Group #18019559 — subgroups

Subgroups — C₁₉×#13600 #18019559

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×#3142	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×#532	C₂²	1	1	yes	characteristic
5	C₂×C₁₉×#125	C₄	1	1	yes	characteristic
6	C₁₉×#491	C₂²	1	2	yes	—
7	C₁₉×#490	C₄	1	2	yes	—
8	C₁₉×#488	—	2	2	no	—
9	C₁₉×#417	—	2	2	no	—
10	C₁₉×#413	—	2	2	no	—
11	C₄×M₁₆×C₁₉	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	—	2	1	no	—
16	C₂×M₁₆×C₁₉	—	4	2	no	—
17	C₁₉×C₄⋊Q₈	C₂³	1	1	yes	characteristic
18	C₁₉×#127	C₂×C₄	1	1	yes	characteristic
19	C₁₉×#125	—	2	1	no	—
20	C₁₉×#125	—	4	1	no	—
21	C₁₉×#125	C₂×C₄	1	2	yes	—
22	C₁₉×#124	D₄	1	2	yes	—
23	C₁₉×#121	—	4	1	no	—
24	C₄×D₄×C₁₉	—	4	1	no	—
25	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
26	C₂×C₄⋊C₄×C₁₉	—	4	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	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₁₉	—	2	1	no	—
34	#41×C₁₉	—	4	2	no	—
35	#41×C₁₉	—	4	2	no	—
36	C₂×Q₈×C₁₉	C₂×D₄	1	2	yes	—
37	C₂×D₄×C₁₉	—	2	2	no	—
38	C₂×D₄×C₁₉	—	4	1	no	—
39	C₂×D₄×C₁₉	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₁₉	—	2	1	no	—
41	C₂²×C₄×C₁₉	—	2	2	no	—
42	C₂²×C₄×C₁₉	—	4	1	no	—
43	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
44	M₁₆×C₁₉	—	4	2	no	—
45	M₁₆×C₁₉	—	4	4	no	—
46	M₁₆×C₁₉	—	8	2	no	—
47	C₂×C₈×C₁₉	—	2	2	no	—
48	C₂×C₈×C₁₉	—	8	2	no	—
49	C₄⋊C₄×C₁₉	—	2	2	no	—
50	C₄⋊C₄×C₁₉	—	4	1	no	—
51	C₄⋊C₄×C₁₉	—	4	2	no	—
52	C₄⋊C₄×C₁₉	—	4	2	no	—
53	C₄⋊C₄×C₁₉	—	4	1	no	—
54	C₂²⋊C₄×C₁₉	—	4	2	no	—
55	C₄²×C₁₉	—	2	1	no	—
56	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
58	#13600	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂³×C₁₉	—	2	2	no	—
60	C₂³×C₁₉	#99	1	1	yes	characteristic
61	Q₈×C₁₉	—	4	2	no	—
62	D₄×C₁₉	—	4	2	no	—
63	D₄×C₁₉	—	8	2	no	—
64	D₄×C₁₉	—	8	2	no	—
65	C₂×C₄×C₁₉	—	2	2	no	—
66	C₂×C₄×C₁₉	—	2	1	no	—
67	C₂×C₄×C₁₉	—	2	1	no	—
68	C₂×C₄×C₁₉	—	2	2	no	—
69	C₂×C₄×C₁₉	—	4	2	no	—
70	C₂×C₄×C₁₉	—	4	2	no	—
71	C₂×C₄×C₁₉	—	4	2	no	—
72	C₂×C₄×C₁₉	—	8	1	no	—
73	C₂×C₄×C₁₉	—	8	2	no	—
74	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
75	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₈×C₁₉	—	4	2	no	—
77	C₈×C₁₉	—	8	4	no	—
78	#3142	C₂×C₁₉	1	1	yes	characteristic
79	#2132	C₂×C₁₉	1	2	yes	—
80	#1413	C₂×C₁₉	1	4	yes	complemented
81	C₂²×C₁₉	—	2	1	no	—
82	C₂²×C₁₉	—	4	1	no	—
83	C₂²×C₁₉	—	4	2	no	—
84	C₂²×C₁₉	—	8	2	no	—
85	C₂²×C₁₉	C₂×#99	1	1	yes	characteristic
86	C₄×C₁₉	—	2	1	no	—
87	C₄×C₁₉	—	4	1	no	—
88	C₄×C₁₉	—	4	1	no	—
89	C₄×C₁₉	—	4	2	no	—
90	C₄×C₁₉	—	8	2	no	—
91	C₄×C₁₉	#354	1	2	yes	—
92	#532	C₂²×C₁₉	1	1	yes	characteristic
93	C₂×#125	C₄×C₁₉	1	1	yes	characteristic
94	#491	C₂²×C₁₉	1	2	yes	—
95	#490	C₄×C₁₉	1	2	yes	—
96	#488	—	2	2	no	—
97	#417	—	2	2	no	—
98	#413	—	2	2	no	—
99	C₄×M₁₆	C₄×C₁₉	1	1	yes	characteristic
100	#356	C₂²×C₁₉	1	4	yes	—
101	#355	—	2	2	no	—
102	C₂×C₁₉	—	2	1	no	—
103	C₂×C₁₉	—	4	1	no	—
104	C₂×C₁₉	—	8	2	no	—
105	C₂×C₁₉	C₂×#354	1	1	yes	characteristic
106	C₂×#41	D₄×C₁₉	1	2	yes	—
107	C₂×M₁₆	—	2	1	no	—
108	C₂×M₁₆	—	4	2	no	—
109	C₄⋊Q₈	C₂³×C₁₉	1	1	yes	characteristic
110	#127	C₂×C₄×C₁₉	1	1	yes	characteristic
111	#125	—	2	1	no	—
112	#125	—	4	1	no	—
113	#125	C₂×C₄×C₁₉	1	2	yes	—
114	#124	D₄×C₁₉	1	2	yes	—
115	#121	—	4	1	no	—
116	C₄×D₄	—	4	1	no	—
117	C₂×C₄⋊C₄	—	2	1	no	—
118	C₂×C₄⋊C₄	—	4	1	no	—
119	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
120	#103	—	2	4	no	—
121	#102	—	2	4	no	—
122	#101	—	2	4	no	—
123	#100	—	4	2	no	—
124	#97	C₂×C₄×C₁₉	1	2	yes	—
125	C₄×C₈	—	2	1	no	—
126	C₁₉	#13600	1	1	yes	characteristic, direct product factor, Sylow
127	#41	—	4	2	no	—
128	#41	—	4	2	no	—
129	C₂×Q₈	C₂×D₄×C₁₉	1	2	yes	—
130	C₂×D₄	—	2	2	no	—
131	C₂×D₄	—	4	1	no	—
132	C₂×D₄	C₂²⋊C₄×C₁₉	1	2	yes	—
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	4	1	no	—
136	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
137	M₁₆	—	4	2	no	—
138	M₁₆	—	4	4	no	—
139	M₁₆	—	8	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	8	2	no	—
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	—	4	1	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	4	1	no	—
147	C₂²⋊C₄	—	4	2	no	—
148	C₄²	—	2	1	no	—
149	C₄²	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
150	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
151	C₂³	—	2	2	no	—
152	C₂³	C₁₉×#99	1	1	yes	characteristic
153	Q₈	—	4	2	no	—
154	D₄	—	4	2	no	—
155	D₄	—	8	2	no	—
156	D₄	—	8	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	8	1	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	C₁₉×#99	1	1	yes	characteristic
167	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
168	C₈	—	4	2	no	—
169	C₈	—	8	4	no	—
170	C₂²	—	2	1	no	—
171	C₂²	—	4	1	no	—
172	C₂²	—	4	2	no	—
173	C₂²	—	8	2	no	—
174	C₂²	C₂×C₁₉×#99	1	1	yes	characteristic
175	C₄	—	2	1	no	—
176	C₄	—	4	1	no	—
177	C₄	—	4	1	no	—
178	C₄	—	4	2	no	—
179	C₄	—	8	2	no	—
180	C₄	C₁₉×#354	1	2	yes	—
181	C₂	—	2	1	no	—
182	C₂	—	4	1	no	—
183	C₂	—	8	2	no	—
184	C₂	C₂×C₁₉×#354	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#3142

C₂

yes

maximal, characteristic

C₁₉×#2132

C₂

yes

maximal

C₁₉×#1413

C₂

yes

maximal, complemented

C₁₉×#532

C₂²

yes

characteristic

C₂×C₁₉×#125

C₄

yes

characteristic

C₁₉×#491

C₂²

yes

—

C₁₉×#490

C₄

yes

—

C₁₉×#488

—

C₁₉×#417

—

C₁₉×#413

—

C₄×M₁₆×C₁₉

C₄

yes

characteristic

C₁₉×#356

C₂²

yes

—

C₁₉×#355

—

C₂×#41×C₁₉

D₄

yes

—

C₂×M₁₆×C₁₉

—

C₂×M₁₆×C₁₉

—

C₁₉×C₄⋊Q₈

C₂³

yes

characteristic

C₁₉×#127

C₂×C₄

yes

characteristic

C₁₉×#125

—

C₁₉×#125

—

C₁₉×#125

C₂×C₄

yes

—

C₁₉×#124

D₄

yes

—

C₁₉×#121

—

C₄×D₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₁₉×#103

—

C₁₉×#102

—

C₁₉×#101

—

C₁₉×#100

—

C₁₉×#97

C₂×C₄

yes

—

C₄×C₈×C₁₉

—

#41×C₁₉

—

#41×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×C₁₉

C₂²⋊C₄

yes

characteristic

M₁₆×C₁₉

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

C₂²⋊C₄

yes

characteristic

C₄²×C₁₉

C₂²×C₄

yes

characteristic

#13600

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

#99

yes

characteristic

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

—

#3142

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

#354

yes

—

#532

C₂²×C₁₉

yes

characteristic

C₂×#125

C₄×C₁₉

yes

characteristic

#491

C₂²×C₁₉

yes

—

#490

C₄×C₁₉

yes

—

#488

—

#417

—

#413

—

C₄×M₁₆

C₄×C₁₉

yes

characteristic

100

#356

C₂²×C₁₉

yes

—

101

#355

—

102

C₂×C₁₉

—

103

C₂×C₁₉

—

104

C₂×C₁₉

—

105

C₂×C₁₉

C₂×#354

yes

characteristic

106

C₂×#41

D₄×C₁₉

yes

—

107

C₂×M₁₆

—

108

C₂×M₁₆

—

109

C₄⋊Q₈

C₂³×C₁₉

yes

characteristic

110

#127

C₂×C₄×C₁₉

yes

characteristic

111

#125

—

112

#125

—

113

#125

C₂×C₄×C₁₉

yes

—

114

#124

D₄×C₁₉

yes

—

115

#121

—

116

C₄×D₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄²

C₂×C₄×C₁₉

yes

characteristic

120

#103

—

121

#102

—

122

#101

—

123

#100

—

124

#97

C₂×C₄×C₁₉

yes

—

125

C₄×C₈

—

126

C₁₉

#13600

yes

characteristic, direct product factor, Sylow

127

#41

—

128

#41

—

129

C₂×Q₈

C₂×D₄×C₁₉

yes

—

130

C₂×D₄

—

131

C₂×D₄

—

132

C₂×D₄

C₂²⋊C₄×C₁₉

yes

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

C₂²⋊C₄×C₁₉

yes

characteristic

137

M₁₆

—

138

M₁₆

—

139

M₁₆

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₂²⋊C₄

—

148

C₄²

—

149

C₄²

C₂²⋊C₄×C₁₉

yes

characteristic

150

C₄²

C₂²×C₄×C₁₉

yes

characteristic

151

C₂³

—

152

C₂³

C₁₉×#99

yes

characteristic

153

Q₈

—

154

D₄

—

155

D₄

—

156

D₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

C₁₉×#99

yes

characteristic

167

C₂×C₄

C₂×C₂²⋊C₄×C₁₉

yes

characteristic

168

C₈

—

169

C₈

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

C₂×C₁₉×#99

yes

characteristic

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

C₁₉×#354

yes

—

181

C₂

—

182

C₂

—

183

C₂

—

184

C₂

C₂×C₁₉×#354

yes

characteristic

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

subgroup

G/H

center, socle

C₂×C₁₉

C₂×#354

commutator

C₄²

C₂²×C₄×C₁₉

Frattini, cosocle

C₄⋊Q₈

C₂³×C₁₉

prime radical

C₁₉×#532

C₂²

LCS

G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

UCS

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

N	H ≅ G/N	type
3. C₁₉×#1413	C₂	semidirect
58. #13600	C₁₉	direct (N × H)
80. #1413	C₂×C₁₉	semidirect
126. C₁₉	#13600	direct (N × H)

H ≅ G/N

type

3. C₁₉×#1413

C₂

semidirect

58. #13600

C₁₉

direct (N × H)

80. #1413

C₂×C₁₉

semidirect

126. C₁₉

#13600

direct (N × H)

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

subgroup	index
C₁₉×#3142	2
C₁₉×#2132	2
C₁₉×#1413	2
#13600	19

subgroup

index

C₁₉×#3142

C₁₉×#2132

C₁₉×#1413

#13600

Subgroups — C₁₉×#13600 #18019559

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#13600 #18019559

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13600 #18019559