Subgroups — C₃³×#10587023 #28356652

Order 13851.

7208 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁴×C₁₉⋊C₃	C₃	1	1	yes	maximal, characteristic
2	C₃²×#10587023	C₃	1	117	yes	maximal, direct product factor
3	C₃³×#4294	C₃	1	2	yes	maximal, complemented
4	C₃³×C₉×C₁₉	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃³×C₁₉⋊C₃	—	3	27	no	—
6	C₃³×C₁₉⋊C₃	C₃²	1	39	yes	—
7	C₃×#10587023	C₃²	1	1053	yes	direct product factor
8	C₃²×#4294	C₃²	1	78	yes	complemented
9	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
10	C₃²×C₉×C₁₉	C₃²	1	39	yes	complemented
11	C₃³×3₋¹⁺²	—	19	1	no	maximal, Sylow
12	C₃³×C₁₉	3₋¹⁺²	1	27	yes	complemented
13	C₃³×C₁₉	C₃³	1	13	yes	—
14	C₃²×C₁₉⋊C₃	—	3	351	no	—
15	C₃²×C₁₉⋊C₃	C₃³	1	117	yes	—
16	C₃×C₉×C₁₉	C₃³	1	117	yes	complemented
17	C₃×#4294	C₃³	1	234	yes	complemented
18	#10587023	C₃³	1	729	yes	direct product factor
19	C₃⁵	—	19	1	no	—
20	C₃²×3₋¹⁺²	—	19	117	no	—
21	C₃³×C₉	—	19	2	no	—
22	C₃³×C₉	C₁₉⋊C₃	1	1	yes	characteristic, complemented
23	C₃²×C₁₉	C₃×3₋¹⁺²	1	117	yes	complemented
24	C₃²×C₁₉	C₃⁴	1	13	yes	—
25	C₃×C₁₉⋊C₃	—	3	351	no	—
26	C₃×C₁₉⋊C₃	C₃⁴	1	27	yes	—
27	C₉×C₁₉	C₃⁴	1	27	yes	complemented
28	#4294	C₃⁴	1	54	yes	complemented
29	C₃⁴	—	19	39	no	—
30	C₃⁴	—	57	27	no	—
31	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
32	C₃×3₋¹⁺²	—	19	1053	no	—
33	C₃²×C₉	—	19	78	no	—
34	C₃²×C₉	C₃×C₁₉⋊C₃	1	39	yes	complemented
35	C₃×C₁₉	C₃²×3₋¹⁺²	1	39	yes	complemented
36	C₃×C₁₉	C₃⁵	1	1	yes	characteristic
37	C₁₉⋊C₃	—	3	27	no	—
38	C₃³	—	19	117	no	—
39	C₃³	—	57	351	no	—
40	C₃³	#10587023	1	27	yes	direct product factor
41	C₃³	C₃²×C₁₉⋊C₃	1	13	yes	—
42	3₋¹⁺²	—	19	729	no	—
43	C₃×C₉	—	19	234	no	—
44	C₃×C₉	C₃²×C₁₉⋊C₃	1	117	yes	complemented
45	C₁₉	C₃³×3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
46	C₃²	—	19	27	no	—
47	C₃²	—	57	351	no	—
48	C₃²	C₃×#10587023	1	117	yes	direct product factor
49	C₃²	C₃³×C₁₉⋊C₃	1	13	yes	—
50	C₉	—	19	54	no	—
51	C₉	C₃³×C₁₉⋊C₃	1	27	yes	complemented
52	C₃	—	57	27	no	—
53	C₃	C₃²×#10587023	1	39	yes	direct product factor
54	C₃	C₃⁴×C₁₉⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃⁴	C₃×C₁₉⋊C₃
commutator, cosocle	C₃×C₁₉	C₃⁵
Frattini	C₃	C₃⁴×C₁₉⋊C₃
Fitting	C₃³×C₉×C₁₉	C₃
socle	C₃⁴×C₁₉	C₃²
prime radical	C₃⁴×C₁₉⋊C₃	C₃
hypercenter	C₃³×C₉	C₁₉⋊C₃
hypocenter	C₁₉	C₃³×3₋¹⁺²
LCS	G ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃⁴ ◁ C₃³×C₉

Complements

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

N	H ≅ G/N	type
2. C₃²×#10587023	C₃	direct (N × H)
3. C₃³×#4294	C₃	semidirect
4. C₃³×C₉×C₁₉	C₃	semidirect
7. C₃×#10587023	C₃²	direct (N × H)
8. C₃²×#4294	C₃²	semidirect
10. C₃²×C₉×C₁₉	C₃²	semidirect
12. C₃³×C₁₉	3₋¹⁺²	semidirect
16. C₃×C₉×C₁₉	C₃³	semidirect
17. C₃×#4294	C₃³	semidirect
18. #10587023	C₃³	direct (N × H)
22. C₃³×C₉	C₁₉⋊C₃	semidirect
23. C₃²×C₁₉	C₃×3₋¹⁺²	semidirect
27. C₉×C₁₉	C₃⁴	semidirect
28. #4294	C₃⁴	semidirect
34. C₃²×C₉	C₃×C₁₉⋊C₃	semidirect
35. C₃×C₁₉	C₃²×3₋¹⁺²	semidirect
40. C₃³	#10587023	direct (N × H)
44. C₃×C₉	C₃²×C₁₉⋊C₃	semidirect
45. C₁₉	C₃³×3₋¹⁺²	semidirect
48. C₃²	C₃×#10587023	direct (N × H)
51. C₉	C₃³×C₁₉⋊C₃	semidirect
53. C₃	C₃²×#10587023	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃⁴×C₁₉⋊C₃	3
C₃²×#10587023	3
C₃³×#4294	3
C₃³×C₉×C₁₉	3
C₃³×3₋¹⁺²	19

Subgroups — C33×#10587023 #28356652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃³×#10587023 #28356652