Group #18018470 — subgroups

Subgroups — C₁₉×#12511 #18018470

Order 4864.

260 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 94 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×#2241	C₂	1	2	yes	maximal, complemented
2	C₁₉×#2240	C₂	1	2	yes	maximal
3	C₁₉×#1710	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1709	C₂	1	1	yes	maximal, characteristic
5	C₈×C₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#500	C₂²	1	2	yes	—
7	C₁₉×#494	C₂²	1	1	yes	characteristic
8	C₁₉×#493	C₂²	1	1	yes	characteristic
9	C₁₉×#487	—	2	2	no	—
10	C₁₉×#486	—	2	2	no	—
11	C₁₉×#358	—	2	4	no	—
12	C₁₉×#357	—	2	4	no	—
13	C₄×C₁₆×C₁₉	C₂²	1	2	yes	—
14	C₈²×C₁₉	C₂²	1	1	yes	characteristic
15	C₂×Q₁₆×C₁₉	—	2	2	no	—
16	C₂×D₈×C₁₉	—	2	2	no	—
17	C₁₉×C₄⋊Q₈	—	2	1	no	—
18	C₁₉×C₄⋊Q₈	—	2	2	no	—
19	C₁₉×#127	—	2	1	no	—
20	C₂×C₁₆×C₁₉	D₄	1	4	yes	—
21	C₁₉×#107	—	2	2	no	—
22	C₁₉×#107	—	2	2	no	—
23	C₁₉×#103	—	4	4	no	—
24	C₁₉×#102	—	4	4	no	—
25	C₄×C₈×C₁₉	D₄	1	2	yes	—
26	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
27	C₂×Q₈×C₁₉	—	4	1	no	—
28	C₂×Q₈×C₁₉	—	4	2	no	—
29	C₂×D₄×C₁₉	—	4	1	no	—
30	C₂×D₄×C₁₉	—	4	2	no	—
31	Q₁₆×C₁₉	—	4	4	no	—
32	D₈×C₁₉	—	4	4	no	—
33	C₂×C₈×C₁₉	D₈	1	4	yes	—
34	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
35	C₄⋊C₄×C₁₉	—	4	2	no	—
36	C₄⋊C₄×C₁₉	—	4	2	no	—
37	C₄⋊C₄×C₁₉	—	4	2	no	—
38	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
39	C₁₆×C₁₉	—	2	4	no	—
40	#12511	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂³×C₁₉	—	8	1	no	—
42	Q₈×C₁₉	—	8	2	no	—
43	Q₈×C₁₉	—	8	4	no	—
44	D₄×C₁₉	—	8	2	no	—
45	D₄×C₁₉	—	8	4	no	—
46	C₂×C₄×C₁₉	—	8	1	no	—
47	C₂×C₄×C₁₉	—	8	2	no	—
48	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
49	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
50	C₈×C₁₉	—	2	4	no	—
51	C₈×C₁₉	#135	1	4	yes	—
52	#2241	C₂×C₁₉	1	2	yes	complemented
53	#2240	C₂×C₁₉	1	2	yes	—
54	#1710	C₂×C₁₉	1	1	yes	characteristic, complemented
55	#1709	C₂×C₁₉	1	1	yes	characteristic
56	C₈×C₁₆	C₂×C₁₉	1	1	yes	characteristic, complemented
57	C₂²×C₁₉	—	16	1	no	—
58	C₂²×C₁₉	—	16	2	no	—
59	C₂²×C₁₉	#493	1	1	yes	characteristic
60	C₄×C₁₉	—	16	1	no	—
61	C₄×C₁₉	—	16	2	no	—
62	C₄×C₁₉	#495	1	2	yes	—
63	C₄×C₁₉	#508	1	4	yes	—
64	#500	C₂²×C₁₉	1	2	yes	—
65	#494	C₂²×C₁₉	1	1	yes	characteristic
66	#493	C₂²×C₁₉	1	1	yes	characteristic
67	#487	—	2	2	no	—
68	#486	—	2	2	no	—
69	#358	—	2	4	no	—
70	#357	—	2	4	no	—
71	C₄×C₁₆	C₂²×C₁₉	1	2	yes	—
72	C₈²	C₂²×C₁₉	1	1	yes	characteristic
73	C₂×C₁₉	—	32	1	no	—
74	C₂×C₁₉	#1709	1	1	yes	characteristic
75	C₂×C₁₉	#2249	1	2	yes	—
76	C₂×Q₁₆	—	2	2	no	—
77	C₂×D₈	—	2	2	no	—
78	C₄⋊Q₈	—	2	1	no	—
79	C₄⋊Q₈	—	2	2	no	—
80	#127	—	2	1	no	—
81	C₂×C₁₆	D₄×C₁₉	1	4	yes	—
82	#107	—	2	2	no	—
83	#107	—	2	2	no	—
84	#103	—	4	4	no	—
85	#102	—	4	4	no	—
86	C₄×C₈	D₄×C₁₉	1	2	yes	—
87	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
88	C₁₉	#12511	1	1	yes	characteristic, direct product factor, Sylow
89	C₂×Q₈	—	4	1	no	—
90	C₂×Q₈	—	4	2	no	—
91	C₂×D₄	—	4	1	no	—
92	C₂×D₄	—	4	2	no	—
93	Q₁₆	—	4	4	no	—
94	D₈	—	4	4	no	—
95	C₂×C₈	D₈×C₁₉	1	4	yes	—
96	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	4	2	no	—
99	C₄⋊C₄	—	4	2	no	—
100	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
101	C₁₆	—	2	4	no	—
102	C₂³	—	8	1	no	—
103	Q₈	—	8	2	no	—
104	Q₈	—	8	4	no	—
105	D₄	—	8	2	no	—
106	D₄	—	8	4	no	—
107	C₂×C₄	—	8	1	no	—
108	C₂×C₄	—	8	2	no	—
109	C₂×C₄	C₁₉×#127	1	1	yes	characteristic
110	C₂×C₄	C₂×D₈×C₁₉	1	2	yes	—
111	C₈	—	2	4	no	—
112	C₈	C₁₉×#135	1	4	yes	—
113	C₂²	—	16	1	no	—
114	C₂²	—	16	2	no	—
115	C₂²	C₁₉×#493	1	1	yes	characteristic
116	C₄	—	16	1	no	—
117	C₄	—	16	2	no	—
118	C₄	C₁₉×#495	1	2	yes	—
119	C₄	C₁₉×#508	1	4	yes	—
120	C₂	—	32	1	no	—
121	C₂	C₁₉×#1709	1	1	yes	characteristic
122	C₂	C₁₉×#2249	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#2241

C₂

yes

maximal, complemented

C₁₉×#2240

C₂

yes

maximal

C₁₉×#1710

C₂

yes

maximal, characteristic, complemented

C₁₉×#1709

C₂

yes

maximal, characteristic

C₈×C₁₆×C₁₉

C₂

yes

maximal, characteristic, complemented

C₁₉×#500

C₂²

yes

—

C₁₉×#494

C₂²

yes

characteristic

C₁₉×#493

C₂²

yes

characteristic

C₁₉×#487

—

C₁₉×#486

—

C₁₉×#358

—

C₁₉×#357

—

C₄×C₁₆×C₁₉

C₂²

yes

—

C₈²×C₁₉

C₂²

yes

characteristic

C₂×Q₁₆×C₁₉

—

C₂×D₈×C₁₉

—

C₁₉×C₄⋊Q₈

—

C₁₉×C₄⋊Q₈

—

C₁₉×#127

—

C₂×C₁₆×C₁₉

D₄

yes

—

C₁₉×#107

—

C₁₉×#107

—

C₁₉×#103

—

C₁₉×#102

—

C₄×C₈×C₁₉

D₄

yes

—

C₄×C₈×C₁₉

C₂³

yes

characteristic

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

Q₁₆×C₁₉

—

D₈×C₁₉

—

C₂×C₈×C₁₉

D₈

yes

—

C₂×C₈×C₁₉

C₂×D₄

yes

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₁₆×C₁₉

—

#12511

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

—

Q₈×C₁₉

—

Q₈×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#127

yes

characteristic

C₂×C₄×C₁₉

C₂×D₈

yes

—

C₈×C₁₉

—

C₈×C₁₉

#135

yes

—

#2241

C₂×C₁₉

yes

complemented

#2240

C₂×C₁₉

yes

—

#1710

C₂×C₁₉

yes

characteristic, complemented

#1709

C₂×C₁₉

yes

characteristic

C₈×C₁₆

C₂×C₁₉

yes

characteristic, complemented

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#493

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#495

yes

—

C₄×C₁₉

#508

yes

—

#500

C₂²×C₁₉

yes

—

#494

C₂²×C₁₉

yes

characteristic

#493

C₂²×C₁₉

yes

—

—

—

—

C₄×C₁₆

C₂²×C₁₉

yes

—

C₈²

C₂²×C₁₉

yes

characteristic

C₂×C₁₉

—

C₂×C₁₉

#1709

yes

characteristic

C₂×C₁₉

#2249

yes

—

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

D₄×C₁₉

yes

—

—

—

—

—

C₄×C₈

D₄×C₁₉

yes

—

C₄×C₈

C₂³×C₁₉

yes

characteristic

C₁₉

#12511

yes

characteristic, direct product factor, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

D₈

—

C₂×C₈

D₈×C₁₉

yes

—

C₂×C₈

C₂×D₄×C₁₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄²

C₂×D₄×C₁₉

yes

characteristic

101

C₁₆

—

102

C₂³

—

103

Q₈

—

104

Q₈

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₁₉×#127

yes

characteristic

110

C₂×C₄

C₂×D₈×C₁₉

yes

—

111

C₈

—

112

C₈

C₁₉×#135

yes

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₁₉×#493

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

C₁₉×#495

yes

—

119

C₄

C₁₉×#508

yes

—

120

C₂

—

121

C₂

C₁₉×#1709

yes

characteristic

122

C₂

C₁₉×#2249

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₉	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₉
prime radical	C₁₉×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₄²×C₁₉ ◁ C₈²×C₁₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₉

#493

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₁₉

prime radical

C₁₉×#493

C₂²

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₉ ◁ C₄²×C₁₉ ◁ C₈²×C₁₉ ◁ G

N	H ≅ G/N	type
1. C₁₉×#2241	C₂	semidirect
3. C₁₉×#1710	C₂	semidirect
5. C₈×C₁₆×C₁₉	C₂	semidirect
40. #12511	C₁₉	direct (N × H)
52. #2241	C₂×C₁₉	semidirect
54. #1710	C₂×C₁₉	semidirect
56. C₈×C₁₆	C₂×C₁₉	semidirect
88. C₁₉	#12511	direct (N × H)

H ≅ G/N

type

1. C₁₉×#2241

C₂

semidirect

3. C₁₉×#1710

C₂

semidirect

5. C₈×C₁₆×C₁₉

C₂

semidirect

40. #12511

C₁₉

direct (N × H)

52. #2241

C₂×C₁₉

semidirect

54. #1710

C₂×C₁₉

semidirect

56. C₈×C₁₆

C₂×C₁₉

semidirect

88. C₁₉

#12511

direct (N × H)

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

subgroup	index
C₁₉×#2241	2
C₁₉×#2240	2
C₁₉×#1710	2
C₁₉×#1709	2
C₈×C₁₆×C₁₉	2
#12511	19

subgroup

index

C₁₉×#2241

C₁₉×#2240

C₁₉×#1710

C₁₉×#1709

C₈×C₁₆×C₁₉

#12511

Subgroups — C₁₉×#12511 #18018470

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#12511 #18018470

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12511 #18018470