Subgroups — C₅₃×#7472 #27214984

Order 13568.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#368	C₂	1	2	yes	maximal
3	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
4	C₂×#108×C₅₃	—	2	2	no	—
5	C₂×C₃₂×C₅₃	—	2	2	no	—
6	C₂×C₃₂×C₅₃	C₄	1	2	yes	—
7	C₅₃×#368	—	2	2	no	—
8	C₅₃×#368	C₄	1	4	yes	—
9	C₂×M₁₆×C₅₃	—	4	2	no	—
10	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
11	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
13	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
14	C₂×C₁₆×C₅₃	Q₈	1	1	yes	characteristic
15	#108×C₅₃	—	2	4	no	—
16	#108×C₅₃	—	4	2	no	—
17	C₃₂×C₅₃	—	2	2	no	—
18	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
19	M₁₆×C₅₃	—	4	4	no	—
20	M₁₆×C₅₃	—	8	2	no	—
21	C₂×C₈×C₅₃	—	8	2	no	—
22	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
23	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
24	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
25	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
26	C₁₆×C₅₃	C₄²	1	1	yes	characteristic
27	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
29	C₂³×C₅₃	Q₃₂	1	1	yes	characteristic
30	C₂×C₄×C₅₃	#102	1	2	yes	—
31	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
32	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
33	C₂×C₄×C₅₃	SD₃₂	1	2	yes	—
34	C₈×C₅₃	—	8	4	no	—
35	C₈×C₅₃	#95	1	1	yes	characteristic
36	C₈×C₅₃	#103	1	2	yes	—
37	C₈×C₅₃	#106	1	1	yes	characteristic
38	#7472	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂²×C₅₃	—	2	2	no	—
40	C₂²×C₅₃	#358	1	2	yes	—
41	C₂²×C₅₃	#366	1	1	yes	characteristic
42	C₄×C₅₃	#340	1	1	yes	characteristic
43	C₄×C₅₃	#357	1	2	yes	—
44	C₄×C₅₃	#367	1	1	yes	characteristic
45	C₂²×C₃₂	C₂×C₅₃	1	1	yes	characteristic
46	C₂×#368	C₂×C₅₃	1	2	yes	—
47	C₂×C₅₃	—	2	2	no	—
48	C₂×C₅₃	#1380	1	1	yes	characteristic
49	C₂×C₅₃	#1425	1	2	yes	—
50	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
51	C₂×#108	—	2	2	no	—
52	C₂×C₃₂	—	2	2	no	—
53	C₂×C₃₂	C₄×C₅₃	1	2	yes	—
54	#368	—	2	2	no	—
55	#368	C₄×C₅₃	1	4	yes	—
56	C₅₃	#7472	1	1	yes	characteristic, direct product factor, Sylow
57	C₂×M₁₆	—	4	2	no	—
58	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
59	C₂×C₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
60	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
61	C₂×C₁₆	D₄×C₅₃	1	2	yes	—
62	C₂×C₁₆	Q₈×C₅₃	1	1	yes	characteristic
63	#108	—	2	4	no	—
64	#108	—	4	2	no	—
65	C₃₂	—	2	2	no	—
66	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
67	M₁₆	—	4	4	no	—
68	M₁₆	—	8	2	no	—
69	C₂×C₈	—	8	2	no	—
70	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
71	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
72	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
73	C₂×C₈	Q₁₆×C₅₃	1	1	yes	characteristic
74	C₁₆	C₄²×C₅₃	1	1	yes	characteristic
75	C₁₆	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
76	C₁₆	C₄⋊C₄×C₅₃	1	2	yes	—
77	C₂³	Q₃₂×C₅₃	1	1	yes	characteristic
78	C₂×C₄	#102×C₅₃	1	2	yes	—
79	C₂×C₄	#107×C₅₃	1	1	yes	characteristic
80	C₂×C₄	D₁₆×C₅₃	1	1	yes	characteristic
81	C₂×C₄	SD₃₂×C₅₃	1	2	yes	—
82	C₈	—	8	4	no	—
83	C₈	#95×C₅₃	1	1	yes	characteristic
84	C₈	#103×C₅₃	1	2	yes	—
85	C₈	#106×C₅₃	1	1	yes	characteristic
86	C₂²	—	2	2	no	—
87	C₂²	C₅₃×#358	1	2	yes	—
88	C₂²	C₅₃×#366	1	1	yes	characteristic
89	C₄	C₅₃×#340	1	1	yes	characteristic
90	C₄	C₅₃×#357	1	2	yes	—
91	C₄	C₅₃×#367	1	1	yes	characteristic
92	C₂	—	2	2	no	—
93	C₂	C₅₃×#1380	1	1	yes	characteristic
94	C₂	C₅₃×#1425	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	D₁₆
commutator	C₁₆	C₄²×C₅₃
Frattini, cosocle	C₂²×C₁₆	C₂²×C₅₃
socle	C₂²×C₅₃	#366
prime radical	C₂³×C₅₃	Q₃₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂²×C₈×C₅₃ ◁ C₂²×C₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
38. #7472	C₅₃	direct (N × H)
56. C₅₃	#7472	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₂×C₅₃	2
C₂×C₅₃×#368	2
#7472	53

Subgroups — C53×#7472 #27214984

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7472 #27214984