Subgroups — C₂×C₃×C₄₁×C₅⋊D₅ #27134331

Order 12300.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄₁×C₅⋊D₅	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₃×C₅²×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄₁×C₅⋊D₅	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₃×C₅²×C₄₁	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₃×D₅×C₄₁	—	5	6	no	maximal
6	C₄₁×C₅⋊D₅	C₂×C₃	1	2	yes	direct product factor
7	C₂×C₅²×C₄₁	C₂×C₃	1	1	yes	characteristic, complemented
8	C₂×C₃×C₅×C₄₁	D₅	1	6	yes	complemented
9	C₃×D₅×C₄₁	—	5	12	no	—
10	C₅²×C₄₁	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
11	C₂×D₅×C₄₁	—	5	6	no	—
12	C₃×C₅×C₄₁	C₂×D₅	1	6	yes	complemented
13	C₂²×C₃×C₄₁	—	25	1	no	Hall
14	C₂×C₅×C₄₁	C₃×D₅	1	6	yes	complemented
15	D₅×C₄₁	—	5	12	no	—
16	C₂×C₃×C₅⋊D₅	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₃×C₄₁	—	25	2	no	—
18	C₂×C₃×C₄₁	C₅⋊D₅	1	1	yes	characteristic, direct product factor
19	C₅×C₄₁	C₂×C₃×D₅	1	6	yes	complemented
20	C₂²×C₄₁	—	25	1	no	Hall
21	C₂×C₃×C₅²	C₂×C₄₁	1	1	yes	characteristic, complemented
22	C₃×C₅⋊D₅	C₂×C₄₁	1	2	yes	direct product factor
23	C₃×C₄₁	C₂×C₅⋊D₅	1	1	yes	characteristic, direct product factor, Hall
24	C₂×C₅⋊D₅	C₃×C₄₁	1	1	yes	characteristic, direct product factor, Hall
25	C₂×C₄₁	—	25	2	no	—
26	C₂×C₄₁	C₃×C₅⋊D₅	1	1	yes	characteristic, direct product factor
27	C₃×C₅²	C₂²×C₄₁	1	1	yes	characteristic, complemented, Hall
28	C₂×C₃×D₅	—	5	6	no	—
29	C₂×C₅²	C₂×C₃×C₄₁	1	1	yes	characteristic, complemented
30	C₅⋊D₅	C₂×C₃×C₄₁	1	2	yes	direct product factor
31	C₄₁	C₂×C₃×C₅⋊D₅	1	1	yes	characteristic, direct product factor, Sylow
32	C₂×C₃×C₅	D₅×C₄₁	1	6	yes	complemented
33	C₃×D₅	—	5	12	no	—
34	C₅²	C₂²×C₃×C₄₁	1	1	yes	characteristic, complemented, Sylow
35	C₂×D₅	—	5	6	no	—
36	C₃×C₅	C₂×D₅×C₄₁	1	6	yes	complemented
37	C₂²×C₃	—	25	1	no	Hall
38	C₂×C₅	C₃×D₅×C₄₁	1	6	yes	complemented
39	D₅	—	5	12	no	—
40	C₂×C₃	—	25	2	no	—
41	C₂×C₃	C₄₁×C₅⋊D₅	1	1	yes	characteristic, direct product factor
42	C₅	C₂×C₃×D₅×C₄₁	1	6	yes	complemented
43	C₂²	—	25	1	no	Sylow
44	C₃	C₂×C₄₁×C₅⋊D₅	1	1	yes	characteristic, direct product factor, Sylow
45	C₂	—	25	2	no	—
46	C₂	C₃×C₄₁×C₅⋊D₅	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃×C₄₁	C₅⋊D₅
commutator, cosocle, hypocenter	C₅²	C₂²×C₃×C₄₁
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅²×C₄₁	C₂
LCS	G ▷ C₅²
UCS	C₁ ◁ C₂×C₃×C₄₁

Complements

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

N	H ≅ G/N	type
1. C₃×C₄₁×C₅⋊D₅	C₂	direct (N × H)
2. C₂×C₃×C₅²×C₄₁	C₂	semidirect
3. C₂×C₄₁×C₅⋊D₅	C₃	direct (N × H)
4. C₃×C₅²×C₄₁	C₂²	semidirect
6. C₄₁×C₅⋊D₅	C₂×C₃	direct (N × H)
7. C₂×C₅²×C₄₁	C₂×C₃	semidirect
8. C₂×C₃×C₅×C₄₁	D₅	semidirect
10. C₅²×C₄₁	C₂²×C₃	semidirect
12. C₃×C₅×C₄₁	C₂×D₅	semidirect
14. C₂×C₅×C₄₁	C₃×D₅	semidirect
16. C₂×C₃×C₅⋊D₅	C₄₁	direct (N × H)
18. C₂×C₃×C₄₁	C₅⋊D₅	direct (N × H)
19. C₅×C₄₁	C₂×C₃×D₅	semidirect
21. C₂×C₃×C₅²	C₂×C₄₁	semidirect
22. C₃×C₅⋊D₅	C₂×C₄₁	direct (N × H)
23. C₃×C₄₁	C₂×C₅⋊D₅	direct (N × H)
24. C₂×C₅⋊D₅	C₃×C₄₁	direct (N × H)
26. C₂×C₄₁	C₃×C₅⋊D₅	direct (N × H)
27. C₃×C₅²	C₂²×C₄₁	semidirect
29. C₂×C₅²	C₂×C₃×C₄₁	semidirect
30. C₅⋊D₅	C₂×C₃×C₄₁	direct (N × H)
31. C₄₁	C₂×C₃×C₅⋊D₅	direct (N × H)
32. C₂×C₃×C₅	D₅×C₄₁	semidirect
34. C₅²	C₂²×C₃×C₄₁	semidirect
36. C₃×C₅	C₂×D₅×C₄₁	semidirect
38. C₂×C₅	C₃×D₅×C₄₁	semidirect
41. C₂×C₃	C₄₁×C₅⋊D₅	direct (N × H)
42. C₅	C₂×C₃×D₅×C₄₁	semidirect
44. C₃	C₂×C₄₁×C₅⋊D₅	direct (N × H)
46. C₂	C₃×C₄₁×C₅⋊D₅	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₄₁×C₅⋊D₅	2
C₂×C₃×C₅²×C₄₁	2
C₂×C₄₁×C₅⋊D₅	3
C₂×C₃×D₅×C₄₁	5
C₂×C₃×C₅⋊D₅	41

Subgroups — C2×C3×C41×C5⋊D5 #27134331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₄₁×C₅⋊D₅ #27134331