Subgroups — C₃×C₅×ΓL(1,𝔽₂₅) #10623851

Order 720.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×D₃	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×M₁₆	—	3	1	no	maximal
5	C₅×ΓL(1,𝔽₂₅)	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂×C₃×C₅×D₃	C₄	1	1	yes	characteristic
7	C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₃×C₅×Dic₃	C₄	1	1	yes	characteristic
9	C₃×ΓL(1,𝔽₂₅)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₃×C₄×C₅	—	3	1	no	—
11	C₄×C₅×D₃	C₂×C₃	1	1	yes	characteristic
12	C₃×C₅×C₈	—	2	1	no	—
13	C₃×C₅×C₈	—	3	1	no	—
14	C₃×C₅×C₈	D₃	1	1	yes	characteristic, complemented
15	C₃×C₅×C₈	C₂×C₃	1	1	yes	characteristic, complemented
16	C₅×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
17	C₂×C₃²×C₅	C₂×C₄	1	1	yes	characteristic
18	C₃×C₅×D₃	—	2	1	no	—
19	C₅×M₁₆	—	3	1	no	Hall
20	C₃×C₄×D₃	C₂×C₅	1	1	yes	characteristic
21	C₃²×C₈	C₂×C₅	1	1	yes	characteristic, complemented
22	C₃×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
23	C₂²×C₃×C₅	—	3	1	no	—
24	C₂×C₅×D₃	C₃×C₄	1	1	yes	characteristic
25	C₃×C₄×C₅	—	2	1	no	—
26	C₃×C₄×C₅	—	3	1	no	—
27	C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
28	C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
29	C₅×Dic₃	C₃×C₄	1	1	yes	characteristic
30	C₃×M₁₆	—	3	1	no	—
31	ΓL(1,𝔽₂₅)	C₃×C₅	1	1	yes	characteristic, direct product factor
32	C₃²×C₅	M₁₆	1	1	yes	characteristic, complemented, Hall
33	C₂×C₄×C₅	—	3	1	no	—
34	C₅×C₈	—	3	1	no	—
35	C₅×C₈	C₃×D₃	1	1	yes	characteristic, complemented
36	C₂×C₃×D₃	C₄×C₅	1	1	yes	characteristic
37	C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
38	C₃×Dic₃	C₄×C₅	1	1	yes	characteristic
39	C₂×C₃×C₅	—	2	1	no	—
40	C₂×C₃×C₅	—	6	1	no	—
41	C₂×C₃×C₅	C₄×D₃	1	1	yes	characteristic
42	C₂×C₃×C₅	C₂×C₃×C₄	1	1	yes	characteristic
43	C₅×D₃	—	2	1	no	—
44	C₂×C₃×C₄	—	3	1	no	—
45	C₄×D₃	C₂×C₃×C₅	1	1	yes	characteristic
46	C₃×C₈	—	2	1	no	—
47	C₃×C₈	—	3	1	no	—
48	C₃×C₈	C₅×D₃	1	1	yes	characteristic, complemented
49	C₃×C₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
50	C₃⋊C₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
51	C₂²×C₅	—	3	1	no	—
52	C₄×C₅	—	3	1	no	—
53	C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
54	C₂×C₃²	C₂×C₄×C₅	1	1	yes	characteristic
55	C₃×D₃	—	2	1	no	—
56	M₁₆	—	3	1	no	Sylow
57	C₃×C₅	—	2	1	no	—
58	C₃×C₅	ΓL(1,𝔽₂₅)	1	1	yes	characteristic, direct product factor
59	C₃×C₅	C₃×M₁₆	1	1	yes	characteristic, complemented
60	C₂²×C₃	—	3	1	no	—
61	C₂×D₃	C₃×C₄×C₅	1	1	yes	characteristic
62	C₃×C₄	—	2	1	no	—
63	C₃×C₄	—	3	1	no	—
64	C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
65	C₃×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
66	Dic₃	C₃×C₄×C₅	1	1	yes	characteristic
67	C₂×C₅	—	6	1	no	—
68	C₂×C₅	C₃×C₄×D₃	1	1	yes	characteristic
69	C₃²	C₅×M₁₆	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₄	—	3	1	no	—
71	C₈	—	3	1	no	—
72	C₈	C₃×C₅×D₃	1	1	yes	characteristic, complemented
73	C₂×C₃	—	2	1	no	—
74	C₂×C₃	—	6	1	no	—
75	C₂×C₃	C₄×C₅×D₃	1	1	yes	characteristic
76	C₂×C₃	C₂×C₃×C₄×C₅	1	1	yes	characteristic
77	D₃	—	2	1	no	—
78	C₅	C₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic, direct product factor, Sylow
79	C₂²	—	3	1	no	—
80	C₄	—	3	1	no	—
81	C₄	C₂×C₃×C₅×D₃	1	1	yes	characteristic
82	C₃	—	2	1	no	—
83	C₃	C₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic, direct product factor
84	C₃	C₃×C₅×M₁₆	1	1	yes	characteristic, complemented
85	C₂	—	6	1	no	—
86	C₂	C₃×C₄×C₅×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄×C₅	C₂×D₃
commutator	C₂×C₃	C₂×C₃×C₄×C₅
Frattini	C₄	C₂×C₃×C₅×D₃
Fitting	C₃²×C₅×C₈	C₂
socle	C₂×C₃²×C₅	C₂×C₄
prime radical	C₂×C₃×C₅×D₃	C₄
cosocle	C₃×C₄	C₂²×C₃×C₅
hypercenter	C₃×C₅×C₈	D₃
hypocenter	C₃	C₃×C₅×M₁₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃×C₄×C₅ ◁ C₃×C₅×C₈

Complements

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

N	H ≅ G/N	type
2. C₃²×C₅×C₈	C₂	semidirect
3. C₃×C₅×C₃⋊C₈	C₂	semidirect
5. C₅×ΓL(1,𝔽₂₅)	C₃	direct (N × H)
9. C₃×ΓL(1,𝔽₂₅)	C₅	direct (N × H)
14. C₃×C₅×C₈	D₃	semidirect
15. C₃×C₅×C₈	C₂×C₃	semidirect
16. C₅×C₃⋊C₈	C₂×C₃	semidirect
21. C₃²×C₈	C₂×C₅	semidirect
22. C₃×C₃⋊C₈	C₂×C₅	semidirect
31. ΓL(1,𝔽₂₅)	C₃×C₅	direct (N × H)
32. C₃²×C₅	M₁₆	semidirect
35. C₅×C₈	C₃×D₃	semidirect
48. C₃×C₈	C₅×D₃	semidirect
49. C₃×C₈	C₂×C₃×C₅	semidirect
50. C₃⋊C₈	C₂×C₃×C₅	semidirect
58. C₃×C₅	ΓL(1,𝔽₂₅)	direct (N × H)
59. C₃×C₅	C₃×M₁₆	semidirect
69. C₃²	C₅×M₁₆	semidirect
72. C₈	C₃×C₅×D₃	semidirect
78. C₅	C₃×ΓL(1,𝔽₂₅)	direct (N × H)
83. C₃	C₅×ΓL(1,𝔽₂₅)	direct (N × H)
84. C₃	C₃×C₅×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₅×D₃	2
C₃²×C₅×C₈	2
C₃×C₅×C₃⋊C₈	2
C₃×C₅×M₁₆	3
C₅×ΓL(1,𝔽₂₅)	3
C₃×ΓL(1,𝔽₂₅)	5

Subgroups — C3×C5×ΓL(1,𝔽25) #10623851

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×ΓL(1,𝔽₂₅) #10623851