Subgroups — C₅×ΓL(1,𝔽₂₅) #6689

Order 240.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×D₃	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×M₁₆	—	3	1	no	maximal, Hall
5	C₂×C₅×D₃	C₄	1	1	yes	characteristic
6	C₃×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₅×Dic₃	C₄	1	1	yes	characteristic
8	ΓL(1,𝔽₂₅)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₄×C₅	—	3	1	no	—
10	C₅×C₈	—	3	1	no	—
11	C₅×C₈	D₃	1	1	yes	characteristic, complemented
12	C₂×C₃×C₅	C₂×C₄	1	1	yes	characteristic
13	C₅×D₃	—	2	1	no	—
14	C₄×D₃	C₂×C₅	1	1	yes	characteristic
15	C₃×C₈	C₂×C₅	1	1	yes	characteristic, complemented
16	C₃⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
17	C₂²×C₅	—	3	1	no	—
18	C₄×C₅	—	3	1	no	—
19	C₄×C₅	C₂×D₃	1	1	yes	characteristic
20	M₁₆	—	3	1	no	Sylow
21	C₃×C₅	M₁₆	1	1	yes	characteristic, complemented, Hall
22	C₂×D₃	C₄×C₅	1	1	yes	characteristic
23	C₃×C₄	C₂²×C₅	1	1	yes	characteristic
24	Dic₃	C₄×C₅	1	1	yes	characteristic
25	C₂×C₅	—	6	1	no	—
26	C₂×C₅	C₄×D₃	1	1	yes	characteristic
27	C₂×C₄	—	3	1	no	—
28	C₈	—	3	1	no	—
29	C₈	C₅×D₃	1	1	yes	characteristic, complemented
30	C₂×C₃	C₂×C₄×C₅	1	1	yes	characteristic
31	D₃	—	2	1	no	—
32	C₅	ΓL(1,𝔽₂₅)	1	1	yes	characteristic, direct product factor, Sylow
33	C₂²	—	3	1	no	—
34	C₄	—	3	1	no	—
35	C₄	C₂×C₅×D₃	1	1	yes	characteristic
36	C₃	C₅×M₁₆	1	1	yes	characteristic, complemented, Sylow
37	C₂	—	6	1	no	—
38	C₂	C₄×C₅×D₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×C₈	C₂	semidirect
3. C₅×C₃⋊C₈	C₂	semidirect
8. ΓL(1,𝔽₂₅)	C₅	direct (N × H)
11. C₅×C₈	D₃	semidirect
15. C₃×C₈	C₂×C₅	semidirect
16. C₃⋊C₈	C₂×C₅	semidirect
21. C₃×C₅	M₁₆	semidirect
29. C₈	C₅×D₃	semidirect
32. C₅	ΓL(1,𝔽₂₅)	direct (N × H)
36. C₃	C₅×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×D₃	2
C₃×C₅×C₈	2
C₅×C₃⋊C₈	2
C₅×M₁₆	3
ΓL(1,𝔽₂₅)	5

Subgroups — C5×ΓL(1,𝔽25) #6689

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×ΓL(1,𝔽₂₅) #6689