Subgroups — #13133586

Order 1701.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
2	C₃²⋊(C₇⋊C₉)	C₃	1	3	yes	maximal
3	#10588036	C₃	1	1	yes	maximal, characteristic, complemented
4	#10588035	C₃	1	1	yes	maximal, characteristic, complemented
5	(C₇⋊C₉)⋊C₉	C₃	1	6	yes	maximal, complemented
6	C₉×C₇⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
7	#6899	—	7	1	no	maximal, Sylow
8	C₃³×C₇	C₃²	1	1	yes	characteristic
9	C₇×3₋¹⁺²	—	3	3	no	—
10	C₃×C₇×C₉	C₃²	1	2	yes	—
11	C₃×C₇×C₉	C₃²	1	1	yes	characteristic
12	C₃×C₇⋊C₉	C₃²	1	3	yes	—
13	C₃×C₇⋊C₉	C₃²	1	3	yes	—
14	C₃×C₇⋊C₉	C₃²	1	3	yes	—
15	C₃×3₋¹⁺²	ΓL(1,𝔽₈)	1	1	yes	characteristic
16	C₉⋊C₉	—	7	6	no	—
17	C₉⋊C₉	—	7	1	no	—
18	C₉⋊C₉	—	7	1	no	—
19	C₃²⋊C₉	—	7	3	no	—
20	C₉²	—	7	1	no	—
21	C₃²×C₇	—	3	1	no	—
22	C₃²×C₇	—	3	3	no	—
23	C₃²×C₇	C₃³	1	1	yes	characteristic
24	C₇×C₉	—	3	2	no	—
25	C₇×C₉	3₋¹⁺²	1	3	yes	—
26	C₇⋊C₉	—	3	3	no	—
27	C₇⋊C₉	—	3	3	no	—
28	C₇⋊C₉	—	3	3	no	—
29	C₃³	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
30	3₋¹⁺²	—	3	3	no	—
31	C₃×C₉	—	7	3	no	—
32	C₃×C₉	—	7	3	no	—
33	C₃×C₉	—	7	3	no	—
34	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	2	yes	—
35	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
36	C₃×C₇	—	9	1	no	—
37	C₃×C₇	C₃×3₋¹⁺²	1	1	yes	characteristic
38	C₃×C₇	#737	1	3	yes	—
39	C₃²	—	3	3	no	—
40	C₃²	—	3	1	no	—
41	C₃²	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
42	C₉	—	3	2	no	—
43	C₉	—	21	3	no	—
44	C₉	—	21	3	no	—
45	C₉	—	21	3	no	—
46	C₉	#4430	1	3	yes	—
47	C₇	#6899	1	1	yes	characteristic, complemented, Sylow
48	C₃	—	9	1	no	—
49	C₃	#10588065	1	3	yes	—
50	C₃	C₃×#4430	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃²×ΓL(1,𝔽₈)
commutator, socle, cosocle	C₃²×C₇	C₃³
Fitting	C₃×C₇×3₋¹⁺²	C₃
prime radical	C₃³×C₇	C₃²
hypercenter	C₃×3₋¹⁺²	ΓL(1,𝔽₈)
hypocenter	C₇	#6899
LCS	G ▷ C₃²×C₇ ▷ C₇
UCS	C₁ ◁ C₃² ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
3. #10588036	C₃	semidirect
4. #10588035	C₃	semidirect
5. (C₇⋊C₉)⋊C₉	C₃	semidirect
6. C₉×C₇⋊C₉	C₃	semidirect
47. 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×C₇×3₋¹⁺²	3
C₃²⋊(C₇⋊C₉)	3
#10588036	3
#10588035	3
(C₇⋊C₉)⋊C₉	3
C₉×C₇⋊C₉	3
#6899	7