Subgroups — C₂²×C₅×C₃⋊D₃ #67230

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×C₅×C₃⋊D₃	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₅×D₃	—	3	4	no	maximal
4	C₂×C₃²×C₅	C₂²	1	3	yes	complemented
5	C₅×C₃⋊D₃	C₂²	1	4	yes	direct product factor
6	C₂²×C₃⋊D₃	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅	D₃	1	4	yes	complemented
8	C₂×C₅×D₃	—	3	24	no	—
9	C₃²×C₅	C₂³	1	1	yes	characteristic, complemented, Hall
10	C₂³×C₅	—	9	1	no	Hall
11	C₂²×C₃²	C₂×C₅	1	1	yes	characteristic, complemented
12	C₂×C₃⋊D₃	C₂×C₅	1	6	yes	direct product factor
13	C₂×C₃×C₅	C₂×D₃	1	12	yes	complemented
14	C₅×D₃	—	3	16	no	—
15	C₂²×D₃	—	3	4	no	—
16	C₂²×C₅	—	9	6	no	—
17	C₂²×C₅	C₃⋊D₃	1	1	yes	characteristic, direct product factor
18	C₂×C₃²	C₂²×C₅	1	3	yes	complemented
19	C₃⋊D₃	C₂²×C₅	1	4	yes	direct product factor
20	C₃×C₅	C₂²×D₃	1	4	yes	complemented
21	C₂²×C₃	C₅×D₃	1	4	yes	complemented
22	C₂×D₃	—	3	24	no	—
23	C₂×C₅	—	9	4	no	—
24	C₂×C₅	C₂×C₃⋊D₃	1	3	yes	direct product factor
25	C₃²	C₂³×C₅	1	1	yes	characteristic, complemented, Sylow
26	C₂³	—	9	1	no	Sylow
27	C₂×C₃	C₂×C₅×D₃	1	12	yes	complemented
28	D₃	—	3	16	no	—
29	C₅	C₂²×C₃⋊D₃	1	1	yes	characteristic, direct product factor, Sylow
30	C₂²	—	9	6	no	—
31	C₂²	C₅×C₃⋊D₃	1	1	yes	characteristic, direct product factor
32	C₃	C₂²×C₅×D₃	1	4	yes	complemented
33	C₂	—	9	4	no	—
34	C₂	C₂×C₅×C₃⋊D₃	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₅	C₃⋊D₃
commutator, cosocle, hypocenter	C₃²	C₂³×C₅
Frattini	C₁	G
Fitting, socle	C₂²×C₃²×C₅	C₂
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂²×C₅

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃²×C₅	C₂	semidirect
2. C₂×C₅×C₃⋊D₃	C₂	direct (N × H)
4. C₂×C₃²×C₅	C₂²	semidirect
5. C₅×C₃⋊D₃	C₂²	direct (N × H)
6. C₂²×C₃⋊D₃	C₅	direct (N × H)
7. C₂²×C₃×C₅	D₃	semidirect
9. C₃²×C₅	C₂³	semidirect
11. C₂²×C₃²	C₂×C₅	semidirect
12. C₂×C₃⋊D₃	C₂×C₅	direct (N × H)
13. C₂×C₃×C₅	C₂×D₃	semidirect
17. C₂²×C₅	C₃⋊D₃	direct (N × H)
18. C₂×C₃²	C₂²×C₅	semidirect
19. C₃⋊D₃	C₂²×C₅	direct (N × H)
20. C₃×C₅	C₂²×D₃	semidirect
21. C₂²×C₃	C₅×D₃	semidirect
24. C₂×C₅	C₂×C₃⋊D₃	direct (N × H)
25. C₃²	C₂³×C₅	semidirect
27. C₂×C₃	C₂×C₅×D₃	semidirect
29. C₅	C₂²×C₃⋊D₃	direct (N × H)
31. C₂²	C₅×C₃⋊D₃	direct (N × H)
32. C₃	C₂²×C₅×D₃	semidirect
34. C₂	C₂×C₅×C₃⋊D₃	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×C₅×C₃⋊D₃	2
C₂²×C₅×D₃	3
C₂²×C₃⋊D₃	5

Subgroups — C22×C5×C3⋊D3 #67230

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×C₃⋊D₃ #67230