Subgroups — C₁₉×#10580 #18016539

Order 4864.

442 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2552	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1948	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1947	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1844	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1296	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1279	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1279	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₉×#125	—	2	1	no	—
9	C₂×C₁₉×#107	—	2	1	no	—
10	C₂×C₁₉×#106	—	2	1	no	—
11	C₂×C₁₉×#106	—	2	2	no	—
12	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₁₉×#398	—	2	2	no	—
16	C₁₉×#389	—	2	1	no	—
17	C₁₉×#389	—	2	1	no	—
18	C₁₉×#384	—	2	1	no	—
19	C₁₉×#384	C₂²	1	1	yes	characteristic
20	C₁₉×#384	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
23	C₁₉×#340	—	2	1	no	—
24	C₁₉×#340	—	2	1	no	—
25	C₂²×C₈×C₁₉	—	2	1	no	—
26	C₂²×C₈×C₁₉	—	2	1	no	—
27	C₂²×C₈×C₁₉	—	2	1	no	—
28	C₁₉×#125	—	4	2	no	—
29	C₁₉×#125	—	4	2	no	—
30	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
31	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
32	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
33	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
34	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
35	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
36	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
37	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
38	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
43	C₂×C₄²×C₁₉	—	2	1	no	—
44	C₂×C₄²×C₁₉	—	2	1	no	—
45	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
46	C₁₉×#107	—	4	2	no	—
47	C₁₉×#106	—	4	2	no	—
48	C₁₉×#106	—	4	4	no	—
49	C₁₉×C₄⋊C₈	—	2	1	no	—
50	C₁₉×C₄⋊C₈	—	2	1	no	—
51	C₁₉×C₄⋊C₈	Q₈	1	2	yes	—
52	C₁₉×C₄⋊C₈	Q₈	1	2	yes	—
53	C₄×C₈×C₁₉	—	2	1	no	—
54	C₄×C₈×C₁₉	Q₈	1	2	yes	—
55	C₁₉×#95	—	2	2	no	—
56	C₁₉×#95	—	2	1	no	—
57	C₁₉×#95	—	2	1	no	—
58	C₂²×C₄×C₁₉	—	2	1	no	—
59	C₂²×C₄×C₁₉	—	2	1	no	—
60	C₂²×C₄×C₁₉	—	2	1	no	—
61	C₂²×C₄×C₁₉	—	2	1	no	—
62	C₂²×C₄×C₁₉	—	2	2	no	—
63	C₂²×C₄×C₁₉	—	4	1	no	—
64	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₁₉	—	2	2	no	—
68	C₂×C₈×C₁₉	—	2	2	no	—
69	C₂×C₈×C₁₉	—	2	2	no	—
70	C₂×C₈×C₁₉	—	2	2	no	—
71	C₂×C₈×C₁₉	—	2	2	no	—
72	C₂×C₈×C₁₉	—	4	1	no	—
73	C₂×C₈×C₁₉	—	4	1	no	—
74	C₂×C₈×C₁₉	—	4	1	no	—
75	C₂×C₈×C₁₉	—	4	1	no	—
76	C₄⋊C₄×C₁₉	—	2	1	no	—
77	C₄⋊C₄×C₁₉	—	2	1	no	—
78	C₄⋊C₄×C₁₉	—	2	1	no	—
79	C₄⋊C₄×C₁₉	—	2	1	no	—
80	C₄⋊C₄×C₁₉	—	2	1	no	—
81	C₄⋊C₄×C₁₉	—	2	1	no	—
82	C₄⋊C₄×C₁₉	—	4	2	no	—
83	C₄⋊C₄×C₁₉	—	4	4	no	—
84	C₄⋊C₄×C₁₉	—	4	4	no	—
85	C₄⋊C₄×C₁₉	—	4	2	no	—
86	C₄⋊C₄×C₁₉	—	4	1	no	—
87	C₄⋊C₄×C₁₉	—	4	1	no	—
88	C₄⋊C₄×C₁₉	—	4	1	no	—
89	C₄⋊C₄×C₁₉	—	4	1	no	—
90	C₄⋊C₄×C₁₉	—	4	1	no	—
91	C₄⋊C₄×C₁₉	—	4	1	no	—
92	C₄⋊C₄×C₁₉	—	4	1	no	—
93	C₄⋊C₄×C₁₉	—	4	1	no	—
94	C₄⋊C₄×C₁₉	—	8	2	no	—
95	C₄⋊C₄×C₁₉	SD₁₆	1	2	yes	—
96	C₄⋊C₄×C₁₉	SD₁₆	1	2	yes	—
97	C₄²×C₁₉	—	4	1	no	—
98	C₄²×C₁₉	—	4	1	no	—
99	C₄²×C₁₉	—	4	1	no	—
100	C₄²×C₁₉	—	4	1	no	—
101	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
102	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
103	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
104	C₄²×C₁₉	#41	1	1	yes	characteristic
105	#10580	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂³×C₁₉	#120	1	1	yes	characteristic
107	C₂×C₄×C₁₉	—	2	1	no	—
108	C₂×C₄×C₁₉	—	2	1	no	—
109	C₂×C₄×C₁₉	—	2	2	no	—
110	C₂×C₄×C₁₉	—	2	2	no	—
111	C₂×C₄×C₁₉	—	4	1	no	—
112	C₂×C₄×C₁₉	—	4	1	no	—
113	C₂×C₄×C₁₉	—	4	1	no	—
114	C₂×C₄×C₁₉	—	4	1	no	—
115	C₂×C₄×C₁₉	—	4	1	no	—
116	C₂×C₄×C₁₉	—	4	1	no	—
117	C₂×C₄×C₁₉	—	4	1	no	—
118	C₂×C₄×C₁₉	—	4	2	no	—
119	C₂×C₄×C₁₉	—	4	1	no	—
120	C₂×C₄×C₁₉	—	4	2	no	—
121	C₂×C₄×C₁₉	—	4	2	no	—
122	C₂×C₄×C₁₉	—	4	1	no	—
123	C₂×C₄×C₁₉	—	4	1	no	—
124	C₂×C₄×C₁₉	—	8	1	no	—
125	C₂×C₄×C₁₉	—	8	1	no	—
126	C₂×C₄×C₁₉	—	8	1	no	—
127	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
129	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
130	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
131	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
133	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
136	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
137	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
140	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
141	C₈×C₁₉	—	2	4	no	—
142	C₈×C₁₉	—	4	2	no	—
143	C₈×C₁₉	—	4	2	no	—
144	#2552	C₂×C₁₉	1	1	yes	characteristic
145	#1948	C₂×C₁₉	1	1	yes	characteristic
146	#1947	C₂×C₁₉	1	1	yes	characteristic
147	#1844	C₂×C₁₉	1	1	yes	characteristic
148	#1296	C₂×C₁₉	1	1	yes	characteristic
149	#1279	C₂×C₁₉	1	1	yes	characteristic
150	#1279	C₂×C₁₉	1	1	yes	characteristic
151	C₂²×C₁₉	#395	1	1	yes	characteristic
152	C₂²×C₁₉	#448	1	1	yes	characteristic
153	C₂²×C₁₉	#449	1	1	yes	characteristic
154	C₂²×C₁₉	#450	1	1	yes	characteristic
155	C₂²×C₁₉	#452	1	1	yes	characteristic
156	C₂²×C₁₉	#453	1	1	yes	characteristic
157	C₂²×C₁₉	#456	1	1	yes	characteristic
158	C₄×C₁₉	—	2	1	no	—
159	C₄×C₁₉	—	2	1	no	—
160	C₄×C₁₉	—	4	2	no	—
161	C₄×C₁₉	—	4	2	no	—
162	C₄×C₁₉	—	8	1	no	—
163	C₄×C₁₉	—	8	2	no	—
164	C₄×C₁₉	—	8	1	no	—
165	C₄×C₁₉	—	16	1	no	—
166	C₄×C₁₉	#475	1	1	yes	characteristic
167	C₄×C₁₉	#476	1	1	yes	characteristic
168	C₄×C₁₉	#478	1	1	yes	characteristic
169	C₄×C₁₉	#479	1	1	yes	characteristic
170	C₄×C₁₉	#498	1	1	yes	characteristic
171	C₄×C₁₉	#499	1	1	yes	characteristic
172	C₄×C₁₉	#501	1	1	yes	characteristic
173	C₄×C₁₉	#501	1	1	yes	characteristic
174	C₂×#125	—	2	1	no	—
175	C₂×#107	—	2	1	no	—
176	C₂×#106	—	2	2	no	—
177	C₂×#106	—	2	1	no	—
178	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
179	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
180	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
181	#398	—	2	2	no	—
182	#389	—	2	1	no	—
183	#389	—	2	1	no	—
184	#384	—	2	1	no	—
185	#384	C₂²×C₁₉	1	1	yes	characteristic
186	#384	C₂²×C₁₉	1	1	yes	characteristic
187	C₄×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
188	C₄×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
189	#340	—	2	1	no	—
190	#340	—	2	1	no	—
191	C₂×C₁₉	#1622	1	1	yes	characteristic
192	C₂×C₁₉	#1631	1	1	yes	characteristic
193	C₂×C₁₉	#1653	1	1	yes	characteristic
194	C₂×C₁₉	#1654	1	1	yes	characteristic
195	C₂×C₁₉	#2058	1	1	yes	characteristic
196	C₂×C₁₉	#2059	1	1	yes	characteristic
197	C₂×C₁₉	#2060	1	1	yes	characteristic
198	C₂²×C₈	—	2	1	no	—
199	C₂²×C₈	—	2	1	no	—
200	C₂²×C₈	—	2	1	no	—
201	#125	—	4	2	no	—
202	#125	—	4	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	—	2	1	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
212	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
213	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
214	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
215	C₂×C₄⋊C₄	D₄×C₁₉	1	2	yes	—
216	C₂×C₄²	—	2	1	no	—
217	C₂×C₄²	—	2	1	no	—
218	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
219	#107	—	4	2	no	—
220	#106	—	4	2	no	—
221	#106	—	4	4	no	—
222	C₄⋊C₈	—	2	1	no	—
223	C₄⋊C₈	—	2	1	no	—
224	C₄⋊C₈	Q₈×C₁₉	1	2	yes	—
225	C₄⋊C₈	Q₈×C₁₉	1	2	yes	—
226	C₄×C₈	—	2	1	no	—
227	C₄×C₈	Q₈×C₁₉	1	2	yes	—
228	#95	—	2	2	no	—
229	#95	—	2	1	no	—
230	#95	—	2	1	no	—
231	C₁₉	#10580	1	1	yes	characteristic, direct product factor, Sylow
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	4	1	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	4	no	—
258	C₄⋊C₄	—	4	4	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	8	2	no	—
269	C₄⋊C₄	SD₁₆×C₁₉	1	2	yes	—
270	C₄⋊C₄	SD₁₆×C₁₉	1	2	yes	—
271	C₄²	—	4	1	no	—
272	C₄²	—	4	1	no	—
273	C₄²	—	4	1	no	—
274	C₄²	—	4	1	no	—
275	C₄²	C₂×Q₈×C₁₉	1	1	yes	characteristic
276	C₄²	C₂×Q₈×C₁₉	1	1	yes	characteristic
277	C₄²	C₂×Q₈×C₁₉	1	1	yes	characteristic
278	C₄²	#41×C₁₉	1	1	yes	characteristic
279	C₂³	C₁₉×#120	1	1	yes	characteristic
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	8	1	no	—
298	C₂×C₄	—	8	1	no	—
299	C₂×C₄	—	8	1	no	—
300	C₂×C₄	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
301	C₂×C₄	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
302	C₂×C₄	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
303	C₂×C₄	C₁₉×C₄⋊Q₈	1	1	yes	characteristic
304	C₂×C₄	C₁₉×C₄⋊Q₈	1	1	yes	characteristic
305	C₂×C₄	C₁₉×C₄⋊Q₈	1	1	yes	characteristic
306	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
307	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
308	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
309	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
310	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
311	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
312	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
313	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
314	C₈	—	2	4	no	—
315	C₈	—	4	2	no	—
316	C₈	—	4	2	no	—
317	C₂²	C₁₉×#395	1	1	yes	characteristic
318	C₂²	C₁₉×#448	1	1	yes	characteristic
319	C₂²	C₁₉×#449	1	1	yes	characteristic
320	C₂²	C₁₉×#450	1	1	yes	characteristic
321	C₂²	C₁₉×#452	1	1	yes	characteristic
322	C₂²	C₁₉×#453	1	1	yes	characteristic
323	C₂²	C₁₉×#456	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	4	2	no	—
327	C₄	—	4	2	no	—
328	C₄	—	8	1	no	—
329	C₄	—	8	2	no	—
330	C₄	—	8	1	no	—
331	C₄	—	16	1	no	—
332	C₄	C₁₉×#475	1	1	yes	characteristic
333	C₄	C₁₉×#476	1	1	yes	characteristic
334	C₄	C₁₉×#478	1	1	yes	characteristic
335	C₄	C₁₉×#479	1	1	yes	characteristic
336	C₄	C₁₉×#498	1	1	yes	characteristic
337	C₄	C₁₉×#499	1	1	yes	characteristic
338	C₄	C₁₉×#501	1	1	yes	characteristic
339	C₄	C₁₉×#501	1	1	yes	characteristic
340	C₂	C₁₉×#1622	1	1	yes	characteristic
341	C₂	C₁₉×#1631	1	1	yes	characteristic
342	C₂	C₁₉×#1653	1	1	yes	characteristic
343	C₂	C₁₉×#1654	1	1	yes	characteristic
344	C₂	C₁₉×#2058	1	1	yes	characteristic
345	C₂	C₁₉×#2059	1	1	yes	characteristic
346	C₂	C₁₉×#2060	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₉	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
LCS	G ▷ C₂×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
105. #10580	C₁₉	direct (N × H)
231. C₁₉	#10580	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₉×#2552	2
C₁₉×#1948	2
C₁₉×#1947	2
C₁₉×#1844	2
C₁₉×#1296	2
C₁₉×#1279	2
#10580	19

Subgroups — C19×#10580 #18016539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#10580 #18016539