Subgroups — C₂×C₃×C₃⋊(D₅₃⋊C₁₃) #37613673

Order 24804.

88 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)	C₃	1	1	yes	maximal, characteristic, direct product factor
4	C₂×C₃×D₅₃⋊C₁₃	—	3	1	no	maximal
5	C₃²×C₅₃⋊C₁₃	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₃⋊(D₅₃⋊C₁₃)	C₂×C₃	1	2	yes	direct product factor
7	C₃×D₅₃⋊C₁₃	—	3	2	no	—
8	C₂×C₃×C₅₃⋊C₁₃	—	2	1	no	—
9	C₂×C₃×C₅₃⋊C₁₃	D₃	1	1	yes	characteristic, complemented
10	C₂×C₃×C₅₃⋊C₁₃	C₂×C₃	1	1	yes	characteristic, complemented
11	C₂×D₅₃⋊C₁₃	—	3	1	no	Hall
12	C₃×C₅₃⋊C₁₃	—	2	1	no	—
13	C₃×C₅₃⋊C₁₃	C₂×D₃	1	1	yes	characteristic, complemented
14	C₃×C₅₃⋊C₁₃	C₂²×C₃	1	1	yes	characteristic, complemented
15	C₂×C₃×D₁₅₉	C₁₃	1	1	yes	maximal, characteristic, complemented, Hall
16	C₂×C₅₃⋊C₁₃	C₃×D₃	1	1	yes	characteristic, complemented
17	D₅₃⋊C₁₃	—	3	2	no	—
18	C₂×C₃²×C₅₃	C₂×C₁₃	1	1	yes	characteristic, complemented
19	C₃×D₁₅₉	C₂×C₁₃	1	2	yes	complemented
20	C₅₃⋊C₁₃	C₂×C₃×D₃	1	1	yes	characteristic, complemented, Hall
21	C₂×D₁₅₉	C₃×C₁₃	1	1	yes	characteristic, complemented
22	C₂×C₃×D₅₃	—	3	1	no	—
23	C₃²×C₅₃	C₂²×C₁₃	1	1	yes	characteristic, complemented, Hall
24	C₂×C₃×D₃×C₁₃	—	53	1	no	maximal, Hall
25	C₂×C₃×C₅₃	—	2	1	no	—
26	C₂×C₃×C₅₃	D₃×C₁₃	1	1	yes	characteristic, complemented
27	C₂×C₃×C₅₃	C₂×C₃×C₁₃	1	1	yes	characteristic, complemented
28	D₁₅₉	C₂×C₃×C₁₃	1	2	yes	complemented
29	C₃×D₅₃	—	3	2	no	—
30	C₂×C₃²×C₁₃	—	53	1	no	—
31	C₃×D₃×C₁₃	—	53	2	no	—
32	C₂×D₅₃	—	3	1	no	Hall
33	C₃×C₅₃	—	2	1	no	—
34	C₃×C₅₃	C₂×D₃×C₁₃	1	1	yes	characteristic, complemented
35	C₃×C₅₃	C₂²×C₃×C₁₃	1	1	yes	characteristic, complemented
36	C₂²×C₃×C₁₃	—	159	1	no	—
37	C₂×D₃×C₁₃	—	53	1	no	—
38	C₃²×C₁₃	—	53	1	no	Hall
39	C₂×C₅₃	C₃×D₃×C₁₃	1	1	yes	characteristic, complemented
40	D₅₃	—	3	2	no	—
41	C₂×C₃×C₁₃	—	53	1	no	—
42	C₂×C₃×C₁₃	—	53	1	no	—
43	C₂×C₃×C₁₃	—	106	1	no	—
44	C₂×C₃×C₁₃	—	159	2	no	—
45	D₃×C₁₃	—	53	2	no	—
46	C₅₃	C₂×C₃×D₃×C₁₃	1	1	yes	characteristic, complemented, Sylow
47	C₂²×C₁₃	—	159	1	no	Hall
48	C₃×C₁₃	—	53	1	no	—
49	C₃×C₁₃	—	53	1	no	—
50	C₃×C₁₃	—	106	1	no	—
51	C₂×C₃×D₃	—	689	1	no	Hall
52	C₂×C₁₃	—	53	1	no	—
53	C₂×C₁₃	—	159	2	no	—
54	C₂×C₃²	D₅₃⋊C₁₃	1	1	yes	characteristic, complemented
55	C₃×D₃	—	689	2	no	—
56	C₁₃	—	53	1	no	Sylow
57	C₂²×C₃	—	2067	1	no	—
58	C₂×D₃	—	689	1	no	—
59	C₃²	C₂×D₅₃⋊C₁₃	1	1	yes	characteristic, complemented, Sylow
60	C₂×C₃	—	2	1	no	—
61	C₂×C₃	—	2067	2	no	—
62	C₂×C₃	C₃×D₅₃⋊C₁₃	1	1	yes	characteristic, complemented
63	C₂×C₃	C₃⋊(D₅₃⋊C₁₃)	1	1	yes	characteristic, direct product factor
64	D₃	—	689	2	no	—
65	C₂²	—	2067	1	no	Sylow
66	C₃	—	2	1	no	—
67	C₃	C₂×C₃×D₅₃⋊C₁₃	1	1	yes	characteristic, complemented
68	C₃	C₂×C₃⋊(D₅₃⋊C₁₃)	1	1	yes	characteristic, direct product factor
69	C₂	—	2067	2	no	—
70	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	C₃×C₅₃	C₂²×C₃×C₁₃
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₅₃	C₂×C₁₃
LCS	G ▷ C₃×C₅₃
UCS	C₁ ◁ C₂×C₃

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)
3. C₂×C₃⋊(D₅₃⋊C₁₃)	C₃	direct (N × H)
5. C₃²×C₅₃⋊C₁₃	C₂²	semidirect
6. C₃⋊(D₅₃⋊C₁₃)	C₂×C₃	direct (N × H)
9. C₂×C₃×C₅₃⋊C₁₃	D₃	semidirect
10. C₂×C₃×C₅₃⋊C₁₃	C₂×C₃	semidirect
13. C₃×C₅₃⋊C₁₃	C₂×D₃	semidirect
14. C₃×C₅₃⋊C₁₃	C₂²×C₃	semidirect
15. C₂×C₃×D₁₅₉	C₁₃	semidirect
16. C₂×C₅₃⋊C₁₃	C₃×D₃	semidirect
18. C₂×C₃²×C₅₃	C₂×C₁₃	semidirect
19. C₃×D₁₅₉	C₂×C₁₃	semidirect
20. C₅₃⋊C₁₃	C₂×C₃×D₃	semidirect
21. C₂×D₁₅₉	C₃×C₁₃	semidirect
23. C₃²×C₅₃	C₂²×C₁₃	semidirect
26. C₂×C₃×C₅₃	D₃×C₁₃	semidirect
27. C₂×C₃×C₅₃	C₂×C₃×C₁₃	semidirect
28. D₁₅₉	C₂×C₃×C₁₃	semidirect
34. C₃×C₅₃	C₂×D₃×C₁₃	semidirect
35. C₃×C₅₃	C₂²×C₃×C₁₃	semidirect
39. C₂×C₅₃	C₃×D₃×C₁₃	semidirect
46. C₅₃	C₂×C₃×D₃×C₁₃	semidirect
54. C₂×C₃²	D₅₃⋊C₁₃	semidirect
59. C₃²	C₂×D₅₃⋊C₁₃	semidirect
62. C₂×C₃	C₃×D₅₃⋊C₁₃	semidirect
63. C₂×C₃	C₃⋊(D₅₃⋊C₁₃)	direct (N × H)
67. C₃	C₂×C₃×D₅₃⋊C₁₃	semidirect
68. C₃	C₂×C₃⋊(D₅₃⋊C₁₃)	direct (N × H)
70. 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×D₅₃⋊C₁₃	3
C₂×C₃×D₁₅₉	13
C₂×C₃×D₃×C₁₃	53

Subgroups — C2×C3×C3⋊(D53⋊C13) #37613673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₃⋊(D₅₃⋊C₁₃) #37613673