁₉

semidirect

85. #1341

C₂×C₁₉

semidirect

86. #1328

C₂×C₁₉

semidirect

135. C₁₉

#12304

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×#2238	2
C₁₉×#2103	2
C₁₉×#1341	2
C₁₉×#1328	2
#12304	19

subgroup

index

C₁₉×#3259

C₁₉×#2238

C₁₉×#2103

C₁₉×#1341

C₁₉×#1328

#12304

Subgroups — C₁₉×#12304 #18018263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#12304 #18018263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12304 #18018263