Subgroups — #46798657

Order 33777.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26014630	C₃	1	2	yes	maximal, complemented
2	#26014628	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×#16836730	C₃	1	1	yes	maximal, characteristic, complemented
4	#26014618	C₃	1	2	yes	maximal, complemented
5	#26014616	C₃	1	1	yes	maximal, characteristic, complemented
6	#26014615	C₃	1	2	yes	maximal, complemented
7	#26014613	C₃	1	1	yes	maximal, characteristic, complemented
8	#26014611	C₃	1	2	yes	maximal, complemented
9	AΣL(1,𝔽₂₇)×C₁₃₉	C₃	1	1	yes	maximal, characteristic, complemented
10	#16836730	—	3	2	no	—
11	#16836730	—	3	1	no	—
12	#16836730	—	3	2	no	—
13	#16836730	C₃²	1	2	yes	—
14	#16836730	C₃²	1	2	yes	—
15	#16836729	—	3	2	no	—
16	#16836728	—	3	2	no	—
17	C₃²⋊(C₁₃₉⋊C₃)	C₃²	1	2	yes	—
18	C₃×C₁₃₉⋊C₉	—	3	1	no	—
19	C₃×C₁₃₉⋊C₉	C₃²	1	1	yes	characteristic
20	C₃×C₁₃₉⋊C₉	C₃²	1	1	yes	characteristic
21	C₃×C₁₃₉⋊C₉	C₃²	1	1	yes	characteristic
22	C₉×C₁₃₉⋊C₃	—	3	2	no	—
23	C₃³×C₁₃₉	C₃²	1	1	yes	characteristic
24	3₋¹⁺²×C₁₃₉	C₃²	1	2	yes	—
25	H(3,𝔽₃)×C₁₃₉	C₃²	1	1	yes	characteristic
26	C₃²×C₁₃₉	—	3	1	no	—
27	C₃²×C₁₃₉	—	3	1	no	—
28	C₃²×C₁₃₉	—	9	1	no	—
29	C₃²×C₁₃₉	C₃³	1	1	yes	characteristic
30	C₃×C₁₃₉⋊C₃	—	3	2	no	—
31	C₉×C₁₃₉	—	3	2	no	—
32	C₁₃₉⋊C₉	—	3	2	no	—
33	C₁₃₉⋊C₉	—	3	2	no	—
34	C₁₃₉⋊C₉	—	3	1	no	—
35	C₁₃₉⋊C₉	—	3	1	no	—
36	C₁₃₉⋊C₉	H(3,𝔽₃)	1	2	yes	—
37	C₁₃₉⋊C₉	H(3,𝔽₃)	1	1	yes	characteristic
38	C₃×C₁₃₉	—	3	1	no	—
39	C₃×C₁₃₉	—	9	1	no	—
40	C₃×C₁₃₉	—	9	1	no	—
41	C₃×C₁₃₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
42	C₁₃₉⋊C₃	—	9	2	no	—
43	#6913	—	139	1	no	maximal, Sylow
44	C₁₃₉	#6913	1	1	yes	characteristic, complemented, Sylow
45	#737	—	139	2	no	—
46	#737	—	139	1	no	—
47	C₃×3₋¹⁺²	—	139	1	no	—
48	#733	—	139	2	no	—
49	#732	—	139	1	no	—
50	#731	—	139	2	no	—
51	#731	—	139	1	no	—
52	AΣL(1,𝔽₂₇)	—	139	2	no	—
53	AΣL(1,𝔽₂₇)	C₁₃₉⋊C₃	1	1	yes	characteristic, complemented
54	C₃³	C₃×C₁₃₉⋊C₃	1	1	yes	characteristic
55	3₋¹⁺²	—	139	2	no	—
56	3₋¹⁺²	—	139	2	no	—
57	3₋¹⁺²	—	417	2	no	—
58	3₋¹⁺²	—	417	2	no	—
59	3₋¹⁺²	—	417	2	no	—
60	3₋¹⁺²	—	417	1	no	—
61	3₋¹⁺²	—	417	2	no	—
62	3₋¹⁺²	C₃×C₁₃₉⋊C₃	1	2	yes	—
63	H(3,𝔽₃)	—	139	2	no	—
64	H(3,𝔽₃)	C₃×C₁₃₉⋊C₃	1	1	yes	characteristic
65	C₃×C₉	—	139	1	no	—
66	C₃×C₉	—	139	1	no	—
67	C₃×C₉	—	139	1	no	—
68	C₃×C₉	—	417	2	no	—
69	C₃×C₉	—	417	1	no	—
70	C₃²	—	3	1	no	—
71	C₃²	—	3	1	no	—
72	C₃²	—	9	1	no	—
73	C₃²	—	417	2	no	—
74	C₃²	C₃²×C₁₃₉⋊C₃	1	1	yes	characteristic
75	C₉	—	3	2	no	—
76	C₉	—	139	2	no	—
77	C₉	—	139	1	no	—
78	C₉	—	417	1	no	—
79	C₉	—	417	1	no	—
80	C₉	—	417	2	no	—
81	C₉	—	417	2	no	—
82	C₃	—	3	1	no	—
83	C₃	—	9	1	no	—
84	C₃	—	9	1	no	—
85	C₃	—	1251	2	no	—
86	C₃	H(3,𝔽₃)×C₁₃₉⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃	H(3,𝔽₃)×C₁₃₉⋊C₃
commutator, cosocle	C₃²×C₁₃₉	C₃³
Frattini	C₃²	C₃²×C₁₃₉⋊C₃
Fitting	AΣL(1,𝔽₂₇)×C₁₃₉	C₃
socle	C₃×C₁₃₉	C₃×H(3,𝔽₃)
hypercenter	AΣL(1,𝔽₂₇)	C₁₃₉⋊C₃
hypocenter	C₁₃₉	#6913
LCS	G ▷ C₃²×C₁₃₉ ▷ C₃×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₃ ◁ C₃² ◁ AΣL(1,𝔽₂₇)

Complements

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

N	H ≅ G/N	type
1. #26014630	C₃	semidirect
2. #26014628	C₃	semidirect
3. C₃×#16836730	C₃	semidirect
4. #26014618	C₃	semidirect
5. #26014616	C₃	semidirect
6. #26014615	C₃	semidirect
7. #26014613	C₃	semidirect
8. #26014611	C₃	semidirect
9. AΣL(1,𝔽₂₇)×C₁₃₉	C₃	semidirect
44. C₁₃₉	#6913	semidirect
53. AΣL(1,𝔽₂₇)	C₁₃₉⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
#26014630	3
#26014628	3
C₃×#16836730	3
#26014618	3
#26014616	3
#26014615	3
#26014613	3
#26014611	3
AΣL(1,𝔽₂₇)×C₁₃₉	3
#6913	139