Subgroups — #27272002

Order 13568.

44 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₄×D₅₃	C₂	1	1	yes	maximal, characteristic
2	#20272534	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×C₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₂×D₅₃	C₄	1	1	yes	characteristic
5	#16810342	C₄	1	1	yes	characteristic
6	C₅₃×C₆₄	C₂²	1	1	yes	characteristic
7	C₁₆×D₅₃	C₈	1	1	yes	characteristic
8	#13133123	C₈	1	1	yes	characteristic
9	C₃₂×C₅₃	C₂×C₄	1	1	yes	characteristic
10	C₈×D₅₃	C₁₆	1	1	yes	characteristic
11	C₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
12	#11719908	C₁₆	1	1	yes	characteristic
13	C₄×D₅₃	C₃₂	1	1	yes	characteristic
14	C₈×C₅₃	C₂×C₁₆	1	1	yes	characteristic
15	#88322	C₃₂	1	1	yes	characteristic
16	M₂₅₆	—	53	1	no	maximal, Sylow
17	C₂×D₅₃	C₆₄	1	1	yes	characteristic
18	C₄×C₅₃	C₂×C₃₂	1	1	yes	characteristic
19	Dic₅₃	C₆₄	1	1	yes	characteristic
20	C₂×C₆₄	—	53	1	no	—
21	C₁₂₈	—	53	1	no	—
22	C₁₂₈	D₅₃	1	1	yes	characteristic, complemented
23	C₂×C₅₃	C₂×C₆₄	1	1	yes	characteristic
24	D₅₃	—	2	1	no	—
25	C₂×C₃₂	—	53	1	no	—
26	C₆₄	—	53	1	no	—
27	C₆₄	C₂×D₅₃	1	1	yes	characteristic
28	C₅₃	M₂₅₆	1	1	yes	characteristic, complemented, Sylow
29	C₂×C₁₆	—	53	1	no	—
30	C₃₂	—	53	1	no	—
31	C₃₂	C₄×D₅₃	1	1	yes	characteristic
32	C₂×C₈	—	53	1	no	—
33	C₁₆	—	53	1	no	—
34	C₁₆	C₈×D₅₃	1	1	yes	characteristic
35	C₂×C₄	—	53	1	no	—
36	C₈	—	53	1	no	—
37	C₈	C₁₆×D₅₃	1	1	yes	characteristic
38	C₂²	—	53	1	no	—
39	C₄	—	53	1	no	—
40	C₄	C₃₂×D₅₃	1	1	yes	characteristic
41	C₂	—	106	1	no	—
42	C₂	C₆₄×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₆₄	C₂×D₅₃
commutator, socle	C₂×C₅₃	C₂×C₆₄
Fitting	C₅₃×C₁₂₈	C₂
prime radical	C₂×D₅₃	C₆₄
cosocle	C₅₃×C₆₄	C₂²
hypercenter	C₁₂₈	D₅₃
hypocenter	C₅₃	M₂₅₆
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₆₄ ◁ C₁₂₈

Complements

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

N	H ≅ G/N	type
2. #20272534	C₂	semidirect
3. C₅₃×C₁₂₈	C₂	semidirect
22. C₁₂₈	D₅₃	semidirect
28. C₅₃	M₂₅₆	semidirect

Maximal subgroups

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

subgroup	index
C₆₄×D₅₃	2
#20272534	2
C₅₃×C₁₂₈	2
M₂₅₆	53