Subgroups — C₁₉×#9819 #18015778

Order 4864.

526 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#498	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1990	C₂	1	2	yes	maximal
3	C₁₉×#1956	C₂	1	2	yes	maximal
4	C₁₉×#1844	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈²×C₁₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₉×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₁₉×#498	—	2	2	no	—
8	C₁₉×#498	C₄	1	4	yes	complemented
9	C₂×C₁₉×#106	—	2	2	no	—
10	C₂×C₁₉×#106	—	2	2	no	—
11	C₂×C₁₉×#103	—	2	4	no	—
12	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
14	C₁₉×#384	C₂²	1	1	yes	characteristic
15	C₁₉×#384	C₂²	1	1	yes	characteristic
16	C₁₉×#340	—	2	4	no	—
17	C₈²×C₁₉	—	2	2	no	—
18	C₂²×Q₈×C₁₉	—	2	2	no	—
19	C₂²×C₈×C₁₉	D₄	1	4	yes	—
20	C₂²×C₈×C₁₉	D₄	1	2	yes	—
21	C₁₉×C₄⋊Q₈	—	2	2	no	—
22	C₁₉×C₄⋊Q₈	—	2	2	no	—
23	C₁₉×C₄⋊Q₈	—	2	2	no	—
24	C₁₉×C₄⋊Q₈	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
26	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
27	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
28	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
31	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
32	C₁₉×#106	—	2	8	no	—
33	C₁₉×#106	—	4	4	no	—
34	C₁₉×#103	—	4	8	no	—
35	C₄×C₈×C₁₉	—	2	2	no	—
36	C₄×C₈×C₁₉	—	2	2	no	—
37	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₁₉	D₄	1	2	yes	—
39	C₂×Q₈×C₁₉	—	2	4	no	—
40	C₂×Q₈×C₁₉	—	4	2	no	—
41	C₂×Q₈×C₁₉	—	4	4	no	—
42	C₂×Q₈×C₁₉	—	4	4	no	—
43	C₂²×C₄×C₁₉	—	4	2	no	—
44	C₂²×C₄×C₁₉	—	4	1	no	—
45	C₂²×C₄×C₁₉	—	4	1	no	—
46	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
48	C₂×C₈×C₁₉	—	2	4	no	—
49	C₂×C₈×C₁₉	—	2	4	no	—
50	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
51	C₂×C₈×C₁₉	SD₁₆	1	8	yes	—
52	C₂×C₈×C₁₉	SD₁₆	1	4	yes	—
53	C₂×C₈×C₁₉	Q₁₆	1	4	yes	—
54	C₄⋊C₄×C₁₉	—	2	4	no	—
55	C₄⋊C₄×C₁₉	—	2	4	no	—
56	C₄⋊C₄×C₁₉	—	4	2	no	—
57	C₄⋊C₄×C₁₉	—	4	2	no	—
58	C₄⋊C₄×C₁₉	—	4	4	no	—
59	C₄⋊C₄×C₁₉	—	4	2	no	—
60	C₄⋊C₄×C₁₉	—	4	2	no	—
61	C₄⋊C₄×C₁₉	—	4	4	no	—
62	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₁₉	#41	1	2	yes	—
65	#9819	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂³×C₁₉	#127	1	1	yes	characteristic
67	Q₈×C₁₉	—	4	8	no	—
68	Q₈×C₁₉	—	8	4	no	—
69	C₂×C₄×C₁₉	—	4	4	no	—
70	C₂×C₄×C₁₉	—	8	2	no	—
71	C₂×C₄×C₁₉	—	8	2	no	—
72	C₂×C₄×C₁₉	—	8	1	no	—
73	C₂×C₄×C₁₉	—	8	2	no	—
74	C₂×C₄×C₁₉	—	8	1	no	—
75	C₂×C₄×C₁₉	—	8	2	no	—
76	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
78	C₂×C₄×C₁₉	#121	1	2	yes	—
79	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
80	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
81	C₂×C₄×C₁₉	C₂×SD₁₆	1	4	yes	—
82	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
83	C₂×C₄×C₁₉	#135	1	4	yes	—
84	C₈×C₁₉	—	2	8	no	—
85	C₈×C₁₉	#103	1	8	yes	complemented
86	C₂×#498	C₂×C₁₉	1	1	yes	characteristic
87	#1990	C₂×C₁₉	1	2	yes	—
88	#1956	C₂×C₁₉	1	2	yes	—
89	#1844	C₂×C₁₉	1	1	yes	characteristic
90	C₂×C₈²	C₂×C₁₉	1	1	yes	characteristic
91	C₂²×C₁₉	#390	1	1	yes	characteristic
92	C₂²×C₁₉	#492	1	1	yes	characteristic
93	C₂²×C₁₉	#492	1	2	yes	—
94	C₂²×C₁₉	#494	1	1	yes	characteristic
95	C₂²×C₁₉	#495	1	2	yes	—
96	C₄×C₁₉	—	8	4	no	—
97	C₄×C₁₉	—	16	1	no	—
98	C₄×C₁₉	—	16	1	no	—
99	C₄×C₁₉	C₂×#103	1	2	yes	—
100	C₄×C₁₉	C₄×SD₁₆	1	4	yes	—
101	C₄×C₁₉	#465	1	4	yes	—
102	C₄×C₁₉	#487	1	2	yes	—
103	C₂×C₄⋊Q₈	C₂²×C₁₉	1	2	yes	—
104	#498	—	2	2	no	—
105	#498	C₄×C₁₉	1	4	yes	complemented
106	C₂×#106	—	2	2	no	—
107	C₂×#106	—	2	2	no	—
108	C₂×#103	—	2	4	no	—
109	C₂×C₄×C₈	C₂²×C₁₉	1	2	yes	—
110	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
111	#384	C₂²×C₁₉	1	1	yes	characteristic
112	#384	C₂²×C₁₉	1	1	yes	characteristic
113	#340	—	2	4	no	—
114	C₈²	—	2	2	no	—
115	C₂×C₁₉	#1705	1	1	yes	characteristic
116	C₂×C₁₉	#1707	1	2	yes	—
117	C₂×C₁₉	#1708	1	1	yes	characteristic
118	C₂×C₁₉	#1956	1	1	yes	characteristic
119	C₂×C₁₉	#1968	1	2	yes	—
120	C₂²×Q₈	—	2	2	no	—
121	C₂²×C₈	D₄×C₁₉	1	4	yes	—
122	C₂²×C₈	D₄×C₁₉	1	2	yes	—
123	C₄⋊Q₈	—	2	2	no	—
124	C₄⋊Q₈	—	2	2	no	—
125	C₄⋊Q₈	—	2	2	no	—
126	C₄⋊Q₈	C₂×C₄×C₁₉	1	4	yes	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	2	1	no	—
132	C₂×C₄⋊C₄	—	2	1	no	—
133	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
134	#106	—	2	8	no	—
135	#106	—	4	4	no	—
136	#103	—	4	8	no	—
137	C₄×C₈	—	2	2	no	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	C₂×C₄×C₁₉	1	2	yes	—
140	C₄×C₈	D₄×C₁₉	1	2	yes	—
141	C₁₉	#9819	1	1	yes	characteristic, direct product factor, Sylow
142	C₂×Q₈	—	2	4	no	—
143	C₂×Q₈	—	4	2	no	—
144	C₂×Q₈	—	4	4	no	—
145	C₂×Q₈	—	4	4	no	—
146	C₂²×C₄	—	4	2	no	—
147	C₂²×C₄	—	4	1	no	—
148	C₂²×C₄	—	4	1	no	—
149	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
150	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	C₂²⋊C₄×C₁₉	1	4	yes	—
154	C₂×C₈	SD₁₆×C₁₉	1	8	yes	—
155	C₂×C₈	SD₁₆×C₁₉	1	4	yes	—
156	C₂×C₈	Q₁₆×C₁₉	1	4	yes	—
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	4	4	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄⋊C₄	—	4	4	no	—
165	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
166	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
167	C₄²	#41×C₁₉	1	2	yes	—
168	C₂³	C₁₉×#127	1	1	yes	characteristic
169	Q₈	—	4	8	no	—
170	Q₈	—	8	4	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	8	1	no	—
175	C₂×C₄	—	8	2	no	—
176	C₂×C₄	—	8	1	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
179	C₂×C₄	C₄×D₄×C₁₉	1	2	yes	—
180	C₂×C₄	C₁₉×#121	1	2	yes	—
181	C₂×C₄	C₁₉×#124	1	1	yes	characteristic
182	C₂×C₄	C₂×SD₁₆×C₁₉	1	2	yes	—
183	C₂×C₄	C₂×SD₁₆×C₁₉	1	4	yes	—
184	C₂×C₄	C₂×Q₁₆×C₁₉	1	2	yes	—
185	C₂×C₄	C₁₉×#135	1	4	yes	—
186	C₈	—	2	8	no	—
187	C₈	C₁₉×#103	1	8	yes	complemented
188	C₂²	C₁₉×#390	1	1	yes	characteristic
189	C₂²	C₁₉×#492	1	1	yes	characteristic
190	C₂²	C₁₉×#492	1	2	yes	—
191	C₂²	C₁₉×#494	1	1	yes	characteristic
192	C₂²	C₁₉×#495	1	2	yes	—
193	C₄	—	8	4	no	—
194	C₄	—	16	1	no	—
195	C₄	—	16	1	no	—
196	C₄	C₂×C₁₉×#103	1	2	yes	—
197	C₄	C₄×SD₁₆×C₁₉	1	4	yes	—
198	C₄	C₁₉×#465	1	4	yes	—
199	C₄	C₁₉×#487	1	2	yes	—
200	C₂	C₁₉×#1705	1	1	yes	characteristic
201	C₂	C₁₉×#1707	1	2	yes	—
202	C₂	C₁₉×#1708	1	1	yes	characteristic
203	C₂	C₁₉×#1956	1	1	yes	characteristic
204	C₂	C₁₉×#1968	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₉	#127
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₁₉×#498	C₄	semidirect
65. #9819	C₁₉	direct (N × H)
85. C₈×C₁₉	#103	semidirect
105. #498	C₄×C₁₉	semidirect
141. C₁₉	#9819	direct (N × H)
187. C₈	C₁₉×#103	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₉×#498	2
C₁₉×#1990	2
C₁₉×#1956	2
C₁₉×#1844	2
C₂×C₈²×C₁₉	2
#9819	19

Subgroups — C19×#9819 #18015778

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#9819 #18015778