Subgroups — C₃²×C₉×#4296 #28357039

Order 13851.

2132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×#4296	C₃	1	3	yes	maximal
2	C₃×C₉×#4296	C₃	1	36	yes	maximal, direct product factor
3	C₃²×C₉×C₁₉⋊C₃	C₃	1	1	yes	maximal, characteristic
4	C₃²×#4296	C₉	1	81	yes	direct product factor
5	C₃²×#4296	C₃²	1	36	yes	—
6	C₉×#4296	C₃²	1	81	yes	direct product factor
7	C₃³×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
8	C₃²×#4294	C₉	1	2	yes	complemented
9	C₃×C₉×C₁₉⋊C₃	C₉	1	24	yes	complemented
10	C₃×C₉×C₁₉⋊C₃	C₃²	1	12	yes	—
11	C₃²×C₉×C₁₉	C₉	1	1	yes	characteristic, complemented
12	C₃²×C₉²	—	19	1	no	maximal, Sylow
13	C₃³×C₁₉	C₃×C₉	1	1	yes	characteristic
14	C₃²×C₁₉⋊C₃	C₃×C₉	1	8	yes	—
15	C₃²×C₁₉⋊C₃	C₃×C₉	1	27	yes	—
16	C₃²×C₁₉⋊C₃	C₃³	1	4	yes	—
17	C₃×#4296	C₃×C₉	1	324	yes	direct product factor
18	C₃×#4296	C₃³	1	27	yes	—
19	C₃×C₉×C₁₉	C₃×C₉	1	12	yes	complemented
20	C₃×#4294	C₃×C₉	1	24	yes	complemented
21	C₉×C₁₉⋊C₃	C₃×C₉	1	72	yes	complemented
22	C₉×C₁₉⋊C₃	C₃³	1	9	yes	—
23	C₃³×C₉	—	19	3	no	—
24	C₃³×C₉	—	19	1	no	—
25	C₃×C₉²	—	19	36	no	—
26	C₃²×C₁₉	C₉²	1	9	yes	complemented
27	C₃²×C₁₉	C₃²×C₉	1	4	yes	—
28	C₃×C₁₉⋊C₃	C₉²	1	72	yes	complemented
29	C₃×C₁₉⋊C₃	C₃²×C₉	1	36	yes	—
30	C₃×C₁₉⋊C₃	C₃²×C₉	1	8	yes	—
31	C₃×C₁₉⋊C₃	C₃⁴	1	1	yes	characteristic
32	#4296	C₃²×C₉	1	81	yes	direct product factor
33	C₉×C₁₉	C₃²×C₉	1	9	yes	complemented
34	#4294	C₃²×C₉	1	18	yes	complemented
35	C₃⁴	—	19	1	no	—
36	C₃²×C₉	—	19	81	no	—
37	C₃²×C₉	—	19	36	no	—
38	C₃²×C₉	—	19	2	no	—
39	C₃²×C₉	—	19	24	no	—
40	C₃²×C₉	—	19	12	no	—
41	C₃²×C₉	#4296	1	1	yes	characteristic, direct product factor
42	C₉²	—	19	81	no	—
43	C₃×C₁₉	C₃×C₉²	1	12	yes	complemented
44	C₃×C₁₉	C₃³×C₉	1	1	yes	characteristic
45	C₁₉⋊C₃	C₃×C₉²	1	24	yes	complemented
46	C₁₉⋊C₃	C₃³×C₉	1	3	yes	—
47	C₃³	—	19	27	no	—
48	C₃³	—	19	8	no	—
49	C₃³	—	19	4	no	—
50	C₃³	C₃×#4296	1	1	yes	characteristic
51	C₃×C₉	—	19	324	no	—
52	C₃×C₉	—	19	27	no	—
53	C₃×C₉	—	19	24	no	—
54	C₃×C₉	—	19	72	no	—
55	C₃×C₉	—	19	9	no	—
56	C₃×C₉	C₃×#4296	1	12	yes	direct product factor
57	C₁₉	C₃²×C₉²	1	1	yes	characteristic, complemented, Sylow
58	C₃²	—	19	36	no	—
59	C₃²	—	19	72	no	—
60	C₃²	—	19	1	no	—
61	C₃²	—	19	8	no	—
62	C₃²	C₉×#4296	1	9	yes	direct product factor
63	C₃²	C₃²×#4296	1	4	yes	—
64	C₉	—	19	81	no	—
65	C₉	—	19	18	no	—
66	C₉	C₃²×#4296	1	9	yes	direct product factor
67	C₃	—	19	3	no	—
68	C₃	—	19	24	no	—
69	C₃	C₃×C₉×#4296	1	12	yes	direct product factor
70	C₃	C₃³×#4296	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₃²×C₉	#4296
commutator, hypocenter	C₁₉	C₃²×C₉²
Frattini	C₃	C₃³×#4296
Fitting	C₃²×C₉×C₁₉	C₉
socle	C₃³×C₁₉	C₃×C₉
prime radical	C₃³×C₁₉⋊C₃	C₃²
cosocle	C₃×C₁₉⋊C₃	C₃⁴
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₃²×C₉

Complements

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

N	H ≅ G/N	type
2. C₃×C₉×#4296	C₃	direct (N × H)
4. C₃²×#4296	C₉	direct (N × H)
6. C₉×#4296	C₃²	direct (N × H)
8. C₃²×#4294	C₉	semidirect
9. C₃×C₉×C₁₉⋊C₃	C₉	semidirect
11. C₃²×C₉×C₁₉	C₉	semidirect
17. C₃×#4296	C₃×C₉	direct (N × H)
19. C₃×C₉×C₁₉	C₃×C₉	semidirect
20. C₃×#4294	C₃×C₉	semidirect
21. C₉×C₁₉⋊C₃	C₃×C₉	semidirect
26. C₃²×C₁₉	C₉²	semidirect
28. C₃×C₁₉⋊C₃	C₉²	semidirect
32. #4296	C₃²×C₉	direct (N × H)
33. C₉×C₁₉	C₃²×C₉	semidirect
34. #4294	C₃²×C₉	semidirect
41. C₃²×C₉	#4296	direct (N × H)
43. C₃×C₁₉	C₃×C₉²	semidirect
45. C₁₉⋊C₃	C₃×C₉²	semidirect
56. C₃×C₉	C₃×#4296	direct (N × H)
57. C₁₉	C₃²×C₉²	semidirect
62. C₃²	C₉×#4296	direct (N × H)
66. C₉	C₃²×#4296	direct (N × H)
69. C₃	C₃×C₉×#4296	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×#4296	3
C₃×C₉×#4296	3
C₃²×C₉×C₁₉⋊C₃	3
C₃²×C₉²	19

Subgroups — C32×C9×#4296 #28357039

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₉×#4296 #28357039