Subgroups — C₂³×C₅×C₁₉×C₃₂ #72201612

Order 24320.

1084 conjugacy classes of subgroups.

The subgroup lattice has 132 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×C₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×C₁₉×C₃₂	C₂	1	14	yes	maximal, direct product factor
3	C₂³×C₅×C₈×C₁₉	C₄	1	1	yes	characteristic
4	C₂²×C₅×C₁₆×C₁₉	C₄	1	7	yes	—
5	C₂²×C₅×C₁₆×C₁₉	C₂²	1	7	yes	—
6	C₂×C₅×C₁₉×C₃₂	C₂²	1	28	yes	direct product factor
7	C₂³×C₄×C₅×C₁₉	C₈	1	1	yes	characteristic
8	C₂²×C₅×C₈×C₁₉	C₈	1	7	yes	—
9	C₂²×C₅×C₈×C₁₉	C₂×C₄	1	7	yes	—
10	C₂×C₅×C₁₆×C₁₉	C₂×C₄	1	21	yes	—
11	C₂×C₅×C₁₆×C₁₉	C₂³	1	7	yes	—
12	C₅×C₁₉×C₃₂	C₂³	1	8	yes	direct product factor
13	C₂³×C₁₉×C₃₂	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂³×C₁₆×C₁₉	C₂×C₅	1	1	yes	characteristic
15	C₂²×C₁₉×C₃₂	C₂×C₅	1	14	yes	direct product factor
16	C₂⁴×C₅×C₁₉	C₁₆	1	1	yes	characteristic
17	C₂²×C₄×C₅×C₁₉	C₁₆	1	7	yes	—
18	C₂²×C₄×C₅×C₁₉	C₂×C₈	1	7	yes	—
19	C₂×C₅×C₈×C₁₉	C₂×C₈	1	21	yes	—
20	C₂×C₅×C₈×C₁₉	C₂²×C₄	1	7	yes	—
21	C₅×C₁₆×C₁₉	C₂²×C₄	1	7	yes	—
22	C₅×C₁₆×C₁₉	C₂⁴	1	1	yes	characteristic
23	C₂³×C₅×C₃₂	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
24	C₂³×C₈×C₁₉	C₄×C₅	1	1	yes	characteristic
25	C₂²×C₁₆×C₁₉	C₄×C₅	1	7	yes	—
26	C₂²×C₁₆×C₁₉	C₂²×C₅	1	7	yes	—
27	C₂×C₁₉×C₃₂	C₂²×C₅	1	28	yes	direct product factor
28	C₂³×C₅×C₁₉	C₃₂	1	8	yes	direct product factor
29	C₂³×C₅×C₁₉	C₂×C₁₆	1	7	yes	—
30	C₂×C₄×C₅×C₁₉	C₂×C₁₆	1	21	yes	—
31	C₂×C₄×C₅×C₁₉	C₂²×C₈	1	7	yes	—
32	C₅×C₈×C₁₉	C₂²×C₈	1	7	yes	—
33	C₅×C₈×C₁₉	C₂³×C₄	1	1	yes	characteristic
34	C₂³×C₅×C₁₆	C₂×C₁₉	1	1	yes	characteristic
35	C₂²×C₅×C₃₂	C₂×C₁₉	1	14	yes	direct product factor
36	C₂³×C₄×C₁₉	C₅×C₈	1	1	yes	characteristic
37	C₂²×C₈×C₁₉	C₅×C₈	1	7	yes	—
38	C₂²×C₈×C₁₉	C₂×C₄×C₅	1	7	yes	—
39	C₂×C₁₆×C₁₉	C₂×C₄×C₅	1	21	yes	—
40	C₂×C₁₆×C₁₉	C₂³×C₅	1	7	yes	—
41	C₁₉×C₃₂	C₂³×C₅	1	8	yes	direct product factor
42	C₂²×C₅×C₁₉	C₂×C₃₂	1	28	yes	direct product factor
43	C₂²×C₅×C₁₉	C₂²×C₁₆	1	7	yes	—
44	C₄×C₅×C₁₉	C₂²×C₁₆	1	7	yes	—
45	C₄×C₅×C₁₉	C₂³×C₈	1	1	yes	characteristic
46	C₂³×C₅×C₈	C₄×C₁₉	1	1	yes	characteristic
47	C₂²×C₅×C₁₆	C₄×C₁₉	1	7	yes	—
48	C₂²×C₅×C₁₆	C₂²×C₁₉	1	7	yes	—
49	C₂×C₅×C₃₂	C₂²×C₁₉	1	28	yes	direct product factor
50	C₂⁴×C₁₉	C₅×C₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	C₅×C₁₆	1	7	yes	—
52	C₂²×C₄×C₁₉	C₂×C₅×C₈	1	7	yes	—
53	C₂×C₈×C₁₉	C₂×C₅×C₈	1	21	yes	—
54	C₂×C₈×C₁₉	C₂²×C₄×C₅	1	7	yes	—
55	C₁₆×C₁₉	C₂²×C₄×C₅	1	7	yes	—
56	C₁₆×C₁₉	C₂⁴×C₅	1	1	yes	characteristic
57	C₂³×C₃₂	C₅×C₁₉	1	1	yes	characteristic, direct product factor, Sylow
58	C₂×C₅×C₁₉	C₂²×C₃₂	1	14	yes	direct product factor
59	C₂×C₅×C₁₉	C₂³×C₁₆	1	1	yes	characteristic
60	C₂³×C₄×C₅	C₈×C₁₉	1	1	yes	characteristic
61	C₂²×C₅×C₈	C₈×C₁₉	1	7	yes	—
62	C₂²×C₅×C₈	C₂×C₄×C₁₉	1	7	yes	—
63	C₂×C₅×C₁₆	C₂×C₄×C₁₉	1	21	yes	—
64	C₂×C₅×C₁₆	C₂³×C₁₉	1	7	yes	—
65	C₅×C₃₂	C₂³×C₁₉	1	8	yes	direct product factor
66	C₂³×C₁₉	C₅×C₃₂	1	8	yes	direct product factor
67	C₂³×C₁₉	C₂×C₅×C₁₆	1	7	yes	—
68	C₂×C₄×C₁₉	C₂×C₅×C₁₆	1	21	yes	—
69	C₂×C₄×C₁₉	C₂²×C₅×C₈	1	7	yes	—
70	C₈×C₁₉	C₂²×C₅×C₈	1	7	yes	—
71	C₈×C₁₉	C₂³×C₄×C₅	1	1	yes	characteristic
72	C₂³×C₁₆	C₂×C₅×C₁₉	1	1	yes	characteristic
73	C₂²×C₃₂	C₂×C₅×C₁₉	1	14	yes	direct product factor
74	C₅×C₁₉	C₂³×C₃₂	1	1	yes	characteristic, direct product factor, Hall
75	C₂⁴×C₅	C₁₆×C₁₉	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₁₆×C₁₉	1	7	yes	—
77	C₂²×C₄×C₅	C₂×C₈×C₁₉	1	7	yes	—
78	C₂×C₅×C₈	C₂×C₈×C₁₉	1	21	yes	—
79	C₂×C₅×C₈	C₂²×C₄×C₁₉	1	7	yes	—
80	C₅×C₁₆	C₂²×C₄×C₁₉	1	7	yes	—
81	C₅×C₁₆	C₂⁴×C₁₉	1	1	yes	characteristic
82	C₂²×C₁₉	C₂×C₅×C₃₂	1	28	yes	direct product factor
83	C₂²×C₁₉	C₂²×C₅×C₁₆	1	7	yes	—
84	C₄×C₁₉	C₂²×C₅×C₁₆	1	7	yes	—
85	C₄×C₁₉	C₂³×C₅×C₈	1	1	yes	characteristic
86	C₂³×C₈	C₄×C₅×C₁₉	1	1	yes	characteristic
87	C₂²×C₁₆	C₄×C₅×C₁₉	1	7	yes	—
88	C₂²×C₁₆	C₂²×C₅×C₁₉	1	7	yes	—
89	C₂×C₃₂	C₂²×C₅×C₁₉	1	28	yes	direct product factor
90	C₂³×C₅	C₁₉×C₃₂	1	8	yes	direct product factor
91	C₂³×C₅	C₂×C₁₆×C₁₉	1	7	yes	—
92	C₂×C₄×C₅	C₂×C₁₆×C₁₉	1	21	yes	—
93	C₂×C₄×C₅	C₂²×C₈×C₁₉	1	7	yes	—
94	C₅×C₈	C₂²×C₈×C₁₉	1	7	yes	—
95	C₅×C₈	C₂³×C₄×C₁₉	1	1	yes	characteristic
96	C₂×C₁₉	C₂²×C₅×C₃₂	1	14	yes	direct product factor
97	C₂×C₁₉	C₂³×C₅×C₁₆	1	1	yes	characteristic
98	C₂³×C₄	C₅×C₈×C₁₉	1	1	yes	characteristic
99	C₂²×C₈	C₅×C₈×C₁₉	1	7	yes	—
100	C₂²×C₈	C₂×C₄×C₅×C₁₉	1	7	yes	—
101	C₂×C₁₆	C₂×C₄×C₅×C₁₉	1	21	yes	—
102	C₂×C₁₆	C₂³×C₅×C₁₉	1	7	yes	—
103	C₃₂	C₂³×C₅×C₁₉	1	8	yes	direct product factor
104	C₂²×C₅	C₂×C₁₉×C₃₂	1	28	yes	direct product factor
105	C₂²×C₅	C₂²×C₁₆×C₁₉	1	7	yes	—
106	C₄×C₅	C₂²×C₁₆×C₁₉	1	7	yes	—
107	C₄×C₅	C₂³×C₈×C₁₉	1	1	yes	characteristic
108	C₁₉	C₂³×C₅×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
109	C₂⁴	C₅×C₁₆×C₁₉	1	1	yes	characteristic
110	C₂²×C₄	C₅×C₁₆×C₁₉	1	7	yes	—
111	C₂²×C₄	C₂×C₅×C₈×C₁₉	1	7	yes	—
112	C₂×C₈	C₂×C₅×C₈×C₁₉	1	21	yes	—
113	C₂×C₈	C₂²×C₄×C₅×C₁₉	1	7	yes	—
114	C₁₆	C₂²×C₄×C₅×C₁₉	1	7	yes	—
115	C₁₆	C₂⁴×C₅×C₁₉	1	1	yes	characteristic
116	C₂×C₅	C₂²×C₁₉×C₃₂	1	14	yes	direct product factor
117	C₂×C₅	C₂³×C₁₆×C₁₉	1	1	yes	characteristic
118	C₂³	C₅×C₁₉×C₃₂	1	8	yes	direct product factor
119	C₂³	C₂×C₅×C₁₆×C₁₉	1	7	yes	—
120	C₂×C₄	C₂×C₅×C₁₆×C₁₉	1	21	yes	—
121	C₂×C₄	C₂²×C₅×C₈×C₁₉	1	7	yes	—
122	C₈	C₂²×C₅×C₈×C₁₉	1	7	yes	—
123	C₈	C₂³×C₄×C₅×C₁₉	1	1	yes	characteristic
124	C₅	C₂³×C₁₉×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
125	C₂²	C₂×C₅×C₁₉×C₃₂	1	28	yes	direct product factor
126	C₂²	C₂²×C₅×C₁₆×C₁₉	1	7	yes	—
127	C₄	C₂²×C₅×C₁₆×C₁₉	1	7	yes	—
128	C₄	C₂³×C₅×C₈×C₁₉	1	1	yes	characteristic
129	C₂	C₂²×C₅×C₁₉×C₃₂	1	14	yes	direct product factor
130	C₂	C₂³×C₅×C₁₆×C₁₉	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₁₆	C₂⁴×C₅×C₁₉
socle, prime radical	C₂⁴×C₅×C₁₉	C₁₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅×C₁₉×C₃₂	C₂	direct (N × H)
6. C₂×C₅×C₁₉×C₃₂	C₂²	direct (N × H)
12. C₅×C₁₉×C₃₂	C₂³	direct (N × H)
13. C₂³×C₁₉×C₃₂	C₅	direct (N × H)
15. C₂²×C₁₉×C₃₂	C₂×C₅	direct (N × H)
23. C₂³×C₅×C₃₂	C₁₉	direct (N × H)
27. C₂×C₁₉×C₃₂	C₂²×C₅	direct (N × H)
28. C₂³×C₅×C₁₉	C₃₂	direct (N × H)
35. C₂²×C₅×C₃₂	C₂×C₁₉	direct (N × H)
41. C₁₉×C₃₂	C₂³×C₅	direct (N × H)
42. C₂²×C₅×C₁₉	C₂×C₃₂	direct (N × H)
49. C₂×C₅×C₃₂	C₂²×C₁₉	direct (N × H)
57. C₂³×C₃₂	C₅×C₁₉	direct (N × H)
58. C₂×C₅×C₁₉	C₂²×C₃₂	direct (N × H)
65. C₅×C₃₂	C₂³×C₁₉	direct (N × H)
66. C₂³×C₁₉	C₅×C₃₂	direct (N × H)
73. C₂²×C₃₂	C₂×C₅×C₁₉	direct (N × H)
74. C₅×C₁₉	C₂³×C₃₂	direct (N × H)
82. C₂²×C₁₉	C₂×C₅×C₃₂	direct (N × H)
89. C₂×C₃₂	C₂²×C₅×C₁₉	direct (N × H)
90. C₂³×C₅	C₁₉×C₃₂	direct (N × H)
96. C₂×C₁₉	C₂²×C₅×C₃₂	direct (N × H)
103. C₃₂	C₂³×C₅×C₁₉	direct (N × H)
104. C₂²×C₅	C₂×C₁₉×C₃₂	direct (N × H)
108. C₁₉	C₂³×C₅×C₃₂	direct (N × H)
116. C₂×C₅	C₂²×C₁₉×C₃₂	direct (N × H)
118. C₂³	C₅×C₁₉×C₃₂	direct (N × H)
124. C₅	C₂³×C₁₉×C₃₂	direct (N × H)
125. C₂²	C₂×C₅×C₁₉×C₃₂	direct (N × H)
129. C₂	C₂²×C₅×C₁₉×C₃₂	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₅×C₁₆×C₁₉	2
C₂²×C₅×C₁₉×C₃₂	2
C₂³×C₁₉×C₃₂	5
C₂³×C₅×C₃₂	19

Subgroups — C23×C5×C19×C32 #72201612

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₅×C₁₉×C₃₂ #72201612