Subgroups — C₃³×3₋¹⁺²×C₃₇ #41004728

Order 26973.

7208 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁵×C₃₇	C₃	1	1	yes	maximal, characteristic
2	C₃²×3₋¹⁺²×C₃₇	C₃	1	117	yes	maximal, direct product factor
3	C₃³×C₉×C₃₇	C₃	1	3	yes	maximal, complemented
4	C₃⁴×C₃₇	—	3	27	no	—
5	C₃⁴×C₃₇	C₃²	1	39	yes	—
6	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
7	C₃×3₋¹⁺²×C₃₇	C₃²	1	1053	yes	direct product factor
8	C₃²×C₉×C₃₇	C₃²	1	117	yes	complemented
9	C₃³×C₃₇	—	3	351	no	—
10	C₃³×C₃₇	3₋¹⁺²	1	27	yes	direct product factor
11	C₃³×C₃₇	C₃³	1	117	yes	—
12	C₃³×C₃₇	C₃³	1	13	yes	—
13	3₋¹⁺²×C₃₇	C₃³	1	729	yes	direct product factor
14	C₃×C₉×C₃₇	C₃³	1	351	yes	complemented
15	C₃³×3₋¹⁺²	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₃²×C₃₇	—	3	351	no	—
17	C₃²×C₃₇	C₃×3₋¹⁺²	1	117	yes	direct product factor
18	C₃²×C₃₇	C₃⁴	1	27	yes	—
19	C₃²×C₃₇	C₃⁴	1	13	yes	—
20	C₉×C₃₇	C₃⁴	1	81	yes	complemented
21	C₃⁵	C₃×C₃₇	1	1	yes	characteristic
22	C₃²×3₋¹⁺²	C₃×C₃₇	1	117	yes	direct product factor
23	C₃³×C₉	C₃×C₃₇	1	3	yes	complemented
24	C₃×C₃₇	—	3	27	no	—
25	C₃×C₃₇	C₃²×3₋¹⁺²	1	39	yes	direct product factor
26	C₃×C₃₇	C₃⁵	1	1	yes	characteristic
27	C₃⁴	—	3	27	no	—
28	C₃⁴	C₃²×C₃₇	1	39	yes	—
29	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
30	C₃×3₋¹⁺²	C₃²×C₃₇	1	1053	yes	direct product factor
31	C₃²×C₉	C₃²×C₃₇	1	117	yes	complemented
32	C₃₇	C₃³×3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
33	C₃³	—	3	351	no	—
34	C₃³	3₋¹⁺²×C₃₇	1	27	yes	direct product factor
35	C₃³	C₃³×C₃₇	1	117	yes	—
36	C₃³	C₃³×C₃₇	1	13	yes	—
37	3₋¹⁺²	C₃³×C₃₇	1	729	yes	direct product factor
38	C₃×C₉	C₃³×C₃₇	1	351	yes	complemented
39	C₃²	—	3	351	no	—
40	C₃²	C₃×3₋¹⁺²×C₃₇	1	117	yes	direct product factor
41	C₃²	C₃⁴×C₃₇	1	27	yes	—
42	C₃²	C₃⁴×C₃₇	1	13	yes	—
43	C₉	C₃⁴×C₃₇	1	81	yes	complemented
44	C₃	—	3	27	no	—
45	C₃	C₃²×3₋¹⁺²×C₃₇	1	39	yes	direct product factor
46	C₃	C₃⁵×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃⁴×C₃₇	C₃²
commutator, Frattini, cosocle	C₃	C₃⁵×C₃₇
prime radical	C₃⁵×C₃₇	C₃
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃⁴×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃²×3₋¹⁺²×C₃₇	C₃	direct (N × H)
3. C₃³×C₉×C₃₇	C₃	semidirect
7. C₃×3₋¹⁺²×C₃₇	C₃²	direct (N × H)
8. C₃²×C₉×C₃₇	C₃²	semidirect
10. C₃³×C₃₇	3₋¹⁺²	direct (N × H)
13. 3₋¹⁺²×C₃₇	C₃³	direct (N × H)
14. C₃×C₉×C₃₇	C₃³	semidirect
15. C₃³×3₋¹⁺²	C₃₇	direct (N × H)
17. C₃²×C₃₇	C₃×3₋¹⁺²	direct (N × H)
20. C₉×C₃₇	C₃⁴	semidirect
22. C₃²×3₋¹⁺²	C₃×C₃₇	direct (N × H)
23. C₃³×C₉	C₃×C₃₇	semidirect
25. C₃×C₃₇	C₃²×3₋¹⁺²	direct (N × H)
30. C₃×3₋¹⁺²	C₃²×C₃₇	direct (N × H)
31. C₃²×C₉	C₃²×C₃₇	semidirect
32. C₃₇	C₃³×3₋¹⁺²	direct (N × H)
34. C₃³	3₋¹⁺²×C₃₇	direct (N × H)
37. 3₋¹⁺²	C₃³×C₃₇	direct (N × H)
38. C₃×C₉	C₃³×C₃₇	semidirect
40. C₃²	C₃×3₋¹⁺²×C₃₇	direct (N × H)
43. C₉	C₃⁴×C₃₇	semidirect
45. C₃	C₃²×3₋¹⁺²×C₃₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃⁵×C₃₇	3
C₃²×3₋¹⁺²×C₃₇	3
C₃³×C₉×C₃₇	3
C₃³×3₋¹⁺²	37

Subgroups — C33×3−1+2×C37 #41004728

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃³×3₋¹⁺²×C₃₇ #41004728