Subgroups — C₃×C₅×Dic₆₁⋊C₅ #32912953

Order 18300.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₆₁⋊C₅	C₂	1	1	yes	maximal, characteristic
2	C₅×Dic₆₁⋊C₅	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₃×C₅×C₆₁⋊C₅	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃×C₅×Dic₆₁	C₅	1	1	yes	maximal, characteristic, complemented
5	C₃×Dic₆₁⋊C₅	C₅	1	5	yes	maximal, direct product factor
6	C₂×C₅×C₆₁⋊C₅	C₂×C₃	1	1	yes	characteristic
7	C₂×C₃×C₅×C₆₁	C₂×C₅	1	1	yes	characteristic
8	C₂×C₃×C₆₁⋊C₅	C₂×C₅	1	5	yes	—
9	C₅×C₆₁⋊C₅	C₃×C₄	1	1	yes	characteristic, complemented, Hall
10	C₅×Dic₆₁	C₃×C₅	1	1	yes	characteristic, complemented
11	Dic₆₁⋊C₅	C₃×C₅	1	5	yes	direct product factor
12	C₃×C₅×C₆₁	C₄×C₅	1	1	yes	characteristic, complemented
13	C₃×C₆₁⋊C₅	C₄×C₅	1	5	yes	complemented
14	C₃×Dic₆₁	C₅²	1	1	yes	characteristic, complemented, Hall
15	C₂×C₅×C₆₁	C₂×C₃×C₅	1	1	yes	characteristic
16	C₂×C₆₁⋊C₅	C₂×C₃×C₅	1	5	yes	—
17	C₂×C₃×C₆₁	C₂×C₅²	1	1	yes	characteristic
18	C₅×C₆₁	C₃×C₄×C₅	1	1	yes	characteristic, complemented
19	C₆₁⋊C₅	C₃×C₄×C₅	1	5	yes	complemented
20	C₃×C₄×C₅²	—	61	1	no	maximal, Hall
21	Dic₆₁	C₃×C₅²	1	1	yes	characteristic, complemented, Hall
22	C₃×C₆₁	C₄×C₅²	1	1	yes	characteristic, complemented, Hall
23	C₂×C₃×C₅²	—	61	1	no	—
24	C₂×C₆₁	C₂×C₃×C₅²	1	1	yes	characteristic
25	C₄×C₅²	—	61	1	no	Hall
26	C₃×C₅²	—	61	1	no	Hall
27	C₆₁	C₃×C₄×C₅²	1	1	yes	characteristic, complemented, Sylow
28	C₃×C₄×C₅	—	305	1	no	—
29	C₃×C₄×C₅	—	61	5	no	—
30	C₂×C₅²	—	61	1	no	—
31	C₂×C₃×C₅	—	61	5	no	—
32	C₂×C₃×C₅	D₆₁⋊C₅	1	1	yes	characteristic
33	C₅²	—	61	1	no	Sylow
34	C₄×C₅	—	305	1	no	—
35	C₄×C₅	—	61	5	no	—
36	C₃×C₅	—	61	5	no	—
37	C₃×C₅	Dic₆₁⋊C₅	1	1	yes	characteristic, direct product factor
38	C₃×C₄	—	305	1	no	Hall
39	C₂×C₅	—	61	5	no	—
40	C₂×C₅	C₃×D₆₁⋊C₅	1	1	yes	characteristic
41	C₂×C₃	C₅×D₆₁⋊C₅	1	1	yes	characteristic
42	C₅	—	61	5	no	—
43	C₅	C₃×Dic₆₁⋊C₅	1	1	yes	characteristic, direct product factor
44	C₄	—	305	1	no	Sylow
45	C₃	C₅×Dic₆₁⋊C₅	1	1	yes	characteristic, direct product factor, Sylow
46	C₂	C₃×C₅×D₆₁⋊C₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃×C₅	D₆₁⋊C₅
commutator, hypocenter	C₆₁	C₃×C₄×C₅²
Frattini	C₂	C₃×C₅×D₆₁⋊C₅
Fitting, socle	C₂×C₃×C₅×C₆₁	C₂×C₅
prime radical	C₂×C₃×C₅×C₆₁⋊C₅	C₂
cosocle	C₂×C₆₁	C₂×C₃×C₅²
LCS	G ▷ C₆₁
UCS	C₁ ◁ C₂×C₃×C₅

Complements

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

N	H ≅ G/N	type
2. C₅×Dic₆₁⋊C₅	C₃	direct (N × H)
3. C₃×C₅×C₆₁⋊C₅	C₄	semidirect
4. C₃×C₅×Dic₆₁	C₅	semidirect
5. C₃×Dic₆₁⋊C₅	C₅	direct (N × H)
9. C₅×C₆₁⋊C₅	C₃×C₄	semidirect
10. C₅×Dic₆₁	C₃×C₅	semidirect
11. Dic₆₁⋊C₅	C₃×C₅	direct (N × H)
12. C₃×C₅×C₆₁	C₄×C₅	semidirect
13. C₃×C₆₁⋊C₅	C₄×C₅	semidirect
14. C₃×Dic₆₁	C₅²	semidirect
18. C₅×C₆₁	C₃×C₄×C₅	semidirect
19. C₆₁⋊C₅	C₃×C₄×C₅	semidirect
21. Dic₆₁	C₃×C₅²	semidirect
22. C₃×C₆₁	C₄×C₅²	semidirect
27. C₆₁	C₃×C₄×C₅²	semidirect
37. C₃×C₅	Dic₆₁⋊C₅	direct (N × H)
43. C₅	C₃×Dic₆₁⋊C₅	direct (N × H)
45. C₃	C₅×Dic₆₁⋊C₅	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×C₃×C₅×C₆₁⋊C₅	2
C₅×Dic₆₁⋊C₅	3
C₃×C₅×Dic₆₁	5
C₃×Dic₆₁⋊C₅	5
C₃×C₄×C₅²	61

Subgroups — C3×C5×Dic61⋊C5 #32912953

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×Dic₆₁⋊C₅ #32912953