Subgroups — C₂×C₃⋊(C₃⋊(D₅₃⋊C₁₃)) #37613675

Order 24804.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₅₃⋊C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃⋊(C₃⋊(D₅₃⋊C₁₃))	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃⋊(D₅₃⋊C₁₃)	—	3	4	no	maximal
4	C₃²×C₅₃⋊C₁₃	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₃⋊(D₅₃⋊C₁₃)	—	3	8	no	—
6	C₂×C₃×C₅₃⋊C₁₃	D₃	1	4	yes	complemented
7	C₂×D₅₃⋊C₁₃	—	9	1	no	Hall
8	C₃×C₅₃⋊C₁₃	C₂×D₃	1	4	yes	complemented
9	C₂×C₃⋊D₁₅₉	C₁₃	1	1	yes	maximal, characteristic, complemented, Hall
10	C₂×C₅₃⋊C₁₃	C₃⋊D₃	1	1	yes	characteristic, complemented
11	D₅₃⋊C₁₃	—	9	2	no	—
12	C₂×C₃²×C₅₃	C₂×C₁₃	1	1	yes	characteristic, complemented
13	C₃⋊D₁₅₉	C₂×C₁₃	1	2	yes	complemented
14	C₅₃⋊C₁₃	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented, Hall
15	C₂×D₁₅₉	—	3	4	no	—
16	C₃²×C₅₃	C₂²×C₁₃	1	1	yes	characteristic, complemented, Hall
17	C₂×C₁₃×C₃⋊D₃	—	53	1	no	maximal, Hall
18	C₂×C₃×C₅₃	D₃×C₁₃	1	4	yes	complemented
19	D₁₅₉	—	3	8	no	—
20	C₂×C₃²×C₁₃	—	53	1	no	—
21	C₁₃×C₃⋊D₃	—	53	2	no	—
22	C₂×D₅₃	—	9	1	no	Hall
23	C₃×C₅₃	C₂×D₃×C₁₃	1	4	yes	complemented
24	C₂×D₃×C₁₃	—	159	4	no	—
25	C₃²×C₁₃	—	53	1	no	Hall
26	C₂×C₅₃	C₁₃×C₃⋊D₃	1	1	yes	characteristic, complemented
27	D₅₃	—	9	2	no	—
28	C₂×C₃×C₁₃	—	53	4	no	—
29	D₃×C₁₃	—	159	8	no	—
30	C₅₃	C₂×C₁₃×C₃⋊D₃	1	1	yes	characteristic, complemented, Sylow
31	C₂²×C₁₃	—	477	1	no	Hall
32	C₃×C₁₃	—	53	4	no	—
33	C₂×C₃⋊D₃	—	689	1	no	Hall
34	C₂×C₁₃	—	53	1	no	—
35	C₂×C₁₃	—	477	2	no	—
36	C₂×C₃²	D₅₃⋊C₁₃	1	1	yes	characteristic, complemented
37	C₃⋊D₃	—	689	2	no	—
38	C₁₃	—	53	1	no	Sylow
39	C₂×D₃	—	2067	4	no	—
40	C₃²	C₂×D₅₃⋊C₁₃	1	1	yes	characteristic, complemented, Sylow
41	C₂×C₃	C₃⋊(D₅₃⋊C₁₃)	1	4	yes	complemented
42	D₃	—	2067	8	no	—
43	C₂²	—	6201	1	no	Sylow
44	C₃	C₂×C₃⋊(D₅₃⋊C₁₃)	1	4	yes	complemented
45	C₂	—	6201	2	no	—
46	C₂	C₃⋊(C₃⋊(D₅₃⋊C₁₃))	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	C₃⋊(C₃⋊(D₅₃⋊C₁₃))
commutator, cosocle, hypocenter	C₃²×C₅₃	C₂²×C₁₃
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₅₃	C₂×C₁₃
LCS	G ▷ C₃²×C₅₃
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×C₅₃⋊C₁₃	C₂	semidirect
2. C₃⋊(C₃⋊(D₅₃⋊C₁₃))	C₂	direct (N × H)
4. C₃²×C₅₃⋊C₁₃	C₂²	semidirect
6. C₂×C₃×C₅₃⋊C₁₃	D₃	semidirect
8. C₃×C₅₃⋊C₁₃	C₂×D₃	semidirect
9. C₂×C₃⋊D₁₅₉	C₁₃	semidirect
10. C₂×C₅₃⋊C₁₃	C₃⋊D₃	semidirect
12. C₂×C₃²×C₅₃	C₂×C₁₃	semidirect
13. C₃⋊D₁₅₉	C₂×C₁₃	semidirect
14. C₅₃⋊C₁₃	C₂×C₃⋊D₃	semidirect
16. C₃²×C₅₃	C₂²×C₁₃	semidirect
18. C₂×C₃×C₅₃	D₃×C₁₃	semidirect
23. C₃×C₅₃	C₂×D₃×C₁₃	semidirect
26. C₂×C₅₃	C₁₃×C₃⋊D₃	semidirect
30. C₅₃	C₂×C₁₃×C₃⋊D₃	semidirect
36. C₂×C₃²	D₅₃⋊C₁₃	semidirect
40. C₃²	C₂×D₅₃⋊C₁₃	semidirect
41. C₂×C₃	C₃⋊(D₅₃⋊C₁₃)	semidirect
44. C₃	C₂×C₃⋊(D₅₃⋊C₁₃)	semidirect
46. C₂	C₃⋊(C₃⋊(D₅₃⋊C₁₃))	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×C₃²×C₅₃⋊C₁₃	2
C₃⋊(C₃⋊(D₅₃⋊C₁₃))	2
C₂×C₃⋊(D₅₃⋊C₁₃)	3
C₂×C₃⋊D₁₅₉	13
C₂×C₁₃×C₃⋊D₃	53

Subgroups — C2×C3⋊(C3⋊(D53⋊C13)) #37613675

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃⋊(C₃⋊(D₅₃⋊C₁₃)) #37613675