Subgroups — C₅₃×#7539 #27215051

Order 13568.

88 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	C₂×C₅₃×C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#1425	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#508	—	2	1	no	—
5	C₅₃×Q₆₄	C₄	1	1	yes	characteristic
6	C₅₃×SD₆₄	—	2	1	no	—
7	C₅₃×D₃₂	C₄	1	1	yes	characteristic
8	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#368	—	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	#108×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	M₁₆×C₅₃	—	8	1	no	—
24	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
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	D₁₆	1	1	yes	characteristic
32	C₈×C₅₃	—	16	1	no	—
33	C₈×C₅₃	#102	1	1	yes	characteristic
34	C₈×C₅₃	SD₃₂	1	1	yes	characteristic
35	#7539	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂²×C₅₃	—	16	1	no	—
37	C₂²×C₅₃	SD₆₄	1	1	yes	characteristic
38	C₄×C₅₃	—	16	1	no	—
39	C₄×C₅₃	#357	1	1	yes	characteristic
40	C₄×C₅₃	D₃₂	1	1	yes	characteristic
41	#2262	C₂×C₅₃	1	1	yes	characteristic
42	C₂×C₆₄	C₂×C₅₃	1	1	yes	characteristic, complemented
43	#1425	C₂×C₅₃	1	1	yes	characteristic, complemented
44	C₂×C₅₃	—	2	1	no	—
45	C₂×C₅₃	—	32	1	no	—
46	C₂×C₅₃	#1415	1	1	yes	characteristic
47	#508	—	2	1	no	—
48	Q₆₄	C₄×C₅₃	1	1	yes	characteristic
49	SD₆₄	—	2	1	no	—
50	D₃₂	C₄×C₅₃	1	1	yes	characteristic
51	C₂×C₃₂	C₂²×C₅₃	1	1	yes	characteristic
52	#368	—	2	1	no	—
53	C₆₄	—	2	1	no	—
54	C₅₃	#7539	1	1	yes	characteristic, direct product factor, Sylow
55	#135	—	4	1	no	—
56	Q₃₂	—	2	1	no	—
57	SD₃₂	—	4	1	no	—
58	D₁₆	—	2	1	no	—
59	C₂×C₁₆	D₄×C₅₃	1	1	yes	characteristic
60	#108	—	4	1	no	—
61	C₃₂	C₂×C₄×C₅₃	1	1	yes	characteristic
62	C₃₂	D₄×C₅₃	1	1	yes	characteristic
63	#41	—	8	1	no	—
64	Q₁₆	—	4	1	no	—
65	SD₁₆	—	8	1	no	—
66	D₈	—	4	1	no	—
67	M₁₆	—	8	1	no	—
68	C₂×C₈	D₈×C₅₃	1	1	yes	characteristic
69	C₁₆	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
70	C₁₆	SD₁₆×C₅₃	1	1	yes	characteristic
71	Q₈	—	8	1	no	—
72	D₄	—	8	1	no	—
73	D₄	—	16	1	no	—
74	C₂×C₄	—	16	1	no	—
75	C₂×C₄	D₁₆×C₅₃	1	1	yes	characteristic
76	C₈	—	16	1	no	—
77	C₈	#102×C₅₃	1	1	yes	characteristic
78	C₈	SD₃₂×C₅₃	1	1	yes	characteristic
79	C₂²	—	16	1	no	—
80	C₂²	C₅₃×SD₆₄	1	1	yes	characteristic
81	C₄	—	16	1	no	—
82	C₄	C₅₃×#357	1	1	yes	characteristic
83	C₄	C₅₃×D₃₂	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	32	1	no	—
86	C₂	C₅₃×#1415	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅₃	D₃₂
commutator	C₃₂	C₂×C₄×C₅₃
Frattini, cosocle	C₂×C₃₂	C₂²×C₅₃
socle	C₂×C₅₃	#1415
prime radical	C₅₃×#2262	C₂
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×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. C₂×C₅₃×C₆₄	C₂	semidirect
3. C₅₃×#1425	C₂	semidirect
35. #7539	C₅₃	direct (N × H)
42. C₂×C₆₄	C₂×C₅₃	semidirect
43. #1425	C₂×C₅₃	semidirect
54. C₅₃	#7539	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
C₂×C₅₃×C₆₄	2
C₅₃×#1425	2
#7539	53

Subgroups — C53×#7539 #27215051

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7539 #27215051