Subgroups — #27271990

Order 13568.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20272926	C₂	1	2	yes	maximal
2	C₄×C₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
3	#16810342	—	2	2	no	—
4	C₂×C₃₂×C₅₃	C₄	1	1	yes	characteristic
5	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
6	C₄×C₁₆×C₅₃	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₅₃	C₈	1	1	yes	characteristic
8	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
9	C₄×C₈×C₅₃	C₈	1	1	yes	characteristic
10	C₃₂×C₅₃	—	2	1	no	—
11	C₃₂×C₅₃	D₄	1	1	yes	characteristic
12	C₃₂×C₅₃	Q₈	1	1	yes	characteristic
13	C₂×C₈×C₅₃	C₁₆	1	1	yes	characteristic
14	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
15	C₄²×C₅₃	C₁₆	1	1	yes	characteristic
16	C₁₆×C₅₃	—	2	1	no	—
17	C₁₆×C₅₃	C₄⋊C₄	1	1	yes	characteristic
18	C₁₆×C₅₃	M₁₆	1	1	yes	characteristic
19	C₂×C₄×C₅₃	—	2	1	no	—
20	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
21	C₈×C₅₃	C₃₂	1	2	yes	—
22	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
23	C₈×C₅₃	M₃₂	1	1	yes	characteristic
24	#7544	—	53	1	no	maximal, Sylow
25	C₂²×C₅₃	M₆₄	1	1	yes	characteristic
26	C₄×C₅₃	—	2	2	no	—
27	C₄×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
28	C₄×C₅₃	C₂×C₃₂	1	1	yes	characteristic
29	M₁₂₈	—	53	2	no	—
30	C₄×C₃₂	D₅₃	1	1	yes	characteristic
31	C₂×C₅₃	—	2	1	no	—
32	C₂×C₅₃	C₄⋊C₃₂	1	1	yes	characteristic
33	C₂×C₃₂	Dic₅₃	1	1	yes	characteristic
34	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
35	C₄×C₁₆	Dic₅₃	1	1	yes	characteristic
36	C₆₄	—	106	2	no	—
37	C₅₃	#7544	1	1	yes	characteristic, complemented, Sylow
38	C₂×C₁₆	#88322	1	1	yes	characteristic
39	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
40	C₄×C₈	#88322	1	1	yes	characteristic
41	C₃₂	—	2	1	no	—
42	C₃₂	Dic₁₀₆	1	1	yes	characteristic
43	C₃₂	D₂₁₂	1	1	yes	characteristic
44	C₂×C₈	#11719908	1	1	yes	characteristic
45	C₂×C₈	C₂×#88322	1	1	yes	characteristic
46	C₄²	#11719908	1	1	yes	characteristic
47	C₁₆	—	2	1	no	—
48	C₁₆	#11719917	1	1	yes	characteristic
49	C₁₆	C₄⋊Dic₅₃	1	1	yes	characteristic
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	C₂×#11719908	1	1	yes	characteristic
52	C₈	#13133123	1	2	yes	—
53	C₈	C₄⋊#88322	1	1	yes	characteristic
54	C₈	#13133157	1	1	yes	characteristic
55	C₂²	#16810457	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	C₄⋊#11719908	1	1	yes	characteristic
58	C₄	C₂×#13133123	1	1	yes	characteristic
59	C₂	—	2	1	no	—
60	C₂	C₄⋊#13133123	1	1	yes	characteristic

	subgroup	G/H
center	C₃₂	D₂₁₂
commutator	C₄×C₅₃	C₂×C₃₂
Frattini	C₂×C₃₂	C₂×D₅₃
Fitting	C₄×C₃₂×C₅₃	C₂
socle	C₂×C₅₃	C₄⋊C₃₂
prime radical	C₂²×C₅₃	M₆₄
cosocle	C₂×C₃₂×C₅₃	C₂²
hypercenter	C₄×C₃₂	D₅₃
hypocenter	C₅₃	#7544
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₃₂ ◁ C₂×C₃₂ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
37. C₅₃	#7544	semidirect

Maximal subgroups

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

subgroup	index
#20272926	2
C₄×C₃₂×C₅₃	2
#7544	53