Subgroups — C₃×(C₃×ΓL(1,𝔽₈))⋊C₉ #13133521

Order 1701.

288 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

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

Complements

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

N	H ≅ G/N	type
2. C₃²×C₇×C₉	C₃	semidirect
3. C₃²×C₇⋊C₉	C₃	semidirect
4. (C₃×ΓL(1,𝔽₈))⋊C₉	C₃	direct (N × H)
10. C₃²×ΓL(1,𝔽₈)	C₉	semidirect
13. C₃×C₇×C₉	C₃²	semidirect
15. C₃×C₇⋊C₉	C₃²	semidirect
18. C₃²×C₉	ΓL(1,𝔽₈)	semidirect
29. C₃×ΓL(1,𝔽₈)	C₃×C₉	semidirect
41. C₃×C₉	C₃×ΓL(1,𝔽₈)	semidirect
42. C₃×C₇	C₃²⋊C₉	semidirect
60. C₇	C₃×C₃²⋊C₉	semidirect
62. C₃	(C₃×ΓL(1,𝔽₈))⋊C₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×ΓL(1,𝔽₈)	3
C₃²×C₇×C₉	3
C₃²×C₇⋊C₉	3
(C₃×ΓL(1,𝔽₈))⋊C₉	3
C₃×C₃²⋊C₉	7

Subgroups — C3×(C3×ΓL(1,𝔽8))⋊C9 #13133521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×(C₃×ΓL(1,𝔽₈))⋊C₉ #13133521