Subgroups — C₃×C₅×H(5,𝔽₂) #92108

Order 480.

332 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#41	C₂	1	6	yes	maximal, complemented
2	C₂×C₃×C₅×D₄	C₂	1	9	yes	maximal, complemented
3	C₅×H(5,𝔽₂)	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂³×C₃×C₅	C₂²	1	6	yes	complemented
5	C₃×C₅×Q₈	C₂²	1	2	yes	complemented
6	C₃×C₅×D₄	C₂²	1	18	yes	complemented
7	C₂×C₃×C₄×C₅	C₂²	1	9	yes	complemented
8	C₃×H(5,𝔽₂)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₅×#41	C₂×C₃	1	6	yes	complemented
10	C₂×C₅×D₄	C₂×C₃	1	9	yes	complemented
11	C₂²×C₃×C₅	—	4	6	no	—
12	C₂²×C₃×C₅	C₂³	1	9	yes	—
13	C₃×C₄×C₅	C₂³	1	6	yes	—
14	C₃×#41	C₂×C₅	1	6	yes	complemented
15	C₂×C₃×D₄	C₂×C₅	1	9	yes	complemented
16	C₂³×C₅	C₂²×C₃	1	6	yes	complemented
17	C₅×Q₈	C₂²×C₃	1	2	yes	complemented
18	C₅×D₄	C₂²×C₃	1	18	yes	complemented
19	C₂×C₄×C₅	C₂²×C₃	1	9	yes	complemented
20	H(5,𝔽₂)	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
21	C₂×C₃×C₅	—	2	9	no	—
22	C₂×C₃×C₅	C₂⁴	1	1	yes	characteristic
23	C₂³×C₃	C₂²×C₅	1	6	yes	complemented
24	C₃×Q₈	C₂²×C₅	1	2	yes	complemented
25	C₃×D₄	C₂²×C₅	1	18	yes	complemented
26	C₂×C₃×C₄	C₂²×C₅	1	9	yes	complemented
27	C₂²×C₅	—	4	6	no	—
28	C₂²×C₅	C₂³×C₃	1	9	yes	—
29	C₄×C₅	C₂³×C₃	1	6	yes	—
30	#41	C₂×C₃×C₅	1	6	yes	complemented
31	C₂×D₄	C₂×C₃×C₅	1	9	yes	complemented
32	C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic, direct product factor, Hall
33	C₂²×C₃	—	4	6	no	—
34	C₂²×C₃	C₂³×C₅	1	9	yes	—
35	C₃×C₄	C₂³×C₅	1	6	yes	—
36	C₂×C₅	—	2	9	no	—
37	C₂×C₅	C₂⁴×C₃	1	1	yes	characteristic
38	C₂³	C₂²×C₃×C₅	1	6	yes	complemented
39	Q₈	C₂²×C₃×C₅	1	2	yes	complemented
40	D₄	C₂²×C₃×C₅	1	18	yes	complemented
41	C₂×C₄	C₂²×C₃×C₅	1	9	yes	complemented
42	C₂×C₃	—	2	9	no	—
43	C₂×C₃	C₂⁴×C₅	1	1	yes	characteristic
44	C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic, direct product factor, Sylow
45	C₂²	—	4	6	no	—
46	C₂²	C₂³×C₃×C₅	1	9	yes	—
47	C₄	C₂³×C₃×C₅	1	6	yes	—
48	C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic, direct product factor, Sylow
49	C₂	—	2	9	no	—
50	C₂	C₂⁴×C₃×C₅	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃×C₅	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁴×C₃×C₅
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#41	C₂	semidirect
2. C₂×C₃×C₅×D₄	C₂	semidirect
3. C₅×H(5,𝔽₂)	C₃	direct (N × H)
4. C₂³×C₃×C₅	C₂²	semidirect
5. C₃×C₅×Q₈	C₂²	semidirect
6. C₃×C₅×D₄	C₂²	semidirect
7. C₂×C₃×C₄×C₅	C₂²	semidirect
8. C₃×H(5,𝔽₂)	C₅	direct (N × H)
9. C₅×#41	C₂×C₃	semidirect
10. C₂×C₅×D₄	C₂×C₃	semidirect
14. C₃×#41	C₂×C₅	semidirect
15. C₂×C₃×D₄	C₂×C₅	semidirect
16. C₂³×C₅	C₂²×C₃	semidirect
17. C₅×Q₈	C₂²×C₃	semidirect
18. C₅×D₄	C₂²×C₃	semidirect
19. C₂×C₄×C₅	C₂²×C₃	semidirect
20. H(5,𝔽₂)	C₃×C₅	direct (N × H)
23. C₂³×C₃	C₂²×C₅	semidirect
24. C₃×Q₈	C₂²×C₅	semidirect
25. C₃×D₄	C₂²×C₅	semidirect
26. C₂×C₃×C₄	C₂²×C₅	semidirect
30. #41	C₂×C₃×C₅	semidirect
31. C₂×D₄	C₂×C₃×C₅	semidirect
32. C₃×C₅	H(5,𝔽₂)	direct (N × H)
38. C₂³	C₂²×C₃×C₅	semidirect
39. Q₈	C₂²×C₃×C₅	semidirect
40. D₄	C₂²×C₃×C₅	semidirect
41. C₂×C₄	C₂²×C₃×C₅	semidirect
44. C₅	C₃×H(5,𝔽₂)	direct (N × H)
48. C₃	C₅×H(5,𝔽₂)	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×#41	2
C₂×C₃×C₅×D₄	2
C₅×H(5,𝔽₂)	3
C₃×H(5,𝔽₂)	5

Subgroups — C3×C5×H(5,𝔽2) #92108

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×H(5,𝔽₂) #92108