Subgroups — C₅⋊GL(2,𝔽₃) #6745

Order 240.

32 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	#688	—	3	1	no	maximal, Hall
3	C₂×D₁₅	—	4	1	no	maximal
4	GL(2,𝔽₃)	—	5	1	no	maximal, Hall
5	C₅×Q₈	D₃	1	1	yes	characteristic, complemented
6	D₂₀	—	3	1	no	—
7	#168	—	3	1	no	—
8	C₂×C₃×C₅	—	4	1	no	—
9	D₁₅	—	4	2	no	—
10	SL(2,𝔽₃)	D₅	1	1	yes	characteristic, complemented
11	C₂×D₅	—	6	1	no	—
12	C₄×C₅	—	3	1	no	—
13	SD₁₆	—	15	1	no	Sylow
14	C₃×C₅	—	4	1	no	Hall
15	C₂×D₃	—	20	1	no	—
16	C₂×C₅	S₄	1	1	yes	characteristic
17	D₅	—	12	1	no	—
18	Q₈	D₁₅	1	1	yes	characteristic, complemented
19	D₄	—	15	1	no	—
20	C₈	—	15	1	no	—
21	C₂×C₃	—	4	1	no	—
22	D₃	—	20	2	no	—
23	C₅	GL(2,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
24	C₂²	—	30	1	no	—
25	C₄	—	3	1	no	—
26	C₃	—	4	1	no	Sylow
27	C₂	—	60	1	no	—
28	C₂	C₅⋊S₄	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	C₅⋊S₄
commutator, cosocle, hypocenter	C₅×SL(2,𝔽₃)	C₂
Fitting	C₅×Q₈	D₃
socle	C₂×C₅	S₄
LCS	G ▷ C₅×SL(2,𝔽₃)
UCS	C₁ ◁ C₂

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.