Subgroups — #27271996

Order 13568.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20272919	C₂	1	2	yes	maximal
2	C₂×C₅₃×C₆₄	C₂	1	1	yes	maximal, characteristic
3	#16810453	—	2	2	no	—
4	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
5	C₅₃×C₆₄	C₄	1	2	yes	—
6	#13133153	—	4	2	no	—
7	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
8	C₃₂×C₅₃	C₂×C₄	1	1	yes	characteristic
9	C₃₂×C₅₃	Q₈	1	1	yes	characteristic
10	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
11	#11719917	—	8	2	no	—
12	C₁₆×C₅₃	C₄⋊C₄	1	1	yes	characteristic
13	C₁₆×C₅₃	Q₁₆	1	1	yes	characteristic
14	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
15	C₈×C₅₃	#107	1	1	yes	characteristic
16	C₈×C₅₃	Q₃₂	1	1	yes	characteristic
17	#88322	—	16	2	no	—
18	#7547	—	53	1	no	maximal, Sylow
19	C₂²×C₅₃	Q₆₄	1	1	yes	characteristic
20	C₄×C₅₃	#366	1	1	yes	characteristic
21	C₄×C₅₃	D₃₂	1	1	yes	characteristic
22	C₂×C₆₄	D₅₃	1	1	yes	characteristic
23	#1425	—	53	2	no	—
24	C₂×C₅₃	—	2	1	no	—
25	C₂×C₅₃	#1423	1	1	yes	characteristic
26	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
27	#368	—	106	2	no	—
28	C₆₄	Dic₅₃	1	2	yes	—
29	C₅₃	#7547	1	1	yes	characteristic, complemented, Sylow
30	C₂×C₁₆	D₂₁₂	1	1	yes	characteristic
31	#108	—	212	2	no	—
32	C₃₂	Dic₁₀₆	1	1	yes	characteristic
33	C₃₂	C₂×Dic₅₃	1	1	yes	characteristic
34	M₁₆	—	424	2	no	—
35	C₂×C₈	D₄₂₄	1	1	yes	characteristic
36	C₁₆	Dic₂₁₂	1	1	yes	characteristic
37	C₁₆	C₄⋊Dic₅₃	1	1	yes	characteristic
38	C₂×C₄	D₈₄₈	1	1	yes	characteristic
39	C₈	—	848	2	no	—
40	C₈	#13133151	1	1	yes	characteristic
41	C₈	Dic₄₂₄	1	1	yes	characteristic
42	C₂²	Dic₈₄₈	1	1	yes	characteristic
43	C₄	#16810449	1	1	yes	characteristic
44	C₄	D₁₆₉₆	1	1	yes	characteristic
45	C₂	—	2	1	no	—
46	C₂	#20272915	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₁₆₉₆
commutator	C₃₂×C₅₃	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₅₃
Fitting	C₂×C₅₃×C₆₄	C₂
socle	C₂×C₅₃	#1423
prime radical	C₂²×C₅₃	Q₆₄
cosocle	C₂×C₃₂×C₅₃	C₂²
hypercenter	C₂×C₆₄	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 ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
29. C₅₃	#7547	semidirect

Maximal subgroups

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

subgroup	index
#20272919	2
C₂×C₅₃×C₆₄	2
#7547	53