Subgroups — C₂²×C₈²×C₅₃ #27224822

Order 13568.

1342 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₈×C₅₃	C₂	1	3	yes	maximal
2	C₂×C₈²×C₅₃	C₂	1	12	yes	maximal, direct product factor
3	C₂³×C₈×C₅₃	C₄	1	6	yes	—
4	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
5	C₂×C₄×C₈×C₅₃	C₄	1	18	yes	—
6	C₂×C₄×C₈×C₅₃	C₂²	1	18	yes	—
7	C₈²×C₅₃	C₂²	1	16	yes	direct product factor
8	C₂³×C₄×C₅₃	C₂×C₄	1	3	yes	—
9	C₂²×C₈×C₅₃	C₈	1	48	yes	direct product factor
10	C₂²×C₈×C₅₃	C₂×C₄	1	36	yes	—
11	C₂×C₄²×C₅₃	C₂×C₄	1	9	yes	—
12	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
13	C₄×C₈×C₅₃	C₂×C₄	1	36	yes	—
14	C₄×C₈×C₅₃	C₂³	1	12	yes	—
15	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	C₄²	1	9	yes	—
17	C₂²×C₄×C₅₃	C₂×C₈	1	24	yes	—
18	C₂²×C₄×C₅₃	C₂²×C₄	1	9	yes	—
19	C₂×C₈×C₅₃	C₂×C₈	1	144	yes	direct product factor
20	C₂×C₈×C₅₃	C₂²×C₄	1	24	yes	—
21	C₄²×C₅₃	C₄²	1	6	yes	—
22	C₄²×C₅₃	C₂²×C₄	1	9	yes	—
23	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
24	C₂³×C₅₃	C₄×C₈	1	12	yes	—
25	C₂³×C₅₃	C₂×C₄²	1	3	yes	—
26	C₂×C₄×C₅₃	C₄×C₈	1	36	yes	—
27	C₂×C₄×C₅₃	C₂×C₄²	1	9	yes	—
28	C₂×C₄×C₅₃	C₂²×C₈	1	36	yes	—
29	C₂×C₄×C₅₃	C₂³×C₄	1	3	yes	—
30	C₈×C₅₃	C₂²×C₈	1	48	yes	direct product factor
31	C₂²×C₈²	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂²×C₅₃	C₈²	1	16	yes	direct product factor
33	C₂²×C₅₃	C₂×C₄×C₈	1	18	yes	—
34	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
35	C₄×C₅₃	C₂×C₄×C₈	1	18	yes	—
36	C₄×C₅₃	C₂³×C₈	1	6	yes	—
37	C₂²×C₄×C₈	C₂×C₅₃	1	3	yes	—
38	C₂×C₈²	C₂×C₅₃	1	12	yes	direct product factor
39	C₂×C₅₃	C₂×C₈²	1	12	yes	direct product factor
40	C₂×C₅₃	C₂²×C₄×C₈	1	3	yes	—
41	C₂³×C₈	C₄×C₅₃	1	6	yes	—
42	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
43	C₂×C₄×C₈	C₄×C₅₃	1	18	yes	—
44	C₂×C₄×C₈	C₂²×C₅₃	1	18	yes	—
45	C₈²	C₂²×C₅₃	1	16	yes	direct product factor
46	C₅₃	C₂²×C₈²	1	1	yes	characteristic, direct product factor, Sylow
47	C₂³×C₄	C₂×C₄×C₅₃	1	3	yes	—
48	C₂²×C₈	C₈×C₅₃	1	48	yes	direct product factor
49	C₂²×C₈	C₂×C₄×C₅₃	1	36	yes	—
50	C₂×C₄²	C₂×C₄×C₅₃	1	9	yes	—
51	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
52	C₄×C₈	C₂×C₄×C₅₃	1	36	yes	—
53	C₄×C₈	C₂³×C₅₃	1	12	yes	—
54	C₂⁴	C₄²×C₅₃	1	1	yes	characteristic
55	C₂²×C₄	C₄²×C₅₃	1	9	yes	—
56	C₂²×C₄	C₂×C₈×C₅₃	1	24	yes	—
57	C₂²×C₄	C₂²×C₄×C₅₃	1	9	yes	—
58	C₂×C₈	C₂×C₈×C₅₃	1	144	yes	direct product factor
59	C₂×C₈	C₂²×C₄×C₅₃	1	24	yes	—
60	C₄²	C₄²×C₅₃	1	6	yes	—
61	C₄²	C₂²×C₄×C₅₃	1	9	yes	—
62	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
63	C₂³	C₄×C₈×C₅₃	1	12	yes	—
64	C₂³	C₂×C₄²×C₅₃	1	3	yes	—
65	C₂×C₄	C₄×C₈×C₅₃	1	36	yes	—
66	C₂×C₄	C₂×C₄²×C₅₃	1	9	yes	—
67	C₂×C₄	C₂²×C₈×C₅₃	1	36	yes	—
68	C₂×C₄	C₂³×C₄×C₅₃	1	3	yes	—
69	C₈	C₂²×C₈×C₅₃	1	48	yes	direct product factor
70	C₂²	C₈²×C₅₃	1	16	yes	direct product factor
71	C₂²	C₂×C₄×C₈×C₅₃	1	18	yes	—
72	C₂²	C₂²×C₄²×C₅₃	1	1	yes	characteristic
73	C₄	C₂×C₄×C₈×C₅₃	1	18	yes	—
74	C₄	C₂³×C₈×C₅₃	1	6	yes	—
75	C₂	C₂×C₈²×C₅₃	1	12	yes	direct product factor
76	C₂	C₂²×C₄×C₈×C₅₃	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle, prime radical	C₂⁴×C₅₃	C₄²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₈²×C₅₃	C₂	direct (N × H)
7. C₈²×C₅₃	C₂²	direct (N × H)
9. C₂²×C₈×C₅₃	C₈	direct (N × H)
19. C₂×C₈×C₅₃	C₂×C₈	direct (N × H)
30. C₈×C₅₃	C₂²×C₈	direct (N × H)
31. C₂²×C₈²	C₅₃	direct (N × H)
32. C₂²×C₅₃	C₈²	direct (N × H)
38. C₂×C₈²	C₂×C₅₃	direct (N × H)
39. C₂×C₅₃	C₂×C₈²	direct (N × H)
45. C₈²	C₂²×C₅₃	direct (N × H)
46. C₅₃	C₂²×C₈²	direct (N × H)
48. C₂²×C₈	C₈×C₅₃	direct (N × H)
58. C₂×C₈	C₂×C₈×C₅₃	direct (N × H)
69. C₈	C₂²×C₈×C₅₃	direct (N × H)
70. C₂²	C₈²×C₅₃	direct (N × H)
75. C₂	C₂×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
C₂²×C₄×C₈×C₅₃	2
C₂×C₈²×C₅₃	2
C₂²×C₈²	53

Subgroups — C22×C82×C53 #27224822

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₈²×C₅₃ #27224822