Subgroups — C₅₃×#10624569 #50311454

Order 38637.

332 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅₃×C₃²⋊C₉	C₃	1	4	yes	maximal
2	C₅₃×#6855	C₃	1	9	yes	maximal, complemented
3	C₃⁴×C₅₃	C₃²	1	1	yes	characteristic
4	C₃²×C₉×C₅₃	C₃²	1	12	yes	—
5	C₅₃×C₃²⋊C₉	—	3	12	no	—
6	C₃³×C₅₃	—	3	12	no	—
7	C₃³×C₅₃	H(3,𝔽₃)	1	3	yes	—
8	C₃³×C₅₃	C₃³	1	1	yes	characteristic
9	C₃×C₉×C₅₃	—	3	12	no	—
10	C₃×C₉×C₅₃	—	3	36	no	—
11	#10624569	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
12	C₃²×C₅₃	—	3	12	no	—
13	C₃²×C₅₃	—	9	9	no	—
14	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
15	C₃²×C₅₃	#737	1	12	yes	—
16	C₉×C₅₃	—	9	12	no	—
17	C₃×C₃²⋊C₉	C₃×C₅₃	1	4	yes	—
18	#6855	C₃×C₅₃	1	9	yes	complemented
19	C₃×C₅₃	—	9	3	no	—
20	C₃×C₅₃	#6893	1	4	yes	—
21	C₃×C₅₃	#6898	1	9	yes	—
22	C₃⁴	C₃²×C₅₃	1	1	yes	characteristic
23	C₃²×C₉	C₃²×C₅₃	1	12	yes	—
24	C₃²⋊C₉	—	3	12	no	—
25	C₅₃	#10624569	1	1	yes	characteristic, direct product factor, Sylow
26	C₃³	—	3	12	no	—
27	C₃³	H(3,𝔽₃)×C₅₃	1	3	yes	—
28	C₃³	C₃³×C₅₃	1	1	yes	characteristic
29	C₃×C₉	—	3	12	no	—
30	C₃×C₉	—	3	36	no	—
31	C₃²	—	3	12	no	—
32	C₃²	—	9	9	no	—
33	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
34	C₃²	C₅₃×#737	1	12	yes	—
35	C₉	—	9	12	no	—
36	C₃	—	9	3	no	—
37	C₃	C₅₃×#6893	1	4	yes	—
38	C₃	C₅₃×#6898	1	9	yes	—

	subgroup	G/H
center, socle	C₃³×C₅₃	C₃³
commutator, Frattini, cosocle	C₃³	C₃³×C₅₃
prime radical	C₃⁴×C₅₃	C₃²
LCS	G ▷ C₃³ ▷ C₁
UCS	C₁ ◁ C₃³×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.