Subgroups — C₃×C₅×#41 #6827

Order 240.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×D₄	C₂	1	3	yes	maximal, complemented
3	C₂×C₃×C₄×C₅	C₂	1	3	yes	maximal, complemented
4	C₅×#41	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×C₅	C₂²	1	3	yes	—
6	C₃×C₄×C₅	C₂²	1	3	yes	—
7	C₃×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₃×#41	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₅×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
10	C₅×D₄	C₂×C₃	1	3	yes	complemented
11	C₂×C₄×C₅	C₂×C₃	1	3	yes	complemented
12	C₂×C₃×C₅	—	2	3	no	—
13	C₂×C₃×C₅	C₂³	1	1	yes	characteristic
14	C₃×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
15	C₃×D₄	C₂×C₅	1	3	yes	complemented
16	C₂×C₃×C₄	C₂×C₅	1	3	yes	complemented
17	C₂²×C₅	C₂²×C₃	1	3	yes	—
18	C₄×C₅	C₂²×C₃	1	3	yes	—
19	C₄×C₅	C₂²×C₃	1	1	yes	characteristic
20	#41	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
21	C₃×C₅	#41	1	1	yes	characteristic, direct product factor, Hall
22	C₂²×C₃	C₂²×C₅	1	3	yes	—
23	C₃×C₄	C₂²×C₅	1	3	yes	—
24	C₃×C₄	C₂²×C₅	1	1	yes	characteristic
25	C₂×C₅	—	2	3	no	—
26	C₂×C₅	C₂³×C₃	1	1	yes	characteristic
27	Q₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
28	D₄	C₂×C₃×C₅	1	3	yes	complemented
29	C₂×C₄	C₂×C₃×C₅	1	3	yes	complemented
30	C₂×C₃	—	2	3	no	—
31	C₂×C₃	C₂³×C₅	1	1	yes	characteristic
32	C₅	C₃×#41	1	1	yes	characteristic, direct product factor, Sylow
33	C₂²	C₂²×C₃×C₅	1	3	yes	—
34	C₄	C₂²×C₃×C₅	1	1	yes	characteristic
35	C₄	C₂²×C₃×C₅	1	3	yes	—
36	C₃	C₅×#41	1	1	yes	characteristic, direct product factor, Sylow
37	C₂	—	2	3	no	—
38	C₂	C₂³×C₃×C₅	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄×C₅	C₂²
commutator, Frattini, cosocle	C₂	C₂³×C₃×C₅
socle	C₂×C₃×C₅	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×Q₈	C₂	semidirect
2. C₃×C₅×D₄	C₂	semidirect
3. C₂×C₃×C₄×C₅	C₂	semidirect
4. C₅×#41	C₃	direct (N × H)
8. C₃×#41	C₅	direct (N × H)
9. C₅×Q₈	C₂×C₃	semidirect
10. C₅×D₄	C₂×C₃	semidirect
11. C₂×C₄×C₅	C₂×C₃	semidirect
14. C₃×Q₈	C₂×C₅	semidirect
15. C₃×D₄	C₂×C₅	semidirect
16. C₂×C₃×C₄	C₂×C₅	semidirect
20. #41	C₃×C₅	direct (N × H)
21. C₃×C₅	#41	direct (N × H)
27. Q₈	C₂×C₃×C₅	semidirect
28. D₄	C₂×C₃×C₅	semidirect
29. C₂×C₄	C₂×C₃×C₅	semidirect
32. C₅	C₃×#41	direct (N × H)
36. C₃	C₅×#41	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×C₅×Q₈	2
C₃×C₅×D₄	2
C₂×C₃×C₄×C₅	2
C₅×#41	3
C₃×#41	5

Subgroups — C3×C5×#41 #6827

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#41 #6827