Subgroups — C₂²×C₅×C₆₂₅ #72190846

Order 12500.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₁₂₅	C₅	1	1	yes	maximal, characteristic
2	C₂²×C₆₂₅	C₅	1	5	yes	maximal, direct product factor
3	C₂×C₅×C₆₂₅	C₂	1	3	yes	maximal, direct product factor
4	C₅×C₆₂₅	C₂²	1	1	yes	characteristic, direct product factor, Sylow
5	C₂×C₅×C₁₂₅	C₂×C₅	1	3	yes	—
6	C₂×C₆₂₅	C₂×C₅	1	15	yes	direct product factor
7	C₅×C₁₂₅	C₂²×C₅	1	1	yes	characteristic
8	C₆₂₅	C₂²×C₅	1	5	yes	direct product factor
9	C₂²×C₅×C₂₅	C₂₅	1	1	yes	characteristic
10	C₂²×C₁₂₅	C₂₅	1	4	yes	—
11	C₂²×C₁₂₅	C₅²	1	1	yes	characteristic
12	C₂×C₅×C₂₅	C₂×C₂₅	1	3	yes	—
13	C₂×C₁₂₅	C₂×C₂₅	1	12	yes	—
14	C₂×C₁₂₅	C₂×C₅²	1	3	yes	—
15	C₅×C₂₅	C₂²×C₂₅	1	1	yes	characteristic
16	C₁₂₅	C₂²×C₂₅	1	4	yes	—
17	C₁₂₅	C₂²×C₅²	1	1	yes	characteristic
18	C₂²×C₅²	C₁₂₅	1	1	yes	characteristic
19	C₂²×C₂₅	C₁₂₅	1	4	yes	—
20	C₂²×C₂₅	C₅×C₂₅	1	1	yes	characteristic
21	C₂×C₅²	C₂×C₁₂₅	1	3	yes	—
22	C₂×C₂₅	C₂×C₁₂₅	1	12	yes	—
23	C₂×C₂₅	C₂×C₅×C₂₅	1	3	yes	—
24	C₅²	C₂²×C₁₂₅	1	1	yes	characteristic
25	C₂₅	C₂²×C₁₂₅	1	4	yes	—
26	C₂₅	C₂²×C₅×C₂₅	1	1	yes	characteristic
27	C₂²×C₅	C₆₂₅	1	5	yes	direct product factor
28	C₂²×C₅	C₅×C₁₂₅	1	1	yes	characteristic
29	C₂×C₅	C₂×C₆₂₅	1	15	yes	direct product factor
30	C₂×C₅	C₂×C₅×C₁₂₅	1	3	yes	—
31	C₅	C₂²×C₆₂₅	1	5	yes	direct product factor
32	C₅	C₂²×C₅×C₁₂₅	1	1	yes	characteristic
33	C₂²	C₅×C₆₂₅	1	1	yes	characteristic, direct product factor, Sylow
34	C₂	C₂×C₅×C₆₂₅	1	3	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₁₂₅	C₂²×C₅²
socle, prime radical	C₂²×C₅²	C₁₂₅
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₆₂₅	C₅	direct (N × H)
3. C₂×C₅×C₆₂₅	C₂	direct (N × H)
4. C₅×C₆₂₅	C₂²	direct (N × H)
6. C₂×C₆₂₅	C₂×C₅	direct (N × H)
8. C₆₂₅	C₂²×C₅	direct (N × H)
27. C₂²×C₅	C₆₂₅	direct (N × H)
29. C₂×C₅	C₂×C₆₂₅	direct (N × H)
31. C₅	C₂²×C₆₂₅	direct (N × H)
33. C₂²	C₅×C₆₂₅	direct (N × H)
34. C₂	C₂×C₅×C₆₂₅	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×C₅×C₁₂₅	5
C₂²×C₆₂₅	5
C₂×C₅×C₆₂₅	2

Subgroups — C22×C5×C625 #72190846

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×C₆₂₅ #72190846