Group #18017987 — subgroups

Subgroups — C₁₉×#12028 #18017987

Order 4864.

210 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 118 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	C₁₉×#1568	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#334	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1368	C₂	1	4	yes	maximal, complemented
5	C₁₉×#432	—	2	1	no	—
6	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
7	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
8	C₁₉×C₄⋊C₁₆	C₄	1	2	yes	—
9	C₁₉×C₂²⋊C₁₆	C₄	1	2	yes	—
10	C₁₉×#346	C₂²	1	2	yes	—
11	C₁₉×#335	—	2	1	no	—
12	C₁₉×#334	—	2	1	no	—
13	C₁₉×#334	C₂²	1	2	yes	—
14	C₁₉×#334	C₂²	1	2	yes	—
15	C₂²×C₈×C₁₉	—	4	1	no	—
16	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₉	Q₈	1	1	yes	characteristic
18	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
19	C₂×C₁₆×C₁₉	C₂×C₄	1	4	yes	—
20	C₁₉×C₄⋊C₈	—	2	2	no	—
21	C₁₉×C₄⋊C₈	—	2	2	no	—
22	C₁₉×C₄⋊C₈	—	4	1	no	—
23	C₁₉×C₂²⋊C₈	—	4	1	no	—
24	C₁₉×#97	—	4	1	no	—
25	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
27	C₄×C₈×C₁₉	Q₈	1	1	yes	characteristic
28	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
32	C₂×C₈×C₁₉	—	2	2	no	—
33	C₂×C₈×C₁₉	—	4	1	no	—
34	C₂×C₈×C₁₉	—	4	2	no	—
35	C₂×C₈×C₁₉	—	4	2	no	—
36	C₂×C₈×C₁₉	—	8	2	no	—
37	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
41	C₄²×C₁₉	C₄⋊C₄	1	1	yes	characteristic
42	C₄²×C₁₉	SD₁₆	1	1	yes	characteristic
43	C₄²×C₁₉	SD₁₆	1	1	yes	characteristic
44	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
45	C₁₆×C₁₉	—	2	4	no	—
46	#12028	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	C₂³×C₁₉	#106	1	1	yes	characteristic
48	C₂×C₄×C₁₉	—	2	2	no	—
49	C₂×C₄×C₁₉	—	2	1	no	—
50	C₂×C₄×C₁₉	—	2	1	no	—
51	C₂×C₄×C₁₉	—	4	1	no	—
52	C₂×C₄×C₁₉	#106	1	1	yes	characteristic
53	C₂×C₄×C₁₉	#106	1	1	yes	characteristic
54	C₂×C₄×C₁₉	#106	1	1	yes	characteristic
55	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
58	C₈×C₁₉	—	2	1	no	—
59	C₈×C₁₉	—	2	1	no	—
60	C₈×C₁₉	—	2	1	no	—
61	C₈×C₁₉	—	8	2	no	—
62	C₈×C₁₉	—	8	2	no	—
63	C₈×C₁₉	#131	1	2	yes	—
64	#2163	C₂×C₁₉	1	1	yes	characteristic
65	#1568	C₂×C₁₉	1	1	yes	characteristic
66	C₂×#334	C₂×C₁₉	1	1	yes	characteristic
67	#1368	C₂×C₁₉	1	4	yes	complemented
68	C₂²×C₁₉	—	2	1	no	—
69	C₂²×C₁₉	—	2	1	no	—
70	C₂²×C₁₉	#365	1	2	yes	—
71	C₂²×C₁₉	C₂×#106	1	1	yes	characteristic
72	C₄×C₁₉	—	2	1	no	—
73	C₄×C₁₉	—	2	1	no	—
74	C₄×C₁₉	—	2	1	no	—
75	C₄×C₁₉	—	2	1	no	—
76	C₄×C₁₉	—	2	1	no	—
77	C₄×C₁₉	#424	1	1	yes	characteristic
78	C₄×C₁₉	#430	1	1	yes	characteristic
79	#432	—	2	1	no	—
80	C₂×C₄⋊C₈	—	2	1	no	—
81	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
82	C₄⋊C₁₆	C₄×C₁₉	1	2	yes	—
83	C₂²⋊C₁₆	C₄×C₁₉	1	2	yes	—
84	#346	C₂²×C₁₉	1	2	yes	—
85	#335	—	2	1	no	—
86	#334	—	2	1	no	—
87	#334	C₂²×C₁₉	1	2	yes	—
88	#334	C₂²×C₁₉	1	2	yes	—
89	C₂×C₁₉	—	2	2	no	—
90	C₂×C₁₉	#1570	1	1	yes	characteristic
91	C₂×C₁₉	C₂×#365	1	1	yes	characteristic
92	C₂×C₁₉	#2155	1	1	yes	characteristic
93	C₂²×C₈	—	4	1	no	—
94	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
95	C₂²×C₈	Q₈×C₁₉	1	1	yes	characteristic
96	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
97	C₂×C₁₆	C₂×C₄×C₁₉	1	4	yes	—
98	C₄⋊C₈	—	2	2	no	—
99	C₄⋊C₈	—	2	2	no	—
100	C₄⋊C₈	—	4	1	no	—
101	C₂²⋊C₈	—	4	1	no	—
102	#97	—	4	1	no	—
103	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
104	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
105	C₄×C₈	Q₈×C₁₉	1	1	yes	characteristic
106	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
107	C₁₉	#12028	1	1	yes	characteristic, direct product factor, Sylow
108	C₂²×C₄	C₄⋊C₄×C₁₉	1	1	yes	characteristic
109	C₂²×C₄	SD₁₆×C₁₉	1	1	yes	characteristic
110	C₂²×C₄	SD₁₆×C₁₉	1	1	yes	characteristic
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	4	1	no	—
113	C₂×C₈	—	4	2	no	—
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	—	8	2	no	—
116	C₂×C₈	C₄⋊C₄×C₁₉	1	1	yes	characteristic
117	C₂×C₈	C₄⋊C₄×C₁₉	1	1	yes	characteristic
118	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
119	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
120	C₄²	C₄⋊C₄×C₁₉	1	1	yes	characteristic
121	C₄²	SD₁₆×C₁₉	1	1	yes	characteristic
122	C₄²	SD₁₆×C₁₉	1	1	yes	characteristic
123	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
124	C₁₆	—	2	4	no	—
125	C₂³	C₁₉×#106	1	1	yes	characteristic
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	C₁₉×#106	1	1	yes	characteristic
131	C₂×C₄	C₁₉×#106	1	1	yes	characteristic
132	C₂×C₄	C₁₉×#106	1	1	yes	characteristic
133	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
134	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
135	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
136	C₈	—	2	1	no	—
137	C₈	—	2	1	no	—
138	C₈	—	2	1	no	—
139	C₈	—	8	2	no	—
140	C₈	—	8	2	no	—
141	C₈	C₁₉×#131	1	2	yes	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	1	no	—
144	C₂²	C₁₉×#365	1	2	yes	—
145	C₂²	C₂×C₁₉×#106	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	C₁₉×#424	1	1	yes	characteristic
152	C₄	C₁₉×#430	1	1	yes	characteristic
153	C₂	—	2	2	no	—
154	C₂	C₁₉×#1570	1	1	yes	characteristic
155	C₂	C₂×C₁₉×#365	1	1	yes	characteristic
156	C₂	C₁₉×#2155	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#2163

C₂

yes

maximal, characteristic

C₁₉×#1568

C₂

yes

maximal, characteristic

C₂×C₁₉×#334

C₂

yes

maximal, characteristic

C₁₉×#1368

C₂

yes

maximal, complemented

C₁₉×#432

—

C₂×C₁₉×C₄⋊C₈

—

C₂×C₄×C₈×C₁₉

C₂²

yes

characteristic

C₁₉×C₄⋊C₁₆

C₄

yes

—

C₁₉×C₂²⋊C₁₆

C₄

yes

—

C₁₉×#346

C₂²

yes

—

C₁₉×#335

—

C₁₉×#334

—

C₁₉×#334

C₂²

yes

—

C₁₉×#334

C₂²

yes

—

C₂²×C₈×C₁₉

—

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₉

Q₈

yes

characteristic

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₂×C₁₆×C₁₉

C₂×C₄

yes

—

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₁₉×#97

—

C₄×C₈×C₁₉

C₂×C₄

yes

characteristic

C₄×C₈×C₁₉

D₄

yes

characteristic

C₄×C₈×C₁₉

Q₈

yes

characteristic

C₄×C₈×C₁₉

C₂³

yes

characteristic

C₂²×C₄×C₁₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

SD₁₆

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

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₉

C₂×Q₈

yes

characteristic

C₄²×C₁₉

C₄⋊C₄

yes

characteristic

C₄²×C₁₉

SD₁₆

yes

characteristic

C₄²×C₁₉

SD₁₆

yes

characteristic

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₁₆×C₁₉

—

#12028

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

#106

yes

characteristic

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₉

C₂×SD₁₆

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

#131

yes

—

#2163

C₂×C₁₉

yes

characteristic

#1568

C₂×C₁₉

yes

characteristic

C₂×#334

C₂×C₁₉

yes

characteristic

#1368

C₂×C₁₉

yes

complemented

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#365

yes

—

C₂²×C₁₉

C₂×#106

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂²×C₁₉

yes

characteristic

C₄⋊C₁₆

C₄×C₁₉

yes

—

C₂²⋊C₁₆

C₄×C₁₉

yes

—

#346

C₂²×C₁₉

yes

—

#335

—

#334

—

#334

C₂²×C₁₉

yes

—

#334

C₂²×C₁₉

yes

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂²×C₈

—

C₂²×C₈

C₂×C₄×C₁₉

yes

characteristic

C₂²×C₈

Q₈×C₁₉

yes

characteristic

C₂×C₄²

D₄×C₁₉

yes

characteristic

C₂×C₁₆

C₂×C₄×C₁₉

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₂²⋊C₈

—

102

#97

—

103

C₄×C₈

C₂×C₄×C₁₉

yes

characteristic

104

C₄×C₈

D₄×C₁₉

yes

characteristic

105

C₄×C₈

Q₈×C₁₉

yes

characteristic

106

C₄×C₈

C₂³×C₁₉

yes

characteristic

107

C₁₉

#12028

yes

characteristic, direct product factor, Sylow

108

C₂²×C₄

C₄⋊C₄×C₁₉

yes

characteristic

109

C₂²×C₄

SD₁₆×C₁₉

yes

characteristic

110

C₂²×C₄

SD₁₆×C₁₉

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

C₄⋊C₄×C₁₉

yes

characteristic

117

C₂×C₈

C₄⋊C₄×C₁₉

yes

characteristic

118

C₂×C₈

C₂²×C₄×C₁₉

yes

characteristic

119

C₂×C₈

C₂×Q₈×C₁₉

yes

characteristic

120

C₄²

C₄⋊C₄×C₁₉

yes

characteristic

121

C₄²

SD₁₆×C₁₉

yes

characteristic

122

C₄²

SD₁₆×C₁₉

yes

characteristic

123

C₄²

C₂×D₄×C₁₉

yes

characteristic

124

C₁₆

—

125

C₂³

C₁₉×#106

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₁₉×#106

yes

characteristic

131

C₂×C₄

C₁₉×#106

yes

characteristic

132

C₂×C₄

C₁₉×#106

yes

characteristic

133

C₂×C₄

C₂×C₄⋊C₄×C₁₉

yes

characteristic

134

C₂×C₄

C₂×SD₁₆×C₁₉

yes

characteristic

135

C₂×C₄

C₂×SD₁₆×C₁₉

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

C₁₉×#131

yes

—

142

C₂²

—

143

C₂²

—

144

C₂²

C₁₉×#365

yes

—

145

C₂²

C₂×C₁₉×#106

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

C₁₉×#424

yes

characteristic

152

C₄

C₁₉×#430

yes

characteristic

153

C₂

—

154

C₂

C₁₉×#1570

yes

characteristic

155

C₂

C₂×C₁₉×#365

yes

characteristic

156

C₂

C₁₉×#2155

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#106
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₄×C₈	C₂³×C₁₉
prime radical	C₂³×C₁₉	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₉

C₂×#106

commutator

C₂×C₈

C₂²×C₄×C₁₉

Frattini, cosocle

C₄×C₈

C₂³×C₁₉

prime radical

C₂³×C₁₉

#106

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

N	H ≅ G/N	type
4. C₁₉×#1368	C₂	semidirect
46. #12028	C₁₉	direct (N × H)
67. #1368	C₂×C₁₉	semidirect
107. C₁₉	#12028	direct (N × H)

H ≅ G/N

type

4. C₁₉×#1368

C₂

semidirect

46. #12028

C₁₉

direct (N × H)

67. #1368

C₂×C₁₉

semidirect

107. C₁₉

#12028

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
C₁₉×#1568	2
C₂×C₁₉×#334	2
C₁₉×#1368	2
#12028	19

subgroup

index

C₁₉×#2163

C₁₉×#1568

C₂×C₁₉×#334

C₁₉×#1368

#12028

Subgroups — C₁₉×#12028 #18017987

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#12028 #18017987

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12028 #18017987