Group #18019662 — subgroups

Subgroups — C₁₉×#13703 #18019662

Order 4864.

176 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 78 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×#2252	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₁₉×D₁₆	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1423	C₂	1	2	yes	maximal, complemented
4	C₁₉×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1415	C₂	1	2	yes	maximal
6	C₂×C₁₉×D₁₆	C₂²	1	1	yes	characteristic
7	C₁₉×#500	—	2	1	no	—
8	C₄×D₈×C₁₉	—	2	1	no	—
9	C₂×C₁₉×C₃₂	C₂²	1	2	yes	—
10	C₁₉×#366	—	2	1	no	—
11	C₁₉×#366	C₂²	1	1	yes	characteristic
12	C₁₉×#366	C₂²	1	2	yes	—
13	C₁₉×#357	—	2	1	no	—
14	C₄×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
15	C₂×D₈×C₁₉	—	2	1	no	—
16	C₁₉×C₄⋊Q₈	—	4	1	no	—
17	C₄×D₄×C₁₉	—	4	1	no	—
18	C₁₉×D₁₆	—	2	1	no	—
19	C₁₉×D₁₆	Q₈	1	2	yes	complemented
20	C₂×C₁₆×C₁₉	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₁₉	C₂³	1	1	yes	characteristic
22	C₁₉×#107	—	2	1	no	—
23	C₁₉×#107	—	2	2	no	—
24	C₁₉×#107	—	4	1	no	—
25	C₁₉×#102	—	4	1	no	—
26	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
27	C₁₉×C₃₂	—	2	2	no	—
28	C₂×Q₈×C₁₉	—	8	1	no	—
29	C₂×D₄×C₁₉	—	4	1	no	—
30	C₂²×C₄×C₁₉	—	8	1	no	—
31	D₈×C₁₉	—	2	2	no	—
32	D₈×C₁₉	—	4	1	no	—
33	C₂×C₈×C₁₉	D₈	1	1	yes	characteristic
34	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
35	C₄⋊C₄×C₁₉	—	4	1	no	—
36	C₄⋊C₄×C₁₉	—	4	2	no	—
37	C₄⋊C₄×C₁₉	—	8	1	no	—
38	C₂²⋊C₄×C₁₉	—	8	1	no	—
39	C₄²×C₁₉	D₈	1	1	yes	characteristic
40	C₁₆×C₁₉	—	2	1	no	—
41	C₁₆×C₁₉	C₂×Q₈	1	1	yes	characteristic
42	C₁₆×C₁₉	#41	1	1	yes	characteristic
43	#13703	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂³×C₁₉	—	8	1	no	—
45	Q₈×C₁₉	—	16	2	no	—
46	D₄×C₁₉	—	4	2	no	—
47	D₄×C₁₉	—	8	1	no	—
48	C₂×C₄×C₁₉	—	8	1	no	—
49	C₂×C₄×C₁₉	—	8	2	no	—
50	C₂×C₄×C₁₉	—	16	2	no	—
51	C₂×C₄×C₁₉	D₁₆	1	1	yes	characteristic
52	C₂×C₄×C₁₉	D₁₆	1	1	yes	characteristic
53	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
54	C₈×C₁₉	—	2	1	no	—
55	C₈×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
56	C₈×C₁₉	#137	1	1	yes	characteristic
57	#2252	C₂×C₁₉	1	1	yes	characteristic, complemented
58	C₄×D₁₆	C₂×C₁₉	1	1	yes	characteristic
59	#1423	C₂×C₁₉	1	2	yes	complemented
60	C₄⋊C₃₂	C₂×C₁₉	1	1	yes	characteristic, complemented
61	#1415	C₂×C₁₉	1	2	yes	—
62	C₂²×C₁₉	—	8	2	no	—
63	C₂²×C₁₉	—	16	1	no	—
64	C₂²×C₁₉	—	16	1	no	—
65	C₂²×C₁₉	C₂×D₁₆	1	1	yes	characteristic
66	C₄×C₁₉	—	2	1	no	—
67	C₄×C₁₉	—	16	1	no	—
68	C₄×C₁₉	—	16	2	no	—
69	C₄×C₁₉	D₃₂	1	2	yes	—
70	C₄×C₁₉	#474	1	1	yes	characteristic
71	C₄×C₁₉	#510	1	1	yes	characteristic
72	C₂×D₁₆	C₂²×C₁₉	1	1	yes	characteristic
73	#500	—	2	1	no	—
74	C₄×D₈	—	2	1	no	—
75	C₂×C₃₂	C₂²×C₁₉	1	2	yes	—
76	#366	—	2	1	no	—
77	#366	C₂²×C₁₉	1	1	yes	characteristic
78	#366	C₂²×C₁₉	1	2	yes	—
79	#357	—	2	1	no	—
80	C₄×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
81	C₂×C₁₉	—	16	2	no	—
82	C₂×C₁₉	#2224	1	1	yes	characteristic
83	C₂×C₁₉	C₂×D₃₂	1	1	yes	characteristic
84	C₂×C₁₉	#2264	1	1	yes	characteristic
85	C₂×D₈	—	2	1	no	—
86	C₄⋊Q₈	—	4	1	no	—
87	C₄×D₄	—	4	1	no	—
88	D₁₆	—	2	1	no	—
89	D₁₆	Q₈×C₁₉	1	2	yes	complemented
90	C₂×C₁₆	D₄×C₁₉	1	1	yes	characteristic
91	C₂×C₁₆	C₂³×C₁₉	1	1	yes	characteristic
92	#107	—	2	1	no	—
93	#107	—	2	2	no	—
94	#107	—	4	1	no	—
95	#102	—	4	1	no	—
96	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
97	C₃₂	—	2	2	no	—
98	C₁₉	#13703	1	1	yes	characteristic, direct product factor, Sylow
99	C₂×Q₈	—	8	1	no	—
100	C₂×D₄	—	4	1	no	—
101	C₂²×C₄	—	8	1	no	—
102	D₈	—	2	2	no	—
103	D₈	—	4	1	no	—
104	C₂×C₈	D₈×C₁₉	1	1	yes	characteristic
105	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
106	C₄⋊C₄	—	4	1	no	—
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	—	8	1	no	—
109	C₂²⋊C₄	—	8	1	no	—
110	C₄²	D₈×C₁₉	1	1	yes	characteristic
111	C₁₆	—	2	1	no	—
112	C₁₆	C₂×Q₈×C₁₉	1	1	yes	characteristic
113	C₁₆	#41×C₁₉	1	1	yes	characteristic
114	C₂³	—	8	1	no	—
115	Q₈	—	16	2	no	—
116	D₄	—	4	2	no	—
117	D₄	—	8	1	no	—
118	C₂×C₄	—	8	1	no	—
119	C₂×C₄	—	8	2	no	—
120	C₂×C₄	—	16	2	no	—
121	C₂×C₄	C₁₉×D₁₆	1	1	yes	characteristic
122	C₂×C₄	C₁₉×D₁₆	1	1	yes	characteristic
123	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
124	C₈	—	2	1	no	—
125	C₈	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
126	C₈	C₁₉×#137	1	1	yes	characteristic
127	C₂²	—	8	2	no	—
128	C₂²	—	16	1	no	—
129	C₂²	—	16	1	no	—
130	C₂²	C₂×C₁₉×D₁₆	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	16	1	no	—
133	C₄	—	16	2	no	—
134	C₄	C₁₉×D₃₂	1	2	yes	—
135	C₄	C₁₉×#474	1	1	yes	characteristic
136	C₄	C₁₉×#510	1	1	yes	characteristic
137	C₂	—	16	2	no	—
138	C₂	C₁₉×#2224	1	1	yes	characteristic
139	C₂	C₂×C₁₉×D₃₂	1	1	yes	characteristic
140	C₂	C₁₉×#2264	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#2252

C₂

yes

maximal, characteristic, complemented

C₄×C₁₉×D₁₆

C₂

yes

maximal, characteristic

C₁₉×#1423

C₂

yes

maximal, complemented

C₁₉×C₄⋊C₃₂

C₂

yes

maximal, characteristic, complemented

C₁₉×#1415

C₂

yes

maximal

C₂×C₁₉×D₁₆

C₂²

yes

characteristic

C₁₉×#500

—

C₄×D₈×C₁₉

—

C₂×C₁₉×C₃₂

C₂²

yes

—

C₁₉×#366

—

C₁₉×#366

C₂²

yes

characteristic

C₁₉×#366

C₂²

yes

—

C₁₉×#357

—

C₄×C₁₆×C₁₉

C₂²

yes

characteristic

C₂×D₈×C₁₉

—

C₁₉×C₄⋊Q₈

—

C₄×D₄×C₁₉

—

C₁₉×D₁₆

—

C₁₉×D₁₆

Q₈

yes

complemented

C₂×C₁₆×C₁₉

D₄

yes

characteristic

C₂×C₁₆×C₁₉

C₂³

yes

characteristic

C₁₉×#107

—

C₁₉×#107

—

C₁₉×#107

—

C₁₉×#102

—

C₄×C₈×C₁₉

D₄

yes

characteristic

C₁₉×C₃₂

—

C₂×Q₈×C₁₉

—

C₂×D₄×C₁₉

—

C₂²×C₄×C₁₉

—

D₈×C₁₉

—

D₈×C₁₉

—

C₂×C₈×C₁₉

D₈

yes

characteristic

C₂×C₈×C₁₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

D₈

yes

characteristic

C₁₆×C₁₉

—

C₁₆×C₁₉

C₂×Q₈

yes

C₁₆×C₁₉

yes

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

—

Q₈×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

D₁₆

yes

characteristic

C₂×C₄×C₁₉

D₁₆

yes

characteristic

C₂×C₄×C₁₉

C₂×D₈

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

C₂²⋊Q₈

yes

C₈×C₁₉

yes

C₂×C₁₉

yes

characteristic, complemented

C₄×D₁₆

C₂×C₁₉

yes

characteristic

#1423

C₂×C₁₉

yes

complemented

C₄⋊C₃₂

C₂×C₁₉

yes

characteristic, complemented

#1415

C₂×C₁₉

yes

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

C₂×D₁₆

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

D₃₂

yes

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₂×D₁₆

C₂²×C₁₉

yes

characteristic

#500

—

C₄×D₈

—

C₂×C₃₂

C₂²×C₁₉

yes

—

#366

—

#366

C₂²×C₁₉

yes

characteristic

#366

C₂²×C₁₉

yes

—

#357

—

C₄×C₁₆

C₂²×C₁₉

yes

characteristic

C₂×C₁₉

—

C₂×C₁₉

#2224

yes

characteristic

C₂×C₁₉

C₂×D₃₂

yes

characteristic

C₂×C₁₉

#2264

yes

characteristic

C₂×D₈

—

C₄⋊Q₈

—

C₄×D₄

—

D₁₆

—

D₁₆

Q₈×C₁₉

yes

complemented

C₂×C₁₆

D₄×C₁₉

yes

characteristic

C₂×C₁₆

C₂³×C₁₉

yes

—

—

—

—

C₄×C₈

D₄×C₁₉

yes

characteristic

C₃₂

—

C₁₉

#13703

yes

characteristic, direct product factor, Sylow

C₂×Q₈

—

100

C₂×D₄

—

101

C₂²×C₄

—

102

D₈

—

103

D₈

—

104

C₂×C₈

D₈×C₁₉

yes

characteristic

105

C₂×C₈

C₂×D₄×C₁₉

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₂²⋊C₄

—

110

C₄²

D₈×C₁₉

yes

characteristic

111

C₁₆

—

112

C₁₆

C₂×Q₈×C₁₉

yes

characteristic

113

C₁₆

#41×C₁₉

yes

characteristic

114

C₂³

—

115

Q₈

—

116

D₄

—

117

D₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₁₉×D₁₆

yes

characteristic

122

C₂×C₄

C₁₉×D₁₆

yes

characteristic

123

C₂×C₄

C₂×D₈×C₁₉

yes

characteristic

124

C₈

—

125

C₈

C₁₉×C₂²⋊Q₈

yes

characteristic

126

C₈

C₁₉×#137

yes

characteristic

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

C₂×C₁₉×D₁₆

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

C₁₉×D₃₂

yes

—

135

C₄

C₁₉×#474

yes

characteristic

136

C₄

C₁₉×#510

yes

characteristic

137

C₂

—

138

C₂

C₁₉×#2224

yes

characteristic

139

C₂

C₂×C₁₉×D₃₂

yes

characteristic

140

C₂

C₁₉×#2264

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₁₉
prime radical	C₂×C₁₉×D₁₆	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₄²×C₁₉ ◁ C₄×C₈×C₁₉ ◁ C₄×C₁₆×C₁₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₉

C₂×D₁₆

commutator, Frattini, cosocle

C₂×C₁₆

C₂³×C₁₉

prime radical

C₂×C₁₉×D₁₆

C₂²

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₉ ◁ C₄²×C₁₉ ◁ C₄×C₈×C₁₉ ◁ C₄×C₁₆×C₁₉ ◁ G

N	H ≅ G/N	type
1. C₁₉×#2252	C₂	semidirect
3. C₁₉×#1423	C₂	semidirect
4. C₁₉×C₄⋊C₃₂	C₂	semidirect
19. C₁₉×D₁₆	Q₈	semidirect
43. #13703	C₁₉	direct (N × H)
57. #2252	C₂×C₁₉	semidirect
59. #1423	C₂×C₁₉	semidirect
60. C₄⋊C₃₂	C₂×C₁₉	semidirect
89. D₁₆	Q₈×C₁₉	semidirect
98. C₁₉	#13703	direct (N × H)

H ≅ G/N

type

1. C₁₉×#2252

C₂

semidirect

3. C₁₉×#1423

C₂

semidirect

4. C₁₉×C₄⋊C₃₂

C₂

semidirect

19. C₁₉×D₁₆

Q₈

semidirect

43. #13703

C₁₉

direct (N × H)

57. #2252

C₂×C₁₉

semidirect

59. #1423

C₂×C₁₉

semidirect

60. C₄⋊C₃₂

C₂×C₁₉

semidirect

89. D₁₆

Q₈×C₁₉

semidirect

98. C₁₉

#13703

direct (N × H)

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

subgroup	index
C₁₉×#2252	2
C₄×C₁₉×D₁₆	2
C₁₉×#1423	2
C₁₉×C₄⋊C₃₂	2
C₁₉×#1415	2
#13703	19

subgroup

index

C₁₉×#2252

C₄×C₁₉×D₁₆

C₁₉×#1423

C₁₉×C₄⋊C₃₂

C₁₉×#1415

#13703

Subgroups — C₁₉×#13703 #18019662

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#13703 #18019662

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13703 #18019662