Subgroups — C₃×C₅×Dic₆ #67165

Order 360.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×C₅	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×Dic₃	C₂	1	2	yes	maximal
3	C₃×C₅×Q₈	—	3	1	no	maximal
4	C₅×Dic₆	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₂×C₃²×C₅	C₂²	1	1	yes	characteristic
6	C₃×Dic₆	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×C₄×C₅	—	2	1	no	—
8	C₃×C₄×C₅	—	3	2	no	—
9	C₃×C₄×C₅	D₃	1	1	yes	characteristic
10	C₃×C₄×C₅	C₂×C₃	1	1	yes	characteristic
11	C₅×Dic₃	C₂×C₃	1	2	yes	—
12	C₃²×C₅	Q₈	1	1	yes	characteristic, complemented, Hall
13	C₅×Q₈	—	3	1	no	Hall
14	C₃²×C₄	C₂×C₅	1	1	yes	characteristic
15	C₃×Dic₃	C₂×C₅	1	2	yes	—
16	C₂×C₃×C₅	—	2	1	no	—
17	C₂×C₃×C₅	C₂×D₃	1	1	yes	characteristic
18	C₂×C₃×C₅	C₂²×C₃	1	1	yes	characteristic
19	C₃×Q₈	—	3	1	no	—
20	Dic₆	C₃×C₅	1	1	yes	characteristic, direct product factor
21	C₄×C₅	—	3	2	no	—
22	C₄×C₅	C₃×D₃	1	1	yes	characteristic
23	C₂×C₃²	C₂²×C₅	1	1	yes	characteristic
24	C₃×C₅	—	2	1	no	—
25	C₃×C₅	Dic₆	1	1	yes	characteristic, direct product factor
26	C₃×C₅	C₃×Q₈	1	1	yes	characteristic, complemented
27	C₃×C₄	—	2	1	no	—
28	C₃×C₄	—	3	2	no	—
29	C₃×C₄	C₅×D₃	1	1	yes	characteristic
30	C₃×C₄	C₂×C₃×C₅	1	1	yes	characteristic
31	Dic₃	C₂×C₃×C₅	1	2	yes	—
32	C₂×C₅	C₂×C₃×D₃	1	1	yes	characteristic
33	C₃²	C₅×Q₈	1	1	yes	characteristic, complemented, Sylow
34	Q₈	—	3	1	no	Sylow
35	C₂×C₃	—	2	1	no	—
36	C₂×C₃	C₂×C₅×D₃	1	1	yes	characteristic
37	C₂×C₃	C₂²×C₃×C₅	1	1	yes	characteristic
38	C₅	C₃×Dic₆	1	1	yes	characteristic, direct product factor, Sylow
39	C₄	—	3	2	no	—
40	C₄	C₃×C₅×D₃	1	1	yes	characteristic
41	C₃	—	2	1	no	—
42	C₃	C₅×Dic₆	1	1	yes	characteristic, direct product factor
43	C₃	C₃×C₅×Q₈	1	1	yes	characteristic, complemented
44	C₂	C₂×C₃×C₅×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₅	C₂×D₃
commutator, cosocle	C₂×C₃	C₂²×C₃×C₅
Frattini	C₂	C₂×C₃×C₅×D₃
Fitting	C₃²×C₄×C₅	C₂
socle, prime radical	C₂×C₃²×C₅	C₂²
hypercenter	C₃×C₄×C₅	D₃
hypocenter	C₃	C₃×C₅×Q₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₅ ◁ C₃×C₄×C₅

Complements

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

N	H ≅ G/N	type
4. C₅×Dic₆	C₃	direct (N × H)
6. C₃×Dic₆	C₅	direct (N × H)
12. C₃²×C₅	Q₈	semidirect
20. Dic₆	C₃×C₅	direct (N × H)
25. C₃×C₅	Dic₆	direct (N × H)
26. C₃×C₅	C₃×Q₈	semidirect
33. C₃²	C₅×Q₈	semidirect
38. C₅	C₃×Dic₆	direct (N × H)
42. C₃	C₅×Dic₆	direct (N × H)
43. C₃	C₃×C₅×Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄×C₅	2
C₃×C₅×Dic₃	2
C₃×C₅×Q₈	3
C₅×Dic₆	3
C₃×Dic₆	5

Subgroups — C3×C5×Dic6 #67165

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×Dic₆ #67165