Subgroups — C₅₃×#7538 #27215050

Order 13568.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×Q₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×C₆₄	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1423	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₃₂×C₅₃	—	2	1	no	—
5	C₅₃×Q₆₄	—	2	1	no	—
6	C₅₃×Q₆₄	C₄	1	2	yes	complemented
7	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×#366	—	2	1	no	—
9	C₅₃×C₆₄	—	2	1	no	—
10	C₂×Q₁₆×C₅₃	—	4	1	no	—
11	Q₃₂×C₅₃	—	2	2	no	—
12	Q₃₂×C₅₃	—	4	1	no	—
13	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
14	#107×C₅₃	—	4	1	no	—
15	C₃₂×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₅₃	D₄	1	1	yes	characteristic
17	C₂×Q₈×C₅₃	—	8	1	no	—
18	Q₁₆×C₅₃	—	4	2	no	—
19	Q₁₆×C₅₃	—	8	1	no	—
20	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
21	C₄⋊C₄×C₅₃	—	8	1	no	—
22	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₁₆×C₅₃	SD₁₆	1	1	yes	characteristic
24	Q₈×C₅₃	—	8	2	no	—
25	Q₈×C₅₃	—	16	1	no	—
26	C₂×C₄×C₅₃	—	16	1	no	—
27	C₂×C₄×C₅₃	—	16	1	no	—
28	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
29	C₈×C₅₃	#102	1	1	yes	characteristic
30	C₈×C₅₃	SD₃₂	1	1	yes	characteristic
31	#7538	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂²×C₅₃	D₃₂	1	1	yes	characteristic
33	C₄×C₅₃	—	16	2	no	—
34	C₄×C₅₃	—	32	1	no	—
35	C₄×C₅₃	#357	1	1	yes	characteristic
36	C₄×C₅₃	SD₆₄	1	1	yes	characteristic
37	C₂×Q₆₄	C₂×C₅₃	1	1	yes	characteristic
38	C₂×C₆₄	C₂×C₅₃	1	1	yes	characteristic
39	#1423	C₂×C₅₃	1	1	yes	characteristic
40	C₂×C₅₃	#1415	1	1	yes	characteristic
41	C₂×C₅₃	SD₁₂₈	1	1	yes	characteristic
42	C₂×C₅₃	Q₁₂₈	1	1	yes	characteristic
43	C₂×Q₃₂	—	2	1	no	—
44	Q₆₄	—	2	1	no	—
45	Q₆₄	C₄×C₅₃	1	2	yes	complemented
46	C₂×C₃₂	C₂²×C₅₃	1	1	yes	characteristic
47	#366	—	2	1	no	—
48	C₆₄	—	2	1	no	—
49	C₅₃	#7538	1	1	yes	characteristic, direct product factor, Sylow
50	C₂×Q₁₆	—	4	1	no	—
51	Q₃₂	—	2	2	no	—
52	Q₃₂	—	4	1	no	—
53	C₂×C₁₆	D₄×C₅₃	1	1	yes	characteristic
54	#107	—	4	1	no	—
55	C₃₂	C₂×C₄×C₅₃	1	1	yes	characteristic
56	C₃₂	D₄×C₅₃	1	1	yes	characteristic
57	C₂×Q₈	—	8	1	no	—
58	Q₁₆	—	4	2	no	—
59	Q₁₆	—	8	1	no	—
60	C₂×C₈	D₈×C₅₃	1	1	yes	characteristic
61	C₄⋊C₄	—	8	1	no	—
62	C₁₆	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
63	C₁₆	SD₁₆×C₅₃	1	1	yes	characteristic
64	Q₈	—	8	2	no	—
65	Q₈	—	16	1	no	—
66	C₂×C₄	—	16	1	no	—
67	C₂×C₄	—	16	1	no	—
68	C₂×C₄	D₁₆×C₅₃	1	1	yes	characteristic
69	C₈	#102×C₅₃	1	1	yes	characteristic
70	C₈	SD₃₂×C₅₃	1	1	yes	characteristic
71	C₂²	C₅₃×D₃₂	1	1	yes	characteristic
72	C₄	—	16	2	no	—
73	C₄	—	32	1	no	—
74	C₄	C₅₃×#357	1	1	yes	characteristic
75	C₄	C₅₃×SD₆₄	1	1	yes	characteristic
76	C₂	C₅₃×#1415	1	1	yes	characteristic
77	C₂	C₅₃×SD₁₂₈	1	1	yes	characteristic
78	C₂	C₅₃×Q₁₂₈	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₅₃	D₃₂
commutator	C₃₂	C₂×C₄×C₅₃
Frattini, cosocle	C₂×C₃₂	C₂²×C₅₃
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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
6. C₅₃×Q₆₄	C₄	semidirect
31. #7538	C₅₃	direct (N × H)
45. Q₆₄	C₄×C₅₃	semidirect
49. C₅₃	#7538	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×Q₆₄	2
C₂×C₅₃×C₆₄	2
C₅₃×#1423	2
#7538	53

Subgroups — C53×#7538 #27215050

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7538 #27215050