Subgroups — C₅₃×#8135 #27215647

Order 13568.

382 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#1744	C₂	1	3	yes	maximal
2	C₅₃×#1273	C₂	1	4	yes	maximal
3	C₂×C₄×C₈×C₅₃	C₂²	1	6	yes	—
4	C₄³×C₅₃	C₂²	1	1	yes	characteristic
5	C₅₃×#336	C₄	1	12	yes	—
6	C₂²×C₈×C₅₃	C₂×C₄	1	12	yes	—
7	C₂×C₄²×C₅₃	C₂×C₄	1	6	yes	—
8	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
9	C₄×C₈×C₅₃	—	2	12	no	—
10	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
11	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
12	C₂×C₈×C₅₃	—	2	12	no	—
13	C₂×C₈×C₅₃	—	2	24	no	—
14	C₄²×C₅₃	—	2	12	no	—
15	C₄²×C₅₃	#41	1	4	yes	—
16	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
17	C₂×C₄×C₅₃	—	2	12	no	—
18	C₂×C₄×C₅₃	#117	1	6	yes	—
19	C₂×C₄×C₅₃	#131	1	12	yes	—
20	C₈×C₅₃	—	4	12	no	—
21	#8135	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂²×C₅₃	#376	1	1	yes	characteristic
23	C₂²×C₅₃	#405	1	6	yes	—
24	C₄×C₅₃	—	4	4	no	—
25	C₄×C₅₃	#433	1	12	yes	—
26	#1744	C₂×C₅₃	1	3	yes	—
27	#1273	C₂×C₅₃	1	4	yes	—
28	C₂×C₅₃	#1454	1	4	yes	—
29	C₂×C₅₃	#1746	1	3	yes	—
30	C₂×C₄×C₈	C₂²×C₅₃	1	6	yes	—
31	C₄³	C₂²×C₅₃	1	1	yes	characteristic
32	#336	C₄×C₅₃	1	12	yes	—
33	C₅₃	#8135	1	1	yes	characteristic, direct product factor, Sylow
34	C₂²×C₈	C₂×C₄×C₅₃	1	12	yes	—
35	C₂×C₄²	C₂×C₄×C₅₃	1	6	yes	—
36	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
37	C₄×C₈	—	2	12	no	—
38	C₂²×C₄	C₄²×C₅₃	1	4	yes	—
39	C₂²×C₄	C₂²×C₄×C₅₃	1	3	yes	—
40	C₂×C₈	—	2	12	no	—
41	C₂×C₈	—	2	24	no	—
42	C₄²	—	2	12	no	—
43	C₄²	#41×C₅₃	1	4	yes	—
44	C₂³	C₂×C₄²×C₅₃	1	1	yes	characteristic
45	C₂×C₄	—	2	12	no	—
46	C₂×C₄	#117×C₅₃	1	6	yes	—
47	C₂×C₄	#131×C₅₃	1	12	yes	—
48	C₈	—	4	12	no	—
49	C₂²	C₅₃×#376	1	1	yes	characteristic
50	C₂²	C₅₃×#405	1	6	yes	—
51	C₄	—	4	4	no	—
52	C₄	C₅₃×#433	1	12	yes	—
53	C₂	C₅₃×#1454	1	4	yes	—
54	C₂	C₅₃×#1746	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
21. #8135	C₅₃	direct (N × H)
33. C₅₃	#8135	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#1744	2
C₅₃×#1273	2
#8135	53

Subgroups — C53×#8135 #27215647

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8135 #27215647