Subgroups — C₂²×C₅²×C₁₆×C₁₉ #72207191

Order 30400.

784 conjugacy classes of subgroups.

The subgroup lattice has 114 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅²×C₈×C₁₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅²×C₁₆×C₁₉	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₄×C₅²×C₁₉	C₄	1	1	yes	characteristic
4	C₂×C₅²×C₈×C₁₉	C₄	1	3	yes	—
5	C₂×C₅²×C₈×C₁₉	C₂²	1	3	yes	—
6	C₅²×C₁₆×C₁₉	C₂²	1	4	yes	direct product factor
7	C₂²×C₅×C₁₆×C₁₉	C₅	1	6	yes	maximal, direct product factor
8	C₂³×C₅²×C₁₉	C₈	1	1	yes	characteristic
9	C₂×C₄×C₅²×C₁₉	C₈	1	3	yes	—
10	C₂×C₄×C₅²×C₁₉	C₂×C₄	1	3	yes	—
11	C₅²×C₈×C₁₉	C₂×C₄	1	3	yes	—
12	C₅²×C₈×C₁₉	C₂³	1	1	yes	characteristic
13	C₂²×C₅×C₈×C₁₉	C₂×C₅	1	6	yes	—
14	C₂×C₅×C₁₆×C₁₉	C₂×C₅	1	36	yes	direct product factor
15	C₂²×C₅²×C₁₉	C₁₆	1	4	yes	direct product factor
16	C₂²×C₅²×C₁₉	C₂×C₈	1	3	yes	—
17	C₄×C₅²×C₁₉	C₂×C₈	1	3	yes	—
18	C₄×C₅²×C₁₉	C₂²×C₄	1	1	yes	characteristic
19	C₂²×C₅²×C₁₆	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂²×C₄×C₅×C₁₉	C₄×C₅	1	6	yes	—
21	C₂×C₅×C₈×C₁₉	C₄×C₅	1	18	yes	—
22	C₂×C₅×C₈×C₁₉	C₂²×C₅	1	18	yes	—
23	C₅×C₁₆×C₁₉	C₂²×C₅	1	24	yes	direct product factor
24	C₂²×C₁₆×C₁₉	C₅²	1	1	yes	characteristic, direct product factor, Hall
25	C₂×C₅²×C₁₉	C₂×C₁₆	1	6	yes	direct product factor
26	C₂×C₅²×C₁₉	C₂²×C₈	1	1	yes	characteristic
27	C₂²×C₅²×C₈	C₂×C₁₉	1	1	yes	characteristic
28	C₂×C₅²×C₁₆	C₂×C₁₉	1	6	yes	direct product factor
29	C₂³×C₅×C₁₉	C₅×C₈	1	6	yes	—
30	C₂×C₄×C₅×C₁₉	C₅×C₈	1	18	yes	—
31	C₂×C₄×C₅×C₁₉	C₂×C₄×C₅	1	18	yes	—
32	C₅×C₈×C₁₉	C₂×C₄×C₅	1	18	yes	—
33	C₅×C₈×C₁₉	C₂³×C₅	1	6	yes	—
34	C₂²×C₈×C₁₉	C₂×C₅²	1	1	yes	characteristic
35	C₂×C₁₆×C₁₉	C₂×C₅²	1	6	yes	direct product factor
36	C₅²×C₁₉	C₂²×C₁₆	1	1	yes	characteristic, direct product factor, Hall
37	C₂²×C₄×C₅²	C₄×C₁₉	1	1	yes	characteristic
38	C₂×C₅²×C₈	C₄×C₁₉	1	3	yes	—
39	C₂×C₅²×C₈	C₂²×C₁₉	1	3	yes	—
40	C₅²×C₁₆	C₂²×C₁₉	1	4	yes	direct product factor
41	C₂²×C₅×C₁₉	C₅×C₁₆	1	24	yes	direct product factor
42	C₂²×C₅×C₁₉	C₂×C₅×C₈	1	18	yes	—
43	C₄×C₅×C₁₉	C₂×C₅×C₈	1	18	yes	—
44	C₄×C₅×C₁₉	C₂²×C₄×C₅	1	6	yes	—
45	C₂²×C₅×C₁₆	C₅×C₁₉	1	6	yes	direct product factor
46	C₂²×C₄×C₁₉	C₄×C₅²	1	1	yes	characteristic
47	C₂×C₈×C₁₉	C₄×C₅²	1	3	yes	—
48	C₂×C₈×C₁₉	C₂²×C₅²	1	3	yes	—
49	C₁₆×C₁₉	C₂²×C₅²	1	4	yes	direct product factor
50	C₂³×C₅²	C₈×C₁₉	1	1	yes	characteristic
51	C₂×C₄×C₅²	C₈×C₁₉	1	3	yes	—
52	C₂×C₄×C₅²	C₂×C₄×C₁₉	1	3	yes	—
53	C₅²×C₈	C₂×C₄×C₁₉	1	3	yes	—
54	C₅²×C₈	C₂³×C₁₉	1	1	yes	characteristic
55	C₂×C₅×C₁₉	C₂×C₅×C₁₆	1	36	yes	direct product factor
56	C₂×C₅×C₁₉	C₂²×C₅×C₈	1	6	yes	—
57	C₂²×C₅×C₈	C₂×C₅×C₁₉	1	6	yes	—
58	C₂×C₅×C₁₆	C₂×C₅×C₁₉	1	36	yes	direct product factor
59	C₂³×C₁₉	C₅²×C₈	1	1	yes	characteristic
60	C₂×C₄×C₁₉	C₅²×C₈	1	3	yes	—
61	C₂×C₄×C₁₉	C₂×C₄×C₅²	1	3	yes	—
62	C₈×C₁₉	C₂×C₄×C₅²	1	3	yes	—
63	C₈×C₁₉	C₂³×C₅²	1	1	yes	characteristic
64	C₂²×C₅²	C₁₆×C₁₉	1	4	yes	direct product factor
65	C₂²×C₅²	C₂×C₈×C₁₉	1	3	yes	—
66	C₄×C₅²	C₂×C₈×C₁₉	1	3	yes	—
67	C₄×C₅²	C₂²×C₄×C₁₉	1	1	yes	characteristic
68	C₅×C₁₉	C₂²×C₅×C₁₆	1	6	yes	direct product factor
69	C₂²×C₄×C₅	C₄×C₅×C₁₉	1	6	yes	—
70	C₂×C₅×C₈	C₄×C₅×C₁₉	1	18	yes	—
71	C₂×C₅×C₈	C₂²×C₅×C₁₉	1	18	yes	—
72	C₅×C₁₆	C₂²×C₅×C₁₉	1	24	yes	direct product factor
73	C₂²×C₁₉	C₅²×C₁₆	1	4	yes	direct product factor
74	C₂²×C₁₉	C₂×C₅²×C₈	1	3	yes	—
75	C₄×C₁₉	C₂×C₅²×C₈	1	3	yes	—
76	C₄×C₁₉	C₂²×C₄×C₅²	1	1	yes	characteristic
77	C₂²×C₁₆	C₅²×C₁₉	1	1	yes	characteristic, direct product factor, Sylow
78	C₂×C₅²	C₂×C₁₆×C₁₉	1	6	yes	direct product factor
79	C₂×C₅²	C₂²×C₈×C₁₉	1	1	yes	characteristic
80	C₂³×C₅	C₅×C₈×C₁₉	1	6	yes	—
81	C₂×C₄×C₅	C₅×C₈×C₁₉	1	18	yes	—
82	C₂×C₄×C₅	C₂×C₄×C₅×C₁₉	1	18	yes	—
83	C₅×C₈	C₂×C₄×C₅×C₁₉	1	18	yes	—
84	C₅×C₈	C₂³×C₅×C₁₉	1	6	yes	—
85	C₂×C₁₉	C₂×C₅²×C₁₆	1	6	yes	direct product factor
86	C₂×C₁₉	C₂²×C₅²×C₈	1	1	yes	characteristic
87	C₂²×C₈	C₂×C₅²×C₁₉	1	1	yes	characteristic
88	C₂×C₁₆	C₂×C₅²×C₁₉	1	6	yes	direct product factor
89	C₅²	C₂²×C₁₆×C₁₉	1	1	yes	characteristic, direct product factor, Sylow
90	C₂²×C₅	C₅×C₁₆×C₁₉	1	24	yes	direct product factor
91	C₂²×C₅	C₂×C₅×C₈×C₁₉	1	18	yes	—
92	C₄×C₅	C₂×C₅×C₈×C₁₉	1	18	yes	—
93	C₄×C₅	C₂²×C₄×C₅×C₁₉	1	6	yes	—
94	C₁₉	C₂²×C₅²×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
95	C₂²×C₄	C₄×C₅²×C₁₉	1	1	yes	characteristic
96	C₂×C₈	C₄×C₅²×C₁₉	1	3	yes	—
97	C₂×C₈	C₂²×C₅²×C₁₉	1	3	yes	—
98	C₁₆	C₂²×C₅²×C₁₉	1	4	yes	direct product factor
99	C₂×C₅	C₂×C₅×C₁₆×C₁₉	1	36	yes	direct product factor
100	C₂×C₅	C₂²×C₅×C₈×C₁₉	1	6	yes	—
101	C₂³	C₅²×C₈×C₁₉	1	1	yes	characteristic
102	C₂×C₄	C₅²×C₈×C₁₉	1	3	yes	—
103	C₂×C₄	C₂×C₄×C₅²×C₁₉	1	3	yes	—
104	C₈	C₂×C₄×C₅²×C₁₉	1	3	yes	—
105	C₈	C₂³×C₅²×C₁₉	1	1	yes	characteristic
106	C₅	C₂²×C₅×C₁₆×C₁₉	1	6	yes	direct product factor
107	C₂²	C₅²×C₁₆×C₁₉	1	4	yes	direct product factor
108	C₂²	C₂×C₅²×C₈×C₁₉	1	3	yes	—
109	C₄	C₂×C₅²×C₈×C₁₉	1	3	yes	—
110	C₄	C₂²×C₄×C₅²×C₁₉	1	1	yes	characteristic
111	C₂	C₂×C₅²×C₁₆×C₁₉	1	6	yes	direct product factor
112	C₂	C₂²×C₅²×C₈×C₁₉	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈	C₂³×C₅²×C₁₉
socle, prime radical	C₂³×C₅²×C₁₉	C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅²×C₁₆×C₁₉	C₂	direct (N × H)
6. C₅²×C₁₆×C₁₉	C₂²	direct (N × H)
7. C₂²×C₅×C₁₆×C₁₉	C₅	direct (N × H)
14. C₂×C₅×C₁₆×C₁₉	C₂×C₅	direct (N × H)
15. C₂²×C₅²×C₁₉	C₁₆	direct (N × H)
19. C₂²×C₅²×C₁₆	C₁₉	direct (N × H)
23. C₅×C₁₆×C₁₉	C₂²×C₅	direct (N × H)
24. C₂²×C₁₆×C₁₉	C₅²	direct (N × H)
25. C₂×C₅²×C₁₉	C₂×C₁₆	direct (N × H)
28. C₂×C₅²×C₁₆	C₂×C₁₉	direct (N × H)
35. C₂×C₁₆×C₁₉	C₂×C₅²	direct (N × H)
36. C₅²×C₁₉	C₂²×C₁₆	direct (N × H)
40. C₅²×C₁₆	C₂²×C₁₉	direct (N × H)
41. C₂²×C₅×C₁₉	C₅×C₁₆	direct (N × H)
45. C₂²×C₅×C₁₆	C₅×C₁₉	direct (N × H)
49. C₁₆×C₁₉	C₂²×C₅²	direct (N × H)
55. C₂×C₅×C₁₉	C₂×C₅×C₁₆	direct (N × H)
58. C₂×C₅×C₁₆	C₂×C₅×C₁₉	direct (N × H)
64. C₂²×C₅²	C₁₆×C₁₉	direct (N × H)
68. C₅×C₁₉	C₂²×C₅×C₁₆	direct (N × H)
72. C₅×C₁₆	C₂²×C₅×C₁₉	direct (N × H)
73. C₂²×C₁₉	C₅²×C₁₆	direct (N × H)
77. C₂²×C₁₆	C₅²×C₁₉	direct (N × H)
78. C₂×C₅²	C₂×C₁₆×C₁₉	direct (N × H)
85. C₂×C₁₉	C₂×C₅²×C₁₆	direct (N × H)
88. C₂×C₁₆	C₂×C₅²×C₁₉	direct (N × H)
89. C₅²	C₂²×C₁₆×C₁₉	direct (N × H)
90. C₂²×C₅	C₅×C₁₆×C₁₉	direct (N × H)
94. C₁₉	C₂²×C₅²×C₁₆	direct (N × H)
98. C₁₆	C₂²×C₅²×C₁₉	direct (N × H)
99. C₂×C₅	C₂×C₅×C₁₆×C₁₉	direct (N × H)
106. C₅	C₂²×C₅×C₁₆×C₁₉	direct (N × H)
107. C₂²	C₅²×C₁₆×C₁₉	direct (N × H)
111. C₂	C₂×C₅²×C₁₆×C₁₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅²×C₈×C₁₉	2
C₂×C₅²×C₁₆×C₁₉	2
C₂²×C₅×C₁₆×C₁₉	5
C₂²×C₅²×C₁₆	19

Subgroups — C22×C52×C16×C19 #72207191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅²×C₁₆×C₁₉ #72207191