Subgroups — C₃³×C₉×C₃₇⋊C₃ #41008240

Order 26973.

10232 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁴×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
2	C₃³×#66524	C₃	1	2	yes	maximal, complemented
3	C₃²×C₉×C₃₇⋊C₃	C₃	1	117	yes	maximal, direct product factor
4	C₃³×C₉×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃³×C₃₇⋊C₃	C₉	1	81	yes	direct product factor
6	C₃³×C₃₇⋊C₃	C₃²	1	39	yes	—
7	C₃²×#66524	C₃²	1	78	yes	complemented
8	C₃×C₉×C₃₇⋊C₃	C₃²	1	1053	yes	direct product factor
9	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
10	C₃²×C₉×C₃₇	C₃²	1	39	yes	complemented
11	C₃³×C₃₇	C₃×C₉	1	27	yes	complemented
12	C₃³×C₃₇	C₃³	1	13	yes	—
13	C₃²×C₃₇⋊C₃	C₃×C₉	1	1053	yes	direct product factor
14	C₃²×C₃₇⋊C₃	C₃³	1	117	yes	—
15	C₃×C₉×C₃₇	C₃³	1	117	yes	complemented
16	C₃×#66524	C₃³	1	234	yes	complemented
17	C₉×C₃₇⋊C₃	C₃³	1	729	yes	direct product factor
18	C₃⁴×C₉	—	37	1	no	maximal, Sylow
19	C₃²×C₃₇	C₃²×C₉	1	117	yes	complemented
20	C₃²×C₃₇	C₃⁴	1	13	yes	—
21	C₃×C₃₇⋊C₃	C₃²×C₉	1	1053	yes	direct product factor
22	C₃×C₃₇⋊C₃	C₃⁴	1	27	yes	—
23	C₉×C₃₇	C₃⁴	1	27	yes	complemented
24	#66524	C₃⁴	1	54	yes	complemented
25	C₃⁵	—	37	1	no	—
26	C₃³×C₉	—	37	2	no	—
27	C₃³×C₉	—	37	117	no	—
28	C₃³×C₉	C₃₇⋊C₃	1	1	yes	characteristic, direct product factor
29	C₃×C₃₇	C₃³×C₉	1	39	yes	complemented
30	C₃×C₃₇	C₃⁵	1	1	yes	characteristic
31	C₃₇⋊C₃	C₃³×C₉	1	81	yes	direct product factor
32	C₃⁴	—	37	81	no	—
33	C₃⁴	—	37	39	no	—
34	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
35	C₃²×C₉	—	37	78	no	—
36	C₃²×C₉	—	37	1053	no	—
37	C₃²×C₉	C₃×C₃₇⋊C₃	1	39	yes	direct product factor
38	C₃₇	C₃⁴×C₉	1	1	yes	characteristic, complemented, Sylow
39	C₃³	—	37	1053	no	—
40	C₃³	—	37	117	no	—
41	C₃³	C₉×C₃₇⋊C₃	1	27	yes	direct product factor
42	C₃³	C₃²×C₃₇⋊C₃	1	13	yes	—
43	C₃×C₉	—	37	234	no	—
44	C₃×C₉	—	37	729	no	—
45	C₃×C₉	C₃²×C₃₇⋊C₃	1	117	yes	direct product factor
46	C₃²	—	37	1053	no	—
47	C₃²	—	37	27	no	—
48	C₃²	C₃×C₉×C₃₇⋊C₃	1	117	yes	direct product factor
49	C₃²	C₃³×C₃₇⋊C₃	1	13	yes	—
50	C₉	—	37	54	no	—
51	C₉	C₃³×C₃₇⋊C₃	1	27	yes	direct product factor
52	C₃	—	37	81	no	—
53	C₃	C₃²×C₉×C₃₇⋊C₃	1	39	yes	direct product factor
54	C₃	C₃⁴×C₃₇⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₃³×C₉	C₃₇⋊C₃
commutator, hypocenter	C₃₇	C₃⁴×C₉
Frattini	C₃	C₃⁴×C₃₇⋊C₃
Fitting	C₃³×C₉×C₃₇	C₃
socle	C₃⁴×C₃₇	C₃²
prime radical	C₃⁴×C₃₇⋊C₃	C₃
cosocle	C₃×C₃₇	C₃⁵
LCS	G ▷ C₃₇
UCS	C₁ ◁ C₃³×C₉

Complements

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

N	H ≅ G/N	type
2. C₃³×#66524	C₃	semidirect
3. C₃²×C₉×C₃₇⋊C₃	C₃	direct (N × H)
4. C₃³×C₉×C₃₇	C₃	semidirect
5. C₃³×C₃₇⋊C₃	C₉	direct (N × H)
7. C₃²×#66524	C₃²	semidirect
8. C₃×C₉×C₃₇⋊C₃	C₃²	direct (N × H)
10. C₃²×C₉×C₃₇	C₃²	semidirect
11. C₃³×C₃₇	C₃×C₉	semidirect
13. C₃²×C₃₇⋊C₃	C₃×C₉	direct (N × H)
15. C₃×C₉×C₃₇	C₃³	semidirect
16. C₃×#66524	C₃³	semidirect
17. C₉×C₃₇⋊C₃	C₃³	direct (N × H)
19. C₃²×C₃₇	C₃²×C₉	semidirect
21. C₃×C₃₇⋊C₃	C₃²×C₉	direct (N × H)
23. C₉×C₃₇	C₃⁴	semidirect
24. #66524	C₃⁴	semidirect
28. C₃³×C₉	C₃₇⋊C₃	direct (N × H)
29. C₃×C₃₇	C₃³×C₉	semidirect
31. C₃₇⋊C₃	C₃³×C₉	direct (N × H)
37. C₃²×C₉	C₃×C₃₇⋊C₃	direct (N × H)
38. C₃₇	C₃⁴×C₉	semidirect
41. C₃³	C₉×C₃₇⋊C₃	direct (N × H)
45. C₃×C₉	C₃²×C₃₇⋊C₃	direct (N × H)
48. C₃²	C₃×C₉×C₃₇⋊C₃	direct (N × H)
51. C₉	C₃³×C₃₇⋊C₃	direct (N × H)
53. C₃	C₃²×C₉×C₃₇⋊C₃	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×C₃₇⋊C₃	3
C₃³×#66524	3
C₃²×C₉×C₃₇⋊C₃	3
C₃³×C₉×C₃₇	3
C₃⁴×C₉	37

Subgroups — C33×C9×C37⋊C3 #41008240

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃³×C₉×C₃₇⋊C₃ #41008240