Subgroups — C₃³×#11758606 #41008245

Order 26973.

7208 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	C₃²×#11758606	C₃	1	117	yes	maximal, direct product factor
3	C₃³×#66524	C₃	1	2	yes	maximal, complemented
4	C₃³×C₉×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃³×C₃₇⋊C₃	—	3	27	no	—
6	C₃³×C₃₇⋊C₃	C₃²	1	39	yes	—
7	C₃×#11758606	C₃²	1	1053	yes	direct product factor
8	C₃²×#66524	C₃²	1	78	yes	complemented
9	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
10	C₃²×C₉×C₃₇	C₃²	1	39	yes	complemented
11	C₃³×C₃₇	3₋¹⁺²	1	27	yes	complemented
12	C₃³×C₃₇	C₃³	1	13	yes	—
13	C₃²×C₃₇⋊C₃	—	3	351	no	—
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	#11758606	C₃³	1	729	yes	direct product factor
18	C₃³×3₋¹⁺²	—	37	1	no	maximal, Sylow
19	C₃²×C₃₇	C₃×3₋¹⁺²	1	117	yes	complemented
20	C₃²×C₃₇	C₃⁴	1	13	yes	—
21	C₃×C₃₇⋊C₃	—	3	351	no	—
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	C₃²×3₋¹⁺²	—	37	117	no	—
27	C₃³×C₉	—	37	2	no	—
28	C₃³×C₉	C₃₇⋊C₃	1	1	yes	characteristic, complemented
29	C₃×C₃₇	C₃²×3₋¹⁺²	1	39	yes	complemented
30	C₃×C₃₇	C₃⁵	1	1	yes	characteristic
31	C₃₇⋊C₃	—	3	27	no	—
32	C₃⁴	—	37	39	no	—
33	C₃⁴	—	111	27	no	—
34	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
35	C₃×3₋¹⁺²	—	37	1053	no	—
36	C₃²×C₉	—	37	78	no	—
37	C₃²×C₉	C₃×C₃₇⋊C₃	1	39	yes	complemented
38	C₃₇	C₃³×3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
39	C₃³	—	37	117	no	—
40	C₃³	—	111	351	no	—
41	C₃³	#11758606	1	27	yes	direct product factor
42	C₃³	C₃²×C₃₇⋊C₃	1	13	yes	—
43	3₋¹⁺²	—	37	729	no	—
44	C₃×C₉	—	37	234	no	—
45	C₃×C₉	C₃²×C₃₇⋊C₃	1	117	yes	complemented
46	C₃²	—	37	27	no	—
47	C₃²	—	111	351	no	—
48	C₃²	C₃×#11758606	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	complemented
52	C₃	—	111	27	no	—
53	C₃	C₃²×#11758606	1	39	yes	direct product factor
54	C₃	C₃⁴×C₃₇⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃⁴	C₃×C₃₇⋊C₃
commutator, cosocle	C₃×C₃₇	C₃⁵
Frattini	C₃	C₃⁴×C₃₇⋊C₃
Fitting	C₃³×C₉×C₃₇	C₃
socle	C₃⁴×C₃₇	C₃²
prime radical	C₃⁴×C₃₇⋊C₃	C₃
hypercenter	C₃³×C₉	C₃₇⋊C₃
hypocenter	C₃₇	C₃³×3₋¹⁺²
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃⁴ ◁ C₃³×C₉

Complements

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

N	H ≅ G/N	type
2. C₃²×#11758606	C₃	direct (N × H)
3. C₃³×#66524	C₃	semidirect
4. C₃³×C₉×C₃₇	C₃	semidirect
7. C₃×#11758606	C₃²	direct (N × H)
8. C₃²×#66524	C₃²	semidirect
10. C₃²×C₉×C₃₇	C₃²	semidirect
11. C₃³×C₃₇	3₋¹⁺²	semidirect
15. C₃×C₉×C₃₇	C₃³	semidirect
16. C₃×#66524	C₃³	semidirect
17. #11758606	C₃³	direct (N × H)
19. C₃²×C₃₇	C₃×3₋¹⁺²	semidirect
23. C₉×C₃₇	C₃⁴	semidirect
24. #66524	C₃⁴	semidirect
28. C₃³×C₉	C₃₇⋊C₃	semidirect
29. C₃×C₃₇	C₃²×3₋¹⁺²	semidirect
37. C₃²×C₉	C₃×C₃₇⋊C₃	semidirect
38. C₃₇	C₃³×3₋¹⁺²	semidirect
41. C₃³	#11758606	direct (N × H)
45. C₃×C₉	C₃²×C₃₇⋊C₃	semidirect
48. C₃²	C₃×#11758606	direct (N × H)
51. C₉	C₃³×C₃₇⋊C₃	semidirect
53. C₃	C₃²×#11758606	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
C₃²×#11758606	3
C₃³×#66524	3
C₃³×C₉×C₃₇	3
C₃³×3₋¹⁺²	37

Subgroups — C33×#11758606 #41008245

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃³×#11758606 #41008245