Finite Group Databasebeta

← C₇×D₁₇×C₉₇ #37528137

Subgroups — C₇×D₁₇×C₉₇ #37528137

Order 23086.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₁₇×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	D₁₇×C₉₇	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₁₇×C₉₇	C₂×C₇	1	1	yes	characteristic, complemented, Hall
4	C₂×C₇×C₉₇	—	17	1	no	maximal, Hall
5	C₇×C₉₇	D₁₇	1	1	yes	characteristic, direct product factor, Hall
6	C₇×D₁₇	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₉₇	—	17	1	no	Hall
8	C₇×C₁₇	C₂×C₉₇	1	1	yes	characteristic, complemented, Hall
9	C₉₇	C₇×D₁₇	1	1	yes	characteristic, direct product factor, Sylow
10	D₁₇	C₇×C₉₇	1	1	yes	characteristic, direct product factor, Hall
11	C₁₇	C₂×C₇×C₉₇	1	1	yes	characteristic, complemented, Sylow
12	C₂×C₇	—	17	1	no	Hall
13	C₇	D₁₇×C₉₇	1	1	yes	characteristic, direct product factor, Sylow
14	C₂	—	17	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₇×C₉₇	D₁₇
commutator, cosocle, hypocenter	C₁₇	C₂×C₇×C₉₇
Frattini	C₁	G
Fitting, socle	C₇×C₁₇×C₉₇	C₂
LCS	G ▷ C₁₇
UCS	C₁ ◁ C₇×C₉₇

Complements

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

N	H ≅ G/N	type
1. C₇×C₁₇×C₉₇	C₂	semidirect
2. D₁₇×C₉₇	C₇	direct (N × H)
3. C₁₇×C₉₇	C₂×C₇	semidirect
5. C₇×C₉₇	D₁₇	direct (N × H)
6. C₇×D₁₇	C₉₇	direct (N × H)
8. C₇×C₁₇	C₂×C₉₇	semidirect
9. C₉₇	C₇×D₁₇	direct (N × H)
10. D₁₇	C₇×C₉₇	direct (N × H)
11. C₁₇	C₂×C₇×C₉₇	semidirect
13. C₇	D₁₇×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
C₇×C₁₇×C₉₇	2
D₁₇×C₉₇	7
C₂×C₇×C₉₇	17
C₇×D₁₇	97