Subgroups — C₅₃×#7540 #27215052

Order 13568.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2262	C₂	1	1	yes	maximal, characteristic
2	C₅₃×M₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#1424	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#508	—	2	1	no	—
5	C₅₃×Q₆₄	C₄	1	1	yes	characteristic, complemented
6	C₅₃×SD₆₄	—	2	1	no	—
7	C₅₃×D₃₂	C₄	1	1	yes	characteristic, complemented
8	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#366	—	2	1	no	—
10	C₅₃×C₆₄	—	2	1	no	—
11	#135×C₅₃	—	4	1	no	—
12	Q₃₂×C₅₃	—	2	1	no	—
13	SD₃₂×C₅₃	—	4	1	no	—
14	D₁₆×C₅₃	—	2	1	no	—
15	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
16	#107×C₅₃	—	4	1	no	—
17	C₃₂×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₃₂×C₅₃	D₄	1	1	yes	characteristic
19	#41×C₅₃	—	8	1	no	—
20	Q₁₆×C₅₃	—	4	1	no	—
21	SD₁₆×C₅₃	—	8	1	no	—
22	D₈×C₅₃	—	4	1	no	—
23	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
24	C₄⋊C₄×C₅₃	—	8	1	no	—
25	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₁₆×C₅₃	SD₁₆	1	1	yes	characteristic
27	Q₈×C₅₃	—	8	1	no	—
28	D₄×C₅₃	—	8	1	no	—
29	D₄×C₅₃	—	16	1	no	—
30	C₂×C₄×C₅₃	—	16	1	no	—
31	C₂×C₄×C₅₃	—	16	1	no	—
32	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
33	C₈×C₅₃	#102	1	1	yes	characteristic
34	C₈×C₅₃	SD₃₂	1	1	yes	characteristic
35	#7540	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂²×C₅₃	—	16	1	no	—
37	C₂²×C₅₃	D₃₂	1	1	yes	characteristic
38	C₄×C₅₃	—	16	1	no	—
39	C₄×C₅₃	—	32	1	no	—
40	C₄×C₅₃	#357	1	1	yes	characteristic
41	C₄×C₅₃	SD₆₄	1	1	yes	characteristic
42	#2262	C₂×C₅₃	1	1	yes	characteristic
43	M₁₂₈	C₂×C₅₃	1	1	yes	characteristic, complemented
44	#1424	C₂×C₅₃	1	1	yes	characteristic, complemented
45	C₂×C₅₃	—	2	1	no	—
46	C₂×C₅₃	—	32	1	no	—
47	C₂×C₅₃	#1415	1	1	yes	characteristic
48	#508	—	2	1	no	—
49	Q₆₄	C₄×C₅₃	1	1	yes	characteristic, complemented
50	SD₆₄	—	2	1	no	—
51	D₃₂	C₄×C₅₃	1	1	yes	characteristic, complemented
52	C₂×C₃₂	C₂²×C₅₃	1	1	yes	characteristic
53	#366	—	2	1	no	—
54	C₆₄	—	2	1	no	—
55	C₅₃	#7540	1	1	yes	characteristic, direct product factor, Sylow
56	#135	—	4	1	no	—
57	Q₃₂	—	2	1	no	—
58	SD₃₂	—	4	1	no	—
59	D₁₆	—	2	1	no	—
60	C₂×C₁₆	D₄×C₅₃	1	1	yes	characteristic
61	#107	—	4	1	no	—
62	C₃₂	C₂×C₄×C₅₃	1	1	yes	characteristic
63	C₃₂	D₄×C₅₃	1	1	yes	characteristic
64	#41	—	8	1	no	—
65	Q₁₆	—	4	1	no	—
66	SD₁₆	—	8	1	no	—
67	D₈	—	4	1	no	—
68	C₂×C₈	D₈×C₅₃	1	1	yes	characteristic
69	C₄⋊C₄	—	8	1	no	—
70	C₁₆	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
71	C₁₆	SD₁₆×C₅₃	1	1	yes	characteristic
72	Q₈	—	8	1	no	—
73	D₄	—	8	1	no	—
74	D₄	—	16	1	no	—
75	C₂×C₄	—	16	1	no	—
76	C₂×C₄	—	16	1	no	—
77	C₂×C₄	D₁₆×C₅₃	1	1	yes	characteristic
78	C₈	#102×C₅₃	1	1	yes	characteristic
79	C₈	SD₃₂×C₅₃	1	1	yes	characteristic
80	C₂²	—	16	1	no	—
81	C₂²	C₅₃×D₃₂	1	1	yes	characteristic
82	C₄	—	16	1	no	—
83	C₄	—	32	1	no	—
84	C₄	C₅₃×#357	1	1	yes	characteristic
85	C₄	C₅₃×SD₆₄	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	32	1	no	—
88	C₂	C₅₃×#1415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅₃	#1415
commutator	C₃₂	C₂×C₄×C₅₃
Frattini, cosocle	C₂×C₃₂	C₂²×C₅₃
prime radical	C₅₃×#2262	C₂
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
2. C₅₃×M₁₂₈	C₂	semidirect
3. C₅₃×#1424	C₂	semidirect
5. C₅₃×Q₆₄	C₄	semidirect
7. C₅₃×D₃₂	C₄	semidirect
35. #7540	C₅₃	direct (N × H)
43. M₁₂₈	C₂×C₅₃	semidirect
44. #1424	C₂×C₅₃	semidirect
49. Q₆₄	C₄×C₅₃	semidirect
51. D₃₂	C₄×C₅₃	semidirect
55. C₅₃	#7540	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×#2262	2
C₅₃×M₁₂₈	2
C₅₃×#1424	2
#7540	53

Subgroups — C53×#7540 #27215052

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7540 #27215052