Subgroups — C₉×C₃₇×#737 #41004650

Order 26973.

986 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃₇×#737	C₃	1	1	yes	maximal, characteristic
2	C₉×3₋¹⁺²×C₃₇	C₃	1	24	yes	maximal, complemented
3	C₉×H(3,𝔽₃)×C₃₇	C₃	1	3	yes	maximal
4	C₃₇×#6887	C₃	1	8	yes	maximal, complemented
5	C₃×C₉²×C₃₇	C₃	1	4	yes	maximal, complemented
6	C₃₇×#737	C₉	1	27	yes	direct product factor
7	C₃×3₋¹⁺²×C₃₇	C₃²	1	8	yes	—
8	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
9	C₃²×C₉×C₃₇	C₃²	1	12	yes	—
10	C₃²×C₉×C₃₇	C₃²	1	4	yes	—
11	C₃₇×C₉⋊C₉	C₃²	1	48	yes	—
12	C₃₇×C₃²⋊C₉	C₃²	1	24	yes	—
13	C₉²×C₃₇	C₃²	1	8	yes	—
14	C₉²×C₃₇	C₃²	1	24	yes	—
15	C₉²×C₃₇	C₃²	1	1	yes	characteristic
16	C₃³×C₃₇	C₃³	1	4	yes	—
17	3₋¹⁺²×C₃₇	C₃×C₉	1	72	yes	complemented
18	H(3,𝔽₃)×C₃₇	C₃×C₉	1	9	yes	—
19	C₃×C₉×C₃₇	—	3	36	no	—
20	C₃×C₉×C₃₇	C₃×C₉	1	36	yes	complemented
21	C₃×C₉×C₃₇	C₃³	1	24	yes	—
22	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
23	C₃×C₉×C₃₇	C₃³	1	8	yes	—
24	C₃×C₉×C₃₇	C₃³	1	3	yes	—
25	C₉×#737	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₃²×C₃₇	—	3	12	no	—
27	C₃²×C₃₇	C₃²×C₉	1	12	yes	—
28	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
29	C₉×C₃₇	—	3	24	no	—
30	C₉×C₃₇	C₃²×C₉	1	3	yes	—
31	C₉×C₃₇	C₃²×C₉	1	24	yes	—
32	C₉×C₃₇	#737	1	9	yes	direct product factor
33	C₃×#737	C₃×C₃₇	1	1	yes	characteristic
34	C₉×3₋¹⁺²	C₃×C₃₇	1	24	yes	complemented
35	C₉×H(3,𝔽₃)	C₃×C₃₇	1	3	yes	—
36	#6887	C₃×C₃₇	1	8	yes	complemented
37	C₃×C₉²	C₃×C₃₇	1	4	yes	complemented
38	C₃×C₃₇	—	3	12	no	—
39	C₃×C₃₇	C₃³×C₉	1	1	yes	characteristic
40	C₃×C₃₇	C₃×#737	1	3	yes	—
41	#737	C₉×C₃₇	1	27	yes	direct product factor
42	C₃×3₋¹⁺²	C₃²×C₃₇	1	8	yes	—
43	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	1	yes	characteristic
44	C₃²×C₉	C₃²×C₃₇	1	12	yes	—
45	C₃²×C₉	C₃²×C₃₇	1	4	yes	—
46	C₉⋊C₉	C₃²×C₃₇	1	48	yes	—
47	C₃²⋊C₉	C₃²×C₃₇	1	24	yes	—
48	C₉²	C₃²×C₃₇	1	8	yes	—
49	C₉²	C₃²×C₃₇	1	24	yes	—
50	C₉²	C₃²×C₃₇	1	1	yes	characteristic
51	C₃₇	C₉×#737	1	1	yes	characteristic, direct product factor, Sylow
52	C₃³	C₃³×C₃₇	1	4	yes	—
53	3₋¹⁺²	C₃×C₉×C₃₇	1	72	yes	complemented
54	H(3,𝔽₃)	C₃×C₉×C₃₇	1	9	yes	—
55	C₃×C₉	—	3	36	no	—
56	C₃×C₉	C₃×C₉×C₃₇	1	36	yes	complemented
57	C₃×C₉	C₃³×C₃₇	1	24	yes	—
58	C₃×C₉	C₃³×C₃₇	1	1	yes	characteristic
59	C₃×C₉	C₃³×C₃₇	1	8	yes	—
60	C₃×C₉	C₃³×C₃₇	1	3	yes	—
61	C₃²	—	3	12	no	—
62	C₃²	C₃²×C₉×C₃₇	1	12	yes	—
63	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
64	C₉	—	3	24	no	—
65	C₉	C₃²×C₉×C₃₇	1	3	yes	—
66	C₉	C₃²×C₉×C₃₇	1	24	yes	—
67	C₉	C₃₇×#737	1	9	yes	direct product factor
68	C₃	—	3	12	no	—
69	C₃	C₃³×C₉×C₃₇	1	1	yes	characteristic
70	C₃	C₃×C₃₇×#737	1	3	yes	—

	subgroup	G/H
center	C₉²×C₃₇	C₃²
commutator	C₃	C₃³×C₉×C₃₇
Frattini, cosocle	C₃²	C₃⁴×C₃₇
socle	C₃²×C₃₇	C₃⁴
prime radical	C₃×H(3,𝔽₃)×C₃₇	C₃²
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₉²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₉×3₋¹⁺²×C₃₇	C₃	semidirect
4. C₃₇×#6887	C₃	semidirect
5. C₃×C₉²×C₃₇	C₃	semidirect
6. C₃₇×#737	C₉	direct (N × H)
17. 3₋¹⁺²×C₃₇	C₃×C₉	semidirect
20. C₃×C₉×C₃₇	C₃×C₉	semidirect
25. C₉×#737	C₃₇	direct (N × H)
32. C₉×C₃₇	#737	direct (N × H)
34. C₉×3₋¹⁺²	C₃×C₃₇	semidirect
36. #6887	C₃×C₃₇	semidirect
37. C₃×C₉²	C₃×C₃₇	semidirect
41. #737	C₉×C₃₇	direct (N × H)
51. C₃₇	C₉×#737	direct (N × H)
53. 3₋¹⁺²	C₃×C₉×C₃₇	semidirect
56. C₃×C₉	C₃×C₉×C₃₇	semidirect
67. C₉	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
C₉×3₋¹⁺²×C₃₇	3
C₉×H(3,𝔽₃)×C₃₇	3
C₃₇×#6887	3
C₃×C₉²×C₃₇	3
C₉×#737	37

Subgroups — C9×C37×#737 #41004650

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×C₃₇×#737 #41004650