Subgroups — #27272003

Order 13568.

44 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20272534	C₂	1	2	yes	maximal, complemented
2	C₂×C₅₃×C₆₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₂×C₅₃	C₄	1	1	yes	characteristic
4	C₅₃×C₆₄	C₄	1	1	yes	characteristic
5	C₅₃×C₆₄	C₂²	1	1	yes	characteristic
6	C₂×C₁₆×C₅₃	C₈	1	1	yes	characteristic
7	C₃₂×C₅₃	C₈	1	1	yes	characteristic
8	C₃₂×C₅₃	C₂×C₄	1	1	yes	characteristic
9	C₂×C₈×C₅₃	C₁₆	1	1	yes	characteristic
10	C₁₆×C₅₃	C₁₆	1	1	yes	characteristic
11	C₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
12	C₂×C₄×C₅₃	C₃₂	1	1	yes	characteristic
13	C₈×C₅₃	C₃₂	1	1	yes	characteristic
14	C₈×C₅₃	C₂×C₁₆	1	1	yes	characteristic
15	M₂₅₆	—	53	1	no	maximal, Sylow
16	C₂²×C₅₃	C₆₄	1	1	yes	characteristic
17	C₄×C₅₃	C₆₄	1	1	yes	characteristic
18	C₄×C₅₃	C₂×C₃₂	1	1	yes	characteristic
19	C₂×C₆₄	D₅₃	1	1	yes	characteristic
20	C₁₂₈	—	53	2	no	—
21	C₂×C₅₃	—	2	1	no	—
22	C₂×C₅₃	C₂×C₆₄	1	1	yes	characteristic
23	C₂×C₃₂	Dic₅₃	1	1	yes	characteristic
24	C₆₄	Dic₅₃	1	1	yes	characteristic
25	C₆₄	C₂×D₅₃	1	1	yes	characteristic
26	C₅₃	M₂₅₆	1	1	yes	characteristic, complemented, Sylow
27	C₂×C₁₆	#88322	1	1	yes	characteristic
28	C₃₂	#88322	1	1	yes	characteristic
29	C₃₂	C₂×Dic₅₃	1	1	yes	characteristic
30	C₂×C₈	#11719908	1	1	yes	characteristic
31	C₁₆	#11719908	1	1	yes	characteristic
32	C₁₆	C₂×#88322	1	1	yes	characteristic
33	C₂×C₄	#13133123	1	1	yes	characteristic
34	C₈	#13133123	1	1	yes	characteristic
35	C₈	C₂×#11719908	1	1	yes	characteristic
36	C₂²	#16810342	1	1	yes	characteristic
37	C₄	#16810342	1	1	yes	characteristic
38	C₄	C₂×#13133123	1	1	yes	characteristic
39	C₂	—	2	1	no	—
40	C₂	C₂×#16810342	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₆₄	C₂×D₅₃
commutator, socle	C₂×C₅₃	C₂×C₆₄
Fitting	C₂×C₅₃×C₆₄	C₂
prime radical	C₂²×C₅₃	C₆₄
cosocle	C₅₃×C₆₄	C₂²
hypercenter	C₂×C₆₄	D₅₃
hypocenter	C₅₃	M₂₅₆
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₆₄ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
1. #20272534	C₂	semidirect
26. C₅₃	M₂₅₆	semidirect

Maximal subgroups

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

subgroup	index
#20272534	2
C₂×C₅₃×C₆₄	2
M₂₅₆	53