Subgroups — C₂×C₅²×D₉×C₃₁ #42124587

Order 27900.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×D₉×C₃₁	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₅²×C₉×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅²×D₃×C₃₁	—	3	1	no	maximal
4	C₅²×C₉×C₃₁	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₅×D₉×C₃₁	C₅	1	6	yes	maximal, direct product factor
6	C₅²×D₃×C₃₁	—	3	2	no	—
7	C₂×C₃×C₅²×C₃₁	D₃	1	1	yes	characteristic
8	C₂²×C₅²×C₃₁	—	9	1	no	Hall
9	C₅×D₉×C₃₁	C₂×C₅	1	12	yes	direct product factor
10	C₂×C₅×C₉×C₃₁	C₂×C₅	1	6	yes	complemented
11	C₃×C₅²×C₃₁	C₂×D₃	1	1	yes	characteristic
12	C₂×C₅×D₃×C₃₁	—	3	6	no	—
13	C₂×C₅²×C₃₁	—	9	2	no	—
14	C₂×C₅²×C₃₁	D₉	1	1	yes	characteristic, direct product factor
15	C₅×C₉×C₃₁	C₂²×C₅	1	6	yes	complemented
16	C₂×D₉×C₃₁	C₅²	1	1	yes	characteristic, direct product factor, Hall
17	C₂×C₃×C₅×C₃₁	C₅×D₃	1	6	yes	—
18	C₅×D₃×C₃₁	—	3	12	no	—
19	C₂×C₅²×D₉	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₅²×C₃₁	C₂×D₉	1	1	yes	characteristic, direct product factor, Hall
21	C₂²×C₅×C₃₁	—	9	6	no	—
22	C₂×C₉×C₃₁	C₂×C₅²	1	1	yes	characteristic, complemented
23	D₉×C₃₁	C₂×C₅²	1	2	yes	direct product factor
24	C₃×C₅×C₃₁	C₂×C₅×D₃	1	6	yes	—
25	C₂×C₅²×C₉	C₂×C₃₁	1	1	yes	characteristic, complemented
26	C₅²×D₉	C₂×C₃₁	1	2	yes	direct product factor
27	C₂×D₃×C₃₁	—	3	1	no	—
28	C₂×C₅×C₃₁	—	9	12	no	—
29	C₂×C₅×C₃₁	C₅×D₉	1	6	yes	direct product factor
30	C₂×C₅²×D₃	—	3	1	no	—
31	C₉×C₃₁	C₂²×C₅²	1	1	yes	characteristic, complemented, Hall
32	C₅²×C₉	C₂²×C₃₁	1	1	yes	characteristic, complemented, Hall
33	C₂×C₃×C₃₁	C₅²×D₃	1	1	yes	characteristic
34	D₃×C₃₁	—	3	2	no	—
35	C₂×C₅×D₉	C₅×C₃₁	1	6	yes	direct product factor
36	C₅×C₃₁	C₂×C₅×D₉	1	6	yes	direct product factor
37	C₂×C₃×C₅²	D₃×C₃₁	1	1	yes	characteristic
38	C₅²×D₃	—	3	2	no	—
39	C₂²×C₃₁	—	9	1	no	Hall
40	C₂²×C₅²	—	9	1	no	Hall
41	C₃×C₃₁	C₂×C₅²×D₃	1	1	yes	characteristic
42	C₂×C₅×C₉	C₂×C₅×C₃₁	1	6	yes	complemented
43	C₅×D₉	C₂×C₅×C₃₁	1	12	yes	direct product factor
44	C₃×C₅²	C₂×D₃×C₃₁	1	1	yes	characteristic
45	C₂×C₃₁	—	9	2	no	—
46	C₂×C₃₁	C₅²×D₉	1	1	yes	characteristic, direct product factor
47	C₂×C₅×D₃	—	3	6	no	—
48	C₂×C₅²	—	9	2	no	—
49	C₂×C₅²	D₉×C₃₁	1	1	yes	characteristic, direct product factor
50	C₅×C₉	C₂²×C₅×C₃₁	1	6	yes	complemented
51	C₂×D₉	C₅²×C₃₁	1	1	yes	characteristic, direct product factor, Hall
52	C₃₁	C₂×C₅²×D₉	1	1	yes	characteristic, direct product factor, Sylow
53	C₂×C₃×C₅	C₅×D₃×C₃₁	1	6	yes	—
54	C₅×D₃	—	3	12	no	—
55	C₅²	C₂×D₉×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
56	C₂²×C₅	—	9	6	no	—
57	C₂×C₉	C₂×C₅²×C₃₁	1	1	yes	characteristic, complemented
58	D₉	C₂×C₅²×C₃₁	1	2	yes	direct product factor
59	C₃×C₅	C₂×C₅×D₃×C₃₁	1	6	yes	—
60	C₂×D₃	—	3	1	no	—
61	C₂×C₅	—	9	12	no	—
62	C₂×C₅	C₅×D₉×C₃₁	1	6	yes	direct product factor
63	C₉	C₂²×C₅²×C₃₁	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₃	C₅²×D₃×C₃₁	1	1	yes	characteristic
65	D₃	—	3	2	no	—
66	C₅	C₂×C₅×D₉×C₃₁	1	6	yes	direct product factor
67	C₂²	—	9	1	no	Sylow
68	C₃	C₂×C₅²×D₃×C₃₁	1	1	yes	characteristic
69	C₂	—	9	2	no	—
70	C₂	C₅²×D₉×C₃₁	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₅²×C₃₁	D₉
commutator, cosocle, hypocenter	C₉	C₂²×C₅²×C₃₁
Frattini	C₃	C₂×C₅²×D₃×C₃₁
Fitting	C₂×C₅²×C₉×C₃₁	C₂
socle	C₂×C₃×C₅²×C₃₁	D₃
LCS	G ▷ C₉
UCS	C₁ ◁ C₂×C₅²×C₃₁

Complements

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

N	H ≅ G/N	type
1. C₅²×D₉×C₃₁	C₂	direct (N × H)
2. C₂×C₅²×C₉×C₃₁	C₂	semidirect
4. C₅²×C₉×C₃₁	C₂²	semidirect
5. C₂×C₅×D₉×C₃₁	C₅	direct (N × H)
9. C₅×D₉×C₃₁	C₂×C₅	direct (N × H)
10. C₂×C₅×C₉×C₃₁	C₂×C₅	semidirect
14. C₂×C₅²×C₃₁	D₉	direct (N × H)
15. C₅×C₉×C₃₁	C₂²×C₅	semidirect
16. C₂×D₉×C₃₁	C₅²	direct (N × H)
19. C₂×C₅²×D₉	C₃₁	direct (N × H)
20. C₅²×C₃₁	C₂×D₉	direct (N × H)
22. C₂×C₉×C₃₁	C₂×C₅²	semidirect
23. D₉×C₃₁	C₂×C₅²	direct (N × H)
25. C₂×C₅²×C₉	C₂×C₃₁	semidirect
26. C₅²×D₉	C₂×C₃₁	direct (N × H)
29. C₂×C₅×C₃₁	C₅×D₉	direct (N × H)
31. C₉×C₃₁	C₂²×C₅²	semidirect
32. C₅²×C₉	C₂²×C₃₁	semidirect
35. C₂×C₅×D₉	C₅×C₃₁	direct (N × H)
36. C₅×C₃₁	C₂×C₅×D₉	direct (N × H)
42. C₂×C₅×C₉	C₂×C₅×C₃₁	semidirect
43. C₅×D₉	C₂×C₅×C₃₁	direct (N × H)
46. C₂×C₃₁	C₅²×D₉	direct (N × H)
49. C₂×C₅²	D₉×C₃₁	direct (N × H)
50. C₅×C₉	C₂²×C₅×C₃₁	semidirect
51. C₂×D₉	C₅²×C₃₁	direct (N × H)
52. C₃₁	C₂×C₅²×D₉	direct (N × H)
55. C₅²	C₂×D₉×C₃₁	direct (N × H)
57. C₂×C₉	C₂×C₅²×C₃₁	semidirect
58. D₉	C₂×C₅²×C₃₁	direct (N × H)
62. C₂×C₅	C₅×D₉×C₃₁	direct (N × H)
63. C₉	C₂²×C₅²×C₃₁	semidirect
66. C₅	C₂×C₅×D₉×C₃₁	direct (N × H)
70. C₂	C₅²×D₉×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×D₉×C₃₁	2
C₂×C₅²×C₉×C₃₁	2
C₂×C₅²×D₃×C₃₁	3
C₂×C₅×D₉×C₃₁	5
C₂×C₅²×D₉	31

Subgroups — C2×C52×D9×C31 #42124587

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅²×D₉×C₃₁ #42124587