Subgroups — C₅×C₃₇×5₋¹⁺² #37528885

Order 23125.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅³×C₃₇	C₅	1	1	yes	maximal, characteristic
2	C₃₇×5₋¹⁺²	C₅	1	25	yes	maximal, direct product factor
3	C₅×C₂₅×C₃₇	C₅	1	5	yes	maximal, complemented
4	C₅²×C₃₇	—	5	5	no	—
5	C₅²×C₃₇	C₅²	1	5	yes	—
6	C₅²×C₃₇	C₅²	1	1	yes	characteristic
7	C₂₅×C₃₇	C₅²	1	25	yes	complemented
8	C₅×5₋¹⁺²	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₅×C₃₇	—	5	5	no	—
10	C₅×C₃₇	5₋¹⁺²	1	5	yes	direct product factor
11	C₅×C₃₇	C₅³	1	1	yes	characteristic
12	C₅³	C₅×C₃₇	1	1	yes	characteristic
13	5₋¹⁺²	C₅×C₃₇	1	25	yes	direct product factor
14	C₅×C₂₅	C₅×C₃₇	1	5	yes	complemented
15	C₃₇	C₅×5₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
16	C₅²	—	5	5	no	—
17	C₅²	C₅²×C₃₇	1	5	yes	—
18	C₅²	C₅²×C₃₇	1	1	yes	characteristic
19	C₂₅	C₅²×C₃₇	1	25	yes	complemented
20	C₅	—	5	5	no	—
21	C₅	C₃₇×5₋¹⁺²	1	5	yes	direct product factor
22	C₅	C₅³×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₅²×C₃₇	C₅²
commutator, Frattini, cosocle	C₅	C₅³×C₃₇
prime radical	C₅³×C₃₇	C₅
LCS	G ▷ C₅ ▷ C₁
UCS	C₁ ◁ C₅²×C₃₇ ◁ G

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.