Subgroups — C₁₉⋊C₃×H(5,𝔽₃) #28356659

Order 13851.

6776 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉⋊H(5,𝔽₃)	C₃	1	80	yes	maximal, complemented
2	C₃×H(3,𝔽₃)×C₁₉⋊C₃	C₃	1	40	yes	maximal, complemented
3	C₁₉×H(5,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃³×C₁₉⋊C₃	C₃²	1	40	yes	complemented
5	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	320	yes	complemented
6	H(3,𝔽₃)×C₁₉⋊C₃	C₃²	1	90	yes	complemented
7	H(3,𝔽₃)×C₁₉⋊C₃	C₃²	1	720	yes	complemented
8	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	40	yes	complemented
9	C₃×H(5,𝔽₃)	—	19	1	no	maximal, Sylow
10	C₃³×C₁₉	C₃³	1	40	yes	complemented
11	C₃²×C₁₉⋊C₃	—	9	120	no	—
12	C₃²×C₁₉⋊C₃	C₃³	1	40	yes	—
13	C₃²×C₁₉⋊C₃	C₃³	1	320	yes	complemented
14	C₁₉×H(3,𝔽₃)	C₃³	1	90	yes	complemented
15	C₃²⋊(C₁₉⋊C₃)	C₃³	1	720	yes	complemented
16	H(5,𝔽₃)	—	19	80	no	—
17	H(5,𝔽₃)	C₁₉⋊C₃	1	1	yes	characteristic, direct product factor
18	C₃²×H(3,𝔽₃)	—	19	40	no	—
19	C₃²×C₁₉	—	9	40	no	—
20	C₃²×C₁₉	C₃⁴	1	40	yes	—
21	C₃×C₁₉⋊C₃	—	3	120	no	—
22	C₃×C₁₉⋊C₃	—	9	320	no	—
23	C₃×C₁₉⋊C₃	C₃⁴	1	80	yes	—
24	C₃×C₁₉⋊C₃	C₃⁴	1	1	yes	characteristic
25	C₃⁴	—	19	40	no	—
26	C₃×H(3,𝔽₃)	—	19	320	no	—
27	C₃×H(3,𝔽₃)	—	19	90	no	—
28	C₃×H(3,𝔽₃)	—	19	720	no	—
29	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	40	yes	complemented
30	C₃×C₁₉	—	3	40	no	—
31	C₃×C₁₉	C₃⁵	1	1	yes	characteristic
32	C₁₉⋊C₃	—	3	80	no	—
33	C₁₉⋊C₃	H(5,𝔽₃)	1	3	yes	direct product factor
34	C₃³	—	19	40	no	—
35	C₃³	—	19	320	no	—
36	C₃³	—	171	120	no	—
37	C₃³	C₃²×C₁₉⋊C₃	1	40	yes	complemented
38	H(3,𝔽₃)	—	19	720	no	—
39	H(3,𝔽₃)	C₃²×C₁₉⋊C₃	1	90	yes	complemented
40	C₁₉	C₃×H(5,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
41	C₃²	—	9	40	no	—
42	C₃²	—	19	1	no	—
43	C₃²	—	19	80	no	—
44	C₃²	—	57	120	no	—
45	C₃²	—	171	320	no	—
46	C₃²	C₃³×C₁₉⋊C₃	1	40	yes	—
47	C₃	—	3	40	no	—
48	C₃	—	19	3	no	—
49	C₃	—	57	80	no	—
50	C₃	C₃⁴×C₁₉⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃	C₃⁴×C₁₉⋊C₃
commutator, socle, cosocle	C₃×C₁₉	C₃⁵
Fitting	C₁₉×H(5,𝔽₃)	C₃
hypercenter	H(5,𝔽₃)	C₁₉⋊C₃
hypocenter	C₁₉	C₃×H(5,𝔽₃)
LCS	G ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ H(5,𝔽₃)

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₁₉⋊H(5,𝔽₃)	C₃	semidirect
2. C₃×H(3,𝔽₃)×C₁₉⋊C₃	C₃	semidirect
3. C₁₉×H(5,𝔽₃)	C₃	semidirect
4. C₃³×C₁₉⋊C₃	C₃²	semidirect
5. C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	semidirect
6. H(3,𝔽₃)×C₁₉⋊C₃	C₃²	semidirect
7. H(3,𝔽₃)×C₁₉⋊C₃	C₃²	semidirect
8. C₃×C₁₉×H(3,𝔽₃)	C₃²	semidirect
10. C₃³×C₁₉	C₃³	semidirect
13. C₃²×C₁₉⋊C₃	C₃³	semidirect
14. C₁₉×H(3,𝔽₃)	C₃³	semidirect
15. C₃²⋊(C₁₉⋊C₃)	C₃³	semidirect
17. H(5,𝔽₃)	C₁₉⋊C₃	direct (N × H)
29. C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	semidirect
33. C₁₉⋊C₃	H(5,𝔽₃)	direct (N × H)
37. C₃³	C₃²×C₁₉⋊C₃	semidirect
39. H(3,𝔽₃)	C₃²×C₁₉⋊C₃	semidirect
40. C₁₉	C₃×H(5,𝔽₃)	semidirect

Maximal subgroups

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

subgroup	index
C₁₉⋊H(5,𝔽₃)	3
C₃×H(3,𝔽₃)×C₁₉⋊C₃	3
C₁₉×H(5,𝔽₃)	3
C₃×H(5,𝔽₃)	19

Subgroups — C19⋊C3×H(5,𝔽3) #28356659

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉⋊C₃×H(5,𝔽₃) #28356659