Subgroups — C₁₃₁×Q₂₅₆ #45693437

Order 33536.

42 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₁₂₈×C₁₃₁	C₂	1	2	yes	maximal
2	C₁₂₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	Q₆₄×C₁₃₁	—	2	2	no	—
4	C₆₄×C₁₃₁	C₂²	1	1	yes	characteristic
5	Q₃₂×C₁₃₁	—	4	2	no	—
6	C₃₂×C₁₃₁	D₄	1	1	yes	characteristic
7	Q₁₆×C₁₃₁	—	8	2	no	—
8	C₁₆×C₁₃₁	D₈	1	1	yes	characteristic
9	Q₈×C₁₃₁	—	16	2	no	—
10	C₈×C₁₃₁	D₁₆	1	1	yes	characteristic
11	C₄×C₁₃₁	—	32	2	no	—
12	C₄×C₁₃₁	D₃₂	1	1	yes	characteristic
13	C₂×C₁₃₁	D₆₄	1	1	yes	characteristic
14	Q₂₅₆	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₁₃₁	Q₂₅₆	1	1	yes	characteristic, direct product factor, Sylow
16	Q₁₂₈	C₂×C₁₃₁	1	2	yes	—
17	C₁₂₈	C₂×C₁₃₁	1	1	yes	characteristic
18	Q₆₄	—	2	2	no	—
19	C₆₄	C₂²×C₁₃₁	1	1	yes	characteristic
20	Q₃₂	—	4	2	no	—
21	C₃₂	D₄×C₁₃₁	1	1	yes	characteristic
22	Q₁₆	—	8	2	no	—
23	C₁₆	D₈×C₁₃₁	1	1	yes	characteristic
24	Q₈	—	16	2	no	—
25	C₈	D₁₆×C₁₃₁	1	1	yes	characteristic
26	C₄	—	32	2	no	—
27	C₄	D₃₂×C₁₃₁	1	1	yes	characteristic
28	C₂	D₆₄×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂×C₁₃₁	D₆₄
commutator, Frattini, cosocle	C₆₄	C₂²×C₁₃₁
LCS	G ▷ C₆₄ ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃₁ ◁ C₄×C₁₃₁ ◁ C₈×C₁₃₁ ◁ C₁₆×C₁₃₁ ◁ C₃₂×C₁₃₁ ◁ C₆₄×C₁₃₁ ◁ G

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

N	H ≅ G/N	type
14. Q₂₅₆	C₁₃₁	direct (N × H)
15. C₁₃₁	Q₂₅₆	direct (N × H)

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