Subgroups — C₃²×C₃₁×#1043 #42124893

Order 27900.

288 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₃₁×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
2	C₃×C₃₁×#1043	C₃	1	4	yes	maximal, direct product factor
3	C₃²×C₅²×C₃₁	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃²×AGL(1,𝔽₅)×C₃₁	—	5	6	no	maximal
5	C₃×C₃₁×C₅⋊D₅	C₂×C₃	1	4	yes	—
6	C₃₁×#1043	C₃²	1	1	yes	characteristic, direct product factor, Hall
7	C₃²×D₅×C₃₁	—	5	6	no	—
8	C₃×C₅²×C₃₁	C₃×C₄	1	4	yes	complemented
9	C₃×AGL(1,𝔽₅)×C₃₁	—	5	24	no	—
10	C₃₁×C₅⋊D₅	C₂×C₃²	1	1	yes	characteristic
11	C₃²×C₅×C₃₁	AGL(1,𝔽₅)	1	6	yes	complemented
12	C₃²×C₄×C₃₁	—	25	1	no	Hall
13	C₃×D₅×C₃₁	—	5	24	no	—
14	C₃²×#1043	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₅²×C₃₁	C₃²×C₄	1	1	yes	characteristic, complemented, Hall
16	AGL(1,𝔽₅)×C₃₁	—	5	6	no	—
17	C₂×C₃²×C₃₁	—	25	1	no	—
18	C₃×C₅×C₃₁	C₃×AGL(1,𝔽₅)	1	24	yes	complemented
19	C₃²×C₅⋊D₅	C₂×C₃₁	1	1	yes	characteristic
20	C₃×C₄×C₃₁	—	25	4	no	—
21	D₅×C₃₁	—	5	6	no	—
22	C₃×#1043	C₃×C₃₁	1	4	yes	direct product factor
23	C₃²×C₃₁	#1043	1	1	yes	characteristic, direct product factor, Hall
24	C₃²×C₅²	C₄×C₃₁	1	1	yes	characteristic, complemented, Hall
25	C₂×C₃×C₃₁	—	25	4	no	—
26	C₃²×AGL(1,𝔽₅)	—	5	6	no	—
27	C₅×C₃₁	C₃²×AGL(1,𝔽₅)	1	6	yes	complemented
28	C₃×C₅⋊D₅	C₂×C₃×C₃₁	1	4	yes	—
29	C₄×C₃₁	—	25	1	no	Hall
30	#1043	C₃²×C₃₁	1	1	yes	characteristic, direct product factor, Hall
31	C₃×C₃₁	C₃×#1043	1	4	yes	direct product factor
32	C₃²×D₅	—	5	6	no	—
33	C₃×C₅²	C₃×C₄×C₃₁	1	4	yes	complemented
34	C₂×C₃₁	—	25	1	no	—
35	C₃×AGL(1,𝔽₅)	—	5	24	no	—
36	C₅⋊D₅	C₂×C₃²×C₃₁	1	1	yes	characteristic
37	C₃²×C₅	AGL(1,𝔽₅)×C₃₁	1	6	yes	complemented
38	C₃²×C₄	—	25	1	no	Hall
39	C₃₁	C₃²×#1043	1	1	yes	characteristic, direct product factor, Sylow
40	C₃×D₅	—	5	24	no	—
41	C₅²	C₃²×C₄×C₃₁	1	1	yes	characteristic, complemented, Sylow
42	AGL(1,𝔽₅)	—	5	6	no	—
43	C₂×C₃²	—	25	1	no	—
44	C₃×C₅	C₃×AGL(1,𝔽₅)×C₃₁	1	24	yes	complemented
45	C₃×C₄	—	25	4	no	—
46	D₅	—	5	6	no	—
47	C₃²	C₃₁×#1043	1	1	yes	characteristic, direct product factor, Sylow
48	C₂×C₃	—	25	4	no	—
49	C₅	C₃²×AGL(1,𝔽₅)×C₃₁	1	6	yes	complemented
50	C₄	—	25	1	no	Sylow
51	C₃	C₃×C₃₁×#1043	1	4	yes	direct product factor
52	C₂	—	25	1	no	—

	subgroup	G/H
center, hypercenter	C₃²×C₃₁	#1043
commutator, hypocenter	C₅²	C₃²×C₄×C₃₁
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₃₁	C₄
prime radical	C₃²×C₃₁×C₅⋊D₅	C₂
cosocle	C₅⋊D₅	C₂×C₃²×C₃₁
LCS	G ▷ C₅²
UCS	C₁ ◁ C₃²×C₃₁

Complements

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

N	H ≅ G/N	type
2. C₃×C₃₁×#1043	C₃	direct (N × H)
3. C₃²×C₅²×C₃₁	C₄	semidirect
6. C₃₁×#1043	C₃²	direct (N × H)
8. C₃×C₅²×C₃₁	C₃×C₄	semidirect
11. C₃²×C₅×C₃₁	AGL(1,𝔽₅)	semidirect
14. C₃²×#1043	C₃₁	direct (N × H)
15. C₅²×C₃₁	C₃²×C₄	semidirect
18. C₃×C₅×C₃₁	C₃×AGL(1,𝔽₅)	semidirect
22. C₃×#1043	C₃×C₃₁	direct (N × H)
23. C₃²×C₃₁	#1043	direct (N × H)
24. C₃²×C₅²	C₄×C₃₁	semidirect
27. C₅×C₃₁	C₃²×AGL(1,𝔽₅)	semidirect
30. #1043	C₃²×C₃₁	direct (N × H)
31. C₃×C₃₁	C₃×#1043	direct (N × H)
33. C₃×C₅²	C₃×C₄×C₃₁	semidirect
37. C₃²×C₅	AGL(1,𝔽₅)×C₃₁	semidirect
39. C₃₁	C₃²×#1043	direct (N × H)
41. C₅²	C₃²×C₄×C₃₁	semidirect
44. C₃×C₅	C₃×AGL(1,𝔽₅)×C₃₁	semidirect
47. C₃²	C₃₁×#1043	direct (N × H)
49. C₅	C₃²×AGL(1,𝔽₅)×C₃₁	semidirect
51. C₃	C₃×C₃₁×#1043	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×C₃₁×C₅⋊D₅	2
C₃×C₃₁×#1043	3
C₃²×AGL(1,𝔽₅)×C₃₁	5
C₃²×#1043	31

Subgroups — C32×C31×#1043 #42124893

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₃₁×#1043 #42124893