Subgroups — C₅²×(C₃⋊D₃)⋊D₁₉ #30685922

Order 17100.

352 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×C₃⋊D₃×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅²×D₅₇	C₂	1	2	yes	maximal, complemented
3	C₅²×D₃×D₁₉	—	3	2	no	maximal
4	C₃²×C₅²×C₁₉	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₅×(C₃⋊D₃)⋊D₁₉	C₅	1	6	yes	maximal, direct product factor
6	C₅²×D₅₇	D₃	1	2	yes	complemented
7	C₅²×D₃×C₁₉	—	3	2	no	—
8	C₅²×D₃×C₁₉	—	6	1	no	—
9	C₃×C₅²×D₁₉	—	3	2	no	—
10	C₂×C₅²×D₁₉	—	9	1	no	Hall
11	C₅×C₃⋊D₃×C₁₉	C₂×C₅	1	6	yes	complemented
12	C₃×C₅×D₅₇	C₂×C₅	1	12	yes	complemented
13	C₃×C₅²×C₁₉	—	2	1	no	—
14	C₃×C₅²×C₁₉	C₂×D₃	1	2	yes	complemented
15	C₅×D₃×D₁₉	—	3	12	no	—
16	C₂×C₅²×C₁₉	—	9	1	no	—
17	C₅²×D₁₉	—	3	2	no	—
18	C₅²×D₃²	—	19	1	no	maximal, Hall
19	C₃²×C₅×C₁₉	C₂²×C₅	1	6	yes	complemented
20	(C₃⋊D₃)⋊D₁₉	C₅²	1	1	yes	characteristic, direct product factor, Hall
21	C₅×D₅₇	C₅×D₃	1	12	yes	complemented
22	C₅×D₃×C₁₉	—	3	12	no	—
23	C₅×D₃×C₁₉	—	6	6	no	—
24	C₃×C₅×D₁₉	—	3	12	no	—
25	C₅²×C₁₉	D₃²	1	1	yes	characteristic, complemented, Hall
26	C₅²×C₃⋊D₃	D₁₉	1	1	yes	characteristic, complemented
27	C₃×C₅²×D₃	—	19	2	no	—
28	C₂×C₅×D₁₉	—	9	6	no	—
29	C₃⋊D₃×C₁₉	C₂×C₅²	1	1	yes	characteristic, complemented
30	C₃×D₅₇	C₂×C₅²	1	2	yes	complemented
31	C₂×C₅²×D₃	—	57	2	no	—
32	C₃×C₅×C₁₉	—	2	6	no	—
33	C₃×C₅×C₁₉	C₂×C₅×D₃	1	12	yes	complemented
34	D₃×D₁₉	—	3	2	no	—
35	C₃²×C₅²	C₂×D₁₉	1	1	yes	characteristic, complemented, Hall
36	C₂×C₅×C₁₉	—	9	6	no	—
37	C₅×D₁₉	—	3	12	no	—
38	C₅×D₃²	—	19	6	no	—
39	C₃²×C₁₉	C₂²×C₅²	1	1	yes	characteristic, complemented, Hall
40	C₂×C₃×C₅²	—	57	2	no	—
41	C₅²×D₃	—	3	2	no	—
42	C₅²×D₃	—	6	1	no	—
43	C₅²×D₃	—	19	2	no	—
44	D₅₇	C₅²×D₃	1	2	yes	complemented
45	C₃×D₁₉	—	3	2	no	—
46	D₃×C₁₉	—	3	2	no	—
47	D₃×C₁₉	—	6	1	no	—
48	C₂²×C₅²	—	171	1	no	Hall
49	C₅×C₁₉	C₅×D₃²	1	6	yes	complemented
50	C₅×C₃⋊D₃	C₅×D₁₉	1	6	yes	complemented
51	C₃×C₅×D₃	—	19	12	no	—
52	C₂×D₁₉	—	9	1	no	Hall
53	C₃×C₅²	—	2	1	no	—
54	C₃×C₅²	D₃×D₁₉	1	2	yes	complemented
55	C₂×C₅×D₃	—	57	12	no	—
56	C₃×C₁₉	—	2	1	no	—
57	C₃×C₁₉	C₂×C₅²×D₃	1	2	yes	complemented
58	C₂×C₅²	—	9	1	no	—
59	C₂×C₅²	—	57	2	no	—
60	C₃²×C₅	C₂×C₅×D₁₉	1	6	yes	complemented
61	C₂×C₁₉	—	9	1	no	—
62	D₁₉	—	3	2	no	—
63	D₃²	—	19	1	no	Hall
64	C₂×C₃×C₅	—	57	12	no	—
65	C₅×D₃	—	3	12	no	—
66	C₅×D₃	—	6	6	no	—
67	C₅×D₃	—	19	12	no	—
68	C₅²	(C₃⋊D₃)⋊D₁₉	1	1	yes	characteristic, direct product factor, Sylow
69	C₂²×C₅	—	171	6	no	—
70	C₁₉	C₅²×D₃²	1	1	yes	characteristic, complemented, Sylow
71	C₃⋊D₃	C₅²×D₁₉	1	1	yes	characteristic, complemented
72	C₃×D₃	—	19	2	no	—
73	C₃×C₅	—	2	6	no	—
74	C₃×C₅	C₅×D₃×D₁₉	1	12	yes	complemented
75	C₂×D₃	—	57	2	no	—
76	C₂×C₅	—	9	6	no	—
77	C₂×C₅	—	57	12	no	—
78	C₃²	C₂×C₅²×D₁₉	1	1	yes	characteristic, complemented, Sylow
79	C₂×C₃	—	57	2	no	—
80	D₃	—	3	2	no	—
81	D₃	—	6	1	no	—
82	D₃	—	19	2	no	—
83	C₅	C₅×(C₃⋊D₃)⋊D₁₉	1	6	yes	direct product factor
84	C₂²	—	171	1	no	Sylow
85	C₃	—	2	1	no	—
86	C₃	C₅²×D₃×D₁₉	1	2	yes	complemented
87	C₂	—	9	1	no	—
88	C₂	—	57	2	no	—

	subgroup	G/H
center, hypercenter	C₅²	(C₃⋊D₃)⋊D₁₉
commutator, cosocle, hypocenter	C₃²×C₁₉	C₂²×C₅²
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₁₉	C₂²
LCS	G ▷ C₃²×C₁₉
UCS	C₁ ◁ C₅²

Complements

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

N	H ≅ G/N	type
1. C₅²×C₃⋊D₃×C₁₉	C₂	semidirect
2. C₃×C₅²×D₅₇	C₂	semidirect
4. C₃²×C₅²×C₁₉	C₂²	semidirect
5. C₅×(C₃⋊D₃)⋊D₁₉	C₅	direct (N × H)
6. C₅²×D₅₇	D₃	semidirect
11. C₅×C₃⋊D₃×C₁₉	C₂×C₅	semidirect
12. C₃×C₅×D₅₇	C₂×C₅	semidirect
14. C₃×C₅²×C₁₉	C₂×D₃	semidirect
19. C₃²×C₅×C₁₉	C₂²×C₅	semidirect
20. (C₃⋊D₃)⋊D₁₉	C₅²	direct (N × H)
21. C₅×D₅₇	C₅×D₃	semidirect
25. C₅²×C₁₉	D₃²	semidirect
26. C₅²×C₃⋊D₃	D₁₉	semidirect
29. C₃⋊D₃×C₁₉	C₂×C₅²	semidirect
30. C₃×D₅₇	C₂×C₅²	semidirect
33. C₃×C₅×C₁₉	C₂×C₅×D₃	semidirect
35. C₃²×C₅²	C₂×D₁₉	semidirect
39. C₃²×C₁₉	C₂²×C₅²	semidirect
44. D₅₇	C₅²×D₃	semidirect
49. C₅×C₁₉	C₅×D₃²	semidirect
50. C₅×C₃⋊D₃	C₅×D₁₉	semidirect
54. C₃×C₅²	D₃×D₁₉	semidirect
57. C₃×C₁₉	C₂×C₅²×D₃	semidirect
60. C₃²×C₅	C₂×C₅×D₁₉	semidirect
68. C₅²	(C₃⋊D₃)⋊D₁₉	direct (N × H)
70. C₁₉	C₅²×D₃²	semidirect
71. C₃⋊D₃	C₅²×D₁₉	semidirect
74. C₃×C₅	C₅×D₃×D₁₉	semidirect
78. C₃²	C₂×C₅²×D₁₉	semidirect
83. C₅	C₅×(C₃⋊D₃)⋊D₁₉	direct (N × H)
86. C₃	C₅²×D₃×D₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₅²×C₃⋊D₃×C₁₉	2
C₃×C₅²×D₅₇	2
C₅²×D₃×D₁₉	3
C₅×(C₃⋊D₃)⋊D₁₉	5
C₅²×D₃²	19

Subgroups — C52×(C3⋊D3)⋊D19 #30685922

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×(C₃⋊D₃)⋊D₁₉ #30685922