Subgroups — C₃⋊D₃×C₃₁×C₅⋊D₅ #42124965

Order 27900.

480 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₃₁×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅²×C₃⋊D₃×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×C₃⋊(C₃⋊(C₅⋊D₅))	C₂	1	1	yes	maximal, characteristic, complemented
4	D₃×C₃₁×C₅⋊D₅	—	3	4	no	maximal
5	C₃²×C₅²×C₃₁	C₂²	1	1	yes	characteristic, complemented, Hall
6	D₅×C₃⋊D₃×C₃₁	—	5	6	no	maximal
7	C₃₁×C₃⋊(C₅⋊D₅)	—	3	4	no	—
8	C₅²×D₃×C₃₁	—	3	4	no	—
9	C₃×C₃₁×C₅⋊D₅	D₃	1	4	yes	complemented
10	C₂×C₃₁×C₅⋊D₅	—	9	1	no	Hall
11	C₃₁×C₃⋊D₁₅	—	5	6	no	—
12	C₅×C₃⋊D₃×C₃₁	D₅	1	6	yes	complemented
13	C₃²×D₅×C₃₁	—	5	6	no	—
14	C₃×C₅²×C₃₁	C₂×D₃	1	4	yes	complemented
15	D₃×D₅×C₃₁	—	15	24	no	—
16	C₂×C₅²×C₃₁	—	9	1	no	—
17	C₃₁×C₅⋊D₅	—	9	1	no	—
18	C₃₁×C₅⋊D₅	C₃⋊D₃	1	1	yes	characteristic, direct product factor
19	C₃²×C₅×C₃₁	C₂×D₅	1	6	yes	complemented
20	C₂×C₃⋊D₃×C₃₁	—	25	1	no	Hall
21	D₁₅×C₃₁	—	15	24	no	—
22	C₅×D₃×C₃₁	—	3	24	no	—
23	C₃×D₅×C₃₁	—	5	24	no	—
24	C₃⋊D₃×C₅⋊D₅	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅²×C₃₁	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented, Hall
26	C₂×D₅×C₃₁	—	45	6	no	—
27	C₂×C₃²×C₃₁	—	25	1	no	—
28	C₃⋊D₃×C₃₁	—	25	1	no	—
29	C₃⋊D₃×C₃₁	C₅⋊D₅	1	1	yes	characteristic, direct product factor
30	C₃×C₅×C₃₁	D₃×D₅	1	24	yes	complemented
31	C₃⋊(C₃⋊(C₅⋊D₅))	C₂×C₃₁	1	1	yes	characteristic, complemented
32	C₅²×C₃⋊D₃	C₂×C₃₁	1	1	yes	characteristic, complemented
33	C₃²×C₅⋊D₅	C₂×C₃₁	1	1	yes	characteristic, complemented
34	C₂×D₃×C₃₁	—	75	4	no	—
35	C₂×C₅×C₃₁	—	9	6	no	—
36	D₅×C₃₁	—	5	6	no	—
37	D₅×C₃₁	—	45	6	no	—
38	D₃×C₅⋊D₅	—	3	4	no	—
39	C₃²×C₃₁	C₂×C₅⋊D₅	1	1	yes	characteristic, complemented, Hall
40	C₃²×C₅²	C₂²×C₃₁	1	1	yes	characteristic, complemented, Hall
41	C₂×C₃×C₃₁	—	25	4	no	—
42	D₃×C₃₁	—	3	4	no	—
43	D₃×C₃₁	—	75	4	no	—
44	D₅×C₃⋊D₃	—	5	6	no	—
45	C₅×C₃₁	D₅×C₃⋊D₃	1	6	yes	complemented
46	C₃⋊(C₅⋊D₅)	—	3	4	no	—
47	C₅²×D₃	—	3	4	no	—
48	C₃×C₅⋊D₅	D₃×C₃₁	1	4	yes	complemented
49	C₂²×C₃₁	—	225	1	no	Hall
50	C₂×C₅⋊D₅	—	9	1	no	Hall
51	C₃×C₃₁	D₃×C₅⋊D₅	1	4	yes	complemented
52	C₃⋊D₁₅	—	5	6	no	—
53	C₅×C₃⋊D₃	D₅×C₃₁	1	6	yes	complemented
54	C₃²×D₅	—	5	6	no	—
55	C₃×C₅²	C₂×D₃×C₃₁	1	4	yes	complemented
56	C₂×C₃₁	—	9	1	no	—
57	C₂×C₃₁	—	25	1	no	—
58	C₂×C₃₁	—	225	1	no	—
59	D₃×D₅	—	15	24	no	—
60	C₂×C₅²	—	9	1	no	—
61	C₅⋊D₅	—	9	1	no	—
62	C₅⋊D₅	C₃⋊D₃×C₃₁	1	1	yes	characteristic, direct product factor
63	C₃²×C₅	C₂×D₅×C₃₁	1	6	yes	complemented
64	C₂×C₃⋊D₃	—	25	1	no	Hall
65	C₃₁	C₃⋊D₃×C₅⋊D₅	1	1	yes	characteristic, direct product factor, Sylow
66	D₁₅	—	15	24	no	—
67	C₃×D₅	—	5	24	no	—
68	C₅×D₃	—	3	24	no	—
69	C₅²	C₂×C₃⋊D₃×C₃₁	1	1	yes	characteristic, complemented, Sylow
70	C₂×D₅	—	45	6	no	—
71	C₂×C₃²	—	25	1	no	—
72	C₃⋊D₃	—	25	1	no	—
73	C₃⋊D₃	C₃₁×C₅⋊D₅	1	1	yes	characteristic, direct product factor
74	C₃×C₅	D₃×D₅×C₃₁	1	24	yes	complemented
75	C₂×D₃	—	75	4	no	—
76	C₂×C₅	—	9	6	no	—
77	D₅	—	5	6	no	—
78	D₅	—	45	6	no	—
79	C₃²	C₂×C₃₁×C₅⋊D₅	1	1	yes	characteristic, complemented, Sylow
80	C₂×C₃	—	25	4	no	—
81	D₃	—	3	4	no	—
82	D₃	—	75	4	no	—
83	C₅	D₅×C₃⋊D₃×C₃₁	1	6	yes	complemented
84	C₂²	—	225	1	no	Sylow
85	C₃	D₃×C₃₁×C₅⋊D₅	1	4	yes	complemented
86	C₂	—	9	1	no	—
87	C₂	—	25	1	no	—
88	C₂	—	225	1	no	—

	subgroup	G/H
center, hypercenter	C₃₁	C₃⋊D₃×C₅⋊D₅
commutator, cosocle, hypocenter	C₃²×C₅²	C₂²×C₃₁
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₃₁	C₂²
LCS	G ▷ C₃²×C₅²
UCS	C₁ ◁ C₃₁

Complements

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

N	H ≅ G/N	type
1. C₃²×C₃₁×C₅⋊D₅	C₂	semidirect
2. C₅²×C₃⋊D₃×C₃₁	C₂	semidirect
3. C₃₁×C₃⋊(C₃⋊(C₅⋊D₅))	C₂	semidirect
5. C₃²×C₅²×C₃₁	C₂²	semidirect
9. C₃×C₃₁×C₅⋊D₅	D₃	semidirect
12. C₅×C₃⋊D₃×C₃₁	D₅	semidirect
14. C₃×C₅²×C₃₁	C₂×D₃	semidirect
18. C₃₁×C₅⋊D₅	C₃⋊D₃	direct (N × H)
19. C₃²×C₅×C₃₁	C₂×D₅	semidirect
24. C₃⋊D₃×C₅⋊D₅	C₃₁	direct (N × H)
25. C₅²×C₃₁	C₂×C₃⋊D₃	semidirect
29. C₃⋊D₃×C₃₁	C₅⋊D₅	direct (N × H)
30. C₃×C₅×C₃₁	D₃×D₅	semidirect
31. C₃⋊(C₃⋊(C₅⋊D₅))	C₂×C₃₁	semidirect
32. C₅²×C₃⋊D₃	C₂×C₃₁	semidirect
33. C₃²×C₅⋊D₅	C₂×C₃₁	semidirect
39. C₃²×C₃₁	C₂×C₅⋊D₅	semidirect
40. C₃²×C₅²	C₂²×C₃₁	semidirect
45. C₅×C₃₁	D₅×C₃⋊D₃	semidirect
48. C₃×C₅⋊D₅	D₃×C₃₁	semidirect
51. C₃×C₃₁	D₃×C₅⋊D₅	semidirect
53. C₅×C₃⋊D₃	D₅×C₃₁	semidirect
55. C₃×C₅²	C₂×D₃×C₃₁	semidirect
62. C₅⋊D₅	C₃⋊D₃×C₃₁	direct (N × H)
63. C₃²×C₅	C₂×D₅×C₃₁	semidirect
65. C₃₁	C₃⋊D₃×C₅⋊D₅	direct (N × H)
69. C₅²	C₂×C₃⋊D₃×C₃₁	semidirect
73. C₃⋊D₃	C₃₁×C₅⋊D₅	direct (N × H)
74. C₃×C₅	D₃×D₅×C₃₁	semidirect
79. C₃²	C₂×C₃₁×C₅⋊D₅	semidirect
83. C₅	D₅×C₃⋊D₃×C₃₁	semidirect
85. C₃	D₃×C₃₁×C₅⋊D₅	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₃₁×C₅⋊D₅	2
C₅²×C₃⋊D₃×C₃₁	2
C₃₁×C₃⋊(C₃⋊(C₅⋊D₅))	2
D₃×C₃₁×C₅⋊D₅	3
D₅×C₃⋊D₃×C₃₁	5
C₃⋊D₃×C₅⋊D₅	31

Subgroups — C3⋊D3×C31×C5⋊D5 #42124965

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃⋊D₃×C₃₁×C₅⋊D₅ #42124965