Subgroups — C₅₃×#10691 #27218203

Order 13568.

372 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#1304	C₂	1	7	yes	maximal
2	C₅₃×#379	C₂²	1	7	yes	—
3	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
4	C₂×C₂²⋊C₄×C₅₃	—	2	21	no	—
5	C₂⁴×C₅₃	—	8	3	no	—
6	C₂⁴×C₅₃	#41	1	7	yes	—
7	C₂²×C₄×C₅₃	—	4	7	no	—
8	C₂²⋊C₄×C₅₃	—	8	21	no	—
9	C₂³×C₅₃	—	4	21	no	—
10	C₂³×C₅₃	—	8	8	no	—
11	C₂³×C₅₃	#126	1	7	yes	—
12	C₂×C₄×C₅₃	—	8	21	no	—
13	#10691	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂²×C₅₃	—	2	21	no	—
15	C₂²×C₅₃	—	8	14	no	—
16	C₂²×C₅₃	#401	1	1	yes	characteristic
17	C₄×C₅₃	—	16	7	no	—
18	#1304	C₂×C₅₃	1	7	yes	—
19	C₂×C₅₃	—	4	7	no	—
20	C₂×C₅₃	#2104	1	3	yes	—
21	#379	C₂²×C₅₃	1	7	yes	—
22	C₅₃	#10691	1	1	yes	characteristic, direct product factor, Sylow
23	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
24	C₂×C₂²⋊C₄	—	2	21	no	—
25	C₂⁴	—	8	3	no	—
26	C₂⁴	#41×C₅₃	1	7	yes	—
27	C₂²×C₄	—	4	7	no	—
28	C₂²⋊C₄	—	8	21	no	—
29	C₂³	—	4	21	no	—
30	C₂³	—	8	8	no	—
31	C₂³	#126×C₅₃	1	7	yes	—
32	C₂×C₄	—	8	21	no	—
33	C₂²	—	2	21	no	—
34	C₂²	—	8	14	no	—
35	C₂²	C₅₃×#401	1	1	yes	characteristic
36	C₄	—	16	7	no	—
37	C₂	—	4	7	no	—
38	C₂	C₅₃×#2104	1	3	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	#401
commutator, Frattini, cosocle	C₂⁵	C₂³×C₅₃
prime radical	C₂⁵×C₅₃	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂⁵×C₅₃ ◁ G

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

N	H ≅ G/N	type
13. #10691	C₅₃	direct (N × H)
22. C₅₃	#10691	direct (N × H)

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

subgroup	index
C₅₃×#1304	2
#10691	53