Subgroups — C₅⋊(C₅⋊#272) #13062529

Order 1350.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₃⋊(C₅⋊D₅)	—	3	4	no	maximal
3	C₅⋊#272	—	5	6	no	maximal
4	C₃²×C₅²	D₃	1	4	yes	complemented
5	C₃⋊(C₅⋊D₅)	—	3	4	no	—
6	C₃×C₅⋊D₅	—	9	1	no	—
7	C₅×H(3,𝔽₃)	D₅	1	6	yes	complemented
8	C₃×D₁₅	—	15	24	no	—
9	C₃×C₅²	—	3	4	no	—
10	C₃×C₅²	C₃⋊D₃	1	1	yes	characteristic
11	#272	—	25	1	no	Hall
12	C₅⋊D₅	—	9	1	no	Hall
13	C₃²×C₅	D₁₅	1	24	yes	complemented
14	D₁₅	—	15	24	no	—
15	C₃×D₅	—	45	6	no	—
16	H(3,𝔽₃)	C₅⋊D₅	1	1	yes	characteristic, complemented, Sylow
17	C₅²	#272	1	1	yes	characteristic, complemented, Sylow
18	C₃×D₃	—	75	4	no	—
19	C₃×C₅	—	3	24	no	—
20	C₃×C₅	C₃⋊D₁₅	1	6	yes	—
21	D₅	—	45	6	no	—
22	C₃²	C₃⋊(C₅⋊D₅)	1	4	yes	complemented
23	C₂×C₃	—	225	1	no	—
24	D₃	—	75	4	no	—
25	C₅	C₅⋊#272	1	6	yes	complemented
26	C₃	—	3	4	no	—
27	C₃	C₃⋊(C₃⋊(C₅⋊D₅))	1	1	yes	characteristic
28	C₂	—	225	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₃	C₃⋊(C₃⋊(C₅⋊D₅))
commutator, Fitting, cosocle, hypocenter	C₅²×H(3,𝔽₃)	C₂
socle	C₃×C₅²	C₃⋊D₃
LCS	G ▷ C₅²×H(3,𝔽₃)
UCS	C₁ ◁ C₃

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.