Group #18019557 — subgroups

Subgroups — C₁₉×#13598 #18019557

Order 4864.

318 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 90 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×#3135	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2132	C₂	1	2	yes	maximal
3	C₁₉×#1413	C₂	1	4	yes	maximal, complemented
4	C₁₉×#533	C₄	1	1	yes	characteristic
5	C₁₉×#532	C₂²	1	1	yes	characteristic
6	C₁₉×#491	C₂²	1	2	yes	—
7	C₁₉×#490	C₄	1	2	yes	—
8	C₁₉×#489	—	2	2	no	—
9	C₁₉×#416	—	2	2	no	—
10	C₁₉×#413	—	2	2	no	—
11	C₂×C₁₉×#97	C₄	1	1	yes	characteristic
12	C₁₉×#356	C₂²	1	4	yes	—
13	C₁₉×#355	—	2	2	no	—
14	C₂×#41×C₁₉	D₄	1	2	yes	—
15	C₂×M₁₆×C₁₉	—	4	2	no	—
16	C₂²×C₈×C₁₉	—	2	1	no	—
17	C₁₉×C₄⋊Q₈	—	2	1	no	—
18	C₁₉×C₄⋊Q₈	C₂³	1	1	yes	characteristic
19	C₁₉×#127	C₂×C₄	1	1	yes	characteristic
20	C₁₉×#125	C₂×C₄	1	2	yes	—
21	C₁₉×#124	D₄	1	2	yes	—
22	C₁₉×C₂²⋊Q₈	—	4	1	no	—
23	C₁₉×#121	—	4	1	no	—
24	C₄×Q₈×C₁₉	—	4	1	no	—
25	C₄×D₄×C₁₉	—	4	1	no	—
26	C₁₉×#117	—	2	1	no	—
27	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₁₉×#103	—	2	4	no	—
29	C₁₉×#102	—	2	4	no	—
30	C₁₉×#101	—	2	4	no	—
31	C₁₉×#100	—	4	2	no	—
32	C₁₉×#97	—	2	1	no	—
33	C₁₉×#97	C₂×C₄	1	2	yes	—
34	#41×C₁₉	—	4	2	no	—
35	#41×C₁₉	—	4	2	no	—
36	C₂×Q₈×C₁₉	—	4	1	no	—
37	C₂×Q₈×C₁₉	C₂×D₄	1	2	yes	—
38	C₂×D₄×C₁₉	—	2	2	no	—
39	C₂×D₄×C₁₉	—	4	1	no	—
40	C₂×D₄×C₁₉	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₁₉	—	2	1	no	—
42	C₂²×C₄×C₁₉	—	2	2	no	—
43	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
44	M₁₆×C₁₉	—	4	4	no	—
45	M₁₆×C₁₉	—	8	2	no	—
46	C₂×C₈×C₁₉	—	2	2	no	—
47	C₂×C₈×C₁₉	—	4	1	no	—
48	C₂×C₈×C₁₉	—	4	1	no	—
49	C₂×C₈×C₁₉	—	8	2	no	—
50	C₄⋊C₄×C₁₉	—	2	2	no	—
51	C₄⋊C₄×C₁₉	—	4	1	no	—
52	C₄⋊C₄×C₁₉	—	4	2	no	—
53	C₄⋊C₄×C₁₉	—	4	1	no	—
54	C₂²⋊C₄×C₁₉	—	4	2	no	—
55	C₂²⋊C₄×C₁₉	—	4	1	no	—
56	C₄²×C₁₉	—	2	1	no	—
57	C₄²×C₁₉	—	4	1	no	—
58	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
60	#13598	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂³×C₁₉	—	2	2	no	—
62	C₂³×C₁₉	#99	1	1	yes	characteristic
63	Q₈×C₁₉	—	4	2	no	—
64	Q₈×C₁₉	—	8	2	no	—
65	D₄×C₁₉	—	4	2	no	—
66	D₄×C₁₉	—	8	2	no	—
67	D₄×C₁₉	—	8	2	no	—
68	C₂×C₄×C₁₉	—	2	1	no	—
69	C₂×C₄×C₁₉	—	2	1	no	—
70	C₂×C₄×C₁₉	—	2	1	no	—
71	C₂×C₄×C₁₉	—	2	1	no	—
72	C₂×C₄×C₁₉	—	2	2	no	—
73	C₂×C₄×C₁₉	—	4	2	no	—
74	C₂×C₄×C₁₉	—	4	2	no	—
75	C₂×C₄×C₁₉	—	4	2	no	—
76	C₂×C₄×C₁₉	—	8	1	no	—
77	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
78	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₈×C₁₉	—	4	2	no	—
80	C₈×C₁₉	—	8	4	no	—
81	#3135	C₂×C₁₉	1	1	yes	characteristic
82	#2132	C₂×C₁₉	1	2	yes	—
83	#1413	C₂×C₁₉	1	4	yes	complemented
84	C₂²×C₁₉	—	2	1	no	—
85	C₂²×C₁₉	—	4	1	no	—
86	C₂²×C₁₉	—	4	2	no	—
87	C₂²×C₁₉	—	8	2	no	—
88	C₂²×C₁₉	C₂×#99	1	1	yes	characteristic
89	C₄×C₁₉	—	2	1	no	—
90	C₄×C₁₉	—	4	1	no	—
91	C₄×C₁₉	—	4	1	no	—
92	C₄×C₁₉	—	4	2	no	—
93	C₄×C₁₉	—	8	2	no	—
94	C₄×C₁₉	#354	1	1	yes	characteristic
95	C₄×C₁₉	#354	1	1	yes	characteristic
96	#533	C₄×C₁₉	1	1	yes	characteristic
97	#532	C₂²×C₁₉	1	1	yes	characteristic
98	#491	C₂²×C₁₉	1	2	yes	—
99	#490	C₄×C₁₉	1	2	yes	—
100	#489	—	2	2	no	—
101	#416	—	2	2	no	—
102	#413	—	2	2	no	—
103	C₂×#97	C₄×C₁₉	1	1	yes	characteristic
104	#356	C₂²×C₁₉	1	4	yes	—
105	#355	—	2	2	no	—
106	C₂×C₁₉	—	2	1	no	—
107	C₂×C₁₉	—	4	1	no	—
108	C₂×C₁₉	—	8	2	no	—
109	C₂×C₁₉	C₂×#354	1	1	yes	characteristic
110	C₂×#41	D₄×C₁₉	1	2	yes	—
111	C₂×M₁₆	—	4	2	no	—
112	C₂²×C₈	—	2	1	no	—
113	C₄⋊Q₈	—	2	1	no	—
114	C₄⋊Q₈	C₂³×C₁₉	1	1	yes	characteristic
115	#127	C₂×C₄×C₁₉	1	1	yes	characteristic
116	#125	C₂×C₄×C₁₉	1	2	yes	—
117	#124	D₄×C₁₉	1	2	yes	—
118	C₂²⋊Q₈	—	4	1	no	—
119	#121	—	4	1	no	—
120	C₄×Q₈	—	4	1	no	—
121	C₄×D₄	—	4	1	no	—
122	#117	—	2	1	no	—
123	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
124	#103	—	2	4	no	—
125	#102	—	2	4	no	—
126	#101	—	2	4	no	—
127	#100	—	4	2	no	—
128	#97	—	2	1	no	—
129	#97	C₂×C₄×C₁₉	1	2	yes	—
130	C₁₉	#13598	1	1	yes	characteristic, direct product factor, Sylow
131	#41	—	4	2	no	—
132	#41	—	4	2	no	—
133	C₂×Q₈	—	4	1	no	—
134	C₂×Q₈	C₂×D₄×C₁₉	1	2	yes	—
135	C₂×D₄	—	2	2	no	—
136	C₂×D₄	—	4	1	no	—
137	C₂×D₄	C₂²⋊C₄×C₁₉	1	2	yes	—
138	C₂²×C₄	—	2	1	no	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
141	M₁₆	—	4	4	no	—
142	M₁₆	—	8	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	4	1	no	—
145	C₂×C₈	—	4	1	no	—
146	C₂×C₈	—	8	2	no	—
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	4	1	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	4	1	no	—
151	C₂²⋊C₄	—	4	2	no	—
152	C₂²⋊C₄	—	4	1	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	4	1	no	—
155	C₄²	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
156	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
157	C₂³	—	2	2	no	—
158	C₂³	C₁₉×#99	1	1	yes	characteristic
159	Q₈	—	4	2	no	—
160	Q₈	—	8	2	no	—
161	D₄	—	4	2	no	—
162	D₄	—	8	2	no	—
163	D₄	—	8	2	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	8	1	no	—
173	C₂×C₄	C₁₉×#99	1	1	yes	characteristic
174	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
175	C₈	—	4	2	no	—
176	C₈	—	8	4	no	—
177	C₂²	—	2	1	no	—
178	C₂²	—	4	1	no	—
179	C₂²	—	4	2	no	—
180	C₂²	—	8	2	no	—
181	C₂²	C₂×C₁₉×#99	1	1	yes	characteristic
182	C₄	—	2	1	no	—
183	C₄	—	4	1	no	—
184	C₄	—	4	1	no	—
185	C₄	—	4	2	no	—
186	C₄	—	8	2	no	—
187	C₄	C₁₉×#354	1	1	yes	characteristic
188	C₄	C₁₉×#354	1	1	yes	characteristic
189	C₂	—	2	1	no	—
190	C₂	—	4	1	no	—
191	C₂	—	8	2	no	—
192	C₂	C₂×C₁₉×#354	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#3135

C₂

yes

maximal, characteristic

C₁₉×#2132

C₂

yes

maximal

C₁₉×#1413

C₂

yes

maximal, complemented

C₁₉×#533

C₄

yes

characteristic

C₁₉×#532

C₂²

yes

characteristic

C₁₉×#491

C₂²

yes

—

C₁₉×#490

C₄

yes

—

C₁₉×#489

—

C₁₉×#416

—

C₁₉×#413

—

C₂×C₁₉×#97

C₄

yes

characteristic

C₁₉×#356

C₂²

yes

—

C₁₉×#355

—

C₂×#41×C₁₉

D₄

yes

—

C₂×M₁₆×C₁₉

—

C₂²×C₈×C₁₉

—

C₁₉×C₄⋊Q₈

—

C₁₉×C₄⋊Q₈

C₂³

yes

characteristic

C₁₉×#127

C₂×C₄

yes

characteristic

C₁₉×#125

C₂×C₄

yes

—

C₁₉×#124

D₄

yes

—

C₁₉×C₂²⋊Q₈

—

C₁₉×#121

—

C₄×Q₈×C₁₉

—

C₄×D₄×C₁₉

—

C₁₉×#117

—

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₁₉×#103

—

C₁₉×#102

—

C₁₉×#101

—

C₁₉×#100

—

C₁₉×#97

—

C₁₉×#97

C₂×C₄

yes

—

#41×C₁₉

—

#41×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

C₂×D₄

yes

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

C₂²⋊C₄

yes

characteristic

C₄²×C₁₉

C₂²×C₄

yes

characteristic

#13598

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

#99

yes

characteristic

Q₈×C₁₉

—

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

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#99

yes

characteristic

C₂×C₄×C₁₉

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

—

#3135

C₂×C₁₉

yes

characteristic

#2132

C₂×C₁₉

yes

—

#1413

C₂×C₁₉

yes

complemented

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

C₂×#99

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

characteristic

#532

C₂²×C₁₉

yes

characteristic

#491

C₂²×C₁₉

yes

—

#490

C₄×C₁₉

yes

—

100

#489

—

101

#416

—

102

#413

—

103

C₂×#97

C₄×C₁₉

yes

characteristic

104

#356

C₂²×C₁₉

yes

—

105

#355

—

106

C₂×C₁₉

—

107

C₂×C₁₉

—

108

C₂×C₁₉

—

109

C₂×C₁₉

C₂×#354

yes

characteristic

110

C₂×#41

D₄×C₁₉

yes

—

111

C₂×M₁₆

—

112

C₂²×C₈

—

113

C₄⋊Q₈

—

114

C₄⋊Q₈

C₂³×C₁₉

yes

characteristic

115

#127

C₂×C₄×C₁₉

yes

characteristic

116

#125

C₂×C₄×C₁₉

yes

—

117

#124

D₄×C₁₉

yes

—

118

C₂²⋊Q₈

—

119

#121

—

120

C₄×Q₈

—

121

C₄×D₄

—

122

#117

—

123

C₂×C₄²

C₂×C₄×C₁₉

yes

characteristic

124

#103

—

125

#102

—

126

#101

—

127

#100

—

128

#97

—

129

#97

C₂×C₄×C₁₉

yes

—

130

C₁₉

#13598

yes

characteristic, direct product factor, Sylow

131

#41

—

132

#41

—

133

C₂×Q₈

—

134

C₂×Q₈

C₂×D₄×C₁₉

yes

—

135

C₂×D₄

—

136

C₂×D₄

—

137

C₂×D₄

C₂²⋊C₄×C₁₉

yes

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

C₂²⋊C₄×C₁₉

yes

characteristic

141

M₁₆

—

142

M₁₆

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₂²⋊C₄

—

152

C₂²⋊C₄

—

153

C₄²

—

154

C₄²

—

155

C₄²

C₂²⋊C₄×C₁₉

yes

characteristic

156

C₄²

C₂²×C₄×C₁₉

yes

characteristic

157

C₂³

—

158

C₂³

C₁₉×#99

yes

characteristic

159

Q₈

—

160

Q₈

—

161

D₄

—

162

D₄

—

163

D₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

C₁₉×#99

yes

characteristic

174

C₂×C₄

C₂×C₂²⋊C₄×C₁₉

yes

characteristic

175

C₈

—

176

C₈

—

177

C₂²

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

C₂×C₁₉×#99

yes

characteristic

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

C₁₉×#354

yes

characteristic

188

C₄

C₁₉×#354

yes

characteristic

189

C₂

—

190

C₂

—

191

C₂

—

192

C₂

C₂×C₁₉×#354

yes

characteristic

	subgroup	G/H
center	C₄×C₁₉	#354
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	C₄⋊Q₈	C₂³×C₁₉
socle	C₂×C₁₉	C₂×#354
prime radical	C₁₉×#532	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#532 ◁ G

subgroup

G/H

center

C₄×C₁₉

#354

commutator

C₄²

C₂²×C₄×C₁₉

Frattini, cosocle

C₄⋊Q₈

C₂³×C₁₉

socle

C₂×C₁₉

C₂×#354

prime radical

C₁₉×#532

C₂²

LCS

G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#532 ◁ G

N	H ≅ G/N	type
3. C₁₉×#1413	C₂	semidirect
60. #13598	C₁₉	direct (N × H)
83. #1413	C₂×C₁₉	semidirect
130. C₁₉	#13598	direct (N × H)

H ≅ G/N

type

3. C₁₉×#1413

C₂

semidirect

60. #13598

C₁₉

direct (N × H)

83. #1413

C₂×C₁₉

semidirect

130. C₁₉

#13598

direct (N × H)

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

subgroup	index
C₁₉×#3135	2
C₁₉×#2132	2
C₁₉×#1413	2
#13598	19

subgroup

index

C₁₉×#3135

C₁₉×#2132

C₁₉×#1413

#13598

Subgroups — C₁₉×#13598 #18019557

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#13598 #18019557

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13598 #18019557