Subgroups — C₂×C₅×SL(2,𝔽₃) #6792

Order 240.

36 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, direct product factor
2	C₂×C₅×Q₈	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₂²×C₃×C₅	—	4	1	no	maximal
4	C₂×SL(2,𝔽₃)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₅×Q₈	—	3	1	no	—
6	C₅×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂×C₄×C₅	—	3	1	no	—
8	C₂×C₃×C₅	—	4	2	no	—
9	C₂×C₃×C₅	—	4	1	no	—
10	SL(2,𝔽₃)	C₂×C₅	1	1	yes	characteristic, direct product factor
11	C₂²×C₅	A₄	1	1	yes	characteristic
12	C₄×C₅	—	3	1	no	—
13	C₄×C₅	—	3	1	no	—
14	C₂×Q₈	C₃×C₅	1	1	yes	characteristic, complemented, Sylow
15	C₃×C₅	—	4	1	no	Hall
16	C₂²×C₃	—	4	1	no	—
17	C₂×C₅	SL(2,𝔽₃)	1	2	yes	direct product factor
18	C₂×C₅	C₂×A₄	1	1	yes	characteristic
19	Q₈	—	3	1	no	—
20	Q₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
21	C₂×C₄	—	3	1	no	—
22	C₂×C₃	—	4	2	no	—
23	C₂×C₃	—	4	1	no	—
24	C₅	C₂×SL(2,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
25	C₂²	C₅×A₄	1	1	yes	characteristic
26	C₄	—	3	1	no	—
27	C₄	—	3	1	no	—
28	C₃	—	4	1	no	Sylow
29	C₂	C₅×SL(2,𝔽₃)	1	2	yes	direct product factor
30	C₂	C₂×C₅×A₄	1	1	yes	characteristic

	subgroup	G/H
center, socle, hypercenter	C₂²×C₅	A₄
commutator, cosocle, hypocenter	Q₈	C₂×C₃×C₅
Frattini	C₂	C₂×C₅×A₄
Fitting	C₂×C₅×Q₈	C₃
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂²×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.