Subgroups — C₂×C₃⋊(C₅₃⋊(D₁₉⋊C₃)) #49136291

Order 36252.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅₃×C₁₉⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃⋊(C₅₃⋊(D₁₉⋊C₃))	C₂	1	2	yes	maximal, direct product factor
3	C₂×D₃₀₂₁	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃⋊(D₁₉⋊C₃)	—	3	1	no	maximal
5	C₃×C₅₃×C₁₉⋊C₃	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₅₃⋊(D₁₉⋊C₃)	—	3	2	no	—
7	D₃₀₂₁	C₂×C₃	1	2	yes	complemented
8	C₂×C₅₃×C₁₉⋊C₃	—	2	1	no	—
9	C₂×C₅₃×C₁₉⋊C₃	D₃	1	1	yes	characteristic, complemented
10	C₂×C₃×C₁₉×C₅₃	C₂×C₃	1	1	yes	characteristic, complemented
11	C₂×D₁₀₀₇	—	3	1	no	Hall
12	C₃×C₁₉×C₅₃	C₂²×C₃	1	1	yes	characteristic, complemented
13	C₅₃×C₁₉⋊C₃	—	2	1	no	—
14	C₅₃×C₁₉⋊C₃	C₂×D₃	1	1	yes	characteristic, complemented
15	C₂×C₁₉×C₅₃	C₃×D₃	1	1	yes	characteristic, complemented
16	D₁₀₀₇	—	3	2	no	—
17	C₂×C₃×D₁₅₉	—	19	1	no	maximal, Hall
18	C₁₉×C₅₃	C₂×C₃×D₃	1	1	yes	characteristic, complemented, Hall
19	C₂×C₃²×C₅₃	—	19	1	no	—
20	C₃×D₁₅₉	—	19	2	no	—
21	C₂×C₃⋊(D₁₉⋊C₃)	—	53	1	no	maximal, Hall
22	C₂×D₁₅₉	—	19	1	no	—
23	C₂×C₃×D₅₃	—	57	1	no	—
24	C₃²×C₅₃	—	19	1	no	Hall
25	C₂×C₃×C₁₉⋊C₃	D₅₃	1	1	yes	characteristic, complemented
26	C₃⋊(D₁₉⋊C₃)	—	53	2	no	—
27	C₂×C₃×C₅₃	—	19	1	no	—
28	C₂×C₃×C₅₃	—	38	1	no	—
29	C₂×C₃×C₅₃	D₁₉⋊C₃	1	1	yes	characteristic, complemented
30	D₁₅₉	—	19	2	no	—
31	C₃×D₅₃	—	57	2	no	—
32	C₂×D₅₇	—	53	1	no	—
33	C₂×D₁₉⋊C₃	—	159	1	no	—
34	C₂×D₅₃	—	57	1	no	Hall
35	C₃×C₁₉⋊C₃	C₂×D₅₃	1	1	yes	characteristic, complemented, Hall
36	C₃×C₅₃	—	19	1	no	—
37	C₃×C₅₃	—	38	1	no	—
38	C₃×C₅₃	C₂×D₁₉⋊C₃	1	1	yes	characteristic, complemented
39	C₂×C₃×C₁₉	C₃×D₅₃	1	1	yes	characteristic, complemented
40	D₅₇	—	53	2	no	—
41	C₂×C₁₉⋊C₃	—	2	1	no	—
42	C₂×C₁₉⋊C₃	D₁₅₉	1	1	yes	characteristic, complemented
43	D₁₉⋊C₃	—	159	2	no	—
44	C₂×C₅₃	C₃⋊(D₁₉⋊C₃)	1	1	yes	characteristic, complemented
45	D₅₃	—	57	2	no	—
46	C₂×D₁₉	—	159	1	no	Hall
47	C₃×C₁₉	C₂×C₃×D₅₃	1	1	yes	characteristic, complemented
48	C₁₉⋊C₃	—	2	1	no	—
49	C₁₉⋊C₃	C₂×D₁₅₉	1	1	yes	characteristic, complemented
50	C₅₃	C₂×C₃⋊(D₁₉⋊C₃)	1	1	yes	characteristic, complemented, Sylow
51	C₂×C₁₉	C₃×D₁₅₉	1	1	yes	characteristic, complemented
52	D₁₉	—	159	2	no	—
53	C₂×C₃×D₃	—	1007	1	no	Hall
54	C₁₉	C₂×C₃×D₁₅₉	1	1	yes	characteristic, complemented, Sylow
55	C₂×C₃²	—	19	1	no	—
56	C₃×D₃	—	1007	2	no	—
57	C₂²×C₃	—	3021	1	no	—
58	C₂×D₃	—	1007	1	no	—
59	C₃²	—	19	1	no	Sylow
60	C₂×C₃	—	19	1	no	—
61	C₂×C₃	—	38	1	no	—
62	C₂×C₃	—	3021	2	no	—
63	C₂×C₃	C₅₃⋊(D₁₉⋊C₃)	1	1	yes	characteristic, complemented
64	D₃	—	1007	2	no	—
65	C₂²	—	3021	1	no	Sylow
66	C₃	—	19	1	no	—
67	C₃	—	38	1	no	—
68	C₃	C₂×C₅₃⋊(D₁₉⋊C₃)	1	1	yes	characteristic, complemented
69	C₂	—	3021	2	no	—
70	C₂	C₃⋊(C₅₃⋊(D₁₉⋊C₃))	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	C₃⋊(C₅₃⋊(D₁₉⋊C₃))
commutator, cosocle, hypocenter	C₃×C₁₉×C₅₃	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₁₉×C₅₃	C₂×C₃
LCS	G ▷ C₃×C₁₉×C₅₃
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅₃×C₁₉⋊C₃	C₂	semidirect
2. C₃⋊(C₅₃⋊(D₁₉⋊C₃))	C₂	direct (N × H)
3. C₂×D₃₀₂₁	C₃	semidirect
5. C₃×C₅₃×C₁₉⋊C₃	C₂²	semidirect
7. D₃₀₂₁	C₂×C₃	semidirect
9. C₂×C₅₃×C₁₉⋊C₃	D₃	semidirect
10. C₂×C₃×C₁₉×C₅₃	C₂×C₃	semidirect
12. C₃×C₁₉×C₅₃	C₂²×C₃	semidirect
14. C₅₃×C₁₉⋊C₃	C₂×D₃	semidirect
15. C₂×C₁₉×C₅₃	C₃×D₃	semidirect
18. C₁₉×C₅₃	C₂×C₃×D₃	semidirect
25. C₂×C₃×C₁₉⋊C₃	D₅₃	semidirect
29. C₂×C₃×C₅₃	D₁₉⋊C₃	semidirect
35. C₃×C₁₉⋊C₃	C₂×D₅₃	semidirect
38. C₃×C₅₃	C₂×D₁₉⋊C₃	semidirect
39. C₂×C₃×C₁₉	C₃×D₅₃	semidirect
42. C₂×C₁₉⋊C₃	D₁₅₉	semidirect
44. C₂×C₅₃	C₃⋊(D₁₉⋊C₃)	semidirect
47. C₃×C₁₉	C₂×C₃×D₅₃	semidirect
49. C₁₉⋊C₃	C₂×D₁₅₉	semidirect
50. C₅₃	C₂×C₃⋊(D₁₉⋊C₃)	semidirect
51. C₂×C₁₉	C₃×D₁₅₉	semidirect
54. C₁₉	C₂×C₃×D₁₅₉	semidirect
63. C₂×C₃	C₅₃⋊(D₁₉⋊C₃)	semidirect
68. C₃	C₂×C₅₃⋊(D₁₉⋊C₃)	semidirect
70. C₂	C₃⋊(C₅₃⋊(D₁₉⋊C₃))	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×C₃×C₅₃×C₁₉⋊C₃	2
C₃⋊(C₅₃⋊(D₁₉⋊C₃))	2
C₂×D₃₀₂₁	3
C₂×C₅₃⋊(D₁₉⋊C₃)	3
C₂×C₃×D₁₅₉	19
C₂×C₃⋊(D₁₉⋊C₃)	53

Subgroups — C2×C3⋊(C53⋊(D19⋊C3)) #49136291

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃⋊(C₅₃⋊(D₁₉⋊C₃)) #49136291