Subgroups — C₁₃₉×#6912 #46798435

Order 33777.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×3₋¹⁺²×C₁₃₉	C₃	1	3	yes	maximal
2	C₃×3₋¹⁺²×C₁₃₉	C₃	1	1	yes	maximal, characteristic
3	#733×C₁₃₉	C₃	1	9	yes	maximal, complemented
4	C₃³×C₁₃₉	C₃²	1	1	yes	characteristic
5	3₋¹⁺²×C₁₃₉	—	3	6	no	—
6	3₋¹⁺²×C₁₃₉	—	3	3	no	—
7	3₋¹⁺²×C₁₃₉	C₃²	1	9	yes	—
8	C₃×C₉×C₁₃₉	—	3	3	no	—
9	C₃×C₉×C₁₃₉	C₃²	1	3	yes	—
10	C₃²×C₁₃₉	—	9	1	no	—
11	C₃²×C₁₃₉	H(3,𝔽₃)	1	3	yes	—
12	C₃²×C₁₃₉	C₃³	1	1	yes	characteristic
13	C₉×C₁₃₉	—	3	9	no	—
14	C₉×C₁₃₉	—	3	3	no	—
15	C₃×C₁₃₉	—	3	3	no	—
16	C₃×C₁₃₉	—	3	1	no	—
17	C₃×C₁₃₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
18	#6912	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₁₃₉	#6912	1	1	yes	characteristic, direct product factor, Sylow
20	C₃×3₋¹⁺²	C₃×C₁₃₉	1	3	yes	—
21	C₃×3₋¹⁺²	C₃×C₁₃₉	1	1	yes	characteristic
22	#733	C₃×C₁₃₉	1	9	yes	complemented
23	C₃³	C₃²×C₁₃₉	1	1	yes	characteristic
24	3₋¹⁺²	—	3	6	no	—
25	3₋¹⁺²	—	3	3	no	—
26	3₋¹⁺²	C₃²×C₁₃₉	1	9	yes	—
27	C₃×C₉	—	3	3	no	—
28	C₃×C₉	C₃²×C₁₃₉	1	3	yes	—
29	C₃²	—	9	1	no	—
30	C₃²	H(3,𝔽₃)×C₁₃₉	1	3	yes	—
31	C₃²	C₃³×C₁₃₉	1	1	yes	characteristic
32	C₉	—	3	9	no	—
33	C₉	—	3	3	no	—
34	C₃	—	3	3	no	—
35	C₃	—	3	1	no	—
36	C₃	C₃×H(3,𝔽₃)×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃×C₁₃₉	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃²	C₃³×C₁₃₉
prime radical	C₃³×C₁₃₉	C₃²
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₁₃₉ ◁ C₃³×C₁₃₉ ◁ G

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

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