Subgroups — C₅₃×Q₂₅₆ #27215065

Order 13568.

42 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×Q₁₂₈	C₂	1	2	yes	maximal
2	C₅₃×C₁₂₈	C₂	1	1	yes	maximal, characteristic
3	C₅₃×Q₆₄	—	2	2	no	—
4	C₅₃×C₆₄	C₂²	1	1	yes	characteristic
5	Q₃₂×C₅₃	—	4	2	no	—
6	C₃₂×C₅₃	D₄	1	1	yes	characteristic
7	Q₁₆×C₅₃	—	8	2	no	—
8	C₁₆×C₅₃	D₈	1	1	yes	characteristic
9	Q₈×C₅₃	—	16	2	no	—
10	C₈×C₅₃	D₁₆	1	1	yes	characteristic
11	Q₂₅₆	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
12	C₄×C₅₃	—	32	2	no	—
13	C₄×C₅₃	D₃₂	1	1	yes	characteristic
14	Q₁₂₈	C₂×C₅₃	1	2	yes	—
15	C₁₂₈	C₂×C₅₃	1	1	yes	characteristic
16	C₂×C₅₃	D₆₄	1	1	yes	characteristic
17	Q₆₄	—	2	2	no	—
18	C₆₄	C₂²×C₅₃	1	1	yes	characteristic
19	C₅₃	Q₂₅₆	1	1	yes	characteristic, direct product factor, Sylow
20	Q₃₂	—	4	2	no	—
21	C₃₂	D₄×C₅₃	1	1	yes	characteristic
22	Q₁₆	—	8	2	no	—
23	C₁₆	D₈×C₅₃	1	1	yes	characteristic
24	Q₈	—	16	2	no	—
25	C₈	D₁₆×C₅₃	1	1	yes	characteristic
26	C₄	—	32	2	no	—
27	C₄	C₅₃×D₃₂	1	1	yes	characteristic
28	C₂	C₅₃×D₆₄	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂×C₅₃	D₆₄
commutator, Frattini, cosocle	C₆₄	C₂²×C₅₃
LCS	G ▷ C₆₄ ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₄×C₅₃ ◁ C₈×C₅₃ ◁ C₁₆×C₅₃ ◁ C₃₂×C₅₃ ◁ C₅₃×C₆₄ ◁ G

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

N	H ≅ G/N	type
11. Q₂₅₆	C₅₃	direct (N × H)
19. C₅₃	Q₂₅₆	direct (N × H)

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