Subgroups — #27271762

Order 13568.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20272783	C₂	1	1	yes	maximal, characteristic
2	#20272781	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1398	C₂	1	1	yes	maximal, characteristic
4	#16810444	—	2	1	no	—
5	#16810374	—	2	1	no	—
6	C₅₃×M₆₄	C₄	1	1	yes	characteristic
7	C₅₃×M₆₄	C₄	1	1	yes	characteristic
8	C₅₃×#346	C₂²	1	1	yes	characteristic
9	#13133157	—	4	1	no	—
10	C₄⋊#88322	—	4	1	no	—
11	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₁₆×C₅₃	Q₈	1	1	yes	characteristic
13	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
14	C₃₂×C₅₃	—	2	1	no	—
15	C₃₂×C₅₃	C₈	1	2	yes	complemented
16	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
17	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
18	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
19	C₂×#88322	—	8	1	no	—
20	C₁₆×C₅₃	—	2	1	no	—
21	C₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
22	C₁₆×C₅₃	M₁₆	1	1	yes	characteristic
23	#11719908	—	8	1	no	—
24	C₂×C₄×C₅₃	—	2	1	no	—
25	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
26	C₈×C₅₃	—	2	1	no	—
27	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
28	C₈×C₅₃	#108	1	1	yes	characteristic
29	#88322	—	16	1	no	—
30	#7457	—	53	1	no	maximal, Sylow
31	C₂²×C₅₃	#365	1	1	yes	characteristic
32	C₄×C₅₃	—	4	1	no	—
33	C₄×C₅₃	#334	1	1	yes	characteristic
34	C₄×C₅₃	#365	1	1	yes	characteristic
35	#1398	D₅₃	1	1	yes	characteristic
36	#1369	—	53	1	no	—
37	#1368	—	53	1	no	—
38	C₂×C₅₃	—	2	1	no	—
39	C₂×C₅₃	#1368	1	1	yes	characteristic
40	M₆₄	Dic₅₃	1	1	yes	characteristic
41	M₆₄	Dic₅₃	1	1	yes	characteristic
42	#364	—	106	1	no	—
43	#346	C₂×D₅₃	1	1	yes	characteristic
44	#334	—	106	1	no	—
45	C₅₃	#7457	1	1	yes	characteristic, complemented, Sylow
46	M₃₂	—	212	1	no	—
47	C₂×C₁₆	Dic₁₀₆	1	1	yes	characteristic
48	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
49	C₄⋊C₈	—	212	1	no	—
50	C₄×C₈	D₂₁₂	1	1	yes	characteristic
51	C₃₂	—	2	1	no	—
52	C₃₂	#88322	1	2	yes	complemented
53	C₂×C₈	—	424	1	no	—
54	C₂×C₈	#11719913	1	1	yes	characteristic
55	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
56	C₄²	#11719913	1	1	yes	characteristic
57	C₁₆	—	2	1	no	—
58	C₁₆	—	424	1	no	—
59	C₁₆	C₂×#88322	1	1	yes	characteristic
60	C₁₆	#11719917	1	1	yes	characteristic
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	#13133149	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	848	1	no	—
65	C₈	C₄⋊#88322	1	1	yes	characteristic
66	C₈	#13133153	1	1	yes	characteristic
67	C₂²	#16810446	1	1	yes	characteristic
68	C₄	—	4	1	no	—
69	C₄	#16810374	1	1	yes	characteristic
70	C₄	#16810446	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	#20272781	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20272781
commutator	C₁₆×C₅₃	C₂×C₈
Frattini	#346	C₂×D₅₃
Fitting	C₅₃×#1398	C₂
socle	C₂×C₅₃	#1368
prime radical	C₂²×C₅₃	#365
cosocle	C₅₃×#346	C₂²
hypercenter	#1398	D₅₃
hypocenter	C₅₃	#7457
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346 ◁ #1398

Complements

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

N	H ≅ G/N	type
15. C₃₂×C₅₃	C₈	semidirect
45. C₅₃	#7457	semidirect
52. C₃₂	#88322	semidirect

Maximal subgroups

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

subgroup	index
#20272783	2
#20272781	2
C₅₃×#1398	2
#7457	53