Subgroups — C₃×C₅×D₅×C₃₁⋊C₃ #28358474

Order 13950.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

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

Complements

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

N	H ≅ G/N	type
1. C₃×C₅²×C₃₁⋊C₃	C₂	semidirect
2. C₃×C₅×D₅×C₃₁	C₃	semidirect
3. C₅×D₅×C₃₁⋊C₃	C₃	direct (N × H)
5. C₃×D₅×C₃₁⋊C₃	C₅	direct (N × H)
6. C₅²×C₃₁⋊C₃	C₂×C₃	semidirect
7. C₃×C₅²×C₃₁	C₂×C₃	semidirect
8. C₅×D₅×C₃₁	C₃²	semidirect
10. C₃×C₅×C₃₁⋊C₃	D₅	direct (N × H)
11. C₃×C₅×C₃₁⋊C₃	C₂×C₅	semidirect
13. C₃×D₅×C₃₁	C₃×C₅	semidirect
15. D₅×C₃₁⋊C₃	C₃×C₅	direct (N × H)
16. C₅²×C₃₁	C₂×C₃²	semidirect
19. C₃×C₅×C₃₁	C₃×D₅	semidirect
20. C₃×C₅×C₃₁	C₂×C₃×C₅	semidirect
22. C₅×C₃₁⋊C₃	C₃×D₅	direct (N × H)
23. C₅×C₃₁⋊C₃	C₂×C₃×C₅	semidirect
26. D₅×C₃₁	C₃²×C₅	semidirect
27. C₃×C₃₁⋊C₃	C₅×D₅	direct (N × H)
32. C₅×C₃₁	C₃²×D₅	semidirect
33. C₅×C₃₁	C₂×C₃²×C₅	semidirect
35. C₃×C₅×D₅	C₃₁⋊C₃	direct (N × H)
36. C₃×C₃₁	C₃×C₅×D₅	semidirect
37. C₃₁⋊C₃	C₃×C₅×D₅	direct (N × H)
41. C₃×C₅²	C₂×C₃₁⋊C₃	semidirect
43. C₅×D₅	C₃×C₃₁⋊C₃	direct (N × H)
47. C₃₁	C₃²×C₅×D₅	semidirect
51. C₃×D₅	C₅×C₃₁⋊C₃	direct (N × H)
52. C₅²	C₂×C₃×C₃₁⋊C₃	semidirect
58. C₃×C₅	D₅×C₃₁⋊C₃	direct (N × H)
59. C₃×C₅	C₂×C₅×C₃₁⋊C₃	semidirect
61. D₅	C₃×C₅×C₃₁⋊C₃	direct (N × H)
66. C₅	C₃×D₅×C₃₁⋊C₃	direct (N × H)
67. C₅	C₂×C₃×C₅×C₃₁⋊C₃	semidirect
69. C₃	C₅×D₅×C₃₁⋊C₃	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×C₅²×C₃₁⋊C₃	2
C₃×C₅×D₅×C₃₁	3
C₅×D₅×C₃₁⋊C₃	3
C₂×C₃×C₅×C₃₁⋊C₃	5
C₃×D₅×C₃₁⋊C₃	5
C₃²×C₅×D₅	31

Subgroups — C3×C5×D5×C31⋊C3 #28358474

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×D₅×C₃₁⋊C₃ #28358474