Subgroups — C₂×C₅×D₃² #67223

Order 360.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₅×D₃	C₂	1	2	yes	maximal, complemented
3	C₅×D₃²	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₅×D₃	—	3	2	no	maximal
5	C₂×C₃²×C₅	C₂²	1	1	yes	characteristic, complemented
6	C₅×C₃⋊D₃	C₂²	1	2	yes	complemented
7	C₃×C₅×D₃	C₂²	1	4	yes	complemented
8	C₂×D₃²	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂²×C₃×C₅	—	3	2	no	—
10	C₂×C₅×D₃	—	3	2	no	—
11	C₂×C₅×D₃	—	3	8	no	—
12	C₂×C₅×D₃	—	6	1	no	—
13	C₂×C₅×D₃	D₃	1	2	yes	direct product factor
14	C₃²×C₅	C₂³	1	1	yes	characteristic, complemented, Hall
15	C₂³×C₅	—	9	1	no	Hall
16	C₂×C₃⋊D₃	C₂×C₅	1	1	yes	characteristic, complemented
17	C₂×C₃×D₃	C₂×C₅	1	2	yes	complemented
18	D₃²	C₂×C₅	1	4	yes	direct product factor
19	C₂×C₃×C₅	—	2	1	no	—
20	C₂×C₃×C₅	—	3	4	no	—
21	C₂×C₃×C₅	C₂×D₃	1	2	yes	complemented
22	C₅×D₃	—	3	4	no	—
23	C₅×D₃	—	6	2	no	—
24	C₅×D₃	C₂×D₃	1	4	yes	direct product factor
25	C₂²×D₃	—	3	2	no	—
26	C₂²×C₅	—	3	2	no	—
27	C₂²×C₅	—	9	4	no	—
28	C₂²×C₅	—	9	1	no	—
29	C₂×C₃²	C₂²×C₅	1	1	yes	characteristic, complemented
30	C₃⋊D₃	C₂²×C₅	1	2	yes	complemented
31	C₃×D₃	C₂²×C₅	1	4	yes	complemented
32	C₃×C₅	—	2	1	no	—
33	C₃×C₅	C₂²×D₃	1	2	yes	complemented
34	C₂²×C₃	—	3	2	no	—
35	C₂×D₃	—	3	2	no	—
36	C₂×D₃	—	3	8	no	—
37	C₂×D₃	—	6	1	no	—
38	C₂×D₃	C₅×D₃	1	2	yes	direct product factor
39	C₂×C₅	—	3	4	no	—
40	C₂×C₅	—	9	2	no	—
41	C₂×C₅	D₃²	1	1	yes	characteristic, direct product factor
42	C₃²	C₂³×C₅	1	1	yes	characteristic, complemented, Sylow
43	C₂³	—	9	1	no	Sylow
44	C₂×C₃	—	2	1	no	—
45	C₂×C₃	—	3	4	no	—
46	C₂×C₃	C₂×C₅×D₃	1	2	yes	complemented
47	D₃	—	3	4	no	—
48	D₃	—	6	2	no	—
49	D₃	C₂×C₅×D₃	1	4	yes	direct product factor
50	C₅	C₂×D₃²	1	1	yes	characteristic, direct product factor, Sylow
51	C₂²	—	3	2	no	—
52	C₂²	—	9	4	no	—
53	C₂²	—	9	1	no	—
54	C₃	—	2	1	no	—
55	C₃	C₂²×C₅×D₃	1	2	yes	complemented
56	C₂	—	3	4	no	—
57	C₂	—	9	2	no	—
58	C₂	C₅×D₃²	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₅	D₃²
commutator, cosocle, hypocenter	C₃²	C₂³×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₅	C₂²
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂×C₅

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×C₃⋊D₃	C₂	semidirect
2. C₂×C₃×C₅×D₃	C₂	semidirect
3. C₅×D₃²	C₂	direct (N × H)
5. C₂×C₃²×C₅	C₂²	semidirect
6. C₅×C₃⋊D₃	C₂²	semidirect
7. C₃×C₅×D₃	C₂²	semidirect
8. C₂×D₃²	C₅	direct (N × H)
13. C₂×C₅×D₃	D₃	direct (N × H)
14. C₃²×C₅	C₂³	semidirect
16. C₂×C₃⋊D₃	C₂×C₅	semidirect
17. C₂×C₃×D₃	C₂×C₅	semidirect
18. D₃²	C₂×C₅	direct (N × H)
21. C₂×C₃×C₅	C₂×D₃	semidirect
24. C₅×D₃	C₂×D₃	direct (N × H)
29. C₂×C₃²	C₂²×C₅	semidirect
30. C₃⋊D₃	C₂²×C₅	semidirect
31. C₃×D₃	C₂²×C₅	semidirect
33. C₃×C₅	C₂²×D₃	semidirect
38. C₂×D₃	C₅×D₃	direct (N × H)
41. C₂×C₅	D₃²	direct (N × H)
42. C₃²	C₂³×C₅	semidirect
46. C₂×C₃	C₂×C₅×D₃	semidirect
49. D₃	C₂×C₅×D₃	direct (N × H)
50. C₅	C₂×D₃²	direct (N × H)
55. C₃	C₂²×C₅×D₃	semidirect
58. C₂	C₅×D₃²	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×C₅×C₃⋊D₃	2
C₂×C₃×C₅×D₃	2
C₅×D₃²	2
C₂²×C₅×D₃	3
C₂×D₃²	5

Subgroups — C2×C5×D32 #67223

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×D₃² #67223