Subgroups — C₅×#1200 #10597106

Order 600.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×D₁₅	C₂	1	1	yes	maximal, characteristic
2	C₅²×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₅×D₅	—	3	1	no	maximal, Hall
5	C₂×C₃×C₅²	C₂²	1	1	yes	characteristic
6	C₅×D₁₅	C₄	1	2	yes	complemented
7	C₄×C₅×D₃	—	5	1	no	maximal
8	#1200	C₅	1	1	yes	maximal, characteristic, direct product factor
9	C₂×C₅×D₅	—	3	1	no	—
10	C₄×C₅²	—	3	1	no	—
11	C₅×Dic₅	D₃	1	1	yes	characteristic, complemented
12	C₃×C₅²	C₂×C₄	1	1	yes	characteristic, complemented, Hall
13	C₂×D₁₅	C₂×C₅	1	1	yes	characteristic
14	C₂×C₅×D₃	—	5	1	no	—
15	C₃×C₄×C₅	—	5	1	no	—
16	C₃×Dic₅	C₂×C₅	1	1	yes	characteristic, complemented
17	C₅×Dic₃	—	2	2	no	—
18	C₅×Dic₃	D₅	1	1	yes	characteristic, complemented
19	C₅×Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
20	C₂×C₅²	C₂×D₃	1	1	yes	characteristic
21	C₅×D₅	—	3	2	no	—
22	C₂×C₄×C₅	—	15	1	no	—
23	C₄×D₅	—	3	1	no	—
24	C₂×C₃×C₅	—	2	2	no	—
25	C₂×C₃×C₅	C₂×D₅	1	1	yes	characteristic
26	C₂×C₃×C₅	C₂²×C₅	1	1	yes	characteristic
27	D₁₅	C₄×C₅	1	2	yes	complemented
28	C₅×D₃	—	5	2	no	—
29	C₅²	C₄×D₃	1	1	yes	characteristic, complemented, Sylow
30	C₄×D₃	—	5	1	no	Hall
31	C₂²×C₅	—	15	1	no	—
32	C₂×D₅	—	3	1	no	—
33	C₄×C₅	—	3	1	no	—
34	C₄×C₅	—	3	1	no	—
35	C₄×C₅	—	5	1	no	—
36	C₄×C₅	—	6	2	no	—
37	Dic₅	C₅×D₃	1	1	yes	characteristic, complemented
38	C₃×C₅	—	2	2	no	—
39	C₃×C₅	C₄×D₅	1	1	yes	characteristic, complemented
40	C₃×C₅	C₂×C₄×C₅	1	1	yes	characteristic, complemented
41	C₂×D₃	—	5	1	no	—
42	C₃×C₄	—	5	1	no	—
43	Dic₃	C₅×D₅	1	1	yes	characteristic, complemented
44	C₂×C₅	—	2	2	no	—
45	C₂×C₅	—	15	2	no	—
46	C₂×C₅	D₃×D₅	1	1	yes	characteristic
47	C₂×C₅	C₂×C₅×D₃	1	1	yes	characteristic
48	D₅	—	3	2	no	—
49	C₂×C₄	—	15	1	no	Sylow
50	C₂×C₃	C₂×C₅×D₅	1	1	yes	characteristic
51	D₃	—	5	2	no	—
52	C₅	—	2	2	no	—
53	C₅	#1200	1	1	yes	characteristic, direct product factor
54	C₅	C₄×C₅×D₃	1	1	yes	characteristic, complemented
55	C₂²	—	15	1	no	—
56	C₄	—	3	1	no	—
57	C₄	—	5	1	no	—
58	C₃	C₄×C₅×D₅	1	1	yes	characteristic, complemented, Sylow
59	C₂	—	15	2	no	—
60	C₂	C₅×D₃×D₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₅	D₃×D₅
commutator, hypocenter	C₃×C₅	C₂×C₄×C₅
Frattini	C₂	C₅×D₃×D₅
Fitting, socle	C₂×C₃×C₅²	C₂²
prime radical	C₂×C₅×D₁₅	C₂
cosocle	C₂×C₃×C₅	C₂²×C₅
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₅

Complements

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

N	H ≅ G/N	type
2. C₅²×Dic₃	C₂	semidirect
3. C₃×C₅×Dic₅	C₂	semidirect
6. C₅×D₁₅	C₄	semidirect
8. #1200	C₅	direct (N × H)
11. C₅×Dic₅	D₃	semidirect
12. C₃×C₅²	C₂×C₄	semidirect
16. C₃×Dic₅	C₂×C₅	semidirect
18. C₅×Dic₃	D₅	semidirect
19. C₅×Dic₃	C₂×C₅	semidirect
27. D₁₅	C₄×C₅	semidirect
29. C₅²	C₄×D₃	semidirect
37. Dic₅	C₅×D₃	semidirect
39. C₃×C₅	C₄×D₅	semidirect
40. C₃×C₅	C₂×C₄×C₅	semidirect
43. Dic₃	C₅×D₅	semidirect
53. C₅	#1200	direct (N × H)
54. C₅	C₄×C₅×D₃	semidirect
58. C₃	C₄×C₅×D₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×D₁₅	2
C₅²×Dic₃	2
C₃×C₅×Dic₅	2
C₄×C₅×D₅	3
C₄×C₅×D₃	5
#1200	5

Subgroups — C5×#1200 #10597106

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#1200 #10597106