Finite Group Databasebeta

← C₂⁴×D₇₂₇ #37551597

Subgroups — C₂⁴×D₇₂₇ #37551597

Order 23264.

748 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₇₂₇	C₂	1	30	yes	maximal, direct product factor
2	C₂⁴×C₇₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×D₇₂₇	C₂²	1	140	yes	direct product factor
4	C₂³×C₇₂₇	C₂²	1	15	yes	complemented
5	C₂²×C₇₂₇	C₂³	1	35	yes	complemented
6	C₂×D₇₂₇	C₂³	1	120	yes	direct product factor
7	C₂×C₇₂₇	C₂⁴	1	15	yes	complemented
8	D₇₂₇	C₂⁴	1	16	yes	direct product factor
9	C₇₂₇	C₂⁵	1	1	yes	characteristic, complemented, Sylow
10	C₂⁵	—	727	1	no	maximal, Sylow
11	C₂⁴	—	727	30	no	—
12	C₂⁴	D₇₂₇	1	1	yes	characteristic, direct product factor
13	C₂³	—	727	140	no	—
14	C₂³	C₂×D₇₂₇	1	15	yes	direct product factor
15	C₂²	—	727	120	no	—
16	C₂²	C₂²×D₇₂₇	1	35	yes	direct product factor
17	C₂	—	727	16	no	—
18	C₂	C₂³×D₇₂₇	1	15	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂⁴	D₇₂₇
commutator, cosocle, hypocenter	C₇₂₇	C₂⁵
Frattini	C₁	G
Fitting, socle	C₂⁴×C₇₂₇	C₂
LCS	G ▷ C₇₂₇
UCS	C₁ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
1. C₂³×D₇₂₇	C₂	direct (N × H)
2. C₂⁴×C₇₂₇	C₂	semidirect
3. C₂²×D₇₂₇	C₂²	direct (N × H)
4. C₂³×C₇₂₇	C₂²	semidirect
5. C₂²×C₇₂₇	C₂³	semidirect
6. C₂×D₇₂₇	C₂³	direct (N × H)
7. C₂×C₇₂₇	C₂⁴	semidirect
8. D₇₂₇	C₂⁴	direct (N × H)
9. C₇₂₇	C₂⁵	semidirect
12. C₂⁴	D₇₂₇	direct (N × H)
14. C₂³	C₂×D₇₂₇	direct (N × H)
16. C₂²	C₂²×D₇₂₇	direct (N × H)
18. C₂	C₂³×D₇₂₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₇₂₇	2
C₂⁴×C₇₂₇	2
C₂⁵	727