Subgroups — C₅₃×#7457 #27214969

Order 13568.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#1398	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1369	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1368	C₂	1	1	yes	maximal, characteristic
4	C₅₃×M₆₄	C₄	1	1	yes	characteristic
5	C₅₃×M₆₄	C₄	1	1	yes	characteristic
6	C₅₃×#364	—	2	1	no	—
7	C₅₃×#346	C₂²	1	1	yes	characteristic
8	C₅₃×#334	—	2	1	no	—
9	M₃₂×C₅₃	—	4	1	no	—
10	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
11	C₂×C₁₆×C₅₃	Q₈	1	1	yes	characteristic
12	C₄⋊C₈×C₅₃	—	4	1	no	—
13	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
14	C₃₂×C₅₃	—	2	1	no	—
15	C₃₂×C₅₃	C₈	1	2	yes	complemented
16	C₂×C₈×C₅₃	—	8	1	no	—
17	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
19	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
20	C₁₆×C₅₃	—	2	1	no	—
21	C₁₆×C₅₃	—	8	1	no	—
22	C₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
23	C₁₆×C₅₃	M₁₆	1	1	yes	characteristic
24	C₂×C₄×C₅₃	—	2	1	no	—
25	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
26	C₈×C₅₃	—	2	1	no	—
27	C₈×C₅₃	—	16	1	no	—
28	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
29	C₈×C₅₃	#108	1	1	yes	characteristic
30	#7457	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂²×C₅₃	#365	1	1	yes	characteristic
32	C₄×C₅₃	—	4	1	no	—
33	C₄×C₅₃	#334	1	1	yes	characteristic
34	C₄×C₅₃	#365	1	1	yes	characteristic
35	#1398	C₂×C₅₃	1	1	yes	characteristic
36	#1369	C₂×C₅₃	1	1	yes	characteristic
37	#1368	C₂×C₅₃	1	1	yes	characteristic
38	C₂×C₅₃	—	2	1	no	—
39	C₂×C₅₃	#1368	1	1	yes	characteristic
40	M₆₄	C₄×C₅₃	1	1	yes	characteristic
41	M₆₄	C₄×C₅₃	1	1	yes	characteristic
42	#364	—	2	1	no	—
43	#346	C₂²×C₅₃	1	1	yes	characteristic
44	#334	—	2	1	no	—
45	C₅₃	#7457	1	1	yes	characteristic, direct product factor, Sylow
46	M₃₂	—	4	1	no	—
47	C₂×C₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
48	C₂×C₁₆	Q₈×C₅₃	1	1	yes	characteristic
49	C₄⋊C₈	—	4	1	no	—
50	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
51	C₃₂	—	2	1	no	—
52	C₃₂	C₈×C₅₃	1	2	yes	complemented
53	C₂×C₈	—	8	1	no	—
54	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
55	C₂×C₈	SD₁₆×C₅₃	1	1	yes	characteristic
56	C₄²	SD₁₆×C₅₃	1	1	yes	characteristic
57	C₁₆	—	2	1	no	—
58	C₁₆	—	8	1	no	—
59	C₁₆	C₂×C₈×C₅₃	1	1	yes	characteristic
60	C₁₆	M₁₆×C₅₃	1	1	yes	characteristic
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	#106×C₅₃	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	16	1	no	—
65	C₈	C₄⋊C₈×C₅₃	1	1	yes	characteristic
66	C₈	#108×C₅₃	1	1	yes	characteristic
67	C₂²	C₅₃×#365	1	1	yes	characteristic
68	C₄	—	4	1	no	—
69	C₄	C₅₃×#334	1	1	yes	characteristic
70	C₄	C₅₃×#365	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	C₅₃×#1368	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅₃	#1368
commutator	C₁₆	C₂×C₈×C₅₃
Frattini, cosocle	#346	C₂²×C₅₃
prime radical	C₂²×C₅₃	#365
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₄×C₈×C₅₃ ◁ C₅₃×#346 ◁ G

Complements

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

N	H ≅ G/N	type
15. C₃₂×C₅₃	C₈	semidirect
30. #7457	C₅₃	direct (N × H)
45. C₅₃	#7457	direct (N × H)
52. C₃₂	C₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#1398	2
C₅₃×#1369	2
C₅₃×#1368	2
#7457	53

Subgroups — C53×#7457 #27214969

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7457 #27214969