Subgroups — #13133591

Order 1701.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#4427	C₃	1	1	yes	maximal, characteristic
2	C₇×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
3	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃	1	3	yes	maximal
4	#10588036	C₃	1	1	yes	maximal, characteristic, complemented
5	#10588035	C₃	1	3	yes	maximal, complemented
6	(C₇⋊C₉)⋊C₉	C₃	1	3	yes	maximal, complemented
7	C₉×C₇⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
8	#6899	—	7	1	no	maximal, Sylow
9	C₃²×ΓL(1,𝔽₈)	C₃²	1	1	yes	characteristic
10	C₃×C₇×C₉	C₃²	1	3	yes	—
11	C₃×C₇×C₉	C₃²	1	1	yes	characteristic
12	C₃×C₇⋊C₉	C₃²	1	3	yes	—
13	C₃×C₇⋊C₉	C₃²	1	1	yes	characteristic
14	C₃×C₇⋊C₉	C₃²	1	3	yes	—
15	C₃×C₇⋊C₉	C₃²	1	1	yes	characteristic
16	#4427	—	3	3	no	—
17	C₃×3₋¹⁺²	—	7	1	no	—
18	C₉⋊C₉	—	7	3	no	—
19	C₉⋊C₉	—	7	3	no	—
20	C₉⋊C₉	—	7	1	no	—
21	C₉⋊C₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
22	C₃²⋊C₉	—	7	3	no	—
23	C₉²	—	7	1	no	—
24	C₃²×C₇	C₃³	1	1	yes	characteristic
25	C₃×ΓL(1,𝔽₈)	—	3	1	no	—
26	C₃×ΓL(1,𝔽₈)	—	3	3	no	—
27	C₇×C₉	—	3	3	no	—
28	C₇×C₉	3₋¹⁺²	1	3	yes	—
29	C₇⋊C₉	—	3	1	no	—
30	C₇⋊C₉	—	3	1	no	—
31	C₇⋊C₉	—	3	3	no	—
32	C₇⋊C₉	—	3	3	no	—
33	C₃³	—	7	1	no	—
34	3₋¹⁺²	—	21	3	no	—
35	C₃×C₉	—	7	3	no	—
36	C₃×C₉	—	7	3	no	—
37	C₃×C₉	—	7	1	no	—
38	C₃×C₉	—	7	1	no	—
39	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	3	yes	—
40	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
41	C₃×C₇	C₃×3₋¹⁺²	1	1	yes	characteristic
42	C₃×C₇	#737	1	3	yes	—
43	ΓL(1,𝔽₈)	—	9	1	no	—
44	C₃²	—	21	1	no	—
45	C₃²	—	21	3	no	—
46	C₃²	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
47	C₉	—	3	3	no	—
48	C₉	—	21	1	no	—
49	C₉	—	21	1	no	—
50	C₉	—	21	3	no	—
51	C₉	—	21	3	no	—
52	C₉	#4427	1	3	yes	—
53	C₇	#6899	1	1	yes	characteristic, complemented, Sylow
54	C₃	—	63	1	no	—
55	C₃	C₃×#4427	1	1	yes	characteristic
56	C₃	#10588065	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×ΓL(1,𝔽₈)
commutator, socle, cosocle	C₃²×C₇	C₃³
Fitting	C₇×C₉⋊C₉	C₃
prime radical	C₃²×ΓL(1,𝔽₈)	C₃²
hypercenter	C₉⋊C₉	ΓL(1,𝔽₈)
hypocenter	C₇	#6899
LCS	G ▷ C₃²×C₇ ▷ C₇
UCS	C₁ ◁ C₃² ◁ C₉⋊C₉

Complements

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

N	H ≅ G/N	type
2. C₇×C₉⋊C₉	C₃	semidirect
4. #10588036	C₃	semidirect
5. #10588035	C₃	semidirect
6. (C₇⋊C₉)⋊C₉	C₃	semidirect
7. C₉×C₇⋊C₉	C₃	semidirect
21. C₉⋊C₉	ΓL(1,𝔽₈)	semidirect
53. C₇	#6899	semidirect

Maximal subgroups

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

subgroup	index
C₃×#4427	3
C₇×C₉⋊C₉	3
(C₃×ΓL(1,𝔽₈))⋊C₉	3
#10588036	3
#10588035	3
(C₇⋊C₉)⋊C₉	3
C₉×C₇⋊C₉	3
#6899	7