Subgroups — C₄×C₅×C₂₅² #72190842

Order 12500.

1278 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅²×C₂₅	C₅	1	6	yes	maximal
2	C₄×C₂₅²	C₅	1	25	yes	maximal, direct product factor
3	C₂×C₅×C₂₅²	C₂	1	1	yes	maximal, characteristic
4	C₅×C₂₅²	C₄	1	1	yes	characteristic, direct product factor, Sylow
5	C₂×C₅²×C₂₅	C₂×C₅	1	6	yes	—
6	C₂×C₂₅²	C₂×C₅	1	25	yes	—
7	C₅²×C₂₅	C₄×C₅	1	6	yes	—
8	C₂₅²	C₄×C₅	1	25	yes	direct product factor
9	C₄×C₅³	C₅²	1	1	yes	characteristic
10	C₄×C₅×C₂₅	C₂₅	1	150	yes	direct product factor
11	C₄×C₅×C₂₅	C₅²	1	30	yes	—
12	C₂×C₅³	C₂×C₅²	1	1	yes	characteristic
13	C₂×C₅×C₂₅	C₂×C₂₅	1	150	yes	—
14	C₂×C₅×C₂₅	C₂×C₅²	1	30	yes	—
15	C₅³	C₄×C₅²	1	1	yes	characteristic
16	C₅×C₂₅	C₄×C₂₅	1	150	yes	direct product factor
17	C₅×C₂₅	C₄×C₅²	1	30	yes	—
18	C₄×C₅²	C₅×C₂₅	1	30	yes	—
19	C₄×C₅²	C₅³	1	1	yes	characteristic
20	C₄×C₂₅	C₅×C₂₅	1	150	yes	direct product factor
21	C₂×C₅²	C₂×C₅×C₂₅	1	30	yes	—
22	C₂×C₅²	C₂×C₅³	1	1	yes	characteristic
23	C₂×C₂₅	C₂×C₅×C₂₅	1	150	yes	—
24	C₅²	C₄×C₅×C₂₅	1	30	yes	—
25	C₅²	C₄×C₅³	1	1	yes	characteristic
26	C₂₅	C₄×C₅×C₂₅	1	150	yes	direct product factor
27	C₄×C₅	C₂₅²	1	25	yes	direct product factor
28	C₄×C₅	C₅²×C₂₅	1	6	yes	—
29	C₂×C₅	C₂×C₂₅²	1	25	yes	—
30	C₂×C₅	C₂×C₅²×C₂₅	1	6	yes	—
31	C₅	C₄×C₂₅²	1	25	yes	direct product factor
32	C₅	C₄×C₅²×C₂₅	1	6	yes	—
33	C₄	C₅×C₂₅²	1	1	yes	characteristic, direct product factor, Sylow
34	C₂	C₂×C₅×C₂₅²	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₅²	C₂×C₅³
socle, prime radical	C₂×C₅³	C₂×C₅²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₂₅²	C₅	direct (N × H)
4. C₅×C₂₅²	C₄	direct (N × H)
8. C₂₅²	C₄×C₅	direct (N × H)
10. C₄×C₅×C₂₅	C₂₅	direct (N × H)
16. C₅×C₂₅	C₄×C₂₅	direct (N × H)
20. C₄×C₂₅	C₅×C₂₅	direct (N × H)
26. C₂₅	C₄×C₅×C₂₅	direct (N × H)
27. C₄×C₅	C₂₅²	direct (N × H)
31. C₅	C₄×C₂₅²	direct (N × H)
33. C₄	C₅×C₂₅²	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₅²×C₂₅	5
C₄×C₂₅²	5
C₂×C₅×C₂₅²	2

Subgroups — C4×C5×C252 #72190842

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×C₂₅² #72190842