Subgroups — C₂²×C₁₉⋊C₃×C₅²⋊C₃ #30685817

Order 17100.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉⋊C₃×C₅²⋊C₃	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₁₉⋊(C₅²⋊C₃)	C₃	1	2	yes	maximal, complemented
3	C₂²×C₁₉×C₅²⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₅²×C₁₉⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
5	C₁₉⋊C₃×C₅²⋊C₃	C₂²	1	1	yes	characteristic, direct product factor, Hall
6	C₂×C₁₉⋊(C₅²⋊C₃)	C₂×C₃	1	6	yes	complemented
7	C₂×C₁₉×C₅²⋊C₃	C₂×C₃	1	3	yes	complemented
8	C₂×C₅²×C₁₉⋊C₃	C₂×C₃	1	3	yes	complemented
9	C₂²×C₅²×C₁₉	C₃²	1	1	yes	characteristic, complemented, Hall
10	C₁₉⋊(C₅²⋊C₃)	C₂²×C₃	1	2	yes	complemented
11	C₁₉×C₅²⋊C₃	C₂²×C₃	1	1	yes	characteristic, complemented
12	C₅²×C₁₉⋊C₃	C₂²×C₃	1	1	yes	characteristic, complemented
13	C₂²×C₅×C₁₉⋊C₃	—	3	2	no	—
14	C₂×C₅²×C₁₉	C₂×C₃²	1	3	yes	complemented
15	C₂²×C₃×C₅²⋊C₃	—	19	1	no	maximal, Hall
16	C₂²×C₃×C₁₉⋊C₃	—	25	1	no	maximal, Hall
17	C₂×C₅×C₁₉⋊C₃	—	3	6	no	—
18	C₅²×C₁₉	C₂²×C₃²	1	1	yes	characteristic, complemented, Hall
19	C₂×C₃×C₅²⋊C₃	—	19	3	no	—
20	C₂²×C₅×C₁₉	—	3	2	no	—
21	C₂×C₃×C₁₉⋊C₃	—	25	3	no	—
22	C₂²×C₃×C₅²	—	19	1	no	—
23	C₂²×C₅²⋊C₃	—	19	2	no	—
24	C₂²×C₅²⋊C₃	C₁₉⋊C₃	1	1	yes	characteristic, direct product factor
25	C₅×C₁₉⋊C₃	—	3	2	no	—
26	C₂²×C₃×C₁₉	—	25	1	no	—
27	C₂²×C₁₉⋊C₃	—	25	2	no	—
28	C₂²×C₁₉⋊C₃	C₅²⋊C₃	1	1	yes	characteristic, direct product factor
29	C₃×C₅²⋊C₃	—	19	1	no	Hall
30	C₂×C₅×C₁₉	—	3	6	no	—
31	C₃×C₁₉⋊C₃	—	25	1	no	Hall
32	C₂×C₃×C₅²	—	19	3	no	—
33	C₂×C₅²⋊C₃	—	19	6	no	—
34	C₂×C₅²⋊C₃	C₂×C₁₉⋊C₃	1	3	yes	direct product factor
35	C₂×C₃×C₁₉	—	25	3	no	—
36	C₂×C₁₉⋊C₃	—	25	6	no	—
37	C₂×C₁₉⋊C₃	C₂×C₅²⋊C₃	1	3	yes	direct product factor
38	C₂²×C₅²	C₃×C₁₉⋊C₃	1	1	yes	characteristic, complemented, Hall
39	C₅×C₁₉	—	3	2	no	—
40	C₂²×C₁₉	C₃×C₅²⋊C₃	1	1	yes	characteristic, complemented, Hall
41	C₃×C₅²	—	19	1	no	—
42	C₅²⋊C₃	—	19	2	no	—
43	C₅²⋊C₃	C₂²×C₁₉⋊C₃	1	1	yes	characteristic, direct product factor
44	C₂²×C₃×C₅	—	57	2	no	—
45	C₃×C₁₉	—	25	1	no	—
46	C₁₉⋊C₃	—	25	2	no	—
47	C₁₉⋊C₃	C₂²×C₅²⋊C₃	1	1	yes	characteristic, direct product factor
48	C₂×C₅²	C₂×C₃×C₁₉⋊C₃	1	3	yes	complemented
49	C₂×C₁₉	C₂×C₃×C₅²⋊C₃	1	3	yes	complemented
50	C₂²×C₃²	—	475	1	no	Hall
51	C₂×C₃×C₅	—	57	6	no	—
52	C₅²	C₂²×C₃×C₁₉⋊C₃	1	1	yes	characteristic, complemented, Sylow
53	C₂²×C₅	—	3	2	no	—
54	C₁₉	C₂²×C₃×C₅²⋊C₃	1	1	yes	characteristic, complemented, Sylow
55	C₂×C₃²	—	475	3	no	—
56	C₃×C₅	—	57	2	no	—
57	C₂²×C₃	—	19	1	no	—
58	C₂²×C₃	—	25	1	no	—
59	C₂²×C₃	—	475	2	no	—
60	C₂×C₅	—	3	6	no	—
61	C₃²	—	475	1	no	Sylow
62	C₂×C₃	—	19	3	no	—
63	C₂×C₃	—	25	3	no	—
64	C₂×C₃	—	475	6	no	—
65	C₅	—	3	2	no	—
66	C₂²	C₁₉⋊C₃×C₅²⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
67	C₃	—	19	1	no	—
68	C₃	—	25	1	no	—
69	C₃	—	475	2	no	—
70	C₂	C₂×C₁₉⋊C₃×C₅²⋊C₃	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²	C₁₉⋊C₃×C₅²⋊C₃
commutator, cosocle, hypocenter	C₅²×C₁₉	C₂²×C₃²
Frattini	C₁	G
Fitting, socle	C₂²×C₅²×C₁₉	C₃²
LCS	G ▷ C₅²×C₁₉
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₉⋊C₃×C₅²⋊C₃	C₂	direct (N × H)
2. C₂²×C₁₉⋊(C₅²⋊C₃)	C₃	semidirect
3. C₂²×C₁₉×C₅²⋊C₃	C₃	semidirect
4. C₂²×C₅²×C₁₉⋊C₃	C₃	semidirect
5. C₁₉⋊C₃×C₅²⋊C₃	C₂²	direct (N × H)
6. C₂×C₁₉⋊(C₅²⋊C₃)	C₂×C₃	semidirect
7. C₂×C₁₉×C₅²⋊C₃	C₂×C₃	semidirect
8. C₂×C₅²×C₁₉⋊C₃	C₂×C₃	semidirect
9. C₂²×C₅²×C₁₉	C₃²	semidirect
10. C₁₉⋊(C₅²⋊C₃)	C₂²×C₃	semidirect
11. C₁₉×C₅²⋊C₃	C₂²×C₃	semidirect
12. C₅²×C₁₉⋊C₃	C₂²×C₃	semidirect
14. C₂×C₅²×C₁₉	C₂×C₃²	semidirect
18. C₅²×C₁₉	C₂²×C₃²	semidirect
24. C₂²×C₅²⋊C₃	C₁₉⋊C₃	direct (N × H)
28. C₂²×C₁₉⋊C₃	C₅²⋊C₃	direct (N × H)
34. C₂×C₅²⋊C₃	C₂×C₁₉⋊C₃	direct (N × H)
37. C₂×C₁₉⋊C₃	C₂×C₅²⋊C₃	direct (N × H)
38. C₂²×C₅²	C₃×C₁₉⋊C₃	semidirect
40. C₂²×C₁₉	C₃×C₅²⋊C₃	semidirect
43. C₅²⋊C₃	C₂²×C₁₉⋊C₃	direct (N × H)
47. C₁₉⋊C₃	C₂²×C₅²⋊C₃	direct (N × H)
48. C₂×C₅²	C₂×C₃×C₁₉⋊C₃	semidirect
49. C₂×C₁₉	C₂×C₃×C₅²⋊C₃	semidirect
52. C₅²	C₂²×C₃×C₁₉⋊C₃	semidirect
54. C₁₉	C₂²×C₃×C₅²⋊C₃	semidirect
66. C₂²	C₁₉⋊C₃×C₅²⋊C₃	direct (N × H)
70. C₂	C₂×C₁₉⋊C₃×C₅²⋊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×C₁₉⋊C₃×C₅²⋊C₃	2
C₂²×C₁₉⋊(C₅²⋊C₃)	3
C₂²×C₁₉×C₅²⋊C₃	3
C₂²×C₅²×C₁₉⋊C₃	3
C₂²×C₃×C₅²⋊C₃	19
C₂²×C₃×C₁₉⋊C₃	25

Subgroups — C22×C19⋊C3×C52⋊C3 #30685817

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₉⋊C₃×C₅²⋊C₃ #30685817