Subgroups — #15691654

Order 3159.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	3₋¹⁺²×C₁₃⋊C₃	C₃	1	2	yes	maximal, complemented
2	C₃×C₃²⋊(C₁₃⋊C₃)	C₃	1	1	yes	maximal, characteristic, complemented
3	H(3,𝔽₃)×C₁₃⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
4	#11760436	C₃	1	4	yes	maximal, complemented
5	#11760435	C₃	1	2	yes	maximal, complemented
6	#11760434	C₃	1	2	yes	maximal, complemented
7	C₁₃×AΣL(1,𝔽₂₇)	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃³×C₁₃	C₃²	1	1	yes	characteristic, complemented
9	C₃²×C₁₃⋊C₃	—	3	1	no	—
10	C₃²×C₁₃⋊C₃	C₃²	1	1	yes	characteristic
11	C₃²×C₁₃⋊C₃	C₃²	1	2	yes	complemented
12	C₁₃×3₋¹⁺²	C₃²	1	2	yes	complemented
13	C₁₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
14	C₃²⋊(C₁₃⋊C₃)	—	3	2	no	—
15	C₃²⋊(C₁₃⋊C₃)	—	3	2	no	—
16	C₃²⋊(C₁₃⋊C₃)	—	3	1	no	—
17	C₃²⋊(C₁₃⋊C₃)	C₃²	1	2	yes	—
18	#66851	C₃²	1	4	yes	complemented
19	#66849	—	3	2	no	—
20	#66848	—	3	2	no	—
21	C₉×C₁₃⋊C₃	—	3	2	no	—
22	#6909	—	13	1	no	maximal, Sylow
23	C₃²×C₁₃	—	3	1	no	—
24	C₃²×C₁₃	—	3	1	no	—
25	C₃²×C₁₃	—	9	1	no	—
26	C₃²×C₁₃	C₃³	1	1	yes	characteristic
27	C₃×C₁₃⋊C₃	—	3	2	no	—
28	C₃×C₁₃⋊C₃	—	3	2	no	—
29	C₃×C₁₃⋊C₃	—	9	1	no	—
30	C₃×C₁₃⋊C₃	—	9	3	no	—
31	C₃×C₁₃⋊C₃	—	9	2	no	—
32	C₃×C₁₃⋊C₃	H(3,𝔽₃)	1	2	yes	—
33	C₃×C₁₃⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
34	C₉×C₁₃	—	3	2	no	—
35	C₁₃⋊C₉	—	3	4	no	—
36	C₃×3₋¹⁺²	—	13	2	no	—
37	C₃×H(3,𝔽₃)	—	13	1	no	—
38	C₃×H(3,𝔽₃)	—	13	1	no	—
39	AΣL(1,𝔽₂₇)	—	13	2	no	—
40	AΣL(1,𝔽₂₇)	—	13	4	no	—
41	AΣL(1,𝔽₂₇)	—	13	2	no	—
42	AΣL(1,𝔽₂₇)	C₁₃⋊C₃	1	1	yes	characteristic, complemented
43	C₃×C₁₃	—	3	1	no	—
44	C₃×C₁₃	—	9	1	no	—
45	C₃×C₁₃	—	9	1	no	—
46	C₃×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
47	C₁₃⋊C₃	—	3	1	no	—
48	C₁₃⋊C₃	—	3	2	no	—
49	C₁₃⋊C₃	—	9	2	no	—
50	C₁₃⋊C₃	—	9	2	no	—
51	C₃³	—	13	1	no	—
52	C₃³	—	13	2	no	—
53	C₃³	—	39	1	no	—
54	C₃³	C₃×C₁₃⋊C₃	1	1	yes	characteristic, complemented
55	3₋¹⁺²	—	13	4	no	—
56	3₋¹⁺²	—	39	2	no	—
57	3₋¹⁺²	—	39	2	no	—
58	3₋¹⁺²	C₃×C₁₃⋊C₃	1	2	yes	complemented
59	H(3,𝔽₃)	—	13	2	no	—
60	H(3,𝔽₃)	—	39	2	no	—
61	H(3,𝔽₃)	—	39	1	no	—
62	H(3,𝔽₃)	—	39	2	no	—
63	H(3,𝔽₃)	C₃×C₁₃⋊C₃	1	1	yes	characteristic
64	C₃×C₉	—	39	2	no	—
65	C₁₃	#6909	1	1	yes	characteristic, complemented, Sylow
66	C₃²	—	3	1	no	—
67	C₃²	—	3	1	no	—
68	C₃²	—	9	1	no	—
69	C₃²	—	13	2	no	—
70	C₃²	—	13	1	no	—
71	C₃²	—	39	2	no	—
72	C₃²	—	39	2	no	—
73	C₃²	—	117	1	no	—
74	C₃²	—	117	2	no	—
75	C₃²	—	117	3	no	—
76	C₃²	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
77	C₉	—	3	2	no	—
78	C₉	—	39	4	no	—
79	C₃	—	3	1	no	—
80	C₃	—	9	1	no	—
81	C₃	—	9	1	no	—
82	C₃	—	39	1	no	—
83	C₃	—	39	2	no	—
84	C₃	—	117	2	no	—
85	C₃	—	117	2	no	—
86	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×AΣL(1,𝔽₂₇)	C₃
socle	C₃×C₁₃	C₃×H(3,𝔽₃)
hypercenter	AΣL(1,𝔽₂₇)	C₁₃⋊C₃
hypocenter	C₁₃	#6909
LCS	G ▷ C₃²×C₁₃ ▷ C₃×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₃ ◁ C₃² ◁ AΣL(1,𝔽₂₇)

Complements

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

N	H ≅ G/N	type
1. 3₋¹⁺²×C₁₃⋊C₃	C₃	semidirect
2. C₃×C₃²⋊(C₁₃⋊C₃)	C₃	semidirect
3. H(3,𝔽₃)×C₁₃⋊C₃	C₃	semidirect
4. #11760436	C₃	semidirect
5. #11760435	C₃	semidirect
6. #11760434	C₃	semidirect
7. C₁₃×AΣL(1,𝔽₂₇)	C₃	semidirect
8. C₃³×C₁₃	C₃²	semidirect
11. C₃²×C₁₃⋊C₃	C₃²	semidirect
12. C₁₃×3₋¹⁺²	C₃²	semidirect
18. #66851	C₃²	semidirect
42. AΣL(1,𝔽₂₇)	C₁₃⋊C₃	semidirect
54. C₃³	C₃×C₁₃⋊C₃	semidirect
58. 3₋¹⁺²	C₃×C₁₃⋊C₃	semidirect
65. C₁₃	#6909	semidirect

Maximal subgroups

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

subgroup	index
3₋¹⁺²×C₁₃⋊C₃	3
C₃×C₃²⋊(C₁₃⋊C₃)	3
H(3,𝔽₃)×C₁₃⋊C₃	3
#11760436	3
#11760435	3
#11760434	3
C₁₃×AΣL(1,𝔽₂₇)	3
#6909	13