Subgroups — C₂²×C₅×C₂₅² #72190848

Order 12500.

2130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅²×C₂₅	C₅	1	6	yes	maximal
2	C₂²×C₂₅²	C₅	1	25	yes	maximal, direct product factor
3	C₂×C₅×C₂₅²	C₂	1	3	yes	maximal, direct product factor
4	C₅×C₂₅²	C₂²	1	1	yes	characteristic, direct product factor, Sylow
5	C₂×C₅²×C₂₅	C₂×C₅	1	18	yes	—
6	C₂×C₂₅²	C₂×C₅	1	75	yes	direct product factor
7	C₅²×C₂₅	C₂²×C₅	1	6	yes	—
8	C₂₅²	C₂²×C₅	1	25	yes	direct product factor
9	C₂²×C₅³	C₅²	1	1	yes	characteristic
10	C₂²×C₅×C₂₅	C₂₅	1	150	yes	direct product factor
11	C₂²×C₅×C₂₅	C₅²	1	30	yes	—
12	C₂×C₅³	C₂×C₅²	1	3	yes	—
13	C₂×C₅×C₂₅	C₂×C₂₅	1	450	yes	direct product factor
14	C₂×C₅×C₂₅	C₂×C₅²	1	90	yes	—
15	C₅³	C₂²×C₅²	1	1	yes	characteristic
16	C₅×C₂₅	C₂²×C₂₅	1	150	yes	direct product factor
17	C₅×C₂₅	C₂²×C₅²	1	30	yes	—
18	C₂²×C₅²	C₅×C₂₅	1	30	yes	—
19	C₂²×C₅²	C₅³	1	1	yes	characteristic
20	C₂²×C₂₅	C₅×C₂₅	1	150	yes	direct product factor
21	C₂×C₅²	C₂×C₅×C₂₅	1	90	yes	—
22	C₂×C₅²	C₂×C₅³	1	3	yes	—
23	C₂×C₂₅	C₂×C₅×C₂₅	1	450	yes	direct product factor
24	C₅²	C₂²×C₅×C₂₅	1	30	yes	—
25	C₅²	C₂²×C₅³	1	1	yes	characteristic
26	C₂₅	C₂²×C₅×C₂₅	1	150	yes	direct product factor
27	C₂²×C₅	C₂₅²	1	25	yes	direct product factor
28	C₂²×C₅	C₅²×C₂₅	1	6	yes	—
29	C₂×C₅	C₂×C₂₅²	1	75	yes	direct product factor
30	C₂×C₅	C₂×C₅²×C₂₅	1	18	yes	—
31	C₅	C₂²×C₂₅²	1	25	yes	direct product factor
32	C₅	C₂²×C₅²×C₂₅	1	6	yes	—
33	C₂²	C₅×C₂₅²	1	1	yes	characteristic, direct product factor, Sylow
34	C₂	C₂×C₅×C₂₅²	1	3	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₅²	C₂²×C₅³
socle, prime radical	C₂²×C₅³	C₅²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₂₅²	C₅	direct (N × H)
3. C₂×C₅×C₂₅²	C₂	direct (N × H)
4. C₅×C₂₅²	C₂²	direct (N × H)
6. C₂×C₂₅²	C₂×C₅	direct (N × H)
8. C₂₅²	C₂²×C₅	direct (N × H)
10. C₂²×C₅×C₂₅	C₂₅	direct (N × H)
13. C₂×C₅×C₂₅	C₂×C₂₅	direct (N × H)
16. C₅×C₂₅	C₂²×C₂₅	direct (N × H)
20. C₂²×C₂₅	C₅×C₂₅	direct (N × H)
23. C₂×C₂₅	C₂×C₅×C₂₅	direct (N × H)
26. C₂₅	C₂²×C₅×C₂₅	direct (N × H)
27. C₂²×C₅	C₂₅²	direct (N × H)
29. C₂×C₅	C₂×C₂₅²	direct (N × H)
31. C₅	C₂²×C₂₅²	direct (N × H)
33. C₂²	C₅×C₂₅²	direct (N × H)
34. C₂	C₂×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
C₂²×C₅²×C₂₅	5
C₂²×C₂₅²	5
C₂×C₅×C₂₅²	2

Subgroups — C22×C5×C252 #72190848

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×C₂₅² #72190848