Subgroups — C₄×C₈²×C₅₃ #27215350

Order 13568.

750 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₈×C₅₃	C₂	1	3	yes	maximal
2	C₂×C₈²×C₅₃	C₂	1	4	yes	maximal
3	C₂×C₄×C₈×C₅₃	C₄	1	12	yes	—
4	C₂×C₄×C₈×C₅₃	C₂²	1	6	yes	—
5	C₄³×C₅₃	C₂²	1	1	yes	characteristic
6	C₈²×C₅₃	C₄	1	16	yes	direct product factor
7	C₂²×C₈×C₅₃	C₂×C₄	1	12	yes	—
8	C₂×C₄²×C₅₃	C₂×C₄	1	6	yes	—
9	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
10	C₄×C₈×C₅₃	C₈	1	48	yes	direct product factor
11	C₄×C₈×C₅₃	C₂×C₄	1	24	yes	—
12	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
13	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
14	C₂×C₈×C₅₃	C₄²	1	24	yes	—
15	C₂×C₈×C₅₃	C₂×C₈	1	48	yes	—
16	C₄²×C₅₃	C₂×C₈	1	24	yes	—
17	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
18	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
19	C₂×C₄×C₅₃	C₄×C₈	1	24	yes	—
20	C₂×C₄×C₅₃	C₂×C₄²	1	6	yes	—
21	C₂×C₄×C₅₃	C₂²×C₈	1	12	yes	—
22	C₈×C₅₃	C₄×C₈	1	48	yes	direct product factor
23	C₄×C₈²	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂²×C₅₃	C₄³	1	1	yes	characteristic
25	C₂²×C₅₃	C₂×C₄×C₈	1	6	yes	—
26	C₄×C₅₃	C₈²	1	16	yes	direct product factor
27	C₄×C₅₃	C₂×C₄×C₈	1	12	yes	—
28	C₄²×C₈	C₂×C₅₃	1	3	yes	—
29	C₂×C₈²	C₂×C₅₃	1	4	yes	—
30	C₂×C₅₃	C₂×C₈²	1	4	yes	—
31	C₂×C₅₃	C₄²×C₈	1	3	yes	—
32	C₂×C₄×C₈	C₄×C₅₃	1	12	yes	—
33	C₂×C₄×C₈	C₂²×C₅₃	1	6	yes	—
34	C₄³	C₂²×C₅₃	1	1	yes	characteristic
35	C₈²	C₄×C₅₃	1	16	yes	direct product factor
36	C₅₃	C₄×C₈²	1	1	yes	characteristic, direct product factor, Sylow
37	C₂²×C₈	C₂×C₄×C₅₃	1	12	yes	—
38	C₂×C₄²	C₂×C₄×C₅₃	1	6	yes	—
39	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
40	C₄×C₈	C₈×C₅₃	1	48	yes	direct product factor
41	C₄×C₈	C₂×C₄×C₅₃	1	24	yes	—
42	C₂²×C₄	C₄²×C₅₃	1	4	yes	—
43	C₂²×C₄	C₂²×C₄×C₅₃	1	3	yes	—
44	C₂×C₈	C₄²×C₅₃	1	24	yes	—
45	C₂×C₈	C₂×C₈×C₅₃	1	48	yes	—
46	C₄²	C₂×C₈×C₅₃	1	24	yes	—
47	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
48	C₂³	C₂×C₄²×C₅₃	1	1	yes	characteristic
49	C₂×C₄	C₄×C₈×C₅₃	1	24	yes	—
50	C₂×C₄	C₂×C₄²×C₅₃	1	6	yes	—
51	C₂×C₄	C₂²×C₈×C₅₃	1	12	yes	—
52	C₈	C₄×C₈×C₅₃	1	48	yes	direct product factor
53	C₂²	C₄³×C₅₃	1	1	yes	characteristic
54	C₂²	C₂×C₄×C₈×C₅₃	1	6	yes	—
55	C₄	C₈²×C₅₃	1	16	yes	direct product factor
56	C₄	C₂×C₄×C₈×C₅₃	1	12	yes	—
57	C₂	C₂×C₈²×C₅₃	1	4	yes	—
58	C₂	C₄²×C₈×C₅₃	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
6. C₈²×C₅₃	C₄	direct (N × H)
10. C₄×C₈×C₅₃	C₈	direct (N × H)
22. C₈×C₅₃	C₄×C₈	direct (N × H)
23. C₄×C₈²	C₅₃	direct (N × H)
26. C₄×C₅₃	C₈²	direct (N × H)
35. C₈²	C₄×C₅₃	direct (N × H)
36. C₅₃	C₄×C₈²	direct (N × H)
40. C₄×C₈	C₈×C₅₃	direct (N × H)
52. C₈	C₄×C₈×C₅₃	direct (N × H)
55. C₄	C₈²×C₅₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄²×C₈×C₅₃	2
C₂×C₈²×C₅₃	2
C₄×C₈²	53

Subgroups — C4×C82×C53 #27215350

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₈²×C₅₃ #27215350