Subgroups — C₂×SL(2,𝔽₅) #6733

Order 240.

31 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	SL(2,𝔽₅)	C₂	1	1	yes	maximal, characteristic, direct product factor
2	C₂×SL(2,𝔽₃)	—	5	1	no	maximal, Hall
3	C₂×Dic₅	—	6	1	no	maximal
4	C₂×Dic₃	—	10	1	no	maximal
5	SL(2,𝔽₃)	—	5	1	no	—
6	C₂²×C₅	—	6	1	no	—
7	Dic₅	—	6	1	no	—
8	Dic₅	—	6	1	no	—
9	C₂×Q₈	—	5	1	no	Sylow
10	C₂²×C₃	—	10	1	no	—
11	Dic₃	—	10	1	no	—
12	Dic₃	—	10	1	no	—
13	C₂×C₅	—	6	2	no	—
14	C₂×C₅	—	6	1	no	—
15	Q₈	—	5	1	no	—
16	Q₈	—	15	1	no	—
17	C₂×C₄	—	15	1	no	—
18	C₂×C₃	—	10	2	no	—
19	C₂×C₃	—	10	1	no	—
20	C₅	—	6	1	no	Sylow
21	C₂²	A₅	1	1	yes	maximal normal, characteristic
22	C₄	—	15	1	no	—
23	C₄	—	15	1	no	—
24	C₃	—	10	1	no	Sylow
25	C₂	SL(2,𝔽₅)	1	2	yes	direct product factor
26	C₂	C₂×A₅	1	1	yes	characteristic

	subgroup	G/H
center, Fitting, socle, hypercenter	C₂²	A₅
commutator, hypocenter	SL(2,𝔽₅)	C₂
Frattini, cosocle	C₂	C₂×A₅
LCS	G ▷ SL(2,𝔽₅)
UCS	C₁ ◁ C₂²

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

N	H ≅ G/N	type
1. SL(2,𝔽₅)	C₂	direct (N × H)
25. C₂	SL(2,𝔽₅)	direct (N × H)

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