Subgroups — Q₆₄×C₄₃₁ #42118564

Order 27584.

30 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₃₂×C₄₃₁	C₂	1	2	yes	maximal
2	C₃₂×C₄₃₁	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×C₄₃₁	—	2	2	no	—
4	C₁₆×C₄₃₁	C₂²	1	1	yes	characteristic
5	Q₈×C₄₃₁	—	4	2	no	—
6	C₈×C₄₃₁	D₄	1	1	yes	characteristic
7	C₄×C₄₃₁	—	8	2	no	—
8	C₄×C₄₃₁	D₈	1	1	yes	characteristic
9	C₂×C₄₃₁	D₁₆	1	1	yes	characteristic
10	C₄₃₁	Q₆₄	1	1	yes	characteristic, direct product factor, Sylow
11	Q₆₄	C₄₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
12	Q₃₂	C₂×C₄₃₁	1	2	yes	—
13	C₃₂	C₂×C₄₃₁	1	1	yes	characteristic
14	Q₁₆	—	2	2	no	—
15	C₁₆	C₂²×C₄₃₁	1	1	yes	characteristic
16	Q₈	—	4	2	no	—
17	C₈	D₄×C₄₃₁	1	1	yes	characteristic
18	C₄	—	8	2	no	—
19	C₄	D₈×C₄₃₁	1	1	yes	characteristic
20	C₂	D₁₆×C₄₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂×C₄₃₁	D₁₆
commutator, Frattini, cosocle	C₁₆	C₂²×C₄₃₁
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄₃₁ ◁ C₄×C₄₃₁ ◁ C₈×C₄₃₁ ◁ C₁₆×C₄₃₁ ◁ G

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

N	H ≅ G/N	type
10. C₄₃₁	Q₆₄	direct (N × H)
11. Q₆₄	C₄₃₁	direct (N × H)

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