Subgroups — C₅₃×#2263 #20271200

Order 6784.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#508	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×SD₆₄	C₂	1	2	yes	maximal, complemented
4	C₅₃×D₃₂	C₂	1	2	yes	maximal, complemented
5	C₅₃×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#135×C₅₃	—	2	1	no	—
7	C₂×D₈×C₅₃	—	2	1	no	—
8	Q₃₂×C₅₃	C₂²	1	1	yes	characteristic, complemented
9	SD₃₂×C₅₃	—	2	1	no	—
10	D₁₆×C₅₃	—	2	1	no	—
11	D₁₆×C₅₃	C₂²	1	2	yes	—
12	D₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₃₂×C₅₃	C₂²	1	2	yes	complemented
15	#41×C₅₃	—	4	1	no	—
16	C₂×D₄×C₅₃	—	4	1	no	—
17	Q₁₆×C₅₃	—	2	1	no	—
18	SD₁₆×C₅₃	—	4	1	no	—
19	D₈×C₅₃	—	2	2	no	—
20	D₈×C₅₃	—	2	1	no	—
21	D₈×C₅₃	—	4	1	no	—
22	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₁₆×C₅₃	D₄	1	1	yes	characteristic
24	C₁₆×C₅₃	C₂³	1	1	yes	characteristic
25	C₂³×C₅₃	—	8	1	no	—
26	Q₈×C₅₃	—	4	1	no	—
27	D₄×C₅₃	—	4	2	no	—
28	D₄×C₅₃	—	4	1	no	—
29	D₄×C₅₃	—	8	1	no	—
30	D₄×C₅₃	—	8	1	no	—
31	C₂×C₄×C₅₃	—	8	1	no	—
32	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
33	C₈×C₅₃	D₈	1	1	yes	characteristic
34	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₅₃	—	8	1	no	—
36	C₂²×C₅₃	—	8	2	no	—
37	C₂²×C₅₃	—	16	2	no	—
38	C₂²×C₅₃	D₁₆	1	1	yes	characteristic
39	C₄×C₅₃	—	8	1	no	—
40	C₄×C₅₃	D₁₆	1	1	yes	characteristic
41	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
42	#2263	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂×C₅₃	—	2	1	no	—
44	C₂×C₅₃	—	16	1	no	—
45	C₂×C₅₃	—	16	2	no	—
46	C₂×C₅₃	C₂×D₁₆	1	1	yes	characteristic
47	#508	C₂×C₅₃	1	1	yes	characteristic, complemented
48	C₂×D₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
49	SD₆₄	C₂×C₅₃	1	2	yes	complemented
50	D₃₂	C₂×C₅₃	1	2	yes	complemented
51	M₆₄	C₂×C₅₃	1	1	yes	characteristic, complemented
52	C₅₃	#2263	1	1	yes	characteristic, direct product factor, Sylow
53	#135	—	2	1	no	—
54	C₂×D₈	—	2	1	no	—
55	Q₃₂	C₂²×C₅₃	1	1	yes	characteristic, complemented
56	SD₃₂	—	2	1	no	—
57	D₁₆	—	2	1	no	—
58	D₁₆	C₂²×C₅₃	1	2	yes	—
59	D₁₆	C₂²×C₅₃	1	1	yes	characteristic, complemented
60	C₂×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
61	C₃₂	C₂²×C₅₃	1	2	yes	complemented
62	#41	—	4	1	no	—
63	C₂×D₄	—	4	1	no	—
64	Q₁₆	—	2	1	no	—
65	SD₁₆	—	4	1	no	—
66	D₈	—	2	2	no	—
67	D₈	—	2	1	no	—
68	D₈	—	4	1	no	—
69	C₂×C₈	D₄×C₅₃	1	1	yes	characteristic
70	C₁₆	D₄×C₅₃	1	1	yes	characteristic
71	C₁₆	C₂³×C₅₃	1	1	yes	characteristic
72	C₂³	—	8	1	no	—
73	Q₈	—	4	1	no	—
74	D₄	—	4	2	no	—
75	D₄	—	4	1	no	—
76	D₄	—	8	1	no	—
77	D₄	—	8	1	no	—
78	C₂×C₄	—	8	1	no	—
79	C₂×C₄	D₈×C₅₃	1	1	yes	characteristic
80	C₈	D₈×C₅₃	1	1	yes	characteristic
81	C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
82	C₂²	—	8	1	no	—
83	C₂²	—	8	2	no	—
84	C₂²	—	16	2	no	—
85	C₂²	D₁₆×C₅₃	1	1	yes	characteristic
86	C₄	—	8	1	no	—
87	C₄	D₁₆×C₅₃	1	1	yes	characteristic
88	C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
89	C₂	—	2	1	no	—
90	C₂	—	16	2	no	—
91	C₂	—	16	1	no	—
92	C₂	C₂×D₁₆×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅₃	C₂×D₁₆
commutator, Frattini, cosocle	C₁₆	C₂³×C₅₃
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₂×C₈×C₅₃ ◁ C₂×C₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#508	C₂	semidirect
2. C₂×D₁₆×C₅₃	C₂	semidirect
3. C₅₃×SD₆₄	C₂	semidirect
4. C₅₃×D₃₂	C₂	semidirect
5. C₅₃×M₆₄	C₂	semidirect
8. Q₃₂×C₅₃	C₂²	semidirect
12. D₁₆×C₅₃	C₂²	semidirect
14. C₃₂×C₅₃	C₂²	semidirect
42. #2263	C₅₃	direct (N × H)
47. #508	C₂×C₅₃	semidirect
48. C₂×D₁₆	C₂×C₅₃	semidirect
49. SD₆₄	C₂×C₅₃	semidirect
50. D₃₂	C₂×C₅₃	semidirect
51. M₆₄	C₂×C₅₃	semidirect
52. C₅₃	#2263	direct (N × H)
55. Q₃₂	C₂²×C₅₃	semidirect
59. D₁₆	C₂²×C₅₃	semidirect
61. C₃₂	C₂²×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#508	2
C₂×D₁₆×C₅₃	2
C₅₃×SD₆₄	2
C₅₃×D₃₂	2
C₅₃×M₆₄	2
#2263	53

Subgroups — C53×#2263 #20271200

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2263 #20271200