Subgroups — C₃×C₃₇×3₋¹⁺⁴ #41004731

Order 26973.

5588 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, socle	C₃²×C₃₇	C₃⁴
commutator, Frattini, cosocle	C₃	C₃⁵×C₃₇
prime radical	C₃²×H(3,𝔽₃)×C₃₇	C₃
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₇×3₋¹⁺⁴	C₃	direct (N × H)
2. C₃×C₃₇×#737	C₃	semidirect
3. C₃²×3₋¹⁺²×C₃₇	C₃	semidirect
6. C₃₇×#737	C₃²	semidirect
8. C₃×3₋¹⁺²×C₃₇	C₃²	semidirect
9. C₃×3₋¹⁺²×C₃₇	C₃²	semidirect
12. C₃²×C₉×C₃₇	C₃²	semidirect
18. 3₋¹⁺²×C₃₇	C₃³	semidirect
21. C₃×C₉×C₃₇	C₃³	semidirect
22. C₃×3₋¹⁺⁴	C₃₇	direct (N × H)
30. 3₋¹⁺⁴	C₃×C₃₇	direct (N × H)
31. C₃×#737	C₃×C₃₇	semidirect
32. C₃²×3₋¹⁺²	C₃×C₃₇	semidirect
36. C₃×C₃₇	3₋¹⁺⁴	direct (N × H)
39. #737	C₃²×C₃₇	semidirect
41. C₃×3₋¹⁺²	C₃²×C₃₇	semidirect
42. C₃×3₋¹⁺²	C₃²×C₃₇	semidirect
45. C₃²×C₉	C₃²×C₃₇	semidirect
46. C₃₇	C₃×3₋¹⁺⁴	direct (N × H)
52. 3₋¹⁺²	C₃³×C₃₇	semidirect
55. C₃×C₉	C₃³×C₃₇	semidirect
65. C₃	C₃₇×3₋¹⁺⁴	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₇×3₋¹⁺⁴	3
C₃×C₃₇×#737	3
C₃²×3₋¹⁺²×C₃₇	3
C₃²×H(3,𝔽₃)×C₃₇	3
C₃×3₋¹⁺⁴	37

Subgroups — C3×C37×3−1+4 #41004731

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₃₇×3₋¹⁺⁴ #41004731