Subgroups — C₂×C₃×C₅×Q₈ #6826

Order 240.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×Q₈	C₂	1	4	yes	maximal, direct product factor
2	C₂×C₃×C₄×C₅	C₂	1	3	yes	maximal
3	C₂×C₅×Q₈	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂²×C₃×C₅	C₂²	1	1	yes	characteristic
5	C₃×C₄×C₅	C₂²	1	6	yes	—
6	C₂×C₃×Q₈	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₅×Q₈	C₂×C₃	1	4	yes	direct product factor
8	C₂×C₄×C₅	C₂×C₃	1	3	yes	—
9	C₂×C₃×C₅	Q₈	1	2	yes	direct product factor
10	C₂×C₃×C₅	C₂³	1	1	yes	characteristic
11	C₃×Q₈	C₂×C₅	1	4	yes	direct product factor
12	C₂×C₃×C₄	C₂×C₅	1	3	yes	—
13	C₂²×C₅	C₂²×C₃	1	1	yes	characteristic
14	C₄×C₅	C₂²×C₃	1	6	yes	—
15	C₂×Q₈	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
16	C₃×C₅	C₂×Q₈	1	1	yes	characteristic, direct product factor, Hall
17	C₂²×C₃	C₂²×C₅	1	1	yes	characteristic
18	C₃×C₄	C₂²×C₅	1	6	yes	—
19	C₂×C₅	C₃×Q₈	1	2	yes	direct product factor
20	C₂×C₅	C₂³×C₃	1	1	yes	characteristic
21	Q₈	C₂×C₃×C₅	1	4	yes	direct product factor
22	C₂×C₄	C₂×C₃×C₅	1	3	yes	—
23	C₂×C₃	C₅×Q₈	1	2	yes	direct product factor
24	C₂×C₃	C₂³×C₅	1	1	yes	characteristic
25	C₅	C₂×C₃×Q₈	1	1	yes	characteristic, direct product factor, Sylow
26	C₂²	C₂²×C₃×C₅	1	1	yes	characteristic
27	C₄	C₂²×C₃×C₅	1	6	yes	—
28	C₃	C₂×C₅×Q₈	1	1	yes	characteristic, direct product factor, Sylow
29	C₂	C₃×C₅×Q₈	1	2	yes	direct product factor
30	C₂	C₂³×C₃×C₅	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×Q₈	C₂	direct (N × H)
3. C₂×C₅×Q₈	C₃	direct (N × H)
6. C₂×C₃×Q₈	C₅	direct (N × H)
7. C₅×Q₈	C₂×C₃	direct (N × H)
9. C₂×C₃×C₅	Q₈	direct (N × H)
11. C₃×Q₈	C₂×C₅	direct (N × H)
15. C₂×Q₈	C₃×C₅	direct (N × H)
16. C₃×C₅	C₂×Q₈	direct (N × H)
19. C₂×C₅	C₃×Q₈	direct (N × H)
21. Q₈	C₂×C₃×C₅	direct (N × H)
23. C₂×C₃	C₅×Q₈	direct (N × H)
25. C₅	C₂×C₃×Q₈	direct (N × H)
28. C₃	C₂×C₅×Q₈	direct (N × H)
29. C₂	C₃×C₅×Q₈	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
C₂×C₃×C₄×C₅	2
C₂×C₅×Q₈	3
C₂×C₃×Q₈	5

Subgroups — C2×C3×C5×Q8 #6826

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×Q₈ #6826