Subgroups — C₃²×C₄×D₅ #67161

Order 360.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×D₅	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₄×C₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄×D₅	C₃	1	4	yes	maximal, direct product factor
5	C₂×C₃²×C₅	C₂²	1	1	yes	characteristic
6	C₃²×D₅	C₄	1	2	yes	direct product factor
7	C₂×C₃²×C₄	—	5	1	no	maximal, Hall
8	C₂×C₃×D₅	C₂×C₃	1	4	yes	—
9	C₃×C₄×C₅	C₂×C₃	1	4	yes	complemented
10	C₃×Dic₅	C₂×C₃	1	4	yes	complemented
11	C₃²×C₅	C₂×C₄	1	1	yes	characteristic, complemented, Hall
12	C₄×D₅	C₃²	1	1	yes	characteristic, direct product factor, Hall
13	C₂²×C₃²	—	5	1	no	—
14	C₃²×C₄	—	5	1	no	—
15	C₃²×C₄	D₅	1	1	yes	characteristic, direct product factor
16	C₂×C₃×C₅	C₂²×C₃	1	4	yes	—
17	C₃×D₅	C₃×C₄	1	8	yes	direct product factor
18	C₂×C₃×C₄	—	5	4	no	—
19	C₂×D₅	C₂×C₃²	1	1	yes	characteristic
20	C₄×C₅	C₂×C₃²	1	1	yes	characteristic, complemented
21	Dic₅	C₂×C₃²	1	1	yes	characteristic, complemented
22	C₂×C₃²	—	5	2	no	—
23	C₂×C₃²	C₂×D₅	1	1	yes	characteristic
24	C₃×C₅	C₂×C₃×C₄	1	4	yes	complemented
25	C₂²×C₃	—	5	4	no	—
26	C₃×C₄	—	5	4	no	—
27	C₃×C₄	C₃×D₅	1	4	yes	direct product factor
28	C₂×C₅	C₂²×C₃²	1	1	yes	characteristic
29	D₅	C₃²×C₄	1	2	yes	direct product factor
30	C₃²	C₄×D₅	1	1	yes	characteristic, direct product factor, Sylow
31	C₂×C₄	—	5	1	no	Sylow
32	C₂×C₃	—	5	8	no	—
33	C₂×C₃	C₂×C₃×D₅	1	4	yes	—
34	C₅	C₂×C₃²×C₄	1	1	yes	characteristic, complemented, Sylow
35	C₂²	—	5	1	no	—
36	C₄	—	5	1	no	—
37	C₄	C₃²×D₅	1	1	yes	characteristic, direct product factor
38	C₃	C₃×C₄×D₅	1	4	yes	direct product factor
39	C₂	—	5	2	no	—
40	C₂	C₂×C₃²×D₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₃²×C₄	D₅
commutator, hypocenter	C₅	C₂×C₃²×C₄
Frattini	C₂	C₂×C₃²×D₅
Fitting	C₃²×C₄×C₅	C₂
socle	C₂×C₃²×C₅	C₂²
prime radical	C₂×C₃²×D₅	C₂
cosocle	C₂×C₅	C₂²×C₃²
LCS	G ▷ C₅
UCS	C₁ ◁ C₃²×C₄

Complements

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

N	H ≅ G/N	type
2. C₃²×C₄×C₅	C₂	semidirect
3. C₃²×Dic₅	C₂	semidirect
4. C₃×C₄×D₅	C₃	direct (N × H)
6. C₃²×D₅	C₄	direct (N × H)
9. C₃×C₄×C₅	C₂×C₃	semidirect
10. C₃×Dic₅	C₂×C₃	semidirect
11. C₃²×C₅	C₂×C₄	semidirect
12. C₄×D₅	C₃²	direct (N × H)
15. C₃²×C₄	D₅	direct (N × H)
17. C₃×D₅	C₃×C₄	direct (N × H)
20. C₄×C₅	C₂×C₃²	semidirect
21. Dic₅	C₂×C₃²	semidirect
24. C₃×C₅	C₂×C₃×C₄	semidirect
27. C₃×C₄	C₃×D₅	direct (N × H)
29. D₅	C₃²×C₄	direct (N × H)
30. C₃²	C₄×D₅	direct (N × H)
34. C₅	C₂×C₃²×C₄	semidirect
37. C₄	C₃²×D₅	direct (N × H)
38. C₃	C₃×C₄×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
C₂×C₃²×D₅	2
C₃²×C₄×C₅	2
C₃²×Dic₅	2
C₃×C₄×D₅	3
C₂×C₃²×C₄	5

Subgroups — C32×C4×D5 #67161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₄×D₅ #67161