Subgroups — D₃×H(5,𝔽₃) #13091707

Order 1458.

2573 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×H(5,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₂×H(5,𝔽₃)	—	3	1	no	maximal
3	C₃×D₃×H(3,𝔽₃)	C₃	1	40	yes	maximal, complemented
4	H(5,𝔽₃)	—	2	40	no	—
5	H(5,𝔽₃)	D₃	1	1	yes	characteristic, direct product factor
6	C₃²×H(3,𝔽₃)	C₂×C₃	1	40	yes	complemented
7	C₃³×D₃	C₃²	1	40	yes	complemented
8	C₂×C₃×H(3,𝔽₃)	—	3	40	no	—
9	D₃×H(3,𝔽₃)	C₃²	1	90	yes	complemented
10	C₃⁴	C₂×C₃²	1	40	yes	complemented
11	C₃×H(3,𝔽₃)	—	2	160	no	—
12	C₃×H(3,𝔽₃)	—	2	360	no	—
13	C₃×H(3,𝔽₃)	C₃×D₃	1	40	yes	complemented
14	C₃×H(3,𝔽₃)	C₂×C₃²	1	90	yes	complemented
15	C₂×C₃³	—	3	40	no	—
16	C₃²×D₃	—	9	40	no	—
17	C₃²×D₃	C₃³	1	40	yes	—
18	C₂×H(3,𝔽₃)	—	3	90	no	—
19	C₃³	—	2	160	no	—
20	C₃³	—	9	40	no	—
21	C₃³	—	18	40	no	—
22	C₃³	C₃²×D₃	1	40	yes	complemented
23	C₃³	C₂×C₃³	1	40	yes	—
24	H(3,𝔽₃)	—	2	360	no	—
25	H(3,𝔽₃)	C₃²×D₃	1	90	yes	complemented
26	C₂×C₃²	—	3	40	no	—
27	C₂×C₃²	—	27	40	no	—
28	C₃×D₃	—	3	40	no	—
29	C₃×D₃	C₃⁴	1	1	yes	characteristic
30	C₃²	—	2	40	no	—
31	C₃²	—	3	40	no	—
32	C₃²	—	6	40	no	—
33	C₃²	—	9	40	no	—
34	C₃²	—	18	160	no	—
35	C₃²	C₃³×D₃	1	40	yes	—
36	C₃²	C₂×C₃⁴	1	1	yes	characteristic
37	C₂×C₃	—	3	1	no	—
38	C₂×C₃	—	9	40	no	—
39	D₃	H(5,𝔽₃)	1	1	yes	characteristic, direct product factor
40	C₃	—	2	1	no	—
41	C₃	—	3	40	no	—
42	C₃	—	6	40	no	—
43	C₃	C₂×H(5,𝔽₃)	1	1	yes	characteristic, complemented
44	C₃	C₃⁴×D₃	1	1	yes	characteristic
45	C₂	—	3	1	no	Sylow

	subgroup	G/H
center, Frattini	C₃	C₃⁴×D₃
commutator, socle, cosocle	C₃²	C₂×C₃⁴
Fitting	C₃×H(5,𝔽₃)	C₂
hypercenter	H(5,𝔽₃)	D₃
hypocenter	C₃	C₂×H(5,𝔽₃)
LCS	G ▷ C₃² ▷ C₃
UCS	C₁ ◁ C₃ ◁ H(5,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃×H(5,𝔽₃)	C₂	semidirect
3. C₃×D₃×H(3,𝔽₃)	C₃	semidirect
5. H(5,𝔽₃)	D₃	direct (N × H)
6. C₃²×H(3,𝔽₃)	C₂×C₃	semidirect
7. C₃³×D₃	C₃²	semidirect
9. D₃×H(3,𝔽₃)	C₃²	semidirect
10. C₃⁴	C₂×C₃²	semidirect
13. C₃×H(3,𝔽₃)	C₃×D₃	semidirect
14. C₃×H(3,𝔽₃)	C₂×C₃²	semidirect
22. C₃³	C₃²×D₃	semidirect
25. H(3,𝔽₃)	C₃²×D₃	semidirect
39. D₃	H(5,𝔽₃)	direct (N × H)
43. C₃	C₂×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
C₃×H(5,𝔽₃)	2
C₂×H(5,𝔽₃)	3
C₃×D₃×H(3,𝔽₃)	3

Subgroups — D3×H(5,𝔽3) #13091707

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×H(5,𝔽₃) #13091707