Subgroups — #15691676

Order 3159.

126 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×#66849	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	#66849	—	3	3	no	—
16	#6913	—	13	1	no	maximal, Sylow
17	C₃²×C₁₃	—	3	3	no	—
18	C₃²×C₁₃	C₃³	1	1	yes	characteristic
19	C₃×C₁₃⋊C₃	—	3	1	no	—
20	C₃×C₁₃⋊C₃	—	9	1	no	—
21	C₉×C₁₃	—	3	1	no	—
22	C₁₃⋊C₉	—	3	1	no	—
23	C₁₃⋊C₉	—	3	6	no	—
24	C₁₃⋊C₉	H(3,𝔽₃)	1	3	yes	—
25	#737	—	13	3	no	—
26	C₃×3₋¹⁺²	—	13	1	no	—
27	#733	—	13	2	no	—
28	#732	C₁₃⋊C₃	1	1	yes	characteristic, complemented
29	#731	—	13	3	no	—
30	AΣL(1,𝔽₂₇)	—	13	3	no	—
31	C₃×C₁₃	—	3	1	no	—
32	C₃×C₁₃	—	9	3	no	—
33	C₃×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
34	C₁₃⋊C₃	—	9	1	no	—
35	C₃³	—	13	1	no	—
36	3₋¹⁺²	—	13	6	no	—
37	3₋¹⁺²	—	39	3	no	—
38	3₋¹⁺²	—	39	6	no	—
39	H(3,𝔽₃)	C₃×C₁₃⋊C₃	1	3	yes	—
40	C₃×C₉	—	13	1	no	—
41	C₃×C₉	—	13	1	no	—
42	C₃×C₉	—	39	3	no	—
43	C₃×C₉	C₃×C₁₃⋊C₃	1	1	yes	characteristic
44	C₁₃	#6913	1	1	yes	characteristic, complemented, Sylow
45	C₃²	—	3	3	no	—
46	C₃²	—	39	1	no	—
47	C₃²	—	117	1	no	—
48	C₃²	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
49	C₉	—	3	1	no	—
50	C₉	—	13	3	no	—
51	C₉	—	39	1	no	—
52	C₉	—	39	6	no	—
53	C₃	—	3	1	no	—
54	C₃	—	9	3	no	—
55	C₃	—	117	1	no	—
56	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	#6913
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×#66849	C₃	semidirect
3. #11760443	C₃	semidirect
4. #11760437	C₃	semidirect
5. #11760434	C₃	semidirect
6. C₁₃×#732	C₃	semidirect
28. #732	C₁₃⋊C₃	semidirect
44. C₁₃	#6913	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×#66849	3
#11760443	3
#11760437	3
#11760434	3
C₁₃×#732	3
#6913	13