Subgroups — #13133263

Order 1696.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#11719949	C₂	1	2	yes	maximal, complemented
3	D₄×D₅₃	C₂	1	6	yes	maximal, complemented
4	#11719945	C₂	1	3	yes	maximal, complemented
5	C₂×D₂₁₂	C₂	1	3	yes	maximal, complemented
6	C₂²×D₅₃	C₂²	1	6	yes	complemented
7	Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
8	D₄×C₅₃	C₂²	1	3	yes	complemented
9	C₂×C₄×C₅₃	C₂²	1	3	yes	complemented
10	C₂²⋊D₅₃	C₂²	1	6	yes	complemented
11	D₂₁₂	C₂²	1	3	yes	complemented
12	D₂₁₂	C₂²	1	6	yes	complemented
13	C₄×D₅₃	C₂²	1	6	yes	complemented
14	Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
15	C₂²×C₅₃	C₂³	1	3	yes	—
16	C₂×D₅₃	—	4	6	no	—
17	C₂×D₅₃	C₂³	1	6	yes	—
18	C₄×C₅₃	C₂³	1	1	yes	characteristic
19	C₄×C₅₃	C₂³	1	3	yes	—
20	Dic₅₃	C₂³	1	2	yes	—
21	C₂×C₅₃	—	2	3	no	—
22	C₂×C₅₃	C₂⁴	1	1	yes	characteristic
23	D₅₃	—	2	6	no	—
24	C₅₃	H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
25	H(5,𝔽₂)	—	53	1	no	maximal, Sylow
26	#41	—	53	2	no	—
27	#41	—	53	3	no	—
28	#41	D₅₃	1	1	yes	characteristic, complemented
29	C₂×D₄	—	53	6	no	—
30	C₂×D₄	—	53	3	no	—
31	C₂³	—	53	6	no	—
32	Q₈	—	53	1	no	—
33	Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
34	D₄	—	53	6	no	—
35	D₄	—	53	6	no	—
36	D₄	—	53	3	no	—
37	D₄	C₂×D₅₃	1	3	yes	complemented
38	C₂×C₄	—	53	6	no	—
39	C₂×C₄	C₂×D₅₃	1	3	yes	complemented
40	C₂²	—	53	6	no	—
41	C₂²	—	212	6	no	—
42	C₂²	C₂²×D₅₃	1	3	yes	—
43	C₄	—	53	2	no	—
44	C₄	C₂²×D₅₃	1	3	yes	—
45	C₄	C₂²×D₅₃	1	1	yes	characteristic
46	C₂	—	2	3	no	—
47	C₂	—	106	6	no	—
48	C₂	C₂³×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×D₅₃
commutator, socle, cosocle	C₂×C₅₃	C₂⁴
Fitting	#41×C₅₃	C₂
hypercenter	#41	D₅₃
hypocenter	C₅₃	H(5,𝔽₂)
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ #41

Complements

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

N	H ≅ G/N	type
1. #41×C₅₃	C₂	semidirect
2. #11719949	C₂	semidirect
3. D₄×D₅₃	C₂	semidirect
4. #11719945	C₂	semidirect
5. C₂×D₂₁₂	C₂	semidirect
6. C₂²×D₅₃	C₂²	semidirect
7. Q₈×C₅₃	C₂²	semidirect
8. D₄×C₅₃	C₂²	semidirect
9. C₂×C₄×C₅₃	C₂²	semidirect
10. C₂²⋊D₅₃	C₂²	semidirect
11. D₂₁₂	C₂²	semidirect
12. D₂₁₂	C₂²	semidirect
13. C₄×D₅₃	C₂²	semidirect
14. Dic₁₀₆	C₂²	semidirect
24. C₅₃	H(5,𝔽₂)	semidirect
28. #41	D₅₃	semidirect
33. Q₈	C₂×D₅₃	semidirect
37. D₄	C₂×D₅₃	semidirect
39. C₂×C₄	C₂×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#41×C₅₃	2
#11719949	2
D₄×D₅₃	2
#11719945	2
C₂×D₂₁₂	2
H(5,𝔽₂)	53