Finite Group Databasebeta

← C₃²×C₉×C₃₃₇ #41031069

Subgroups — C₃²×C₉×C₃₃₇ #41031069

Order 27297.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₃₃₇	C₃	1	1	yes	maximal, characteristic
2	C₃×C₉×C₃₃₇	C₃	1	12	yes	maximal, direct product factor
3	C₃²×C₃₃₇	C₉	1	9	yes	direct product factor
4	C₃²×C₃₃₇	C₃²	1	4	yes	—
5	C₉×C₃₃₇	C₃²	1	9	yes	direct product factor
6	C₃×C₃₃₇	C₃×C₉	1	12	yes	direct product factor
7	C₃×C₃₃₇	C₃³	1	1	yes	characteristic
8	C₃₃₇	C₃²×C₉	1	1	yes	characteristic, direct product factor, Sylow
9	C₃²×C₉	C₃₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₃³	C₃×C₃₃₇	1	1	yes	characteristic
11	C₃×C₉	C₃×C₃₃₇	1	12	yes	direct product factor
12	C₃²	C₉×C₃₃₇	1	9	yes	direct product factor
13	C₃²	C₃²×C₃₃₇	1	4	yes	—
14	C₉	C₃²×C₃₃₇	1	9	yes	direct product factor
15	C₃	C₃×C₉×C₃₃₇	1	12	yes	direct product factor
16	C₃	C₃³×C₃₃₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃	C₃³×C₃₃₇
socle, prime radical	C₃³×C₃₃₇	C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₉×C₃₃₇	C₃	direct (N × H)
3. C₃²×C₃₃₇	C₉	direct (N × H)
5. C₉×C₃₃₇	C₃²	direct (N × H)
6. C₃×C₃₃₇	C₃×C₉	direct (N × H)
8. C₃₃₇	C₃²×C₉	direct (N × H)
9. C₃²×C₉	C₃₃₇	direct (N × H)
11. C₃×C₉	C₃×C₃₃₇	direct (N × H)
12. C₃²	C₉×C₃₃₇	direct (N × H)
14. C₉	C₃²×C₃₃₇	direct (N × H)
15. C₃	C₃×C₉×C₃₃₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×C₃₃₇	3
C₃×C₉×C₃₃₇	3
C₃²×C₉	337