Subgroups — C₃²×C₉×AGL(1,𝔽₅) #13108359

Order 1620.

300 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₉×D₅	C₂	1	1	yes	maximal, characteristic
2	C₃³×AGL(1,𝔽₅)	C₃	1	1	yes	maximal, characteristic
3	C₃×C₉×AGL(1,𝔽₅)	C₃	1	12	yes	maximal, direct product factor
4	C₃²×C₅×C₉	C₄	1	1	yes	characteristic, complemented, Hall
5	C₃²×C₄×C₉	—	5	1	no	maximal, Hall
6	C₃³×D₅	C₂×C₃	1	1	yes	characteristic
7	C₃×C₉×D₅	C₂×C₃	1	12	yes	—
8	C₃²×AGL(1,𝔽₅)	C₉	1	9	yes	direct product factor
9	C₃²×AGL(1,𝔽₅)	C₃²	1	4	yes	—
10	C₉×AGL(1,𝔽₅)	C₃²	1	9	yes	direct product factor
11	C₂×C₃²×C₉	—	5	1	no	—
12	C₃³×C₅	C₃×C₄	1	1	yes	characteristic
13	C₃×C₅×C₉	C₃×C₄	1	12	yes	complemented
14	C₃³×C₄	—	5	1	no	—
15	C₃×C₄×C₉	—	5	12	no	—
16	C₃²×D₅	C₂×C₉	1	9	yes	—
17	C₃²×D₅	C₂×C₃²	1	4	yes	—
18	C₉×D₅	C₂×C₃²	1	9	yes	—
19	C₃²×C₉	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor, Sylow
20	C₃×AGL(1,𝔽₅)	C₃×C₉	1	12	yes	direct product factor
21	C₃×AGL(1,𝔽₅)	C₃³	1	1	yes	characteristic
22	C₂×C₃³	—	5	1	no	—
23	C₂×C₃×C₉	—	5	12	no	—
24	C₃²×C₅	C₄×C₉	1	9	yes	complemented
25	C₃²×C₅	C₃²×C₄	1	4	yes	—
26	C₅×C₉	C₃²×C₄	1	9	yes	complemented
27	C₃²×C₄	—	5	9	no	—
28	C₃²×C₄	—	5	4	no	—
29	C₄×C₉	—	5	9	no	—
30	C₃×D₅	C₂×C₃×C₉	1	12	yes	—
31	C₃×D₅	C₂×C₃³	1	1	yes	characteristic
32	C₃³	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
33	C₃×C₉	C₃×AGL(1,𝔽₅)	1	12	yes	direct product factor
34	AGL(1,𝔽₅)	C₃²×C₉	1	1	yes	characteristic, direct product factor, Hall
35	C₂×C₃²	—	5	9	no	—
36	C₂×C₃²	—	5	4	no	—
37	C₂×C₉	—	5	9	no	—
38	C₃×C₅	C₃×C₄×C₉	1	12	yes	complemented
39	C₃×C₅	C₃³×C₄	1	1	yes	characteristic
40	C₃×C₄	—	5	12	no	—
41	C₃×C₄	—	5	1	no	—
42	D₅	C₂×C₃²×C₉	1	1	yes	characteristic
43	C₃²	C₉×AGL(1,𝔽₅)	1	9	yes	direct product factor
44	C₃²	C₃²×AGL(1,𝔽₅)	1	4	yes	—
45	C₉	C₃²×AGL(1,𝔽₅)	1	9	yes	direct product factor
46	C₂×C₃	—	5	1	no	—
47	C₂×C₃	—	5	12	no	—
48	C₅	C₃²×C₄×C₉	1	1	yes	characteristic, complemented, Sylow
49	C₄	—	5	1	no	Sylow
50	C₃	C₃×C₉×AGL(1,𝔽₅)	1	12	yes	direct product factor
51	C₃	C₃³×AGL(1,𝔽₅)	1	1	yes	characteristic
52	C₂	—	5	1	no	—

	subgroup	G/H
center, hypercenter	C₃²×C₉	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₃²×C₄×C₉
Frattini	C₃	C₃³×AGL(1,𝔽₅)
Fitting	C₃²×C₅×C₉	C₄
socle	C₃³×C₅	C₃×C₄
prime radical	C₃³×D₅	C₂×C₃
cosocle	C₃×D₅	C₂×C₃³
LCS	G ▷ C₅
UCS	C₁ ◁ C₃²×C₉

Complements

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

N	H ≅ G/N	type
3. C₃×C₉×AGL(1,𝔽₅)	C₃	direct (N × H)
4. C₃²×C₅×C₉	C₄	semidirect
8. C₃²×AGL(1,𝔽₅)	C₉	direct (N × H)
10. C₉×AGL(1,𝔽₅)	C₃²	direct (N × H)
13. C₃×C₅×C₉	C₃×C₄	semidirect
19. C₃²×C₉	AGL(1,𝔽₅)	direct (N × H)
20. C₃×AGL(1,𝔽₅)	C₃×C₉	direct (N × H)
24. C₃²×C₅	C₄×C₉	semidirect
26. C₅×C₉	C₃²×C₄	semidirect
33. C₃×C₉	C₃×AGL(1,𝔽₅)	direct (N × H)
34. AGL(1,𝔽₅)	C₃²×C₉	direct (N × H)
38. C₃×C₅	C₃×C₄×C₉	semidirect
43. C₃²	C₉×AGL(1,𝔽₅)	direct (N × H)
45. C₉	C₃²×AGL(1,𝔽₅)	direct (N × H)
48. C₅	C₃²×C₄×C₉	semidirect
50. C₃	C₃×C₉×AGL(1,𝔽₅)	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×C₉×D₅	2
C₃³×AGL(1,𝔽₅)	3
C₃×C₉×AGL(1,𝔽₅)	3
C₃²×C₄×C₉	5

Subgroups — C32×C9×AGL(1,𝔽5) #13108359

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₉×AGL(1,𝔽₅) #13108359