Subgroups — C₅₃×#1385 #20270322

Order 6784.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#108×C₅₃	C₂	1	2	yes	maximal
3	C₂×M₁₆×C₅₃	—	2	2	no	—
4	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
5	M₃₂×C₅₃	—	2	2	no	—
6	C₂×C₁₆×C₅₃	C₄	1	2	yes	—
7	#108×C₅₃	—	2	2	no	—
8	#108×C₅₃	C₄	1	4	yes	—
9	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
10	M₁₆×C₅₃	—	2	4	no	—
11	M₁₆×C₅₃	—	4	2	no	—
12	C₂×C₈×C₅₃	—	4	2	no	—
13	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
15	C₂×C₈×C₅₃	D₄	1	2	yes	—
16	C₂×C₈×C₅₃	Q₈	1	1	yes	characteristic
17	C₁₆×C₅₃	—	2	2	no	—
18	C₂³×C₅₃	Q₁₆	1	1	yes	characteristic
19	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
20	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
22	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
23	C₈×C₅₃	—	4	4	no	—
24	C₈×C₅₃	C₄²	1	1	yes	characteristic
25	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₅₃	C₄⋊C₄	1	2	yes	—
27	C₂²×C₅₃	—	4	1	no	—
28	C₂²×C₅₃	#103	1	2	yes	—
29	C₂²×C₅₃	#107	1	1	yes	characteristic
30	C₄×C₅₃	#95	1	1	yes	characteristic
31	C₄×C₅₃	#102	1	2	yes	—
32	C₄×C₅₃	#106	1	1	yes	characteristic
33	#1385	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂×C₅₃	—	2	2	no	—
35	C₂×C₅₃	—	2	1	no	—
36	C₂×C₅₃	#340	1	1	yes	characteristic
37	C₂×M₃₂	C₂×C₅₃	1	1	yes	characteristic
38	C₂×#108	C₂×C₅₃	1	2	yes	—
39	C₅₃	#1385	1	1	yes	characteristic, direct product factor, Sylow
40	C₂×M₁₆	—	2	2	no	—
41	C₂²×C₈	C₂²×C₅₃	1	1	yes	characteristic
42	M₃₂	—	2	2	no	—
43	C₂×C₁₆	C₄×C₅₃	1	2	yes	—
44	#108	—	2	2	no	—
45	#108	C₄×C₅₃	1	4	yes	—
46	C₂²×C₄	D₄×C₅₃	1	1	yes	characteristic
47	M₁₆	—	2	4	no	—
48	M₁₆	—	4	2	no	—
49	C₂×C₈	—	4	2	no	—
50	C₂×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
51	C₂×C₈	C₂×C₄×C₅₃	1	2	yes	—
52	C₂×C₈	D₄×C₅₃	1	2	yes	—
53	C₂×C₈	Q₈×C₅₃	1	1	yes	characteristic
54	C₁₆	—	2	2	no	—
55	C₂³	Q₁₆×C₅₃	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
57	C₂×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
58	C₂×C₄	D₈×C₅₃	1	1	yes	characteristic
59	C₂×C₄	SD₁₆×C₅₃	1	2	yes	—
60	C₈	—	4	4	no	—
61	C₈	C₄²×C₅₃	1	1	yes	characteristic
62	C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
63	C₈	C₄⋊C₄×C₅₃	1	2	yes	—
64	C₂²	—	4	1	no	—
65	C₂²	#103×C₅₃	1	2	yes	—
66	C₂²	#107×C₅₃	1	1	yes	characteristic
67	C₄	#95×C₅₃	1	1	yes	characteristic
68	C₄	#102×C₅₃	1	2	yes	—
69	C₄	#106×C₅₃	1	1	yes	characteristic
70	C₂	—	2	2	no	—
71	C₂	—	2	1	no	—
72	C₂	C₅₃×#340	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅₃	#106
commutator	C₈	C₄²×C₅₃
Frattini, cosocle	C₂²×C₈	C₂²×C₅₃
socle	C₂×C₅₃	#340
prime radical	C₂³×C₅₃	Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂²×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
33. #1385	C₅₃	direct (N × H)
39. C₅₃	#1385	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×M₃₂×C₅₃	2
C₂×#108×C₅₃	2
#1385	53

Subgroups — C53×#1385 #20270322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#1385 #20270322