Subgroups — #13133557

Order 1701.

156 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×C₇⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₃×#4430	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃²×C₇⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
5	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃	1	9	yes	maximal, complemented
6	#6893	—	7	1	no	maximal, Sylow
7	C₃³×C₇	C₃²	1	1	yes	characteristic
8	C₃²×ΓL(1,𝔽₈)	C₃²	1	3	yes	—
9	C₇×3₋¹⁺²	—	3	3	no	—
10	C₃×C₇×C₉	C₃²	1	3	yes	—
11	C₇⋊H(3,𝔽₃)	—	3	3	no	—
12	#4430	—	3	3	no	—
13	C₃×C₇⋊C₉	—	3	3	no	—
14	C₃×C₇⋊C₉	C₃²	1	3	yes	—
15	C₃×C₇⋊C₉	C₃²	1	3	yes	—
16	C₃×3₋¹⁺²	—	7	1	no	—
17	C₃×3₋¹⁺²	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
18	C₃×H(3,𝔽₃)	—	7	1	no	—
19	C₃²×C₉	—	7	1	no	—
20	C₃²⋊C₉	—	7	9	no	—
21	C₃²×C₇	—	3	3	no	—
22	C₃²×C₇	H(3,𝔽₃)	1	3	yes	—
23	C₃²×C₇	C₃³	1	1	yes	characteristic
24	C₃×ΓL(1,𝔽₈)	—	3	3	no	—
25	C₃×ΓL(1,𝔽₈)	—	3	9	no	—
26	C₇×C₉	—	3	3	no	—
27	C₇⋊C₉	—	3	3	no	—
28	C₇⋊C₉	—	3	3	no	—
29	C₃³	—	7	3	no	—
30	C₃³	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
31	3₋¹⁺²	—	3	3	no	—
32	3₋¹⁺²	—	21	3	no	—
33	H(3,𝔽₃)	—	21	3	no	—
34	C₃×C₉	—	7	3	no	—
35	C₃×C₉	—	7	3	no	—
36	C₃×C₉	—	21	3	no	—
37	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	3	yes	—
38	C₃×C₇	—	3	3	no	—
39	C₃×C₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
40	C₃×C₇	#737	1	3	yes	—
41	ΓL(1,𝔽₈)	—	9	3	no	—
42	C₃²	—	3	3	no	—
43	C₃²	—	21	9	no	—
44	C₃²	—	21	3	no	—
45	C₃²	C₇⋊H(3,𝔽₃)	1	3	yes	—
46	C₃²	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
47	C₉	—	3	3	no	—
48	C₉	—	21	3	no	—
49	C₉	—	21	3	no	—
50	C₇	#6893	1	1	yes	characteristic, complemented, Sylow
51	C₃	—	3	3	no	—
52	C₃	—	63	3	no	—
53	C₃	#10588065	1	3	yes	—
54	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	C₃×C₇×3₋¹⁺²	C₃
prime radical	C₃×C₇⋊H(3,𝔽₃)	C₃
hypercenter	C₃×3₋¹⁺²	ΓL(1,𝔽₈)
hypocenter	C₇	#6893
LCS	G ▷ C₃²×C₇ ▷ C₇
UCS	C₁ ◁ C₃² ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
1. C₃×C₇×3₋¹⁺²	C₃	semidirect
3. C₃×#4430	C₃	semidirect
4. C₃²×C₇⋊C₉	C₃	semidirect
5. (C₃×ΓL(1,𝔽₈))⋊C₉	C₃	semidirect
17. C₃×3₋¹⁺²	ΓL(1,𝔽₈)	semidirect
50. C₇	#6893	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₇×3₋¹⁺²	3
C₃×C₇⋊H(3,𝔽₃)	3
C₃×#4430	3
C₃²×C₇⋊C₉	3
(C₃×ΓL(1,𝔽₈))⋊C₉	3
#6893	7