Subgroups — C₃×#17987417 #28355939

Order 13851.

3824 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₃²⋊(C₁₉⋊C₃)	C₃	1	12	yes	maximal, complemented
2	#17987417	C₃	1	27	yes	maximal, direct product factor
3	C₃⁵×C₁₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃³×C₁₉⋊C₃	C₃²	1	9	yes	complemented
5	C₃×C₃²⋊(C₁₉⋊C₃)	—	3	108	no	—
6	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	108	yes	complemented
7	C₃⁴×C₁₉	—	3	36	no	—
8	C₃⁴×C₁₉	C₃²	1	4	yes	—
9	C₃⁴×C₁₉	C₃²	1	9	yes	complemented
10	C₃×C₃²⋊H(3,𝔽₃)	—	19	1	no	maximal, Sylow
11	C₃³×C₁₉	—	3	243	no	—
12	C₃³×C₁₉	—	3	36	no	—
13	C₃³×C₁₉	—	3	108	no	—
14	C₃³×C₁₉	H(3,𝔽₃)	1	36	yes	complemented
15	C₃³×C₁₉	C₃³	1	12	yes	—
16	C₃³×C₁₉	C₃³	1	1	yes	characteristic
17	C₃²×C₁₉⋊C₃	—	3	108	no	—
18	C₃²×C₁₉⋊C₃	C₃³	1	27	yes	complemented
19	C₃²⋊(C₁₉⋊C₃)	—	3	324	no	—
20	C₃⁵	C₁₉⋊C₃	1	1	yes	characteristic, complemented
21	C₃²×H(3,𝔽₃)	—	19	12	no	—
22	C₃²⋊H(3,𝔽₃)	—	19	27	no	—
23	C₃²×C₁₉	—	3	243	no	—
24	C₃²×C₁₉	—	3	36	no	—
25	C₃²×C₁₉	—	3	108	no	—
26	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	12	yes	—
27	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	36	yes	complemented
28	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
29	C₃×C₁₉⋊C₃	—	3	108	no	—
30	C₃×C₁₉⋊C₃	—	9	81	no	—
31	C₃⁴	—	3	36	no	—
32	C₃⁴	—	19	9	no	—
33	C₃⁴	C₃×C₁₉⋊C₃	1	4	yes	—
34	C₃⁴	C₃×C₁₉⋊C₃	1	9	yes	complemented
35	C₃×H(3,𝔽₃)	—	19	108	no	—
36	C₃×H(3,𝔽₃)	—	57	108	no	—
37	C₃×C₁₉	—	3	36	no	—
38	C₃×C₁₉	C₃²⋊H(3,𝔽₃)	1	9	yes	complemented
39	C₃×C₁₉	C₃²×H(3,𝔽₃)	1	4	yes	—
40	C₁₉⋊C₃	—	9	27	no	—
41	C₃³	—	3	243	no	—
42	C₃³	—	3	36	no	—
43	C₃³	—	3	108	no	—
44	C₃³	—	19	27	no	—
45	C₃³	—	57	108	no	—
46	C₃³	C₃²⋊(C₁₉⋊C₃)	1	36	yes	complemented
47	C₃³	C₃²×C₁₉⋊C₃	1	12	yes	—
48	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
49	H(3,𝔽₃)	—	57	324	no	—
50	C₁₉	C₃×C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
51	C₃²	—	3	243	no	—
52	C₃²	—	3	108	no	—
53	C₃²	—	3	36	no	—
54	C₃²	—	57	108	no	—
55	C₃²	—	171	81	no	—
56	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	12	yes	—
57	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	36	yes	complemented
58	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
59	C₃	—	3	36	no	—
60	C₃	—	171	27	no	—
61	C₃	#17987417	1	9	yes	direct product factor
62	C₃	C₃²×C₃²⋊(C₁₉⋊C₃)	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₃²×C₃²⋊(C₁₉⋊C₃)	C₃	semidirect
2. #17987417	C₃	direct (N × H)
3. C₃⁵×C₁₉	C₃	semidirect
4. C₃³×C₁₉⋊C₃	C₃²	semidirect
6. C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	semidirect
9. C₃⁴×C₁₉	C₃²	semidirect
14. C₃³×C₁₉	H(3,𝔽₃)	semidirect
18. C₃²×C₁₉⋊C₃	C₃³	semidirect
20. C₃⁵	C₁₉⋊C₃	semidirect
27. C₃²×C₁₉	C₃×H(3,𝔽₃)	semidirect
34. C₃⁴	C₃×C₁₉⋊C₃	semidirect
38. C₃×C₁₉	C₃²⋊H(3,𝔽₃)	semidirect
46. C₃³	C₃²⋊(C₁₉⋊C₃)	semidirect
50. C₁₉	C₃×C₃²⋊H(3,𝔽₃)	semidirect
57. C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	semidirect
61. C₃	#17987417	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×C₃²⋊(C₁₉⋊C₃)	3
#17987417	3
C₃⁵×C₁₉	3
C₃×C₃²⋊H(3,𝔽₃)	19

Subgroups — C3×#17987417 #28355939

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#17987417 #28355939