Finite Group Databasebeta

← D₇×C₂₃×C₉₇ #44530308

Subgroups — D₇×C₂₃×C₉₇ #44530308

Order 31234.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₂₃×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₂×C₂₃×C₉₇	—	7	1	no	maximal, Hall
3	C₂₃×C₉₇	D₇	1	1	yes	characteristic, direct product factor, Hall
4	D₇×C₉₇	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₇×C₉₇	C₂×C₂₃	1	1	yes	characteristic, complemented, Hall
6	D₇×C₂₃	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₉₇	—	7	1	no	Hall
8	C₇×C₂₃	C₂×C₉₇	1	1	yes	characteristic, complemented, Hall
9	C₉₇	D₇×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
10	C₂×C₂₃	—	7	1	no	Hall
11	C₂₃	D₇×C₉₇	1	1	yes	characteristic, direct product factor, Sylow
12	D₇	C₂₃×C₉₇	1	1	yes	characteristic, direct product factor, Hall
13	C₇	C₂×C₂₃×C₉₇	1	1	yes	characteristic, complemented, Sylow
14	C₂	—	7	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₂₃×C₉₇	D₇
commutator, cosocle, hypocenter	C₇	C₂×C₂₃×C₉₇
Frattini	C₁	G
Fitting, socle	C₇×C₂₃×C₉₇	C₂
LCS	G ▷ C₇
UCS	C₁ ◁ C₂₃×C₉₇

Complements

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

N	H ≅ G/N	type
1. C₇×C₂₃×C₉₇	C₂	semidirect
3. C₂₃×C₉₇	D₇	direct (N × H)
4. D₇×C₉₇	C₂₃	direct (N × H)
5. C₇×C₉₇	C₂×C₂₃	semidirect
6. D₇×C₂₃	C₉₇	direct (N × H)
8. C₇×C₂₃	C₂×C₉₇	semidirect
9. C₉₇	D₇×C₂₃	direct (N × H)
11. C₂₃	D₇×C₉₇	direct (N × H)
12. D₇	C₂₃×C₉₇	direct (N × H)
13. C₇	C₂×C₂₃×C₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₇×C₂₃×C₉₇	2
C₂×C₂₃×C₉₇	7
D₇×C₉₇	23
D₇×C₂₃	97