Subgroups — #13133530

Order 1701.

138 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
2	C₃²⋊(C₇⋊C₉)	C₃	1	3	yes	maximal
3	#10588036	C₃	1	3	yes	maximal, complemented
4	#10588035	C₃	1	3	yes	maximal, complemented
5	C₉×C₇⋊C₉	C₃	1	3	yes	maximal, complemented
6	#6887	—	7	1	no	maximal, Sylow
7	C₃³×C₇	C₃²	1	1	yes	characteristic
8	C₃×C₇×C₉	C₉	1	9	yes	complemented
9	C₃×C₇×C₉	C₃²	1	1	yes	characteristic
10	C₃×C₇×C₉	C₃²	1	1	yes	characteristic
11	C₃×C₇×C₉	C₃²	1	1	yes	characteristic
12	C₃×C₇⋊C₉	C₃²	1	9	yes	—
13	C₃²×C₉	ΓL(1,𝔽₈)	1	1	yes	characteristic
14	C₉⋊C₉	—	7	3	no	—
15	C₉⋊C₉	—	7	3	no	—
16	C₃²⋊C₉	—	7	3	no	—
17	C₉²	—	7	3	no	—
18	C₃²×C₇	—	3	3	no	—
19	C₃²×C₇	C₃×C₉	1	3	yes	—
20	C₃²×C₇	C₃³	1	1	yes	characteristic
21	C₇×C₉	C₃×C₉	1	3	yes	—
22	C₇×C₉	C₃×C₉	1	3	yes	—
23	C₇×C₉	C₃×C₉	1	3	yes	—
24	C₇⋊C₉	—	3	9	no	—
25	C₃³	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
26	C₃×C₉	—	7	9	no	—
27	C₃×C₉	C₇⋊C₉	1	9	yes	complemented
28	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
29	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
30	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
31	C₃×C₇	—	3	3	no	—
32	C₃×C₇	C₃²×C₉	1	1	yes	characteristic
33	C₃×C₇	#737	1	3	yes	—
34	C₃²	—	3	3	no	—
35	C₃²	C₃×C₇⋊C₉	1	3	yes	—
36	C₃²	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
37	C₉	—	21	9	no	—
38	C₉	C₃×C₇⋊C₉	1	3	yes	—
39	C₉	C₃×C₇⋊C₉	1	3	yes	—
40	C₉	C₃×C₇⋊C₉	1	3	yes	—
41	C₇	#6887	1	1	yes	characteristic, complemented, Sylow
42	C₃	—	3	3	no	—
43	C₃	#10588061	1	3	yes	—
44	C₃	C₃²×C₇⋊C₉	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₉	C₃×ΓL(1,𝔽₈)
commutator	C₃×C₇	C₃²×C₉
Frattini	C₃²	C₃²×ΓL(1,𝔽₈)
Fitting	C₃²×C₇×C₉	C₃
socle, cosocle	C₃²×C₇	C₃³
prime radical	C₃³×C₇	C₃²
hypercenter	C₃²×C₉	ΓL(1,𝔽₈)
hypocenter	C₇	#6887
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
3. #10588036	C₃	semidirect
4. #10588035	C₃	semidirect
5. C₉×C₇⋊C₉	C₃	semidirect
8. C₃×C₇×C₉	C₉	semidirect
27. C₃×C₉	C₇⋊C₉	semidirect
41. C₇	#6887	semidirect

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
C₃²⋊(C₇⋊C₉)	3
#10588036	3
#10588035	3
C₉×C₇⋊C₉	3
#6887	7