Subgroups — #13133560

Order 1701.

124 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
2	C₃²×C₇⋊C₉	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊(C₇⋊C₉)	C₃	1	2	yes	maximal
4	(C₇⋊C₉)⋊C₉	C₃	1	6	yes	maximal, complemented
5	(C₇⋊C₉)⋊C₉	C₃	1	3	yes	maximal, complemented
6	#6894	—	7	1	no	maximal, Sylow
7	C₃³×C₇	C₃²	1	1	yes	characteristic
8	C₇×3₋¹⁺²	—	3	3	no	—
9	C₃×C₇×C₉	C₃²	1	3	yes	—
10	C₃×C₇⋊C₉	—	3	3	no	—
11	C₃×C₇⋊C₉	C₃²	1	6	yes	—
12	C₃×C₇⋊C₉	C₃²	1	2	yes	—
13	C₃×C₇⋊C₉	C₃²	1	1	yes	characteristic
14	C₃×3₋¹⁺²	ΓL(1,𝔽₈)	1	1	yes	characteristic
15	C₃²×C₉	—	7	1	no	—
16	C₉⋊C₉	—	7	3	no	—
17	C₉⋊C₉	—	7	6	no	—
18	C₃²⋊C₉	—	7	2	no	—
19	C₃²×C₇	—	3	1	no	—
20	C₃²×C₇	—	3	2	no	—
21	C₃²×C₇	3₋¹⁺²	1	3	yes	—
22	C₃²×C₇	C₃³	1	1	yes	characteristic
23	C₇×C₉	—	3	3	no	—
24	C₇⋊C₉	—	3	2	no	—
25	C₇⋊C₉	—	3	6	no	—
26	C₇⋊C₉	3₋¹⁺²	1	3	yes	complemented
27	C₃³	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
28	3₋¹⁺²	—	3	3	no	—
29	C₃×C₉	—	7	1	no	—
30	C₃×C₉	—	7	6	no	—
31	C₃×C₉	—	7	2	no	—
32	C₃×C₉	—	21	3	no	—
33	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	3	yes	—
34	C₃×C₇	—	3	3	no	—
35	C₃×C₇	C₃×3₋¹⁺²	1	1	yes	characteristic
36	C₃×C₇	C₃×3₋¹⁺²	1	1	yes	characteristic
37	C₃×C₇	#737	1	2	yes	—
38	C₃²	—	3	1	no	—
39	C₃²	—	3	2	no	—
40	C₃²	#4430	1	3	yes	—
41	C₃²	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
42	C₉	—	3	3	no	—
43	C₉	—	7	3	no	—
44	C₉	—	21	2	no	—
45	C₉	—	21	6	no	—
46	C₇	#6894	1	1	yes	characteristic, complemented, Sylow
47	C₃	—	3	3	no	—
48	C₃	ΓL(1,𝔽₈)×3₋¹⁺²	1	1	yes	characteristic
49	C₃	#10588065	1	2	yes	—
50	C₃	C₃×#4430	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	C₃³×C₇	C₃²
hypercenter	C₃×3₋¹⁺²	ΓL(1,𝔽₈)
hypocenter	C₇	#6894
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
4. (C₇⋊C₉)⋊C₉	C₃	semidirect
5. (C₇⋊C₉)⋊C₉	C₃	semidirect
26. C₇⋊C₉	3₋¹⁺²	semidirect
46. C₇	#6894	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
C₃²×C₇⋊C₉	3
C₃²⋊(C₇⋊C₉)	3
(C₇⋊C₉)⋊C₉	3
#6894	7