Subgroups — C₅₃×#7542 #27215054

Order 13568.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×Q₆₄	C₂	1	1	yes	maximal, characteristic
2	C₅₃×M₁₂₈	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1425	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₃₂×C₅₃	—	2	1	no	—
5	C₅₃×Q₆₄	—	2	1	no	—
6	C₅₃×Q₆₄	C₄	1	2	yes	—
7	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×#368	—	2	1	no	—
9	C₅₃×C₆₄	—	2	1	no	—
10	C₂×Q₁₆×C₅₃	—	4	1	no	—
11	Q₃₂×C₅₃	—	2	2	no	—
12	Q₃₂×C₅₃	—	4	1	no	—
13	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
14	#108×C₅₃	—	4	1	no	—
15	C₃₂×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₅₃	D₄	1	1	yes	characteristic
17	C₂×Q₈×C₅₃	—	8	1	no	—
18	Q₁₆×C₅₃	—	4	2	no	—
19	Q₁₆×C₅₃	—	8	1	no	—
20	M₁₆×C₅₃	—	8	1	no	—
21	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
22	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₁₆×C₅₃	SD₁₆	1	1	yes	characteristic
24	Q₈×C₅₃	—	8	2	no	—
25	Q₈×C₅₃	—	16	1	no	—
26	C₂×C₄×C₅₃	—	16	1	no	—
27	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
28	C₈×C₅₃	—	16	1	no	—
29	C₈×C₅₃	#102	1	1	yes	characteristic
30	C₈×C₅₃	SD₃₂	1	1	yes	characteristic
31	#7542	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂²×C₅₃	SD₆₄	1	1	yes	characteristic
33	C₄×C₅₃	—	16	2	no	—
34	C₄×C₅₃	#357	1	1	yes	characteristic
35	C₄×C₅₃	D₃₂	1	1	yes	characteristic
36	C₂×Q₆₄	C₂×C₅₃	1	1	yes	characteristic
37	M₁₂₈	C₂×C₅₃	1	1	yes	characteristic
38	#1425	C₂×C₅₃	1	1	yes	characteristic
39	C₂×C₅₃	—	2	1	no	—
40	C₂×C₅₃	#1415	1	1	yes	characteristic
41	C₂×Q₃₂	—	2	1	no	—
42	Q₆₄	—	2	1	no	—
43	Q₆₄	C₄×C₅₃	1	2	yes	—
44	C₂×C₃₂	C₂²×C₅₃	1	1	yes	characteristic
45	#368	—	2	1	no	—
46	C₆₄	—	2	1	no	—
47	C₅₃	#7542	1	1	yes	characteristic, direct product factor, Sylow
48	C₂×Q₁₆	—	4	1	no	—
49	Q₃₂	—	2	2	no	—
50	Q₃₂	—	4	1	no	—
51	C₂×C₁₆	D₄×C₅₃	1	1	yes	characteristic
52	#108	—	4	1	no	—
53	C₃₂	C₂×C₄×C₅₃	1	1	yes	characteristic
54	C₃₂	D₄×C₅₃	1	1	yes	characteristic
55	C₂×Q₈	—	8	1	no	—
56	Q₁₆	—	4	2	no	—
57	Q₁₆	—	8	1	no	—
58	M₁₆	—	8	1	no	—
59	C₂×C₈	D₈×C₅₃	1	1	yes	characteristic
60	C₁₆	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
61	C₁₆	SD₁₆×C₅₃	1	1	yes	characteristic
62	Q₈	—	8	2	no	—
63	Q₈	—	16	1	no	—
64	C₂×C₄	—	16	1	no	—
65	C₂×C₄	D₁₆×C₅₃	1	1	yes	characteristic
66	C₈	—	16	1	no	—
67	C₈	#102×C₅₃	1	1	yes	characteristic
68	C₈	SD₃₂×C₅₃	1	1	yes	characteristic
69	C₂²	C₅₃×SD₆₄	1	1	yes	characteristic
70	C₄	—	16	2	no	—
71	C₄	C₅₃×#357	1	1	yes	characteristic
72	C₄	C₅₃×D₃₂	1	1	yes	characteristic
73	C₂	—	2	1	no	—
74	C₂	C₅₃×#1415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅₃	#1415
commutator	C₃₂	C₂×C₄×C₅₃
Frattini, cosocle	C₂×C₃₂	C₂²×C₅₃
prime radical	C₂²×C₅₃	SD₆₄
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₂×C₈×C₅₃ ◁ C₂×C₁₆×C₅₃ ◁ C₂×C₃₂×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
31. #7542	C₅₃	direct (N × H)
47. C₅₃	#7542	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×C₅₃×Q₆₄	2
C₅₃×M₁₂₈	2
C₅₃×#1425	2
#7542	53

Subgroups — C53×#7542 #27215054

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7542 #27215054