Subgroups — #27271995

Order 13568.

60 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	#20272919	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1425	C₂	1	1	yes	maximal, characteristic
4	#16810453	—	2	1	no	—
5	#16810342	C₄	1	2	yes	—
6	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#368	—	2	1	no	—
8	#13133153	—	4	1	no	—
9	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
10	#108×C₅₃	—	4	1	no	—
11	C₃₂×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₃₂×C₅₃	Q₈	1	1	yes	characteristic
13	M₁₆×C₅₃	—	8	1	no	—
14	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
15	#11719917	—	8	1	no	—
16	C₁₆×C₅₃	C₄⋊C₄	1	1	yes	characteristic
17	C₁₆×C₅₃	Q₁₆	1	1	yes	characteristic
18	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
19	C₈×C₅₃	—	16	1	no	—
20	C₈×C₅₃	#107	1	1	yes	characteristic
21	C₈×C₅₃	Q₃₂	1	1	yes	characteristic
22	#88322	—	16	1	no	—
23	#7547	—	53	1	no	maximal, Sylow
24	C₂²×C₅₃	Q₆₄	1	1	yes	characteristic
25	C₄×C₅₃	#366	1	1	yes	characteristic
26	C₄×C₅₃	D₃₂	1	1	yes	characteristic
27	C₂×C₆₄	—	53	1	no	—
28	#1425	—	53	1	no	—
29	#1425	D₅₃	1	1	yes	characteristic
30	C₂×C₅₃	—	2	1	no	—
31	C₂×C₅₃	#1423	1	1	yes	characteristic
32	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
33	#368	—	2	1	no	—
34	#368	—	106	1	no	—
35	C₆₄	—	53	2	no	—
36	C₅₃	#7547	1	1	yes	characteristic, complemented, Sylow
37	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
38	#108	—	4	1	no	—
39	#108	—	212	1	no	—
40	C₃₂	Dic₁₀₆	1	1	yes	characteristic
41	C₃₂	C₄×D₅₃	1	1	yes	characteristic
42	M₁₆	—	8	1	no	—
43	M₁₆	—	424	1	no	—
44	C₂×C₈	#11719922	1	1	yes	characteristic
45	C₁₆	#11719919	1	1	yes	characteristic
46	C₁₆	#11719925	1	1	yes	characteristic
47	C₂×C₄	#13133158	1	1	yes	characteristic
48	C₈	—	16	1	no	—
49	C₈	—	848	1	no	—
50	C₈	#13133150	1	1	yes	characteristic
51	C₈	#13133163	1	1	yes	characteristic
52	C₂²	#16810463	1	1	yes	characteristic
53	C₄	#16810448	1	1	yes	characteristic
54	C₄	#16810458	1	1	yes	characteristic
55	C₂	—	2	1	no	—
56	C₂	#20272914	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16810458
commutator	C₃₂×C₅₃	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₅₃
Fitting	C₅₃×#1425	C₂
socle	C₂×C₅₃	#1423
prime radical	C₂²×C₅₃	Q₆₄
cosocle	C₂×C₃₂×C₅₃	C₂²
hypercenter	#1425	D₅₃
hypocenter	C₅₃	#7547
LCS	G ▷ C₃₂×C₅₃ ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ #1425

Complements

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

N	H ≅ G/N	type
36. C₅₃	#7547	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
#20272919	2
C₅₃×#1425	2
#7547	53