Subgroups — C₂²×C₅×C₂₅×C₃₇ #72196313

Order 18500.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₂₅×C₃₇	C₂	1	3	yes	maximal, direct product factor
2	C₅×C₂₅×C₃₇	C₂²	1	1	yes	characteristic, direct product factor, Hall
3	C₂²×C₅²×C₃₇	C₅	1	1	yes	maximal, characteristic
4	C₂²×C₂₅×C₃₇	C₅	1	5	yes	maximal, direct product factor
5	C₂×C₅²×C₃₇	C₂×C₅	1	3	yes	—
6	C₂×C₂₅×C₃₇	C₂×C₅	1	15	yes	direct product factor
7	C₅²×C₃₇	C₂²×C₅	1	1	yes	characteristic
8	C₂₅×C₃₇	C₂²×C₅	1	5	yes	direct product factor
9	C₂²×C₅×C₃₇	C₂₅	1	5	yes	direct product factor
10	C₂²×C₅×C₃₇	C₅²	1	1	yes	characteristic
11	C₂²×C₅×C₂₅	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂×C₅×C₃₇	C₂×C₂₅	1	15	yes	direct product factor
13	C₂×C₅×C₃₇	C₂×C₅²	1	3	yes	—
14	C₂×C₅×C₂₅	C₂×C₃₇	1	3	yes	direct product factor
15	C₅×C₃₇	C₂²×C₂₅	1	5	yes	direct product factor
16	C₅×C₃₇	C₂²×C₅²	1	1	yes	characteristic
17	C₂²×C₃₇	C₅×C₂₅	1	1	yes	characteristic, direct product factor, Hall
18	C₅×C₂₅	C₂²×C₃₇	1	1	yes	characteristic, direct product factor, Sylow
19	C₂²×C₅²	C₅×C₃₇	1	1	yes	characteristic
20	C₂²×C₂₅	C₅×C₃₇	1	5	yes	direct product factor
21	C₂×C₃₇	C₂×C₅×C₂₅	1	3	yes	direct product factor
22	C₂×C₅²	C₂×C₅×C₃₇	1	3	yes	—
23	C₂×C₂₅	C₂×C₅×C₃₇	1	15	yes	direct product factor
24	C₃₇	C₂²×C₅×C₂₅	1	1	yes	characteristic, direct product factor, Sylow
25	C₅²	C₂²×C₅×C₃₇	1	1	yes	characteristic
26	C₂₅	C₂²×C₅×C₃₇	1	5	yes	direct product factor
27	C₂²×C₅	C₂₅×C₃₇	1	5	yes	direct product factor
28	C₂²×C₅	C₅²×C₃₇	1	1	yes	characteristic
29	C₂×C₅	C₂×C₂₅×C₃₇	1	15	yes	direct product factor
30	C₂×C₅	C₂×C₅²×C₃₇	1	3	yes	—
31	C₅	C₂²×C₂₅×C₃₇	1	5	yes	direct product factor
32	C₅	C₂²×C₅²×C₃₇	1	1	yes	characteristic
33	C₂²	C₅×C₂₅×C₃₇	1	1	yes	characteristic, direct product factor, Sylow
34	C₂	C₂×C₅×C₂₅×C₃₇	1	3	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₅	C₂²×C₅²×C₃₇
socle, prime radical	C₂²×C₅²×C₃₇	C₅
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×C₂₅×C₃₇	C₂	direct (N × H)
2. C₅×C₂₅×C₃₇	C₂²	direct (N × H)
4. C₂²×C₂₅×C₃₇	C₅	direct (N × H)
6. C₂×C₂₅×C₃₇	C₂×C₅	direct (N × H)
8. C₂₅×C₃₇	C₂²×C₅	direct (N × H)
9. C₂²×C₅×C₃₇	C₂₅	direct (N × H)
11. C₂²×C₅×C₂₅	C₃₇	direct (N × H)
12. C₂×C₅×C₃₇	C₂×C₂₅	direct (N × H)
14. C₂×C₅×C₂₅	C₂×C₃₇	direct (N × H)
15. C₅×C₃₇	C₂²×C₂₅	direct (N × H)
17. C₂²×C₃₇	C₅×C₂₅	direct (N × H)
18. C₅×C₂₅	C₂²×C₃₇	direct (N × H)
20. C₂²×C₂₅	C₅×C₃₇	direct (N × H)
21. C₂×C₃₇	C₂×C₅×C₂₅	direct (N × H)
23. C₂×C₂₅	C₂×C₅×C₃₇	direct (N × H)
24. C₃₇	C₂²×C₅×C₂₅	direct (N × H)
26. C₂₅	C₂²×C₅×C₃₇	direct (N × H)
27. C₂²×C₅	C₂₅×C₃₇	direct (N × H)
29. C₂×C₅	C₂×C₂₅×C₃₇	direct (N × H)
31. C₅	C₂²×C₂₅×C₃₇	direct (N × H)
33. C₂²	C₅×C₂₅×C₃₇	direct (N × H)
34. C₂	C₂×C₅×C₂₅×C₃₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×C₂₅×C₃₇	2
C₂²×C₅²×C₃₇	5
C₂²×C₂₅×C₃₇	5
C₂²×C₅×C₂₅	37

Subgroups — C22×C5×C25×C37 #72196313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×C₂₅×C₃₇ #72196313