Subgroups — C₃²×C₂²⋊(C₅⋊D₅) #14268327

Order 1800.

384 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₅²	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃²×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#1039	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₂²⋊(C₅⋊D₅)	C₃	1	4	yes	maximal, direct product factor
5	C₂×C₃²×C₅²	—	2	1	no	—
6	C₂×C₃²×C₅²	C₂²	1	1	yes	characteristic
7	C₃²×C₅⋊D₅	—	2	1	no	—
8	C₃²×C₂²⋊D₅	—	5	6	no	maximal
9	C₂²×C₃×C₅²	C₂×C₃	1	4	yes	complemented
10	C₂×C₃×C₅⋊D₅	C₂×C₃	1	4	yes	complemented
11	C₃×#1039	C₂×C₃	1	4	yes	complemented
12	C₃²×C₅²	D₄	1	1	yes	characteristic, complemented, Hall
13	C₂²⋊(C₅⋊D₅)	C₃²	1	1	yes	characteristic, direct product factor, Hall
14	C₂²×C₃²×C₅	D₅	1	6	yes	complemented
15	C₂×C₃²×D₅	—	5	6	no	—
16	C₃²×Dic₅	—	5	6	no	—
17	C₂×C₃×C₅²	—	2	4	no	—
18	C₂×C₃×C₅²	C₂²×C₃	1	4	yes	—
19	C₃×C₅⋊D₅	—	2	4	no	—
20	C₃×C₂²⋊D₅	—	5	24	no	—
21	C₂²×C₅²	C₂×C₃²	1	1	yes	characteristic, complemented
22	C₂×C₅⋊D₅	C₂×C₃²	1	1	yes	characteristic, complemented
23	#1039	C₂×C₃²	1	1	yes	characteristic, complemented
24	C₂×C₃²×C₅	—	2	6	no	—
25	C₂×C₃²×C₅	C₂×D₅	1	6	yes	—
26	C₃²×D₅	—	10	6	no	—
27	C₃×C₅²	C₃×D₄	1	4	yes	complemented
28	C₃²×D₄	—	25	1	no	Hall
29	C₂²×C₃×C₅	C₃×D₅	1	24	yes	complemented
30	C₂×C₃×D₅	—	5	24	no	—
31	C₃×Dic₅	—	5	24	no	—
32	C₂×C₅²	—	2	1	no	—
33	C₂×C₅²	C₂²×C₃²	1	1	yes	characteristic
34	C₅⋊D₅	—	2	1	no	—
35	C₃²×C₅	C₂²⋊D₅	1	6	yes	complemented
36	C₂²⋊D₅	—	5	6	no	—
37	C₂²×C₃²	—	25	1	no	—
38	C₂²×C₃²	C₅⋊D₅	1	1	yes	characteristic, complemented
39	C₃²×C₄	—	25	1	no	—
40	C₂×C₃×C₅	—	2	24	no	—
41	C₂×C₃×C₅	C₂×C₃×D₅	1	24	yes	—
42	C₃×D₅	—	10	24	no	—
43	C₅²	C₃²×D₄	1	1	yes	characteristic, complemented, Sylow
44	C₃×D₄	—	25	4	no	—
45	C₂²×C₅	C₃²×D₅	1	6	yes	complemented
46	C₂×D₅	—	5	6	no	—
47	Dic₅	—	5	6	no	—
48	C₂×C₃²	—	2	1	no	—
49	C₂×C₃²	—	50	1	no	—
50	C₂×C₃²	C₂×C₅⋊D₅	1	1	yes	characteristic
51	C₃×C₅	C₃×C₂²⋊D₅	1	24	yes	complemented
52	C₂²×C₃	—	25	4	no	—
53	C₂²×C₃	C₃×C₅⋊D₅	1	4	yes	complemented
54	C₃×C₄	—	25	4	no	—
55	C₂×C₅	—	2	6	no	—
56	C₂×C₅	C₂×C₃²×D₅	1	6	yes	—
57	D₅	—	10	6	no	—
58	C₃²	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic, direct product factor, Sylow
59	D₄	—	25	1	no	Sylow
60	C₂×C₃	—	2	4	no	—
61	C₂×C₃	—	50	4	no	—
62	C₂×C₃	C₂×C₃×C₅⋊D₅	1	4	yes	—
63	C₅	C₃²×C₂²⋊D₅	1	6	yes	complemented
64	C₂²	—	25	1	no	—
65	C₂²	C₃²×C₅⋊D₅	1	1	yes	characteristic, complemented
66	C₄	—	25	1	no	—
67	C₃	C₃×C₂²⋊(C₅⋊D₅)	1	4	yes	direct product factor
68	C₂	—	2	1	no	—
69	C₂	—	50	1	no	—
70	C₂	C₂×C₃²×C₅⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²	C₂×C₅⋊D₅
commutator, cosocle	C₂×C₅²	C₂²×C₃²
Frattini	C₂	C₂×C₃²×C₅⋊D₅
Fitting	C₂²×C₃²×C₅²	C₂
socle	C₂×C₃²×C₅²	C₂²
hypercenter	C₂²×C₃²	C₅⋊D₅
hypocenter	C₅²	C₃²×D₄
LCS	G ▷ C₂×C₅² ▷ C₅²
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×C₅²	C₂	semidirect
2. C₂×C₃²×C₅⋊D₅	C₂	semidirect
3. C₃²×#1039	C₂	semidirect
4. C₃×C₂²⋊(C₅⋊D₅)	C₃	direct (N × H)
9. C₂²×C₃×C₅²	C₂×C₃	semidirect
10. C₂×C₃×C₅⋊D₅	C₂×C₃	semidirect
11. C₃×#1039	C₂×C₃	semidirect
12. C₃²×C₅²	D₄	semidirect
13. C₂²⋊(C₅⋊D₅)	C₃²	direct (N × H)
14. C₂²×C₃²×C₅	D₅	semidirect
21. C₂²×C₅²	C₂×C₃²	semidirect
22. C₂×C₅⋊D₅	C₂×C₃²	semidirect
23. #1039	C₂×C₃²	semidirect
27. C₃×C₅²	C₃×D₄	semidirect
29. C₂²×C₃×C₅	C₃×D₅	semidirect
35. C₃²×C₅	C₂²⋊D₅	semidirect
38. C₂²×C₃²	C₅⋊D₅	semidirect
43. C₅²	C₃²×D₄	semidirect
45. C₂²×C₅	C₃²×D₅	semidirect
51. C₃×C₅	C₃×C₂²⋊D₅	semidirect
53. C₂²×C₃	C₃×C₅⋊D₅	semidirect
58. C₃²	C₂²⋊(C₅⋊D₅)	direct (N × H)
63. C₅	C₃²×C₂²⋊D₅	semidirect
65. C₂²	C₃²×C₅⋊D₅	semidirect
67. C₃	C₃×C₂²⋊(C₅⋊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×C₅²	2
C₂×C₃²×C₅⋊D₅	2
C₃²×#1039	2
C₃×C₂²⋊(C₅⋊D₅)	3
C₃²×C₂²⋊D₅	5

Subgroups — C32×C22⋊(C5⋊D5) #14268327

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₂²⋊(C₅⋊D₅) #14268327