Subgroups — C₃²×#4428 #13133661

Order 1701.

576 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×ΓL(1,𝔽₈)	C₃	1	1	yes	maximal, characteristic
2	C₃²×C₇×C₉	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃²×C₇⋊C₉	C₃	1	2	yes	maximal, complemented
4	C₃×#4428	C₃	1	36	yes	maximal, direct product factor
5	C₃²×3₋¹⁺²	—	7	1	no	maximal, Sylow
6	C₃³×C₇	C₃²	1	1	yes	characteristic
7	C₃²×ΓL(1,𝔽₈)	—	3	9	no	—
8	C₃²×ΓL(1,𝔽₈)	C₃²	1	12	yes	—
9	C₃×C₇×C₉	C₃²	1	12	yes	complemented
10	C₃×C₇⋊C₉	C₃²	1	24	yes	complemented
11	#4428	C₃²	1	81	yes	direct product factor
12	C₃⁴	—	7	1	no	—
13	C₃×3₋¹⁺²	—	7	36	no	—
14	C₃²×C₉	—	7	2	no	—
15	C₃²×C₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
16	C₃²×C₇	3₋¹⁺²	1	9	yes	complemented
17	C₃²×C₇	C₃³	1	4	yes	—
18	C₃×ΓL(1,𝔽₈)	—	3	36	no	—
19	C₃×ΓL(1,𝔽₈)	C₃³	1	9	yes	—
20	C₇×C₉	C₃³	1	9	yes	complemented
21	C₇⋊C₉	C₃³	1	18	yes	complemented
22	C₃³	—	7	12	no	—
23	C₃³	—	21	9	no	—
24	C₃³	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
25	3₋¹⁺²	—	7	81	no	—
26	C₃×C₉	—	7	24	no	—
27	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	12	yes	complemented
28	C₃×C₇	C₃×3₋¹⁺²	1	12	yes	complemented
29	C₃×C₇	C₃⁴	1	1	yes	characteristic
30	ΓL(1,𝔽₈)	—	3	9	no	—
31	C₃²	—	7	9	no	—
32	C₃²	—	21	36	no	—
33	C₃²	#4428	1	9	yes	direct product factor
34	C₃²	C₃²×ΓL(1,𝔽₈)	1	4	yes	—
35	C₉	—	7	18	no	—
36	C₉	C₃²×ΓL(1,𝔽₈)	1	9	yes	complemented
37	C₇	C₃²×3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
38	C₃	—	21	9	no	—
39	C₃	C₃×#4428	1	12	yes	direct product factor
40	C₃	C₃³×ΓL(1,𝔽₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₃³	C₃×ΓL(1,𝔽₈)
commutator, cosocle	C₃×C₇	C₃⁴
Frattini	C₃	C₃³×ΓL(1,𝔽₈)
Fitting	C₃²×C₇×C₉	C₃
socle	C₃³×C₇	C₃²
prime radical	C₃³×ΓL(1,𝔽₈)	C₃
hypercenter	C₃²×C₉	ΓL(1,𝔽₈)
hypocenter	C₇	C₃²×3₋¹⁺²
LCS	G ▷ C₃×C₇ ▷ C₇
UCS	C₁ ◁ C₃³ ◁ C₃²×C₉

Complements

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

N	H ≅ G/N	type
2. C₃²×C₇×C₉	C₃	semidirect
3. C₃²×C₇⋊C₉	C₃	semidirect
4. C₃×#4428	C₃	direct (N × H)
9. C₃×C₇×C₉	C₃²	semidirect
10. C₃×C₇⋊C₉	C₃²	semidirect
11. #4428	C₃²	direct (N × H)
15. C₃²×C₉	ΓL(1,𝔽₈)	semidirect
16. C₃²×C₇	3₋¹⁺²	semidirect
20. C₇×C₉	C₃³	semidirect
21. C₇⋊C₉	C₃³	semidirect
27. C₃×C₉	C₃×ΓL(1,𝔽₈)	semidirect
28. C₃×C₇	C₃×3₋¹⁺²	semidirect
33. C₃²	#4428	direct (N × H)
36. C₉	C₃²×ΓL(1,𝔽₈)	semidirect
37. C₇	C₃²×3₋¹⁺²	semidirect
39. C₃	C₃×#4428	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×ΓL(1,𝔽₈)	3
C₃²×C₇×C₉	3
C₃²×C₇⋊C₉	3
C₃×#4428	3
C₃²×3₋¹⁺²	7

Subgroups — C32×#4428 #13133661

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4428 #13133661