Subgroups — C₅₃×#7329 #27214841

Order 13568.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₂×C₅₃	C₂	1	2	yes	maximal
2	C₈×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₂×C₅₃	C₄	1	4	yes	—
4	C₄×C₁₆×C₅₃	C₄	1	1	yes	characteristic
5	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
6	C₈²×C₅₃	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₅₃	C₈	1	2	yes	—
8	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
9	C₄×C₈×C₅₃	C₈	1	2	yes	—
10	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
11	C₃₂×C₅₃	C₈	1	8	yes	complemented
12	C₂×C₈×C₅₃	C₁₆	1	4	yes	—
13	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
14	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
15	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
16	C₁₆×C₅₃	C₁₆	1	4	yes	—
17	C₁₆×C₅₃	C₂×C₈	1	4	yes	—
18	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
19	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
20	C₈×C₅₃	—	2	4	no	—
21	C₈×C₅₃	C₄×C₈	1	2	yes	—
22	C₈×C₅₃	C₂×C₁₆	1	2	yes	—
23	#7329	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂²×C₅₃	C₄×C₁₆	1	1	yes	characteristic
25	C₄×C₅₃	C₈²	1	1	yes	characteristic
26	C₄×C₅₃	C₄×C₁₆	1	1	yes	characteristic
27	C₄×C₅₃	M₆₄	1	4	yes	—
28	C₄×C₃₂	C₂×C₅₃	1	2	yes	—
29	C₈×C₁₆	C₂×C₅₃	1	1	yes	characteristic
30	C₂×C₅₃	C₈×C₁₆	1	1	yes	characteristic
31	C₂×C₅₃	#1397	1	2	yes	—
32	C₂×C₃₂	C₄×C₅₃	1	4	yes	—
33	C₄×C₁₆	C₄×C₅₃	1	1	yes	characteristic
34	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
35	C₈²	C₄×C₅₃	1	1	yes	characteristic
36	C₅₃	#7329	1	1	yes	characteristic, direct product factor, Sylow
37	C₂×C₁₆	C₈×C₅₃	1	2	yes	—
38	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
39	C₄×C₈	C₈×C₅₃	1	2	yes	—
40	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
41	C₃₂	C₈×C₅₃	1	8	yes	complemented
42	C₂×C₈	C₁₆×C₅₃	1	4	yes	—
43	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
44	C₂×C₈	C₂×C₈×C₅₃	1	1	yes	characteristic
45	C₄²	C₂×C₈×C₅₃	1	1	yes	characteristic
46	C₁₆	C₁₆×C₅₃	1	4	yes	—
47	C₁₆	C₂×C₈×C₅₃	1	4	yes	—
48	C₂×C₄	C₄×C₈×C₅₃	1	1	yes	characteristic
49	C₂×C₄	C₂×C₁₆×C₅₃	1	2	yes	—
50	C₈	—	2	4	no	—
51	C₈	C₄×C₈×C₅₃	1	2	yes	—
52	C₈	C₂×C₁₆×C₅₃	1	2	yes	—
53	C₂²	C₄×C₁₆×C₅₃	1	1	yes	characteristic
54	C₄	C₈²×C₅₃	1	1	yes	characteristic
55	C₄	C₄×C₁₆×C₅₃	1	1	yes	characteristic
56	C₄	C₅₃×M₆₄	1	4	yes	—
57	C₂	C₈×C₁₆×C₅₃	1	1	yes	characteristic
58	C₂	C₅₃×#1397	1	2	yes	—

	subgroup	G/H
center	C₄×C₁₆×C₅₃	C₂²
commutator	C₂	C₈×C₁₆×C₅₃
Frattini, cosocle	C₄×C₁₆	C₂²×C₅₃
socle, prime radical	C₂²×C₅₃	C₄×C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₃₂×C₅₃	C₈	semidirect
23. #7329	C₅₃	direct (N × H)
36. C₅₃	#7329	direct (N × H)
41. C₃₂	C₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₃₂×C₅₃	2
C₈×C₁₆×C₅₃	2
#7329	53

Subgroups — C53×#7329 #27214841

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7329 #27214841