Subgroups — C₅₃×#7461 #27214973

Order 13568.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1371	C₂	1	2	yes	maximal
3	C₂×C₃₂×C₅₃	C₄	1	1	yes	characteristic
4	C₂×C₃₂×C₅₃	C₄	1	1	yes	characteristic
5	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
6	C₅₃×#335	—	2	2	no	—
7	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
8	C₂×C₁₆×C₅₃	Q₈	1	1	yes	characteristic
9	C₄⋊C₈×C₅₃	—	4	2	no	—
10	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
11	C₃₂×C₅₃	—	2	1	no	—
12	C₃₂×C₅₃	C₈	1	2	yes	complemented
13	C₂×C₈×C₅₃	—	8	2	no	—
14	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
16	C₄²×C₅₃	D₈	1	1	yes	characteristic
17	C₁₆×C₅₃	—	2	1	no	—
18	C₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
19	C₁₆×C₅₃	M₁₆	1	1	yes	characteristic
20	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
21	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
22	C₂×C₄×C₅₃	Q₃₂	1	1	yes	characteristic
23	C₈×C₅₃	—	2	1	no	—
24	C₈×C₅₃	—	16	2	no	—
25	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
26	C₈×C₅₃	#108	1	1	yes	characteristic
27	#7461	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂²×C₅₃	#366	1	1	yes	characteristic
29	C₄×C₅₃	—	2	1	no	—
30	C₄×C₅₃	#335	1	1	yes	characteristic
31	C₄×C₅₃	#368	1	1	yes	characteristic
32	C₄×C₅₃	SD₆₄	1	2	yes	—
33	C₄×C₃₂	C₂×C₅₃	1	1	yes	characteristic
34	#1371	C₂×C₅₃	1	2	yes	—
35	C₂×C₅₃	#1371	1	1	yes	characteristic
36	C₂×C₅₃	#1424	1	1	yes	characteristic
37	C₂×C₅₃	#1425	1	1	yes	characteristic
38	C₂×C₃₂	C₄×C₅₃	1	1	yes	characteristic
39	C₂×C₃₂	C₄×C₅₃	1	1	yes	characteristic
40	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
41	#335	—	2	2	no	—
42	C₅₃	#7461	1	1	yes	characteristic, direct product factor, Sylow
43	C₂×C₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
44	C₂×C₁₆	Q₈×C₅₃	1	1	yes	characteristic
45	C₄⋊C₈	—	4	2	no	—
46	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
47	C₃₂	—	2	1	no	—
48	C₃₂	C₈×C₅₃	1	2	yes	complemented
49	C₂×C₈	—	8	2	no	—
50	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
51	C₂×C₈	Q₁₆×C₅₃	1	1	yes	characteristic
52	C₄²	D₈×C₅₃	1	1	yes	characteristic
53	C₁₆	—	2	1	no	—
54	C₁₆	C₂×C₈×C₅₃	1	1	yes	characteristic
55	C₁₆	M₁₆×C₅₃	1	1	yes	characteristic
56	C₂×C₄	#107×C₅₃	1	1	yes	characteristic
57	C₂×C₄	D₁₆×C₅₃	1	1	yes	characteristic
58	C₂×C₄	Q₃₂×C₅₃	1	1	yes	characteristic
59	C₈	—	2	1	no	—
60	C₈	—	16	2	no	—
61	C₈	C₄⋊C₈×C₅₃	1	1	yes	characteristic
62	C₈	#108×C₅₃	1	1	yes	characteristic
63	C₂²	C₅₃×#366	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	C₅₃×#335	1	1	yes	characteristic
66	C₄	C₅₃×#368	1	1	yes	characteristic
67	C₄	C₅₃×SD₆₄	1	2	yes	—
68	C₂	C₅₃×#1371	1	1	yes	characteristic
69	C₂	C₅₃×#1424	1	1	yes	characteristic
70	C₂	C₅₃×#1425	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	D₁₆
commutator	C₁₆	C₂×C₈×C₅₃
Frattini, cosocle	C₄×C₁₆	C₂²×C₅₃
socle, prime radical	C₂²×C₅₃	#366
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₄²×C₅₃ ◁ C₄×C₈×C₅₃ ◁ C₄×C₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₃₂×C₅₃	C₈	semidirect
27. #7461	C₅₃	direct (N × H)
42. C₅₃	#7461	direct (N × H)
48. 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
C₅₃×#1371	2
#7461	53

Subgroups — C53×#7461 #27214973

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7461 #27214973