Subgroups — C₅₃×D₁₂₈ #27215063

Order 13568.

46 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×D₆₄	C₂	1	2	yes	maximal, complemented
2	C₅₃×C₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×D₃₂	—	2	2	no	—
4	C₅₃×C₆₄	C₂²	1	1	yes	characteristic
5	D₁₆×C₅₃	—	4	2	no	—
6	C₃₂×C₅₃	D₄	1	1	yes	characteristic
7	D₈×C₅₃	—	8	2	no	—
8	C₁₆×C₅₃	D₈	1	1	yes	characteristic
9	D₄×C₅₃	—	16	2	no	—
10	C₈×C₅₃	D₁₆	1	1	yes	characteristic
11	D₁₂₈	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
12	C₂²×C₅₃	—	32	2	no	—
13	C₄×C₅₃	D₃₂	1	1	yes	characteristic
14	D₆₄	C₂×C₅₃	1	2	yes	complemented
15	C₁₂₈	C₂×C₅₃	1	1	yes	characteristic, complemented
16	C₂×C₅₃	—	64	2	no	—
17	C₂×C₅₃	D₆₄	1	1	yes	characteristic
18	D₃₂	—	2	2	no	—
19	C₆₄	C₂²×C₅₃	1	1	yes	characteristic
20	C₅₃	D₁₂₈	1	1	yes	characteristic, direct product factor, Sylow
21	D₁₆	—	4	2	no	—
22	C₃₂	D₄×C₅₃	1	1	yes	characteristic
23	D₈	—	8	2	no	—
24	C₁₆	D₈×C₅₃	1	1	yes	characteristic
25	D₄	—	16	2	no	—
26	C₈	D₁₆×C₅₃	1	1	yes	characteristic
27	C₂²	—	32	2	no	—
28	C₄	C₅₃×D₃₂	1	1	yes	characteristic
29	C₂	—	64	2	no	—
30	C₂	C₅₃×D₆₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	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.

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