Subgroups — C₂²×M₃₂×C₅₃ #20272342

Order 6784.

400 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×M₃₂×C₅₃	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₁₆×C₅₃	C₂	1	2	yes	maximal, complemented
4	C₂³×C₄×C₅₃	C₄	1	1	yes	characteristic
5	C₂²×C₈×C₅₃	C₄	1	6	yes	—
6	C₂²×C₈×C₅₃	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₅₃	C₂²	1	6	yes	—
8	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	M₃₂×C₅₃	C₂²	1	16	yes	direct product factor
10	C₂×C₁₆×C₅₃	C₂²	1	12	yes	complemented
11	C₂⁴×C₅₃	C₈	1	1	yes	characteristic
12	C₂²×C₄×C₅₃	C₈	1	6	yes	—
13	C₂²×C₄×C₅₃	C₈	1	1	yes	characteristic
14	C₂²×C₄×C₅₃	C₂×C₄	1	6	yes	—
15	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₅₃	C₂×C₄	1	6	yes	—
17	C₂×C₈×C₅₃	C₂×C₄	1	12	yes	—
18	C₂×C₈×C₅₃	C₂×C₄	1	3	yes	—
19	C₂×C₈×C₅₃	C₂³	1	3	yes	—
20	C₂×C₈×C₅₃	C₂³	1	4	yes	—
21	C₁₆×C₅₃	C₂³	1	8	yes	complemented
22	C₂³×C₅₃	—	2	4	no	—
23	C₂³×C₅₃	C₂×C₈	1	6	yes	—
24	C₂³×C₅₃	C₂×C₈	1	1	yes	characteristic
25	C₂×C₄×C₅₃	C₂×C₈	1	12	yes	—
26	C₂×C₄×C₅₃	C₂×C₈	1	6	yes	—
27	C₂×C₄×C₅₃	C₂×C₈	1	3	yes	—
28	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
29	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
30	C₈×C₅₃	C₂²×C₄	1	4	yes	—
31	C₈×C₅₃	C₂²×C₄	1	3	yes	—
32	C₈×C₅₃	C₂⁴	1	1	yes	characteristic
33	C₂²×C₅₃	—	2	12	no	—
34	C₂²×C₅₃	M₃₂	1	4	yes	direct product factor
35	C₂²×C₅₃	C₂²×C₈	1	4	yes	—
36	C₂²×C₅₃	C₂²×C₈	1	3	yes	—
37	C₄×C₅₃	C₂²×C₈	1	3	yes	—
38	C₄×C₅₃	C₂²×C₈	1	4	yes	—
39	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
40	C₂²×M₃₂	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂×C₅₃	—	2	4	no	—
42	C₂×C₅₃	C₂×M₃₂	1	6	yes	direct product factor
43	C₂×C₅₃	C₂³×C₈	1	1	yes	characteristic
44	C₂³×C₈	C₂×C₅₃	1	1	yes	characteristic
45	C₂×M₃₂	C₂×C₅₃	1	12	yes	direct product factor
46	C₂²×C₁₆	C₂×C₅₃	1	2	yes	complemented
47	C₅₃	C₂²×M₃₂	1	1	yes	characteristic, direct product factor, Sylow
48	C₂³×C₄	C₄×C₅₃	1	1	yes	characteristic
49	C₂²×C₈	C₄×C₅₃	1	1	yes	characteristic
50	C₂²×C₈	C₄×C₅₃	1	6	yes	—
51	C₂²×C₈	C₂²×C₅₃	1	6	yes	—
52	C₂²×C₈	C₂²×C₅₃	1	1	yes	characteristic
53	M₃₂	C₂²×C₅₃	1	16	yes	direct product factor
54	C₂×C₁₆	C₂²×C₅₃	1	12	yes	complemented
55	C₂⁴	C₈×C₅₃	1	1	yes	characteristic
56	C₂²×C₄	C₈×C₅₃	1	6	yes	—
57	C₂²×C₄	C₈×C₅₃	1	1	yes	characteristic
58	C₂²×C₄	C₂×C₄×C₅₃	1	6	yes	—
59	C₂²×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
60	C₂×C₈	C₂×C₄×C₅₃	1	12	yes	—
61	C₂×C₈	C₂×C₄×C₅₃	1	6	yes	—
62	C₂×C₈	C₂×C₄×C₅₃	1	3	yes	—
63	C₂×C₈	C₂³×C₅₃	1	3	yes	—
64	C₂×C₈	C₂³×C₅₃	1	4	yes	—
65	C₁₆	C₂³×C₅₃	1	8	yes	complemented
66	C₂³	—	2	4	no	—
67	C₂³	C₂×C₈×C₅₃	1	6	yes	—
68	C₂³	C₂×C₈×C₅₃	1	1	yes	characteristic
69	C₂×C₄	C₂×C₈×C₅₃	1	12	yes	—
70	C₂×C₄	C₂×C₈×C₅₃	1	6	yes	—
71	C₂×C₄	C₂×C₈×C₅₃	1	3	yes	—
72	C₂×C₄	C₂²×C₄×C₅₃	1	4	yes	—
73	C₂×C₄	C₂²×C₄×C₅₃	1	3	yes	—
74	C₈	C₂²×C₄×C₅₃	1	3	yes	—
75	C₈	C₂²×C₄×C₅₃	1	4	yes	—
76	C₈	C₂⁴×C₅₃	1	1	yes	characteristic
77	C₂²	—	2	12	no	—
78	C₂²	M₃₂×C₅₃	1	4	yes	direct product factor
79	C₂²	C₂²×C₈×C₅₃	1	4	yes	—
80	C₂²	C₂²×C₈×C₅₃	1	3	yes	—
81	C₄	C₂²×C₈×C₅₃	1	4	yes	—
82	C₄	C₂²×C₈×C₅₃	1	3	yes	—
83	C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
84	C₂	—	2	4	no	—
85	C₂	C₂×M₃₂×C₅₃	1	6	yes	direct product factor
86	C₂	C₂³×C₈×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈×C₅₃	C₂²
commutator	C₂	C₂³×C₈×C₅₃
Frattini, cosocle	C₈	C₂⁴×C₅₃
socle	C₂³×C₅₃	C₂×C₈
prime radical	C₂⁴×C₅₃	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×M₃₂×C₅₃	C₂	direct (N × H)
3. C₂²×C₁₆×C₅₃	C₂	semidirect
9. M₃₂×C₅₃	C₂²	direct (N × H)
10. C₂×C₁₆×C₅₃	C₂²	semidirect
21. C₁₆×C₅₃	C₂³	semidirect
34. C₂²×C₅₃	M₃₂	direct (N × H)
40. C₂²×M₃₂	C₅₃	direct (N × H)
42. C₂×C₅₃	C₂×M₃₂	direct (N × H)
45. C₂×M₃₂	C₂×C₅₃	direct (N × H)
46. C₂²×C₁₆	C₂×C₅₃	semidirect
47. C₅₃	C₂²×M₃₂	direct (N × H)
53. M₃₂	C₂²×C₅₃	direct (N × H)
54. C₂×C₁₆	C₂²×C₅₃	semidirect
65. C₁₆	C₂³×C₅₃	semidirect
78. C₂²	M₃₂×C₅₃	direct (N × H)
85. C₂	C₂×M₃₂×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
C₂³×C₈×C₅₃	2
C₂×M₃₂×C₅₃	2
C₂²×C₁₆×C₅₃	2
C₂²×M₃₂	53

Subgroups — C22×M32×C53 #20272342

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×M₃₂×C₅₃ #20272342