Subgroups — C₅₃×#13732 #27221244

Order 13568.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2253	C₂	1	2	yes	maximal
2	C₅₃×#1424	C₂	1	2	yes	maximal
3	C₅₃×#1423	C₂	1	2	yes	maximal
4	C₄×C₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#499	—	2	2	no	—
6	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×#367	—	2	2	no	—
9	C₅₃×#366	C₂²	1	4	yes	—
10	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
11	#125×C₅₃	—	4	2	no	—
12	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
13	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
14	#107×C₅₃	—	2	4	no	—
15	#106×C₅₃	—	4	2	no	—
16	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
17	C₃₂×C₅₃	Q₈	1	2	yes	—
18	C₃₂×C₅₃	Q₈	1	2	yes	—
19	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
20	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
21	C₄⋊C₄×C₅₃	—	4	4	no	—
22	C₄⋊C₄×C₅₃	—	8	4	no	—
23	C₄²×C₅₃	D₈	1	1	yes	characteristic
24	C₁₆×C₅₃	Q₁₆	1	2	yes	—
25	C₁₆×C₅₃	C₂×Q₈	1	1	yes	characteristic
26	C₁₆×C₅₃	C₂×Q₈	1	1	yes	characteristic
27	C₂×C₄×C₅₃	—	8	4	no	—
28	C₂×C₄×C₅₃	D₁₆	1	2	yes	—
29	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
30	C₈×C₅₃	Q₃₂	1	2	yes	—
31	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
32	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
33	#13732	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
35	C₄×C₅₃	—	2	2	no	—
36	C₄×C₅₃	—	16	4	no	—
37	C₄×C₅₃	#500	1	1	yes	characteristic
38	C₄×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
39	#2253	C₂×C₅₃	1	2	yes	—
40	#1424	C₂×C₅₃	1	2	yes	—
41	#1423	C₂×C₅₃	1	2	yes	—
42	C₄×C₃₂	C₂×C₅₃	1	1	yes	characteristic
43	C₂×C₅₃	#2252	1	1	yes	characteristic
44	C₂×C₅₃	#2262	1	2	yes	—
45	#499	—	2	2	no	—
46	C₂×C₃₂	C₂²×C₅₃	1	1	yes	characteristic
47	C₂×C₃₂	C₂²×C₅₃	1	1	yes	characteristic
48	#367	—	2	2	no	—
49	#366	C₂²×C₅₃	1	4	yes	—
50	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
51	C₅₃	#13732	1	1	yes	characteristic, direct product factor, Sylow
52	#125	—	4	2	no	—
53	C₂×C₁₆	D₄×C₅₃	1	1	yes	characteristic
54	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
55	#107	—	2	4	no	—
56	#106	—	4	2	no	—
57	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
58	C₃₂	Q₈×C₅₃	1	2	yes	—
59	C₃₂	Q₈×C₅₃	1	2	yes	—
60	C₂×C₈	D₈×C₅₃	1	1	yes	characteristic
61	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
62	C₄⋊C₄	—	4	4	no	—
63	C₄⋊C₄	—	8	4	no	—
64	C₄²	D₈×C₅₃	1	1	yes	characteristic
65	C₁₆	Q₁₆×C₅₃	1	2	yes	—
66	C₁₆	C₂×Q₈×C₅₃	1	1	yes	characteristic
67	C₁₆	C₂×Q₈×C₅₃	1	1	yes	characteristic
68	C₂×C₄	—	8	4	no	—
69	C₂×C₄	D₁₆×C₅₃	1	2	yes	—
70	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
71	C₈	Q₃₂×C₅₃	1	2	yes	—
72	C₈	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
73	C₈	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
74	C₂²	C₂×D₁₆×C₅₃	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	16	4	no	—
77	C₄	C₅₃×#500	1	1	yes	characteristic
78	C₄	C₂×Q₃₂×C₅₃	1	1	yes	characteristic
79	C₂	C₅₃×#2252	1	1	yes	characteristic
80	C₂	C₅₃×#2262	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₅₃	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₅₃
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ C₄×C₈×C₅₃ ◁ C₄×C₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
33. #13732	C₅₃	direct (N × H)
51. C₅₃	#13732	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×#2253	2
C₅₃×#1424	2
C₅₃×#1423	2
C₄×C₃₂×C₅₃	2
#13732	53

Subgroups — C53×#13732 #27221244

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#13732 #27221244