Subgroups — C₃×C₃₁⋊C₃×#1043 #42124763

Order 27900.

288 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅⋊D₅×C₃₁⋊C₃	C₂	1	1	yes	maximal, characteristic
2	C₃×C₃₁×#1043	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃₁⋊C₃×#1043	C₃	1	3	yes	maximal, direct product factor
4	C₃×C₅²×C₃₁⋊C₃	C₄	1	1	yes	characteristic, complemented, Hall
5	C₃×AGL(1,𝔽₅)×C₃₁⋊C₃	—	5	6	no	maximal
6	C₃×C₃₁×C₅⋊D₅	C₂×C₃	1	1	yes	characteristic
7	C₅⋊D₅×C₃₁⋊C₃	C₂×C₃	1	3	yes	—
8	C₃₁×#1043	C₃²	1	1	yes	characteristic, complemented, Hall
9	C₃×D₅×C₃₁⋊C₃	—	5	6	no	—
10	C₅²×C₃₁⋊C₃	C₃×C₄	1	3	yes	complemented
11	C₃×C₅²×C₃₁	C₃×C₄	1	1	yes	characteristic, complemented
12	C₃×AGL(1,𝔽₅)×C₃₁	—	5	6	no	—
13	AGL(1,𝔽₅)×C₃₁⋊C₃	—	5	18	no	—
14	C₃₁×C₅⋊D₅	C₂×C₃²	1	1	yes	characteristic
15	C₃×C₅×C₃₁⋊C₃	AGL(1,𝔽₅)	1	6	yes	complemented
16	C₃×C₄×C₃₁⋊C₃	—	25	1	no	Hall
17	C₃×D₅×C₃₁	—	5	6	no	—
18	D₅×C₃₁⋊C₃	—	5	18	no	—
19	C₃²×#1043	—	31	1	no	maximal, Hall
20	C₅²×C₃₁	C₃²×C₄	1	1	yes	characteristic, complemented, Hall
21	AGL(1,𝔽₅)×C₃₁	—	5	6	no	—
22	C₂×C₃×C₃₁⋊C₃	—	25	1	no	—
23	C₃×C₅×C₃₁	C₃×AGL(1,𝔽₅)	1	6	yes	complemented
24	C₅×C₃₁⋊C₃	C₃×AGL(1,𝔽₅)	1	18	yes	complemented
25	C₃²×C₅⋊D₅	—	31	1	no	—
26	C₃×C₄×C₃₁	—	25	1	no	—
27	C₄×C₃₁⋊C₃	—	25	3	no	—
28	D₅×C₃₁	—	5	6	no	—
29	C₃×#1043	—	31	3	no	—
30	C₃×#1043	C₃₁⋊C₃	1	1	yes	characteristic, direct product factor
31	C₃×C₃₁⋊C₃	#1043	1	1	yes	characteristic, direct product factor, Hall
32	C₃²×C₅²	—	31	1	no	Hall
33	C₂×C₃×C₃₁	—	25	1	no	—
34	C₂×C₃₁⋊C₃	—	25	3	no	—
35	C₃²×AGL(1,𝔽₅)	—	155	6	no	—
36	C₅×C₃₁	C₃²×AGL(1,𝔽₅)	1	6	yes	complemented
37	C₃×C₅⋊D₅	—	31	3	no	—
38	C₃×C₅⋊D₅	C₂×C₃₁⋊C₃	1	1	yes	characteristic
39	C₄×C₃₁	—	25	1	no	Hall
40	#1043	C₃×C₃₁⋊C₃	1	1	yes	characteristic, direct product factor, Hall
41	C₃×C₃₁	C₃×#1043	1	1	yes	characteristic, complemented
42	C₃₁⋊C₃	C₃×#1043	1	3	yes	direct product factor
43	C₃²×D₅	—	155	6	no	—
44	C₃×C₅²	—	31	3	no	—
45	C₃×C₅²	C₄×C₃₁⋊C₃	1	1	yes	characteristic, complemented
46	C₂×C₃₁	—	25	1	no	—
47	C₃×AGL(1,𝔽₅)	—	5	6	no	—
48	C₃×AGL(1,𝔽₅)	—	155	18	no	—
49	C₅⋊D₅	C₂×C₃×C₃₁⋊C₃	1	1	yes	characteristic
50	C₃²×C₅	—	31	6	no	—
51	C₃²×C₄	—	775	1	no	Hall
52	C₃₁	C₃²×#1043	1	1	yes	characteristic, complemented, Sylow
53	C₃×D₅	—	5	6	no	—
54	C₃×D₅	—	155	18	no	—
55	C₅²	C₃×C₄×C₃₁⋊C₃	1	1	yes	characteristic, complemented, Sylow
56	AGL(1,𝔽₅)	—	5	6	no	—
57	C₂×C₃²	—	775	1	no	—
58	C₃×C₅	—	31	18	no	—
59	C₃×C₅	AGL(1,𝔽₅)×C₃₁⋊C₃	1	6	yes	complemented
60	C₃×C₄	—	775	3	no	—
61	C₃×C₄	—	25	1	no	—
62	D₅	—	5	6	no	—
63	C₃²	—	31	1	no	Sylow
64	C₂×C₃	—	775	3	no	—
65	C₂×C₃	—	25	1	no	—
66	C₅	C₃×AGL(1,𝔽₅)×C₃₁⋊C₃	1	6	yes	complemented
67	C₄	—	25	1	no	Sylow
68	C₃	—	31	3	no	—
69	C₃	C₃₁⋊C₃×#1043	1	1	yes	characteristic, direct product factor
70	C₂	—	25	1	no	—

	subgroup	G/H
center, hypercenter	C₃	C₃₁⋊C₃×#1043
commutator, hypocenter	C₅²×C₃₁	C₃²×C₄
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₃₁	C₃×C₄
prime radical	C₃×C₅⋊D₅×C₃₁⋊C₃	C₂
cosocle	C₃₁×C₅⋊D₅	C₂×C₃²
LCS	G ▷ C₅²×C₃₁
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
2. C₃×C₃₁×#1043	C₃	semidirect
3. C₃₁⋊C₃×#1043	C₃	direct (N × H)
4. C₃×C₅²×C₃₁⋊C₃	C₄	semidirect
8. C₃₁×#1043	C₃²	semidirect
10. C₅²×C₃₁⋊C₃	C₃×C₄	semidirect
11. C₃×C₅²×C₃₁	C₃×C₄	semidirect
15. C₃×C₅×C₃₁⋊C₃	AGL(1,𝔽₅)	semidirect
20. C₅²×C₃₁	C₃²×C₄	semidirect
23. C₃×C₅×C₃₁	C₃×AGL(1,𝔽₅)	semidirect
24. C₅×C₃₁⋊C₃	C₃×AGL(1,𝔽₅)	semidirect
30. C₃×#1043	C₃₁⋊C₃	direct (N × H)
31. C₃×C₃₁⋊C₃	#1043	direct (N × H)
36. C₅×C₃₁	C₃²×AGL(1,𝔽₅)	semidirect
40. #1043	C₃×C₃₁⋊C₃	direct (N × H)
41. C₃×C₃₁	C₃×#1043	semidirect
42. C₃₁⋊C₃	C₃×#1043	direct (N × H)
45. C₃×C₅²	C₄×C₃₁⋊C₃	semidirect
52. C₃₁	C₃²×#1043	semidirect
55. C₅²	C₃×C₄×C₃₁⋊C₃	semidirect
59. C₃×C₅	AGL(1,𝔽₅)×C₃₁⋊C₃	semidirect
66. C₅	C₃×AGL(1,𝔽₅)×C₃₁⋊C₃	semidirect
69. C₃	C₃₁⋊C₃×#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
C₃×C₅⋊D₅×C₃₁⋊C₃	2
C₃×C₃₁×#1043	3
C₃₁⋊C₃×#1043	3
C₃×AGL(1,𝔽₅)×C₃₁⋊C₃	5
C₃²×#1043	31

Subgroups — C3×C31⋊C3×#1043 #42124763

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₃₁⋊C₃×#1043 #42124763