Subgroups — #15691672

Order 3159.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11760454	C₃	1	2	yes	maximal, complemented
2	#11760452	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×#66851	C₃	1	1	yes	maximal, characteristic, complemented
4	#11760442	C₃	1	2	yes	maximal, complemented
5	#11760440	C₃	1	1	yes	maximal, characteristic, complemented
6	#11760439	C₃	1	2	yes	maximal, complemented
7	#11760437	C₃	1	1	yes	maximal, characteristic, complemented
8	#11760435	C₃	1	2	yes	maximal, complemented
9	C₁₃×AΣL(1,𝔽₂₇)	C₃	1	1	yes	maximal, characteristic, complemented
10	C₃³×C₁₃	C₃²	1	1	yes	characteristic
11	C₁₃×3₋¹⁺²	C₃²	1	2	yes	—
12	C₁₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
13	C₃²⋊(C₁₃⋊C₃)	C₃²	1	2	yes	—
14	#66851	—	3	2	no	—
15	#66851	—	3	1	no	—
16	#66851	—	3	2	no	—
17	#66851	C₃²	1	2	yes	—
18	#66851	C₃²	1	2	yes	—
19	C₃×C₁₃⋊C₉	—	3	1	no	—
20	C₃×C₁₃⋊C₉	C₃²	1	1	yes	characteristic
21	C₃×C₁₃⋊C₉	C₃²	1	1	yes	characteristic
22	C₃×C₁₃⋊C₉	C₃²	1	1	yes	characteristic
23	#66849	—	3	2	no	—
24	#66848	—	3	2	no	—
25	C₉×C₁₃⋊C₃	—	3	2	no	—
26	#6913	—	13	1	no	maximal, Sylow
27	C₃²×C₁₃	—	3	1	no	—
28	C₃²×C₁₃	—	3	1	no	—
29	C₃²×C₁₃	—	9	1	no	—
30	C₃²×C₁₃	C₃³	1	1	yes	characteristic
31	C₃×C₁₃⋊C₃	—	3	2	no	—
32	C₉×C₁₃	—	3	2	no	—
33	C₁₃⋊C₉	—	3	2	no	—
34	C₁₃⋊C₉	—	3	2	no	—
35	C₁₃⋊C₉	—	3	1	no	—
36	C₁₃⋊C₉	—	3	1	no	—
37	C₁₃⋊C₉	H(3,𝔽₃)	1	2	yes	—
38	C₁₃⋊C₉	H(3,𝔽₃)	1	1	yes	characteristic
39	#737	—	13	2	no	—
40	#737	—	13	1	no	—
41	C₃×3₋¹⁺²	—	13	1	no	—
42	#733	—	13	2	no	—
43	#732	—	13	1	no	—
44	#731	—	13	2	no	—
45	#731	—	13	1	no	—
46	AΣL(1,𝔽₂₇)	—	13	2	no	—
47	AΣL(1,𝔽₂₇)	C₁₃⋊C₃	1	1	yes	characteristic, complemented
48	C₃×C₁₃	—	3	1	no	—
49	C₃×C₁₃	—	9	1	no	—
50	C₃×C₁₃	—	9	1	no	—
51	C₃×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
52	C₁₃⋊C₃	—	9	2	no	—
53	C₃³	C₃×C₁₃⋊C₃	1	1	yes	characteristic
54	3₋¹⁺²	—	13	2	no	—
55	3₋¹⁺²	—	13	2	no	—
56	3₋¹⁺²	—	39	2	no	—
57	3₋¹⁺²	—	39	2	no	—
58	3₋¹⁺²	—	39	2	no	—
59	3₋¹⁺²	—	39	1	no	—
60	3₋¹⁺²	—	39	2	no	—
61	3₋¹⁺²	C₃×C₁₃⋊C₃	1	2	yes	—
62	H(3,𝔽₃)	—	13	2	no	—
63	H(3,𝔽₃)	C₃×C₁₃⋊C₃	1	1	yes	characteristic
64	C₃×C₉	—	13	1	no	—
65	C₃×C₉	—	13	1	no	—
66	C₃×C₉	—	13	1	no	—
67	C₃×C₉	—	39	2	no	—
68	C₃×C₉	—	39	1	no	—
69	C₁₃	#6913	1	1	yes	characteristic, complemented, Sylow
70	C₃²	—	3	1	no	—
71	C₃²	—	3	1	no	—
72	C₃²	—	9	1	no	—
73	C₃²	—	39	2	no	—
74	C₃²	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
75	C₉	—	3	2	no	—
76	C₉	—	13	2	no	—
77	C₉	—	13	1	no	—
78	C₉	—	39	1	no	—
79	C₉	—	39	1	no	—
80	C₉	—	39	2	no	—
81	C₉	—	39	2	no	—
82	C₃	—	3	1	no	—
83	C₃	—	9	1	no	—
84	C₃	—	9	1	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	#6913
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. #11760454	C₃	semidirect
2. #11760452	C₃	semidirect
3. C₃×#66851	C₃	semidirect
4. #11760442	C₃	semidirect
5. #11760440	C₃	semidirect
6. #11760439	C₃	semidirect
7. #11760437	C₃	semidirect
8. #11760435	C₃	semidirect
9. C₁₃×AΣL(1,𝔽₂₇)	C₃	semidirect
47. AΣL(1,𝔽₂₇)	C₁₃⋊C₃	semidirect
69. C₁₃	#6913	semidirect

Maximal subgroups

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

subgroup	index
#11760454	3
#11760452	3
C₃×#66851	3
#11760442	3
#11760440	3
#11760439	3
#11760437	3
#11760435	3
C₁₃×AΣL(1,𝔽₂₇)	3
#6913	13