Subgroups — C₅²⋊C₃×C₂²⋊(C₁₉⋊C₃) #30685823

Order 17100.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊(C₁₉⋊(C₅²⋊C₃))	C₃	1	2	yes	maximal, complemented
2	C₅²×C₂²⋊(C₁₉⋊C₃)	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₁₉×C₅²⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
4	C₁₉⋊C₃×C₅²⋊C₃	—	4	1	no	maximal, Hall
5	C₂×C₁₉×C₅²⋊C₃	—	3	1	no	—
6	C₂²×C₅²×C₁₉	C₃²	1	1	yes	characteristic, complemented, Hall
7	C₁₉⋊(C₅²⋊C₃)	—	4	2	no	—
8	C₁₉×C₅²⋊C₃	A₄	1	1	yes	characteristic, complemented
9	C₅²×C₁₉⋊C₃	—	4	1	no	—
10	C₅×C₂²⋊(C₁₉⋊C₃)	—	3	2	no	—
11	C₂×C₅²×C₁₉	—	3	1	no	—
12	A₄×C₅²⋊C₃	—	19	1	no	maximal, Hall
13	C₃×C₂²⋊(C₁₉⋊C₃)	—	25	1	no	maximal, Hall
14	C₅²×C₁₉	C₃×A₄	1	1	yes	characteristic, complemented, Hall
15	C₂²×C₅×C₁₉	—	3	2	no	—
16	C₂²⋊(C₅²⋊C₃)	—	19	2	no	—
17	C₅²×A₄	—	19	1	no	—
18	C₂²×C₅²⋊C₃	C₁₉⋊C₃	1	1	yes	characteristic, complemented
19	C₅×C₁₉⋊C₃	—	12	2	no	—
20	C₂²×C₃×C₁₉	—	25	1	no	—
21	C₂²⋊(C₁₉⋊C₃)	—	25	2	no	—
22	C₂²⋊(C₁₉⋊C₃)	C₅²⋊C₃	1	1	yes	characteristic, direct product factor
23	C₃×C₅²⋊C₃	—	76	1	no	Hall
24	C₂×C₅×C₁₉	—	9	2	no	—
25	C₃×C₁₉⋊C₃	—	100	1	no	Hall
26	C₂×C₅²⋊C₃	—	3	1	no	—
27	C₂×C₃×C₁₉	—	75	1	no	—
28	C₂²×C₅²	C₃×C₁₉⋊C₃	1	1	yes	characteristic, complemented, Hall
29	C₅×C₁₉	—	3	2	no	—
30	C₂²×C₁₉	C₃×C₅²⋊C₃	1	1	yes	characteristic, complemented, Hall
31	C₃×C₅²	—	76	1	no	—
32	C₅²⋊C₃	—	76	2	no	—
33	C₅²⋊C₃	C₂²⋊(C₁₉⋊C₃)	1	1	yes	characteristic, direct product factor
34	C₅×A₄	—	57	2	no	—
35	C₃×C₁₉	—	25	1	no	—
36	C₁₉⋊C₃	—	4	1	no	—
37	C₁₉⋊C₃	—	100	2	no	—
38	C₂×C₅²	—	3	1	no	—
39	C₂×C₁₉	—	3	1	no	—
40	C₃×A₄	—	475	1	no	Hall
41	C₅²	C₃×C₂²⋊(C₁₉⋊C₃)	1	1	yes	characteristic, complemented, Sylow
42	C₂²×C₅	—	3	2	no	—
43	C₁₉	A₄×C₅²⋊C₃	1	1	yes	characteristic, complemented, Sylow
44	C₃×C₅	—	228	2	no	—
45	C₂²×C₃	—	25	1	no	—
46	A₄	—	19	1	no	—
47	A₄	—	475	2	no	—
48	C₂×C₅	—	9	2	no	—
49	C₃²	—	1900	1	no	Sylow
50	C₂×C₃	—	75	1	no	—
51	C₅	—	3	2	no	—
52	C₂²	C₁₉⋊C₃×C₅²⋊C₃	1	1	yes	characteristic, complemented, Sylow
53	C₃	—	25	1	no	—
54	C₃	—	76	1	no	—
55	C₃	—	1900	2	no	—
56	C₂	—	3	1	no	—

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

Complements

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

N	H ≅ G/N	type
1. C₂²⋊(C₁₉⋊(C₅²⋊C₃))	C₃	semidirect
2. C₅²×C₂²⋊(C₁₉⋊C₃)	C₃	semidirect
3. C₂²×C₁₉×C₅²⋊C₃	C₃	semidirect
6. C₂²×C₅²×C₁₉	C₃²	semidirect
8. C₁₉×C₅²⋊C₃	A₄	semidirect
14. C₅²×C₁₉	C₃×A₄	semidirect
18. C₂²×C₅²⋊C₃	C₁₉⋊C₃	semidirect
22. C₂²⋊(C₁₉⋊C₃)	C₅²⋊C₃	direct (N × H)
28. C₂²×C₅²	C₃×C₁₉⋊C₃	semidirect
30. C₂²×C₁₉	C₃×C₅²⋊C₃	semidirect
33. C₅²⋊C₃	C₂²⋊(C₁₉⋊C₃)	direct (N × H)
41. C₅²	C₃×C₂²⋊(C₁₉⋊C₃)	semidirect
43. C₁₉	A₄×C₅²⋊C₃	semidirect
52. C₂²	C₁₉⋊C₃×C₅²⋊C₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²⋊(C₁₉⋊(C₅²⋊C₃))	3
C₅²×C₂²⋊(C₁₉⋊C₃)	3
C₂²×C₁₉×C₅²⋊C₃	3
C₁₉⋊C₃×C₅²⋊C₃	4
A₄×C₅²⋊C₃	19
C₃×C₂²⋊(C₁₉⋊C₃)	25

Subgroups — C52⋊C3×C22⋊(C19⋊C3) #30685823

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²⋊C₃×C₂²⋊(C₁₉⋊C₃) #30685823