Subgroups — C₅₃×#7841 #27215353

Order 13568.

574 conjugacy classes of subgroups.

The subgroup lattice has 118 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#1744	C₂	1	2	yes	maximal
2	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#322	C₂	1	2	yes	maximal
4	C₂×C₈²×C₅₃	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₅₃	C₄	1	4	yes	—
6	C₂×C₄×C₈×C₅₃	C₂²	1	4	yes	—
7	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₄³×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×#336	C₄	1	8	yes	—
11	C₅₃×#322	C₄	1	8	yes	complemented
12	C₈²×C₅₃	C₄	1	8	yes	complemented
13	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
14	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
15	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
17	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
19	C₄×C₈×C₅₃	C₈	1	16	yes	complemented
20	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	—
21	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
22	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	—
23	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
25	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
27	C₂×C₈×C₅₃	—	2	16	no	—
28	C₂×C₈×C₅₃	C₄²	1	16	yes	—
29	C₂×C₈×C₅₃	C₄²	1	4	yes	—
30	C₂×C₈×C₅₃	C₄²	1	4	yes	—
31	C₂×C₈×C₅₃	C₂×C₈	1	8	yes	—
32	C₂×C₈×C₅₃	C₂×C₈	1	8	yes	—
33	C₄²×C₅₃	—	2	8	no	—
34	C₄²×C₅₃	C₂×C₈	1	8	yes	—
35	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
36	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
37	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
38	C₂×C₄×C₅₃	—	2	8	no	—
39	C₂×C₄×C₅₃	C₄×C₈	1	8	yes	—
40	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
43	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
44	C₂×C₄×C₅₃	#131	1	8	yes	—
45	C₈×C₅₃	—	2	16	no	—
46	C₈×C₅₃	C₄×C₈	1	8	yes	—
47	C₈×C₅₃	C₄×C₈	1	8	yes	—
48	#7841	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂²×C₅₃	C₄³	1	1	yes	characteristic
50	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
51	C₂²×C₅₃	#405	1	4	yes	—
52	C₄×C₅₃	—	2	8	no	—
53	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
54	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
55	C₄×C₅₃	#405	1	8	yes	—
56	#1744	C₂×C₅₃	1	2	yes	—
57	C₄²×C₈	C₂×C₅₃	1	1	yes	characteristic
58	C₂×#322	C₂×C₅₃	1	2	yes	—
59	C₂×C₈²	C₂×C₅₃	1	2	yes	—
60	C₂×C₅₃	#1450	1	4	yes	—
61	C₂×C₅₃	C₄²×C₈	1	1	yes	characteristic
62	C₂×C₅₃	#1726	1	2	yes	—
63	C₂×C₄×C₈	C₄×C₅₃	1	4	yes	—
64	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
65	C₂×C₄×C₈	C₂²×C₅₃	1	4	yes	—
66	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
67	C₄³	C₂²×C₅₃	1	1	yes	characteristic
68	#336	C₄×C₅₃	1	8	yes	—
69	#322	C₄×C₅₃	1	8	yes	complemented
70	C₈²	C₄×C₅₃	1	8	yes	complemented
71	C₅₃	#7841	1	1	yes	characteristic, direct product factor, Sylow
72	C₂²×C₈	C₂×C₄×C₅₃	1	8	yes	—
73	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
74	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
75	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
76	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
77	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
78	C₄×C₈	C₈×C₅₃	1	16	yes	complemented
79	C₄×C₈	C₂×C₄×C₅₃	1	8	yes	—
80	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
81	C₄×C₈	C₂×C₄×C₅₃	1	8	yes	—
82	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
83	C₂²×C₄	C₄²×C₅₃	1	4	yes	—
84	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
85	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
86	C₂×C₈	—	2	16	no	—
87	C₂×C₈	C₄²×C₅₃	1	16	yes	—
88	C₂×C₈	C₄²×C₅₃	1	4	yes	—
89	C₂×C₈	C₄²×C₅₃	1	4	yes	—
90	C₂×C₈	C₂×C₈×C₅₃	1	8	yes	—
91	C₂×C₈	C₂×C₈×C₅₃	1	8	yes	—
92	C₄²	—	2	8	no	—
93	C₄²	C₂×C₈×C₅₃	1	8	yes	—
94	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
95	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
96	C₂³	C₂×C₄²×C₅₃	1	1	yes	characteristic
97	C₂×C₄	—	2	8	no	—
98	C₂×C₄	C₄×C₈×C₅₃	1	8	yes	—
99	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
100	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
101	C₂×C₄	C₂×C₄²×C₅₃	1	4	yes	—
102	C₂×C₄	C₂²×C₈×C₅₃	1	4	yes	—
103	C₂×C₄	#131×C₅₃	1	8	yes	—
104	C₈	—	2	16	no	—
105	C₈	C₄×C₈×C₅₃	1	8	yes	—
106	C₈	C₄×C₈×C₅₃	1	8	yes	—
107	C₂²	C₄³×C₅₃	1	1	yes	characteristic
108	C₂²	C₂×C₄×C₈×C₅₃	1	2	yes	—
109	C₂²	C₅₃×#405	1	4	yes	—
110	C₄	—	2	8	no	—
111	C₄	C₂×C₄×C₈×C₅₃	1	2	yes	—
112	C₄	C₂×C₄×C₈×C₅₃	1	2	yes	—
113	C₄	C₅₃×#405	1	8	yes	—
114	C₂	C₅₃×#1450	1	4	yes	—
115	C₂	C₄²×C₈×C₅₃	1	1	yes	characteristic
116	C₂	C₅₃×#1726	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₈×C₅₃	C₂²
commutator	C₂	C₄²×C₈×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle, prime radical	C₂³×C₅₃	C₂×C₄²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₅₃×#322	C₄	semidirect
12. C₈²×C₅₃	C₄	semidirect
19. C₄×C₈×C₅₃	C₈	semidirect
48. #7841	C₅₃	direct (N × H)
69. #322	C₄×C₅₃	semidirect
70. C₈²	C₄×C₅₃	semidirect
71. C₅₃	#7841	direct (N × H)
78. C₄×C₈	C₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#1744	2
C₄²×C₈×C₅₃	2
C₂×C₅₃×#322	2
C₂×C₈²×C₅₃	2
#7841	53

Subgroups — C53×#7841 #27215353

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7841 #27215353