Subgroups — C₂⁴×C₄²×C₅×C₁₉ #72201624

Order 24320.

48060 conjugacy classes of subgroups.

The subgroup lattice has 120 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₄×C₅×C₁₉	C₂	1	3	yes	maximal
2	C₂³×C₄²×C₅×C₁₉	C₂	1	60	yes	maximal, direct product factor
3	C₂⁴×C₄×C₅×C₁₉	C₄	1	96	yes	direct product factor
4	C₂⁴×C₄×C₅×C₁₉	C₂²	1	90	yes	—
5	C₂⁶×C₅×C₁₉	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₅×C₁₉	C₂²	1	560	yes	direct product factor
7	C₂⁵×C₅×C₁₉	C₂×C₄	1	48	yes	—
8	C₂⁵×C₅×C₁₉	C₂³	1	15	yes	—
9	C₂³×C₄×C₅×C₁₉	C₂×C₄	1	1440	yes	direct product factor
10	C₂³×C₄×C₅×C₁₉	C₂³	1	420	yes	—
11	C₂×C₄²×C₅×C₁₉	C₂³	1	960	yes	direct product factor
12	C₂⁴×C₄²×C₁₉	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂⁵×C₄×C₁₉	C₂×C₅	1	3	yes	—
14	C₂³×C₄²×C₁₉	C₂×C₅	1	60	yes	direct product factor
15	C₂⁴×C₅×C₁₉	C₄²	1	256	yes	direct product factor
16	C₂⁴×C₅×C₁₉	C₂²×C₄	1	360	yes	—
17	C₂⁴×C₅×C₁₉	C₂⁴	1	35	yes	—
18	C₂²×C₄×C₅×C₁₉	C₂²×C₄	1	3360	yes	direct product factor
19	C₂²×C₄×C₅×C₁₉	C₂⁴	1	360	yes	—
20	C₄²×C₅×C₁₉	C₂⁴	1	256	yes	direct product factor
21	C₂⁴×C₄²×C₅	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂⁶×C₁₉	C₂²×C₅	1	1	yes	characteristic
23	C₂⁴×C₄×C₁₉	C₄×C₅	1	96	yes	direct product factor
24	C₂⁴×C₄×C₁₉	C₂²×C₅	1	90	yes	—
25	C₂²×C₄²×C₁₉	C₂²×C₅	1	560	yes	direct product factor
26	C₂³×C₅×C₁₉	C₂×C₄²	1	960	yes	direct product factor
27	C₂³×C₅×C₁₉	C₂³×C₄	1	420	yes	—
28	C₂³×C₅×C₁₉	C₂⁵	1	15	yes	—
29	C₂×C₄×C₅×C₁₉	C₂³×C₄	1	1440	yes	direct product factor
30	C₂×C₄×C₅×C₁₉	C₂⁵	1	48	yes	—
31	C₂⁵×C₄×C₅	C₂×C₁₉	1	3	yes	—
32	C₂³×C₄²×C₅	C₂×C₁₉	1	60	yes	direct product factor
33	C₂⁵×C₁₉	C₂×C₄×C₅	1	48	yes	—
34	C₂⁵×C₁₉	C₂³×C₅	1	15	yes	—
35	C₂³×C₄×C₁₉	C₂×C₄×C₅	1	1440	yes	direct product factor
36	C₂³×C₄×C₁₉	C₂³×C₅	1	420	yes	—
37	C₂×C₄²×C₁₉	C₂³×C₅	1	960	yes	direct product factor
38	C₂²×C₅×C₁₉	C₂²×C₄²	1	560	yes	direct product factor
39	C₂²×C₅×C₁₉	C₂⁴×C₄	1	90	yes	—
40	C₂²×C₅×C₁₉	C₂⁶	1	1	yes	characteristic
41	C₄×C₅×C₁₉	C₂⁴×C₄	1	96	yes	direct product factor
42	C₂⁶×C₅	C₂²×C₁₉	1	1	yes	characteristic
43	C₂⁴×C₄×C₅	C₄×C₁₉	1	96	yes	direct product factor
44	C₂⁴×C₄×C₅	C₂²×C₁₉	1	90	yes	—
45	C₂²×C₄²×C₅	C₂²×C₁₉	1	560	yes	direct product factor
46	C₂⁴×C₁₉	C₄²×C₅	1	256	yes	direct product factor
47	C₂⁴×C₁₉	C₂²×C₄×C₅	1	360	yes	—
48	C₂⁴×C₁₉	C₂⁴×C₅	1	35	yes	—
49	C₂²×C₄×C₁₉	C₂²×C₄×C₅	1	3360	yes	direct product factor
50	C₂²×C₄×C₁₉	C₂⁴×C₅	1	360	yes	—
51	C₄²×C₁₉	C₂⁴×C₅	1	256	yes	direct product factor
52	C₂⁴×C₄²	C₅×C₁₉	1	1	yes	characteristic, direct product factor, Sylow
53	C₂×C₅×C₁₉	C₂³×C₄²	1	60	yes	direct product factor
54	C₂×C₅×C₁₉	C₂⁵×C₄	1	3	yes	—
55	C₂⁵×C₅	C₂×C₄×C₁₉	1	48	yes	—
56	C₂⁵×C₅	C₂³×C₁₉	1	15	yes	—
57	C₂³×C₄×C₅	C₂×C₄×C₁₉	1	1440	yes	direct product factor
58	C₂³×C₄×C₅	C₂³×C₁₉	1	420	yes	—
59	C₂×C₄²×C₅	C₂³×C₁₉	1	960	yes	direct product factor
60	C₂³×C₁₉	C₂×C₄²×C₅	1	960	yes	direct product factor
61	C₂³×C₁₉	C₂³×C₄×C₅	1	420	yes	—
62	C₂³×C₁₉	C₂⁵×C₅	1	15	yes	—
63	C₂×C₄×C₁₉	C₂³×C₄×C₅	1	1440	yes	direct product factor
64	C₂×C₄×C₁₉	C₂⁵×C₅	1	48	yes	—
65	C₂⁵×C₄	C₂×C₅×C₁₉	1	3	yes	—
66	C₂³×C₄²	C₂×C₅×C₁₉	1	60	yes	direct product factor
67	C₅×C₁₉	C₂⁴×C₄²	1	1	yes	characteristic, direct product factor, Hall
68	C₂⁴×C₅	C₄²×C₁₉	1	256	yes	direct product factor
69	C₂⁴×C₅	C₂²×C₄×C₁₉	1	360	yes	—
70	C₂⁴×C₅	C₂⁴×C₁₉	1	35	yes	—
71	C₂²×C₄×C₅	C₂²×C₄×C₁₉	1	3360	yes	direct product factor
72	C₂²×C₄×C₅	C₂⁴×C₁₉	1	360	yes	—
73	C₄²×C₅	C₂⁴×C₁₉	1	256	yes	direct product factor
74	C₂²×C₁₉	C₂²×C₄²×C₅	1	560	yes	direct product factor
75	C₂²×C₁₉	C₂⁴×C₄×C₅	1	90	yes	—
76	C₂²×C₁₉	C₂⁶×C₅	1	1	yes	characteristic
77	C₄×C₁₉	C₂⁴×C₄×C₅	1	96	yes	direct product factor
78	C₂⁶	C₂²×C₅×C₁₉	1	1	yes	characteristic
79	C₂⁴×C₄	C₄×C₅×C₁₉	1	96	yes	direct product factor
80	C₂⁴×C₄	C₂²×C₅×C₁₉	1	90	yes	—
81	C₂²×C₄²	C₂²×C₅×C₁₉	1	560	yes	direct product factor
82	C₂³×C₅	C₂×C₄²×C₁₉	1	960	yes	direct product factor
83	C₂³×C₅	C₂³×C₄×C₁₉	1	420	yes	—
84	C₂³×C₅	C₂⁵×C₁₉	1	15	yes	—
85	C₂×C₄×C₅	C₂³×C₄×C₁₉	1	1440	yes	direct product factor
86	C₂×C₄×C₅	C₂⁵×C₁₉	1	48	yes	—
87	C₂×C₁₉	C₂³×C₄²×C₅	1	60	yes	direct product factor
88	C₂×C₁₉	C₂⁵×C₄×C₅	1	3	yes	—
89	C₂⁵	C₂×C₄×C₅×C₁₉	1	48	yes	—
90	C₂⁵	C₂³×C₅×C₁₉	1	15	yes	—
91	C₂³×C₄	C₂×C₄×C₅×C₁₉	1	1440	yes	direct product factor
92	C₂³×C₄	C₂³×C₅×C₁₉	1	420	yes	—
93	C₂×C₄²	C₂³×C₅×C₁₉	1	960	yes	direct product factor
94	C₂²×C₅	C₂²×C₄²×C₁₉	1	560	yes	direct product factor
95	C₂²×C₅	C₂⁴×C₄×C₁₉	1	90	yes	—
96	C₂²×C₅	C₂⁶×C₁₉	1	1	yes	characteristic
97	C₄×C₅	C₂⁴×C₄×C₁₉	1	96	yes	direct product factor
98	C₁₉	C₂⁴×C₄²×C₅	1	1	yes	characteristic, direct product factor, Sylow
99	C₂⁴	C₄²×C₅×C₁₉	1	256	yes	direct product factor
100	C₂⁴	C₂²×C₄×C₅×C₁₉	1	360	yes	—
101	C₂⁴	C₂⁴×C₅×C₁₉	1	35	yes	—
102	C₂²×C₄	C₂²×C₄×C₅×C₁₉	1	3360	yes	direct product factor
103	C₂²×C₄	C₂⁴×C₅×C₁₉	1	360	yes	—
104	C₄²	C₂⁴×C₅×C₁₉	1	256	yes	direct product factor
105	C₂×C₅	C₂³×C₄²×C₁₉	1	60	yes	direct product factor
106	C₂×C₅	C₂⁵×C₄×C₁₉	1	3	yes	—
107	C₂³	C₂×C₄²×C₅×C₁₉	1	960	yes	direct product factor
108	C₂³	C₂³×C₄×C₅×C₁₉	1	420	yes	—
109	C₂³	C₂⁵×C₅×C₁₉	1	15	yes	—
110	C₂×C₄	C₂³×C₄×C₅×C₁₉	1	1440	yes	direct product factor
111	C₂×C₄	C₂⁵×C₅×C₁₉	1	48	yes	—
112	C₅	C₂⁴×C₄²×C₁₉	1	1	yes	characteristic, direct product factor, Sylow
113	C₂²	C₂²×C₄²×C₅×C₁₉	1	560	yes	direct product factor
114	C₂²	C₂⁶×C₅×C₁₉	1	1	yes	characteristic
115	C₂²	C₂⁴×C₄×C₅×C₁₉	1	90	yes	—
116	C₄	C₂⁴×C₄×C₅×C₁₉	1	96	yes	direct product factor
117	C₂	C₂³×C₄²×C₅×C₁₉	1	60	yes	direct product factor
118	C₂	C₂⁵×C₄×C₅×C₁₉	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²	C₂⁶×C₅×C₁₉
socle, prime radical	C₂⁶×C₅×C₁₉	C₂²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₄²×C₅×C₁₉	C₂	direct (N × H)
3. C₂⁴×C₄×C₅×C₁₉	C₄	direct (N × H)
6. C₂²×C₄²×C₅×C₁₉	C₂²	direct (N × H)
9. C₂³×C₄×C₅×C₁₉	C₂×C₄	direct (N × H)
11. C₂×C₄²×C₅×C₁₉	C₂³	direct (N × H)
12. C₂⁴×C₄²×C₁₉	C₅	direct (N × H)
14. C₂³×C₄²×C₁₉	C₂×C₅	direct (N × H)
15. C₂⁴×C₅×C₁₉	C₄²	direct (N × H)
18. C₂²×C₄×C₅×C₁₉	C₂²×C₄	direct (N × H)
20. C₄²×C₅×C₁₉	C₂⁴	direct (N × H)
21. C₂⁴×C₄²×C₅	C₁₉	direct (N × H)
23. C₂⁴×C₄×C₁₉	C₄×C₅	direct (N × H)
25. C₂²×C₄²×C₁₉	C₂²×C₅	direct (N × H)
26. C₂³×C₅×C₁₉	C₂×C₄²	direct (N × H)
29. C₂×C₄×C₅×C₁₉	C₂³×C₄	direct (N × H)
32. C₂³×C₄²×C₅	C₂×C₁₉	direct (N × H)
35. C₂³×C₄×C₁₉	C₂×C₄×C₅	direct (N × H)
37. C₂×C₄²×C₁₉	C₂³×C₅	direct (N × H)
38. C₂²×C₅×C₁₉	C₂²×C₄²	direct (N × H)
41. C₄×C₅×C₁₉	C₂⁴×C₄	direct (N × H)
43. C₂⁴×C₄×C₅	C₄×C₁₉	direct (N × H)
45. C₂²×C₄²×C₅	C₂²×C₁₉	direct (N × H)
46. C₂⁴×C₁₉	C₄²×C₅	direct (N × H)
49. C₂²×C₄×C₁₉	C₂²×C₄×C₅	direct (N × H)
51. C₄²×C₁₉	C₂⁴×C₅	direct (N × H)
52. C₂⁴×C₄²	C₅×C₁₉	direct (N × H)
53. C₂×C₅×C₁₉	C₂³×C₄²	direct (N × H)
57. C₂³×C₄×C₅	C₂×C₄×C₁₉	direct (N × H)
59. C₂×C₄²×C₅	C₂³×C₁₉	direct (N × H)
60. C₂³×C₁₉	C₂×C₄²×C₅	direct (N × H)
63. C₂×C₄×C₁₉	C₂³×C₄×C₅	direct (N × H)
66. C₂³×C₄²	C₂×C₅×C₁₉	direct (N × H)
67. C₅×C₁₉	C₂⁴×C₄²	direct (N × H)
68. C₂⁴×C₅	C₄²×C₁₉	direct (N × H)
71. C₂²×C₄×C₅	C₂²×C₄×C₁₉	direct (N × H)
73. C₄²×C₅	C₂⁴×C₁₉	direct (N × H)
74. C₂²×C₁₉	C₂²×C₄²×C₅	direct (N × H)
77. C₄×C₁₉	C₂⁴×C₄×C₅	direct (N × H)
79. C₂⁴×C₄	C₄×C₅×C₁₉	direct (N × H)
81. C₂²×C₄²	C₂²×C₅×C₁₉	direct (N × H)
82. C₂³×C₅	C₂×C₄²×C₁₉	direct (N × H)
85. C₂×C₄×C₅	C₂³×C₄×C₁₉	direct (N × H)
87. C₂×C₁₉	C₂³×C₄²×C₅	direct (N × H)
91. C₂³×C₄	C₂×C₄×C₅×C₁₉	direct (N × H)
93. C₂×C₄²	C₂³×C₅×C₁₉	direct (N × H)
94. C₂²×C₅	C₂²×C₄²×C₁₉	direct (N × H)
97. C₄×C₅	C₂⁴×C₄×C₁₉	direct (N × H)
98. C₁₉	C₂⁴×C₄²×C₅	direct (N × H)
99. C₂⁴	C₄²×C₅×C₁₉	direct (N × H)
102. C₂²×C₄	C₂²×C₄×C₅×C₁₉	direct (N × H)
104. C₄²	C₂⁴×C₅×C₁₉	direct (N × H)
105. C₂×C₅	C₂³×C₄²×C₁₉	direct (N × H)
107. C₂³	C₂×C₄²×C₅×C₁₉	direct (N × H)
110. C₂×C₄	C₂³×C₄×C₅×C₁₉	direct (N × H)
112. C₅	C₂⁴×C₄²×C₁₉	direct (N × H)
113. C₂²	C₂²×C₄²×C₅×C₁₉	direct (N × H)
116. C₄	C₂⁴×C₄×C₅×C₁₉	direct (N × H)
117. C₂	C₂³×C₄²×C₅×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
C₂⁵×C₄×C₅×C₁₉	2
C₂³×C₄²×C₅×C₁₉	2
C₂⁴×C₄²×C₁₉	5
C₂⁴×C₄²×C₅	19

Subgroups — C24×C42×C5×C19 #72201624

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₄²×C₅×C₁₉ #72201624