Group #18013086 — subgroups

Subgroups — C₁₉×#7127 #18013086

Order 4864.

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

normal subgroups only

This group has 202 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×#1944	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1377	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1275	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₉×M₃₂	—	2	1	no	—
5	C₁₉×#431	—	2	1	no	—
6	C₂×C₁₉×#107	C₄	1	1	yes	characteristic
7	C₂×C₁₉×#106	C₄	1	1	yes	characteristic
8	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
9	C₁₉×#389	—	2	1	no	—
10	C₁₉×#340	—	2	1	no	—
11	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
14	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
15	C₂×C₄⋊C₄×C₁₉	—	4	1	no	—
16	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
17	C₁₉×M₃₂	—	2	2	no	—
18	C₁₉×M₃₂	—	4	1	no	—
19	C₂×C₁₆×C₁₉	—	4	1	no	—
20	C₁₉×#107	—	4	1	no	—
21	C₁₉×#106	—	2	1	no	—
22	C₁₉×#106	C₈	1	2	yes	complemented
23	C₁₉×C₄⋊C₈	—	4	1	no	—
24	C₁₉×C₂²⋊C₈	—	4	1	no	—
25	C₁₉×#97	—	4	1	no	—
26	C₄×C₈×C₁₉	—	2	2	no	—
27	C₁₉×#95	—	4	1	no	—
28	C₂²×C₄×C₁₉	—	4	1	no	—
29	C₂²×C₄×C₁₉	—	4	1	no	—
30	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
32	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
33	C₂×C₈×C₁₉	—	2	1	no	—
34	C₂×C₈×C₁₉	—	2	1	no	—
35	C₂×C₈×C₁₉	—	2	1	no	—
36	C₂×C₈×C₁₉	—	2	1	no	—
37	C₂×C₈×C₁₉	—	4	2	no	—
38	C₂×C₈×C₁₉	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈×C₁₉	M₁₆	1	1	yes	characteristic
40	C₂×C₈×C₁₉	SD₁₆	1	1	yes	characteristic
41	C₂×C₈×C₁₉	Q₁₆	1	1	yes	characteristic
42	C₄⋊C₄×C₁₉	—	2	2	no	—
43	C₄⋊C₄×C₁₉	—	4	1	no	—
44	C₄⋊C₄×C₁₉	—	4	1	no	—
45	C₄⋊C₄×C₁₉	—	4	1	no	—
46	C₄⋊C₄×C₁₉	—	8	2	no	—
47	C₄²×C₁₉	—	2	1	no	—
48	C₄²×C₁₉	—	2	1	no	—
49	C₁₆×C₁₉	—	4	2	no	—
50	#7127	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂³×C₁₉	#102	1	1	yes	characteristic
52	C₂×C₄×C₁₉	—	2	1	no	—
53	C₂×C₄×C₁₉	—	2	1	no	—
54	C₂×C₄×C₁₉	—	4	1	no	—
55	C₂×C₄×C₁₉	—	4	1	no	—
56	C₂×C₄×C₁₉	—	4	2	no	—
57	C₂×C₄×C₁₉	—	8	1	no	—
58	C₂×C₄×C₁₉	—	8	1	no	—
59	C₂×C₄×C₁₉	—	8	1	no	—
60	C₂×C₄×C₁₉	—	8	1	no	—
61	C₂×C₄×C₁₉	—	8	1	no	—
62	C₂×C₄×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
64	C₂×C₄×C₁₉	#101	1	1	yes	characteristic
65	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
66	C₂×C₄×C₁₉	#104	1	1	yes	characteristic
67	C₂×C₄×C₁₉	#104	1	1	yes	characteristic
68	C₈×C₁₉	—	2	1	no	—
69	C₈×C₁₉	—	2	1	no	—
70	C₈×C₁₉	—	2	1	no	—
71	C₈×C₁₉	—	2	1	no	—
72	C₈×C₁₉	—	8	2	no	—
73	#1944	C₂×C₁₉	1	1	yes	characteristic
74	#1377	C₂×C₁₉	1	1	yes	characteristic
75	#1275	C₂×C₁₉	1	1	yes	characteristic
76	C₂²×C₁₉	—	2	1	no	—
77	C₂²×C₁₉	—	2	1	no	—
78	C₂²×C₁₉	D₄⋊C₈	1	1	yes	characteristic
79	C₂²×C₁₉	#327	1	1	yes	characteristic
80	C₂²×C₁₉	#332	1	1	yes	characteristic
81	C₄×C₁₉	—	4	1	no	—
82	C₄×C₁₉	—	4	1	no	—
83	C₄×C₁₉	—	8	2	no	—
84	C₄×C₁₉	—	16	1	no	—
85	C₄×C₁₉	#324	1	1	yes	characteristic
86	C₄×C₁₉	Q₈⋊C₈	1	1	yes	characteristic
87	C₄×C₁₉	#328	1	1	yes	characteristic
88	C₄×C₁₉	#330	1	1	yes	characteristic
89	C₂×M₃₂	—	2	1	no	—
90	#431	—	2	1	no	—
91	C₂×#107	C₄×C₁₉	1	1	yes	characteristic
92	C₂×#106	C₄×C₁₉	1	1	yes	characteristic
93	C₂×#97	C₂²×C₁₉	1	1	yes	characteristic
94	#389	—	2	1	no	—
95	#340	—	2	1	no	—
96	C₂×C₁₉	—	2	1	no	—
97	C₂×C₁₉	—	2	1	no	—
98	C₂×C₁₉	#1271	1	1	yes	characteristic
99	C₂×C₁₉	#1339	1	1	yes	characteristic
100	C₂×C₁₉	#1342	1	1	yes	characteristic
101	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
102	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
103	C₂×C₄⋊C₄	—	2	1	no	—
104	C₂×C₄⋊C₄	—	2	1	no	—
105	C₂×C₄⋊C₄	—	4	1	no	—
106	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
107	M₃₂	—	2	2	no	—
108	M₃₂	—	4	1	no	—
109	C₂×C₁₆	—	4	1	no	—
110	#107	—	4	1	no	—
111	#106	—	2	1	no	—
112	#106	C₈×C₁₉	1	2	yes	complemented
113	C₄⋊C₈	—	4	1	no	—
114	C₂²⋊C₈	—	4	1	no	—
115	#97	—	4	1	no	—
116	C₄×C₈	—	2	2	no	—
117	#95	—	4	1	no	—
118	C₁₉	#7127	1	1	yes	characteristic, direct product factor, Sylow
119	C₂²×C₄	—	4	1	no	—
120	C₂²×C₄	—	4	1	no	—
121	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
122	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
123	C₂²×C₄	SD₁₆×C₁₉	1	1	yes	characteristic
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	C₂×C₈×C₁₉	1	1	yes	characteristic
130	C₂×C₈	M₁₆×C₁₉	1	1	yes	characteristic
131	C₂×C₈	SD₁₆×C₁₉	1	1	yes	characteristic
132	C₂×C₈	Q₁₆×C₁₉	1	1	yes	characteristic
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	—	4	1	no	—
135	C₄⋊C₄	—	4	1	no	—
136	C₄⋊C₄	—	4	1	no	—
137	C₄⋊C₄	—	8	2	no	—
138	C₄²	—	2	1	no	—
139	C₄²	—	2	1	no	—
140	C₁₆	—	4	2	no	—
141	C₂³	C₁₉×#102	1	1	yes	characteristic
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	8	1	no	—
148	C₂×C₄	—	8	1	no	—
149	C₂×C₄	—	8	1	no	—
150	C₂×C₄	—	8	1	no	—
151	C₂×C₄	—	8	1	no	—
152	C₂×C₄	C₁₉×C₂²⋊C₈	1	1	yes	characteristic
153	C₂×C₄	C₁₉×#99	1	1	yes	characteristic
154	C₂×C₄	C₁₉×#101	1	1	yes	characteristic
155	C₂×C₄	C₁₉×#103	1	1	yes	characteristic
156	C₂×C₄	C₁₉×#104	1	1	yes	characteristic
157	C₂×C₄	C₁₉×#104	1	1	yes	characteristic
158	C₈	—	2	1	no	—
159	C₈	—	2	1	no	—
160	C₈	—	2	1	no	—
161	C₈	—	2	1	no	—
162	C₈	—	8	2	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	C₁₉×D₄⋊C₈	1	1	yes	characteristic
166	C₂²	C₁₉×#327	1	1	yes	characteristic
167	C₂²	C₁₉×#332	1	1	yes	characteristic
168	C₄	—	4	1	no	—
169	C₄	—	4	1	no	—
170	C₄	—	8	2	no	—
171	C₄	—	16	1	no	—
172	C₄	C₁₉×#324	1	1	yes	characteristic
173	C₄	C₁₉×Q₈⋊C₈	1	1	yes	characteristic
174	C₄	C₁₉×#328	1	1	yes	characteristic
175	C₄	C₁₉×#330	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	2	1	no	—
178	C₂	C₁₉×#1271	1	1	yes	characteristic
179	C₂	C₁₉×#1339	1	1	yes	characteristic
180	C₂	C₁₉×#1342	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#1944

C₂

yes

maximal, characteristic

C₁₉×#1377

C₂

yes

maximal, characteristic

C₁₉×#1275

C₂

yes

maximal, characteristic

C₂×C₁₉×M₃₂

—

C₁₉×#431

—

C₂×C₁₉×#107

C₄

yes

characteristic

C₂×C₁₉×#106

C₄

yes

characteristic

C₂×C₁₉×#97

C₂²

yes

characteristic

C₁₉×#389

—

C₁₉×#340

—

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₁₉×M₃₂

—

C₁₉×M₃₂

—

C₂×C₁₆×C₁₉

—

C₁₉×#107

—

C₁₉×#106

—

C₁₉×#106

C₈

yes

complemented

C₁₉×C₄⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₁₉×#97

—

C₄×C₈×C₁₉

—

C₁₉×#95

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

D₈

yes

characteristic

C₂²×C₄×C₁₉

SD₁₆

yes

characteristic

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₂×C₈

yes

characteristic

C₂×C₈×C₁₉

M₁₆

yes

characteristic

C₂×C₈×C₁₉

SD₁₆

yes

characteristic

C₂×C₈×C₁₉

Q₁₆

yes

characteristic

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₁₆×C₁₉

—

#7127

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

#102

yes

characteristic

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

—

C₂×C₄×C₁₉

C₂²⋊C₈

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

#1944

C₂×C₁₉

yes

characteristic

#1377

C₂×C₁₉

yes

characteristic

#1275

C₂×C₁₉

yes

characteristic

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

D₄⋊C₈

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#324

yes

characteristic

C₄×C₁₉

Q₈⋊C₈

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

C₂×M₃₂

—

#431

—

C₂×#107

C₄×C₁₉

yes

characteristic

C₂×#106

C₄×C₁₉

yes

characteristic

C₂×#97

C₂²×C₁₉

yes

characteristic

#389

—

#340

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

100

C₂×C₁₉

#1342

yes

characteristic

101

C₂²×C₈

C₂×C₄×C₁₉

yes

characteristic

102

C₂²×C₈

D₄×C₁₉

yes

characteristic

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄²

D₄×C₁₉

yes

characteristic

107

M₃₂

—

108

M₃₂

—

109

C₂×C₁₆

—

110

#107

—

111

#106

—

112

#106

C₈×C₁₉

yes

complemented

113

C₄⋊C₈

—

114

C₂²⋊C₈

—

115

#97

—

116

C₄×C₈

—

117

#95

—

118

C₁₉

#7127

yes

characteristic, direct product factor, Sylow

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

C₂²⋊C₄×C₁₉

yes

characteristic

122

C₂²×C₄

D₈×C₁₉

yes

characteristic

123

C₂²×C₄

SD₁₆×C₁₉

yes

characteristic

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

C₂×C₈×C₁₉

yes

characteristic

130

C₂×C₈

M₁₆×C₁₉

yes

characteristic

131

C₂×C₈

SD₁₆×C₁₉

yes

characteristic

132

C₂×C₈

Q₁₆×C₁₉

yes

characteristic

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄²

—

139

C₄²

—

140

C₁₆

—

141

C₂³

C₁₉×#102

yes

characteristic

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

C₁₉×C₂²⋊C₈

yes

characteristic

153

C₂×C₄

C₁₉×#99

yes

characteristic

154

C₂×C₄

C₁₉×#101

yes

characteristic

155

C₂×C₄

C₁₉×#103

yes

characteristic

156

C₂×C₄

C₁₉×#104

yes

characteristic

157

C₂×C₄

C₁₉×#104

yes

characteristic

158

C₈

—

159

C₈

—

160

C₈

—

161

C₈

—

162

C₈

—

163

C₂²

—

164

C₂²

—

165

C₂²

C₁₉×D₄⋊C₈

yes

characteristic

166

C₂²

C₁₉×#327

yes

characteristic

167

C₂²

C₁₉×#332

yes

characteristic

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

C₁₉×#324

yes

characteristic

173

C₄

C₁₉×Q₈⋊C₈

yes

characteristic

174

C₄

C₁₉×#328

yes

characteristic

175

C₄

C₁₉×#330

yes

characteristic

176

C₂

—

177

C₂

—

178

C₂

C₁₉×#1271

yes

characteristic

179

C₂

C₁₉×#1339

yes

characteristic

180

C₂

C₁₉×#1342

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#327
commutator	C₂×C₈	C₂×C₈×C₁₉
Frattini, cosocle	C₂×#97	C₂²×C₁₉
prime radical	C₂³×C₁₉	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₁₉×#97 ◁ G

subgroup

G/H

center, socle

C₂²×C₁₉

#327

commutator

C₂×C₈

C₂×C₈×C₁₉

Frattini, cosocle

C₂×#97

C₂²×C₁₉

prime radical

C₂³×C₁₉

#102

LCS

G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₁₉×#97 ◁ G

N	H ≅ G/N	type
22. C₁₉×#106	C₈	semidirect
50. #7127	C₁₉	direct (N × H)
112. #106	C₈×C₁₉	semidirect
118. C₁₉	#7127	direct (N × H)

H ≅ G/N

type

22. C₁₉×#106

C₈

semidirect

50. #7127

C₁₉

direct (N × H)

112. #106

C₈×C₁₉

semidirect

118. C₁₉

#7127

direct (N × H)

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

subgroup	index
C₁₉×#1944	2
C₁₉×#1377	2
C₁₉×#1275	2
#7127	19

subgroup

index

C₁₉×#1944

C₁₉×#1377

C₁₉×#1275

#7127

Subgroups — C₁₉×#7127 #18013086

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#7127 #18013086

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#7127 #18013086