Subgroups — C₅₃×#7463 #27214975

Order 13568.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#1397	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1372	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	SD₃₂	1	2	yes	—
22	C₈×C₅₃	—	2	1	no	—
23	C₈×C₅₃	—	16	2	no	—
24	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
25	C₈×C₅₃	#108	1	1	yes	characteristic
26	#7463	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂²×C₅₃	#367	1	1	yes	characteristic
28	C₄×C₅₃	—	2	2	no	—
29	C₄×C₅₃	#335	1	1	yes	characteristic
30	C₄×C₅₃	#368	1	1	yes	characteristic
31	#1397	C₂×C₅₃	1	1	yes	characteristic
32	#1372	C₂×C₅₃	1	2	yes	—
33	C₂×C₅₃	#1372	1	1	yes	characteristic
34	C₂×C₅₃	#1426	1	2	yes	—
35	C₂×C₃₂	C₄×C₅₃	1	1	yes	characteristic
36	C₂×C₃₂	C₄×C₅₃	1	1	yes	characteristic
37	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
38	#335	—	2	2	no	—
39	C₅₃	#7463	1	1	yes	characteristic, direct product factor, Sylow
40	C₂×C₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
41	C₂×C₁₆	Q₈×C₅₃	1	1	yes	characteristic
42	C₄⋊C₈	—	4	2	no	—
43	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
44	C₃₂	—	2	1	no	—
45	C₃₂	C₈×C₅₃	1	2	yes	complemented
46	C₂×C₈	—	8	2	no	—
47	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
48	C₂×C₈	Q₁₆×C₅₃	1	1	yes	characteristic
49	C₄²	D₈×C₅₃	1	1	yes	characteristic
50	C₁₆	—	2	1	no	—
51	C₁₆	C₂×C₈×C₅₃	1	1	yes	characteristic
52	C₁₆	M₁₆×C₅₃	1	1	yes	characteristic
53	C₂×C₄	#107×C₅₃	1	1	yes	characteristic
54	C₂×C₄	SD₃₂×C₅₃	1	2	yes	—
55	C₈	—	2	1	no	—
56	C₈	—	16	2	no	—
57	C₈	C₄⋊C₈×C₅₃	1	1	yes	characteristic
58	C₈	#108×C₅₃	1	1	yes	characteristic
59	C₂²	C₅₃×#367	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	C₅₃×#335	1	1	yes	characteristic
62	C₄	C₅₃×#368	1	1	yes	characteristic
63	C₂	C₅₃×#1372	1	1	yes	characteristic
64	C₂	C₅₃×#1426	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₅₃	#367
commutator	C₁₆	C₂×C₈×C₅₃
Frattini, cosocle	C₄×C₁₆	C₂²×C₅₃
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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
26. #7463	C₅₃	direct (N × H)
39. C₅₃	#7463	direct (N × H)
45. 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
C₅₃×#1397	2
C₅₃×#1372	2
#7463	53

Subgroups — C53×#7463 #27214975

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7463 #27214975