Subgroups — C₃×C₃₇×#6900 #41004667

Order 26973.

1076 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×3₋¹⁺²×C₃₇	C₃	1	4	yes	maximal
2	C₃₇×#6900	C₃	1	27	yes	maximal, direct product factor
3	C₃×C₃₇×C₉⋊C₉	C₃	1	8	yes	maximal, complemented
4	C₃×C₉²×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
6	C₃×3₋¹⁺²×C₃₇	—	3	36	no	—
7	C₃×3₋¹⁺²×C₃₇	C₃²	1	36	yes	—
8	C₃²×C₉×C₃₇	C₃²	1	8	yes	—
9	C₃²×C₉×C₃₇	C₃²	1	4	yes	—
10	C₃₇×C₉⋊C₉	C₃²	1	72	yes	complemented
11	C₉²×C₃₇	C₃²	1	9	yes	complemented
12	C₃³×C₃₇	—	3	12	no	—
13	C₃³×C₃₇	C₃³	1	3	yes	—
14	C₃³×C₃₇	C₃³	1	1	yes	characteristic
15	3₋¹⁺²×C₃₇	—	3	108	no	—
16	C₃×C₉×C₃₇	—	3	24	no	—
17	C₃×C₉×C₃₇	3₋¹⁺²	1	36	yes	complemented
18	C₃×C₉×C₃₇	C₃³	1	24	yes	—
19	C₃×C₉×C₃₇	C₃³	1	12	yes	—
20	C₃×#6900	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₃²×C₃₇	—	3	12	no	—
22	C₃²×C₃₇	—	9	9	no	—
23	C₃²×C₃₇	C₃×3₋¹⁺²	1	12	yes	—
24	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
25	C₉×C₃₇	—	3	24	no	—
26	C₉×C₃₇	C₃×3₋¹⁺²	1	36	yes	complemented
27	C₃²×3₋¹⁺²	C₃×C₃₇	1	4	yes	—
28	#6900	C₃×C₃₇	1	27	yes	direct product factor
29	C₃×C₉⋊C₉	C₃×C₃₇	1	8	yes	complemented
30	C₃×C₉²	C₃×C₃₇	1	1	yes	characteristic, complemented
31	C₃×C₃₇	—	9	3	no	—
32	C₃×C₃₇	#6900	1	9	yes	direct product factor
33	C₃×C₃₇	C₃²×3₋¹⁺²	1	4	yes	—
34	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
35	C₃×3₋¹⁺²	—	3	36	no	—
36	C₃×3₋¹⁺²	C₃²×C₃₇	1	36	yes	—
37	C₃²×C₉	C₃²×C₃₇	1	8	yes	—
38	C₃²×C₉	C₃²×C₃₇	1	4	yes	—
39	C₉⋊C₉	C₃²×C₃₇	1	72	yes	complemented
40	C₉²	C₃²×C₃₇	1	9	yes	complemented
41	C₃₇	C₃×#6900	1	1	yes	characteristic, direct product factor, Sylow
42	C₃³	—	3	12	no	—
43	C₃³	C₃³×C₃₇	1	3	yes	—
44	C₃³	C₃³×C₃₇	1	1	yes	characteristic
45	3₋¹⁺²	—	3	108	no	—
46	C₃×C₉	—	3	24	no	—
47	C₃×C₉	3₋¹⁺²×C₃₇	1	36	yes	complemented
48	C₃×C₉	C₃³×C₃₇	1	24	yes	—
49	C₃×C₉	C₃³×C₃₇	1	12	yes	—
50	C₃²	—	3	12	no	—
51	C₃²	—	9	9	no	—
52	C₃²	C₃×3₋¹⁺²×C₃₇	1	12	yes	—
53	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
54	C₉	—	3	24	no	—
55	C₉	C₃×3₋¹⁺²×C₃₇	1	36	yes	complemented
56	C₃	—	9	3	no	—
57	C₃	C₃₇×#6900	1	9	yes	direct product factor
58	C₃	C₃²×3₋¹⁺²×C₃₇	1	4	yes	—

	subgroup	G/H
center, socle	C₃³×C₃₇	C₃³
commutator, Frattini, cosocle	C₃²	C₃⁴×C₃₇
prime radical	C₃⁴×C₃₇	C₃²
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃³×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₇×#6900	C₃	direct (N × H)
3. C₃×C₃₇×C₉⋊C₉	C₃	semidirect
4. C₃×C₉²×C₃₇	C₃	semidirect
10. C₃₇×C₉⋊C₉	C₃²	semidirect
11. C₉²×C₃₇	C₃²	semidirect
17. C₃×C₉×C₃₇	3₋¹⁺²	semidirect
20. C₃×#6900	C₃₇	direct (N × H)
26. C₉×C₃₇	C₃×3₋¹⁺²	semidirect
28. #6900	C₃×C₃₇	direct (N × H)
29. C₃×C₉⋊C₉	C₃×C₃₇	semidirect
30. C₃×C₉²	C₃×C₃₇	semidirect
32. C₃×C₃₇	#6900	direct (N × H)
39. C₉⋊C₉	C₃²×C₃₇	semidirect
40. C₉²	C₃²×C₃₇	semidirect
41. C₃₇	C₃×#6900	direct (N × H)
47. C₃×C₉	3₋¹⁺²×C₃₇	semidirect
55. C₉	C₃×3₋¹⁺²×C₃₇	semidirect
57. C₃	C₃₇×#6900	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×3₋¹⁺²×C₃₇	3
C₃₇×#6900	3
C₃×C₃₇×C₉⋊C₉	3
C₃×C₉²×C₃₇	3
C₃×#6900	37

Subgroups — C3×C37×#6900 #41004667

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₃₇×#6900 #41004667