Finite Group Databasebeta

← C₇²×C₉×C₆₁ #41002006

Subgroups — C₇²×C₉×C₆₁ #41002006

Order 26901.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇²×C₆₁	C₃	1	1	yes	maximal, characteristic
2	C₇×C₉×C₆₁	C₇	1	8	yes	maximal, direct product factor
3	C₇²×C₆₁	C₉	1	1	yes	characteristic, direct product factor, Hall
4	C₃×C₇×C₆₁	C₃×C₇	1	8	yes	—
5	C₉×C₆₁	C₇²	1	1	yes	characteristic, direct product factor, Hall
6	C₇²×C₉	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₇×C₆₁	C₇×C₉	1	8	yes	direct product factor
8	C₃×C₆₁	C₃×C₇²	1	1	yes	characteristic
9	C₃×C₇²	C₃×C₆₁	1	1	yes	characteristic
10	C₇×C₉	C₇×C₆₁	1	8	yes	direct product factor
11	C₆₁	C₇²×C₉	1	1	yes	characteristic, direct product factor, Sylow
12	C₇²	C₉×C₆₁	1	1	yes	characteristic, direct product factor, Sylow
13	C₃×C₇	C₃×C₇×C₆₁	1	8	yes	—
14	C₉	C₇²×C₆₁	1	1	yes	characteristic, direct product factor, Sylow
15	C₇	C₇×C₉×C₆₁	1	8	yes	direct product factor
16	C₃	C₃×C₇²×C₆₁	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃	C₃×C₇²×C₆₁
socle, prime radical	C₃×C₇²×C₆₁	C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×C₉×C₆₁	C₇	direct (N × H)
3. C₇²×C₆₁	C₉	direct (N × H)
5. C₉×C₆₁	C₇²	direct (N × H)
6. C₇²×C₉	C₆₁	direct (N × H)
7. C₇×C₆₁	C₇×C₉	direct (N × H)
10. C₇×C₉	C₇×C₆₁	direct (N × H)
11. C₆₁	C₇²×C₉	direct (N × H)
12. C₇²	C₉×C₆₁	direct (N × H)
14. C₉	C₇²×C₆₁	direct (N × H)
15. C₇	C₇×C₉×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×C₇²×C₆₁	3
C₇×C₉×C₆₁	7
C₇²×C₉	61