Subgroups — #27271989

Order 13568.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20272926	C₂	1	1	yes	maximal, characteristic
2	C₃₂×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×M₁₂₈	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133123	C₄	1	1	yes	characteristic
5	C₁₆×Dic₅₃	C₄	1	1	yes	characteristic
6	#16810342	—	2	1	no	—
7	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×C₆₄	—	2	1	no	—
9	C₂×#11719908	C₈	1	1	yes	characteristic
10	C₈×Dic₅₃	C₈	1	1	yes	characteristic
11	#13133123	—	2	1	no	—
12	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₃₂×C₅₃	D₄	1	1	yes	characteristic
14	C₃₂×C₅₃	Q₈	1	1	yes	characteristic
15	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
16	C₄×Dic₅₃	C₁₆	1	1	yes	characteristic
17	C₂×#88322	C₁₆	1	1	yes	characteristic
18	C₁₆×C₅₃	C₄⋊C₄	1	1	yes	characteristic
19	C₁₆×C₅₃	M₁₆	1	1	yes	characteristic
20	#11719908	—	2	1	no	—
21	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
22	C₂×Dic₅₃	—	2	1	no	—
23	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
24	C₈×C₅₃	M₃₂	1	1	yes	characteristic
25	#88322	C₃₂	1	2	yes	—
26	#7544	—	53	1	no	maximal, Sylow
27	C₂²×C₅₃	M₆₄	1	1	yes	characteristic
28	C₄×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
29	C₄×C₅₃	C₂×C₃₂	1	1	yes	characteristic
30	Dic₅₃	—	2	2	no	—
31	M₁₂₈	—	53	1	no	—
32	M₁₂₈	D₅₃	1	1	yes	characteristic
33	C₄×C₃₂	—	53	1	no	—
34	C₂×C₅₃	—	2	1	no	—
35	C₂×C₅₃	C₄⋊C₃₂	1	1	yes	characteristic
36	C₂×C₃₂	—	53	1	no	—
37	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
38	C₄×C₁₆	—	53	1	no	—
39	C₆₄	—	2	1	no	—
40	C₆₄	—	106	1	no	—
41	C₅₃	#7544	1	1	yes	characteristic, complemented, Sylow
42	C₂×C₁₆	—	53	1	no	—
43	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
44	C₄×C₈	—	53	1	no	—
45	C₃₂	—	106	1	no	—
46	C₃₂	Dic₁₀₆	1	1	yes	characteristic
47	C₃₂	C₂²⋊D₅₃	1	1	yes	characteristic
48	C₂×C₈	—	53	1	no	—
49	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
50	C₄²	—	53	1	no	—
51	C₁₆	—	106	1	no	—
52	C₁₆	#11719912	1	1	yes	characteristic
53	C₁₆	#11719919	1	1	yes	characteristic
54	C₂×C₄	—	106	1	no	—
55	C₂×C₄	C₁₆×D₅₃	1	1	yes	characteristic
56	C₈	—	53	2	no	—
57	C₈	#13133146	1	1	yes	characteristic
58	C₈	#13133156	1	1	yes	characteristic
59	C₂²	#16810456	1	1	yes	characteristic
60	C₄	—	106	2	no	—
61	C₄	#16810441	1	1	yes	characteristic
62	C₄	C₃₂×D₅₃	1	1	yes	characteristic
63	C₂	—	2	1	no	—
64	C₂	#20272910	1	1	yes	characteristic

	subgroup	G/H
center	C₃₂	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂×C₃₂
Frattini	C₂×C₃₂	C₂×D₅₃
Fitting	C₅₃×M₁₂₈	C₂
socle	C₂×C₅₃	C₄⋊C₃₂
prime radical	C₂²×C₅₃	M₆₄
cosocle	C₂×C₃₂×C₅₃	C₂²
hypercenter	M₁₂₈	D₅₃
hypocenter	C₅₃	#7544
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₃₂ ◁ C₂×C₃₂ ◁ M₁₂₈

Complements

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

N	H ≅ G/N	type
41. 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
C₃₂×Dic₅₃	2
C₅₃×M₁₂₈	2
#7544	53