Subgroups — #13133573

Order 1701.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#4428	C₃	1	1	yes	maximal, characteristic
2	C₃×#4427	C₃	1	1	yes	maximal, characteristic
3	C₇×C₉²	C₃	1	1	yes	maximal, characteristic, complemented
4	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃	1	2	yes	maximal
5	#10588036	C₃	1	2	yes	maximal, complemented
6	#10588035	C₃	1	2	yes	maximal, complemented
7	(C₇⋊C₉)⋊C₉	C₃	1	4	yes	maximal, complemented
8	#6896	—	7	1	no	maximal, Sylow
9	C₃²×ΓL(1,𝔽₈)	C₃²	1	1	yes	characteristic
10	C₃×C₇×C₉	C₃²	1	2	yes	—
11	C₃×C₇×C₉	C₃²	1	1	yes	characteristic
12	C₃×C₇×C₉	C₃²	1	1	yes	characteristic
13	C₃×C₇⋊C₉	C₃²	1	2	yes	—
14	C₃×C₇⋊C₉	C₃²	1	2	yes	—
15	C₃×C₇⋊C₉	C₃²	1	4	yes	—
16	#4428	—	3	3	no	—
17	#4427	—	3	3	no	—
18	C₃×3₋¹⁺²	—	7	1	no	—
19	C₃×3₋¹⁺²	—	7	1	no	—
20	C₉⋊C₉	—	7	4	no	—
21	C₉⋊C₉	—	7	2	no	—
22	C₉⋊C₉	—	7	2	no	—
23	C₃²⋊C₉	—	7	2	no	—
24	C₉²	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
25	C₃²×C₇	C₃³	1	1	yes	characteristic
26	C₃×ΓL(1,𝔽₈)	—	3	1	no	—
27	C₃×ΓL(1,𝔽₈)	—	3	1	no	—
28	C₃×ΓL(1,𝔽₈)	—	3	2	no	—
29	C₇×C₉	—	3	2	no	—
30	C₇×C₉	3₋¹⁺²	1	3	yes	complemented
31	C₇×C₉	3₋¹⁺²	1	3	yes	complemented
32	C₇⋊C₉	—	3	2	no	—
33	C₇⋊C₉	—	3	2	no	—
34	C₇⋊C₉	—	3	4	no	—
35	C₃³	—	7	1	no	—
36	3₋¹⁺²	—	21	3	no	—
37	3₋¹⁺²	—	21	3	no	—
38	C₃×C₉	—	7	4	no	—
39	C₃×C₉	—	7	2	no	—
40	C₃×C₉	—	7	2	no	—
41	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	2	yes	—
42	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
43	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
44	C₃×C₇	C₃×3₋¹⁺²	1	1	yes	characteristic
45	C₃×C₇	C₃×3₋¹⁺²	1	1	yes	characteristic
46	C₃×C₇	#737	1	2	yes	—
47	ΓL(1,𝔽₈)	—	9	1	no	—
48	C₃²	—	21	1	no	—
49	C₃²	—	21	1	no	—
50	C₃²	—	21	2	no	—
51	C₃²	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
52	C₉	—	3	2	no	—
53	C₉	—	21	2	no	—
54	C₉	—	21	2	no	—
55	C₉	—	21	4	no	—
56	C₉	#4427	1	3	yes	complemented
57	C₉	#4428	1	3	yes	complemented
58	C₇	#6896	1	1	yes	characteristic, complemented, Sylow
59	C₃	—	63	1	no	—
60	C₃	C₃×#4427	1	1	yes	characteristic
61	C₃	C₃×#4428	1	1	yes	characteristic
62	C₃	#10588061	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×ΓL(1,𝔽₈)
commutator, socle, cosocle	C₃²×C₇	C₃³
Fitting	C₇×C₉²	C₃
prime radical	C₃²×ΓL(1,𝔽₈)	C₃²
hypercenter	C₉²	ΓL(1,𝔽₈)
hypocenter	C₇	#6896
LCS	G ▷ C₃²×C₇ ▷ C₇
UCS	C₁ ◁ C₃² ◁ C₉²

Complements

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

N	H ≅ G/N	type
3. C₇×C₉²	C₃	semidirect
5. #10588036	C₃	semidirect
6. #10588035	C₃	semidirect
7. (C₇⋊C₉)⋊C₉	C₃	semidirect
24. C₉²	ΓL(1,𝔽₈)	semidirect
30. C₇×C₉	3₋¹⁺²	semidirect
31. C₇×C₉	3₋¹⁺²	semidirect
56. C₉	#4427	semidirect
57. C₉	#4428	semidirect
58. C₇	#6896	semidirect

Maximal subgroups

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

subgroup	index
C₃×#4428	3
C₃×#4427	3
C₇×C₉²	3
(C₃×ΓL(1,𝔽₈))⋊C₉	3
#10588036	3
#10588035	3
(C₇⋊C₉)⋊C₉	3
#6896	7