Subgroups — C₃²×C₂²⋊D₅ #67164

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, complemented
2	C₂×C₃²×D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₂²⋊D₅	C₃	1	4	yes	maximal, direct product factor
5	C₂×C₃²×C₅	—	2	1	no	—
6	C₂×C₃²×C₅	C₂²	1	1	yes	characteristic
7	C₃²×D₅	—	2	1	no	—
8	C₃²×D₄	—	5	1	no	maximal, Hall
9	C₂²×C₃×C₅	C₂×C₃	1	4	yes	complemented
10	C₂×C₃×D₅	C₂×C₃	1	4	yes	complemented
11	C₃×Dic₅	C₂×C₃	1	4	yes	complemented
12	C₃²×C₅	D₄	1	1	yes	characteristic, complemented, Hall
13	C₂²⋊D₅	C₃²	1	1	yes	characteristic, direct product factor, Hall
14	C₂²×C₃²	—	5	1	no	—
15	C₂²×C₃²	D₅	1	1	yes	characteristic, complemented
16	C₃²×C₄	—	5	1	no	—
17	C₂×C₃×C₅	—	2	4	no	—
18	C₂×C₃×C₅	C₂²×C₃	1	4	yes	—
19	C₃×D₅	—	2	4	no	—
20	C₃×D₄	—	5	4	no	—
21	C₂²×C₅	C₂×C₃²	1	1	yes	characteristic, complemented
22	C₂×D₅	C₂×C₃²	1	1	yes	characteristic, complemented
23	Dic₅	C₂×C₃²	1	1	yes	characteristic, complemented
24	C₂×C₃²	—	2	1	no	—
25	C₂×C₃²	—	10	1	no	—
26	C₂×C₃²	C₂×D₅	1	1	yes	characteristic
27	C₃×C₅	C₃×D₄	1	4	yes	complemented
28	C₂²×C₃	—	5	4	no	—
29	C₂²×C₃	C₃×D₅	1	4	yes	complemented
30	C₃×C₄	—	5	4	no	—
31	C₂×C₅	—	2	1	no	—
32	C₂×C₅	C₂²×C₃²	1	1	yes	characteristic
33	D₅	—	2	1	no	—
34	C₃²	C₂²⋊D₅	1	1	yes	characteristic, direct product factor, Sylow
35	D₄	—	5	1	no	Sylow
36	C₂×C₃	—	2	4	no	—
37	C₂×C₃	—	10	4	no	—
38	C₂×C₃	C₂×C₃×D₅	1	4	yes	—
39	C₅	C₃²×D₄	1	1	yes	characteristic, complemented, Sylow
40	C₂²	—	5	1	no	—
41	C₂²	C₃²×D₅	1	1	yes	characteristic, complemented
42	C₄	—	5	1	no	—
43	C₃	C₃×C₂²⋊D₅	1	4	yes	direct product factor
44	C₂	—	2	1	no	—
45	C₂	—	10	1	no	—
46	C₂	C₂×C₃²×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²	C₂×D₅
commutator, cosocle	C₂×C₅	C₂²×C₃²
Frattini	C₂	C₂×C₃²×D₅
Fitting	C₂²×C₃²×C₅	C₂
socle	C₂×C₃²×C₅	C₂²
hypercenter	C₂²×C₃²	D₅
hypocenter	C₅	C₃²×D₄
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃² ◁ 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	semidirect
3. C₃²×Dic₅	C₂	semidirect
4. C₃×C₂²⋊D₅	C₃	direct (N × H)
9. C₂²×C₃×C₅	C₂×C₃	semidirect
10. C₂×C₃×D₅	C₂×C₃	semidirect
11. C₃×Dic₅	C₂×C₃	semidirect
12. C₃²×C₅	D₄	semidirect
13. C₂²⋊D₅	C₃²	direct (N × H)
15. C₂²×C₃²	D₅	semidirect
21. C₂²×C₅	C₂×C₃²	semidirect
22. C₂×D₅	C₂×C₃²	semidirect
23. Dic₅	C₂×C₃²	semidirect
27. C₃×C₅	C₃×D₄	semidirect
29. C₂²×C₃	C₃×D₅	semidirect
34. C₃²	C₂²⋊D₅	direct (N × H)
39. C₅	C₃²×D₄	semidirect
41. C₂²	C₃²×D₅	semidirect
43. C₃	C₃×C₂²⋊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
C₃²×Dic₅	2
C₃×C₂²⋊D₅	3
C₃²×D₄	5

Subgroups — C32×C22⋊D5 #67164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₂²⋊D₅ #67164