Subgroups — C₅₃×#1747 #20270684

Order 6784.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#405	C₂	1	3	yes	maximal
2	C₅₃×#338	C₂	1	1	yes	maximal, characteristic
3	C₅₃×H(3,ℤ/4)	C₂	1	3	yes	maximal
4	C₂×M₁₆×C₅₃	C₂²	1	3	yes	—
5	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
6	#117×C₅₃	C₂²	1	3	yes	—
7	#97×C₅₃	C₄	1	6	yes	complemented
8	C₄×C₈×C₅₃	C₄	1	6	yes	complemented
9	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
10	M₁₆×C₅₃	—	2	6	no	—
11	C₂×C₈×C₅₃	C₂×C₄	1	6	yes	—
12	C₂×C₈×C₅₃	C₂×C₄	1	3	yes	—
13	C₂×C₈×C₅₃	C₂×C₄	1	3	yes	—
14	C₄⋊C₄×C₅₃	—	2	3	no	—
15	C₂²⋊C₄×C₅₃	—	2	3	no	—
16	C₄²×C₅₃	C₂×C₄	1	6	yes	—
17	C₂³×C₅₃	#41	1	1	yes	characteristic
18	C₂×C₄×C₅₃	—	2	6	no	—
19	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
20	C₂×C₄×C₅₃	#41	1	3	yes	—
21	C₈×C₅₃	—	2	6	no	—
22	C₈×C₅₃	C₄²	1	3	yes	—
23	C₈×C₅₃	C₄²	1	1	yes	characteristic
24	C₂²×C₅₃	—	4	1	no	—
25	C₂²×C₅₃	#117	1	3	yes	—
26	C₄×C₅₃	—	4	6	no	—
27	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
28	C₄×C₅₃	#117	1	3	yes	—
29	#1747	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂×C₅₃	—	2	3	no	—
31	C₂×C₅₃	#376	1	1	yes	characteristic
32	#405	C₂×C₅₃	1	3	yes	—
33	#338	C₂×C₅₃	1	1	yes	characteristic
34	H(3,ℤ/4)	C₂×C₅₃	1	3	yes	—
35	C₅₃	#1747	1	1	yes	characteristic, direct product factor, Sylow
36	C₂×M₁₆	C₂²×C₅₃	1	3	yes	—
37	C₂²×C₈	C₂²×C₅₃	1	1	yes	characteristic
38	#117	C₂²×C₅₃	1	3	yes	—
39	#97	C₄×C₅₃	1	6	yes	complemented
40	C₄×C₈	C₄×C₅₃	1	6	yes	complemented
41	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
42	M₁₆	—	2	6	no	—
43	C₂×C₈	C₂×C₄×C₅₃	1	6	yes	—
44	C₂×C₈	C₂×C₄×C₅₃	1	3	yes	—
45	C₂×C₈	C₂×C₄×C₅₃	1	3	yes	—
46	C₄⋊C₄	—	2	3	no	—
47	C₂²⋊C₄	—	2	3	no	—
48	C₄²	C₂×C₄×C₅₃	1	6	yes	—
49	C₂³	#41×C₅₃	1	1	yes	characteristic
50	C₂×C₄	—	2	6	no	—
51	C₂×C₄	C₂²×C₄×C₅₃	1	3	yes	—
52	C₂×C₄	#41×C₅₃	1	3	yes	—
53	C₈	—	2	6	no	—
54	C₈	C₄²×C₅₃	1	3	yes	—
55	C₈	C₄²×C₅₃	1	1	yes	characteristic
56	C₂²	—	4	1	no	—
57	C₂²	#117×C₅₃	1	3	yes	—
58	C₄	—	4	6	no	—
59	C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
60	C₄	#117×C₅₃	1	3	yes	—
61	C₂	—	2	3	no	—
62	C₂	C₅₃×#376	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₅₃	C₄²
commutator	C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂²×C₄	C₂³×C₅₃
socle	C₂×C₅₃	#376
prime radical	C₂³×C₅₃	#41
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
7. #97×C₅₃	C₄	semidirect
8. C₄×C₈×C₅₃	C₄	semidirect
29. #1747	C₅₃	direct (N × H)
35. C₅₃	#1747	direct (N × H)
39. #97	C₄×C₅₃	semidirect
40. C₄×C₈	C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#405	2
C₅₃×#338	2
C₅₃×H(3,ℤ/4)	2
#1747	53

Subgroups — C53×#1747 #20270684

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#1747 #20270684