Subgroups — #13133556

Order 1701.

156 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, complemented
2	C₃×C₇⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₃×#4430	C₃	1	2	yes	maximal, complemented
4	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃	1	9	yes	maximal, complemented
5	#6893	—	7	1	no	maximal, Sylow
6	C₃³×C₇	C₃²	1	1	yes	characteristic
7	C₃²×ΓL(1,𝔽₈)	C₃²	1	3	yes	—
8	C₃×C₇×C₉	—	3	3	no	—
9	C₃×C₇×C₉	C₃²	1	3	yes	—
10	C₇⋊H(3,𝔽₃)	—	3	3	no	—
11	#4430	—	3	6	no	—
12	C₃×C₇⋊C₉	C₃²	1	6	yes	—
13	C₃×3₋¹⁺²	—	7	2	no	—
14	C₃×H(3,𝔽₃)	—	7	1	no	—
15	C₃²×C₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
16	C₃²⋊C₉	—	7	9	no	—
17	C₃²×C₇	—	3	3	no	—
18	C₃²×C₇	H(3,𝔽₃)	1	3	yes	—
19	C₃²×C₇	C₃³	1	1	yes	characteristic
20	C₃×ΓL(1,𝔽₈)	—	3	3	no	—
21	C₃×ΓL(1,𝔽₈)	—	3	9	no	—
22	C₇×C₉	—	3	3	no	—
23	C₇⋊C₉	—	3	6	no	—
24	C₃³	—	7	3	no	—
25	C₃³	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
26	3₋¹⁺²	—	21	6	no	—
27	H(3,𝔽₃)	—	21	3	no	—
28	C₃×C₉	—	3	3	no	—
29	C₃×C₉	—	7	6	no	—
30	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	3	yes	—
31	C₃×C₇	—	3	3	no	—
32	C₃×C₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
33	C₃×C₇	#737	1	3	yes	—
34	ΓL(1,𝔽₈)	—	9	3	no	—
35	C₃²	—	3	3	no	—
36	C₃²	—	21	9	no	—
37	C₃²	—	21	3	no	—
38	C₃²	C₇⋊H(3,𝔽₃)	1	3	yes	—
39	C₃²	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
40	C₉	—	3	3	no	—
41	C₉	—	21	6	no	—
42	C₇	#6893	1	1	yes	characteristic, complemented, Sylow
43	C₃	—	3	3	no	—
44	C₃	—	63	3	no	—
45	C₃	#10588061	1	3	yes	—
46	C₃	C₃×C₇⋊H(3,𝔽₃)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃²×ΓL(1,𝔽₈)
commutator, socle, cosocle	C₃²×C₇	C₃³
Fitting	C₃²×C₇×C₉	C₃
prime radical	C₃×C₇⋊H(3,𝔽₃)	C₃
hypercenter	C₃²×C₉	ΓL(1,𝔽₈)
hypocenter	C₇	#6893
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
1. C₃²×C₇×C₉	C₃	semidirect
3. C₃×#4430	C₃	semidirect
4. (C₃×ΓL(1,𝔽₈))⋊C₉	C₃	semidirect
15. C₃²×C₉	ΓL(1,𝔽₈)	semidirect
42. C₇	#6893	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
C₃×C₇⋊H(3,𝔽₃)	3
C₃×#4430	3
(C₃×ΓL(1,𝔽₈))⋊C₉	3
#6893	7