Subgroups — C₅₃×#13741 #27221253

Order 13568.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2262	C₂	1	2	yes	maximal, complemented
2	C₅₃×Q₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×SD₁₂₈	C₂	1	2	yes	maximal, complemented
4	C₅₃×D₆₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅₃×C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#508	—	2	2	no	—
7	C₅₃×Q₆₄	C₂²	1	2	yes	—
8	C₅₃×SD₆₄	—	2	2	no	—
9	C₅₃×D₃₂	C₂²	1	2	yes	—
10	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×C₆₄	C₂²	1	1	yes	characteristic
12	C₅₃×C₆₄	C₂²	1	1	yes	characteristic
13	#135×C₅₃	—	4	2	no	—
14	Q₃₂×C₅₃	—	2	2	no	—
15	SD₃₂×C₅₃	—	4	2	no	—
16	D₁₆×C₅₃	—	2	2	no	—
17	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
18	C₃₂×C₅₃	D₄	1	1	yes	characteristic
19	C₃₂×C₅₃	C₂³	1	1	yes	characteristic
20	#41×C₅₃	—	8	2	no	—
21	Q₁₆×C₅₃	—	4	2	no	—
22	SD₁₆×C₅₃	—	8	2	no	—
23	D₈×C₅₃	—	4	2	no	—
24	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
25	C₁₆×C₅₃	D₈	1	1	yes	characteristic
26	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
27	Q₈×C₅₃	—	8	2	no	—
28	D₄×C₅₃	—	8	2	no	—
29	D₄×C₅₃	—	16	2	no	—
30	C₂×C₄×C₅₃	—	16	2	no	—
31	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
32	C₈×C₅₃	D₁₆	1	1	yes	characteristic
33	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
34	#13741	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂²×C₅₃	—	16	2	no	—
36	C₂²×C₅₃	D₃₂	1	1	yes	characteristic
37	C₄×C₅₃	—	16	2	no	—
38	C₄×C₅₃	D₃₂	1	1	yes	characteristic
39	C₄×C₅₃	C₂×D₁₆	1	1	yes	characteristic
40	#2262	C₂×C₅₃	1	2	yes	complemented
41	Q₁₂₈	C₂×C₅₃	1	1	yes	characteristic, complemented
42	SD₁₂₈	C₂×C₅₃	1	2	yes	complemented
43	D₆₄	C₂×C₅₃	1	1	yes	characteristic, complemented
44	C₂×C₆₄	C₂×C₅₃	1	1	yes	characteristic, complemented
45	C₂×C₅₃	—	2	1	no	—
46	C₂×C₅₃	—	32	2	no	—
47	C₂×C₅₃	C₂×D₃₂	1	1	yes	characteristic
48	#508	—	2	2	no	—
49	Q₆₄	C₂²×C₅₃	1	2	yes	—
50	SD₆₄	—	2	2	no	—
51	D₃₂	C₂²×C₅₃	1	2	yes	—
52	C₂×C₃₂	C₂²×C₅₃	1	1	yes	characteristic
53	C₆₄	C₂²×C₅₃	1	1	yes	characteristic
54	C₆₄	C₂²×C₅₃	1	1	yes	characteristic
55	C₅₃	#13741	1	1	yes	characteristic, direct product factor, Sylow
56	#135	—	4	2	no	—
57	Q₃₂	—	2	2	no	—
58	SD₃₂	—	4	2	no	—
59	D₁₆	—	2	2	no	—
60	C₂×C₁₆	D₄×C₅₃	1	1	yes	characteristic
61	C₃₂	D₄×C₅₃	1	1	yes	characteristic
62	C₃₂	C₂³×C₅₃	1	1	yes	characteristic
63	#41	—	8	2	no	—
64	Q₁₆	—	4	2	no	—
65	SD₁₆	—	8	2	no	—
66	D₈	—	4	2	no	—
67	C₂×C₈	D₈×C₅₃	1	1	yes	characteristic
68	C₁₆	D₈×C₅₃	1	1	yes	characteristic
69	C₁₆	C₂×D₄×C₅₃	1	1	yes	characteristic
70	Q₈	—	8	2	no	—
71	D₄	—	8	2	no	—
72	D₄	—	16	2	no	—
73	C₂×C₄	—	16	2	no	—
74	C₂×C₄	D₁₆×C₅₃	1	1	yes	characteristic
75	C₈	D₁₆×C₅₃	1	1	yes	characteristic
76	C₈	C₂×D₈×C₅₃	1	1	yes	characteristic
77	C₂²	—	16	2	no	—
78	C₂²	C₅₃×D₃₂	1	1	yes	characteristic
79	C₄	—	16	2	no	—
80	C₄	C₅₃×D₃₂	1	1	yes	characteristic
81	C₄	C₂×D₁₆×C₅₃	1	1	yes	characteristic
82	C₂	—	2	1	no	—
83	C₂	—	32	2	no	—
84	C₂	C₂×C₅₃×D₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅₃	D₃₂
commutator, Frattini, cosocle	C₃₂	C₂³×C₅₃
socle	C₂×C₅₃	C₂×D₃₂
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×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
1. C₅₃×#2262	C₂	semidirect
2. C₅₃×Q₁₂₈	C₂	semidirect
3. C₅₃×SD₁₂₈	C₂	semidirect
4. C₅₃×D₆₄	C₂	semidirect
5. C₂×C₅₃×C₆₄	C₂	semidirect
34. #13741	C₅₃	direct (N × H)
40. #2262	C₂×C₅₃	semidirect
41. Q₁₂₈	C₂×C₅₃	semidirect
42. SD₁₂₈	C₂×C₅₃	semidirect
43. D₆₄	C₂×C₅₃	semidirect
44. C₂×C₆₄	C₂×C₅₃	semidirect
55. C₅₃	#13741	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×#2262	2
C₅₃×Q₁₂₈	2
C₅₃×SD₁₂₈	2
C₅₃×D₆₄	2
C₂×C₅₃×C₆₄	2
#13741	53

Subgroups — C53×#13741 #27221253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#13741 #27221253