Subgroups — C₅₃×#1371 #20270308

Order 6784.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#335	C₂	1	2	yes	maximal
3	C₂×C₁₆×C₅₃	C₄	1	1	yes	characteristic
4	C₂×C₁₆×C₅₃	C₄	1	1	yes	characteristic
5	C₄⋊C₈×C₅₃	—	2	2	no	—
6	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×C₈×C₅₃	—	4	2	no	—
8	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
9	C₂×C₈×C₅₃	Q₈	1	1	yes	characteristic
10	C₄²×C₅₃	D₄	1	1	yes	characteristic
11	C₁₆×C₅₃	—	2	1	no	—
12	C₁₆×C₅₃	C₈	1	2	yes	complemented
13	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
14	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
15	C₂×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
16	C₈×C₅₃	—	2	1	no	—
17	C₈×C₅₃	—	8	2	no	—
18	C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
19	C₈×C₅₃	M₁₆	1	1	yes	characteristic
20	C₂²×C₅₃	#107	1	1	yes	characteristic
21	C₄×C₅₃	—	2	1	no	—
22	C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
23	C₄×C₅₃	#108	1	1	yes	characteristic
24	C₄×C₅₃	D₁₆	1	1	yes	characteristic
25	C₄×C₅₃	Q₃₂	1	1	yes	characteristic
26	#1371	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂×C₅₃	#335	1	1	yes	characteristic
28	C₂×C₅₃	#366	1	1	yes	characteristic
29	C₂×C₅₃	#368	1	1	yes	characteristic
30	C₄×C₁₆	C₂×C₅₃	1	1	yes	characteristic
31	#335	C₂×C₅₃	1	2	yes	—
32	C₅₃	#1371	1	1	yes	characteristic, direct product factor, Sylow
33	C₂×C₁₆	C₄×C₅₃	1	1	yes	characteristic
34	C₂×C₁₆	C₄×C₅₃	1	1	yes	characteristic
35	C₄⋊C₈	—	2	2	no	—
36	C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
37	C₂×C₈	—	4	2	no	—
38	C₂×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
39	C₂×C₈	Q₈×C₅₃	1	1	yes	characteristic
40	C₄²	D₄×C₅₃	1	1	yes	characteristic
41	C₁₆	—	2	1	no	—
42	C₁₆	C₈×C₅₃	1	2	yes	complemented
43	C₂×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
44	C₂×C₄	D₈×C₅₃	1	1	yes	characteristic
45	C₂×C₄	Q₁₆×C₅₃	1	1	yes	characteristic
46	C₈	—	2	1	no	—
47	C₈	—	8	2	no	—
48	C₈	C₂×C₈×C₅₃	1	1	yes	characteristic
49	C₈	M₁₆×C₅₃	1	1	yes	characteristic
50	C₂²	#107×C₅₃	1	1	yes	characteristic
51	C₄	—	2	1	no	—
52	C₄	C₄⋊C₈×C₅₃	1	1	yes	characteristic
53	C₄	#108×C₅₃	1	1	yes	characteristic
54	C₄	D₁₆×C₅₃	1	1	yes	characteristic
55	C₄	Q₃₂×C₅₃	1	1	yes	characteristic
56	C₂	C₅₃×#335	1	1	yes	characteristic
57	C₂	C₅₃×#366	1	1	yes	characteristic
58	C₂	C₅₃×#368	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	D₈
commutator	C₈	C₂×C₈×C₅₃
Frattini, cosocle	C₄×C₈	C₂²×C₅₃
socle, prime radical	C₂²×C₅₃	#107
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₄²×C₅₃ ◁ C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₁₆×C₅₃	C₈	semidirect
26. #1371	C₅₃	direct (N × H)
32. C₅₃	#1371	direct (N × H)
42. C₁₆	C₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₁₆×C₅₃	2
C₅₃×#335	2
#1371	53

Subgroups — C53×#1371 #20270308

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#1371 #20270308