Subgroups — C₃²×#16811780 #44501786

Order 30690.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#13133724	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×#16811780	C₃	1	4	yes	maximal, direct product factor
3	C₃²×D₁₁×C₃₁	C₅	1	1	yes	maximal, characteristic, complemented, Hall
4	C₃×#13133724	C₂×C₃	1	4	yes	complemented
5	#16811780	C₃²	1	1	yes	characteristic, direct product factor, Hall
6	C₃²×C₁₁×C₃₁	C₂×C₅	1	1	yes	characteristic, complemented, Hall
7	C₂×C₃²×ΓL(1,𝔽₃₂)	—	11	1	no	maximal, Hall
8	C₃×D₁₁×C₃₁	C₃×C₅	1	4	yes	complemented
9	#13133724	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
10	C₃²×ΓL(1,𝔽₃₂)	—	11	1	no	Hall
11	C₃×C₁₁×C₃₁	C₂×C₃×C₅	1	4	yes	complemented
12	C₃²×AGL(1,𝔽₁₁)	—	31	1	no	maximal, Hall
13	C₂×C₃×ΓL(1,𝔽₃₂)	—	11	4	no	—
14	D₁₁×C₃₁	C₃²×C₅	1	1	yes	characteristic, complemented, Hall
15	C₂×C₃²×C₃₁	—	11	1	no	Hall
16	C₃²×C₁₁⋊C₅	—	31	1	no	Hall
17	C₃×ΓL(1,𝔽₃₂)	—	11	4	no	—
18	C₁₁×C₃₁	C₂×C₃²×C₅	1	1	yes	characteristic, complemented, Hall
19	C₃×AGL(1,𝔽₁₁)	—	31	4	no	—
20	C₂×ΓL(1,𝔽₃₂)	—	11	1	no	Hall
21	C₃²×C₃₁	AGL(1,𝔽₁₁)	1	1	yes	characteristic, complemented, Hall
22	C₃²×D₁₁	ΓL(1,𝔽₃₂)	1	1	yes	characteristic, complemented, Hall
23	C₂×C₃×C₃₁	—	11	4	no	—
24	C₃×C₁₁⋊C₅	—	31	4	no	—
25	ΓL(1,𝔽₃₂)	—	11	1	no	Hall
26	AGL(1,𝔽₁₁)	—	31	1	no	Hall
27	C₃²×C₁₁	C₂×ΓL(1,𝔽₃₂)	1	1	yes	characteristic, complemented, Hall
28	C₃×C₃₁	C₃×AGL(1,𝔽₁₁)	1	4	yes	complemented
29	C₂×C₃²×C₅	—	341	1	no	Hall
30	C₃×D₁₁	C₃×ΓL(1,𝔽₃₂)	1	4	yes	complemented
31	C₂×C₃₁	—	11	1	no	Hall
32	C₁₁⋊C₅	—	31	1	no	Hall
33	C₃²×C₅	—	341	1	no	Hall
34	C₃×C₁₁	C₂×C₃×ΓL(1,𝔽₃₂)	1	4	yes	complemented
35	C₃₁	C₃²×AGL(1,𝔽₁₁)	1	1	yes	characteristic, complemented, Sylow
36	C₂×C₃×C₅	—	341	4	no	—
37	D₁₁	C₃²×ΓL(1,𝔽₃₂)	1	1	yes	characteristic, complemented, Hall
38	C₂×C₃²	—	11	1	no	Hall
39	C₃×C₅	—	341	4	no	—
40	C₁₁	C₂×C₃²×ΓL(1,𝔽₃₂)	1	1	yes	characteristic, complemented, Sylow
41	C₂×C₅	—	341	1	no	Hall
42	C₃²	#16811780	1	1	yes	characteristic, direct product factor, Sylow
43	C₂×C₃	—	11	4	no	—
44	C₅	—	341	1	no	Sylow
45	C₃	C₃×#16811780	1	4	yes	direct product factor
46	C₂	—	11	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃²	#16811780
commutator, cosocle, hypocenter	C₁₁×C₃₁	C₂×C₃²×C₅
Frattini	C₁	G
Fitting, socle	C₃²×C₁₁×C₃₁	C₂×C₅
LCS	G ▷ C₁₁×C₃₁
UCS	C₁ ◁ C₃²

Complements

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

N	H ≅ G/N	type
1. C₃²×#13133724	C₂	semidirect
2. C₃×#16811780	C₃	direct (N × H)
3. C₃²×D₁₁×C₃₁	C₅	semidirect
4. C₃×#13133724	C₂×C₃	semidirect
5. #16811780	C₃²	direct (N × H)
6. C₃²×C₁₁×C₃₁	C₂×C₅	semidirect
8. C₃×D₁₁×C₃₁	C₃×C₅	semidirect
9. #13133724	C₂×C₃²	semidirect
11. C₃×C₁₁×C₃₁	C₂×C₃×C₅	semidirect
14. D₁₁×C₃₁	C₃²×C₅	semidirect
18. C₁₁×C₃₁	C₂×C₃²×C₅	semidirect
21. C₃²×C₃₁	AGL(1,𝔽₁₁)	semidirect
22. C₃²×D₁₁	ΓL(1,𝔽₃₂)	semidirect
27. C₃²×C₁₁	C₂×ΓL(1,𝔽₃₂)	semidirect
28. C₃×C₃₁	C₃×AGL(1,𝔽₁₁)	semidirect
30. C₃×D₁₁	C₃×ΓL(1,𝔽₃₂)	semidirect
34. C₃×C₁₁	C₂×C₃×ΓL(1,𝔽₃₂)	semidirect
35. C₃₁	C₃²×AGL(1,𝔽₁₁)	semidirect
37. D₁₁	C₃²×ΓL(1,𝔽₃₂)	semidirect
40. C₁₁	C₂×C₃²×ΓL(1,𝔽₃₂)	semidirect
42. C₃²	#16811780	direct (N × H)
45. C₃	C₃×#16811780	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×#13133724	2
C₃×#16811780	3
C₃²×D₁₁×C₃₁	5
C₂×C₃²×ΓL(1,𝔽₃₂)	11
C₃²×AGL(1,𝔽₁₁)	31

Subgroups — C32×#16811780 #44501786

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#16811780 #44501786