Subgroups — C₂²×C₅×C₈²×C₁₉ #72201618

Order 24320.

2684 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅×C₈×C₁₉	C₂	1	3	yes	maximal
2	C₂×C₅×C₈²×C₁₉	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₄²×C₅×C₁₉	C₂²	1	1	yes	characteristic
4	C₂³×C₅×C₈×C₁₉	C₄	1	6	yes	—
5	C₂×C₄×C₅×C₈×C₁₉	C₄	1	18	yes	—
6	C₂×C₄×C₅×C₈×C₁₉	C₂²	1	18	yes	—
7	C₅×C₈²×C₁₉	C₂²	1	16	yes	direct product factor
8	C₂³×C₄×C₅×C₁₉	C₂×C₄	1	3	yes	—
9	C₂×C₄²×C₅×C₁₉	C₂×C₄	1	9	yes	—
10	C₂×C₄²×C₅×C₁₉	C₂³	1	3	yes	—
11	C₂²×C₅×C₈×C₁₉	C₈	1	48	yes	direct product factor
12	C₂²×C₅×C₈×C₁₉	C₂×C₄	1	36	yes	—
13	C₄×C₅×C₈×C₁₉	C₂×C₄	1	36	yes	—
14	C₄×C₅×C₈×C₁₉	C₂³	1	12	yes	—
15	C₂²×C₈²×C₁₉	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂²×C₄×C₈×C₁₉	C₂×C₅	1	3	yes	—
17	C₂×C₈²×C₁₉	C₂×C₅	1	12	yes	direct product factor
18	C₂⁴×C₅×C₁₉	C₄²	1	1	yes	characteristic
19	C₂²×C₄×C₅×C₁₉	C₄²	1	9	yes	—
20	C₂²×C₄×C₅×C₁₉	C₂×C₈	1	24	yes	—
21	C₂²×C₄×C₅×C₁₉	C₂²×C₄	1	9	yes	—
22	C₂×C₅×C₈×C₁₉	C₂×C₈	1	144	yes	direct product factor
23	C₂×C₅×C₈×C₁₉	C₂²×C₄	1	24	yes	—
24	C₄²×C₅×C₁₉	C₄²	1	6	yes	—
25	C₄²×C₅×C₁₉	C₂²×C₄	1	9	yes	—
26	C₄²×C₅×C₁₉	C₂⁴	1	1	yes	characteristic
27	C₂²×C₅×C₈²	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
28	C₂³×C₈×C₁₉	C₄×C₅	1	6	yes	—
29	C₂²×C₄²×C₁₉	C₂²×C₅	1	1	yes	characteristic
30	C₂×C₄×C₈×C₁₉	C₄×C₅	1	18	yes	—
31	C₂×C₄×C₈×C₁₉	C₂²×C₅	1	18	yes	—
32	C₈²×C₁₉	C₂²×C₅	1	16	yes	direct product factor
33	C₂³×C₅×C₁₉	C₄×C₈	1	12	yes	—
34	C₂³×C₅×C₁₉	C₂×C₄²	1	3	yes	—
35	C₂×C₄×C₅×C₁₉	C₄×C₈	1	36	yes	—
36	C₂×C₄×C₅×C₁₉	C₂×C₄²	1	9	yes	—
37	C₂×C₄×C₅×C₁₉	C₂²×C₈	1	36	yes	—
38	C₂×C₄×C₅×C₁₉	C₂³×C₄	1	3	yes	—
39	C₅×C₈×C₁₉	C₂²×C₈	1	48	yes	direct product factor
40	C₂²×C₄×C₅×C₈	C₂×C₁₉	1	3	yes	—
41	C₂×C₅×C₈²	C₂×C₁₉	1	12	yes	direct product factor
42	C₂³×C₄×C₁₉	C₂×C₄×C₅	1	3	yes	—
43	C₂²×C₈×C₁₉	C₅×C₈	1	48	yes	direct product factor
44	C₂²×C₈×C₁₉	C₂×C₄×C₅	1	36	yes	—
45	C₂×C₄²×C₁₉	C₂×C₄×C₅	1	9	yes	—
46	C₂×C₄²×C₁₉	C₂³×C₅	1	3	yes	—
47	C₄×C₈×C₁₉	C₂×C₄×C₅	1	36	yes	—
48	C₄×C₈×C₁₉	C₂³×C₅	1	12	yes	—
49	C₂²×C₅×C₁₉	C₈²	1	16	yes	direct product factor
50	C₂²×C₅×C₁₉	C₂×C₄×C₈	1	18	yes	—
51	C₂²×C₅×C₁₉	C₂²×C₄²	1	1	yes	characteristic
52	C₄×C₅×C₁₉	C₂×C₄×C₈	1	18	yes	—
53	C₄×C₅×C₁₉	C₂³×C₈	1	6	yes	—
54	C₂³×C₅×C₈	C₄×C₁₉	1	6	yes	—
55	C₂²×C₄²×C₅	C₂²×C₁₉	1	1	yes	characteristic
56	C₂×C₄×C₅×C₈	C₄×C₁₉	1	18	yes	—
57	C₂×C₄×C₅×C₈	C₂²×C₁₉	1	18	yes	—
58	C₅×C₈²	C₂²×C₁₉	1	16	yes	direct product factor
59	C₂⁴×C₁₉	C₄²×C₅	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₄²×C₅	1	9	yes	—
61	C₂²×C₄×C₁₉	C₂×C₅×C₈	1	24	yes	—
62	C₂²×C₄×C₁₉	C₂²×C₄×C₅	1	9	yes	—
63	C₂×C₈×C₁₉	C₂×C₅×C₈	1	144	yes	direct product factor
64	C₂×C₈×C₁₉	C₂²×C₄×C₅	1	24	yes	—
65	C₄²×C₁₉	C₄²×C₅	1	6	yes	—
66	C₄²×C₁₉	C₂²×C₄×C₅	1	9	yes	—
67	C₄²×C₁₉	C₂⁴×C₅	1	1	yes	characteristic
68	C₂²×C₈²	C₅×C₁₉	1	1	yes	characteristic, direct product factor, Sylow
69	C₂×C₅×C₁₉	C₂×C₈²	1	12	yes	direct product factor
70	C₂×C₅×C₁₉	C₂²×C₄×C₈	1	3	yes	—
71	C₂³×C₄×C₅	C₂×C₄×C₁₉	1	3	yes	—
72	C₂²×C₅×C₈	C₈×C₁₉	1	48	yes	direct product factor
73	C₂²×C₅×C₈	C₂×C₄×C₁₉	1	36	yes	—
74	C₂×C₄²×C₅	C₂×C₄×C₁₉	1	9	yes	—
75	C₂×C₄²×C₅	C₂³×C₁₉	1	3	yes	—
76	C₄×C₅×C₈	C₂×C₄×C₁₉	1	36	yes	—
77	C₄×C₅×C₈	C₂³×C₁₉	1	12	yes	—
78	C₂³×C₁₉	C₄×C₅×C₈	1	12	yes	—
79	C₂³×C₁₉	C₂×C₄²×C₅	1	3	yes	—
80	C₂×C₄×C₁₉	C₄×C₅×C₈	1	36	yes	—
81	C₂×C₄×C₁₉	C₂×C₄²×C₅	1	9	yes	—
82	C₂×C₄×C₁₉	C₂²×C₅×C₈	1	36	yes	—
83	C₂×C₄×C₁₉	C₂³×C₄×C₅	1	3	yes	—
84	C₈×C₁₉	C₂²×C₅×C₈	1	48	yes	direct product factor
85	C₂²×C₄×C₈	C₂×C₅×C₁₉	1	3	yes	—
86	C₂×C₈²	C₂×C₅×C₁₉	1	12	yes	direct product factor
87	C₅×C₁₉	C₂²×C₈²	1	1	yes	characteristic, direct product factor, Hall
88	C₂⁴×C₅	C₄²×C₁₉	1	1	yes	characteristic
89	C₂²×C₄×C₅	C₄²×C₁₉	1	9	yes	—
90	C₂²×C₄×C₅	C₂×C₈×C₁₉	1	24	yes	—
91	C₂²×C₄×C₅	C₂²×C₄×C₁₉	1	9	yes	—
92	C₂×C₅×C₈	C₂×C₈×C₁₉	1	144	yes	direct product factor
93	C₂×C₅×C₈	C₂²×C₄×C₁₉	1	24	yes	—
94	C₄²×C₅	C₄²×C₁₉	1	6	yes	—
95	C₄²×C₅	C₂²×C₄×C₁₉	1	9	yes	—
96	C₄²×C₅	C₂⁴×C₁₉	1	1	yes	characteristic
97	C₂²×C₁₉	C₅×C₈²	1	16	yes	direct product factor
98	C₂²×C₁₉	C₂×C₄×C₅×C₈	1	18	yes	—
99	C₂²×C₁₉	C₂²×C₄²×C₅	1	1	yes	characteristic
100	C₄×C₁₉	C₂×C₄×C₅×C₈	1	18	yes	—
101	C₄×C₁₉	C₂³×C₅×C₈	1	6	yes	—
102	C₂³×C₈	C₄×C₅×C₁₉	1	6	yes	—
103	C₂²×C₄²	C₂²×C₅×C₁₉	1	1	yes	characteristic
104	C₂×C₄×C₈	C₄×C₅×C₁₉	1	18	yes	—
105	C₂×C₄×C₈	C₂²×C₅×C₁₉	1	18	yes	—
106	C₈²	C₂²×C₅×C₁₉	1	16	yes	direct product factor
107	C₂³×C₅	C₄×C₈×C₁₉	1	12	yes	—
108	C₂³×C₅	C₂×C₄²×C₁₉	1	3	yes	—
109	C₂×C₄×C₅	C₄×C₈×C₁₉	1	36	yes	—
110	C₂×C₄×C₅	C₂×C₄²×C₁₉	1	9	yes	—
111	C₂×C₄×C₅	C₂²×C₈×C₁₉	1	36	yes	—
112	C₂×C₄×C₅	C₂³×C₄×C₁₉	1	3	yes	—
113	C₅×C₈	C₂²×C₈×C₁₉	1	48	yes	direct product factor
114	C₂×C₁₉	C₂×C₅×C₈²	1	12	yes	direct product factor
115	C₂×C₁₉	C₂²×C₄×C₅×C₈	1	3	yes	—
116	C₂³×C₄	C₂×C₄×C₅×C₁₉	1	3	yes	—
117	C₂²×C₈	C₅×C₈×C₁₉	1	48	yes	direct product factor
118	C₂²×C₈	C₂×C₄×C₅×C₁₉	1	36	yes	—
119	C₂×C₄²	C₂×C₄×C₅×C₁₉	1	9	yes	—
120	C₂×C₄²	C₂³×C₅×C₁₉	1	3	yes	—
121	C₄×C₈	C₂×C₄×C₅×C₁₉	1	36	yes	—
122	C₄×C₈	C₂³×C₅×C₁₉	1	12	yes	—
123	C₂²×C₅	C₈²×C₁₉	1	16	yes	direct product factor
124	C₂²×C₅	C₂×C₄×C₈×C₁₉	1	18	yes	—
125	C₂²×C₅	C₂²×C₄²×C₁₉	1	1	yes	characteristic
126	C₄×C₅	C₂×C₄×C₈×C₁₉	1	18	yes	—
127	C₄×C₅	C₂³×C₈×C₁₉	1	6	yes	—
128	C₁₉	C₂²×C₅×C₈²	1	1	yes	characteristic, direct product factor, Sylow
129	C₂⁴	C₄²×C₅×C₁₉	1	1	yes	characteristic
130	C₂²×C₄	C₄²×C₅×C₁₉	1	9	yes	—
131	C₂²×C₄	C₂×C₅×C₈×C₁₉	1	24	yes	—
132	C₂²×C₄	C₂²×C₄×C₅×C₁₉	1	9	yes	—
133	C₂×C₈	C₂×C₅×C₈×C₁₉	1	144	yes	direct product factor
134	C₂×C₈	C₂²×C₄×C₅×C₁₉	1	24	yes	—
135	C₄²	C₄²×C₅×C₁₉	1	6	yes	—
136	C₄²	C₂²×C₄×C₅×C₁₉	1	9	yes	—
137	C₄²	C₂⁴×C₅×C₁₉	1	1	yes	characteristic
138	C₂×C₅	C₂×C₈²×C₁₉	1	12	yes	direct product factor
139	C₂×C₅	C₂²×C₄×C₈×C₁₉	1	3	yes	—
140	C₂³	C₄×C₅×C₈×C₁₉	1	12	yes	—
141	C₂³	C₂×C₄²×C₅×C₁₉	1	3	yes	—
142	C₂×C₄	C₄×C₅×C₈×C₁₉	1	36	yes	—
143	C₂×C₄	C₂²×C₅×C₈×C₁₉	1	36	yes	—
144	C₂×C₄	C₂×C₄²×C₅×C₁₉	1	9	yes	—
145	C₂×C₄	C₂³×C₄×C₅×C₁₉	1	3	yes	—
146	C₈	C₂²×C₅×C₈×C₁₉	1	48	yes	direct product factor
147	C₅	C₂²×C₈²×C₁₉	1	1	yes	characteristic, direct product factor, Sylow
148	C₂²	C₅×C₈²×C₁₉	1	16	yes	direct product factor
149	C₂²	C₂×C₄×C₅×C₈×C₁₉	1	18	yes	—
150	C₂²	C₂²×C₄²×C₅×C₁₉	1	1	yes	characteristic
151	C₄	C₂×C₄×C₅×C₈×C₁₉	1	18	yes	—
152	C₄	C₂³×C₅×C₈×C₁₉	1	6	yes	—
153	C₂	C₂×C₅×C₈²×C₁₉	1	12	yes	direct product factor
154	C₂	C₂²×C₄×C₅×C₈×C₁₉	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄²	C₂⁴×C₅×C₁₉
socle, prime radical	C₂⁴×C₅×C₁₉	C₄²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×C₈²×C₁₉	C₂	direct (N × H)
7. C₅×C₈²×C₁₉	C₂²	direct (N × H)
11. C₂²×C₅×C₈×C₁₉	C₈	direct (N × H)
15. C₂²×C₈²×C₁₉	C₅	direct (N × H)
17. C₂×C₈²×C₁₉	C₂×C₅	direct (N × H)
22. C₂×C₅×C₈×C₁₉	C₂×C₈	direct (N × H)
27. C₂²×C₅×C₈²	C₁₉	direct (N × H)
32. C₈²×C₁₉	C₂²×C₅	direct (N × H)
39. C₅×C₈×C₁₉	C₂²×C₈	direct (N × H)
41. C₂×C₅×C₈²	C₂×C₁₉	direct (N × H)
43. C₂²×C₈×C₁₉	C₅×C₈	direct (N × H)
49. C₂²×C₅×C₁₉	C₈²	direct (N × H)
58. C₅×C₈²	C₂²×C₁₉	direct (N × H)
63. C₂×C₈×C₁₉	C₂×C₅×C₈	direct (N × H)
68. C₂²×C₈²	C₅×C₁₉	direct (N × H)
69. C₂×C₅×C₁₉	C₂×C₈²	direct (N × H)
72. C₂²×C₅×C₈	C₈×C₁₉	direct (N × H)
84. C₈×C₁₉	C₂²×C₅×C₈	direct (N × H)
86. C₂×C₈²	C₂×C₅×C₁₉	direct (N × H)
87. C₅×C₁₉	C₂²×C₈²	direct (N × H)
92. C₂×C₅×C₈	C₂×C₈×C₁₉	direct (N × H)
97. C₂²×C₁₉	C₅×C₈²	direct (N × H)
106. C₈²	C₂²×C₅×C₁₉	direct (N × H)
113. C₅×C₈	C₂²×C₈×C₁₉	direct (N × H)
114. C₂×C₁₉	C₂×C₅×C₈²	direct (N × H)
117. C₂²×C₈	C₅×C₈×C₁₉	direct (N × H)
123. C₂²×C₅	C₈²×C₁₉	direct (N × H)
128. C₁₉	C₂²×C₅×C₈²	direct (N × H)
133. C₂×C₈	C₂×C₅×C₈×C₁₉	direct (N × H)
138. C₂×C₅	C₂×C₈²×C₁₉	direct (N × H)
146. C₈	C₂²×C₅×C₈×C₁₉	direct (N × H)
147. C₅	C₂²×C₈²×C₁₉	direct (N × H)
148. C₂²	C₅×C₈²×C₁₉	direct (N × H)
153. C₂	C₂×C₅×C₈²×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×C₄×C₅×C₈×C₁₉	2
C₂×C₅×C₈²×C₁₉	2
C₂²×C₈²×C₁₉	5
C₂²×C₅×C₈²	19

Subgroups — C22×C5×C82×C19 #72201618

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×C₈²×C₁₉ #72201618