Finite Group Databasebeta

← C₂⁴×D₇ #6564

Subgroups — C₂⁴×D₇ #6564

Order 224.

748 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×D₇	C₂	1	30	yes	maximal, direct product factor
3	C₂³×C₇	C₂²	1	15	yes	complemented
4	C₂²×D₇	C₂²	1	140	yes	direct product factor
5	C₂⁵	—	7	1	no	maximal, Sylow
6	C₂²×C₇	C₂³	1	35	yes	complemented
7	C₂×D₇	C₂³	1	120	yes	direct product factor
8	C₂⁴	—	7	30	no	—
9	C₂⁴	D₇	1	1	yes	characteristic, direct product factor
10	C₂×C₇	C₂⁴	1	15	yes	complemented
11	D₇	C₂⁴	1	16	yes	direct product factor
12	C₂³	—	7	140	no	—
13	C₂³	C₂×D₇	1	15	yes	direct product factor
14	C₇	C₂⁵	1	1	yes	characteristic, complemented, Sylow
15	C₂²	—	7	120	no	—
16	C₂²	C₂²×D₇	1	35	yes	direct product factor
17	C₂	—	7	16	no	—
18	C₂	C₂³×D₇	1	15	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂⁴	D₇
commutator, cosocle, hypocenter	C₇	C₂⁵
Frattini	C₁	G
Fitting, socle	C₂⁴×C₇	C₂
LCS	G ▷ C₇
UCS	C₁ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₇	C₂	semidirect
2. C₂³×D₇	C₂	direct (N × H)
3. C₂³×C₇	C₂²	semidirect
4. C₂²×D₇	C₂²	direct (N × H)
6. C₂²×C₇	C₂³	semidirect
7. C₂×D₇	C₂³	direct (N × H)
9. C₂⁴	D₇	direct (N × H)
10. C₂×C₇	C₂⁴	semidirect
11. D₇	C₂⁴	direct (N × H)
13. C₂³	C₂×D₇	direct (N × H)
14. C₇	C₂⁵	semidirect
16. C₂²	C₂²×D₇	direct (N × H)
18. C₂	C₂³×D₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₇	2
C₂³×D₇	2
C₂⁵	7