Group #18013123 — subgroups

Subgroups — C₁₉×#7164 #18013123

Order 4864.

212 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2163	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₉×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1451	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₉×M₃₂	—	2	1	no	—
5	C₁₉×#431	C₄	1	1	yes	characteristic
6	C₄×M₁₆×C₁₉	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
8	C₁₉×C₄⋊C₁₆	C₄	1	2	yes	—
9	C₁₉×C₂²⋊C₁₆	—	2	1	no	—
10	C₁₉×#346	—	2	1	no	—
11	C₁₉×#346	—	2	1	no	—
12	C₄×C₁₆×C₁₉	C₄	1	2	yes	—
13	C₁₉×#322	—	2	1	no	—
14	C₁₉×#322	—	2	1	no	—
15	C₂×M₁₆×C₁₉	—	2	1	no	—
16	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₉	Q₈	1	1	yes	characteristic
18	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
19	C₁₉×M₃₂	—	2	4	no	—
20	C₂×C₁₆×C₁₉	—	2	2	no	—
21	C₂×C₁₆×C₁₉	—	2	2	no	—
22	C₁₉×C₄⋊C₈	—	2	1	no	—
23	C₁₉×C₂²⋊C₈	C₈	1	2	yes	—
24	C₁₉×#97	C₈	1	2	yes	—
25	C₄×C₈×C₁₉	—	2	1	no	—
26	C₄×C₈×C₁₉	—	2	1	no	—
27	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
30	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₉	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄×C₁₉	M₁₆	1	1	yes	characteristic
34	M₁₆×C₁₉	—	4	2	no	—
35	C₂×C₈×C₁₉	—	2	2	no	—
36	C₂×C₈×C₁₉	—	2	2	no	—
37	C₂×C₈×C₁₉	—	2	2	no	—
38	C₂×C₈×C₁₉	—	2	2	no	—
39	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₁₉	C₄²	1	1	yes	characteristic
44	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₄²×C₁₉	C₂×C₈	1	1	yes	characteristic
46	C₄²×C₁₉	M₁₆	1	1	yes	characteristic
47	C₁₆×C₁₉	—	2	4	no	—
48	C₁₆×C₁₉	—	4	2	no	—
49	#7164	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂³×C₁₉	C₄⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₁₉	—	2	1	no	—
52	C₂×C₄×C₁₉	—	2	1	no	—
53	C₂×C₄×C₁₉	—	2	2	no	—
54	C₂×C₄×C₁₉	—	4	1	no	—
55	C₂×C₄×C₁₉	#95	1	1	yes	characteristic
56	C₂×C₄×C₁₉	C₄×C₈	1	1	yes	characteristic
57	C₂×C₄×C₁₉	#97	1	1	yes	characteristic
58	C₂×C₄×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₁₉	C₄⋊C₈	1	1	yes	characteristic
61	C₈×C₁₉	—	2	1	no	—
62	C₈×C₁₉	—	2	1	no	—
63	C₈×C₁₉	—	4	2	no	—
64	C₈×C₁₉	—	4	2	no	—
65	C₈×C₁₉	#104	1	2	yes	—
66	C₈×C₁₉	#108	1	2	yes	—
67	#2163	C₂×C₁₉	1	1	yes	characteristic
68	C₄×M₃₂	C₂×C₁₉	1	1	yes	characteristic
69	#1451	C₂×C₁₉	1	1	yes	characteristic
70	C₂²×C₁₉	—	2	1	no	—
71	C₂²×C₁₉	—	2	1	no	—
72	C₂²×C₁₉	—	2	1	no	—
73	C₂²×C₁₉	#336	1	1	yes	characteristic
74	C₄×C₁₉	—	2	1	no	—
75	C₄×C₁₉	—	2	1	no	—
76	C₄×C₁₉	—	2	1	no	—
77	C₄×C₁₉	—	2	1	no	—
78	C₄×C₁₉	#339	1	1	yes	characteristic
79	C₄×C₁₉	#341	1	1	yes	characteristic
80	C₄×C₁₉	#347	1	2	yes	—
81	C₂×M₃₂	—	2	1	no	—
82	#431	C₄×C₁₉	1	1	yes	characteristic
83	C₄×M₁₆	C₄×C₁₉	1	1	yes	characteristic
84	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
85	C₄⋊C₁₆	C₄×C₁₉	1	2	yes	—
86	C₂²⋊C₁₆	—	2	1	no	—
87	#346	—	2	1	no	—
88	#346	—	2	1	no	—
89	C₄×C₁₆	C₄×C₁₉	1	2	yes	—
90	#322	—	2	1	no	—
91	#322	—	2	1	no	—
92	C₂×C₁₉	—	4	1	no	—
93	C₂×C₁₉	#1278	1	1	yes	characteristic
94	C₂×C₁₉	#1375	1	1	yes	characteristic
95	C₂×C₁₉	#1378	1	1	yes	characteristic
96	C₂×M₁₆	—	2	1	no	—
97	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
98	C₂²×C₈	Q₈×C₁₉	1	1	yes	characteristic
99	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
100	M₃₂	—	2	4	no	—
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	—	2	2	no	—
103	C₄⋊C₈	—	2	1	no	—
104	C₂²⋊C₈	C₈×C₁₉	1	2	yes	—
105	#97	C₈×C₁₉	1	2	yes	—
106	C₄×C₈	—	2	1	no	—
107	C₄×C₈	—	2	1	no	—
108	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
109	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
110	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
111	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
112	C₁₉	#7164	1	1	yes	characteristic, direct product factor, Sylow
113	C₂²×C₄	C₄⋊C₄×C₁₉	1	1	yes	characteristic
114	C₂²×C₄	C₂×C₈×C₁₉	1	1	yes	characteristic
115	C₂²×C₄	M₁₆×C₁₉	1	1	yes	characteristic
116	M₁₆	—	4	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
122	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
123	C₂×C₈	C₄⋊C₄×C₁₉	1	1	yes	characteristic
124	C₂×C₈	C₄⋊C₄×C₁₉	1	1	yes	characteristic
125	C₄²	C₄²×C₁₉	1	1	yes	characteristic
126	C₄²	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
127	C₄²	C₂×C₈×C₁₉	1	1	yes	characteristic
128	C₄²	M₁₆×C₁₉	1	1	yes	characteristic
129	C₁₆	—	2	4	no	—
130	C₁₆	—	4	2	no	—
131	C₂³	C₁₉×C₄⋊C₈	1	1	yes	characteristic
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	C₁₉×#95	1	1	yes	characteristic
137	C₂×C₄	C₄×C₈×C₁₉	1	1	yes	characteristic
138	C₂×C₄	C₁₉×#97	1	1	yes	characteristic
139	C₂×C₄	C₁₉×C₂²⋊C₈	1	1	yes	characteristic
140	C₂×C₄	C₁₉×C₂²⋊C₈	1	1	yes	characteristic
141	C₂×C₄	C₁₉×C₄⋊C₈	1	1	yes	characteristic
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	4	2	no	—
145	C₈	—	4	2	no	—
146	C₈	C₁₉×#104	1	2	yes	—
147	C₈	C₁₉×#108	1	2	yes	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	C₁₉×#336	1	1	yes	characteristic
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	C₁₉×#339	1	1	yes	characteristic
157	C₄	C₁₉×#341	1	1	yes	characteristic
158	C₄	C₁₉×#347	1	2	yes	—
159	C₂	—	4	1	no	—
160	C₂	C₁₉×#1278	1	1	yes	characteristic
161	C₂	C₁₉×#1375	1	1	yes	characteristic
162	C₂	C₁₉×#1378	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#2163

C₂

yes

maximal, characteristic

C₄×C₁₉×M₃₂

C₂

yes

maximal, characteristic

C₁₉×#1451

C₂

yes

maximal, characteristic

C₂×C₁₉×M₃₂

—

C₁₉×#431

C₄

yes

characteristic

C₄×M₁₆×C₁₉

C₄

yes

characteristic

C₂×C₄×C₈×C₁₉

C₂²

yes

characteristic

C₁₉×C₄⋊C₁₆

C₄

yes

—

C₁₉×C₂²⋊C₁₆

—

C₁₉×#346

—

C₁₉×#346

—

C₄×C₁₆×C₁₉

C₄

yes

—

C₁₉×#322

—

C₁₉×#322

—

C₂×M₁₆×C₁₉

—

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂²×C₈×C₁₉

Q₈

yes

characteristic

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₁₉×M₃₂

—

C₂×C₁₆×C₁₉

—

C₂×C₁₆×C₁₉

—

C₁₉×C₄⋊C₈

—

C₁₉×C₂²⋊C₈

C₈

yes

—

C₁₉×#97

C₈

yes

—

C₄×C₈×C₁₉

—

C₄×C₈×C₁₉

—

C₄×C₈×C₁₉

C₂×C₄

yes

characteristic

C₄×C₈×C₁₉

C₂×C₄

yes

characteristic

C₄×C₈×C₁₉

D₄

yes

characteristic

C₄×C₈×C₁₉

D₄

yes

characteristic

C₂²×C₄×C₁₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₉

M₁₆

yes

characteristic

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₄⋊C₄

yes

characteristic

C₄²×C₁₉

C₄²

yes

characteristic

C₄²×C₁₉

C₂²⋊C₄

yes

characteristic

C₄²×C₁₉

C₂×C₈

yes

characteristic

C₄²×C₁₉

M₁₆

yes

characteristic

C₁₆×C₁₉

—

C₁₆×C₁₉

—

#7164

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

C₄⋊C₈

yes

characteristic

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#95

yes

characteristic

C₂×C₄×C₁₉

C₄×C₈

yes

characteristic

C₂×C₄×C₁₉

#97

yes

characteristic

C₂×C₄×C₁₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₉

C₄⋊C₈

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

#104

yes

—

C₈×C₁₉

#108

yes

—

#2163

C₂×C₁₉

yes

characteristic

C₄×M₃₂

C₂×C₁₉

yes

characteristic

#1451

C₂×C₁₉

yes

characteristic

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#336

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

—

C₂×M₃₂

—

#431

C₄×C₁₉

yes

characteristic

C₄×M₁₆

C₄×C₁₉

yes

characteristic

C₂×C₄×C₈

C₂²×C₁₉

yes

characteristic

C₄⋊C₁₆

C₄×C₁₉

yes

—

C₂²⋊C₁₆

—

#346

—

#346

—

C₄×C₁₆

C₄×C₁₉

yes

—

#322

—

#322

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×M₁₆

—

C₂²×C₈

D₄×C₁₉

yes

characteristic

C₂²×C₈

Q₈×C₁₉

yes

characteristic

C₂×C₄²

C₂×C₄×C₁₉

yes

characteristic

100

M₃₂

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

C₄⋊C₈

—

104

C₂²⋊C₈

C₈×C₁₉

yes

—

105

#97

C₈×C₁₉

yes

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

C₂×C₄×C₁₉

yes

characteristic

109

C₄×C₈

C₂×C₄×C₁₉

yes

characteristic

110

C₄×C₈

D₄×C₁₉

yes

characteristic

111

C₄×C₈

D₄×C₁₉

yes

characteristic

112

C₁₉

#7164

yes

characteristic, direct product factor, Sylow

113

C₂²×C₄

C₄⋊C₄×C₁₉

yes

characteristic

114

C₂²×C₄

C₂×C₈×C₁₉

yes

characteristic

115

C₂²×C₄

M₁₆×C₁₉

yes

characteristic

116

M₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

C₂²⋊C₄×C₁₉

yes

characteristic

122

C₂×C₈

C₂²⋊C₄×C₁₉

yes

characteristic

123

C₂×C₈

C₄⋊C₄×C₁₉

yes

characteristic

124

C₂×C₈

C₄⋊C₄×C₁₉

yes

characteristic

125

C₄²

C₄²×C₁₉

yes

characteristic

126

C₄²

C₂²⋊C₄×C₁₉

yes

characteristic

127

C₄²

C₂×C₈×C₁₉

yes

characteristic

128

C₄²

M₁₆×C₁₉

yes

characteristic

129

C₁₆

—

130

C₁₆

—

131

C₂³

C₁₉×C₄⋊C₈

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

C₁₉×#95

yes

characteristic

137

C₂×C₄

C₄×C₈×C₁₉

yes

characteristic

138

C₂×C₄

C₁₉×#97

yes

characteristic

139

C₂×C₄

C₁₉×C₂²⋊C₈

yes

characteristic

140

C₂×C₄

C₁₉×C₂²⋊C₈

yes

characteristic

141

C₂×C₄

C₁₉×C₄⋊C₈

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

C₁₉×#104

yes

—

147

C₈

C₁₉×#108

yes

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

C₁₉×#336

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

C₁₉×#339

yes

characteristic

157

C₄

C₁₉×#341

yes

characteristic

158

C₄

C₁₉×#347

yes

—

159

C₂

—

160

C₂

C₁₉×#1278

yes

characteristic

161

C₂

C₁₉×#1375

yes

characteristic

162

C₂

C₁₉×#1378

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₁₉	#95
commutator	C₂×C₄	C₄×C₈×C₁₉
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₉
socle	C₂²×C₁₉	#336
prime radical	C₂³×C₁₉	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

subgroup

G/H

center

C₂×C₄×C₁₉

#95

commutator

C₂×C₄

C₄×C₈×C₁₉

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₁₉

socle

C₂²×C₁₉

#336

prime radical

C₂³×C₁₉

C₄⋊C₈

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

N	H ≅ G/N	type
49. #7164	C₁₉	direct (N × H)
112. C₁₉	#7164	direct (N × H)

H ≅ G/N

type

49. #7164

C₁₉

direct (N × H)

112. C₁₉

#7164

direct (N × H)

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

subgroup	index
C₁₉×#2163	2
C₄×C₁₉×M₃₂	2
C₁₉×#1451	2
#7164	19

subgroup

index

C₁₉×#2163

C₄×C₁₉×M₃₂

C₁₉×#1451

#7164

Subgroups — C₁₉×#7164 #18013123

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#7164 #18013123

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#7164 #18013123