Subgroups — C₃×SL(2,𝔽₅) #67121

Order 360.

28 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	SL(2,𝔽₅)	C₃	1	1	yes	maximal, characteristic, direct product factor
2	C₃×SL(2,𝔽₃)	—	5	1	no	maximal, Hall
3	C₃×Dic₅	—	6	1	no	maximal
4	C₃×Dic₃	—	10	1	no	maximal
5	C₂×C₃×C₅	—	6	1	no	—
6	C₃×Q₈	—	5	1	no	—
7	SL(2,𝔽₃)	—	5	1	no	—
8	SL(2,𝔽₃)	—	5	2	no	—
9	Dic₅	—	6	1	no	—
10	C₂×C₃²	—	10	1	no	—
11	C₃×C₅	—	6	1	no	—
12	C₃×C₄	—	15	1	no	—
13	Dic₃	—	10	1	no	—
14	C₂×C₅	—	6	1	no	—
15	C₃²	—	10	1	no	Sylow
16	Q₈	—	5	1	no	Sylow
17	C₂×C₃	—	10	1	no	—
18	C₂×C₃	—	20	1	no	—
19	C₂×C₃	A₅	1	1	yes	maximal normal, characteristic
20	C₅	—	6	1	no	Sylow
21	C₄	—	15	1	no	—
22	C₃	—	10	1	no	—
23	C₃	—	20	1	no	—
24	C₃	SL(2,𝔽₅)	1	1	yes	characteristic, direct product factor
25	C₂	C₃×A₅	1	1	yes	characteristic

	subgroup	G/H
center, Fitting, socle, hypercenter	C₂×C₃	A₅
commutator, hypocenter	SL(2,𝔽₅)	C₃
Frattini, cosocle	C₂	C₃×A₅
LCS	G ▷ SL(2,𝔽₅)
UCS	C₁ ◁ C₂×C₃

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

N	H ≅ G/N	type
1. SL(2,𝔽₅)	C₃	direct (N × H)
24. C₃	SL(2,𝔽₅)	direct (N × H)

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