Subgroups — C₅₃×#7513 #27215025

Order 13568.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×M₆₄	C₂	1	1	yes	maximal, characteristic
2	C₅₃×M₁₂₈	C₂	1	2	yes	maximal, complemented
3	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
4	C₅₃×M₆₄	—	2	2	no	—
5	C₂×C₃₂×C₅₃	C₄	1	1	yes	characteristic
6	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×C₆₄	—	2	2	no	—
8	C₂²×C₈×C₅₃	C₈	1	1	yes	characteristic
9	C₂×C₁₆×C₅₃	—	2	1	no	—
10	C₂×C₁₆×C₅₃	—	2	1	no	—
11	C₂×C₁₆×C₅₃	C₈	1	1	yes	characteristic
12	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₃₂×C₅₃	—	2	1	no	—
14	C₃₂×C₅₃	D₄	1	2	yes	—
15	C₂²×C₄×C₅₃	C₁₆	1	1	yes	characteristic
16	C₂×C₈×C₅₃	—	2	1	no	—
17	C₂×C₈×C₅₃	—	2	1	no	—
18	C₂×C₈×C₅₃	C₁₆	1	1	yes	characteristic
19	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
20	C₁₆×C₅₃	—	2	1	no	—
21	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₁₆×C₅₃	M₁₆	1	1	yes	characteristic
23	C₂³×C₅₃	C₃₂	1	1	yes	characteristic
24	C₂×C₄×C₅₃	—	2	1	no	—
25	C₂×C₄×C₅₃	—	2	1	no	—
26	C₂×C₄×C₅₃	C₃₂	1	1	yes	characteristic
27	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
28	C₈×C₅₃	—	2	1	no	—
29	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
30	C₈×C₅₃	M₃₂	1	1	yes	characteristic
31	#7513	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂²×C₅₃	—	2	1	no	—
33	C₂²×C₅₃	—	4	1	no	—
34	C₂²×C₅₃	C₂×C₃₂	1	1	yes	characteristic
35	C₄×C₅₃	—	2	1	no	—
36	C₄×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
37	C₄×C₅₃	M₆₄	1	1	yes	characteristic
38	C₂×M₆₄	C₂×C₅₃	1	1	yes	characteristic
39	M₁₂₈	C₂×C₅₃	1	2	yes	complemented
40	C₂×C₅₃	—	2	1	no	—
41	C₂×C₅₃	—	4	1	no	—
42	C₂×C₅₃	C₂²⋊C₃₂	1	1	yes	characteristic
43	C₂²×C₁₆	C₄×C₅₃	1	1	yes	characteristic
44	M₆₄	—	2	2	no	—
45	C₂×C₃₂	C₄×C₅₃	1	1	yes	characteristic
46	C₂×C₃₂	C₂²×C₅₃	1	1	yes	characteristic
47	C₆₄	—	2	2	no	—
48	C₅₃	#7513	1	1	yes	characteristic, direct product factor, Sylow
49	C₂²×C₈	C₈×C₅₃	1	1	yes	characteristic
50	C₂×C₁₆	—	2	1	no	—
51	C₂×C₁₆	—	2	1	no	—
52	C₂×C₁₆	C₈×C₅₃	1	1	yes	characteristic
53	C₂×C₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
54	C₃₂	—	2	1	no	—
55	C₃₂	D₄×C₅₃	1	2	yes	—
56	C₂²×C₄	C₁₆×C₅₃	1	1	yes	characteristic
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	—	2	1	no	—
59	C₂×C₈	C₁₆×C₅₃	1	1	yes	characteristic
60	C₂×C₈	C₂×C₈×C₅₃	1	1	yes	characteristic
61	C₁₆	—	2	1	no	—
62	C₁₆	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
63	C₁₆	M₁₆×C₅₃	1	1	yes	characteristic
64	C₂³	C₃₂×C₅₃	1	1	yes	characteristic
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	C₃₂×C₅₃	1	1	yes	characteristic
68	C₂×C₄	C₂×C₁₆×C₅₃	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₈	C₂²⋊C₈×C₅₃	1	1	yes	characteristic
71	C₈	M₃₂×C₅₃	1	1	yes	characteristic
72	C₂²	—	2	1	no	—
73	C₂²	—	4	1	no	—
74	C₂²	C₂×C₃₂×C₅₃	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	C₅₃×C₂²⋊C₁₆	1	1	yes	characteristic
77	C₄	C₅₃×M₆₄	1	1	yes	characteristic
78	C₂	—	2	1	no	—
79	C₂	—	4	1	no	—
80	C₂	C₅₃×C₂²⋊C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆×C₅₃	C₂²⋊C₄
commutator	C₂²	C₂×C₃₂×C₅₃
Frattini, cosocle	C₂×C₃₂	C₂²×C₅₃
socle	C₂×C₅₃	C₂²⋊C₃₂
prime radical	C₂³×C₅₃	C₃₂
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆×C₅₃ ◁ C₂×C₃₂×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×M₁₂₈	C₂	semidirect
31. #7513	C₅₃	direct (N × H)
39. M₁₂₈	C₂×C₅₃	semidirect
48. C₅₃	#7513	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×M₆₄	2
C₅₃×M₁₂₈	2
#7513	53

Subgroups — C53×#7513 #27215025

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7513 #27215025