Subgroups — C₂²×M₁₆×D₁₉ #14578335

Order 2432.

1720 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×D₁₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×#64520	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#64515	C₂	1	2	yes	maximal, complemented
4	C₂×M₁₆×D₁₉	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₈×D₁₉	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁴×D₁₉	C₄	1	1	yes	characteristic
8	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic
9	C₂³×Dic₁₉	C₄	1	1	yes	characteristic
10	C₂²×C₄×D₁₉	C₄	1	1	yes	characteristic
11	C₂²×C₄×D₁₉	C₄	1	12	yes	—
12	C₂²×C₄×D₁₉	C₄	1	1	yes	characteristic
13	C₂²×C₄×D₁₉	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₁₉	C₂²	1	12	yes	—
15	C₂²×C₄×D₁₉	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₁₉	C₂²	1	12	yes	complemented
17	C₂²×C₈×C₁₉	C₂²	1	2	yes	complemented
18	C₂×#64520	C₂²	1	12	yes	complemented
19	C₂²×C₁₉⋊C₈	C₂²	1	2	yes	complemented
20	M₁₆×D₁₉	C₂²	1	64	yes	direct product factor
21	C₂×#64515	C₂²	1	24	yes	complemented
22	C₂×C₈×D₁₉	C₂²	1	24	yes	complemented
23	C₂⁴×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂³×D₁₉	—	2	8	no	—
25	C₂³×D₁₉	C₂×C₄	1	1	yes	characteristic
26	C₂³×D₁₉	C₂×C₄	1	12	yes	—
27	C₂³×D₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₉	C₂×C₄	1	6	yes	—
29	C₂²×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₉	C₂³	1	6	yes	—
31	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
32	C₂²×Dic₁₉	C₂×C₄	1	12	yes	—
33	C₂²×Dic₁₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×Dic₁₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×D₁₉	C₂×C₄	1	48	yes	—
36	C₂×C₄×D₁₉	C₂×C₄	1	12	yes	—
37	C₂×C₄×D₁₉	C₂×C₄	1	6	yes	—
38	C₂×C₄×D₁₉	C₂×C₄	1	6	yes	—
39	C₂×C₄×D₁₉	C₂×C₄	1	12	yes	—
40	C₂×C₄×D₁₉	C₂³	1	6	yes	—
41	C₂×C₄×D₁₉	C₂³	1	6	yes	—
42	C₂×C₄×D₁₉	C₂³	1	16	yes	—
43	M₁₆×C₁₉	C₂³	1	16	yes	complemented
44	C₂×C₈×C₁₉	C₂³	1	12	yes	complemented
45	#64520	C₂³	1	16	yes	complemented
46	C₂×C₁₉⋊C₈	C₂³	1	12	yes	complemented
47	#64515	C₂³	1	32	yes	complemented
48	C₈×D₁₉	C₂³	1	32	yes	complemented
49	C₂³×C₁₉	—	2	4	no	—
50	C₂³×C₁₉	C₂²×C₄	1	6	yes	—
51	C₂³×C₁₉	C₂²×C₄	1	1	yes	characteristic
52	C₂²×D₁₉	—	2	4	no	—
53	C₂²×D₁₉	—	2	48	no	—
54	C₂²×D₁₉	M₁₆	1	8	yes	direct product factor
55	C₂²×D₁₉	C₂²×C₄	1	6	yes	—
56	C₂²×D₁₉	C₂²×C₄	1	16	yes	—
57	C₂²×D₁₉	C₂²×C₄	1	6	yes	—
58	C₂×C₄×C₁₉	C₂²×C₄	1	12	yes	—
59	C₂×C₄×C₁₉	C₂²×C₄	1	6	yes	—
60	C₂×C₄×C₁₉	C₂²×C₄	1	3	yes	—
61	C₂×C₄×C₁₉	C₂⁴	1	3	yes	—
62	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
63	C₂×Dic₁₉	C₂²×C₄	1	16	yes	—
64	C₂×Dic₁₉	C₂²×C₄	1	6	yes	—
65	C₂×Dic₁₉	C₂²×C₄	1	6	yes	—
66	C₄×D₁₉	C₂²×C₄	1	12	yes	—
67	C₄×D₁₉	C₂²×C₄	1	16	yes	—
68	C₄×D₁₉	C₂²×C₄	1	12	yes	—
69	C₄×D₁₉	C₂²×C₄	1	16	yes	—
70	C₄×D₁₉	C₂⁴	1	4	yes	—
71	C₄×D₁₉	C₂⁴	1	4	yes	—
72	C₈×C₁₉	C₂⁴	1	8	yes	complemented
73	C₁₉⋊C₈	C₂⁴	1	8	yes	complemented
74	C₂³×M₁₆	—	19	1	no	maximal, Sylow
75	C₂²×C₁₉	—	2	12	no	—
76	C₂²×C₁₉	C₂×M₁₆	1	4	yes	complemented
77	C₂²×C₁₉	C₂³×C₄	1	4	yes	—
78	C₂²×C₁₉	C₂³×C₄	1	3	yes	—
79	C₂×D₁₉	—	2	12	no	—
80	C₂×D₁₉	—	2	32	no	—
81	C₂×D₁₉	C₂×M₁₆	1	24	yes	direct product factor
82	C₂×D₁₉	C₂³×C₄	1	4	yes	—
83	C₂×D₁₉	C₂³×C₄	1	4	yes	—
84	C₄×C₁₉	C₂³×C₄	1	3	yes	—
85	C₄×C₁₉	C₂³×C₄	1	4	yes	—
86	C₄×C₁₉	C₂⁵	1	1	yes	characteristic
87	Dic₁₉	C₂³×C₄	1	4	yes	—
88	Dic₁₉	C₂³×C₄	1	4	yes	—
89	C₂⁴×C₄	—	19	1	no	—
90	C₂²×M₁₆	—	19	1	no	—
91	C₂²×M₁₆	—	19	24	no	—
92	C₂²×M₁₆	—	19	2	no	—
93	C₂²×M₁₆	D₁₉	1	1	yes	characteristic, direct product factor
94	C₂³×C₈	—	19	2	no	—
95	C₂×C₁₉	—	2	4	no	—
96	C₂×C₁₉	C₂²×M₁₆	1	6	yes	complemented
97	C₂×C₁₉	C₂⁴×C₄	1	1	yes	characteristic
98	D₁₉	—	2	4	no	—
99	D₁₉	C₂²×M₁₆	1	8	yes	direct product factor
100	C₂⁵	—	19	1	no	—
101	C₂³×C₄	—	19	1	no	—
102	C₂³×C₄	—	19	1	no	—
103	C₂³×C₄	—	19	12	no	—
104	C₂³×C₄	—	19	1	no	—
105	C₂³×C₄	—	19	12	no	—
106	C₂³×C₄	—	19	1	no	—
107	C₂³×C₄	—	19	1	no	—
108	C₂³×C₄	C₂×D₁₉	1	1	yes	characteristic
109	C₂×M₁₆	—	19	12	no	—
110	C₂×M₁₆	—	19	64	no	—
111	C₂×M₁₆	—	19	24	no	—
112	C₂×M₁₆	C₂×D₁₉	1	12	yes	direct product factor
113	C₂²×C₈	—	19	2	no	—
114	C₂²×C₈	—	19	24	no	—
115	C₂²×C₈	C₂×D₁₉	1	2	yes	complemented
116	C₁₉	C₂³×M₁₆	1	1	yes	characteristic, complemented, Sylow
117	C₂⁴	—	19	1	no	—
118	C₂⁴	—	19	12	no	—
119	C₂⁴	—	19	1	no	—
120	C₂⁴	—	38	8	no	—
121	C₂⁴	C₄×D₁₉	1	1	yes	characteristic
122	C₂²×C₄	—	19	12	no	—
123	C₂²×C₄	—	19	1	no	—
124	C₂²×C₄	—	19	1	no	—
125	C₂²×C₄	—	19	12	no	—
126	C₂²×C₄	—	19	48	no	—
127	C₂²×C₄	—	19	12	no	—
128	C₂²×C₄	—	19	6	no	—
129	C₂²×C₄	—	19	6	no	—
130	C₂²×C₄	—	19	6	no	—
131	C₂²×C₄	—	19	16	no	—
132	C₂²×C₄	—	19	6	no	—
133	C₂²×C₄	C₄×D₁₉	1	1	yes	characteristic
134	C₂²×C₄	C₄×D₁₉	1	6	yes	—
135	C₂²×C₄	C₂²×D₁₉	1	6	yes	—
136	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
137	M₁₆	—	19	16	no	—
138	M₁₆	—	19	32	no	—
139	M₁₆	C₂²×D₁₉	1	16	yes	direct product factor
140	C₂×C₈	—	19	12	no	—
141	C₂×C₈	—	19	32	no	—
142	C₂×C₈	C₂²×D₁₉	1	12	yes	complemented
143	C₂³	—	2	4	no	—
144	C₂³	—	19	6	no	—
145	C₂³	—	19	8	no	—
146	C₂³	—	19	16	no	—
147	C₂³	—	19	6	no	—
148	C₂³	—	38	48	no	—
149	C₂³	—	38	4	no	—
150	C₂³	C₂×C₄×D₁₉	1	6	yes	—
151	C₂³	C₂×C₄×D₁₉	1	1	yes	characteristic
152	C₂×C₄	—	19	16	no	—
153	C₂×C₄	—	19	6	no	—
154	C₂×C₄	—	19	6	no	—
155	C₂×C₄	—	19	12	no	—
156	C₂×C₄	—	19	16	no	—
157	C₂×C₄	—	19	16	no	—
158	C₂×C₄	—	19	12	no	—
159	C₂×C₄	—	19	4	no	—
160	C₂×C₄	—	19	4	no	—
161	C₂×C₄	C₂×C₄×D₁₉	1	12	yes	—
162	C₂×C₄	C₂×C₄×D₁₉	1	6	yes	—
163	C₂×C₄	C₂×C₄×D₁₉	1	3	yes	—
164	C₂×C₄	C₂³×D₁₉	1	3	yes	—
165	C₂×C₄	C₂³×D₁₉	1	4	yes	—
166	C₈	—	19	8	no	—
167	C₈	C₂³×D₁₉	1	8	yes	complemented
168	C₂²	—	2	12	no	—
169	C₂²	—	19	24	no	—
170	C₂²	—	19	4	no	—
171	C₂²	—	19	4	no	—
172	C₂²	—	38	32	no	—
173	C₂²	—	38	12	no	—
174	C₂²	M₁₆×D₁₉	1	4	yes	direct product factor
175	C₂²	C₂²×C₄×D₁₉	1	4	yes	—
176	C₂²	C₂²×C₄×D₁₉	1	3	yes	—
177	C₄	—	19	4	no	—
178	C₄	—	19	4	no	—
179	C₄	C₂²×C₄×D₁₉	1	4	yes	—
180	C₄	C₂²×C₄×D₁₉	1	3	yes	—
181	C₄	C₂⁴×D₁₉	1	1	yes	characteristic
182	C₂	—	2	4	no	—
183	C₂	—	19	8	no	—
184	C₂	—	38	4	no	—
185	C₂	C₂×M₁₆×D₁₉	1	6	yes	direct product factor
186	C₂	C₂³×C₄×D₁₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₉
commutator	C₂×C₁₉	C₂⁴×C₄
Frattini	C₄	C₂⁴×D₁₉
Fitting	C₂²×M₁₆×C₁₉	C₂
socle	C₂³×C₁₉	C₂²×C₄
prime radical	C₂⁴×D₁₉	C₄
cosocle	C₄×C₁₉	C₂⁵
hypercenter	C₂²×M₁₆	D₁₉
hypocenter	C₁₉	C₂³×M₁₆
LCS	G ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂²×#64520	C₂	semidirect
3. C₂²×#64515	C₂	semidirect
4. C₂×M₁₆×D₁₉	C₂	direct (N × H)
5. C₂²×C₈×D₁₉	C₂	semidirect
6. C₂²×M₁₆×C₁₉	C₂	semidirect
16. C₂×M₁₆×C₁₉	C₂²	semidirect
17. C₂²×C₈×C₁₉	C₂²	semidirect
18. C₂×#64520	C₂²	semidirect
19. C₂²×C₁₉⋊C₈	C₂²	semidirect
20. M₁₆×D₁₉	C₂²	direct (N × H)
21. C₂×#64515	C₂²	semidirect
22. C₂×C₈×D₁₉	C₂²	semidirect
43. M₁₆×C₁₉	C₂³	semidirect
44. C₂×C₈×C₁₉	C₂³	semidirect
45. #64520	C₂³	semidirect
46. C₂×C₁₉⋊C₈	C₂³	semidirect
47. #64515	C₂³	semidirect
48. C₈×D₁₉	C₂³	semidirect
54. C₂²×D₁₉	M₁₆	direct (N × H)
72. C₈×C₁₉	C₂⁴	semidirect
73. C₁₉⋊C₈	C₂⁴	semidirect
76. C₂²×C₁₉	C₂×M₁₆	semidirect
81. C₂×D₁₉	C₂×M₁₆	direct (N × H)
93. C₂²×M₁₆	D₁₉	direct (N × H)
96. C₂×C₁₉	C₂²×M₁₆	semidirect
99. D₁₉	C₂²×M₁₆	direct (N × H)
112. C₂×M₁₆	C₂×D₁₉	direct (N × H)
115. C₂²×C₈	C₂×D₁₉	semidirect
116. C₁₉	C₂³×M₁₆	semidirect
139. M₁₆	C₂²×D₁₉	direct (N × H)
142. C₂×C₈	C₂²×D₁₉	semidirect
167. C₈	C₂³×D₁₉	semidirect
174. C₂²	M₁₆×D₁₉	direct (N × H)
185. C₂	C₂×M₁₆×D₁₉	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×C₄×D₁₉	2
C₂²×#64520	2
C₂²×#64515	2
C₂×M₁₆×D₁₉	2
C₂²×C₈×D₁₉	2
C₂²×M₁₆×C₁₉	2
C₂³×M₁₆	19

Subgroups — C22×M16×D19 #14578335

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×M₁₆×D₁₉ #14578335