Subgroups — C₅₃×#2810 #20271747

Order 6784.

344 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#400	C₂	1	6	yes	maximal
2	C₅₃×#389	C₂	1	3	yes	maximal
3	C₅₃×#387	C₂	1	6	yes	maximal
4	C₂×C₄⋊C₄×C₅₃	C₂²	1	6	yes	—
5	C₂×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
6	C₂×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
7	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
8	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
9	#95×C₅₃	C₂²	1	6	yes	—
10	#95×C₅₃	C₂²	1	6	yes	—
11	#95×C₅₃	C₂²	1	6	yes	—
12	#95×C₅₃	C₂²	1	2	yes	—
13	C₂²×C₄×C₅₃	C₂³	1	6	yes	—
14	C₂²×C₄×C₅₃	C₂³	1	6	yes	—
15	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
16	C₄⋊C₄×C₅₃	—	2	12	no	—
17	C₄⋊C₄×C₅₃	—	2	6	no	—
18	C₄⋊C₄×C₅₃	—	4	3	no	—
19	C₄²×C₅₃	—	4	1	no	—
20	C₄²×C₅₃	—	4	2	no	—
21	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
22	C₂×C₄×C₅₃	—	2	3	no	—
23	C₂×C₄×C₅₃	—	2	3	no	—
24	C₂×C₄×C₅₃	—	2	6	no	—
25	C₂×C₄×C₅₃	—	2	6	no	—
26	C₂×C₄×C₅₃	—	2	6	no	—
27	C₂×C₄×C₅₃	—	2	6	no	—
28	C₂×C₄×C₅₃	—	2	6	no	—
29	C₂×C₄×C₅₃	—	2	6	no	—
30	C₂×C₄×C₅₃	#41	1	6	yes	—
31	C₂²×C₅₃	C₂×#41	1	3	yes	—
32	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
33	C₂²×C₅₃	2₋¹⁺⁴	1	3	yes	—
34	C₄×C₅₃	—	2	6	no	—
35	C₄×C₅₃	—	4	6	no	—
36	C₄×C₅₃	—	4	6	no	—
37	#2810	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂×C₅₃	#539	1	3	yes	—
39	C₂×C₅₃	#556	1	3	yes	—
40	C₂×C₅₃	#564	1	1	yes	characteristic
41	#400	C₂×C₅₃	1	6	yes	—
42	#389	C₂×C₅₃	1	3	yes	—
43	#387	C₂×C₅₃	1	6	yes	—
44	C₅₃	#2810	1	1	yes	characteristic, direct product factor, Sylow
45	C₂×C₄⋊C₄	C₂²×C₅₃	1	6	yes	—
46	C₂×C₄⋊C₄	C₂²×C₅₃	1	3	yes	—
47	C₂×C₄⋊C₄	C₂²×C₅₃	1	3	yes	—
48	C₂×C₄²	C₂²×C₅₃	1	2	yes	—
49	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
50	#95	C₂²×C₅₃	1	6	yes	—
51	#95	C₂²×C₅₃	1	6	yes	—
52	#95	C₂²×C₅₃	1	6	yes	—
53	#95	C₂²×C₅₃	1	2	yes	—
54	C₂²×C₄	C₂³×C₅₃	1	6	yes	—
55	C₂²×C₄	C₂³×C₅₃	1	6	yes	—
56	C₂²×C₄	C₂³×C₅₃	1	3	yes	—
57	C₄⋊C₄	—	2	12	no	—
58	C₄⋊C₄	—	2	6	no	—
59	C₄⋊C₄	—	4	3	no	—
60	C₄²	—	4	1	no	—
61	C₄²	—	4	2	no	—
62	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
63	C₂×C₄	—	2	3	no	—
64	C₂×C₄	—	2	3	no	—
65	C₂×C₄	—	2	6	no	—
66	C₂×C₄	—	2	6	no	—
67	C₂×C₄	—	2	6	no	—
68	C₂×C₄	—	2	6	no	—
69	C₂×C₄	—	2	6	no	—
70	C₂×C₄	—	2	6	no	—
71	C₂×C₄	#41×C₅₃	1	6	yes	—
72	C₂²	C₂×#41×C₅₃	1	3	yes	—
73	C₂²	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
74	C₂²	2₋¹⁺⁴×C₅₃	1	3	yes	—
75	C₄	—	2	6	no	—
76	C₄	—	4	6	no	—
77	C₄	—	4	6	no	—
78	C₂	C₅₃×#539	1	3	yes	—
79	C₂	C₅₃×#556	1	3	yes	—
80	C₂	C₅₃×#564	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
37. #2810	C₅₃	direct (N × H)
44. C₅₃	#2810	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×#400	2
C₅₃×#389	2
C₅₃×#387	2
#2810	53

Subgroups — C53×#2810 #20271747

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2810 #20271747