Subgroups — C₃²×ΓL(1,𝔽₂₅) #88818

Order 432.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×D₃	C₂	1	1	yes	maximal, characteristic
2	C₃³×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×M₁₆	—	3	1	no	maximal
5	C₃×ΓL(1,𝔽₂₅)	C₃	1	4	yes	maximal, direct product factor
6	C₂×C₃²×D₃	C₄	1	1	yes	characteristic
7	C₃³×C₄	C₂²	1	1	yes	characteristic
8	C₃²×Dic₃	C₄	1	1	yes	characteristic
9	C₂×C₃²×C₄	—	3	1	no	—
10	C₃×C₄×D₃	C₂×C₃	1	4	yes	—
11	C₃²×C₈	—	2	4	no	—
12	C₃²×C₈	—	3	1	no	—
13	C₃²×C₈	D₃	1	1	yes	characteristic, complemented
14	C₃²×C₈	C₂×C₃	1	4	yes	complemented
15	C₃×C₃⋊C₈	C₂×C₃	1	4	yes	complemented
16	C₂×C₃³	C₂×C₄	1	1	yes	characteristic
17	C₃²×D₃	—	2	1	no	—
18	C₃×M₁₆	—	3	4	no	—
19	ΓL(1,𝔽₂₅)	C₃²	1	1	yes	characteristic, direct product factor
20	C₂²×C₃²	—	3	1	no	—
21	C₂×C₃×D₃	C₃×C₄	1	4	yes	—
22	C₃²×C₄	—	2	4	no	—
23	C₃²×C₄	—	3	1	no	—
24	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
25	C₃²×C₄	C₂²×C₃	1	4	yes	—
26	C₃×Dic₃	C₃×C₄	1	4	yes	—
27	C₃³	M₁₆	1	1	yes	characteristic, complemented, Sylow
28	C₂×C₃×C₄	—	3	4	no	—
29	C₄×D₃	C₂×C₃²	1	1	yes	characteristic
30	C₃×C₈	—	2	4	no	—
31	C₃×C₈	—	3	4	no	—
32	C₃×C₈	C₃×D₃	1	4	yes	complemented
33	C₃×C₈	C₂×C₃²	1	1	yes	characteristic, complemented
34	C₃⋊C₈	C₂×C₃²	1	1	yes	characteristic, complemented
35	C₂×C₃²	—	2	4	no	—
36	C₂×C₃²	—	6	1	no	—
37	C₂×C₃²	C₄×D₃	1	1	yes	characteristic
38	C₂×C₃²	C₂×C₃×C₄	1	4	yes	—
39	C₃×D₃	—	2	4	no	—
40	M₁₆	—	3	1	no	Sylow
41	C₂²×C₃	—	3	4	no	—
42	C₂×D₃	C₃²×C₄	1	1	yes	characteristic
43	C₃×C₄	—	2	4	no	—
44	C₃×C₄	—	3	4	no	—
45	C₃×C₄	C₂×C₃×D₃	1	4	yes	—
46	C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
47	Dic₃	C₃²×C₄	1	1	yes	characteristic
48	C₃²	—	2	4	no	—
49	C₃²	ΓL(1,𝔽₂₅)	1	1	yes	characteristic, direct product factor
50	C₃²	C₃×M₁₆	1	4	yes	complemented
51	C₂×C₄	—	3	1	no	—
52	C₈	—	3	1	no	—
53	C₈	C₃²×D₃	1	1	yes	characteristic, complemented
54	C₂×C₃	—	2	4	no	—
55	C₂×C₃	—	6	4	no	—
56	C₂×C₃	C₃×C₄×D₃	1	4	yes	—
57	C₂×C₃	C₂×C₃²×C₄	1	1	yes	characteristic
58	D₃	—	2	1	no	—
59	C₂²	—	3	1	no	—
60	C₄	—	3	1	no	—
61	C₄	C₂×C₃²×D₃	1	1	yes	characteristic
62	C₃	—	2	4	no	—
63	C₃	C₃×ΓL(1,𝔽₂₅)	1	4	yes	direct product factor
64	C₃	C₃²×M₁₆	1	1	yes	characteristic, complemented
65	C₂	—	6	1	no	—
66	C₂	C₃²×C₄×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃²×C₄	C₂×D₃
commutator	C₂×C₃	C₂×C₃²×C₄
Frattini	C₄	C₂×C₃²×D₃
Fitting	C₃³×C₈	C₂
socle	C₂×C₃³	C₂×C₄
prime radical	C₂×C₃²×D₃	C₄
cosocle	C₃×C₄	C₂²×C₃²
hypercenter	C₃²×C₈	D₃
hypocenter	C₃	C₃²×M₁₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃²×C₄ ◁ C₃²×C₈

Complements

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

N	H ≅ G/N	type
2. C₃³×C₈	C₂	semidirect
3. C₃²×C₃⋊C₈	C₂	semidirect
5. C₃×ΓL(1,𝔽₂₅)	C₃	direct (N × H)
13. C₃²×C₈	D₃	semidirect
14. C₃²×C₈	C₂×C₃	semidirect
15. C₃×C₃⋊C₈	C₂×C₃	semidirect
19. ΓL(1,𝔽₂₅)	C₃²	direct (N × H)
27. C₃³	M₁₆	semidirect
32. C₃×C₈	C₃×D₃	semidirect
33. C₃×C₈	C₂×C₃²	semidirect
34. C₃⋊C₈	C₂×C₃²	semidirect
49. C₃²	ΓL(1,𝔽₂₅)	direct (N × H)
50. C₃²	C₃×M₁₆	semidirect
53. C₈	C₃²×D₃	semidirect
63. C₃	C₃×ΓL(1,𝔽₂₅)	direct (N × H)
64. C₃	C₃²×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄×D₃	2
C₃³×C₈	2
C₃²×C₃⋊C₈	2
C₃²×M₁₆	3
C₃×ΓL(1,𝔽₂₅)	3

Subgroups — C32×ΓL(1,𝔽25) #88818

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×ΓL(1,𝔽₂₅) #88818