Subgroups — C₂²×#13133123 #20277004

Order 6784.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133123	C₂	1	6	yes	maximal, direct product factor
2	C₂²×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	#13133123	C₂²	1	4	yes	direct product factor
4	C₂²×C₈×C₅₃	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₅₃	C₄	1	3	yes	—
6	C₂×C₁₆×C₅₃	C₂²	1	3	yes	—
7	C₂²×C₄×C₅₃	C₈	1	1	yes	characteristic
8	C₂×C₈×C₅₃	C₈	1	3	yes	—
9	C₂×C₈×C₅₃	C₂×C₄	1	3	yes	—
10	C₁₆×C₅₃	C₂×C₄	1	3	yes	—
11	C₁₆×C₅₃	C₂³	1	1	yes	characteristic
12	C₂³×C₅₃	C₁₆	1	1	yes	characteristic
13	C₂×C₄×C₅₃	C₁₆	1	3	yes	—
14	C₂×C₄×C₅₃	C₂×C₈	1	3	yes	—
15	C₈×C₅₃	C₂×C₈	1	3	yes	—
16	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
17	C₂²×C₅₃	C₃₂	1	4	yes	complemented
18	C₂²×C₅₃	C₂×C₁₆	1	3	yes	—
19	C₄×C₅₃	C₂×C₁₆	1	3	yes	—
20	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
21	C₂²×C₃₂	—	53	1	no	maximal, Sylow
22	C₂×C₅₃	C₂×C₃₂	1	6	yes	complemented
23	C₂×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
24	C₂²×C₁₆	D₅₃	1	1	yes	characteristic
25	C₂×C₃₂	—	53	6	no	—
26	C₅₃	C₂²×C₃₂	1	1	yes	characteristic, complemented, Sylow
27	C₂²×C₈	Dic₅₃	1	1	yes	characteristic
28	C₂×C₁₆	Dic₅₃	1	3	yes	—
29	C₂×C₁₆	C₂×D₅₃	1	3	yes	—
30	C₃₂	—	53	4	no	—
31	C₂²×C₄	#88322	1	1	yes	characteristic
32	C₂×C₈	#88322	1	3	yes	—
33	C₂×C₈	C₂×Dic₅₃	1	3	yes	—
34	C₁₆	C₂×Dic₅₃	1	3	yes	—
35	C₁₆	C₂²×D₅₃	1	1	yes	characteristic
36	C₂³	#11719908	1	1	yes	characteristic
37	C₂×C₄	#11719908	1	3	yes	—
38	C₂×C₄	C₂×#88322	1	3	yes	—
39	C₈	C₂×#88322	1	3	yes	—
40	C₈	C₂²×Dic₅₃	1	1	yes	characteristic
41	C₂²	#13133123	1	4	yes	direct product factor
42	C₂²	C₂×#11719908	1	3	yes	—
43	C₄	C₂×#11719908	1	3	yes	—
44	C₄	C₂²×#88322	1	1	yes	characteristic
45	C₂	C₂×#13133123	1	6	yes	direct product factor
46	C₂	C₂²×#11719908	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₁₆	D₅₃
commutator, hypocenter	C₅₃	C₂²×C₃₂
Frattini	C₁₆	C₂²×D₅₃
Fitting	C₂²×C₁₆×C₅₃	C₂
socle	C₂³×C₅₃	C₁₆
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#13133123	C₂	direct (N × H)
3. #13133123	C₂²	direct (N × H)
17. C₂²×C₅₃	C₃₂	semidirect
22. C₂×C₅₃	C₂×C₃₂	semidirect
26. C₅₃	C₂²×C₃₂	semidirect
41. C₂²	#13133123	direct (N × H)
45. C₂	C₂×#13133123	direct (N × H)

Subgroups — C22×#13133123 #20277004

Conjugacy classes of subgroups

Complements

Subgroups — C₂²×#13133123 #20277004