Subgroups — C₃×#10588061 #13133666

Order 1701.

480 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₇×C₉	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×C₇⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₃×#4430	C₃	1	2	yes	maximal, complemented
4	#10588061	C₃	1	27	yes	maximal, direct product factor
5	C₃×#4428	C₃	1	3	yes	maximal, complemented
6	C₃×#4427	C₃	1	3	yes	maximal, complemented
7	C₃×C₉×ΓL(1,𝔽₈)	C₃	1	3	yes	maximal, complemented
8	C₃×#737	—	7	1	no	maximal, Sylow
9	C₃³×C₇	C₃²	1	1	yes	characteristic
10	C₃²×ΓL(1,𝔽₈)	C₃²	1	3	yes	—
11	C₃×C₇×C₉	C₃²	1	1	yes	characteristic
12	C₃×C₇×C₉	C₃²	1	1	yes	characteristic
13	C₃×C₇×C₉	C₃²	1	1	yes	characteristic
14	C₃×C₇×C₉	C₃²	1	9	yes	complemented
15	C₇⋊H(3,𝔽₃)	C₃²	1	9	yes	—
16	#4430	C₃²	1	18	yes	complemented
17	C₃×C₇⋊C₉	C₃²	1	6	yes	—
18	#4428	C₃²	1	27	yes	complemented
19	#4427	C₃²	1	27	yes	complemented
20	C₉×ΓL(1,𝔽₈)	C₃²	1	27	yes	complemented
21	#737	—	7	27	no	—
22	C₃×3₋¹⁺²	—	7	2	no	—
23	C₃×3₋¹⁺²	—	7	3	no	—
24	C₃×3₋¹⁺²	—	7	3	no	—
25	C₃×H(3,𝔽₃)	—	7	1	no	—
26	C₃²×C₉	—	7	3	no	—
27	C₃²×C₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
28	C₃²×C₇	—	3	3	no	—
29	C₃²×C₇	C₃³	1	3	yes	—
30	C₃²×C₇	C₃³	1	1	yes	characteristic
31	C₃×ΓL(1,𝔽₈)	—	3	9	no	—
32	C₃×ΓL(1,𝔽₈)	C₃³	1	9	yes	—
33	C₇×C₉	C₃³	1	3	yes	—
34	C₇×C₉	C₃³	1	3	yes	—
35	C₇×C₉	C₃³	1	3	yes	—
36	C₇⋊C₉	C₃³	1	18	yes	—
37	C₃³	—	7	3	no	—
38	C₃³	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
39	3₋¹⁺²	—	7	18	no	—
40	3₋¹⁺²	—	7	27	no	—
41	3₋¹⁺²	—	7	27	no	—
42	H(3,𝔽₃)	—	7	9	no	—
43	C₃×C₉	—	7	6	no	—
44	C₃×C₉	—	7	27	no	—
45	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
46	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
47	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
48	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	9	yes	complemented
49	C₃×C₇	—	3	3	no	—
50	C₃×C₇	#737	1	3	yes	complemented
51	C₃×C₇	C₃⁴	1	1	yes	characteristic
52	ΓL(1,𝔽₈)	—	3	9	no	—
53	C₃²	—	3	3	no	—
54	C₃²	—	7	9	no	—
55	C₃²	—	21	9	no	—
56	C₃²	C₃²×ΓL(1,𝔽₈)	1	3	yes	—
57	C₃²	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
58	C₉	—	7	18	no	—
59	C₉	C₃²×ΓL(1,𝔽₈)	1	3	yes	—
60	C₉	C₃²×ΓL(1,𝔽₈)	1	3	yes	—
61	C₉	C₃²×ΓL(1,𝔽₈)	1	3	yes	—
62	C₇	C₃×#737	1	1	yes	characteristic, complemented, Sylow
63	C₃	—	3	3	no	—
64	C₃	—	21	9	no	—
65	C₃	#10588061	1	3	yes	direct product factor
66	C₃	C₃³×ΓL(1,𝔽₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₉	C₃×ΓL(1,𝔽₈)
commutator, cosocle	C₃×C₇	C₃⁴
Frattini	C₃	C₃³×ΓL(1,𝔽₈)
Fitting	C₃²×C₇×C₉	C₃
socle	C₃²×C₇	C₃³
prime radical	C₃×C₇⋊H(3,𝔽₃)	C₃
hypercenter	C₃²×C₉	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#737
LCS	G ▷ C₃×C₇ ▷ C₇
UCS	C₁ ◁ C₃×C₉ ◁ C₃²×C₉

Complements

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

N	H ≅ G/N	type
1. C₃²×C₇×C₉	C₃	semidirect
3. C₃×#4430	C₃	semidirect
4. #10588061	C₃	direct (N × H)
5. C₃×#4428	C₃	semidirect
6. C₃×#4427	C₃	semidirect
7. C₃×C₉×ΓL(1,𝔽₈)	C₃	semidirect
14. C₃×C₇×C₉	C₃²	semidirect
16. #4430	C₃²	semidirect
18. #4428	C₃²	semidirect
19. #4427	C₃²	semidirect
20. C₉×ΓL(1,𝔽₈)	C₃²	semidirect
27. C₃²×C₉	ΓL(1,𝔽₈)	semidirect
48. C₃×C₉	C₃×ΓL(1,𝔽₈)	semidirect
50. C₃×C₇	#737	semidirect
62. C₇	C₃×#737	semidirect
65. C₃	#10588061	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×C₇×C₉	3
C₃×C₇⋊H(3,𝔽₃)	3
C₃×#4430	3
#10588061	3
C₃×#4428	3
C₃×#4427	3
C₃×C₉×ΓL(1,𝔽₈)	3
C₃×#737	7

Subgroups — C3×#10588061 #13133666

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588061 #13133666