Subgroups — C₅×GL(2,𝔽₃) #6742

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	C₅×SD₁₆	—	3	1	no	maximal, Hall
3	C₂×C₅×D₃	—	4	1	no	maximal
4	GL(2,𝔽₃)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₅×Q₈	D₃	1	1	yes	characteristic, complemented
6	C₅×D₄	—	3	1	no	—
7	C₅×C₈	—	3	1	no	—
8	C₂×C₃×C₅	—	4	1	no	—
9	C₅×D₃	—	4	2	no	—
10	SL(2,𝔽₃)	C₂×C₅	1	1	yes	characteristic, complemented
11	C₂²×C₅	—	6	1	no	—
12	C₄×C₅	—	3	1	no	—
13	SD₁₆	—	3	1	no	Sylow
14	C₃×C₅	—	4	1	no	Hall
15	C₂×D₃	—	4	1	no	—
16	C₂×C₅	—	12	1	no	—
17	C₂×C₅	S₄	1	1	yes	characteristic
18	Q₈	C₅×D₃	1	1	yes	characteristic, complemented
19	D₄	—	3	1	no	—
20	C₈	—	3	1	no	—
21	C₂×C₃	—	4	1	no	—
22	D₃	—	4	2	no	—
23	C₅	GL(2,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
24	C₂²	—	6	1	no	—
25	C₄	—	3	1	no	—
26	C₃	—	4	1	no	Sylow
27	C₂	—	12	1	no	—
28	C₂	C₅×S₄	1	1	yes	characteristic

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