Subgroups — C₅₃×#2645 #20271582

Order 6784.

352 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#401	C₂	1	2	yes	maximal
2	C₅₃×#400	C₂	1	6	yes	maximal
3	C₅₃×#391	C₂	1	3	yes	maximal
4	C₅₃×#389	C₂	1	3	yes	maximal
5	C₅₃×#376	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
8	C₂×C₄⋊C₄×C₅₃	C₂²	1	6	yes	—
9	C₂×C₄²×C₅₃	C₂²	1	3	yes	—
10	#95×C₅₃	C₂²	1	3	yes	—
11	#95×C₅₃	C₂²	1	1	yes	characteristic
12	#95×C₅₃	C₂²	1	6	yes	—
13	#95×C₅₃	C₂²	1	6	yes	—
14	#95×C₅₃	C₂²	1	6	yes	—
15	C₂×Q₈×C₅₃	—	2	6	no	—
16	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
17	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
18	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
19	C₂²×C₄×C₅₃	C₂³	1	6	yes	—
20	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
21	C₄⋊C₄×C₅₃	—	2	6	no	—
22	C₄⋊C₄×C₅₃	—	4	6	no	—
23	C₄²×C₅₃	—	2	6	no	—
24	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
25	Q₈×C₅₃	—	4	4	no	—
26	C₂×C₄×C₅₃	—	2	6	no	—
27	C₂×C₄×C₅₃	—	2	3	no	—
28	C₂×C₄×C₅₃	—	2	3	no	—
29	C₂×C₄×C₅₃	—	2	6	no	—
30	C₂×C₄×C₅₃	—	2	3	no	—
31	C₂×C₄×C₅₃	—	2	6	no	—
32	C₂×C₄×C₅₃	—	2	6	no	—
33	C₂×C₄×C₅₃	—	2	3	no	—
34	C₂×C₄×C₅₃	—	2	6	no	—
35	C₂×C₄×C₅₃	#41	1	6	yes	—
36	C₂²×C₅₃	C₂×#41	1	3	yes	—
37	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
38	C₂²×C₅₃	2₋¹⁺⁴	1	3	yes	—
39	C₄×C₅₃	—	4	2	no	—
40	C₄×C₅₃	—	4	3	no	—
41	C₄×C₅₃	—	4	3	no	—
42	C₄×C₅₃	—	4	6	no	—
43	C₄×C₅₃	#126	1	4	yes	—
44	#2645	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂×C₅₃	C₂×#126	1	1	yes	characteristic
46	C₂×C₅₃	#541	1	3	yes	—
47	C₂×C₅₃	#542	1	3	yes	—
48	#401	C₂×C₅₃	1	2	yes	—
49	#400	C₂×C₅₃	1	6	yes	—
50	#391	C₂×C₅₃	1	3	yes	—
51	#389	C₂×C₅₃	1	3	yes	—
52	#376	C₂×C₅₃	1	1	yes	characteristic
53	C₅₃	#2645	1	1	yes	characteristic, direct product factor, Sylow
54	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic
55	C₂×C₄⋊C₄	C₂²×C₅₃	1	3	yes	—
56	C₂×C₄⋊C₄	C₂²×C₅₃	1	6	yes	—
57	C₂×C₄²	C₂²×C₅₃	1	3	yes	—
58	#95	C₂²×C₅₃	1	3	yes	—
59	#95	C₂²×C₅₃	1	1	yes	characteristic
60	#95	C₂²×C₅₃	1	6	yes	—
61	#95	C₂²×C₅₃	1	6	yes	—
62	#95	C₂²×C₅₃	1	6	yes	—
63	C₂×Q₈	—	2	6	no	—
64	C₂²×C₄	C₂³×C₅₃	1	3	yes	—
65	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
66	C₂²×C₄	C₂³×C₅₃	1	3	yes	—
67	C₂²×C₄	C₂³×C₅₃	1	6	yes	—
68	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
69	C₄⋊C₄	—	2	6	no	—
70	C₄⋊C₄	—	4	6	no	—
71	C₄²	—	2	6	no	—
72	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
73	Q₈	—	4	4	no	—
74	C₂×C₄	—	2	3	no	—
75	C₂×C₄	—	2	3	no	—
76	C₂×C₄	—	2	6	no	—
77	C₂×C₄	—	2	6	no	—
78	C₂×C₄	—	2	3	no	—
79	C₂×C₄	—	2	6	no	—
80	C₂×C₄	—	2	6	no	—
81	C₂×C₄	—	2	3	no	—
82	C₂×C₄	—	2	6	no	—
83	C₂×C₄	#41×C₅₃	1	6	yes	—
84	C₂²	C₂×#41×C₅₃	1	3	yes	—
85	C₂²	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
86	C₂²	2₋¹⁺⁴×C₅₃	1	3	yes	—
87	C₄	—	4	2	no	—
88	C₄	—	4	3	no	—
89	C₄	—	4	3	no	—
90	C₄	—	4	6	no	—
91	C₄	#126×C₅₃	1	4	yes	—
92	C₂	C₂×#126×C₅₃	1	1	yes	characteristic
93	C₂	C₅₃×#541	1	3	yes	—
94	C₂	C₅₃×#542	1	3	yes	—

	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
44. #2645	C₅₃	direct (N × H)
53. C₅₃	#2645	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×#401	2
C₅₃×#400	2
C₅₃×#391	2
C₅₃×#389	2
C₅₃×#376	2
#2645	53

Subgroups — C53×#2645 #20271582

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2645 #20271582