Finite Group Databasebeta

← C₅₃×C₂²⋊(C₃₇⋊C₃) #37555236

Subgroups — C₅₃×C₂²⋊(C₃₇⋊C₃) #37555236

Order 23532.

20 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₇×C₅₃	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅₃×C₃₇⋊C₃	—	4	1	no	maximal, Hall
3	C₂×C₃₇×C₅₃	—	3	1	no	—
4	C₃₇×C₅₃	A₄	1	1	yes	characteristic, complemented, Hall
5	A₄×C₅₃	—	37	1	no	maximal, Hall
6	C₂²⋊(C₃₇⋊C₃)	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₅₃	C₃₇⋊C₃	1	1	yes	characteristic, complemented, Hall
8	C₃×C₅₃	—	148	1	no	Hall
9	C₂²×C₃₇	C₃×C₅₃	1	1	yes	characteristic, complemented, Hall
10	C₃₇⋊C₃	—	4	1	no	Hall
11	C₂×C₅₃	—	3	1	no	—
12	C₂×C₃₇	—	3	1	no	—
13	C₅₃	C₂²⋊(C₃₇⋊C₃)	1	1	yes	characteristic, direct product factor, Sylow
14	C₃₇	A₄×C₅₃	1	1	yes	characteristic, complemented, Sylow
15	A₄	—	37	1	no	Hall
16	C₂²	C₅₃×C₃₇⋊C₃	1	1	yes	characteristic, complemented, Sylow
17	C₃	—	148	1	no	Sylow
18	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₅₃	C₂²⋊(C₃₇⋊C₃)
commutator, cosocle, hypocenter	C₂²×C₃₇	C₃×C₅₃
Frattini	C₁	G
Fitting, socle	C₂²×C₃₇×C₅₃	C₃
LCS	G ▷ C₂²×C₃₇
UCS	C₁ ◁ C₅₃

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃₇×C₅₃	C₃	semidirect
4. C₃₇×C₅₃	A₄	semidirect
6. C₂²⋊(C₃₇⋊C₃)	C₅₃	direct (N × H)
7. C₂²×C₅₃	C₃₇⋊C₃	semidirect
9. C₂²×C₃₇	C₃×C₅₃	semidirect
13. C₅₃	C₂²⋊(C₃₇⋊C₃)	direct (N × H)
14. C₃₇	A₄×C₅₃	semidirect
16. C₂²	C₅₃×C₃₇⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃₇×C₅₃	3
C₅₃×C₃₇⋊C₃	4
A₄×C₅₃	37
C₂²⋊(C₃₇⋊C₃)	53