Subgroups — C₂⁴×C₈×C₉×C₁₉ #72199366

Order 21888.

5928 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄×C₉×C₁₉	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₈×C₉×C₁₉	C₂	1	30	yes	maximal, direct product factor
3	C₂⁴×C₃×C₈×C₁₉	C₃	1	1	yes	maximal, characteristic
4	C₂⁵×C₉×C₁₉	C₄	1	1	yes	characteristic
5	C₂³×C₄×C₉×C₁₉	C₄	1	15	yes	—
6	C₂³×C₄×C₉×C₁₉	C₂²	1	15	yes	—
7	C₂²×C₈×C₉×C₁₉	C₂²	1	140	yes	direct product factor
8	C₂⁴×C₃×C₄×C₁₉	C₂×C₃	1	1	yes	characteristic
9	C₂³×C₃×C₈×C₁₉	C₂×C₃	1	30	yes	—
10	C₂⁴×C₉×C₁₉	C₈	1	16	yes	direct product factor
11	C₂⁴×C₉×C₁₉	C₂×C₄	1	15	yes	—
12	C₂²×C₄×C₉×C₁₉	C₂×C₄	1	105	yes	—
13	C₂²×C₄×C₉×C₁₉	C₂³	1	35	yes	—
14	C₂×C₈×C₉×C₁₉	C₂³	1	120	yes	direct product factor
15	C₂⁴×C₈×C₁₉	C₉	1	1	yes	characteristic, direct product factor, Hall
16	C₂⁵×C₃×C₁₉	C₃×C₄	1	1	yes	characteristic
17	C₂³×C₃×C₄×C₁₉	C₃×C₄	1	15	yes	—
18	C₂³×C₃×C₄×C₁₉	C₂²×C₃	1	15	yes	—
19	C₂²×C₃×C₈×C₁₉	C₂²×C₃	1	140	yes	—
20	C₂³×C₉×C₁₉	C₂×C₈	1	120	yes	direct product factor
21	C₂³×C₉×C₁₉	C₂²×C₄	1	35	yes	—
22	C₂×C₄×C₉×C₁₉	C₂²×C₄	1	105	yes	—
23	C₂×C₄×C₉×C₁₉	C₂⁴	1	15	yes	—
24	C₈×C₉×C₁₉	C₂⁴	1	16	yes	direct product factor
25	C₂⁴×C₄×C₁₉	C₂×C₉	1	1	yes	characteristic
26	C₂³×C₈×C₁₉	C₂×C₉	1	30	yes	direct product factor
27	C₂⁴×C₈×C₉	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
28	C₂⁴×C₃×C₁₉	C₃×C₈	1	16	yes	—
29	C₂⁴×C₃×C₁₉	C₂×C₃×C₄	1	15	yes	—
30	C₂²×C₃×C₄×C₁₉	C₂×C₃×C₄	1	105	yes	—
31	C₂²×C₃×C₄×C₁₉	C₂³×C₃	1	35	yes	—
32	C₂×C₃×C₈×C₁₉	C₂³×C₃	1	120	yes	—
33	C₂²×C₉×C₁₉	C₂²×C₈	1	140	yes	direct product factor
34	C₂²×C₉×C₁₉	C₂³×C₄	1	15	yes	—
35	C₄×C₉×C₁₉	C₂³×C₄	1	15	yes	—
36	C₄×C₉×C₁₉	C₂⁵	1	1	yes	characteristic
37	C₂⁵×C₁₉	C₄×C₉	1	1	yes	characteristic
38	C₂³×C₄×C₁₉	C₄×C₉	1	15	yes	—
39	C₂³×C₄×C₁₉	C₂²×C₉	1	15	yes	—
40	C₂²×C₈×C₁₉	C₂²×C₉	1	140	yes	direct product factor
41	C₂⁴×C₄×C₉	C₂×C₁₉	1	1	yes	characteristic
42	C₂³×C₈×C₉	C₂×C₁₉	1	30	yes	direct product factor
43	C₂³×C₃×C₁₉	C₂×C₃×C₈	1	120	yes	—
44	C₂³×C₃×C₁₉	C₂²×C₃×C₄	1	35	yes	—
45	C₂×C₃×C₄×C₁₉	C₂²×C₃×C₄	1	105	yes	—
46	C₂×C₃×C₄×C₁₉	C₂⁴×C₃	1	15	yes	—
47	C₃×C₈×C₁₉	C₂⁴×C₃	1	16	yes	—
48	C₂⁴×C₃×C₈	C₃×C₁₉	1	1	yes	characteristic
49	C₂×C₉×C₁₉	C₂³×C₈	1	30	yes	direct product factor
50	C₂×C₉×C₁₉	C₂⁴×C₄	1	1	yes	characteristic
51	C₂⁴×C₁₉	C₈×C₉	1	16	yes	direct product factor
52	C₂⁴×C₁₉	C₂×C₄×C₉	1	15	yes	—
53	C₂²×C₄×C₁₉	C₂×C₄×C₉	1	105	yes	—
54	C₂²×C₄×C₁₉	C₂³×C₉	1	35	yes	—
55	C₂×C₈×C₁₉	C₂³×C₉	1	120	yes	direct product factor
56	C₂⁵×C₉	C₄×C₁₉	1	1	yes	characteristic
57	C₂³×C₄×C₉	C₄×C₁₉	1	15	yes	—
58	C₂³×C₄×C₉	C₂²×C₁₉	1	15	yes	—
59	C₂²×C₈×C₉	C₂²×C₁₉	1	140	yes	direct product factor
60	C₂²×C₃×C₁₉	C₂²×C₃×C₈	1	140	yes	—
61	C₂²×C₃×C₁₉	C₂³×C₃×C₄	1	15	yes	—
62	C₃×C₄×C₁₉	C₂³×C₃×C₄	1	15	yes	—
63	C₃×C₄×C₁₉	C₂⁵×C₃	1	1	yes	characteristic
64	C₂⁴×C₃×C₄	C₂×C₃×C₁₉	1	1	yes	characteristic
65	C₂³×C₃×C₈	C₂×C₃×C₁₉	1	30	yes	—
66	C₉×C₁₉	C₂⁴×C₈	1	1	yes	characteristic, direct product factor, Hall
67	C₂³×C₁₉	C₂×C₈×C₉	1	120	yes	direct product factor
68	C₂³×C₁₉	C₂²×C₄×C₉	1	35	yes	—
69	C₂×C₄×C₁₉	C₂²×C₄×C₉	1	105	yes	—
70	C₂×C₄×C₁₉	C₂⁴×C₉	1	15	yes	—
71	C₈×C₁₉	C₂⁴×C₉	1	16	yes	direct product factor
72	C₂⁴×C₉	C₈×C₁₉	1	16	yes	direct product factor
73	C₂⁴×C₉	C₂×C₄×C₁₉	1	15	yes	—
74	C₂²×C₄×C₉	C₂×C₄×C₁₉	1	105	yes	—
75	C₂²×C₄×C₉	C₂³×C₁₉	1	35	yes	—
76	C₂×C₈×C₉	C₂³×C₁₉	1	120	yes	direct product factor
77	C₂⁴×C₈	C₉×C₁₉	1	1	yes	characteristic, direct product factor, Sylow
78	C₂×C₃×C₁₉	C₂³×C₃×C₈	1	30	yes	—
79	C₂×C₃×C₁₉	C₂⁴×C₃×C₄	1	1	yes	characteristic
80	C₂⁵×C₃	C₃×C₄×C₁₉	1	1	yes	characteristic
81	C₂³×C₃×C₄	C₃×C₄×C₁₉	1	15	yes	—
82	C₂³×C₃×C₄	C₂²×C₃×C₁₉	1	15	yes	—
83	C₂²×C₃×C₈	C₂²×C₃×C₁₉	1	140	yes	—
84	C₂²×C₁₉	C₂²×C₈×C₉	1	140	yes	direct product factor
85	C₂²×C₁₉	C₂³×C₄×C₉	1	15	yes	—
86	C₄×C₁₉	C₂³×C₄×C₉	1	15	yes	—
87	C₄×C₁₉	C₂⁵×C₉	1	1	yes	characteristic
88	C₂³×C₉	C₂×C₈×C₁₉	1	120	yes	direct product factor
89	C₂³×C₉	C₂²×C₄×C₁₉	1	35	yes	—
90	C₂×C₄×C₉	C₂²×C₄×C₁₉	1	105	yes	—
91	C₂×C₄×C₉	C₂⁴×C₁₉	1	15	yes	—
92	C₈×C₉	C₂⁴×C₁₉	1	16	yes	direct product factor
93	C₂⁴×C₄	C₂×C₉×C₁₉	1	1	yes	characteristic
94	C₂³×C₈	C₂×C₉×C₁₉	1	30	yes	direct product factor
95	C₃×C₁₉	C₂⁴×C₃×C₈	1	1	yes	characteristic
96	C₂⁴×C₃	C₃×C₈×C₁₉	1	16	yes	—
97	C₂⁴×C₃	C₂×C₃×C₄×C₁₉	1	15	yes	—
98	C₂²×C₃×C₄	C₂×C₃×C₄×C₁₉	1	105	yes	—
99	C₂²×C₃×C₄	C₂³×C₃×C₁₉	1	35	yes	—
100	C₂×C₃×C₈	C₂³×C₃×C₁₉	1	120	yes	—
101	C₂×C₁₉	C₂³×C₈×C₉	1	30	yes	direct product factor
102	C₂×C₁₉	C₂⁴×C₄×C₉	1	1	yes	characteristic
103	C₂²×C₉	C₂²×C₈×C₁₉	1	140	yes	direct product factor
104	C₂²×C₉	C₂³×C₄×C₁₉	1	15	yes	—
105	C₄×C₉	C₂³×C₄×C₁₉	1	15	yes	—
106	C₄×C₉	C₂⁵×C₁₉	1	1	yes	characteristic
107	C₂⁵	C₄×C₉×C₁₉	1	1	yes	characteristic
108	C₂³×C₄	C₄×C₉×C₁₉	1	15	yes	—
109	C₂³×C₄	C₂²×C₉×C₁₉	1	15	yes	—
110	C₂²×C₈	C₂²×C₉×C₁₉	1	140	yes	direct product factor
111	C₂³×C₃	C₂×C₃×C₈×C₁₉	1	120	yes	—
112	C₂³×C₃	C₂²×C₃×C₄×C₁₉	1	35	yes	—
113	C₂×C₃×C₄	C₂²×C₃×C₄×C₁₉	1	105	yes	—
114	C₂×C₃×C₄	C₂⁴×C₃×C₁₉	1	15	yes	—
115	C₃×C₈	C₂⁴×C₃×C₁₉	1	16	yes	—
116	C₁₉	C₂⁴×C₈×C₉	1	1	yes	characteristic, direct product factor, Sylow
117	C₂×C₉	C₂³×C₈×C₁₉	1	30	yes	direct product factor
118	C₂×C₉	C₂⁴×C₄×C₁₉	1	1	yes	characteristic
119	C₂⁴	C₈×C₉×C₁₉	1	16	yes	direct product factor
120	C₂⁴	C₂×C₄×C₉×C₁₉	1	15	yes	—
121	C₂²×C₄	C₂×C₄×C₉×C₁₉	1	105	yes	—
122	C₂²×C₄	C₂³×C₉×C₁₉	1	35	yes	—
123	C₂×C₈	C₂³×C₉×C₁₉	1	120	yes	direct product factor
124	C₂²×C₃	C₂²×C₃×C₈×C₁₉	1	140	yes	—
125	C₂²×C₃	C₂³×C₃×C₄×C₁₉	1	15	yes	—
126	C₃×C₄	C₂³×C₃×C₄×C₁₉	1	15	yes	—
127	C₃×C₄	C₂⁵×C₃×C₁₉	1	1	yes	characteristic
128	C₉	C₂⁴×C₈×C₁₉	1	1	yes	characteristic, direct product factor, Sylow
129	C₂³	C₂×C₈×C₉×C₁₉	1	120	yes	direct product factor
130	C₂³	C₂²×C₄×C₉×C₁₉	1	35	yes	—
131	C₂×C₄	C₂²×C₄×C₉×C₁₉	1	105	yes	—
132	C₂×C₄	C₂⁴×C₉×C₁₉	1	15	yes	—
133	C₈	C₂⁴×C₉×C₁₉	1	16	yes	direct product factor
134	C₂×C₃	C₂³×C₃×C₈×C₁₉	1	30	yes	—
135	C₂×C₃	C₂⁴×C₃×C₄×C₁₉	1	1	yes	characteristic
136	C₂²	C₂²×C₈×C₉×C₁₉	1	140	yes	direct product factor
137	C₂²	C₂³×C₄×C₉×C₁₉	1	15	yes	—
138	C₄	C₂³×C₄×C₉×C₁₉	1	15	yes	—
139	C₄	C₂⁵×C₉×C₁₉	1	1	yes	characteristic
140	C₃	C₂⁴×C₃×C₈×C₁₉	1	1	yes	characteristic
141	C₂	C₂³×C₈×C₉×C₁₉	1	30	yes	direct product factor
142	C₂	C₂⁴×C₄×C₉×C₁₉	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃×C₄	C₂⁵×C₃×C₁₉
socle, prime radical	C₂⁵×C₃×C₁₉	C₃×C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₈×C₉×C₁₉	C₂	direct (N × H)
7. C₂²×C₈×C₉×C₁₉	C₂²	direct (N × H)
10. C₂⁴×C₉×C₁₉	C₈	direct (N × H)
14. C₂×C₈×C₉×C₁₉	C₂³	direct (N × H)
15. C₂⁴×C₈×C₁₉	C₉	direct (N × H)
20. C₂³×C₉×C₁₉	C₂×C₈	direct (N × H)
24. C₈×C₉×C₁₉	C₂⁴	direct (N × H)
26. C₂³×C₈×C₁₉	C₂×C₉	direct (N × H)
27. C₂⁴×C₈×C₉	C₁₉	direct (N × H)
33. C₂²×C₉×C₁₉	C₂²×C₈	direct (N × H)
40. C₂²×C₈×C₁₉	C₂²×C₉	direct (N × H)
42. C₂³×C₈×C₉	C₂×C₁₉	direct (N × H)
49. C₂×C₉×C₁₉	C₂³×C₈	direct (N × H)
51. C₂⁴×C₁₉	C₈×C₉	direct (N × H)
55. C₂×C₈×C₁₉	C₂³×C₉	direct (N × H)
59. C₂²×C₈×C₉	C₂²×C₁₉	direct (N × H)
66. C₉×C₁₉	C₂⁴×C₈	direct (N × H)
67. C₂³×C₁₉	C₂×C₈×C₉	direct (N × H)
71. C₈×C₁₉	C₂⁴×C₉	direct (N × H)
72. C₂⁴×C₉	C₈×C₁₉	direct (N × H)
76. C₂×C₈×C₉	C₂³×C₁₉	direct (N × H)
77. C₂⁴×C₈	C₉×C₁₉	direct (N × H)
84. C₂²×C₁₉	C₂²×C₈×C₉	direct (N × H)
88. C₂³×C₉	C₂×C₈×C₁₉	direct (N × H)
92. C₈×C₉	C₂⁴×C₁₉	direct (N × H)
94. C₂³×C₈	C₂×C₉×C₁₉	direct (N × H)
101. C₂×C₁₉	C₂³×C₈×C₉	direct (N × H)
103. C₂²×C₉	C₂²×C₈×C₁₉	direct (N × H)
110. C₂²×C₈	C₂²×C₉×C₁₉	direct (N × H)
116. C₁₉	C₂⁴×C₈×C₉	direct (N × H)
117. C₂×C₉	C₂³×C₈×C₁₉	direct (N × H)
119. C₂⁴	C₈×C₉×C₁₉	direct (N × H)
123. C₂×C₈	C₂³×C₉×C₁₉	direct (N × H)
128. C₉	C₂⁴×C₈×C₁₉	direct (N × H)
129. C₂³	C₂×C₈×C₉×C₁₉	direct (N × H)
133. C₈	C₂⁴×C₉×C₁₉	direct (N × H)
136. C₂²	C₂²×C₈×C₉×C₁₉	direct (N × H)
141. C₂	C₂³×C₈×C₉×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
C₂⁴×C₄×C₉×C₁₉	2
C₂³×C₈×C₉×C₁₉	2
C₂⁴×C₃×C₈×C₁₉	3
C₂⁴×C₈×C₉	19

Subgroups — C24×C8×C9×C19 #72199366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₈×C₉×C₁₉ #72199366