Subgroups — #13133568

Order 1701.

114 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	C₇×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊(C₇⋊C₉)	C₃	1	2	yes	maximal
4	#10588035	C₃	1	3	yes	maximal, complemented
5	(C₇⋊C₉)⋊C₉	C₃	1	6	yes	maximal, complemented
6	#6895	—	7	1	no	maximal, Sylow
7	C₃³×C₇	C₃²	1	1	yes	characteristic
8	C₃×C₇×C₉	C₃²	1	3	yes	—
9	C₃×C₇⋊C₉	—	3	3	no	—
10	C₃×C₇⋊C₉	C₃²	1	2	yes	—
11	C₃×C₇⋊C₉	C₃²	1	6	yes	—
12	C₃×C₇⋊C₉	C₃²	1	1	yes	characteristic
13	C₃²×C₉	—	7	1	no	—
14	C₉⋊C₉	—	7	6	no	—
15	C₉⋊C₉	—	7	3	no	—
16	C₃²⋊C₉	—	7	2	no	—
17	C₃²⋊C₉	ΓL(1,𝔽₈)	1	1	yes	characteristic
18	C₃²×C₇	—	3	1	no	—
19	C₃²×C₇	—	3	2	no	—
20	C₃²×C₇	3₋¹⁺²	1	3	yes	—
21	C₃²×C₇	C₃³	1	1	yes	characteristic
22	C₇×C₉	—	3	3	no	—
23	C₇⋊C₉	—	3	2	no	—
24	C₇⋊C₉	—	3	1	no	—
25	C₇⋊C₉	—	3	6	no	—
26	C₃³	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
27	C₃×C₉	—	7	6	no	—
28	C₃×C₉	—	7	2	no	—
29	C₃×C₉	—	7	1	no	—
30	C₃×C₉	—	21	3	no	—
31	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	3	yes	—
32	C₃×C₇	—	3	3	no	—
33	C₃×C₇	C₃×3₋¹⁺²	1	1	yes	characteristic
34	C₃×C₇	#737	1	1	yes	characteristic
35	C₃×C₇	#737	1	2	yes	—
36	C₃²	—	3	1	no	—
37	C₃²	—	3	2	no	—
38	C₃²	#4427	1	3	yes	—
39	C₃²	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
40	C₉	—	3	3	no	—
41	C₉	—	21	2	no	—
42	C₉	—	21	1	no	—
43	C₉	—	21	6	no	—
44	C₇	#6895	1	1	yes	characteristic, complemented, Sylow
45	C₃	—	3	3	no	—
46	C₃	C₃×#4427	1	1	yes	characteristic
47	C₃	#10588062	1	1	yes	characteristic
48	C₃	#10588065	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×ΓL(1,𝔽₈)
commutator, socle, cosocle	C₃²×C₇	C₃³
Fitting	C₇×C₃²⋊C₉	C₃
prime radical	C₃³×C₇	C₃²
hypercenter	C₃²⋊C₉	ΓL(1,𝔽₈)
hypocenter	C₇	#6895
LCS	G ▷ C₃²×C₇ ▷ C₇
UCS	C₁ ◁ C₃² ◁ C₃²⋊C₉

Complements

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

N	H ≅ G/N	type
4. #10588035	C₃	semidirect
5. (C₇⋊C₉)⋊C₉	C₃	semidirect
44. C₇	#6895	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
C₇×C₃²⋊C₉	3
C₃²⋊(C₇⋊C₉)	3
#10588035	3
(C₇⋊C₉)⋊C₉	3
#6895	7