Subgroups — ΓL(1,𝔽₈)×C₉⋊C₉ #13133523

Order 1701.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×ΓL(1,𝔽₈)	C₃	1	3	yes	maximal
2	C₃×C₉×ΓL(1,𝔽₈)	C₃	1	1	yes	maximal, characteristic
3	C₇×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
4	#10588036	C₃	1	1	yes	maximal, characteristic, complemented
5	#10588035	C₃	1	1	yes	maximal, characteristic, complemented
6	(C₇⋊C₉)⋊C₉	C₃	1	6	yes	maximal, complemented
7	C₃×C₉⋊C₉	—	7	1	no	maximal, Sylow
8	C₃²×ΓL(1,𝔽₈)	C₃²	1	1	yes	characteristic
9	C₃×C₇×C₉	C₃²	1	3	yes	—
10	C₃×C₇×C₉	C₃²	1	1	yes	characteristic
11	C₃×C₇⋊C₉	C₃²	1	3	yes	—
12	C₃×C₇⋊C₉	C₃²	1	3	yes	—
13	C₃×C₇⋊C₉	C₃²	1	2	yes	—
14	C₉×ΓL(1,𝔽₈)	—	3	9	no	—
15	C₉×ΓL(1,𝔽₈)	C₉	1	9	yes	complemented
16	C₃²×C₉	—	7	1	no	—
17	C₃²×C₉	—	7	3	no	—
18	C₉⋊C₉	—	7	6	no	—
19	C₉⋊C₉	—	7	1	no	—
20	C₉⋊C₉	—	7	1	no	—
21	C₉⋊C₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
22	C₃²×C₇	C₃³	1	1	yes	characteristic
23	C₃×ΓL(1,𝔽₈)	C₃×C₉	1	3	yes	—
24	C₃×ΓL(1,𝔽₈)	3₋¹⁺²	1	9	yes	—
25	C₇×C₉	—	3	3	no	—
26	C₇×C₉	C₃×C₉	1	3	yes	complemented
27	C₇⋊C₉	—	3	3	no	—
28	C₇⋊C₉	—	3	3	no	—
29	C₇⋊C₉	C₃×C₉	1	6	yes	complemented
30	C₃³	—	7	1	no	—
31	C₃×C₉	—	7	2	no	—
32	C₃×C₉	—	7	9	no	—
33	C₃×C₉	—	7	3	no	—
34	C₃×C₉	—	7	3	no	—
35	C₃×C₉	—	21	9	no	—
36	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	3	yes	—
37	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
38	C₃×C₇	C₃²×C₉	1	1	yes	characteristic
39	C₃×C₇	C₃×3₋¹⁺²	1	3	yes	—
40	ΓL(1,𝔽₈)	C₉⋊C₉	1	9	yes	direct product factor
41	C₃²	—	7	3	no	—
42	C₃²	—	7	9	no	—
43	C₃²	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
44	C₉	—	3	3	no	—
45	C₉	—	7	6	no	—
46	C₉	—	21	3	no	—
47	C₉	—	21	3	no	—
48	C₉	C₉×ΓL(1,𝔽₈)	1	3	yes	complemented
49	C₇	C₃×C₉⋊C₉	1	1	yes	characteristic, complemented, Sylow
50	C₃	—	7	9	no	—
51	C₃	C₃×C₉×ΓL(1,𝔽₈)	1	1	yes	characteristic
52	C₃	ΓL(1,𝔽₈)×3₋¹⁺²	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₇×C₉⋊C₉	C₃	semidirect
4. #10588036	C₃	semidirect
5. #10588035	C₃	semidirect
6. (C₇⋊C₉)⋊C₉	C₃	semidirect
15. C₉×ΓL(1,𝔽₈)	C₉	semidirect
21. C₉⋊C₉	ΓL(1,𝔽₈)	direct (N × H)
26. C₇×C₉	C₃×C₉	semidirect
29. C₇⋊C₉	C₃×C₉	semidirect
40. ΓL(1,𝔽₈)	C₉⋊C₉	direct (N × H)
48. C₉	C₉×ΓL(1,𝔽₈)	semidirect
49. C₇	C₃×C₉⋊C₉	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×ΓL(1,𝔽₈)	3
C₇×C₉⋊C₉	3
#10588036	3
#10588035	3
(C₇⋊C₉)⋊C₉	3
C₃×C₉⋊C₉	7

Subgroups — ΓL(1,𝔽8)×C9⋊C9 #13133523

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×C₉⋊C₉ #13133523