Finite Group Databasebeta

← Q₃₂×C₇₅₇ #37564261

Subgroups — Q₃₂×C₇₅₇ #37564261

Order 24224.

24 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₁₆×C₇₅₇	C₂	1	2	yes	maximal
2	C₁₆×C₇₅₇	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₇₅₇	—	2	2	no	—
4	C₈×C₇₅₇	C₂²	1	1	yes	characteristic
5	C₄×C₇₅₇	—	4	2	no	—
6	C₄×C₇₅₇	D₄	1	1	yes	characteristic
7	C₂×C₇₅₇	D₈	1	1	yes	characteristic
8	C₇₅₇	Q₃₂	1	1	yes	characteristic, direct product factor, Sylow
9	Q₃₂	C₇₅₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	Q₁₆	C₂×C₇₅₇	1	2	yes	—
11	C₁₆	C₂×C₇₅₇	1	1	yes	characteristic
12	Q₈	—	2	2	no	—
13	C₈	C₂²×C₇₅₇	1	1	yes	characteristic
14	C₄	—	4	2	no	—
15	C₄	D₄×C₇₅₇	1	1	yes	characteristic
16	C₂	D₈×C₇₅₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂×C₇₅₇	D₈
commutator, Frattini, cosocle	C₈	C₂²×C₇₅₇
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₅₇ ◁ C₄×C₇₅₇ ◁ C₈×C₇₅₇ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₇₅₇	Q₃₂	direct (N × H)
9. Q₃₂	C₇₅₇	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₁₆×C₇₅₇	2
C₁₆×C₇₅₇	2
Q₃₂	757