Subgroups — C₉×ΓL(1,𝔽₂₅) #88463

Order 432.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₃×C₉	C₂	1	1	yes	maximal, characteristic
2	C₃×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×ΓL(1,𝔽₂₅)	C₃	1	1	yes	maximal, characteristic
5	C₉×M₁₆	—	3	1	no	maximal
6	C₂×D₃×C₉	C₄	1	1	yes	characteristic
7	C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₉×Dic₃	C₄	1	1	yes	characteristic
9	C₃×C₄×D₃	C₂×C₃	1	1	yes	characteristic
10	C₃²×C₈	C₂×C₃	1	1	yes	characteristic
11	C₃×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic
12	C₂×C₄×C₉	—	3	1	no	—
13	C₈×C₉	—	2	1	no	—
14	C₈×C₉	—	3	1	no	—
15	C₈×C₉	D₃	1	1	yes	characteristic, complemented
16	C₂×C₃×C₉	C₂×C₄	1	1	yes	characteristic
17	D₃×C₉	—	2	1	no	—
18	C₃×M₁₆	—	3	1	no	—
19	ΓL(1,𝔽₂₅)	C₉	1	1	yes	characteristic, direct product factor
20	C₂×C₃×D₃	C₃×C₄	1	1	yes	characteristic
21	C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
22	C₃×Dic₃	C₃×C₄	1	1	yes	characteristic
23	C₂²×C₉	—	3	1	no	—
24	C₄×C₉	—	2	1	no	—
25	C₄×C₉	—	3	1	no	—
26	C₄×C₉	C₂×D₃	1	1	yes	characteristic
27	C₃×C₉	M₁₆	1	1	yes	characteristic, complemented, Sylow
28	C₂×C₃×C₄	—	3	1	no	—
29	C₄×D₃	C₂×C₉	1	1	yes	characteristic
30	C₃×C₈	—	2	1	no	—
31	C₃×C₈	—	3	1	no	—
32	C₃×C₈	C₂×C₉	1	1	yes	characteristic, complemented
33	C₃×C₈	C₃×D₃	1	1	yes	characteristic
34	C₃⋊C₈	C₂×C₉	1	1	yes	characteristic, complemented
35	C₂×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
36	C₃×D₃	—	2	1	no	—
37	C₂×C₉	—	2	1	no	—
38	C₂×C₉	—	6	1	no	—
39	C₂×C₉	C₄×D₃	1	1	yes	characteristic
40	M₁₆	—	3	1	no	Sylow
41	C₂²×C₃	—	3	1	no	—
42	C₂×D₃	C₄×C₉	1	1	yes	characteristic
43	C₃×C₄	—	2	1	no	—
44	C₃×C₄	—	3	1	no	—
45	C₃×C₄	C₂²×C₉	1	1	yes	characteristic
46	C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
47	Dic₃	C₄×C₉	1	1	yes	characteristic
48	C₃²	C₃×M₁₆	1	1	yes	characteristic
49	C₉	—	2	1	no	—
50	C₉	ΓL(1,𝔽₂₅)	1	1	yes	characteristic, direct product factor
51	C₂×C₄	—	3	1	no	—
52	C₈	—	3	1	no	—
53	C₈	D₃×C₉	1	1	yes	characteristic, complemented
54	C₂×C₃	—	2	1	no	—
55	C₂×C₃	—	6	1	no	—
56	C₂×C₃	C₂×C₄×C₉	1	1	yes	characteristic
57	C₂×C₃	C₃×C₄×D₃	1	1	yes	characteristic
58	D₃	—	2	1	no	—
59	C₂²	—	3	1	no	—
60	C₄	—	3	1	no	—
61	C₄	C₂×D₃×C₉	1	1	yes	characteristic
62	C₃	—	2	1	no	—
63	C₃	C₉×M₁₆	1	1	yes	characteristic, complemented
64	C₃	C₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
65	C₂	—	6	1	no	—
66	C₂	C₄×D₃×C₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₉	C₂×D₃
commutator	C₂×C₃	C₂×C₄×C₉
Frattini	C₃×C₄	C₂×C₃×D₃
Fitting	C₃×C₈×C₉	C₂
socle	C₂×C₃²	C₂×C₃×C₄
prime radical	C₂×C₃×D₃	C₃×C₄
cosocle	C₃²×C₄	C₂²×C₃
hypercenter	C₈×C₉	D₃
hypocenter	C₃	C₉×M₁₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₉ ◁ C₈×C₉

Complements

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

N	H ≅ G/N	type
2. C₃×C₈×C₉	C₂	semidirect
3. C₉×C₃⋊C₈	C₂	semidirect
15. C₈×C₉	D₃	semidirect
19. ΓL(1,𝔽₂₅)	C₉	direct (N × H)
27. C₃×C₉	M₁₆	semidirect
32. C₃×C₈	C₂×C₉	semidirect
34. C₃⋊C₈	C₂×C₉	semidirect
50. C₉	ΓL(1,𝔽₂₅)	direct (N × H)
53. C₈	D₃×C₉	semidirect
63. C₃	C₉×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₃×C₉	2
C₃×C₈×C₉	2
C₉×C₃⋊C₈	2
C₃×ΓL(1,𝔽₂₅)	3
C₉×M₁₆	3

Subgroups — C9×ΓL(1,𝔽25) #88463

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×ΓL(1,𝔽₂₅) #88463