Subgroups — C₃×ΓL(1,𝔽₁₆)×C₁₅₇ #43246366

Order 28260.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×D₅×C₁₅₇	C₂	1	1	yes	maximal, characteristic
2	ΓL(1,𝔽₁₆)×C₁₅₇	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×AGL(1,𝔽₅)×C₁₅₇	—	3	1	no	maximal
4	C₃²×C₅×C₁₅₇	C₄	1	1	yes	characteristic, complemented, Hall
5	C₃×Dic₃×C₁₅₇	—	5	1	no	maximal, Hall
6	C₃×D₅×C₁₅₇	—	2	1	no	—
7	C₃×D₅×C₁₅₇	D₃	1	1	yes	characteristic
8	C₃×D₅×C₁₅₇	C₂×C₃	1	1	yes	characteristic
9	AGL(1,𝔽₅)×C₁₅₇	—	3	1	no	Hall
10	C₂×C₃²×C₁₅₇	—	5	1	no	—
11	C₃×C₅×C₁₅₇	—	2	1	no	—
12	C₃×C₅×C₁₅₇	Dic₃	1	1	yes	characteristic, complemented
13	C₃×C₅×C₁₅₇	C₃×C₄	1	1	yes	characteristic, complemented
14	C₃×C₄×C₁₅₇	—	15	1	no	—
15	Dic₃×C₁₅₇	—	5	1	no	—
16	D₅×C₁₅₇	C₃×D₃	1	1	yes	characteristic
17	C₃²×C₁₅₇	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
18	C₂×C₃×C₁₅₇	—	5	1	no	—
19	C₂×C₃×C₁₅₇	—	5	1	no	—
20	C₂×C₃×C₁₅₇	—	10	1	no	—
21	C₅×C₁₅₇	C₃×Dic₃	1	1	yes	characteristic, complemented, Hall
22	C₄×C₁₅₇	—	15	1	no	Hall
23	C₃×C₁₅₇	—	2	1	no	—
24	C₃×C₁₅₇	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
25	C₃×C₁₅₇	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor
26	C₂×C₁₅₇	—	5	1	no	—
27	C₃×ΓL(1,𝔽₁₆)	C₁₅₇	1	1	yes	maximal, characteristic, direct product factor, Hall
28	C₁₅₇	C₃×ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor, Sylow
29	C₃²×D₅	C₂×C₁₅₇	1	1	yes	characteristic
30	ΓL(1,𝔽₁₆)	C₃×C₁₅₇	1	1	yes	characteristic, direct product factor
31	C₃×AGL(1,𝔽₅)	—	3	1	no	—
32	C₃²×C₅	C₄×C₁₅₇	1	1	yes	characteristic, complemented, Hall
33	C₃×Dic₃	—	5	1	no	Hall
34	C₃×D₅	—	2	1	no	—
35	C₃×D₅	D₃×C₁₅₇	1	1	yes	characteristic
36	C₃×D₅	C₂×C₃×C₁₅₇	1	1	yes	characteristic
37	AGL(1,𝔽₅)	—	3	1	no	Hall
38	C₂×C₃²	—	5	1	no	—
39	C₃×C₅	—	2	1	no	—
40	C₃×C₅	Dic₃×C₁₅₇	1	1	yes	characteristic, complemented
41	C₃×C₅	C₃×C₄×C₁₅₇	1	1	yes	characteristic, complemented
42	C₃×C₄	—	15	1	no	—
43	Dic₃	—	5	1	no	—
44	D₅	C₃×D₃×C₁₅₇	1	1	yes	characteristic
45	C₃²	AGL(1,𝔽₅)×C₁₅₇	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₃	—	5	1	no	—
47	C₂×C₃	—	5	1	no	—
48	C₂×C₃	—	10	1	no	—
49	C₅	C₃×Dic₃×C₁₅₇	1	1	yes	characteristic, complemented, Sylow
50	C₄	—	15	1	no	Sylow
51	C₃	—	2	1	no	—
52	C₃	C₃×AGL(1,𝔽₅)×C₁₅₇	1	1	yes	characteristic, complemented
53	C₃	ΓL(1,𝔽₁₆)×C₁₅₇	1	1	yes	characteristic, direct product factor
54	C₂	—	5	1	no	—

	subgroup	G/H
center, hypercenter	C₃×C₁₅₇	ΓL(1,𝔽₁₆)
commutator, hypocenter	C₃×C₅	C₃×C₄×C₁₅₇
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₁₅₇	C₄
prime radical	C₃²×D₅×C₁₅₇	C₂
cosocle	C₃×D₅	C₂×C₃×C₁₅₇
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₃×C₁₅₇

Complements

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

N	H ≅ G/N	type
2. ΓL(1,𝔽₁₆)×C₁₅₇	C₃	direct (N × H)
4. C₃²×C₅×C₁₅₇	C₄	semidirect
12. C₃×C₅×C₁₅₇	Dic₃	semidirect
13. C₃×C₅×C₁₅₇	C₃×C₄	semidirect
17. C₃²×C₁₅₇	AGL(1,𝔽₅)	semidirect
21. C₅×C₁₅₇	C₃×Dic₃	semidirect
24. C₃×C₁₅₇	C₃×AGL(1,𝔽₅)	semidirect
25. C₃×C₁₅₇	ΓL(1,𝔽₁₆)	direct (N × H)
27. C₃×ΓL(1,𝔽₁₆)	C₁₅₇	direct (N × H)
28. C₁₅₇	C₃×ΓL(1,𝔽₁₆)	direct (N × H)
30. ΓL(1,𝔽₁₆)	C₃×C₁₅₇	direct (N × H)
32. C₃²×C₅	C₄×C₁₅₇	semidirect
40. C₃×C₅	Dic₃×C₁₅₇	semidirect
41. C₃×C₅	C₃×C₄×C₁₅₇	semidirect
45. C₃²	AGL(1,𝔽₅)×C₁₅₇	semidirect
49. C₅	C₃×Dic₃×C₁₅₇	semidirect
52. C₃	C₃×AGL(1,𝔽₅)×C₁₅₇	semidirect
53. C₃	ΓL(1,𝔽₁₆)×C₁₅₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×D₅×C₁₅₇	2
ΓL(1,𝔽₁₆)×C₁₅₇	3
C₃×AGL(1,𝔽₅)×C₁₅₇	3
C₃×Dic₃×C₁₅₇	5
C₃×ΓL(1,𝔽₁₆)	157

Subgroups — C3×ΓL(1,𝔽16)×C157 #43246366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×ΓL(1,𝔽₁₆)×C₁₅₇ #43246366