Finite Group Databasebeta

← C₄⋊C₄×C₄₃₉ #20296293

Subgroups — C₄⋊C₄×C₄₃₉ #20296293

Order 7024.

26 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₄₃₉	C₂	1	2	yes	maximal
2	C₂×C₄×C₄₃₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄₃₉	C₂²	1	1	yes	characteristic
4	C₄×C₄₃₉	—	2	2	no	—
5	C₄×C₄₃₉	C₄	1	2	yes	complemented
6	C₂×C₄₃₉	C₂×C₄	1	1	yes	characteristic
7	C₂×C₄₃₉	D₄	1	1	yes	characteristic
8	C₂×C₄₃₉	Q₈	1	1	yes	characteristic
9	C₄₃₉	C₄⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
10	C₄⋊C₄	C₄₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂×C₄	C₂×C₄₃₉	1	2	yes	—
12	C₂×C₄	C₂×C₄₃₉	1	1	yes	characteristic
13	C₂²	C₂²×C₄₃₉	1	1	yes	characteristic
14	C₄	—	2	2	no	—
15	C₄	C₄×C₄₃₉	1	2	yes	complemented
16	C₂	C₂×C₄×C₄₃₉	1	1	yes	characteristic
17	C₂	D₄×C₄₃₉	1	1	yes	characteristic
18	C₂	Q₈×C₄₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₄₃₉	C₂²
commutator	C₂	C₂×C₄×C₄₃₉
Frattini, cosocle	C₂²	C₂²×C₄₃₉
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₄×C₄₃₉	C₄	semidirect
9. C₄₃₉	C₄⋊C₄	direct (N × H)
10. C₄⋊C₄	C₄₃₉	direct (N × H)
15. C₄	C₄×C₄₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₄₃₉	2
C₄⋊C₄	439