Subgroups — C₃²×D₅² #11744711

Order 900.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃²×C₅×D₅	C₂	1	2	yes	maximal, complemented
3	C₃×D₅²	C₃	1	4	yes	maximal, direct product factor
4	C₃²×C₅²	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₃²×D₅	—	5	2	no	maximal
6	C₃×C₅⋊D₅	C₂×C₃	1	4	yes	complemented
7	C₃×C₅×D₅	C₂×C₃	1	8	yes	complemented
8	D₅²	C₃²	1	1	yes	characteristic, direct product factor, Hall
9	C₂×C₃²×C₅	—	5	2	no	—
10	C₃²×D₅	—	5	2	no	—
11	C₃²×D₅	—	10	2	no	—
12	C₃²×D₅	D₅	1	2	yes	direct product factor
13	C₃×C₅²	C₂²×C₃	1	4	yes	complemented
14	C₂×C₃×D₅	—	5	8	no	—
15	C₅⋊D₅	C₂×C₃²	1	1	yes	characteristic, complemented
16	C₅×D₅	C₂×C₃²	1	2	yes	complemented
17	C₃²×C₅	—	2	2	no	—
18	C₃²×C₅	C₂×D₅	1	2	yes	complemented
19	C₂²×C₃²	—	25	1	no	Hall
20	C₂×C₃×C₅	—	5	8	no	—
21	C₃×D₅	—	5	8	no	—
22	C₃×D₅	—	10	8	no	—
23	C₃×D₅	C₃×D₅	1	8	yes	direct product factor
24	C₅²	C₂²×C₃²	1	1	yes	characteristic, complemented, Sylow
25	C₂×D₅	—	5	2	no	—
26	C₂×C₃²	—	5	2	no	—
27	C₂×C₃²	—	25	1	no	—
28	C₃×C₅	—	2	8	no	—
29	C₃×C₅	C₂×C₃×D₅	1	8	yes	complemented
30	C₂²×C₃	—	25	4	no	—
31	C₂×C₅	—	5	2	no	—
32	D₅	—	5	2	no	—
33	D₅	—	10	2	no	—
34	D₅	C₃²×D₅	1	2	yes	direct product factor
35	C₃²	D₅²	1	1	yes	characteristic, direct product factor, Sylow
36	C₂×C₃	—	5	8	no	—
37	C₂×C₃	—	25	4	no	—
38	C₅	—	2	2	no	—
39	C₅	C₂×C₃²×D₅	1	2	yes	complemented
40	C₂²	—	25	1	no	Sylow
41	C₃	C₃×D₅²	1	4	yes	direct product factor
42	C₂	—	5	2	no	—
43	C₂	—	25	1	no	—

	subgroup	G/H
center, hypercenter	C₃²	D₅²
commutator, cosocle, hypocenter	C₅²	C₂²×C₃²
Frattini	C₁	G
Fitting, socle	C₃²×C₅²	C₂²
LCS	G ▷ C₅²
UCS	C₁ ◁ C₃²

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅⋊D₅	C₂	semidirect
2. C₃²×C₅×D₅	C₂	semidirect
3. C₃×D₅²	C₃	direct (N × H)
4. C₃²×C₅²	C₂²	semidirect
6. C₃×C₅⋊D₅	C₂×C₃	semidirect
7. C₃×C₅×D₅	C₂×C₃	semidirect
8. D₅²	C₃²	direct (N × H)
12. C₃²×D₅	D₅	direct (N × H)
13. C₃×C₅²	C₂²×C₃	semidirect
15. C₅⋊D₅	C₂×C₃²	semidirect
16. C₅×D₅	C₂×C₃²	semidirect
18. C₃²×C₅	C₂×D₅	semidirect
23. C₃×D₅	C₃×D₅	direct (N × H)
24. C₅²	C₂²×C₃²	semidirect
29. C₃×C₅	C₂×C₃×D₅	semidirect
34. D₅	C₃²×D₅	direct (N × H)
35. C₃²	D₅²	direct (N × H)
39. C₅	C₂×C₃²×D₅	semidirect
41. C₃	C₃×D₅²	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃²×C₅⋊D₅	2
C₃²×C₅×D₅	2
C₃×D₅²	3
C₂×C₃²×D₅	5

Subgroups — C32×D52 #11744711

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×D₅² #11744711