Subgroups — C₃²×#13095331 #28356656

Order 13851.

5804 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#13095331	C₃	1	108	yes	maximal, direct product factor
2	C₃²×#10587025	C₃	1	8	yes	maximal, complemented
3	C₃³×#4294	C₃	1	4	yes	maximal, complemented
4	C₃²×C₁₉×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
5	#13095331	C₃²	1	729	yes	direct product factor
6	C₃×#10587025	C₃²	1	288	yes	complemented
7	C₃²×#4294	C₃²	1	1	yes	characteristic
8	C₃²×#4294	C₃²	1	8	yes	—
9	C₃²×#4294	C₃²	1	144	yes	complemented
10	C₃⁴×C₁₉	C₃²	1	4	yes	—
11	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	36	yes	—
12	C₃²×#737	—	19	1	no	maximal, Sylow
13	C₃³×C₁₉	—	3	36	no	—
14	C₃³×C₁₉	C₃³	1	48	yes	—
15	C₃³×C₁₉	C₃³	1	1	yes	characteristic
16	C₁₉×H(3,𝔽₃)	C₃³	1	81	yes	—
17	#10587025	C₃³	1	648	yes	complemented
18	C₃×#4294	C₃³	1	12	yes	—
19	C₃×#4294	C₃³	1	96	yes	—
20	C₃×#4294	C₃³	1	324	yes	complemented
21	C₃×#737	—	19	108	no	—
22	C₃²×3₋¹⁺²	—	19	8	no	—
23	C₃²×H(3,𝔽₃)	C₁₉⋊C₃	1	1	yes	characteristic
24	C₃³×C₉	—	19	4	no	—
25	C₃²×C₁₉	—	3	144	no	—
26	C₃²×C₁₉	#737	1	9	yes	complemented
27	C₃²×C₁₉	C₃⁴	1	36	yes	—
28	C₃²×C₁₉	C₃⁴	1	4	yes	—
29	#4294	C₃⁴	1	9	yes	—
30	#4294	C₃⁴	1	72	yes	—
31	C₃⁴	C₃×C₁₉⋊C₃	1	4	yes	—
32	#737	—	19	729	no	—
33	C₃×3₋¹⁺²	—	19	288	no	—
34	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	36	yes	—
35	C₃²×C₉	—	19	144	no	—
36	C₃²×C₉	—	19	1	no	—
37	C₃²×C₉	—	19	8	no	—
38	C₃×C₁₉	—	3	36	no	—
39	C₃×C₁₉	C₃×#737	1	12	yes	complemented
40	C₃×C₁₉	C₃⁵	1	1	yes	characteristic
41	C₃³	—	3	36	no	—
42	C₃³	C₃²×C₁₉⋊C₃	1	48	yes	—
43	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
44	3₋¹⁺²	—	19	648	no	—
45	H(3,𝔽₃)	C₃²×C₁₉⋊C₃	1	81	yes	—
46	C₃×C₉	—	19	12	no	—
47	C₃×C₉	—	19	96	no	—
48	C₃×C₉	—	19	324	no	—
49	C₁₉	C₃²×#737	1	1	yes	characteristic, complemented, Sylow
50	C₃²	—	3	144	no	—
51	C₃²	#13095331	1	9	yes	direct product factor
52	C₃²	C₃³×C₁₉⋊C₃	1	36	yes	—
53	C₃²	C₃³×C₁₉⋊C₃	1	4	yes	—
54	C₉	—	19	9	no	—
55	C₉	—	19	72	no	—
56	C₃	—	3	36	no	—
57	C₃	C₃×#13095331	1	12	yes	direct product factor
58	C₃	C₃⁴×C₁₉⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃³	C₃²×C₁₉⋊C₃
commutator, cosocle	C₃×C₁₉	C₃⁵
Frattini	C₃	C₃⁴×C₁₉⋊C₃
Fitting, prime radical	C₃²×C₁₉×H(3,𝔽₃)	C₃
socle	C₃³×C₁₉	C₃³
hypercenter	C₃²×H(3,𝔽₃)	C₁₉⋊C₃
hypocenter	C₁₉	C₃²×#737
LCS	G ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃³ ◁ C₃²×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃×#13095331	C₃	direct (N × H)
2. C₃²×#10587025	C₃	semidirect
3. C₃³×#4294	C₃	semidirect
5. #13095331	C₃²	direct (N × H)
6. C₃×#10587025	C₃²	semidirect
9. C₃²×#4294	C₃²	semidirect
17. #10587025	C₃³	semidirect
20. C₃×#4294	C₃³	semidirect
26. C₃²×C₁₉	#737	semidirect
39. C₃×C₁₉	C₃×#737	semidirect
49. C₁₉	C₃²×#737	semidirect
51. C₃²	#13095331	direct (N × H)
57. C₃	C₃×#13095331	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#13095331	3
C₃²×#10587025	3
C₃³×#4294	3
C₃²×C₁₉×H(3,𝔽₃)	3
C₃²×#737	19

Subgroups — C32×#13095331 #28356656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#13095331 #28356656