Subgroups — #13133575

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×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
4	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃	1	2	yes	maximal
5	#10588035	C₃	1	2	yes	maximal, complemented
6	#10588035	C₃	1	1	yes	maximal, characteristic, complemented
7	(C₇⋊C₉)⋊C₉	C₃	1	2	yes	maximal, complemented
8	(C₇⋊C₉)⋊C₉	C₃	1	2	yes	maximal, complemented
9	C₉×C₇⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
10	#6896	—	7	1	no	maximal, Sylow
11	C₃²×ΓL(1,𝔽₈)	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	1	yes	characteristic
15	C₃×C₇⋊C₉	C₃²	1	2	yes	—
16	C₃×C₇⋊C₉	C₃²	1	2	yes	—
17	C₃×C₇⋊C₉	C₃²	1	2	yes	—
18	C₃×C₇⋊C₉	C₃²	1	1	yes	characteristic
19	C₃×C₇⋊C₉	C₃²	1	1	yes	characteristic
20	#4428	—	3	3	no	—
21	#4427	—	3	3	no	—
22	C₃×3₋¹⁺²	—	7	1	no	—
23	C₃×3₋¹⁺²	—	7	1	no	—
24	C₉⋊C₉	—	7	2	no	—
25	C₉⋊C₉	—	7	2	no	—
26	C₉⋊C₉	—	7	2	no	—
27	C₉⋊C₉	—	7	1	no	—
28	C₉⋊C₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
29	C₃²⋊C₉	—	7	2	no	—
30	C₉²	—	7	1	no	—
31	C₃²×C₇	C₃³	1	1	yes	characteristic
32	C₃×ΓL(1,𝔽₈)	—	3	1	no	—
33	C₃×ΓL(1,𝔽₈)	—	3	1	no	—
34	C₃×ΓL(1,𝔽₈)	—	3	2	no	—
35	C₇×C₉	—	3	1	no	—
36	C₇×C₉	—	3	2	no	—
37	C₇×C₉	3₋¹⁺²	1	3	yes	complemented
38	C₇⋊C₉	—	3	1	no	—
39	C₇⋊C₉	—	3	2	no	—
40	C₇⋊C₉	—	3	2	no	—
41	C₇⋊C₉	—	3	2	no	—
42	C₇⋊C₉	3₋¹⁺²	1	3	yes	complemented
43	C₃³	—	7	1	no	—
44	3₋¹⁺²	—	21	3	no	—
45	3₋¹⁺²	—	21	3	no	—
46	C₃×C₉	—	7	1	no	—
47	C₃×C₉	—	7	2	no	—
48	C₃×C₉	—	7	2	no	—
49	C₃×C₉	—	7	2	no	—
50	C₃×C₉	—	7	1	no	—
51	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	2	yes	—
52	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
53	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
54	C₃×C₇	C₃×3₋¹⁺²	1	1	yes	characteristic
55	C₃×C₇	C₃×3₋¹⁺²	1	1	yes	characteristic
56	C₃×C₇	#737	1	2	yes	—
57	ΓL(1,𝔽₈)	—	9	1	no	—
58	C₃²	—	21	1	no	—
59	C₃²	—	21	1	no	—
60	C₃²	—	21	2	no	—
61	C₃²	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
62	C₉	—	3	1	no	—
63	C₉	—	3	2	no	—
64	C₉	—	7	3	no	—
65	C₉	—	21	2	no	—
66	C₉	—	21	1	no	—
67	C₉	—	21	2	no	—
68	C₉	—	21	2	no	—
69	C₉	#4427	1	3	yes	complemented
70	C₇	#6896	1	1	yes	characteristic, complemented, Sylow
71	C₃	—	63	1	no	—
72	C₃	C₃×#4427	1	1	yes	characteristic
73	C₃	ΓL(1,𝔽₈)×3₋¹⁺²	1	1	yes	characteristic
74	C₃	#10588065	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×ΓL(1,𝔽₈)
commutator, socle, cosocle	C₃²×C₇	C₃³
Fitting	C₇×C₉⋊C₉	C₃
prime radical	C₃²×ΓL(1,𝔽₈)	C₃²
hypercenter	C₉⋊C₉	ΓL(1,𝔽₈)
hypocenter	C₇	#6896
LCS	G ▷ C₃²×C₇ ▷ C₇
UCS	C₁ ◁ C₃² ◁ C₉⋊C₉

Complements

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

N	H ≅ G/N	type
3. C₇×C₉⋊C₉	C₃	semidirect
5. #10588035	C₃	semidirect
6. #10588035	C₃	semidirect
7. (C₇⋊C₉)⋊C₉	C₃	semidirect
8. (C₇⋊C₉)⋊C₉	C₃	semidirect
9. C₉×C₇⋊C₉	C₃	semidirect
28. C₉⋊C₉	ΓL(1,𝔽₈)	semidirect
37. C₇×C₉	3₋¹⁺²	semidirect
42. C₇⋊C₉	3₋¹⁺²	semidirect
69. C₉	#4427	semidirect
70. 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×C₉⋊C₉	3
(C₃×ΓL(1,𝔽₈))⋊C₉	3
#10588035	3
(C₇⋊C₉)⋊C₉	3
C₉×C₇⋊C₉	3
#6896	7