Subgroups — #13133528

Order 1701.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×ΓL(1,𝔽₈)	C₃	1	1	yes	maximal, characteristic
2	C₇×C₉²	C₃	1	1	yes	maximal, characteristic, complemented
3	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃	1	3	yes	maximal
4	(C₇⋊C₉)⋊C₉	C₃	1	6	yes	maximal, complemented
5	C₉×C₇⋊C₉	C₃	1	2	yes	maximal, complemented
6	#6887	—	7	1	no	maximal, Sylow
7	C₃²×ΓL(1,𝔽₈)	C₃²	1	1	yes	characteristic
8	C₃×C₇×C₉	C₃²	1	3	yes	—
9	C₃×C₇×C₉	C₃²	1	1	yes	characteristic
10	C₃×C₇⋊C₉	C₃²	1	6	yes	—
11	C₃×C₇⋊C₉	C₃²	1	2	yes	—
12	C₉×ΓL(1,𝔽₈)	C₉	1	9	yes	complemented
13	C₃²×C₉	—	7	1	no	—
14	C₉⋊C₉	—	7	6	no	—
15	C₃²⋊C₉	—	7	3	no	—
16	C₉²	—	7	2	no	—
17	C₉²	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
18	C₃²×C₇	C₃³	1	1	yes	characteristic
19	C₃×ΓL(1,𝔽₈)	—	3	3	no	—
20	C₃×ΓL(1,𝔽₈)	C₃×C₉	1	3	yes	—
21	C₇×C₉	—	3	3	no	—
22	C₇×C₉	C₃×C₉	1	3	yes	—
23	C₇⋊C₉	—	3	6	no	—
24	C₇⋊C₉	C₃×C₉	1	6	yes	—
25	C₃³	—	7	1	no	—
26	C₃×C₉	—	7	2	no	—
27	C₃×C₉	—	7	9	no	—
28	C₃×C₉	—	7	6	no	—
29	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	3	yes	—
30	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
31	C₃×C₇	C₃²×C₉	1	1	yes	characteristic
32	C₃×C₇	#737	1	3	yes	—
33	ΓL(1,𝔽₈)	—	3	3	no	—
34	C₃²	—	7	3	no	—
35	C₃²	—	21	3	no	—
36	C₃²	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
37	C₉	—	3	3	no	—
38	C₉	—	7	6	no	—
39	C₉	—	21	6	no	—
40	C₉	C₉×ΓL(1,𝔽₈)	1	3	yes	—
41	C₇	#6887	1	1	yes	characteristic, complemented, Sylow
42	C₃	—	21	3	no	—
43	C₃	C₃×C₉×ΓL(1,𝔽₈)	1	1	yes	characteristic
44	C₃	#10588061	1	3	yes	—

	subgroup	G/H
center	C₃×C₉	C₃×ΓL(1,𝔽₈)
commutator	C₃×C₇	C₃²×C₉
Frattini	C₃²	C₃²×ΓL(1,𝔽₈)
Fitting	C₇×C₉²	C₃
socle, cosocle	C₃²×C₇	C₃³
prime radical	C₃²×ΓL(1,𝔽₈)	C₃²
hypercenter	C₉²	ΓL(1,𝔽₈)
hypocenter	C₇	#6887
LCS	G ▷ C₃×C₇ ▷ C₇
UCS	C₁ ◁ C₃×C₉ ◁ C₉²

Complements

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

N	H ≅ G/N	type
2. C₇×C₉²	C₃	semidirect
4. (C₇⋊C₉)⋊C₉	C₃	semidirect
5. C₉×C₇⋊C₉	C₃	semidirect
12. C₉×ΓL(1,𝔽₈)	C₉	semidirect
17. C₉²	ΓL(1,𝔽₈)	semidirect
41. C₇	#6887	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×ΓL(1,𝔽₈)	3
C₇×C₉²	3
(C₃×ΓL(1,𝔽₈))⋊C₉	3
(C₇⋊C₉)⋊C₉	3
C₉×C₇⋊C₉	3
#6887	7