Subgroups — C₃×C₅×D₁₅×C₃₁ #28358514

Order 13950.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅²×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅×D₁₅×C₃₁	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×C₅×D₅×C₃₁	—	3	1	no	maximal
4	C₃×D₁₅×C₃₁	C₅	1	1	yes	maximal, characteristic, direct product factor
5	C₃×C₅×D₃×C₃₁	—	5	1	no	maximal
6	C₃×C₅²×C₃₁	—	2	1	no	—
7	C₃×C₅²×C₃₁	D₃	1	1	yes	characteristic, complemented
8	C₃×C₅²×C₃₁	C₂×C₃	1	1	yes	characteristic, complemented
9	C₅×D₅×C₃₁	—	3	1	no	Hall
10	C₃²×C₅×C₃₁	—	2	2	no	—
11	C₃²×C₅×C₃₁	D₅	1	1	yes	characteristic, complemented
12	C₃²×C₅×C₃₁	C₂×C₅	1	1	yes	characteristic, complemented
13	C₂×C₃×C₅×C₃₁	—	15	1	no	—
14	D₁₅×C₃₁	C₃×C₅	1	1	yes	characteristic, direct product factor
15	C₅×D₃×C₃₁	—	5	1	no	—
16	C₃×D₅×C₃₁	—	3	1	no	—
17	C₅²×C₃₁	C₃×D₃	1	1	yes	characteristic, complemented, Hall
18	C₃×D₃×C₃₁	—	5	1	no	Hall
19	C₃×C₅×C₃₁	—	2	4	no	—
20	C₃×C₅×C₃₁	—	2	1	no	—
21	C₃×C₅×C₃₁	—	2	1	no	—
22	C₃×C₅×C₃₁	—	2	2	no	—
23	C₃×C₅×C₃₁	—	2	2	no	—
24	C₃×C₅×C₃₁	C₅×D₃	1	1	yes	characteristic, complemented
25	C₃×C₅×C₃₁	C₃×D₅	1	1	yes	characteristic, complemented
26	C₃×C₅×C₃₁	D₁₅	1	1	yes	characteristic, direct product factor
27	C₃×C₅×C₃₁	C₂×C₃×C₅	1	1	yes	characteristic, complemented
28	C₃×C₅×D₁₅	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
29	C₂×C₅×C₃₁	—	15	1	no	—
30	D₅×C₃₁	—	3	1	no	—
31	C₃²×C₃₁	C₅×D₅	1	1	yes	characteristic, complemented, Hall
32	C₃²×C₅²	C₂×C₃₁	1	1	yes	characteristic, complemented, Hall
33	C₂×C₃×C₃₁	—	15	1	no	—
34	D₃×C₃₁	—	5	1	no	—
35	C₅×C₃₁	—	2	2	no	—
36	C₅×C₃₁	C₃×C₅×D₃	1	1	yes	characteristic, complemented
37	C₅×C₃₁	C₃×D₁₅	1	1	yes	characteristic, direct product factor
38	C₅×D₁₅	C₃×C₃₁	1	1	yes	characteristic, direct product factor
39	C₃×C₅×D₅	—	3	1	no	—
40	C₃×C₃₁	—	2	1	no	—
41	C₃×C₃₁	C₃×C₅×D₅	1	1	yes	characteristic, complemented
42	C₃×C₃₁	C₅×D₁₅	1	1	yes	characteristic, direct product factor
43	C₃×D₁₅	C₅×C₃₁	1	1	yes	characteristic, direct product factor
44	C₃×C₅×D₃	—	5	1	no	—
45	C₃×C₅²	—	2	1	no	—
46	C₃×C₅²	D₃×C₃₁	1	1	yes	characteristic, complemented
47	C₃×C₅²	C₂×C₃×C₃₁	1	1	yes	characteristic, complemented
48	C₂×C₃₁	—	15	1	no	Hall
49	C₅×D₅	—	3	1	no	Hall
50	C₃²×C₅	—	2	2	no	—
51	C₃²×C₅	D₅×C₃₁	1	1	yes	characteristic, complemented
52	C₃²×C₅	C₂×C₅×C₃₁	1	1	yes	characteristic, complemented
53	C₃₁	C₃×C₅×D₁₅	1	1	yes	characteristic, direct product factor, Sylow
54	C₂×C₃×C₅	—	15	1	no	—
55	D₁₅	C₃×C₅×C₃₁	1	1	yes	characteristic, direct product factor
56	C₃×D₅	—	3	1	no	—
57	C₅×D₃	—	5	1	no	—
58	C₅²	C₃×D₃×C₃₁	1	1	yes	characteristic, complemented, Sylow
59	C₃×D₃	—	5	1	no	Hall
60	C₃×C₅	—	2	4	no	—
61	C₃×C₅	—	2	1	no	—
62	C₃×C₅	—	2	1	no	—
63	C₃×C₅	—	2	2	no	—
64	C₃×C₅	—	2	2	no	—
65	C₃×C₅	C₃×D₅×C₃₁	1	1	yes	characteristic, complemented
66	C₃×C₅	C₅×D₃×C₃₁	1	1	yes	characteristic, complemented
67	C₃×C₅	D₁₅×C₃₁	1	1	yes	characteristic, direct product factor
68	C₃×C₅	C₂×C₃×C₅×C₃₁	1	1	yes	characteristic, complemented
69	C₂×C₅	—	15	1	no	—
70	D₅	—	3	1	no	—
71	C₃²	C₅×D₅×C₃₁	1	1	yes	characteristic, complemented, Sylow
72	C₂×C₃	—	15	1	no	—
73	D₃	—	5	1	no	—
74	C₅	—	2	2	no	—
75	C₅	C₃×C₅×D₃×C₃₁	1	1	yes	characteristic, complemented
76	C₅	C₃×D₁₅×C₃₁	1	1	yes	characteristic, direct product factor
77	C₃	—	2	1	no	—
78	C₃	C₃×C₅×D₅×C₃₁	1	1	yes	characteristic, complemented
79	C₃	C₅×D₁₅×C₃₁	1	1	yes	characteristic, direct product factor
80	C₂	—	15	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃×C₅×C₃₁	D₁₅
commutator, cosocle, hypocenter	C₃×C₅	C₂×C₃×C₅×C₃₁
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₃₁	C₂
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₃×C₅×C₃₁

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅²×C₃₁	C₂	semidirect
2. C₅×D₁₅×C₃₁	C₃	direct (N × H)
4. C₃×D₁₅×C₃₁	C₅	direct (N × H)
7. C₃×C₅²×C₃₁	D₃	semidirect
8. C₃×C₅²×C₃₁	C₂×C₃	semidirect
11. C₃²×C₅×C₃₁	D₅	semidirect
12. C₃²×C₅×C₃₁	C₂×C₅	semidirect
14. D₁₅×C₃₁	C₃×C₅	direct (N × H)
17. C₅²×C₃₁	C₃×D₃	semidirect
24. C₃×C₅×C₃₁	C₅×D₃	semidirect
25. C₃×C₅×C₃₁	C₃×D₅	semidirect
26. C₃×C₅×C₃₁	D₁₅	direct (N × H)
27. C₃×C₅×C₃₁	C₂×C₃×C₅	semidirect
28. C₃×C₅×D₁₅	C₃₁	direct (N × H)
31. C₃²×C₃₁	C₅×D₅	semidirect
32. C₃²×C₅²	C₂×C₃₁	semidirect
36. C₅×C₃₁	C₃×C₅×D₃	semidirect
37. C₅×C₃₁	C₃×D₁₅	direct (N × H)
38. C₅×D₁₅	C₃×C₃₁	direct (N × H)
41. C₃×C₃₁	C₃×C₅×D₅	semidirect
42. C₃×C₃₁	C₅×D₁₅	direct (N × H)
43. C₃×D₁₅	C₅×C₃₁	direct (N × H)
46. C₃×C₅²	D₃×C₃₁	semidirect
47. C₃×C₅²	C₂×C₃×C₃₁	semidirect
51. C₃²×C₅	D₅×C₃₁	semidirect
52. C₃²×C₅	C₂×C₅×C₃₁	semidirect
53. C₃₁	C₃×C₅×D₁₅	direct (N × H)
55. D₁₅	C₃×C₅×C₃₁	direct (N × H)
58. C₅²	C₃×D₃×C₃₁	semidirect
65. C₃×C₅	C₃×D₅×C₃₁	semidirect
66. C₃×C₅	C₅×D₃×C₃₁	semidirect
67. C₃×C₅	D₁₅×C₃₁	direct (N × H)
68. C₃×C₅	C₂×C₃×C₅×C₃₁	semidirect
71. C₃²	C₅×D₅×C₃₁	semidirect
75. C₅	C₃×C₅×D₃×C₃₁	semidirect
76. C₅	C₃×D₁₅×C₃₁	direct (N × H)
78. C₃	C₃×C₅×D₅×C₃₁	semidirect
79. C₃	C₅×D₁₅×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
C₃²×C₅²×C₃₁	2
C₅×D₁₅×C₃₁	3
C₃×C₅×D₅×C₃₁	3
C₃×D₁₅×C₃₁	5
C₃×C₅×D₃×C₃₁	5
C₃×C₅×D₁₅	31

Subgroups — C3×C5×D15×C31 #28358514

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×D₁₅×C₃₁ #28358514