Subgroups — C₅₃×#13733 #27221245

Order 13568.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2252	C₂	1	2	yes	maximal
2	C₅₃×#1424	C₂	1	4	yes	maximal
3	C₄×C₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#500	—	2	2	no	—
5	C₂×C₃₂×C₅₃	C₂²	1	2	yes	—
6	C₅₃×#366	—	2	2	no	—
7	C₅₃×#366	C₂²	1	4	yes	—
8	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
9	C₄⋊Q₈×C₅₃	—	4	2	no	—
10	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
11	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
12	#107×C₅₃	—	2	4	no	—
13	#107×C₅₃	—	4	2	no	—
14	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
15	C₃₂×C₅₃	Q₈	1	4	yes	complemented
16	C₂×Q₈×C₅₃	—	8	2	no	—
17	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
18	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
19	C₄⋊C₄×C₅₃	—	4	4	no	—
20	C₄⋊C₄×C₅₃	—	8	2	no	—
21	C₄²×C₅₃	D₈	1	1	yes	characteristic
22	C₁₆×C₅₃	Q₁₆	1	2	yes	—
23	C₁₆×C₅₃	C₂×Q₈	1	2	yes	—
24	Q₈×C₅₃	—	16	4	no	—
25	C₂×C₄×C₅₃	—	8	4	no	—
26	C₂×C₄×C₅₃	D₁₆	1	2	yes	—
27	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
28	C₈×C₅₃	Q₃₂	1	2	yes	—
29	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
30	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
31	#13733	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
33	C₄×C₅₃	—	16	4	no	—
34	C₄×C₅₃	SD₆₄	1	4	yes	—
35	C₄×C₅₃	#500	1	1	yes	characteristic
36	C₄×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
37	#2252	C₂×C₅₃	1	2	yes	—
38	#1424	C₂×C₅₃	1	4	yes	—
39	C₄×C₃₂	C₂×C₅₃	1	1	yes	characteristic
40	C₂×C₅₃	#2252	1	1	yes	characteristic
41	C₂×C₅₃	C₂×SD₆₄	1	2	yes	—
42	#500	—	2	2	no	—
43	C₂×C₃₂	C₂²×C₅₃	1	2	yes	—
44	#366	—	2	2	no	—
45	#366	C₂²×C₅₃	1	4	yes	—
46	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
47	C₅₃	#13733	1	1	yes	characteristic, direct product factor, Sylow
48	C₄⋊Q₈	—	4	2	no	—
49	C₂×C₁₆	D₄×C₅₃	1	1	yes	characteristic
50	C₂×C₁₆	C₂³×C₅₃	1	1	yes	characteristic
51	#107	—	2	4	no	—
52	#107	—	4	2	no	—
53	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
54	C₃₂	Q₈×C₅₃	1	4	yes	complemented
55	C₂×Q₈	—	8	2	no	—
56	C₂×C₈	D₈×C₅₃	1	1	yes	characteristic
57	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
58	C₄⋊C₄	—	4	4	no	—
59	C₄⋊C₄	—	8	2	no	—
60	C₄²	D₈×C₅₃	1	1	yes	characteristic
61	C₁₆	Q₁₆×C₅₃	1	2	yes	—
62	C₁₆	C₂×Q₈×C₅₃	1	2	yes	—
63	Q₈	—	16	4	no	—
64	C₂×C₄	—	8	4	no	—
65	C₂×C₄	D₁₆×C₅₃	1	2	yes	—
66	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
67	C₈	Q₃₂×C₅₃	1	2	yes	—
68	C₈	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
69	C₈	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
70	C₂²	C₂×D₁₆×C₅₃	1	1	yes	characteristic
71	C₄	—	16	4	no	—
72	C₄	C₅₃×SD₆₄	1	4	yes	—
73	C₄	C₅₃×#500	1	1	yes	characteristic
74	C₄	C₂×Q₃₂×C₅₃	1	1	yes	characteristic
75	C₂	C₅₃×#2252	1	1	yes	characteristic
76	C₂	C₂×C₅₃×SD₆₄	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
15. C₃₂×C₅₃	Q₈	semidirect
31. #13733	C₅₃	direct (N × H)
47. C₅₃	#13733	direct (N × H)
54. C₃₂	Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#2252	2
C₅₃×#1424	2
C₄×C₃₂×C₅₃	2
#13733	53

Subgroups — C53×#13733 #27221245

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#13733 #27221245