Subgroups — C₄⋊ΓL(1,𝔽₁₆) #6760

Order 240.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×ΓL(1,𝔽₁₆)	C₂	1	2	yes	maximal
2	C₃×C₄×D₅	C₂	1	1	yes	maximal, characteristic
3	C₄⋊AGL(1,𝔽₅)	—	3	1	no	maximal, Hall
4	C₂×C₃×D₅	C₂²	1	1	yes	characteristic
5	ΓL(1,𝔽₁₆)	—	2	2	no	—
6	C₃×C₄×C₅	C₄	1	1	yes	characteristic, complemented
7	C₃×Dic₅	C₄	1	1	yes	characteristic, complemented
8	C₄⋊Dic₃	—	5	1	no	maximal, Hall
9	C₂×AGL(1,𝔽₅)	—	3	2	no	—
10	C₄×D₅	D₃	1	1	yes	characteristic
11	C₂×C₃×C₅	C₂×C₄	1	1	yes	characteristic
12	C₃×D₅	D₄	1	1	yes	characteristic
13	C₃×D₅	Q₈	1	1	yes	characteristic
14	C₂×C₃×C₄	—	5	1	no	—
15	C₂×Dic₃	—	5	2	no	—
16	C₂×D₅	C₂×D₃	1	1	yes	characteristic
17	AGL(1,𝔽₅)	—	6	2	no	—
18	C₄×C₅	Dic₃	1	1	yes	characteristic, complemented
19	Dic₅	Dic₃	1	1	yes	characteristic, complemented
20	C₄⋊C₄	—	15	1	no	Sylow
21	C₃×C₅	C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
22	C₂²×C₃	—	5	1	no	—
23	C₃×C₄	—	5	1	no	—
24	C₃×C₄	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
25	Dic₃	—	10	2	no	—
26	C₂×C₅	C₂×Dic₃	1	1	yes	characteristic
27	D₅	Dic₆	1	1	yes	characteristic
28	D₅	D₁₂	1	1	yes	characteristic
29	C₂×C₄	—	5	1	no	—
30	C₂×C₄	—	15	2	no	—
31	C₂×C₃	—	5	1	no	—
32	C₂×C₃	—	5	1	no	—
33	C₂×C₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
34	C₅	C₄⋊Dic₃	1	1	yes	characteristic, complemented, Sylow
35	C₂²	—	5	1	no	—
36	C₄	—	5	1	no	—
37	C₄	—	30	2	no	—
38	C₄	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
39	C₃	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
40	C₂	—	5	1	no	—
41	C₂	—	5	1	no	—
42	C₂	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×ΓL(1,𝔽₁₆)
commutator, socle	C₂×C₃×C₅	C₂×C₄
Fitting	C₃×C₄×C₅	C₄
prime radical, cosocle	C₂×C₃×D₅	C₂²
hypercenter	C₄	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	C₄⋊C₄
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
6. C₃×C₄×C₅	C₄	semidirect
7. C₃×Dic₅	C₄	semidirect
18. C₄×C₅	Dic₃	semidirect
19. Dic₅	Dic₃	semidirect
21. C₃×C₅	C₄⋊C₄	semidirect
24. C₃×C₄	AGL(1,𝔽₅)	semidirect
34. C₅	C₄⋊Dic₃	semidirect
38. C₄	ΓL(1,𝔽₁₆)	semidirect
39. C₃	C₄⋊AGL(1,𝔽₅)	semidirect

Maximal subgroups

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

subgroup	index
C₂×ΓL(1,𝔽₁₆)	2
C₃×C₄×D₅	2
C₄⋊AGL(1,𝔽₅)	3
C₄⋊Dic₃	5

Subgroups — C4⋊ΓL(1,𝔽16) #6760

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊ΓL(1,𝔽₁₆) #6760