Subgroups — C₅₃×SD₂₅₆ #27215064

Order 13568.

44 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×Q₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×D₆₄	C₂	1	1	yes	maximal, characteristic
3	C₅₃×C₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×Q₆₄	—	2	1	no	—
5	C₅₃×D₃₂	—	2	1	no	—
6	C₅₃×C₆₄	C₂²	1	1	yes	characteristic
7	Q₃₂×C₅₃	—	4	1	no	—
8	D₁₆×C₅₃	—	4	1	no	—
9	C₃₂×C₅₃	D₄	1	1	yes	characteristic
10	Q₁₆×C₅₃	—	8	1	no	—
11	D₈×C₅₃	—	8	1	no	—
12	C₁₆×C₅₃	D₈	1	1	yes	characteristic
13	Q₈×C₅₃	—	16	1	no	—
14	D₄×C₅₃	—	16	1	no	—
15	C₈×C₅₃	D₁₆	1	1	yes	characteristic
16	SD₂₅₆	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂²×C₅₃	—	32	1	no	—
18	C₄×C₅₃	—	32	1	no	—
19	C₄×C₅₃	D₃₂	1	1	yes	characteristic
20	Q₁₂₈	C₂×C₅₃	1	1	yes	characteristic, complemented
21	D₆₄	C₂×C₅₃	1	1	yes	characteristic
22	C₁₂₈	C₂×C₅₃	1	1	yes	characteristic, complemented
23	C₂×C₅₃	—	64	1	no	—
24	C₂×C₅₃	D₆₄	1	1	yes	characteristic
25	Q₆₄	—	2	1	no	—
26	D₃₂	—	2	1	no	—
27	C₆₄	C₂²×C₅₃	1	1	yes	characteristic
28	C₅₃	SD₂₅₆	1	1	yes	characteristic, direct product factor, Sylow
29	Q₃₂	—	4	1	no	—
30	D₁₆	—	4	1	no	—
31	C₃₂	D₄×C₅₃	1	1	yes	characteristic
32	Q₁₆	—	8	1	no	—
33	D₈	—	8	1	no	—
34	C₁₆	D₈×C₅₃	1	1	yes	characteristic
35	Q₈	—	16	1	no	—
36	D₄	—	16	1	no	—
37	C₈	D₁₆×C₅₃	1	1	yes	characteristic
38	C₂²	—	32	1	no	—
39	C₄	—	32	1	no	—
40	C₄	C₅₃×D₃₂	1	1	yes	characteristic
41	C₂	—	64	1	no	—
42	C₂	C₅₃×D₆₄	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₅₃×Q₁₂₈	C₂	semidirect
3. C₅₃×C₁₂₈	C₂	semidirect
16. SD₂₅₆	C₅₃	direct (N × H)
20. Q₁₂₈	C₂×C₅₃	semidirect
22. C₁₂₈	C₂×C₅₃	semidirect
28. C₅₃	SD₂₅₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×Q₁₂₈	2
C₅₃×D₆₄	2
C₅₃×C₁₂₈	2
SD₂₅₆	53

Subgroups — C53×SD256 #27215064

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×SD₂₅₆ #27215064