Subgroups — C₃×C₅×C₃⋊(C₃⋊Dic₃) #13108507

Order 1620.

648 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃⁴×C₅	C₂	1	1	yes	maximal, characteristic
2	C₅×C₃⋊(C₃⋊Dic₃)	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×C₅×C₃⋊Dic₃	—	3	13	no	maximal
4	C₃⁴×C₅	C₄	1	1	yes	characteristic, complemented, Hall
5	C₃×C₃⋊(C₃⋊Dic₃)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃³×C₅	—	2	13	no	—
7	C₂×C₃³×C₅	D₃	1	13	yes	—
8	C₂×C₃³×C₅	C₂×C₃	1	1	yes	characteristic
9	C₅×C₃⋊Dic₃	—	3	13	no	—
10	C₃×C₅×Dic₃	—	9	13	no	—
11	C₂×C₃⁴	C₂×C₅	1	1	yes	characteristic
12	C₃³×C₅	—	2	13	no	—
13	C₃³×C₅	Dic₃	1	13	yes	complemented
14	C₃³×C₅	C₃×C₄	1	1	yes	characteristic, complemented
15	C₃⋊(C₃⋊Dic₃)	C₃×C₅	1	1	yes	characteristic, direct product factor
16	C₃×C₃⋊Dic₃	—	3	13	no	—
17	C₂×C₃²×C₅	—	2	52	no	—
18	C₂×C₃²×C₅	C₃×D₃	1	13	yes	—
19	C₂×C₃²×C₅	C₃⋊D₃	1	13	yes	—
20	C₃⁴	C₄×C₅	1	1	yes	characteristic, complemented, Sylow
21	C₃×C₄×C₅	—	27	1	no	—
22	C₅×Dic₃	—	9	13	no	—
23	C₂×C₃³	—	2	13	no	—
24	C₂×C₃³	C₅×D₃	1	13	yes	—
25	C₂×C₃³	C₂×C₃×C₅	1	1	yes	characteristic
26	C₃²×C₅	—	2	52	no	—
27	C₃²×C₅	C₃×Dic₃	1	13	yes	complemented
28	C₃²×C₅	C₃⋊Dic₃	1	13	yes	complemented
29	C₃⋊Dic₃	—	3	13	no	—
30	C₃×Dic₃	—	9	13	no	—
31	C₂×C₃×C₅	—	2	13	no	—
32	C₂×C₃×C₅	C₃×C₃⋊D₃	1	13	yes	—
33	C₂×C₃×C₅	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
34	C₃³	—	2	13	no	—
35	C₃³	C₅×Dic₃	1	13	yes	complemented
36	C₃³	C₃×C₄×C₅	1	1	yes	characteristic, complemented
37	C₄×C₅	—	27	1	no	Hall
38	C₂×C₃²	—	2	52	no	—
39	C₂×C₃²	C₃×C₅×D₃	1	13	yes	—
40	C₂×C₃²	C₅×C₃⋊D₃	1	13	yes	—
41	C₃×C₅	—	2	13	no	—
42	C₃×C₅	C₃×C₃⋊Dic₃	1	13	yes	complemented
43	C₃×C₅	C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic, direct product factor
44	C₃×C₄	—	27	1	no	—
45	Dic₃	—	9	13	no	—
46	C₂×C₅	C₃×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
47	C₃²	—	2	52	no	—
48	C₃²	C₃×C₅×Dic₃	1	13	yes	complemented
49	C₃²	C₅×C₃⋊Dic₃	1	13	yes	complemented
50	C₂×C₃	—	2	13	no	—
51	C₂×C₃	C₃×C₅×C₃⋊D₃	1	13	yes	—
52	C₂×C₃	C₅×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
53	C₅	C₃×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic, direct product factor, Sylow
54	C₄	—	27	1	no	Sylow
55	C₃	—	2	13	no	—
56	C₃	C₃×C₅×C₃⋊Dic₃	1	13	yes	complemented
57	C₃	C₅×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic, direct product factor
58	C₂	C₃×C₅×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃×C₅	C₃⋊(C₃⋊D₃)
commutator, hypocenter	C₃³	C₃×C₄×C₅
Frattini	C₂	C₃×C₅×C₃⋊(C₃⋊D₃)
Fitting, socle, prime radical	C₂×C₃⁴×C₅	C₂
cosocle	C₂×C₃³	C₂×C₃×C₅
LCS	G ▷ C₃³
UCS	C₁ ◁ C₂×C₃×C₅

Complements

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

N	H ≅ G/N	type
2. C₅×C₃⋊(C₃⋊Dic₃)	C₃	direct (N × H)
4. C₃⁴×C₅	C₄	semidirect
5. C₃×C₃⋊(C₃⋊Dic₃)	C₅	direct (N × H)
13. C₃³×C₅	Dic₃	semidirect
14. C₃³×C₅	C₃×C₄	semidirect
15. C₃⋊(C₃⋊Dic₃)	C₃×C₅	direct (N × H)
20. C₃⁴	C₄×C₅	semidirect
27. C₃²×C₅	C₃×Dic₃	semidirect
28. C₃²×C₅	C₃⋊Dic₃	semidirect
35. C₃³	C₅×Dic₃	semidirect
36. C₃³	C₃×C₄×C₅	semidirect
42. C₃×C₅	C₃×C₃⋊Dic₃	semidirect
43. C₃×C₅	C₃⋊(C₃⋊Dic₃)	direct (N × H)
48. C₃²	C₃×C₅×Dic₃	semidirect
49. C₃²	C₅×C₃⋊Dic₃	semidirect
53. C₅	C₃×C₃⋊(C₃⋊Dic₃)	direct (N × H)
56. C₃	C₃×C₅×C₃⋊Dic₃	semidirect
57. C₃	C₅×C₃⋊(C₃⋊Dic₃)	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×C₃⁴×C₅	2
C₅×C₃⋊(C₃⋊Dic₃)	3
C₃×C₅×C₃⋊Dic₃	3
C₃×C₃⋊(C₃⋊Dic₃)	5

Subgroups — C3×C5×C3⋊(C3⋊Dic3) #13108507

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×C₃⋊(C₃⋊Dic₃) #13108507