Subgroups — C₃₁×Q₂₅₆ #21464312

Order 7936.

42 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×Q₁₂₈	C₂	1	2	yes	maximal
2	C₃₁×C₁₂₈	C₂	1	1	yes	maximal, characteristic
3	C₃₁×Q₆₄	—	2	2	no	—
4	C₃₁×C₆₄	C₂²	1	1	yes	characteristic
5	C₃₁×Q₃₂	—	4	2	no	—
6	C₃₁×C₃₂	D₄	1	1	yes	characteristic
7	Q₁₆×C₃₁	—	8	2	no	—
8	C₁₆×C₃₁	D₈	1	1	yes	characteristic
9	Q₂₅₆	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	Q₈×C₃₁	—	16	2	no	—
11	C₈×C₃₁	D₁₆	1	1	yes	characteristic
12	Q₁₂₈	C₂×C₃₁	1	2	yes	—
13	C₁₂₈	C₂×C₃₁	1	1	yes	characteristic
14	C₄×C₃₁	—	32	2	no	—
15	C₄×C₃₁	D₃₂	1	1	yes	characteristic
16	Q₆₄	—	2	2	no	—
17	C₆₄	C₂²×C₃₁	1	1	yes	characteristic
18	C₂×C₃₁	D₆₄	1	1	yes	characteristic
19	Q₃₂	—	4	2	no	—
20	C₃₂	D₄×C₃₁	1	1	yes	characteristic
21	C₃₁	Q₂₅₆	1	1	yes	characteristic, direct product factor, Sylow
22	Q₁₆	—	8	2	no	—
23	C₁₆	D₈×C₃₁	1	1	yes	characteristic
24	Q₈	—	16	2	no	—
25	C₈	C₃₁×D₁₆	1	1	yes	characteristic
26	C₄	—	32	2	no	—
27	C₄	C₃₁×D₃₂	1	1	yes	characteristic
28	C₂	C₃₁×D₆₄	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂×C₃₁	D₆₄
commutator, Frattini, cosocle	C₆₄	C₂²×C₃₁
LCS	G ▷ C₆₄ ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₄×C₃₁ ◁ C₈×C₃₁ ◁ C₁₆×C₃₁ ◁ C₃₁×C₃₂ ◁ C₃₁×C₆₄ ◁ G

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

N	H ≅ G/N	type
9. Q₂₅₆	C₃₁	direct (N × H)
21. C₃₁	Q₂₅₆	direct (N × H)

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