Subgroups — C₅₃×#2264 #20271201

Order 6784.

104 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
2	C₂×Q₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×Q₆₄	C₂	1	2	yes	maximal, complemented
4	C₅₃×SD₆₄	C₂	1	2	yes	maximal, complemented
5	C₅₃×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#135×C₅₃	—	2	1	no	—
7	C₂×Q₁₆×C₅₃	—	2	1	no	—
8	Q₃₂×C₅₃	—	2	1	no	—
9	Q₃₂×C₅₃	C₂²	1	2	yes	—
10	Q₃₂×C₅₃	C₂²	1	1	yes	characteristic
11	SD₃₂×C₅₃	—	2	1	no	—
12	D₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₃₂×C₅₃	C₂²	1	2	yes	—
15	#41×C₅₃	—	4	1	no	—
16	C₂×Q₈×C₅₃	—	4	1	no	—
17	Q₁₆×C₅₃	—	2	1	no	—
18	Q₁₆×C₅₃	—	2	2	no	—
19	Q₁₆×C₅₃	—	4	1	no	—
20	SD₁₆×C₅₃	—	4	1	no	—
21	D₈×C₅₃	—	2	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	Q₈×C₅₃	—	4	2	no	—
26	Q₈×C₅₃	—	4	1	no	—
27	Q₈×C₅₃	—	8	1	no	—
28	D₄×C₅₃	—	4	1	no	—
29	D₄×C₅₃	—	8	1	no	—
30	C₂×C₄×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	D₁₆	1	1	yes	characteristic
37	C₄×C₅₃	—	8	1	no	—
38	C₄×C₅₃	—	8	2	no	—
39	C₄×C₅₃	D₁₆	1	1	yes	characteristic
40	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
41	#2264	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂×C₅₃	—	2	1	no	—
43	C₂×C₅₃	—	16	1	no	—
44	C₂×C₅₃	C₂×D₁₆	1	1	yes	characteristic
45	#508	C₂×C₅₃	1	1	yes	characteristic
46	C₂×Q₃₂	C₂×C₅₃	1	1	yes	characteristic, complemented
47	Q₆₄	C₂×C₅₃	1	2	yes	complemented
48	SD₆₄	C₂×C₅₃	1	2	yes	complemented
49	M₆₄	C₂×C₅₃	1	1	yes	characteristic, complemented
50	C₅₃	#2264	1	1	yes	characteristic, direct product factor, Sylow
51	#135	—	2	1	no	—
52	C₂×Q₁₆	—	2	1	no	—
53	Q₃₂	—	2	1	no	—
54	Q₃₂	C₂²×C₅₃	1	2	yes	—
55	Q₃₂	C₂²×C₅₃	1	1	yes	characteristic
56	SD₃₂	—	2	1	no	—
57	D₁₆	C₂²×C₅₃	1	1	yes	characteristic
58	C₂×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
59	C₃₂	C₂²×C₅₃	1	2	yes	—
60	#41	—	4	1	no	—
61	C₂×Q₈	—	4	1	no	—
62	Q₁₆	—	2	1	no	—
63	Q₁₆	—	2	2	no	—
64	Q₁₆	—	4	1	no	—
65	SD₁₆	—	4	1	no	—
66	D₈	—	2	1	no	—
67	C₂×C₈	D₄×C₅₃	1	1	yes	characteristic
68	C₁₆	D₄×C₅₃	1	1	yes	characteristic
69	C₁₆	C₂³×C₅₃	1	1	yes	characteristic
70	Q₈	—	4	1	no	—
71	Q₈	—	4	2	no	—
72	Q₈	—	8	1	no	—
73	D₄	—	4	1	no	—
74	D₄	—	8	1	no	—
75	C₂×C₄	—	8	1	no	—
76	C₂×C₄	—	8	1	no	—
77	C₂×C₄	D₈×C₅₃	1	1	yes	characteristic
78	C₈	D₈×C₅₃	1	1	yes	characteristic
79	C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
80	C₂²	—	8	1	no	—
81	C₂²	D₁₆×C₅₃	1	1	yes	characteristic
82	C₄	—	8	1	no	—
83	C₄	—	8	2	no	—
84	C₄	D₁₆×C₅₃	1	1	yes	characteristic
85	C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	16	1	no	—
88	C₂	C₂×D₁₆×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅₃	C₂×D₁₆
commutator, Frattini, cosocle	C₁₆	C₂³×C₅₃
prime radical	C₅₃×#508	C₂
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
2. C₂×Q₃₂×C₅₃	C₂	semidirect
3. C₅₃×Q₆₄	C₂	semidirect
4. C₅₃×SD₆₄	C₂	semidirect
5. C₅₃×M₆₄	C₂	semidirect
41. #2264	C₅₃	direct (N × H)
46. C₂×Q₃₂	C₂×C₅₃	semidirect
47. Q₆₄	C₂×C₅₃	semidirect
48. SD₆₄	C₂×C₅₃	semidirect
49. M₆₄	C₂×C₅₃	semidirect
50. C₅₃	#2264	direct (N × H)

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×Q₃₂×C₅₃	2
C₅₃×Q₆₄	2
C₅₃×SD₆₄	2
C₅₃×M₆₄	2
#2264	53

Subgroups — C53×#2264 #20271201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2264 #20271201