Subgroups — C₂²×C₃²×D₅ #67227

Order 360.

192 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, complemented
2	C₂×C₃²×D₅	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃×D₅	C₃	1	4	yes	maximal, direct product factor
4	C₂×C₃²×C₅	C₂²	1	3	yes	complemented
5	C₃²×D₅	C₂²	1	4	yes	direct product factor
6	C₂³×C₃²	—	5	1	no	maximal, Hall
7	C₂²×C₃×C₅	C₂×C₃	1	4	yes	complemented
8	C₂×C₃×D₅	C₂×C₃	1	24	yes	direct product factor
9	C₃²×C₅	C₂³	1	1	yes	characteristic, complemented, Hall
10	C₂²×D₅	C₃²	1	1	yes	characteristic, direct product factor, Hall
11	C₂²×C₃²	—	5	6	no	—
12	C₂²×C₃²	D₅	1	1	yes	characteristic, direct product factor
13	C₂×C₃×C₅	C₂²×C₃	1	12	yes	complemented
14	C₃×D₅	C₂²×C₃	1	16	yes	direct product factor
15	C₂³×C₃	—	5	4	no	—
16	C₂²×C₅	C₂×C₃²	1	1	yes	characteristic, complemented
17	C₂×D₅	C₂×C₃²	1	6	yes	direct product factor
18	C₂×C₃²	—	5	4	no	—
19	C₂×C₃²	C₂×D₅	1	3	yes	direct product factor
20	C₃×C₅	C₂³×C₃	1	4	yes	complemented
21	C₂²×C₃	—	5	24	no	—
22	C₂²×C₃	C₃×D₅	1	4	yes	direct product factor
23	C₂×C₅	C₂²×C₃²	1	3	yes	complemented
24	D₅	C₂²×C₃²	1	4	yes	direct product factor
25	C₃²	C₂²×D₅	1	1	yes	characteristic, direct product factor, Sylow
26	C₂³	—	5	1	no	Sylow
27	C₂×C₃	—	5	16	no	—
28	C₂×C₃	C₂×C₃×D₅	1	12	yes	direct product factor
29	C₅	C₂³×C₃²	1	1	yes	characteristic, complemented, Sylow
30	C₂²	—	5	6	no	—
31	C₂²	C₃²×D₅	1	1	yes	characteristic, direct product factor
32	C₃	C₂²×C₃×D₅	1	4	yes	direct product factor
33	C₂	—	5	4	no	—
34	C₂	C₂×C₃²×D₅	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₃²	D₅
commutator, cosocle, hypocenter	C₅	C₂³×C₃²
Frattini	C₁	G
Fitting, socle	C₂²×C₃²×C₅	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₂²×C₃²

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃²×C₅	C₂	semidirect
2. C₂×C₃²×D₅	C₂	direct (N × H)
3. C₂²×C₃×D₅	C₃	direct (N × H)
4. C₂×C₃²×C₅	C₂²	semidirect
5. C₃²×D₅	C₂²	direct (N × H)
7. C₂²×C₃×C₅	C₂×C₃	semidirect
8. C₂×C₃×D₅	C₂×C₃	direct (N × H)
9. C₃²×C₅	C₂³	semidirect
10. C₂²×D₅	C₃²	direct (N × H)
12. C₂²×C₃²	D₅	direct (N × H)
13. C₂×C₃×C₅	C₂²×C₃	semidirect
14. C₃×D₅	C₂²×C₃	direct (N × H)
16. C₂²×C₅	C₂×C₃²	semidirect
17. C₂×D₅	C₂×C₃²	direct (N × H)
19. C₂×C₃²	C₂×D₅	direct (N × H)
20. C₃×C₅	C₂³×C₃	semidirect
22. C₂²×C₃	C₃×D₅	direct (N × H)
23. C₂×C₅	C₂²×C₃²	semidirect
24. D₅	C₂²×C₃²	direct (N × H)
25. C₃²	C₂²×D₅	direct (N × H)
28. C₂×C₃	C₂×C₃×D₅	direct (N × H)
29. C₅	C₂³×C₃²	semidirect
31. C₂²	C₃²×D₅	direct (N × H)
32. C₃	C₂²×C₃×D₅	direct (N × H)
34. C₂	C₂×C₃²×D₅	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
C₂×C₃²×D₅	2
C₂²×C₃×D₅	3
C₂³×C₃²	5

Subgroups — C22×C32×D5 #67227

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃²×D₅ #67227