Subgroups — C₅₃×#1424 #20270361

Order 6784.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#366	C₂	1	2	yes	maximal
3	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
4	#107×C₅₃	—	2	2	no	—
5	C₃₂×C₅₃	C₄	1	2	yes	complemented
6	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
7	C₄⋊C₄×C₅₃	—	4	2	no	—
8	C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
9	C₁₆×C₅₃	Q₈	1	1	yes	characteristic
10	C₂×C₄×C₅₃	—	8	2	no	—
11	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
12	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
13	C₈×C₅₃	Q₁₆	1	1	yes	characteristic
14	C₂²×C₅₃	D₁₆	1	1	yes	characteristic
15	C₄×C₅₃	—	16	2	no	—
16	C₄×C₅₃	#107	1	1	yes	characteristic
17	C₄×C₅₃	Q₃₂	1	1	yes	characteristic
18	#1424	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂×C₅₃	#366	1	1	yes	characteristic
20	C₂×C₅₃	SD₆₄	1	2	yes	—
21	C₂×C₃₂	C₂×C₅₃	1	1	yes	characteristic
22	#366	C₂×C₅₃	1	2	yes	—
23	C₅₃	#1424	1	1	yes	characteristic, direct product factor, Sylow
24	C₂×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
25	#107	—	2	2	no	—
26	C₃₂	C₄×C₅₃	1	2	yes	complemented
27	C₂×C₈	D₄×C₅₃	1	1	yes	characteristic
28	C₄⋊C₄	—	4	2	no	—
29	C₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
30	C₁₆	Q₈×C₅₃	1	1	yes	characteristic
31	C₂×C₄	—	8	2	no	—
32	C₂×C₄	D₈×C₅₃	1	1	yes	characteristic
33	C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
34	C₈	Q₁₆×C₅₃	1	1	yes	characteristic
35	C₂²	D₁₆×C₅₃	1	1	yes	characteristic
36	C₄	—	16	2	no	—
37	C₄	#107×C₅₃	1	1	yes	characteristic
38	C₄	Q₃₂×C₅₃	1	1	yes	characteristic
39	C₂	C₅₃×#366	1	1	yes	characteristic
40	C₂	C₅₃×SD₆₄	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₅₃	D₁₆
commutator	C₁₆	C₂×C₄×C₅₃
Frattini, cosocle	C₂×C₁₆	C₂²×C₅₃
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₂×C₈×C₅₃ ◁ C₂×C₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₃₂×C₅₃	C₄	semidirect
18. #1424	C₅₃	direct (N × H)
23. C₅₃	#1424	direct (N × H)
26. C₃₂	C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₂×C₅₃	2
C₅₃×#366	2
#1424	53

Subgroups — C53×#1424 #20270361

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#1424 #20270361