Finite Group Databasebeta

← C₂×C₂₃×C₉₇⋊C₃ #27211461

Subgroups — C₂×C₂₃×C₉₇⋊C₃ #27211461

Order 13386.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×C₉₇⋊C₃	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂×C₂₃×C₉₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₂₃×C₉₇	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	C₂×C₉₇⋊C₃	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₉₇⋊C₃	C₂×C₂₃	1	1	yes	characteristic, direct product factor, Hall
6	C₂×C₉₇	C₃×C₂₃	1	1	yes	characteristic, complemented, Hall
7	C₂×C₃×C₂₃	—	97	1	no	maximal, Hall
8	C₉₇	C₂×C₃×C₂₃	1	1	yes	characteristic, complemented, Sylow
9	C₃×C₂₃	—	97	1	no	Hall
10	C₂×C₂₃	C₉₇⋊C₃	1	1	yes	characteristic, direct product factor, Hall
11	C₂₃	C₂×C₉₇⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
12	C₂×C₃	—	97	1	no	Hall
13	C₃	—	97	1	no	Sylow
14	C₂	C₂₃×C₉₇⋊C₃	1	1	yes	characteristic, direct product factor, Sylow

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

Complements

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

N	H ≅ G/N	type
1. C₂₃×C₉₇⋊C₃	C₂	direct (N × H)
2. C₂×C₂₃×C₉₇	C₃	semidirect
3. C₂₃×C₉₇	C₂×C₃	semidirect
4. C₂×C₉₇⋊C₃	C₂₃	direct (N × H)
5. C₉₇⋊C₃	C₂×C₂₃	direct (N × H)
6. C₂×C₉₇	C₃×C₂₃	semidirect
8. C₉₇	C₂×C₃×C₂₃	semidirect
10. C₂×C₂₃	C₉₇⋊C₃	direct (N × H)
11. C₂₃	C₂×C₉₇⋊C₃	direct (N × H)
14. C₂	C₂₃×C₉₇⋊C₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×C₉₇⋊C₃	2
C₂×C₂₃×C₉₇	3
C₂×C₉₇⋊C₃	23
C₂×C₃×C₂₃	97