Subgroups — C₅₃×#2273 #20271210

Order 6784.

912 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊C₄×C₅₃	C₂	1	3	yes	maximal
2	C₄×C₂²⋊C₄×C₅₃	C₂	1	12	yes	maximal, complemented
3	C₂⁵×C₅₃	C₂²	1	1	yes	characteristic
4	C₂³×C₄×C₅₃	C₂²	1	6	yes	—
5	C₂×C₂²⋊C₄×C₅₃	C₄	1	24	yes	complemented
6	C₂×C₂²⋊C₄×C₅₃	C₂²	1	12	yes	—
7	C₂×C₄²×C₅₃	C₂²	1	8	yes	complemented
8	#95×C₅₃	C₂²	1	8	yes	complemented
9	C₂⁴×C₅₃	—	4	4	no	—
10	C₂⁴×C₅₃	C₂×C₄	1	12	yes	—
11	C₂⁴×C₅₃	C₂³	1	3	yes	—
12	C₂²×C₄×C₅₃	—	4	12	no	—
13	C₂²×C₄×C₅₃	C₂×C₄	1	24	yes	complemented
14	C₂²×C₄×C₅₃	C₂³	1	12	yes	—
15	C₂²⋊C₄×C₅₃	C₂×C₄	1	48	yes	complemented
16	C₄²×C₅₃	—	4	8	no	—
17	C₂³×C₅₃	—	2	12	no	—
18	C₂³×C₅₃	—	4	24	no	—
19	C₂³×C₅₃	C₄²	1	16	yes	complemented
20	C₂³×C₅₃	C₂²×C₄	1	6	yes	—
21	C₂³×C₅₃	C₂²×C₄	1	12	yes	—
22	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
23	C₂×C₄×C₅₃	—	2	24	no	—
24	C₂×C₄×C₅₃	—	4	24	no	—
25	C₂×C₄×C₅₃	C₂²×C₄	1	24	yes	—
26	C₂²×C₅₃	—	2	12	no	—
27	C₂²×C₅₃	—	2	24	no	—
28	C₂²×C₅₃	—	4	16	no	—
29	C₂²×C₅₃	C₂×C₄²	1	12	yes	—
30	C₂²×C₅₃	C₂³×C₄	1	3	yes	—
31	C₂²×C₅₃	H(5,𝔽₂)	1	4	yes	—
32	C₄×C₅₃	—	2	24	no	—
33	#2273	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂×C₅₃	—	2	12	no	—
35	C₂×C₅₃	C₂²×C₄²	1	1	yes	characteristic
36	C₂×C₅₃	#518	1	6	yes	—
37	C₂²×C₂²⋊C₄	C₂×C₅₃	1	3	yes	—
38	C₄×C₂²⋊C₄	C₂×C₅₃	1	12	yes	complemented
39	C₅₃	#2273	1	1	yes	characteristic, direct product factor, Sylow
40	C₂⁵	C₂²×C₅₃	1	1	yes	characteristic
41	C₂³×C₄	C₂²×C₅₃	1	6	yes	—
42	C₂×C₂²⋊C₄	C₄×C₅₃	1	24	yes	complemented
43	C₂×C₂²⋊C₄	C₂²×C₅₃	1	12	yes	—
44	C₂×C₄²	C₂²×C₅₃	1	8	yes	complemented
45	#95	C₂²×C₅₃	1	8	yes	complemented
46	C₂⁴	—	4	4	no	—
47	C₂⁴	C₂×C₄×C₅₃	1	12	yes	—
48	C₂⁴	C₂³×C₅₃	1	3	yes	—
49	C₂²×C₄	—	4	12	no	—
50	C₂²×C₄	C₂×C₄×C₅₃	1	24	yes	complemented
51	C₂²×C₄	C₂³×C₅₃	1	12	yes	—
52	C₂²⋊C₄	C₂×C₄×C₅₃	1	48	yes	complemented
53	C₄²	—	4	8	no	—
54	C₂³	—	2	12	no	—
55	C₂³	—	4	24	no	—
56	C₂³	C₄²×C₅₃	1	16	yes	complemented
57	C₂³	C₂²×C₄×C₅₃	1	12	yes	—
58	C₂³	C₂²×C₄×C₅₃	1	6	yes	—
59	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
60	C₂×C₄	—	2	24	no	—
61	C₂×C₄	—	4	24	no	—
62	C₂×C₄	C₂²×C₄×C₅₃	1	24	yes	—
63	C₂²	—	2	12	no	—
64	C₂²	—	2	24	no	—
65	C₂²	—	4	16	no	—
66	C₂²	C₂×C₄²×C₅₃	1	12	yes	—
67	C₂²	C₂³×C₄×C₅₃	1	3	yes	—
68	C₂²	H(5,𝔽₂)×C₅₃	1	4	yes	—
69	C₄	—	2	24	no	—
70	C₂	—	2	12	no	—
71	C₂	C₂²×C₄²×C₅₃	1	1	yes	characteristic
72	C₂	C₅₃×#518	1	6	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁴
commutator	C₂	C₂²×C₄²×C₅₃
Frattini, cosocle	C₂³	C₂⁴×C₅₃
prime radical	C₂⁵×C₅₃	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₂²⋊C₄×C₅₃	C₂	semidirect
5. C₂×C₂²⋊C₄×C₅₃	C₄	semidirect
7. C₂×C₄²×C₅₃	C₂²	semidirect
8. #95×C₅₃	C₂²	semidirect
13. C₂²×C₄×C₅₃	C₂×C₄	semidirect
15. C₂²⋊C₄×C₅₃	C₂×C₄	semidirect
19. C₂³×C₅₃	C₄²	semidirect
33. #2273	C₅₃	direct (N × H)
38. C₄×C₂²⋊C₄	C₂×C₅₃	semidirect
39. C₅₃	#2273	direct (N × H)
42. C₂×C₂²⋊C₄	C₄×C₅₃	semidirect
44. C₂×C₄²	C₂²×C₅₃	semidirect
45. #95	C₂²×C₅₃	semidirect
50. C₂²×C₄	C₂×C₄×C₅₃	semidirect
52. C₂²⋊C₄	C₂×C₄×C₅₃	semidirect
56. C₂³	C₄²×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂²⋊C₄×C₅₃	2
C₄×C₂²⋊C₄×C₅₃	2
#2273	53

Subgroups — C53×#2273 #20271210

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2273 #20271210