Subgroups — C₅×(C₃⋊Dic₃)⋊C₃ #10587358

Order 540.

50 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₅×C₃⋊Dic₃	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×Dic₃	—	3	1	no	maximal
4	C₅×H(3,𝔽₃)	C₄	1	1	yes	characteristic, complemented, Hall
5	(C₃⋊Dic₃)⋊C₃	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃²×C₅	—	2	1	no	—
7	C₂×C₃²×C₅	D₃	1	1	yes	characteristic
8	C₂×C₃²×C₅	C₂×C₃	1	1	yes	characteristic
9	C₃×C₄×C₅	—	9	1	no	—
10	C₅×Dic₃	—	3	1	no	—
11	C₅×Dic₃	—	9	1	no	—
12	C₂×H(3,𝔽₃)	C₂×C₅	1	1	yes	characteristic
13	C₃²×C₅	—	2	1	no	—
14	C₃²×C₅	Dic₃	1	1	yes	characteristic, complemented
15	C₃²×C₅	C₃×C₄	1	1	yes	characteristic, complemented
16	C₃⋊Dic₃	C₃×C₅	1	1	yes	characteristic, complemented
17	C₃×Dic₃	—	3	1	no	—
18	C₂×C₃×C₅	—	3	1	no	—
19	C₂×C₃×C₅	—	3	1	no	—
20	C₂×C₃×C₅	—	6	1	no	—
21	C₂×C₃×C₅	C₃×D₃	1	1	yes	characteristic
22	H(3,𝔽₃)	C₄×C₅	1	1	yes	characteristic, complemented, Sylow
23	C₄×C₅	—	9	1	no	Hall
24	C₂×C₃²	—	2	1	no	—
25	C₂×C₃²	C₅×D₃	1	1	yes	characteristic
26	C₂×C₃²	C₂×C₃×C₅	1	1	yes	characteristic
27	C₃×C₅	—	3	1	no	—
28	C₃×C₅	—	3	1	no	—
29	C₃×C₅	—	6	1	no	—
30	C₃×C₅	C₃×Dic₃	1	1	yes	characteristic
31	C₃×C₄	—	9	1	no	—
32	Dic₃	—	3	1	no	—
33	Dic₃	—	9	1	no	—
34	C₂×C₅	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
35	C₃²	—	2	1	no	—
36	C₃²	C₅×Dic₃	1	1	yes	characteristic, complemented
37	C₃²	C₃×C₄×C₅	1	1	yes	characteristic, complemented
38	C₂×C₃	—	3	1	no	—
39	C₂×C₃	—	3	1	no	—
40	C₂×C₃	—	6	1	no	—
41	C₂×C₃	C₃×C₅×D₃	1	1	yes	characteristic
42	C₅	(C₃⋊Dic₃)⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
43	C₄	—	9	1	no	Sylow
44	C₃	—	3	1	no	—
45	C₃	—	3	1	no	—
46	C₃	—	6	1	no	—
47	C₃	C₃×C₅×Dic₃	1	1	yes	characteristic
48	C₂	C₅×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₅	(C₃⋊D₃)⋊C₃
commutator, hypocenter	C₃²	C₃×C₄×C₅
Frattini	C₂×C₃	C₃×C₅×D₃
Fitting, prime radical	C₂×C₅×H(3,𝔽₃)	C₂
socle	C₂×C₃×C₅	C₃×D₃
cosocle	C₂×C₃²	C₂×C₃×C₅
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂×C₅

Complements

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

N	H ≅ G/N	type
2. C₅×C₃⋊Dic₃	C₃	semidirect
4. C₅×H(3,𝔽₃)	C₄	semidirect
5. (C₃⋊Dic₃)⋊C₃	C₅	direct (N × H)
14. C₃²×C₅	Dic₃	semidirect
15. C₃²×C₅	C₃×C₄	semidirect
16. C₃⋊Dic₃	C₃×C₅	semidirect
22. H(3,𝔽₃)	C₄×C₅	semidirect
36. C₃²	C₅×Dic₃	semidirect
37. C₃²	C₃×C₄×C₅	semidirect
42. C₅	(C₃⋊Dic₃)⋊C₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×H(3,𝔽₃)	2
C₅×C₃⋊Dic₃	3
C₃×C₅×Dic₃	3
(C₃⋊Dic₃)⋊C₃	5

Subgroups — C5×(C3⋊Dic3)⋊C3 #10587358

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×(C₃⋊Dic₃)⋊C₃ #10587358