Subgroups — #731×C₁₉₉ #30633113

Order 16119.

28 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	3₋¹⁺²×C₁₉₉	C₃	1	2	yes	maximal, complemented
2	H(3,𝔽₃)×C₁₉₉	C₃	1	1	yes	maximal, characteristic
3	C₃×C₉×C₁₉₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃²×C₁₉₉	—	3	1	no	—
5	C₃²×C₁₉₉	C₃²	1	1	yes	characteristic
6	C₉×C₁₉₉	—	3	1	no	—
7	C₉×C₁₉₉	—	3	2	no	—
8	C₃×C₁₉₉	—	3	1	no	—
9	C₃×C₁₉₉	—	9	1	no	—
10	C₃×C₁₉₉	H(3,𝔽₃)	1	1	yes	characteristic
11	C₁₉₉	#731	1	1	yes	characteristic, direct product factor, Sylow
12	#731	C₁₉₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	3₋¹⁺²	C₃×C₁₉₉	1	2	yes	complemented
14	H(3,𝔽₃)	C₃×C₁₉₉	1	1	yes	characteristic
15	C₃×C₉	C₃×C₁₉₉	1	1	yes	characteristic, complemented
16	C₃²	—	3	1	no	—
17	C₃²	C₃²×C₁₉₉	1	1	yes	characteristic
18	C₉	—	3	1	no	—
19	C₉	—	3	2	no	—
20	C₃	—	3	1	no	—
21	C₃	—	9	1	no	—
22	C₃	H(3,𝔽₃)×C₁₉₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃×C₁₉₉	H(3,𝔽₃)
commutator, Frattini, cosocle	C₃²	C₃²×C₁₉₉
prime radical	H(3,𝔽₃)×C₁₉₉	C₃
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₁₉₉ ◁ C₃²×C₁₉₉ ◁ 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.