Subgroups — C₅₃×#8136 #27215648

Order 13568.

478 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#97×C₅₃	C₂	1	3	yes	maximal
2	C₅₃×#1273	C₂	1	4	yes	maximal
3	C₂×#97×C₅₃	C₄	1	12	yes	—
4	C₂×C₄×C₈×C₅₃	C₂²	1	6	yes	—
5	C₄³×C₅₃	C₂²	1	1	yes	characteristic
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	#97×C₅₃	—	2	24	no	—
10	C₄×C₈×C₅₃	—	2	12	no	—
11	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
12	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
13	C₂×C₈×C₅₃	—	2	12	no	—
14	C₂×C₈×C₅₃	—	2	12	no	—
15	C₂×C₈×C₅₃	M₁₆	1	24	yes	—
16	C₄²×C₅₃	—	2	12	no	—
17	C₄²×C₅₃	#41	1	4	yes	—
18	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
19	C₂×C₄×C₅₃	—	2	12	no	—
20	C₂×C₄×C₅₃	#117	1	6	yes	—
21	C₂×C₄×C₅₃	C₂×M₁₆	1	12	yes	—
22	C₈×C₅₃	—	2	24	no	—
23	#8136	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂²×C₅₃	#376	1	1	yes	characteristic
25	C₂²×C₅₃	C₄×M₁₆	1	6	yes	—
26	C₄×C₅₃	—	4	4	no	—
27	C₄×C₅₃	#432	1	12	yes	—
28	C₄×#97	C₂×C₅₃	1	3	yes	—
29	#1273	C₂×C₅₃	1	4	yes	—
30	C₂×C₅₃	#1455	1	4	yes	—
31	C₂×C₅₃	#1745	1	3	yes	—
32	C₂×#97	C₄×C₅₃	1	12	yes	—
33	C₂×C₄×C₈	C₂²×C₅₃	1	6	yes	—
34	C₄³	C₂²×C₅₃	1	1	yes	characteristic
35	C₅₃	#8136	1	1	yes	characteristic, direct product factor, Sylow
36	C₂²×C₈	C₂×C₄×C₅₃	1	12	yes	—
37	C₂×C₄²	C₂×C₄×C₅₃	1	6	yes	—
38	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
39	#97	—	2	24	no	—
40	C₄×C₈	—	2	12	no	—
41	C₂²×C₄	C₄²×C₅₃	1	4	yes	—
42	C₂²×C₄	C₂²×C₄×C₅₃	1	3	yes	—
43	C₂×C₈	—	2	12	no	—
44	C₂×C₈	—	2	12	no	—
45	C₂×C₈	M₁₆×C₅₃	1	24	yes	—
46	C₄²	—	2	12	no	—
47	C₄²	#41×C₅₃	1	4	yes	—
48	C₂³	C₂×C₄²×C₅₃	1	1	yes	characteristic
49	C₂×C₄	—	2	12	no	—
50	C₂×C₄	#117×C₅₃	1	6	yes	—
51	C₂×C₄	C₂×M₁₆×C₅₃	1	12	yes	—
52	C₈	—	2	24	no	—
53	C₂²	C₅₃×#376	1	1	yes	characteristic
54	C₂²	C₄×M₁₆×C₅₃	1	6	yes	—
55	C₄	—	4	4	no	—
56	C₄	C₅₃×#432	1	12	yes	—
57	C₂	C₅₃×#1455	1	4	yes	—
58	C₂	C₅₃×#1745	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
23. #8136	C₅₃	direct (N × H)
35. C₅₃	#8136	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#97×C₅₃	2
C₅₃×#1273	2
#8136	53

Subgroups — C53×#8136 #27215648

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8136 #27215648