Subgroups — C₅₃×#2811 #20271748

Order 6784.

432 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#395	C₂	1	6	yes	maximal
2	C₅₃×#391	C₂	1	9	yes	maximal
3	C₂²×Q₈×C₅₃	C₂²	1	2	yes	—
4	C₂²×Q₈×C₅₃	C₂²	1	3	yes	—
5	C₂×C₄⋊C₄×C₅₃	C₂²	1	9	yes	—
6	C₂×C₄²×C₅₃	C₂²	1	3	yes	—
7	#95×C₅₃	C₂²	1	18	yes	—
8	C₂×Q₈×C₅₃	—	4	3	no	—
9	C₂×Q₈×C₅₃	—	4	3	no	—
10	C₂×Q₈×C₅₃	—	4	6	no	—
11	C₂×Q₈×C₅₃	Q₈	1	12	yes	complemented
12	C₂²×C₄×C₅₃	C₂³	1	6	yes	—
13	C₂²×C₄×C₅₃	C₂³	1	9	yes	—
14	C₄⋊C₄×C₅₃	—	4	9	no	—
15	C₄²×C₅₃	—	4	3	no	—
16	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
17	Q₈×C₅₃	—	4	12	no	—
18	Q₈×C₅₃	—	4	8	no	—
19	C₂×C₄×C₅₃	—	2	9	no	—
20	C₂×C₄×C₅₃	—	2	9	no	—
21	C₂×C₄×C₅₃	—	2	18	no	—
22	C₂×C₄×C₅₃	C₂×Q₈	1	18	yes	—
23	C₂²×C₅₃	C₂²×Q₈	1	3	yes	—
24	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
25	C₂²×C₅₃	H(5,𝔽₂)	1	3	yes	—
26	C₄×C₅₃	—	2	18	no	—
27	C₄×C₅₃	—	4	6	no	—
28	#2811	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₂×C₅₃	#543	1	3	yes	—
30	C₂×C₅₃	Q₈²	1	3	yes	—
31	C₂×C₅₃	H(3,𝔽₄)	1	1	yes	characteristic
32	#395	C₂×C₅₃	1	6	yes	—
33	#391	C₂×C₅₃	1	9	yes	—
34	C₅₃	#2811	1	1	yes	characteristic, direct product factor, Sylow
35	C₂²×Q₈	C₂²×C₅₃	1	2	yes	—
36	C₂²×Q₈	C₂²×C₅₃	1	3	yes	—
37	C₂×C₄⋊C₄	C₂²×C₅₃	1	9	yes	—
38	C₂×C₄²	C₂²×C₅₃	1	3	yes	—
39	#95	C₂²×C₅₃	1	18	yes	—
40	C₂×Q₈	—	4	3	no	—
41	C₂×Q₈	—	4	3	no	—
42	C₂×Q₈	—	4	6	no	—
43	C₂×Q₈	Q₈×C₅₃	1	12	yes	complemented
44	C₂²×C₄	C₂³×C₅₃	1	6	yes	—
45	C₂²×C₄	C₂³×C₅₃	1	9	yes	—
46	C₄⋊C₄	—	4	9	no	—
47	C₄²	—	4	3	no	—
48	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
49	Q₈	—	4	12	no	—
50	Q₈	—	4	8	no	—
51	C₂×C₄	—	2	9	no	—
52	C₂×C₄	—	2	9	no	—
53	C₂×C₄	—	2	18	no	—
54	C₂×C₄	C₂×Q₈×C₅₃	1	18	yes	—
55	C₂²	C₂²×Q₈×C₅₃	1	3	yes	—
56	C₂²	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
57	C₂²	H(5,𝔽₂)×C₅₃	1	3	yes	—
58	C₄	—	2	18	no	—
59	C₄	—	4	6	no	—
60	C₂	C₅₃×#543	1	3	yes	—
61	C₂	Q₈²×C₅₃	1	3	yes	—
62	C₂	C₅₃×H(3,𝔽₄)	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
11. C₂×Q₈×C₅₃	Q₈	semidirect
28. #2811	C₅₃	direct (N × H)
34. C₅₃	#2811	direct (N × H)
43. C₂×Q₈	Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#395	2
C₅₃×#391	2
#2811	53

Subgroups — C53×#2811 #20271748

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2811 #20271748