Subgroups — #27271993

Order 13568.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810342	C₂	1	1	yes	maximal, characteristic
2	#20272915	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1423	C₂	1	1	yes	maximal, characteristic
4	#16810449	—	2	1	no	—
5	#16810342	C₄	1	2	yes	complemented
6	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#366	—	2	1	no	—
8	#13133151	—	4	1	no	—
9	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
10	#107×C₅₃	—	4	1	no	—
11	C₃₂×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₃₂×C₅₃	Q₈	1	1	yes	characteristic
13	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
14	C₄⋊C₄×C₅₃	—	8	1	no	—
15	C₄⋊Dic₅₃	—	8	1	no	—
16	C₁₆×C₅₃	C₄⋊C₄	1	1	yes	characteristic
17	C₁₆×C₅₃	Q₁₆	1	1	yes	characteristic
18	C₂×C₄×C₅₃	—	16	1	no	—
19	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
20	C₂×Dic₅₃	—	16	1	no	—
21	C₈×C₅₃	#107	1	1	yes	characteristic
22	C₈×C₅₃	Q₃₂	1	1	yes	characteristic
23	#7546	—	53	1	no	maximal, Sylow
24	C₂²×C₅₃	D₃₂	1	1	yes	characteristic
25	C₄×C₅₃	—	32	1	no	—
26	C₄×C₅₃	#366	1	1	yes	characteristic
27	C₄×C₅₃	Q₆₄	1	1	yes	characteristic
28	Dic₅₃	—	32	1	no	—
29	C₂×C₆₄	—	53	1	no	—
30	#1423	—	53	1	no	—
31	#1423	D₅₃	1	1	yes	characteristic
32	C₂×C₅₃	#1423	1	1	yes	characteristic
33	C₂×C₅₃	SD₁₂₈	1	1	yes	characteristic
34	C₂×C₅₃	SD₁₂₈	1	1	yes	characteristic
35	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
36	#366	—	2	1	no	—
37	#366	—	106	1	no	—
38	C₆₄	—	53	2	no	—
39	C₅₃	#7546	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
41	#107	—	4	1	no	—
42	#107	—	212	1	no	—
43	C₃₂	Dic₁₀₆	1	1	yes	characteristic
44	C₃₂	C₄×D₅₃	1	1	yes	characteristic
45	C₂×C₈	#11719922	1	1	yes	characteristic
46	C₄⋊C₄	—	8	1	no	—
47	C₄⋊C₄	—	424	1	no	—
48	C₁₆	#11719919	1	1	yes	characteristic
49	C₁₆	#11719925	1	1	yes	characteristic
50	C₂×C₄	—	16	1	no	—
51	C₂×C₄	—	848	1	no	—
52	C₂×C₄	#13133158	1	1	yes	characteristic
53	C₈	#13133150	1	1	yes	characteristic
54	C₈	#13133163	1	1	yes	characteristic
55	C₂²	#16810458	1	1	yes	characteristic
56	C₄	—	32	1	no	—
57	C₄	—	1696	1	no	—
58	C₄	#16810448	1	1	yes	characteristic
59	C₄	#16810463	1	1	yes	characteristic
60	C₂	#20272914	1	1	yes	characteristic
61	C₂	#20272929	1	1	yes	characteristic
62	C₂	#20272930	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16810458
commutator	C₃₂×C₅₃	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₅₃
Fitting	C₅₃×#1423	C₂
socle, prime radical	C₂²×C₅₃	D₃₂
cosocle	C₂×C₃₂×C₅₃	C₂²
hypercenter	#1423	D₅₃
hypocenter	C₅₃	#7546
LCS	G ▷ C₃₂×C₅₃ ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ #1423

Complements

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

N	H ≅ G/N	type
5. #16810342	C₄	semidirect
39. C₅₃	#7546	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810342	2
#20272915	2
C₅₃×#1423	2
#7546	53