Subgroups — C₅×#3663 #10623582

Order 720.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×D₉	C₂	1	1	yes	maximal, characteristic
2	C₅×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×ΓL(1,𝔽₂₅)	—	3	1	no	maximal
5	C₂×C₅×D₉	C₄	1	1	yes	characteristic
6	C₄×C₅×C₉	C₂²	1	1	yes	characteristic
7	C₅×Dic₉	C₄	1	1	yes	characteristic
8	#3663	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₄×C₅×D₃	—	3	1	no	—
10	C₃×C₅×C₈	D₃	1	1	yes	characteristic
11	C₅×C₃⋊C₈	—	3	1	no	—
12	C₂×C₅×C₉	C₂×C₄	1	1	yes	characteristic
13	C₅×D₉	—	2	1	no	—
14	C₅×M₁₆	—	9	1	no	Hall
15	C₄×D₉	C₂×C₅	1	1	yes	characteristic
16	C₈×C₉	C₂×C₅	1	1	yes	characteristic, complemented
17	C₉⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
18	C₂×C₅×D₃	—	3	1	no	—
19	C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
20	C₅×Dic₃	—	3	1	no	—
21	ΓL(1,𝔽₂₅)	—	3	1	no	—
22	C₅×C₉	M₁₆	1	1	yes	characteristic, complemented, Hall
23	C₂×C₄×C₅	—	9	1	no	—
24	C₅×C₈	—	9	1	no	—
25	C₅×C₈	D₉	1	1	yes	characteristic, complemented
26	C₂×D₉	C₄×C₅	1	1	yes	characteristic
27	C₄×C₉	C₂²×C₅	1	1	yes	characteristic
28	Dic₉	C₄×C₅	1	1	yes	characteristic
29	C₂×C₃×C₅	C₄×D₃	1	1	yes	characteristic
30	C₅×D₃	—	6	1	no	—
31	C₄×D₃	—	3	1	no	—
32	C₃×C₈	C₅×D₃	1	1	yes	characteristic
33	C₃⋊C₈	—	3	1	no	—
34	C₂²×C₅	—	9	1	no	—
35	C₄×C₅	—	9	1	no	—
36	C₄×C₅	C₂×D₉	1	1	yes	characteristic
37	C₂×C₉	C₂×C₄×C₅	1	1	yes	characteristic
38	D₉	—	2	1	no	—
39	M₁₆	—	9	1	no	Sylow
40	C₃×C₅	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
41	C₂×D₃	—	3	1	no	—
42	C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
43	Dic₃	—	3	1	no	—
44	C₂×C₅	—	18	1	no	—
45	C₂×C₅	C₄×D₉	1	1	yes	characteristic
46	C₉	C₅×M₁₆	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₄	—	9	1	no	—
48	C₈	—	9	1	no	—
49	C₈	C₅×D₉	1	1	yes	characteristic, complemented
50	C₂×C₃	C₄×C₅×D₃	1	1	yes	characteristic
51	D₃	—	6	1	no	—
52	C₅	#3663	1	1	yes	characteristic, direct product factor, Sylow
53	C₂²	—	9	1	no	—
54	C₄	—	9	1	no	—
55	C₄	C₂×C₅×D₉	1	1	yes	characteristic
56	C₃	C₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
57	C₂	—	18	1	no	—
58	C₂	C₄×C₅×D₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	C₂×D₉
commutator	C₂×C₉	C₂×C₄×C₅
Frattini	C₃×C₄	C₂×C₅×D₃
Fitting	C₅×C₈×C₉	C₂
socle	C₂×C₃×C₅	C₄×D₃
prime radical	C₂×C₅×D₉	C₄
cosocle	C₄×C₉	C₂²×C₅
hypercenter	C₅×C₈	D₉
hypocenter	C₉	C₅×M₁₆
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₄×C₅ ◁ C₅×C₈

Complements

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

N	H ≅ G/N	type
2. C₅×C₈×C₉	C₂	semidirect
3. C₅×C₉⋊C₈	C₂	semidirect
8. #3663	C₅	direct (N × H)
16. C₈×C₉	C₂×C₅	semidirect
17. C₉⋊C₈	C₂×C₅	semidirect
22. C₅×C₉	M₁₆	semidirect
25. C₅×C₈	D₉	semidirect
46. C₉	C₅×M₁₆	semidirect
49. C₈	C₅×D₉	semidirect
52. C₅	#3663	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₅×D₉	2
C₅×C₈×C₉	2
C₅×C₉⋊C₈	2
C₅×ΓL(1,𝔽₂₅)	3
#3663	5

Subgroups — C5×#3663 #10623582

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#3663 #10623582