Subgroups — #28299142

Order 13568.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20289742	C₂	1	1	yes	maximal, characteristic
2	#20289740	C₂	1	1	yes	maximal, characteristic
3	#20272850	C₂	1	1	yes	maximal, characteristic
4	#16811545	—	2	1	no	—
5	#16811484	—	2	1	no	—
6	#16810457	C₄	1	1	yes	characteristic
7	#16810398	C₂²	1	1	yes	characteristic
8	C₅₃×M₆₄	C₄	1	1	yes	characteristic
9	#13133291	—	4	1	no	—
10	C₄⋊#88324	—	4	1	no	—
11	C₂×#11719908	Q₈	1	1	yes	characteristic
12	C₈×Dic₅₃	D₄	1	1	yes	characteristic
13	#13133123	C₈	1	2	yes	complemented
14	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₃₂×C₅₃	—	2	1	no	—
16	C₂×#88324	—	8	1	no	—
17	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
18	C₄×Dic₅₃	SD₁₆	1	1	yes	characteristic
19	C₂×#88322	SD₁₆	1	1	yes	characteristic
20	#11719910	—	8	1	no	—
21	C₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
22	C₁₆×C₅₃	M₁₆	1	1	yes	characteristic
23	#11719908	—	2	1	no	—
24	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
25	C₂×Dic₅₃	—	2	1	no	—
26	#88324	—	16	1	no	—
27	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
28	C₈×C₅₃	#108	1	1	yes	characteristic
29	#88322	—	2	1	no	—
30	#7457	—	53	1	no	maximal, Sylow
31	C₂²×C₅₃	#365	1	1	yes	characteristic
32	C₄×C₅₃	#334	1	1	yes	characteristic
33	C₄×C₅₃	#365	1	1	yes	characteristic
34	Dic₅₃	—	4	1	no	—
35	#1398	—	53	1	no	—
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	—	53	1	no	—
41	M₆₄	#6157	1	1	yes	characteristic
42	#364	—	106	1	no	—
43	#346	—	53	1	no	—
44	#334	—	106	1	no	—
45	C₅₃	#7457	1	1	yes	characteristic, complemented, Sylow
46	M₃₂	—	212	1	no	—
47	C₂×C₁₆	—	53	1	no	—
48	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
49	C₄⋊C₈	—	212	1	no	—
50	C₄×C₈	—	53	1	no	—
51	C₃₂	—	2	1	no	—
52	C₃₂	—	53	2	no	—
53	C₂×C₈	—	53	1	no	—
54	C₂×C₈	—	424	1	no	—
55	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
56	C₄²	—	53	1	no	—
57	C₁₆	—	106	1	no	—
58	C₁₆	—	424	1	no	—
59	C₁₆	D₅₃⋊C₈	1	1	yes	characteristic
60	C₁₆	#11719940	1	1	yes	characteristic
61	C₂×C₄	—	106	1	no	—
62	C₂×C₄	#13133285	1	1	yes	characteristic
63	C₈	—	848	1	no	—
64	C₈	—	106	1	no	—
65	C₈	#13133284	1	1	yes	characteristic
66	C₈	#13133288	1	1	yes	characteristic
67	C₂²	#16811547	1	1	yes	characteristic
68	C₄	—	212	1	no	—
69	C₄	#16811485	1	1	yes	characteristic
70	C₄	#16811548	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	#20289741	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20289741
commutator	C₁₆×C₅₃	C₂×C₈
Frattini	C₂×C₁₆	C₂×#6157
Fitting	C₅₃×M₆₄	C₄
socle	C₂×C₅₃	#1368
prime radical	C₂²×C₅₃	#365
cosocle	#16810398	C₂²
hypercenter	M₆₄	#6157
hypocenter	C₅₃	#7457
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
13. #13133123	C₈	semidirect
45. C₅₃	#7457	semidirect

Maximal subgroups

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

subgroup	index
#20289742	2
#20289740	2
#20272850	2
#7457	53