Finite Group Databasebeta

← D₇×C₃₁×C₅₃ #37527439

Subgroups — D₇×C₃₁×C₅₃ #37527439

Order 23002.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₃₁×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₂×C₃₁×C₅₃	—	7	1	no	maximal, Hall
3	C₃₁×C₅₃	D₇	1	1	yes	characteristic, direct product factor, Hall
4	D₇×C₅₃	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
5	D₇×C₃₁	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₇×C₅₃	C₂×C₃₁	1	1	yes	characteristic, complemented, Hall
7	C₇×C₃₁	C₂×C₅₃	1	1	yes	characteristic, complemented, Hall
8	C₂×C₅₃	—	7	1	no	Hall
9	C₂×C₃₁	—	7	1	no	Hall
10	C₅₃	D₇×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
11	C₃₁	D₇×C₅₃	1	1	yes	characteristic, direct product factor, Sylow
12	D₇	C₃₁×C₅₃	1	1	yes	characteristic, direct product factor, Hall
13	C₇	C₂×C₃₁×C₅₃	1	1	yes	characteristic, complemented, Sylow
14	C₂	—	7	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃₁×C₅₃	D₇
commutator, cosocle, hypocenter	C₇	C₂×C₃₁×C₅₃
Frattini	C₁	G
Fitting, socle	C₇×C₃₁×C₅₃	C₂
LCS	G ▷ C₇
UCS	C₁ ◁ C₃₁×C₅₃

Complements

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

N	H ≅ G/N	type
1. C₇×C₃₁×C₅₃	C₂	semidirect
3. C₃₁×C₅₃	D₇	direct (N × H)
4. D₇×C₅₃	C₃₁	direct (N × H)
5. D₇×C₃₁	C₅₃	direct (N × H)
6. C₇×C₅₃	C₂×C₃₁	semidirect
7. C₇×C₃₁	C₂×C₅₃	semidirect
10. C₅₃	D₇×C₃₁	direct (N × H)
11. C₃₁	D₇×C₅₃	direct (N × H)
12. D₇	C₃₁×C₅₃	direct (N × H)
13. C₇	C₂×C₃₁×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₇×C₃₁×C₅₃	2
C₂×C₃₁×C₅₃	7
D₇×C₅₃	31
D₇×C₃₁	53