Subgroups — C₂×C₃²×C₄×C₅ #67185

Order 360.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₅	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₄×C₅	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₄×C₅	C₃	1	4	yes	maximal, direct product factor
4	C₂×C₃²×C₅	C₄	1	2	yes	direct product factor
5	C₂×C₃²×C₅	C₂²	1	1	yes	characteristic
6	C₂×C₃²×C₄	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅	C₂×C₃	1	4	yes	—
8	C₃×C₄×C₅	C₂×C₃	1	8	yes	direct product factor
9	C₃²×C₅	C₂×C₄	1	1	yes	characteristic, direct product factor, Hall
10	C₂×C₄×C₅	C₃²	1	1	yes	characteristic, direct product factor, Hall
11	C₂²×C₃²	C₂×C₅	1	1	yes	characteristic
12	C₃²×C₄	C₂×C₅	1	2	yes	direct product factor
13	C₂×C₃×C₅	C₃×C₄	1	8	yes	direct product factor
14	C₂×C₃×C₅	C₂²×C₃	1	4	yes	—
15	C₂×C₃×C₄	C₃×C₅	1	4	yes	direct product factor
16	C₂²×C₅	C₂×C₃²	1	1	yes	characteristic
17	C₄×C₅	C₂×C₃²	1	2	yes	direct product factor
18	C₂×C₃²	C₄×C₅	1	2	yes	direct product factor
19	C₂×C₃²	C₂²×C₅	1	1	yes	characteristic
20	C₃×C₅	C₂×C₃×C₄	1	4	yes	direct product factor
21	C₂²×C₃	C₂×C₃×C₅	1	4	yes	—
22	C₃×C₄	C₂×C₃×C₅	1	8	yes	direct product factor
23	C₂×C₅	C₃²×C₄	1	2	yes	direct product factor
24	C₂×C₅	C₂²×C₃²	1	1	yes	characteristic
25	C₃²	C₂×C₄×C₅	1	1	yes	characteristic, direct product factor, Sylow
26	C₂×C₄	C₃²×C₅	1	1	yes	characteristic, direct product factor, Sylow
27	C₂×C₃	C₃×C₄×C₅	1	8	yes	direct product factor
28	C₂×C₃	C₂²×C₃×C₅	1	4	yes	—
29	C₅	C₂×C₃²×C₄	1	1	yes	characteristic, direct product factor, Sylow
30	C₂²	C₂×C₃²×C₅	1	1	yes	characteristic
31	C₄	C₂×C₃²×C₅	1	2	yes	direct product factor
32	C₃	C₂×C₃×C₄×C₅	1	4	yes	direct product factor
33	C₂	C₃²×C₄×C₅	1	2	yes	direct product factor
34	C₂	C₂²×C₃²×C₅	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂²×C₃²×C₅
socle, prime radical	C₂²×C₃²×C₅	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃²×C₄×C₅	C₂	direct (N × H)
3. C₂×C₃×C₄×C₅	C₃	direct (N × H)
4. C₂×C₃²×C₅	C₄	direct (N × H)
6. C₂×C₃²×C₄	C₅	direct (N × H)
8. C₃×C₄×C₅	C₂×C₃	direct (N × H)
9. C₃²×C₅	C₂×C₄	direct (N × H)
10. C₂×C₄×C₅	C₃²	direct (N × H)
12. C₃²×C₄	C₂×C₅	direct (N × H)
13. C₂×C₃×C₅	C₃×C₄	direct (N × H)
15. C₂×C₃×C₄	C₃×C₅	direct (N × H)
17. C₄×C₅	C₂×C₃²	direct (N × H)
18. C₂×C₃²	C₄×C₅	direct (N × H)
20. C₃×C₅	C₂×C₃×C₄	direct (N × H)
22. C₃×C₄	C₂×C₃×C₅	direct (N × H)
23. C₂×C₅	C₃²×C₄	direct (N × H)
25. C₃²	C₂×C₄×C₅	direct (N × H)
26. C₂×C₄	C₃²×C₅	direct (N × H)
27. C₂×C₃	C₃×C₄×C₅	direct (N × H)
29. C₅	C₂×C₃²×C₄	direct (N × H)
31. C₄	C₂×C₃²×C₅	direct (N × H)
32. C₃	C₂×C₃×C₄×C₅	direct (N × H)
33. C₂	C₃²×C₄×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
C₂²×C₃²×C₅	2
C₃²×C₄×C₅	2
C₂×C₃×C₄×C₅	3
C₂×C₃²×C₄	5

Subgroups — C2×C32×C4×C5 #67185

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×C₄×C₅ #67185