Subgroups — #13133540

Order 1701.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#4430	C₃	1	1	yes	maximal, characteristic
2	C₃²×C₇⋊C₉	C₃	1	1	yes	maximal, characteristic
3	C₇×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
4	C₃²⋊(C₇⋊C₉)	C₃	1	1	yes	maximal, characteristic
5	(C₇⋊C₉)⋊C₉	C₃	1	6	yes	maximal, complemented
6	C₉×C₇⋊C₉	C₃	1	3	yes	maximal, complemented
7	C₉×3₋¹⁺²	—	7	1	no	maximal, Sylow
8	C₃³×C₇	C₃²	1	1	yes	characteristic
9	C₃×C₇×C₉	C₃²	1	3	yes	—
10	#4430	C₉	1	9	yes	complemented
11	C₃×C₇⋊C₉	—	3	3	no	—
12	C₃×C₇⋊C₉	C₃²	1	3	yes	—
13	C₃×C₇⋊C₉	C₃²	1	2	yes	—
14	C₃×C₇⋊C₉	C₃²	1	3	yes	—
15	C₃×C₇⋊C₉	C₃²	1	1	yes	characteristic
16	C₃×3₋¹⁺²	—	7	1	no	—
17	C₃²×C₉	—	7	1	no	—
18	C₉⋊C₉	—	7	6	no	—
19	C₃²⋊C₉	—	7	1	no	—
20	C₃²⋊C₉	ΓL(1,𝔽₈)	1	1	yes	characteristic
21	C₉²	—	7	3	no	—
22	C₃²×C₇	—	3	1	no	—
23	C₃²×C₇	—	3	1	no	—
24	C₃²×C₇	—	3	1	no	—
25	C₃²×C₇	C₃×C₉	1	3	yes	—
26	C₃²×C₇	C₃³	1	1	yes	characteristic
27	C₇×C₉	—	3	3	no	—
28	C₇⋊C₉	—	3	2	no	—
29	C₇⋊C₉	—	3	3	no	—
30	C₇⋊C₉	C₃×C₉	1	9	yes	complemented
31	C₇⋊C₉	3₋¹⁺²	1	3	yes	complemented
32	C₃³	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
33	3₋¹⁺²	—	7	9	no	—
34	C₃×C₉	—	7	1	no	—
35	C₃×C₉	—	7	3	no	—
36	C₃×C₉	—	7	3	no	—
37	C₃×C₉	—	7	2	no	—
38	C₃×C₉	—	21	3	no	—
39	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	3	yes	—
40	C₃×C₇	—	3	3	no	—
41	C₃×C₇	C₃²×C₉	1	1	yes	characteristic
42	C₃×C₇	C₃×3₋¹⁺²	1	1	yes	characteristic
43	C₃×C₇	#737	1	1	yes	characteristic
44	C₃×C₇	#737	1	1	yes	characteristic
45	C₃²	—	3	1	no	—
46	C₃²	—	3	1	no	—
47	C₃²	—	3	1	no	—
48	C₃²	C₉×ΓL(1,𝔽₈)	1	3	yes	—
49	C₃²	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
50	C₉	—	3	3	no	—
51	C₉	—	7	9	no	—
52	C₉	—	7	3	no	—
53	C₉	—	21	2	no	—
54	C₉	—	21	3	no	—
55	C₇	C₉×3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
56	C₃	—	3	3	no	—
57	C₃	C₃×C₉×ΓL(1,𝔽₈)	1	1	yes	characteristic
58	C₃	#10588062	1	1	yes	characteristic
59	C₃	ΓL(1,𝔽₈)×3₋¹⁺²	1	1	yes	characteristic
60	C₃	#10588065	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃²×ΓL(1,𝔽₈)
commutator	C₃×C₇	C₃²×C₉
Fitting	C₇×C₃²⋊C₉	C₃
socle, cosocle	C₃²×C₇	C₃³
prime radical	C₃³×C₇	C₃²
hypercenter	C₃²⋊C₉	ΓL(1,𝔽₈)
hypocenter	C₇	C₉×3₋¹⁺²
LCS	G ▷ C₃×C₇ ▷ C₇
UCS	C₁ ◁ C₃² ◁ C₃²⋊C₉

Complements

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

N	H ≅ G/N	type
5. (C₇⋊C₉)⋊C₉	C₃	semidirect
6. C₉×C₇⋊C₉	C₃	semidirect
10. #4430	C₉	semidirect
30. C₇⋊C₉	C₃×C₉	semidirect
31. C₇⋊C₉	3₋¹⁺²	semidirect
55. C₇	C₉×3₋¹⁺²	semidirect

Maximal subgroups

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

subgroup	index
C₃×#4430	3
C₃²×C₇⋊C₉	3
C₇×C₃²⋊C₉	3
C₃²⋊(C₇⋊C₉)	3
(C₇⋊C₉)⋊C₉	3
C₉×C₇⋊C₉	3
C₉×3₋¹⁺²	7