Finite Group Databasebeta

← C₂₃×D₆₉₇ #44545454

Subgroups — C₂₃×D₆₉₇ #44545454

Order 32062.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×C₂₃×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₂₃×D₄₁	—	17	1	no	maximal, Hall
3	D₆₉₇	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂₃×C₄₁	D₁₇	1	1	yes	characteristic, complemented, Hall
5	C₂₃×D₁₇	—	41	1	no	maximal, Hall
6	C₁₇×C₄₁	C₂×C₂₃	1	1	yes	characteristic, complemented, Hall
7	C₁₇×C₂₃	D₄₁	1	1	yes	characteristic, complemented, Hall
8	D₄₁	—	17	1	no	Hall
9	C₂×C₂₃	—	697	1	no	Hall
10	C₄₁	C₂₃×D₁₇	1	1	yes	characteristic, complemented, Sylow
11	D₁₇	—	41	1	no	Hall
12	C₂₃	D₆₉₇	1	1	yes	characteristic, direct product factor, Sylow
13	C₁₇	C₂₃×D₄₁	1	1	yes	characteristic, complemented, Sylow
14	C₂	—	697	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₂₃	D₆₉₇
commutator, cosocle, hypocenter	C₁₇×C₄₁	C₂×C₂₃
Frattini	C₁	G
Fitting, socle	C₁₇×C₂₃×C₄₁	C₂
LCS	G ▷ C₁₇×C₄₁
UCS	C₁ ◁ C₂₃

Complements

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

N	H ≅ G/N	type
1. C₁₇×C₂₃×C₄₁	C₂	semidirect
3. D₆₉₇	C₂₃	direct (N × H)
4. C₂₃×C₄₁	D₁₇	semidirect
6. C₁₇×C₄₁	C₂×C₂₃	semidirect
7. C₁₇×C₂₃	D₄₁	semidirect
10. C₄₁	C₂₃×D₁₇	semidirect
12. C₂₃	D₆₉₇	direct (N × H)
13. C₁₇	C₂₃×D₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₁₇×C₂₃×C₄₁	2
C₂₃×D₄₁	17
D₆₉₇	23
C₂₃×D₁₇	41