Subgroups — C₃×Dic₃₀₅⋊C₅ #32912954

Order 18300.

64 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	Dic₃₀₅⋊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×Dic₃₀₅	C₅	1	1	yes	maximal, characteristic, complemented
5	C₃×Dic₆₁⋊C₅	—	5	1	no	maximal
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	—	2	2	no	—
9	C₂×C₃×C₆₁⋊C₅	D₅	1	1	yes	characteristic
10	C₅×C₆₁⋊C₅	C₃×C₄	1	1	yes	characteristic, complemented, Hall
11	Dic₃₀₅	C₃×C₅	1	1	yes	characteristic, complemented
12	Dic₆₁⋊C₅	—	5	1	no	—
13	C₃×C₅×C₆₁	C₄×C₅	1	1	yes	characteristic, complemented
14	C₃×C₆₁⋊C₅	—	2	2	no	—
15	C₃×C₆₁⋊C₅	Dic₅	1	1	yes	characteristic, complemented
16	C₃×Dic₆₁	—	5	1	no	Hall
17	C₂×C₅×C₆₁	C₂×C₃×C₅	1	1	yes	characteristic
18	C₂×C₆₁⋊C₅	—	2	2	no	—
19	C₂×C₆₁⋊C₅	C₃×D₅	1	1	yes	characteristic
20	C₂×C₃×C₆₁	C₅×D₅	1	1	yes	characteristic
21	C₅×C₆₁	C₃×C₄×C₅	1	1	yes	characteristic, complemented
22	C₆₁⋊C₅	—	2	2	no	—
23	C₆₁⋊C₅	C₃×Dic₅	1	1	yes	characteristic, complemented
24	C₃×C₅×Dic₅	—	61	1	no	maximal, Hall
25	Dic₆₁	—	5	1	no	Hall
26	C₃×C₆₁	C₅×Dic₅	1	1	yes	characteristic, complemented, Hall
27	C₂×C₃×C₅²	—	122	1	no	—
28	C₂×C₆₁	C₃×C₅×D₅	1	1	yes	characteristic
29	C₅×Dic₅	—	61	1	no	Hall
30	C₃×C₅²	—	122	1	no	Hall
31	C₆₁	C₃×C₅×Dic₅	1	1	yes	characteristic, complemented, Sylow
32	C₃×C₄×C₅	—	305	1	no	—
33	C₃×Dic₅	—	61	1	no	—
34	C₂×C₅²	—	122	1	no	—
35	C₂×C₃×C₅	—	122	1	no	—
36	C₂×C₃×C₅	—	122	2	no	—
37	C₂×C₃×C₅	D₆₁⋊C₅	1	1	yes	characteristic
38	C₅²	—	122	1	no	Sylow
39	C₄×C₅	—	305	1	no	—
40	Dic₅	—	61	1	no	—
41	C₃×C₅	—	122	2	no	—
42	C₃×C₅	—	122	1	no	—
43	C₃×C₅	Dic₆₁⋊C₅	1	1	yes	characteristic, complemented
44	C₃×C₄	—	305	1	no	Hall
45	C₂×C₅	—	122	1	no	—
46	C₂×C₅	—	122	2	no	—
47	C₂×C₅	C₃×D₆₁⋊C₅	1	1	yes	characteristic
48	C₂×C₃	C₅⋊(D₆₁⋊C₅)	1	1	yes	characteristic
49	C₅	—	122	2	no	—
50	C₅	—	122	1	no	—
51	C₅	C₃×Dic₆₁⋊C₅	1	1	yes	characteristic, complemented
52	C₄	—	305	1	no	Sylow
53	C₃	Dic₃₀₅⋊C₅	1	1	yes	characteristic, direct product factor, Sylow
54	C₂	C₃×C₅⋊(D₆₁⋊C₅)	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. Dic₃₀₅⋊C₅	C₃	direct (N × H)
3. C₃×C₅×C₆₁⋊C₅	C₄	semidirect
4. C₃×Dic₃₀₅	C₅	semidirect
10. C₅×C₆₁⋊C₅	C₃×C₄	semidirect
11. Dic₃₀₅	C₃×C₅	semidirect
13. C₃×C₅×C₆₁	C₄×C₅	semidirect
15. C₃×C₆₁⋊C₅	Dic₅	semidirect
21. C₅×C₆₁	C₃×C₄×C₅	semidirect
23. C₆₁⋊C₅	C₃×Dic₅	semidirect
26. C₃×C₆₁	C₅×Dic₅	semidirect
31. C₆₁	C₃×C₅×Dic₅	semidirect
43. C₃×C₅	Dic₆₁⋊C₅	semidirect
51. C₅	C₃×Dic₆₁⋊C₅	semidirect
53. C₃	Dic₃₀₅⋊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
Dic₃₀₅⋊C₅	3
C₃×Dic₃₀₅	5
C₃×Dic₆₁⋊C₅	5
C₃×C₅×Dic₅	61

Subgroups — C3×Dic305⋊C5 #32912954

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×Dic₃₀₅⋊C₅ #32912954