Subgroups — C₅×SL(2,𝔽₅) #10597073

Order 600.

28 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×SL(2,𝔽₃)	—	5	1	no	maximal
2	SL(2,𝔽₅)	C₅	1	1	yes	maximal, characteristic, direct product factor
3	C₅×Dic₅	—	6	1	no	maximal
4	C₅×Dic₃	—	10	1	no	maximal
5	C₂×C₅²	—	6	1	no	—
6	C₅×Q₈	—	5	1	no	—
7	C₂×C₃×C₅	—	10	1	no	—
8	C₅²	—	6	1	no	Sylow
9	SL(2,𝔽₃)	—	5	1	no	Hall
10	C₄×C₅	—	15	1	no	—
11	Dic₅	—	6	1	no	—
12	C₃×C₅	—	10	1	no	—
13	Dic₃	—	10	1	no	—
14	C₂×C₅	—	6	1	no	—
15	C₂×C₅	—	12	2	no	—
16	C₂×C₅	A₅	1	1	yes	maximal normal, characteristic
17	Q₈	—	5	1	no	Sylow
18	C₂×C₃	—	10	1	no	—
19	C₅	—	6	1	no	—
20	C₅	—	12	2	no	—
21	C₅	SL(2,𝔽₅)	1	1	yes	characteristic, direct product factor
22	C₄	—	15	1	no	—
23	C₃	—	10	1	no	Sylow
24	C₂	C₅×A₅	1	1	yes	characteristic

	subgroup	G/H
center, Fitting, socle, hypercenter	C₂×C₅	A₅
commutator, hypocenter	SL(2,𝔽₅)	C₅
Frattini, cosocle	C₂	C₅×A₅
LCS	G ▷ SL(2,𝔽₅)
UCS	C₁ ◁ C₂×C₅

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

N	H ≅ G/N	type
2. SL(2,𝔽₅)	C₅	direct (N × H)
21. C₅	SL(2,𝔽₅)	direct (N × H)

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