Subgroups — C₅₃×#7548 #27215060

Order 13568.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×M₁₂₈	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1425	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1424	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
5	C₅₃×#368	—	2	1	no	—
6	C₅₃×#366	—	2	1	no	—
7	C₅₃×C₆₄	C₄	1	2	yes	complemented
8	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
9	#108×C₅₃	—	4	1	no	—
10	#107×C₅₃	—	4	1	no	—
11	C₃₂×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₃₂×C₅₃	Q₈	1	1	yes	characteristic
13	M₁₆×C₅₃	—	8	1	no	—
14	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
15	C₄⋊C₄×C₅₃	—	8	1	no	—
16	C₁₆×C₅₃	C₄⋊C₄	1	1	yes	characteristic
17	C₁₆×C₅₃	Q₁₆	1	1	yes	characteristic
18	C₂×C₄×C₅₃	—	16	1	no	—
19	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
20	C₈×C₅₃	—	16	1	no	—
21	C₈×C₅₃	#107	1	1	yes	characteristic
22	C₈×C₅₃	Q₃₂	1	1	yes	characteristic
23	#7548	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂²×C₅₃	SD₆₄	1	1	yes	characteristic
25	C₄×C₅₃	—	32	1	no	—
26	C₄×C₅₃	#366	1	1	yes	characteristic
27	C₄×C₅₃	SD₆₄	1	1	yes	characteristic
28	M₁₂₈	C₂×C₅₃	1	1	yes	characteristic
29	#1425	C₂×C₅₃	1	1	yes	characteristic
30	#1424	C₂×C₅₃	1	1	yes	characteristic
31	C₂×C₅₃	—	2	1	no	—
32	C₂×C₅₃	#1424	1	1	yes	characteristic
33	C₂×C₃₂	C₂²×C₅₃	1	1	yes	characteristic
34	#368	—	2	1	no	—
35	#366	—	2	1	no	—
36	C₆₄	C₄×C₅₃	1	2	yes	complemented
37	C₅₃	#7548	1	1	yes	characteristic, direct product factor, Sylow
38	C₂×C₁₆	D₄×C₅₃	1	1	yes	characteristic
39	#108	—	4	1	no	—
40	#107	—	4	1	no	—
41	C₃₂	C₂×C₄×C₅₃	1	1	yes	characteristic
42	C₃₂	Q₈×C₅₃	1	1	yes	characteristic
43	M₁₆	—	8	1	no	—
44	C₂×C₈	D₈×C₅₃	1	1	yes	characteristic
45	C₄⋊C₄	—	8	1	no	—
46	C₁₆	C₄⋊C₄×C₅₃	1	1	yes	characteristic
47	C₁₆	Q₁₆×C₅₃	1	1	yes	characteristic
48	C₂×C₄	—	16	1	no	—
49	C₂×C₄	D₁₆×C₅₃	1	1	yes	characteristic
50	C₈	—	16	1	no	—
51	C₈	#107×C₅₃	1	1	yes	characteristic
52	C₈	Q₃₂×C₅₃	1	1	yes	characteristic
53	C₂²	C₅₃×SD₆₄	1	1	yes	characteristic
54	C₄	—	32	1	no	—
55	C₄	C₅₃×#366	1	1	yes	characteristic
56	C₄	C₅₃×SD₆₄	1	1	yes	characteristic
57	C₂	—	2	1	no	—
58	C₂	C₅₃×#1424	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅₃	#1424
commutator	C₃₂	C₂×C₄×C₅₃
Frattini, cosocle	C₂×C₃₂	C₂²×C₅₃
prime radical	C₂²×C₅₃	SD₆₄
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₂×C₈×C₅₃ ◁ C₂×C₁₆×C₅₃ ◁ C₂×C₃₂×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₅₃×C₆₄	C₄	semidirect
23. #7548	C₅₃	direct (N × H)
36. C₆₄	C₄×C₅₃	semidirect
37. C₅₃	#7548	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×M₁₂₈	2
C₅₃×#1425	2
C₅₃×#1424	2
#7548	53

Subgroups — C53×#7548 #27215060

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7548 #27215060