Finite Group Databasebeta

← C₃²×C₄₉×C₆₁ #41002000

Subgroups — C₃²×C₄₉×C₆₁ #41002000

Order 26901.

36 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄₉×C₆₁	C₃	1	4	yes	maximal, direct product factor
2	C₃²×C₇×C₆₁	C₇	1	1	yes	maximal, characteristic
3	C₄₉×C₆₁	C₃²	1	1	yes	characteristic, direct product factor, Hall
4	C₃×C₇×C₆₁	C₃×C₇	1	4	yes	—
5	C₃²×C₆₁	C₄₉	1	1	yes	characteristic, direct product factor, Hall
6	C₃²×C₄₉	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₇×C₆₁	C₃²×C₇	1	1	yes	characteristic
8	C₃×C₆₁	C₃×C₄₉	1	4	yes	direct product factor
9	C₃×C₄₉	C₃×C₆₁	1	4	yes	direct product factor
10	C₃²×C₇	C₇×C₆₁	1	1	yes	characteristic
11	C₆₁	C₃²×C₄₉	1	1	yes	characteristic, direct product factor, Sylow
12	C₄₉	C₃²×C₆₁	1	1	yes	characteristic, direct product factor, Sylow
13	C₃×C₇	C₃×C₇×C₆₁	1	4	yes	—
14	C₃²	C₄₉×C₆₁	1	1	yes	characteristic, direct product factor, Sylow
15	C₇	C₃²×C₇×C₆₁	1	1	yes	characteristic
16	C₃	C₃×C₄₉×C₆₁	1	4	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₇	C₃²×C₇×C₆₁
socle, prime radical	C₃²×C₇×C₆₁	C₇
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₄₉×C₆₁	C₃	direct (N × H)
3. C₄₉×C₆₁	C₃²	direct (N × H)
5. C₃²×C₆₁	C₄₉	direct (N × H)
6. C₃²×C₄₉	C₆₁	direct (N × H)
8. C₃×C₆₁	C₃×C₄₉	direct (N × H)
9. C₃×C₄₉	C₃×C₆₁	direct (N × H)
11. C₆₁	C₃²×C₄₉	direct (N × H)
12. C₄₉	C₃²×C₆₁	direct (N × H)
14. C₃²	C₄₉×C₆₁	direct (N × H)
16. C₃	C₃×C₄₉×C₆₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₄₉×C₆₁	3
C₃²×C₇×C₆₁	7
C₃²×C₄₉	61