Subgroups — C₃×C₅×SL(2,𝔽₃) #67159

Order 360.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×Q₈	C₃	1	1	yes	maximal, characteristic, complemented
2	C₅×SL(2,𝔽₃)	C₃	1	3	yes	maximal, direct product factor
3	C₂×C₃²×C₅	—	4	1	no	maximal
4	C₃×SL(2,𝔽₃)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₃×C₄×C₅	—	3	1	no	—
6	C₃²×C₅	—	4	1	no	Hall
7	C₅×Q₈	C₃²	1	1	yes	characteristic, complemented, Hall
8	C₂×C₃×C₅	—	4	3	no	—
9	C₂×C₃×C₅	A₄	1	1	yes	characteristic
10	C₃×Q₈	C₃×C₅	1	1	yes	characteristic, complemented
11	SL(2,𝔽₃)	C₃×C₅	1	3	yes	direct product factor
12	C₄×C₅	—	3	1	no	—
13	C₂×C₃²	—	4	1	no	—
14	C₃×C₅	—	4	3	no	—
15	C₃×C₅	SL(2,𝔽₃)	1	1	yes	characteristic, direct product factor
16	C₃×C₄	—	3	1	no	—
17	C₂×C₅	C₃×A₄	1	1	yes	characteristic
18	C₃²	—	4	1	no	Sylow
19	Q₈	C₃²×C₅	1	1	yes	characteristic, complemented, Sylow
20	C₂×C₃	—	4	3	no	—
21	C₂×C₃	C₅×A₄	1	1	yes	characteristic
22	C₅	C₃×SL(2,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
23	C₄	—	3	1	no	—
24	C₃	—	4	3	no	—
25	C₃	C₅×SL(2,𝔽₃)	1	1	yes	characteristic, direct product factor
26	C₂	C₃×C₅×A₄	1	1	yes	characteristic

	subgroup	G/H
center, socle, hypercenter	C₂×C₃×C₅	A₄
commutator, cosocle, hypocenter	Q₈	C₃²×C₅
Frattini	C₂	C₃×C₅×A₄
Fitting	C₃×C₅×Q₈	C₃
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂×C₃×C₅

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

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