Subgroups — C₁₃₉×C₃²⋊H(3,𝔽₃) #46798413

Order 33777.

378 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁴×C₁₃₉	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×H(3,𝔽₃)×C₁₃₉	C₃	1	12	yes	maximal, complemented
3	C₃³×C₁₃₉	—	3	12	no	—
4	C₃³×C₁₃₉	C₃²	1	4	yes	—
5	C₃³×C₁₃₉	C₃²	1	9	yes	complemented
6	H(3,𝔽₃)×C₁₃₉	—	3	36	no	—
7	C₃²×C₁₃₉	—	3	27	no	—
8	C₃²×C₁₃₉	—	3	12	no	—
9	C₃²×C₁₃₉	—	3	36	no	—
10	C₃²×C₁₃₉	H(3,𝔽₃)	1	12	yes	complemented
11	C₃²×C₁₃₉	C₃³	1	1	yes	characteristic
12	C₃×C₁₃₉	—	3	12	no	—
13	C₃×C₁₃₉	—	9	9	no	—
14	C₃×C₁₃₉	C₃×H(3,𝔽₃)	1	4	yes	—
15	C₃²⋊H(3,𝔽₃)	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₁₃₉	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
17	C₃⁴	C₃×C₁₃₉	1	1	yes	characteristic, complemented
18	C₃×H(3,𝔽₃)	C₃×C₁₃₉	1	12	yes	complemented
19	C₃³	—	3	12	no	—
20	C₃³	C₃²×C₁₃₉	1	4	yes	—
21	C₃³	C₃²×C₁₃₉	1	9	yes	complemented
22	H(3,𝔽₃)	—	3	36	no	—
23	C₃²	—	3	27	no	—
24	C₃²	—	3	12	no	—
25	C₃²	—	3	36	no	—
26	C₃²	H(3,𝔽₃)×C₁₃₉	1	12	yes	complemented
27	C₃²	C₃³×C₁₃₉	1	1	yes	characteristic
28	C₃	—	3	12	no	—
29	C₃	—	9	9	no	—
30	C₃	C₃×H(3,𝔽₃)×C₁₃₉	1	4	yes	—

	subgroup	G/H
center, socle	C₃²×C₁₃₉	C₃³
commutator, Frattini, cosocle	C₃²	C₃³×C₁₃₉
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×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.