Finite Group Databasebeta

← C₃×D₂₃×C₉₇ #27211463

Subgroups — C₃×D₂₃×C₉₇ #27211463

Order 13386.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₂₃×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	D₂₃×C₉₇	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₂₃×C₉₇	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	C₂×C₃×C₉₇	—	23	1	no	maximal, Hall
5	C₃×C₉₇	D₂₃	1	1	yes	characteristic, direct product factor, Hall
6	C₂×C₉₇	—	23	1	no	Hall
7	C₃×D₂₃	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₉₇	C₃×D₂₃	1	1	yes	characteristic, direct product factor, Sylow
9	C₃×C₂₃	C₂×C₉₇	1	1	yes	characteristic, complemented, Hall
10	D₂₃	C₃×C₉₇	1	1	yes	characteristic, direct product factor, Hall
11	C₂₃	C₂×C₃×C₉₇	1	1	yes	characteristic, complemented, Sylow
12	C₂×C₃	—	23	1	no	Hall
13	C₃	D₂₃×C₉₇	1	1	yes	characteristic, direct product factor, Sylow
14	C₂	—	23	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃×C₉₇	D₂₃
commutator, cosocle, hypocenter	C₂₃	C₂×C₃×C₉₇
Frattini	C₁	G
Fitting, socle	C₃×C₂₃×C₉₇	C₂
LCS	G ▷ C₂₃
UCS	C₁ ◁ C₃×C₉₇

Complements

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

N	H ≅ G/N	type
1. C₃×C₂₃×C₉₇	C₂	semidirect
2. D₂₃×C₉₇	C₃	direct (N × H)
3. C₂₃×C₉₇	C₂×C₃	semidirect
5. C₃×C₉₇	D₂₃	direct (N × H)
7. C₃×D₂₃	C₉₇	direct (N × H)
8. C₉₇	C₃×D₂₃	direct (N × H)
9. C₃×C₂₃	C₂×C₉₇	semidirect
10. D₂₃	C₃×C₉₇	direct (N × H)
11. C₂₃	C₂×C₃×C₉₇	semidirect
13. C₃	D₂₃×C₉₇	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×C₂₃×C₉₇	2
D₂₃×C₉₇	3
C₂×C₃×C₉₇	23
C₃×D₂₃	97