Subgroups — C₅₃×#7459 #27214971

Order 13568.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#1422	C₂	1	2	yes	maximal
2	C₈×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×M₆₄	—	2	2	no	—
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	1	yes	characteristic
8	C₂×C₁₆×C₅₃	C₈	1	1	yes	characteristic
9	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
10	C₂×C₁₆×C₅₃	Q₈	1	1	yes	characteristic
11	C₄×C₈×C₅₃	—	2	1	no	—
12	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₃₂×C₅₃	—	4	2	no	—
14	C₂×C₈×C₅₃	—	2	2	no	—
15	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
16	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
17	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
18	C₁₆×C₅₃	—	2	1	no	—
19	C₁₆×C₅₃	—	2	1	no	—
20	C₁₆×C₅₃	C₁₆	1	2	yes	—
21	C₁₆×C₅₃	D₈	1	1	yes	characteristic
22	C₁₆×C₅₃	Q₁₆	1	1	yes	characteristic
23	C₂×C₄×C₅₃	—	2	1	no	—
24	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
25	C₈×C₅₃	—	2	4	no	—
26	C₈×C₅₃	#107	1	1	yes	characteristic
27	C₈×C₅₃	#108	1	1	yes	characteristic
28	C₈×C₅₃	C₂×C₁₆	1	1	yes	characteristic
29	C₈×C₅₃	M₃₂	1	1	yes	characteristic
30	#7459	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂²×C₅₃	#364	1	1	yes	characteristic
32	C₄×C₅₃	—	2	2	no	—
33	C₄×C₅₃	#335	1	1	yes	characteristic
34	C₄×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
35	#1422	C₂×C₅₃	1	2	yes	—
36	C₈×C₁₆	C₂×C₅₃	1	1	yes	characteristic
37	C₂×C₅₃	—	2	1	no	—
38	C₂×C₅₃	#1370	1	1	yes	characteristic
39	M₆₄	—	2	2	no	—
40	C₄×C₁₆	C₄×C₅₃	1	1	yes	characteristic
41	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
42	C₈²	C₄×C₅₃	1	1	yes	characteristic
43	C₅₃	#7459	1	1	yes	characteristic, direct product factor, Sylow
44	C₂×C₁₆	C₈×C₅₃	1	1	yes	characteristic
45	C₂×C₁₆	C₈×C₅₃	1	1	yes	characteristic
46	C₂×C₁₆	D₄×C₅₃	1	1	yes	characteristic
47	C₂×C₁₆	Q₈×C₅₃	1	1	yes	characteristic
48	C₄×C₈	—	2	1	no	—
49	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
50	C₃₂	—	4	2	no	—
51	C₂×C₈	—	2	2	no	—
52	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
53	C₂×C₈	C₂×C₈×C₅₃	1	1	yes	characteristic
54	C₄²	M₁₆×C₅₃	1	1	yes	characteristic
55	C₁₆	—	2	1	no	—
56	C₁₆	—	2	1	no	—
57	C₁₆	C₁₆×C₅₃	1	2	yes	—
58	C₁₆	D₈×C₅₃	1	1	yes	characteristic
59	C₁₆	Q₁₆×C₅₃	1	1	yes	characteristic
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	C₄⋊C₈×C₅₃	1	1	yes	characteristic
62	C₈	—	2	4	no	—
63	C₈	#107×C₅₃	1	1	yes	characteristic
64	C₈	#108×C₅₃	1	1	yes	characteristic
65	C₈	C₂×C₁₆×C₅₃	1	1	yes	characteristic
66	C₈	M₃₂×C₅₃	1	1	yes	characteristic
67	C₂²	C₅₃×#364	1	1	yes	characteristic
68	C₄	—	2	2	no	—
69	C₄	C₅₃×#335	1	1	yes	characteristic
70	C₄	C₅₃×C₄⋊C₁₆	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	C₅₃×#1370	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆×C₅₃	D₈
commutator	C₈	C₂×C₁₆×C₅₃
Frattini, cosocle	C₄×C₁₆	C₂²×C₅₃
socle	C₂×C₅₃	#1370
prime radical	C₂²×C₅₃	#364
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆×C₅₃ ◁ C₂×C₁₆×C₅₃ ◁ C₄×C₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
30. #7459	C₅₃	direct (N × H)
43. C₅₃	#7459	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×#1422	2
C₈×C₁₆×C₅₃	2
#7459	53

Subgroups — C53×#7459 #27214971

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7459 #27214971