Subgroups — #15691651

Order 3159.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11760453	C₃	1	3	yes	maximal, complemented
2	#11760443	C₃	1	2	yes	maximal, complemented
3	#11760441	C₃	1	1	yes	maximal, characteristic, complemented
4	#11760438	C₃	1	3	yes	maximal, complemented
5	#11760435	C₃	1	3	yes	maximal, complemented
6	C₃²×C₉×C₁₃	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃³×C₁₃	C₃²	1	1	yes	characteristic
8	C₃×C₉×C₁₃	—	3	1	no	—
9	C₃×C₉×C₁₃	—	3	1	no	—
10	C₃×C₉×C₁₃	—	3	1	no	—
11	C₃×C₉×C₁₃	C₃²	1	1	yes	characteristic
12	C₃×C₉×C₁₃	C₃²	1	1	yes	characteristic
13	C₃×C₉×C₁₃	C₃²	1	1	yes	characteristic
14	C₃²⋊(C₁₃⋊C₃)	C₃²	1	3	yes	—
15	#66851	C₃²	1	6	yes	—
16	#66849	—	3	3	no	—
17	#66848	—	3	3	no	—
18	C₉×C₁₃⋊C₃	—	3	3	no	—
19	#6908	—	13	1	no	maximal, Sylow
20	C₃²×C₁₃	—	3	3	no	—
21	C₃²×C₁₃	—	3	1	no	—
22	C₃²×C₁₃	C₃³	1	1	yes	characteristic
23	C₃×C₁₃⋊C₃	—	3	3	no	—
24	C₉×C₁₃	—	3	1	no	—
25	C₉×C₁₃	—	3	1	no	—
26	C₉×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
27	C₉×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
28	C₉×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
29	C₁₃⋊C₉	—	3	6	no	—
30	#737	—	13	3	no	—
31	C₃²×C₉	C₁₃⋊C₃	1	1	yes	characteristic, complemented
32	#733	—	13	2	no	—
33	#732	—	13	1	no	—
34	#731	—	13	3	no	—
35	AΣL(1,𝔽₂₇)	—	13	3	no	—
36	C₃×C₁₃	—	3	3	no	—
37	C₃×C₁₃	—	3	1	no	—
38	C₃×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
39	C₁₃⋊C₃	—	9	3	no	—
40	C₃³	C₃×C₁₃⋊C₃	1	1	yes	characteristic
41	3₋¹⁺²	—	13	6	no	—
42	3₋¹⁺²	—	39	3	no	—
43	3₋¹⁺²	—	39	3	no	—
44	H(3,𝔽₃)	—	13	3	no	—
45	C₃×C₉	—	3	1	no	—
46	C₃×C₉	—	3	1	no	—
47	C₃×C₉	—	3	1	no	—
48	C₃×C₉	—	39	3	no	—
49	C₃×C₉	C₃×C₁₃⋊C₃	1	1	yes	characteristic
50	C₃×C₉	C₃×C₁₃⋊C₃	1	1	yes	characteristic
51	C₃×C₉	C₃×C₁₃⋊C₃	1	1	yes	characteristic
52	C₁₃	#6908	1	1	yes	characteristic, complemented, Sylow
53	C₃²	—	3	3	no	—
54	C₃²	—	3	1	no	—
55	C₃²	—	39	3	no	—
56	C₃²	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
57	C₉	—	3	1	no	—
58	C₉	—	3	1	no	—
59	C₉	—	39	6	no	—
60	C₉	C₃²⋊(C₁₃⋊C₃)	1	1	yes	characteristic
61	C₉	C₃²⋊(C₁₃⋊C₃)	1	1	yes	characteristic
62	C₉	C₃²⋊(C₁₃⋊C₃)	1	1	yes	characteristic
63	C₃	—	3	1	no	—
64	C₃	—	3	3	no	—
65	C₃	—	117	3	no	—
66	C₃	C₃×C₃²⋊(C₁₃⋊C₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₉	C₃²⋊(C₁₃⋊C₃)
commutator, cosocle	C₃²×C₁₃	C₃³
Frattini	C₃²	C₃²×C₁₃⋊C₃
Fitting	C₃²×C₉×C₁₃	C₃
socle	C₃×C₁₃	C₃×H(3,𝔽₃)
hypercenter	C₃²×C₉	C₁₃⋊C₃
hypocenter	C₁₃	#6908
LCS	G ▷ C₃²×C₁₃ ▷ C₃×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₉ ◁ C₃×C₉ ◁ C₃²×C₉

Complements

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

N	H ≅ G/N	type
1. #11760453	C₃	semidirect
2. #11760443	C₃	semidirect
3. #11760441	C₃	semidirect
4. #11760438	C₃	semidirect
5. #11760435	C₃	semidirect
6. C₃²×C₉×C₁₃	C₃	semidirect
31. C₃²×C₉	C₁₃⋊C₃	semidirect
52. C₁₃	#6908	semidirect

Maximal subgroups

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

subgroup	index
#11760453	3
#11760443	3
#11760441	3
#11760438	3
#11760435	3
C₃²×C₉×C₁₃	3
#6908	13