Subgroups — C₃×C₅²×A₄×C₃₁ #42124975

Order 27900.

224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×A₄×C₃₁	C₃	1	3	yes	maximal, direct product factor
2	C₂²×C₃×C₅²×C₃₁	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃²×C₅²×C₃₁	—	4	1	no	maximal, Hall
4	C₃×C₅×A₄×C₃₁	C₅	1	6	yes	maximal, direct product factor
5	C₂×C₃×C₅²×C₃₁	—	3	1	no	—
6	C₂²×C₅²×C₃₁	C₃²	1	1	yes	characteristic, complemented, Hall
7	C₃×C₅²×C₃₁	—	4	3	no	—
8	C₃×C₅²×C₃₁	A₄	1	1	yes	characteristic, direct product factor
9	C₂²×C₃×C₅×C₃₁	C₃×C₅	1	6	yes	complemented
10	C₅×A₄×C₃₁	C₃×C₅	1	18	yes	direct product factor
11	C₂×C₅²×C₃₁	—	3	1	no	—
12	C₃²×C₅×C₃₁	—	4	6	no	—
13	C₃×A₄×C₃₁	C₅²	1	1	yes	characteristic, direct product factor, Hall
14	C₂×C₃×C₅×C₃₁	—	3	6	no	—
15	C₃×C₅²×A₄	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₅²×C₃₁	C₃×A₄	1	1	yes	characteristic, direct product factor, Hall
17	C₂²×C₅×C₃₁	C₃²×C₅	1	6	yes	complemented
18	C₃×C₅×C₃₁	—	4	18	no	—
19	C₃×C₅×C₃₁	C₅×A₄	1	6	yes	direct product factor
20	C₂²×C₃×C₃₁	C₃×C₅²	1	1	yes	characteristic, complemented
21	A₄×C₃₁	C₃×C₅²	1	3	yes	direct product factor
22	C₂×C₅×C₃₁	—	3	6	no	—
23	C₂²×C₃×C₅²	C₃×C₃₁	1	1	yes	characteristic, complemented
24	C₅²×A₄	C₃×C₃₁	1	3	yes	direct product factor
25	C₃²×C₃₁	—	4	1	no	Hall
26	C₃²×C₅²	—	4	1	no	Hall
27	C₂×C₃×C₃₁	—	3	1	no	—
28	C₃×C₅×A₄	C₅×C₃₁	1	6	yes	direct product factor
29	C₅×C₃₁	C₃×C₅×A₄	1	6	yes	direct product factor
30	C₂×C₃×C₅²	—	3	1	no	—
31	C₂²×C₃₁	C₃²×C₅²	1	1	yes	characteristic, complemented, Hall
32	C₂²×C₅²	C₃²×C₃₁	1	1	yes	characteristic, complemented, Hall
33	C₃×C₃₁	—	4	3	no	—
34	C₃×C₃₁	C₅²×A₄	1	1	yes	characteristic, direct product factor
35	C₃×C₅²	—	4	3	no	—
36	C₃×C₅²	A₄×C₃₁	1	1	yes	characteristic, direct product factor
37	C₂×C₃₁	—	3	1	no	—
38	C₂²×C₃×C₅	C₃×C₅×C₃₁	1	6	yes	complemented
39	C₅×A₄	C₃×C₅×C₃₁	1	18	yes	direct product factor
40	C₂×C₅²	—	3	1	no	—
41	C₃²×C₅	—	4	6	no	—
42	C₃×A₄	C₅²×C₃₁	1	1	yes	characteristic, direct product factor, Hall
43	C₃₁	C₃×C₅²×A₄	1	1	yes	characteristic, direct product factor, Sylow
44	C₂×C₃×C₅	—	3	6	no	—
45	C₅²	C₃×A₄×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
46	C₂²×C₅	C₃²×C₅×C₃₁	1	6	yes	complemented
47	C₃×C₅	—	4	18	no	—
48	C₃×C₅	C₅×A₄×C₃₁	1	6	yes	direct product factor
49	C₂²×C₃	C₃×C₅²×C₃₁	1	1	yes	characteristic, complemented
50	A₄	C₃×C₅²×C₃₁	1	3	yes	direct product factor
51	C₂×C₅	—	3	6	no	—
52	C₃²	—	4	1	no	Sylow
53	C₂×C₃	—	3	1	no	—
54	C₅	C₃×C₅×A₄×C₃₁	1	6	yes	direct product factor
55	C₂²	C₃²×C₅²×C₃₁	1	1	yes	characteristic, complemented, Sylow
56	C₃	—	4	3	no	—
57	C₃	C₅²×A₄×C₃₁	1	1	yes	characteristic, direct product factor
58	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₃×C₅²×C₃₁	A₄
commutator, cosocle, hypocenter	C₂²	C₃²×C₅²×C₃₁
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₅²×C₃₁	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₃×C₅²×C₃₁

Complements

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

N	H ≅ G/N	type
1. C₅²×A₄×C₃₁	C₃	direct (N × H)
2. C₂²×C₃×C₅²×C₃₁	C₃	semidirect
4. C₃×C₅×A₄×C₃₁	C₅	direct (N × H)
6. C₂²×C₅²×C₃₁	C₃²	semidirect
8. C₃×C₅²×C₃₁	A₄	direct (N × H)
9. C₂²×C₃×C₅×C₃₁	C₃×C₅	semidirect
10. C₅×A₄×C₃₁	C₃×C₅	direct (N × H)
13. C₃×A₄×C₃₁	C₅²	direct (N × H)
15. C₃×C₅²×A₄	C₃₁	direct (N × H)
16. C₅²×C₃₁	C₃×A₄	direct (N × H)
17. C₂²×C₅×C₃₁	C₃²×C₅	semidirect
19. C₃×C₅×C₃₁	C₅×A₄	direct (N × H)
20. C₂²×C₃×C₃₁	C₃×C₅²	semidirect
21. A₄×C₃₁	C₃×C₅²	direct (N × H)
23. C₂²×C₃×C₅²	C₃×C₃₁	semidirect
24. C₅²×A₄	C₃×C₃₁	direct (N × H)
28. C₃×C₅×A₄	C₅×C₃₁	direct (N × H)
29. C₅×C₃₁	C₃×C₅×A₄	direct (N × H)
31. C₂²×C₃₁	C₃²×C₅²	semidirect
32. C₂²×C₅²	C₃²×C₃₁	semidirect
34. C₃×C₃₁	C₅²×A₄	direct (N × H)
36. C₃×C₅²	A₄×C₃₁	direct (N × H)
38. C₂²×C₃×C₅	C₃×C₅×C₃₁	semidirect
39. C₅×A₄	C₃×C₅×C₃₁	direct (N × H)
42. C₃×A₄	C₅²×C₃₁	direct (N × H)
43. C₃₁	C₃×C₅²×A₄	direct (N × H)
45. C₅²	C₃×A₄×C₃₁	direct (N × H)
46. C₂²×C₅	C₃²×C₅×C₃₁	semidirect
48. C₃×C₅	C₅×A₄×C₃₁	direct (N × H)
49. C₂²×C₃	C₃×C₅²×C₃₁	semidirect
50. A₄	C₃×C₅²×C₃₁	direct (N × H)
54. C₅	C₃×C₅×A₄×C₃₁	direct (N × H)
55. C₂²	C₃²×C₅²×C₃₁	semidirect
57. C₃	C₅²×A₄×C₃₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅²×A₄×C₃₁	3
C₂²×C₃×C₅²×C₃₁	3
C₃²×C₅²×C₃₁	4
C₃×C₅×A₄×C₃₁	5
C₃×C₅²×A₄	31

Subgroups — C3×C52×A4×C31 #42124975

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅²×A₄×C₃₁ #42124975