Subgroups — C₃⋊(C₅⋊(#41⋊C₃)) #10624075

Order 720.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	#6821	C₃	1	1	yes	maximal, characteristic, complemented
3	C₅⋊(#41⋊C₃)	—	3	1	no	maximal
4	C₃×Dic₁₅	—	4	1	no	maximal
5	C₃⋊(#41⋊C₃)	—	5	1	no	maximal, Hall
6	C₃×C₅×Q₈	C₂×C₃	1	1	yes	characteristic
7	D₆₀	—	3	1	no	—
8	C₄×D₁₅	—	3	1	no	—
9	C₅×SL(2,𝔽₃)	—	2	1	no	—
10	C₅×SL(2,𝔽₃)	D₃	1	1	yes	characteristic, complemented
11	C₂×C₃²×C₅	—	4	1	no	—
12	#713	—	3	1	no	Hall
13	C₃×SL(2,𝔽₃)	D₅	1	1	yes	characteristic, complemented
14	C₂×D₁₅	—	3	1	no	—
15	C₃×C₄×C₅	—	3	1	no	—
16	Dic₁₅	A₄	1	1	yes	characteristic
17	C₃×Dic₅	—	12	1	no	—
18	#239	—	5	1	no	—
19	#41⋊C₃	—	15	1	no	—
20	C₃²×C₅	—	4	1	no	Hall
21	C₅×Q₈	C₃×D₃	1	1	yes	characteristic
22	D₂₀	—	9	1	no	—
23	C₄×D₅	—	9	1	no	—
24	C₃×Dic₃	—	20	1	no	—
25	C₂×C₃×C₅	—	4	1	no	—
26	C₂×C₃×C₅	—	8	1	no	—
27	C₂×C₃×C₅	C₂×A₄	1	1	yes	characteristic
28	D₁₅	—	6	1	no	—
29	C₃×Q₈	C₃×D₅	1	1	yes	characteristic
30	D₁₂	—	15	1	no	—
31	C₄×D₃	—	15	1	no	—
32	SL(2,𝔽₃)	—	2	1	no	—
33	SL(2,𝔽₃)	D₁₅	1	1	yes	characteristic, complemented
34	C₂×D₅	—	9	1	no	—
35	C₄×C₅	—	3	1	no	—
36	Dic₅	—	3	1	no	—
37	C₂×C₃²	—	4	1	no	—
38	#41	—	15	1	no	Sylow
39	C₃×C₅	—	4	1	no	—
40	C₃×C₅	—	8	1	no	—
41	C₃×C₅	#41⋊C₃	1	1	yes	characteristic, complemented
42	C₂×D₃	—	15	1	no	—
43	C₃×C₄	—	3	1	no	—
44	C₃×C₄	—	60	1	no	—
45	Dic₃	—	5	1	no	—
46	C₂×C₅	D₃×A₄	1	1	yes	characteristic
47	D₅	—	18	1	no	—
48	C₃²	—	4	1	no	Sylow
49	Q₈	C₃×D₁₅	1	1	yes	characteristic
50	D₄	—	45	1	no	—
51	C₂×C₄	—	45	1	no	—
52	C₂×C₃	—	4	1	no	—
53	C₂×C₃	—	8	1	no	—
54	C₂×C₃	D₅×A₄	1	1	yes	characteristic
55	D₃	—	30	1	no	—
56	C₅	C₃⋊(#41⋊C₃)	1	1	yes	characteristic, complemented, Sylow
57	C₂²	—	45	1	no	—
58	C₄	—	3	1	no	—
59	C₄	—	15	1	no	—
60	C₃	—	4	1	no	—
61	C₃	—	8	1	no	—
62	C₃	C₅⋊(#41⋊C₃)	1	1	yes	characteristic, complemented
63	C₂	—	90	1	no	—
64	C₂	A₄×D₁₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	A₄×D₁₅
commutator, Fitting, cosocle, hypocenter	C₃×C₅×Q₈	C₂×C₃
socle	C₂×C₃×C₅	C₂×A₄
LCS	G ▷ C₃×C₅×Q₈
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×SL(2,𝔽₃)	C₂	semidirect
2. #6821	C₃	semidirect
10. C₅×SL(2,𝔽₃)	D₃	semidirect
13. C₃×SL(2,𝔽₃)	D₅	semidirect
33. SL(2,𝔽₃)	D₁₅	semidirect
41. C₃×C₅	#41⋊C₃	semidirect
56. C₅	C₃⋊(#41⋊C₃)	semidirect
62. C₃	C₅⋊(#41⋊C₃)	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×SL(2,𝔽₃)	2
#6821	3
C₅⋊(#41⋊C₃)	3
C₃×Dic₁₅	4
C₃⋊(#41⋊C₃)	5

Subgroups — C3⋊(C5⋊(#41⋊C3)) #10624075

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃⋊(C₅⋊(#41⋊C₃)) #10624075