Subgroups — C₅₃×#1455 #20270392

Order 6784.

214 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×M₁₆×C₅₃	C₂	1	3	yes	maximal
2	C₅₃×#322	C₂	1	3	yes	maximal, complemented
3	C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×M₁₆×C₅₃	C₄	1	6	yes	—
5	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
6	#97×C₅₃	C₄	1	6	yes	—
7	C₄×C₈×C₅₃	C₂²	1	3	yes	—
8	C₄×C₈×C₅₃	C₂²	1	3	yes	—
9	C₂²×C₄×C₅₃	C₂×C₄	1	3	yes	—
10	M₁₆×C₅₃	—	2	12	no	—
11	C₂×C₈×C₅₃	C₂×C₄	1	6	yes	—
12	C₂×C₈×C₅₃	C₂×C₄	1	6	yes	—
13	C₄²×C₅₃	C₂×C₄	1	3	yes	—
14	C₄²×C₅₃	C₂³	1	1	yes	characteristic
15	C₂³×C₅₃	C₄²	1	1	yes	characteristic
16	C₂×C₄×C₅₃	—	2	6	no	—
17	C₂×C₄×C₅₃	C₄²	1	3	yes	—
18	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
19	C₈×C₅₃	—	2	6	no	—
20	C₈×C₅₃	M₁₆	1	12	yes	complemented
21	C₂²×C₅₃	—	2	3	no	—
22	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
23	C₄×C₅₃	—	2	3	no	—
24	C₄×C₅₃	C₂×M₁₆	1	6	yes	—
25	#1455	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂×C₅₃	—	4	1	no	—
27	C₂×C₅₃	C₄×M₁₆	1	3	yes	—
28	C₄×M₁₆	C₂×C₅₃	1	3	yes	—
29	#322	C₂×C₅₃	1	3	yes	complemented
30	C₈²	C₂×C₅₃	1	1	yes	characteristic, complemented
31	C₅₃	#1455	1	1	yes	characteristic, direct product factor, Sylow
32	C₂×M₁₆	C₄×C₅₃	1	6	yes	—
33	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
34	#97	C₄×C₅₃	1	6	yes	—
35	C₄×C₈	C₂²×C₅₃	1	3	yes	—
36	C₄×C₈	C₂²×C₅₃	1	3	yes	—
37	C₂²×C₄	C₂×C₄×C₅₃	1	3	yes	—
38	M₁₆	—	2	12	no	—
39	C₂×C₈	C₂×C₄×C₅₃	1	6	yes	—
40	C₂×C₈	C₂×C₄×C₅₃	1	6	yes	—
41	C₄²	C₂×C₄×C₅₃	1	3	yes	—
42	C₄²	C₂³×C₅₃	1	1	yes	characteristic
43	C₂³	C₄²×C₅₃	1	1	yes	characteristic
44	C₂×C₄	—	2	6	no	—
45	C₂×C₄	C₄²×C₅₃	1	3	yes	—
46	C₂×C₄	C₂²×C₄×C₅₃	1	3	yes	—
47	C₈	—	2	6	no	—
48	C₈	M₁₆×C₅₃	1	12	yes	complemented
49	C₂²	—	2	3	no	—
50	C₂²	C₂×C₄²×C₅₃	1	1	yes	characteristic
51	C₄	—	2	3	no	—
52	C₄	C₂×M₁₆×C₅₃	1	6	yes	—
53	C₂	—	4	1	no	—
54	C₂	C₄×M₁₆×C₅₃	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₅₃×#322	C₂	semidirect
3. C₈²×C₅₃	C₂	semidirect
20. C₈×C₅₃	M₁₆	semidirect
25. #1455	C₅₃	direct (N × H)
29. #322	C₂×C₅₃	semidirect
30. C₈²	C₂×C₅₃	semidirect
31. C₅₃	#1455	direct (N × H)
48. C₈	M₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×M₁₆×C₅₃	2
C₅₃×#322	2
C₈²×C₅₃	2
#1455	53

Subgroups — C53×#1455 #20270392

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#1455 #20270392