Subgroups — #15691649

Order 3159.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11760452	C₃	1	3	yes	maximal, complemented
2	C₃×C₉×C₁₃⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
3	#11760443	C₃	1	2	yes	maximal, complemented
4	#11760437	C₃	1	3	yes	maximal, complemented
5	#11760434	C₃	1	3	yes	maximal, complemented
6	C₁₃×#732	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃²×C₁₃⋊C₃	C₃²	1	1	yes	characteristic
8	C₁₃×H(3,𝔽₃)	C₃²	1	3	yes	—
9	C₃×C₉×C₁₃	C₃²	1	1	yes	characteristic
10	#66851	—	3	6	no	—
11	#66851	C₃²	1	6	yes	—
12	C₃×C₁₃⋊C₉	—	3	3	no	—
13	C₃×C₁₃⋊C₉	C₃²	1	1	yes	characteristic
14	C₃×C₁₃⋊C₉	C₃²	1	1	yes	characteristic
15	C₉×C₁₃⋊C₃	—	3	1	no	—
16	C₉×C₁₃⋊C₃	—	3	2	no	—
17	#6908	—	13	1	no	maximal, Sylow
18	C₃²×C₁₃	—	3	3	no	—
19	C₃²×C₁₃	C₃³	1	1	yes	characteristic
20	C₃×C₁₃⋊C₃	—	3	1	no	—
21	C₃×C₁₃⋊C₃	—	3	3	no	—
22	C₉×C₁₃	—	3	1	no	—
23	C₁₃⋊C₉	—	3	1	no	—
24	C₁₃⋊C₉	—	3	6	no	—
25	C₁₃⋊C₉	H(3,𝔽₃)	1	1	yes	characteristic
26	C₁₃⋊C₉	H(3,𝔽₃)	1	2	yes	—
27	#737	—	13	3	no	—
28	C₃²×C₉	—	13	1	no	—
29	#733	—	13	2	no	—
30	#732	C₁₃⋊C₃	1	1	yes	characteristic, complemented
31	#731	—	13	3	no	—
32	AΣL(1,𝔽₂₇)	—	13	3	no	—
33	C₃×C₁₃	—	3	1	no	—
34	C₃×C₁₃	—	9	3	no	—
35	C₃×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
36	C₁₃⋊C₃	—	3	3	no	—
37	C₃³	—	13	1	no	—
38	3₋¹⁺²	—	13	6	no	—
39	3₋¹⁺²	—	39	6	no	—
40	H(3,𝔽₃)	C₃×C₁₃⋊C₃	1	3	yes	—
41	C₃×C₉	—	13	1	no	—
42	C₃×C₉	—	13	1	no	—
43	C₃×C₉	—	39	1	no	—
44	C₃×C₉	—	39	2	no	—
45	C₃×C₉	—	39	3	no	—
46	C₃×C₉	C₃×C₁₃⋊C₃	1	1	yes	characteristic
47	C₁₃	#6908	1	1	yes	characteristic, complemented, Sylow
48	C₃²	—	3	3	no	—
49	C₃²	—	39	3	no	—
50	C₃²	—	39	1	no	—
51	C₃²	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
52	C₉	—	3	1	no	—
53	C₉	—	13	1	no	—
54	C₉	—	13	2	no	—
55	C₉	—	39	1	no	—
56	C₉	—	39	6	no	—
57	C₃	—	3	1	no	—
58	C₃	—	9	3	no	—
59	C₃	—	39	3	no	—
60	C₃	H(3,𝔽₃)×C₁₃⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃	H(3,𝔽₃)×C₁₃⋊C₃
commutator, cosocle	C₃²×C₁₃	C₃³
Frattini	C₃²	C₃²×C₁₃⋊C₃
Fitting	C₁₃×#732	C₃
socle	C₃×C₁₃	C₃×H(3,𝔽₃)
hypercenter	#732	C₁₃⋊C₃
hypocenter	C₁₃	#6908
LCS	G ▷ C₃²×C₁₃ ▷ C₃×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₃ ◁ C₃² ◁ #732

Complements

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

N	H ≅ G/N	type
1. #11760452	C₃	semidirect
2. C₃×C₉×C₁₃⋊C₃	C₃	semidirect
3. #11760443	C₃	semidirect
4. #11760437	C₃	semidirect
5. #11760434	C₃	semidirect
6. C₁₃×#732	C₃	semidirect
30. #732	C₁₃⋊C₃	semidirect
47. C₁₃	#6908	semidirect

Maximal subgroups

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

subgroup	index
#11760452	3
C₃×C₉×C₁₃⋊C₃	3
#11760443	3
#11760437	3
#11760434	3
C₁₃×#732	3
#6908	13