Group #28352813 — subgroups

Subgroups — C₁₉×#10624617 #28352813

Order 13851.

260 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 86 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×#6891	C₃	1	1	yes	maximal, characteristic
2	C₁₉×#6891	C₃	1	1	yes	maximal, characteristic
3	C₃×C₁₉×C₃²⋊C₉	C₃	1	2	yes	maximal
4	C₁₉×#6861	C₃	1	9	yes	maximal, complemented
5	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
6	C₃×C₁₉×3₋¹⁺²	—	3	1	no	—
7	C₃×C₁₉×3₋¹⁺²	—	3	1	no	—
8	C₃²×C₉×C₁₉	—	3	2	no	—
9	C₁₉×C₃²⋊C₉	—	3	2	no	—
10	C₁₉×C₃²⋊C₉	—	3	2	no	—
11	C₁₉×C₃²⋊C₉	—	3	2	no	—
12	C₁₉×C₃²⋊C₉	—	3	2	no	—
13	C₁₉×C₃²⋊C₉	C₃²	1	6	yes	—
14	C₁₉×C₃²⋊C₉	C₃²	1	3	yes	—
15	C₁₉×C₃²⋊C₉	C₃²	1	3	yes	—
16	#10624617	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₃³×C₁₉	—	3	6	no	—
18	C₃³×C₁₉	—	9	1	no	—
19	C₃³×C₁₉	—	9	1	no	—
20	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
21	C₃³×C₁₉	H(3,𝔽₃)	1	2	yes	—
22	C₃³×C₁₉	C₃³	1	1	yes	characteristic
23	C₁₉×3₋¹⁺²	—	9	3	no	—
24	C₁₉×3₋¹⁺²	—	9	3	no	—
25	C₃×C₉×C₁₉	—	3	6	no	—
26	C₃×C₉×C₁₉	—	3	3	no	—
27	C₃×C₉×C₁₉	—	3	3	no	—
28	C₃×C₉×C₁₉	—	9	6	no	—
29	#6891	C₃×C₁₉	1	1	yes	characteristic
30	#6891	C₃×C₁₉	1	1	yes	characteristic
31	C₃×C₃²⋊C₉	C₃×C₁₉	1	2	yes	—
32	#6861	C₃×C₁₉	1	9	yes	complemented
33	C₃²×C₁₉	—	3	2	no	—
34	C₃²×C₁₉	—	3	2	no	—
35	C₃²×C₁₉	—	3	2	no	—
36	C₃²×C₁₉	—	3	2	no	—
37	C₃²×C₁₉	—	3	2	no	—
38	C₃²×C₁₉	—	3	1	no	—
39	C₃²×C₁₉	—	3	1	no	—
40	C₃²×C₁₉	—	3	1	no	—
41	C₃²×C₁₉	—	3	1	no	—
42	C₃²×C₁₉	—	9	9	no	—
43	C₃²×C₁₉	#733	1	6	yes	—
44	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
45	C₉×C₁₉	—	9	3	no	—
46	C₉×C₁₉	—	9	6	no	—
47	C₉×C₁₉	—	9	3	no	—
48	C₃⁴	C₃²×C₁₉	1	1	yes	characteristic
49	C₃×3₋¹⁺²	—	3	1	no	—
50	C₃×3₋¹⁺²	—	3	1	no	—
51	C₃²×C₉	—	3	2	no	—
52	C₃²⋊C₉	—	3	2	no	—
53	C₃²⋊C₉	—	3	2	no	—
54	C₃²⋊C₉	—	3	2	no	—
55	C₃²⋊C₉	—	3	2	no	—
56	C₃²⋊C₉	C₃²×C₁₉	1	6	yes	—
57	C₃²⋊C₉	C₃²×C₁₉	1	3	yes	—
58	C₃²⋊C₉	C₃²×C₁₉	1	3	yes	—
59	C₃×C₁₉	—	3	6	no	—
60	C₃×C₁₉	—	9	1	no	—
61	C₃×C₁₉	—	9	1	no	—
62	C₃×C₁₉	C₃×#733	1	2	yes	—
63	C₃×C₁₉	#6910	1	1	yes	characteristic
64	C₃×C₁₉	#6911	1	1	yes	characteristic
65	C₃³	—	3	6	no	—
66	C₃³	—	9	1	no	—
67	C₃³	—	9	1	no	—
68	C₃³	C₁₉×H(3,𝔽₃)	1	1	yes	characteristic
69	C₃³	C₁₉×H(3,𝔽₃)	1	2	yes	—
70	C₃³	C₃³×C₁₉	1	1	yes	characteristic
71	3₋¹⁺²	—	9	3	no	—
72	3₋¹⁺²	—	9	3	no	—
73	C₃×C₉	—	3	6	no	—
74	C₃×C₉	—	3	3	no	—
75	C₃×C₉	—	3	3	no	—
76	C₃×C₉	—	9	6	no	—
77	C₁₉	#10624617	1	1	yes	characteristic, direct product factor, Sylow
78	C₃²	—	3	2	no	—
79	C₃²	—	3	2	no	—
80	C₃²	—	3	2	no	—
81	C₃²	—	3	2	no	—
82	C₃²	—	3	2	no	—
83	C₃²	—	3	1	no	—
84	C₃²	—	3	1	no	—
85	C₃²	—	3	1	no	—
86	C₃²	—	3	1	no	—
87	C₃²	—	9	9	no	—
88	C₃²	C₁₉×#733	1	6	yes	—
89	C₃²	C₃×C₁₉×H(3,𝔽₃)	1	1	yes	characteristic
90	C₉	—	9	3	no	—
91	C₉	—	9	6	no	—
92	C₉	—	9	3	no	—
93	C₃	—	3	6	no	—
94	C₃	—	9	1	no	—
95	C₃	—	9	1	no	—
96	C₃	C₃×C₁₉×#733	1	2	yes	—
97	C₃	C₁₉×#6910	1	1	yes	characteristic
98	C₃	C₁₉×#6911	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#6891

C₃

yes

maximal, characteristic

C₁₉×#6891

C₃

yes

maximal, characteristic

C₃×C₁₉×C₃²⋊C₉

C₃

yes

maximal

C₁₉×#6861

C₃

yes

maximal, complemented

C₃⁴×C₁₉

C₃²

yes

characteristic

C₃×C₁₉×3₋¹⁺²

—

C₃×C₁₉×3₋¹⁺²

—

C₃²×C₉×C₁₉

—

C₁₉×C₃²⋊C₉

—

C₁₉×C₃²⋊C₉

—

C₁₉×C₃²⋊C₉

—

C₁₉×C₃²⋊C₉

—

C₁₉×C₃²⋊C₉

C₃²

yes

—

C₁₉×C₃²⋊C₉

C₃²

yes

—

C₁₉×C₃²⋊C₉

C₃²

yes

—

#10624617

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

H(3,𝔽₃)

yes

characteristic

C₃³×C₁₉

H(3,𝔽₃)

yes

—

C₃³×C₁₉

C₃³

yes

characteristic

C₁₉×3₋¹⁺²

—

C₁₉×3₋¹⁺²

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

#6891

C₃×C₁₉

yes

characteristic

#6891

C₃×C₁₉

yes

characteristic

C₃×C₃²⋊C₉

C₃×C₁₉

yes

—

#6861

C₃×C₁₉

yes

complemented

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

#733

yes

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₁₉

—

C₉×C₁₉

—

C₉×C₁₉

—

C₃⁴

C₃²×C₁₉

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃²×C₁₉

yes

—

C₃²⋊C₉

C₃²×C₁₉

yes

—

C₃²⋊C₉

C₃²×C₁₉

yes

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

C₃×#733

yes

—

C₃×C₁₉

yes

C₃×C₁₉

yes

C₃³

—

C₃³

—

C₃³

—

C₃³

C₁₉×H(3,𝔽₃)

yes

characteristic

C₃³

C₁₉×H(3,𝔽₃)

yes

—

C₃³

C₃³×C₁₉

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₁₉

#10624617

yes

characteristic, direct product factor, Sylow

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₁₉×#733

yes

—

C₃²

C₃×C₁₉×H(3,𝔽₃)

yes

characteristic

C₉

—

C₉

—

C₉

—

C₃

—

C₃

—

C₃

—

C₃

C₃×C₁₉×#733

yes

—

C₃

C₁₉×#6910

yes

characteristic

C₃

C₁₉×#6911

yes

characteristic

	subgroup	G/H
center, socle	C₃²×C₁₉	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₁₉
prime radical	C₃⁴×C₁₉	C₃²
LCS	G ▷ C₃³ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₁₉ ◁ C₃⁴×C₁₉ ◁ G

subgroup

G/H

center, socle

C₃²×C₁₉

C₃×H(3,𝔽₃)

commutator, Frattini, cosocle

C₃³

C₃³×C₁₉

prime radical

C₃⁴×C₁₉

C₃²

LCS

G ▷ C₃³ ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃²×C₁₉ ◁ C₃⁴×C₁₉ ◁ G

N	H ≅ G/N	type
4. C₁₉×#6861	C₃	semidirect
16. #10624617	C₁₉	direct (N × H)
32. #6861	C₃×C₁₉	semidirect
77. C₁₉	#10624617	direct (N × H)

H ≅ G/N

type

4. C₁₉×#6861

C₃

semidirect

16. #10624617

C₁₉

direct (N × H)

32. #6861

C₃×C₁₉

semidirect

77. C₁₉

#10624617

direct (N × H)

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

subgroup	index
C₁₉×#6891	3
C₃×C₁₉×C₃²⋊C₉	3
C₁₉×#6861	3
#10624617	19

subgroup

index

C₁₉×#6891

C₃×C₁₉×C₃²⋊C₉

C₁₉×#6861

#10624617

Subgroups — C₁₉×#10624617 #28352813

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#10624617 #28352813

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#10624617 #28352813