Subgroups — #27271998

Order 13568.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20272919	C₂	1	1	yes	maximal, characteristic
2	#20272917	C₂	1	1	yes	maximal, characteristic
3	C₅₃×M₁₂₈	C₂	1	1	yes	maximal, characteristic
4	#16810453	—	2	1	no	—
5	#16810449	—	2	1	no	—
6	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×C₆₄	C₄	1	2	yes	complemented
8	#13133153	—	4	1	no	—
9	#13133151	—	4	1	no	—
10	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
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⋊Dic₅₃	—	8	1	no	—
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	C₂×Dic₅₃	—	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	#7548	—	53	1	no	maximal, Sylow
24	C₂²×C₅₃	SD₆₄	1	1	yes	characteristic
25	C₄×C₅₃	#366	1	1	yes	characteristic
26	C₄×C₅₃	SD₆₄	1	1	yes	characteristic
27	Dic₅₃	—	32	1	no	—
28	M₁₂₈	D₅₃	1	1	yes	characteristic
29	#1425	—	53	1	no	—
30	#1424	—	53	1	no	—
31	C₂×C₅₃	—	2	1	no	—
32	C₂×C₅₃	#1424	1	1	yes	characteristic
33	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
34	#368	—	106	1	no	—
35	#366	—	106	1	no	—
36	C₆₄	Dic₅₃	1	2	yes	complemented
37	C₅₃	#7548	1	1	yes	characteristic, complemented, Sylow
38	C₂×C₁₆	D₂₁₂	1	1	yes	characteristic
39	#108	—	212	1	no	—
40	#107	—	212	1	no	—
41	C₃₂	Dic₁₀₆	1	1	yes	characteristic
42	C₃₂	C₂×Dic₅₃	1	1	yes	characteristic
43	M₁₆	—	424	1	no	—
44	C₂×C₈	D₄₂₄	1	1	yes	characteristic
45	C₄⋊C₄	—	424	1	no	—
46	C₁₆	Dic₂₁₂	1	1	yes	characteristic
47	C₁₆	C₄⋊Dic₅₃	1	1	yes	characteristic
48	C₂×C₄	—	848	1	no	—
49	C₂×C₄	D₈₄₈	1	1	yes	characteristic
50	C₈	—	848	1	no	—
51	C₈	#13133151	1	1	yes	characteristic
52	C₈	Dic₄₂₄	1	1	yes	characteristic
53	C₂²	#16810462	1	1	yes	characteristic
54	C₄	—	1696	1	no	—
55	C₄	#16810449	1	1	yes	characteristic
56	C₄	#16810462	1	1	yes	characteristic
57	C₂	—	2	1	no	—
58	C₂	#20272917	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20272917
commutator	C₃₂×C₅₃	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₅₃
Fitting	C₅₃×M₁₂₈	C₂
socle	C₂×C₅₃	#1424
prime radical	C₂²×C₅₃	SD₆₄
cosocle	C₂×C₃₂×C₅₃	C₂²
hypercenter	M₁₂₈	D₅₃
hypocenter	C₅₃	#7548
LCS	G ▷ C₃₂×C₅₃ ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ M₁₂₈

Complements

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

N	H ≅ G/N	type
7. C₅₃×C₆₄	C₄	semidirect
36. C₆₄	Dic₅₃	semidirect
37. C₅₃	#7548	semidirect

Maximal subgroups

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

subgroup	index
#20272919	2
#20272917	2
C₅₃×M₁₂₈	2
#7548	53