Subgroups — C₃²×C₃₇×#737 #41004729

Order 26973.

5804 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃₇×#737	C₃	1	108	yes	maximal, direct product factor
2	C₃²×3₋¹⁺²×C₃₇	C₃	1	8	yes	maximal, complemented
3	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic
4	C₃³×C₉×C₃₇	C₃	1	4	yes	maximal, complemented
5	C₃⁴×C₃₇	C₃²	1	4	yes	—
6	C₃₇×#737	C₃²	1	729	yes	direct product factor
7	C₃×3₋¹⁺²×C₃₇	C₃²	1	288	yes	complemented
8	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	36	yes	—
9	C₃²×C₉×C₃₇	C₃²	1	8	yes	—
10	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
11	C₃²×C₉×C₃₇	C₃²	1	144	yes	complemented
12	C₃³×C₃₇	—	3	36	no	—
13	C₃³×C₃₇	C₃³	1	48	yes	—
14	C₃³×C₃₇	C₃³	1	1	yes	characteristic
15	3₋¹⁺²×C₃₇	C₃³	1	648	yes	complemented
16	H(3,𝔽₃)×C₃₇	C₃³	1	81	yes	—
17	C₃×C₉×C₃₇	C₃³	1	96	yes	—
18	C₃×C₉×C₃₇	C₃³	1	12	yes	—
19	C₃×C₉×C₃₇	C₃³	1	324	yes	complemented
20	C₃²×#737	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₃²×C₃₇	—	3	144	no	—
22	C₃²×C₃₇	#737	1	9	yes	direct product factor
23	C₃²×C₃₇	C₃⁴	1	36	yes	—
24	C₃²×C₃₇	C₃⁴	1	4	yes	—
25	C₉×C₃₇	C₃⁴	1	9	yes	—
26	C₉×C₃₇	C₃⁴	1	72	yes	—
27	C₃×#737	C₃×C₃₇	1	108	yes	direct product factor
28	C₃²×3₋¹⁺²	C₃×C₃₇	1	8	yes	complemented
29	C₃²×H(3,𝔽₃)	C₃×C₃₇	1	1	yes	characteristic
30	C₃³×C₉	C₃×C₃₇	1	4	yes	complemented
31	C₃×C₃₇	—	3	36	no	—
32	C₃×C₃₇	C₃×#737	1	12	yes	direct product factor
33	C₃×C₃₇	C₃⁵	1	1	yes	characteristic
34	C₃⁴	C₃²×C₃₇	1	4	yes	—
35	#737	C₃²×C₃₇	1	729	yes	direct product factor
36	C₃×3₋¹⁺²	C₃²×C₃₇	1	288	yes	complemented
37	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	36	yes	—
38	C₃²×C₉	C₃²×C₃₇	1	1	yes	characteristic
39	C₃²×C₉	C₃²×C₃₇	1	8	yes	—
40	C₃²×C₉	C₃²×C₃₇	1	144	yes	complemented
41	C₃₇	C₃²×#737	1	1	yes	characteristic, direct product factor, Sylow
42	C₃³	—	3	36	no	—
43	C₃³	C₃³×C₃₇	1	48	yes	—
44	C₃³	C₃³×C₃₇	1	1	yes	characteristic
45	3₋¹⁺²	C₃³×C₃₇	1	648	yes	complemented
46	H(3,𝔽₃)	C₃³×C₃₇	1	81	yes	—
47	C₃×C₉	C₃³×C₃₇	1	12	yes	—
48	C₃×C₉	C₃³×C₃₇	1	96	yes	—
49	C₃×C₉	C₃³×C₃₇	1	324	yes	complemented
50	C₃²	—	3	144	no	—
51	C₃²	C₃₇×#737	1	9	yes	direct product factor
52	C₃²	C₃⁴×C₃₇	1	36	yes	—
53	C₃²	C₃⁴×C₃₇	1	4	yes	—
54	C₉	C₃⁴×C₃₇	1	9	yes	—
55	C₉	C₃⁴×C₃₇	1	72	yes	—
56	C₃	—	3	36	no	—
57	C₃	C₃×C₃₇×#737	1	12	yes	direct product factor
58	C₃	C₃⁵×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₃²×C₉×C₃₇	C₃²
commutator, Frattini, cosocle	C₃	C₃⁵×C₃₇
socle	C₃³×C₃₇	C₃³
prime radical	C₃²×H(3,𝔽₃)×C₃₇	C₃
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₉×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₃₇×#737	C₃	direct (N × H)
2. C₃²×3₋¹⁺²×C₃₇	C₃	semidirect
4. C₃³×C₉×C₃₇	C₃	semidirect
6. C₃₇×#737	C₃²	direct (N × H)
7. C₃×3₋¹⁺²×C₃₇	C₃²	semidirect
11. C₃²×C₉×C₃₇	C₃²	semidirect
15. 3₋¹⁺²×C₃₇	C₃³	semidirect
19. C₃×C₉×C₃₇	C₃³	semidirect
20. C₃²×#737	C₃₇	direct (N × H)
22. C₃²×C₃₇	#737	direct (N × H)
27. C₃×#737	C₃×C₃₇	direct (N × H)
28. C₃²×3₋¹⁺²	C₃×C₃₇	semidirect
30. C₃³×C₉	C₃×C₃₇	semidirect
32. C₃×C₃₇	C₃×#737	direct (N × H)
35. #737	C₃²×C₃₇	direct (N × H)
36. C₃×3₋¹⁺²	C₃²×C₃₇	semidirect
40. C₃²×C₉	C₃²×C₃₇	semidirect
41. C₃₇	C₃²×#737	direct (N × H)
45. 3₋¹⁺²	C₃³×C₃₇	semidirect
49. C₃×C₉	C₃³×C₃₇	semidirect
51. C₃²	C₃₇×#737	direct (N × H)
57. C₃	C₃×C₃₇×#737	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₃₇×#737	3
C₃²×3₋¹⁺²×C₃₇	3
C₃²×H(3,𝔽₃)×C₃₇	3
C₃³×C₉×C₃₇	3
C₃²×#737	37

Subgroups — C32×C37×#737 #41004729

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₃₇×#737 #41004729