Subgroups — C₅₃×#2867 #20271804

Order 6784.

1272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#127×C₅₃	C₂	1	14	yes	maximal, complemented
2	C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×D₄×C₅₃	C₂²	1	28	yes	—
4	#127×C₅₃	—	2	56	no	—
5	C₂×C₄²×C₅₃	C₂²	1	7	yes	—
6	C₂⁴×C₅₃	C₂³	1	8	yes	—
7	C₂×D₄×C₅₃	—	2	168	no	—
8	C₂²×C₄×C₅₃	C₂³	1	7	yes	—
9	C₄²×C₅₃	D₄	1	28	yes	complemented
10	C₂³×C₅₃	—	2	56	no	—
11	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
12	D₄×C₅₃	—	4	112	no	—
13	C₂×C₄×C₅₃	C₂×D₄	1	42	yes	—
14	C₂²×C₅₃	—	4	56	no	—
15	C₂²×C₅₃	C₂²×D₄	1	7	yes	—
16	C₄×C₅₃	#127	1	28	yes	complemented
17	#2867	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂×C₅₃	—	8	8	no	—
19	C₂×C₅₃	C₂×#127	1	7	yes	—
20	C₂×#127	C₂×C₅₃	1	14	yes	complemented
21	C₄³	C₂×C₅₃	1	1	yes	characteristic, complemented
22	C₅₃	#2867	1	1	yes	characteristic, direct product factor, Sylow
23	C₂²×D₄	C₂²×C₅₃	1	28	yes	—
24	#127	—	2	56	no	—
25	C₂×C₄²	C₂²×C₅₃	1	7	yes	—
26	C₂⁴	C₂³×C₅₃	1	8	yes	—
27	C₂×D₄	—	2	168	no	—
28	C₂²×C₄	C₂³×C₅₃	1	7	yes	—
29	C₄²	D₄×C₅₃	1	28	yes	complemented
30	C₂³	—	2	56	no	—
31	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
32	D₄	—	4	112	no	—
33	C₂×C₄	C₂×D₄×C₅₃	1	42	yes	—
34	C₂²	—	4	56	no	—
35	C₂²	C₂²×D₄×C₅₃	1	7	yes	—
36	C₄	#127×C₅₃	1	28	yes	complemented
37	C₂	—	8	8	no	—
38	C₂	C₂×#127×C₅₃	1	7	yes	—

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

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

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