Subgroups — C₁₉×#10201 #18016160

Order 4864.

528 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3246	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#3227	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#3165	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1981	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1980	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1503	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1283	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₉×#533	C₂²	1	1	yes	characteristic
9	C₁₉×#532	C₂²	1	1	yes	characteristic
10	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₁₉×#501	—	2	2	no	—
12	C₁₉×#500	—	2	1	no	—
13	C₁₉×#498	—	2	1	no	—
14	C₁₉×#498	—	2	1	no	—
15	C₁₉×#486	—	2	1	no	—
16	C₁₉×#481	—	2	2	no	—
17	C₁₉×#476	—	2	2	no	—
18	C₁₉×#476	—	2	1	no	—
19	C₁₉×#475	—	2	1	no	—
20	C₁₉×#465	—	2	2	no	—
21	C₁₉×#461	—	2	1	no	—
22	C₁₉×#460	—	2	1	no	—
23	C₁₉×#431	C₂²	1	1	yes	characteristic
24	C₁₉×#428	—	2	2	no	—
25	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₁₉×#104	—	2	2	no	—
27	C₂×C₁₉×#103	—	2	2	no	—
28	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
29	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
30	C₁₉×#328	—	2	2	no	—
31	C₂×#41×C₁₉	—	2	1	no	—
32	C₂²×Q₈×C₁₉	—	2	1	no	—
33	C₂×SD₁₆×C₁₉	—	2	2	no	—
34	C₂×M₁₆×C₁₉	D₄	1	2	yes	—
35	C₂²×C₈×C₁₉	D₄	1	2	yes	—
36	C₁₉×C₄⋊Q₈	—	2	1	no	—
37	C₁₉×C₄⋊Q₈	—	2	1	no	—
38	C₁₉×C₄⋊Q₈	—	2	1	no	—
39	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
40	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
41	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
42	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
43	C₁₉×#127	D₄	1	1	yes	characteristic
44	C₁₉×#125	—	2	1	no	—
45	C₁₉×#124	—	2	1	no	—
46	C₁₉×C₂²⋊Q₈	—	2	1	no	—
47	C₁₉×C₂²⋊Q₈	—	2	1	no	—
48	C₁₉×#121	—	2	1	no	—
49	C₁₉×#121	—	2	1	no	—
50	C₄×Q₈×C₁₉	—	2	1	no	—
51	C₄×Q₈×C₁₉	—	2	1	no	—
52	C₄×D₄×C₁₉	—	2	1	no	—
53	C₄×D₄×C₁₉	—	2	1	no	—
54	C₁₉×#117	—	2	1	no	—
55	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
56	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
57	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
58	C₂×C₄²×C₁₉	—	2	1	no	—
59	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
60	C₁₉×#107	—	2	4	no	—
61	C₁₉×#106	—	2	4	no	—
62	C₁₉×#106	—	2	2	no	—
63	C₁₉×#106	—	2	4	no	—
64	C₁₉×C₄⋊C₈	—	2	1	no	—
65	C₁₉×C₄⋊C₈	—	2	1	no	—
66	C₁₉×C₄⋊C₈	—	2	1	no	—
67	C₁₉×#104	—	4	4	no	—
68	C₁₉×#103	—	2	2	no	—
69	C₁₉×#103	—	4	4	no	—
70	C₁₉×#102	—	2	2	no	—
71	C₁₉×#102	—	2	4	no	—
72	C₁₉×C₂²⋊C₈	D₄	1	2	yes	—
73	C₁₉×#97	—	2	1	no	—
74	C₄×C₈×C₁₉	—	2	1	no	—
75	C₄×C₈×C₁₉	—	2	1	no	—
76	C₁₉×#95	—	2	1	no	—
77	#41×C₁₉	—	4	2	no	—
78	#41×C₁₉	—	4	2	no	—
79	C₂×Q₈×C₁₉	—	2	1	no	—
80	C₂×Q₈×C₁₉	—	2	1	no	—
81	C₂×Q₈×C₁₉	—	2	2	no	—
82	C₂×Q₈×C₁₉	—	2	1	no	—
83	C₂×Q₈×C₁₉	—	4	2	no	—
84	C₂×Q₈×C₁₉	—	4	2	no	—
85	C₂×Q₈×C₁₉	—	4	1	no	—
86	C₂×D₄×C₁₉	—	2	1	no	—
87	C₂×D₄×C₁₉	—	2	1	no	—
88	C₂×D₄×C₁₉	—	2	1	no	—
89	C₂×D₄×C₁₉	—	4	1	no	—
90	C₂²×C₄×C₁₉	—	4	1	no	—
91	C₂²×C₄×C₁₉	—	4	1	no	—
92	C₂²×C₄×C₁₉	—	4	1	no	—
93	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
96	SD₁₆×C₁₉	—	8	2	no	—
97	M₁₆×C₁₉	—	2	2	no	—
98	M₁₆×C₁₉	SD₁₆	1	4	yes	complemented
99	C₂×C₈×C₁₉	—	2	2	no	—
100	C₂×C₈×C₁₉	—	2	2	no	—
101	C₂×C₈×C₁₉	—	2	2	no	—
102	C₂×C₈×C₁₉	—	4	2	no	—
103	C₄⋊C₄×C₁₉	—	2	2	no	—
104	C₄⋊C₄×C₁₉	—	2	2	no	—
105	C₄⋊C₄×C₁₉	—	2	2	no	—
106	C₄⋊C₄×C₁₉	—	2	1	no	—
107	C₄⋊C₄×C₁₉	—	2	1	no	—
108	C₄⋊C₄×C₁₉	—	2	1	no	—
109	C₄⋊C₄×C₁₉	—	2	1	no	—
110	C₄⋊C₄×C₁₉	—	2	1	no	—
111	C₄⋊C₄×C₁₉	—	2	1	no	—
112	C₄⋊C₄×C₁₉	—	4	1	no	—
113	C₄⋊C₄×C₁₉	—	4	1	no	—
114	C₄⋊C₄×C₁₉	—	4	1	no	—
115	C₄⋊C₄×C₁₉	—	4	1	no	—
116	C₄⋊C₄×C₁₉	—	4	1	no	—
117	C₂²⋊C₄×C₁₉	—	4	1	no	—
118	C₂²⋊C₄×C₁₉	—	4	1	no	—
119	C₂²⋊C₄×C₁₉	—	4	1	no	—
120	C₄²×C₁₉	—	4	1	no	—
121	C₄²×C₁₉	—	4	2	no	—
122	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₁₉	#41	1	1	yes	characteristic
126	#10201	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂³×C₁₉	—	4	1	no	—
128	C₂³×C₁₉	#127	1	1	yes	characteristic
129	Q₈×C₁₉	—	4	4	no	—
130	Q₈×C₁₉	—	4	1	no	—
131	Q₈×C₁₉	—	4	1	no	—
132	Q₈×C₁₉	—	4	2	no	—
133	Q₈×C₁₉	—	8	1	no	—
134	Q₈×C₁₉	—	8	1	no	—
135	Q₈×C₁₉	—	8	1	no	—
136	D₄×C₁₉	—	4	2	no	—
137	D₄×C₁₉	—	4	1	no	—
138	D₄×C₁₉	—	4	1	no	—
139	D₄×C₁₉	—	4	2	no	—
140	D₄×C₁₉	—	8	1	no	—
141	D₄×C₁₉	—	8	1	no	—
142	C₂×C₄×C₁₉	—	2	1	no	—
143	C₂×C₄×C₁₉	—	2	1	no	—
144	C₂×C₄×C₁₉	—	4	1	no	—
145	C₂×C₄×C₁₉	—	4	2	no	—
146	C₂×C₄×C₁₉	—	4	1	no	—
147	C₂×C₄×C₁₉	—	4	1	no	—
148	C₂×C₄×C₁₉	—	4	1	no	—
149	C₂×C₄×C₁₉	—	4	2	no	—
150	C₂×C₄×C₁₉	—	8	1	no	—
151	C₂×C₄×C₁₉	—	8	2	no	—
152	C₂×C₄×C₁₉	—	8	1	no	—
153	C₂×C₄×C₁₉	—	8	2	no	—
154	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
155	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
156	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
157	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
158	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
159	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
160	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
161	C₂×C₄×C₁₉	#137	1	2	yes	—
162	C₈×C₁₉	—	2	4	no	—
163	C₈×C₁₉	—	4	2	no	—
164	C₈×C₁₉	—	4	2	no	—
165	#3246	C₂×C₁₉	1	1	yes	characteristic
166	#3227	C₂×C₁₉	1	1	yes	characteristic
167	#3165	C₂×C₁₉	1	1	yes	characteristic, complemented
168	#1981	C₂×C₁₉	1	1	yes	characteristic, complemented
169	#1980	C₂×C₁₉	1	1	yes	characteristic
170	#1503	C₂×C₁₉	1	1	yes	characteristic, complemented
171	#1283	C₂×C₁₉	1	1	yes	characteristic, complemented
172	C₂²×C₁₉	—	2	1	no	—
173	C₂²×C₁₉	—	2	1	no	—
174	C₂²×C₁₉	—	8	1	no	—
175	C₂²×C₁₉	—	8	1	no	—
176	C₂²×C₁₉	—	8	1	no	—
177	C₂²×C₁₉	#392	1	1	yes	characteristic
178	C₂²×C₁₉	#492	1	1	yes	characteristic
179	C₂²×C₁₉	#497	1	1	yes	characteristic
180	C₄×C₁₉	—	2	1	no	—
181	C₄×C₁₉	—	2	1	no	—
182	C₄×C₁₉	—	2	1	no	—
183	C₄×C₁₉	—	8	2	no	—
184	C₄×C₁₉	—	8	2	no	—
185	C₄×C₁₉	—	8	1	no	—
186	C₄×C₁₉	—	8	1	no	—
187	C₄×C₁₉	—	8	1	no	—
188	C₄×C₁₉	#448	1	2	yes	—
189	C₄×C₁₉	#453	1	1	yes	characteristic
190	C₄×C₁₉	#454	1	1	yes	characteristic
191	C₄×C₁₉	#461	1	2	yes	—
192	#533	C₂²×C₁₉	1	1	yes	characteristic
193	#532	C₂²×C₁₉	1	1	yes	characteristic
194	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
195	#501	—	2	2	no	—
196	#500	—	2	1	no	—
197	#498	—	2	1	no	—
198	#498	—	2	1	no	—
199	#486	—	2	1	no	—
200	#481	—	2	2	no	—
201	#476	—	2	2	no	—
202	#476	—	2	1	no	—
203	#475	—	2	1	no	—
204	#465	—	2	2	no	—
205	#461	—	2	1	no	—
206	#460	—	2	1	no	—
207	#431	C₂²×C₁₉	1	1	yes	characteristic
208	#428	—	2	2	no	—
209	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
210	C₂×#104	—	2	2	no	—
211	C₂×#103	—	2	2	no	—
212	C₄×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
213	C₄×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
214	#328	—	2	2	no	—
215	C₂×C₁₉	—	2	1	no	—
216	C₂×C₁₉	—	2	1	no	—
217	C₂×C₁₉	—	16	1	no	—
218	C₂×C₁₉	#2006	1	1	yes	characteristic
219	C₂×C₁₉	#2009	1	1	yes	characteristic
220	C₂×C₁₉	#2013	1	1	yes	characteristic
221	C₂×#41	—	2	1	no	—
222	C₂²×Q₈	—	2	1	no	—
223	C₂×SD₁₆	—	2	2	no	—
224	C₂×M₁₆	D₄×C₁₉	1	2	yes	—
225	C₂²×C₈	D₄×C₁₉	1	2	yes	—
226	C₄⋊Q₈	—	2	1	no	—
227	C₄⋊Q₈	—	2	1	no	—
228	C₄⋊Q₈	—	2	1	no	—
229	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
230	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
231	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
232	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
233	#127	D₄×C₁₉	1	1	yes	characteristic
234	#125	—	2	1	no	—
235	#124	—	2	1	no	—
236	C₂²⋊Q₈	—	2	1	no	—
237	C₂²⋊Q₈	—	2	1	no	—
238	#121	—	2	1	no	—
239	#121	—	2	1	no	—
240	C₄×Q₈	—	2	1	no	—
241	C₄×Q₈	—	2	1	no	—
242	C₄×D₄	—	2	1	no	—
243	C₄×D₄	—	2	1	no	—
244	#117	—	2	1	no	—
245	C₂×C₄⋊C₄	—	2	1	no	—
246	C₂×C₄⋊C₄	—	2	1	no	—
247	C₂×C₄⋊C₄	—	2	1	no	—
248	C₂×C₄²	—	2	1	no	—
249	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
250	#107	—	2	4	no	—
251	#106	—	2	4	no	—
252	#106	—	2	2	no	—
253	#106	—	2	4	no	—
254	C₄⋊C₈	—	2	1	no	—
255	C₄⋊C₈	—	2	1	no	—
256	C₄⋊C₈	—	2	1	no	—
257	#104	—	4	4	no	—
258	#103	—	2	2	no	—
259	#103	—	4	4	no	—
260	#102	—	2	2	no	—
261	#102	—	2	4	no	—
262	C₂²⋊C₈	D₄×C₁₉	1	2	yes	—
263	#97	—	2	1	no	—
264	C₄×C₈	—	2	1	no	—
265	C₄×C₈	—	2	1	no	—
266	#95	—	2	1	no	—
267	C₁₉	#10201	1	1	yes	characteristic, direct product factor, Sylow
268	#41	—	4	2	no	—
269	#41	—	4	2	no	—
270	C₂×Q₈	—	2	1	no	—
271	C₂×Q₈	—	2	1	no	—
272	C₂×Q₈	—	2	2	no	—
273	C₂×Q₈	—	2	1	no	—
274	C₂×Q₈	—	4	2	no	—
275	C₂×Q₈	—	4	2	no	—
276	C₂×Q₈	—	4	1	no	—
277	C₂×D₄	—	2	1	no	—
278	C₂×D₄	—	2	1	no	—
279	C₂×D₄	—	2	1	no	—
280	C₂×D₄	—	4	1	no	—
281	C₂²×C₄	—	4	1	no	—
282	C₂²×C₄	—	4	1	no	—
283	C₂²×C₄	—	4	1	no	—
284	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
285	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
286	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
287	SD₁₆	—	8	2	no	—
288	M₁₆	—	2	2	no	—
289	M₁₆	SD₁₆×C₁₉	1	4	yes	complemented
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	4	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	2	1	no	—
297	C₄⋊C₄	—	2	1	no	—
298	C₄⋊C₄	—	2	1	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	2	1	no	—
302	C₄⋊C₄	—	2	1	no	—
303	C₄⋊C₄	—	4	1	no	—
304	C₄⋊C₄	—	4	1	no	—
305	C₄⋊C₄	—	4	1	no	—
306	C₄⋊C₄	—	4	1	no	—
307	C₄⋊C₄	—	4	1	no	—
308	C₂²⋊C₄	—	4	1	no	—
309	C₂²⋊C₄	—	4	1	no	—
310	C₂²⋊C₄	—	4	1	no	—
311	C₄²	—	4	1	no	—
312	C₄²	—	4	2	no	—
313	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
314	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
315	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
316	C₄²	#41×C₁₉	1	1	yes	characteristic
317	C₂³	—	4	1	no	—
318	C₂³	C₁₉×#127	1	1	yes	characteristic
319	Q₈	—	4	4	no	—
320	Q₈	—	4	1	no	—
321	Q₈	—	4	1	no	—
322	Q₈	—	4	2	no	—
323	Q₈	—	8	1	no	—
324	Q₈	—	8	1	no	—
325	Q₈	—	8	1	no	—
326	D₄	—	4	2	no	—
327	D₄	—	4	1	no	—
328	D₄	—	4	1	no	—
329	D₄	—	4	2	no	—
330	D₄	—	8	1	no	—
331	D₄	—	8	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	8	2	no	—
341	C₂×C₄	—	8	1	no	—
342	C₂×C₄	—	8	1	no	—
343	C₂×C₄	—	8	2	no	—
344	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
345	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
346	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
347	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
348	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
349	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
350	C₂×C₄	C₂×SD₁₆×C₁₉	1	2	yes	—
351	C₂×C₄	C₁₉×#137	1	2	yes	—
352	C₈	—	2	4	no	—
353	C₈	—	4	2	no	—
354	C₈	—	4	2	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	8	1	no	—
358	C₂²	—	8	1	no	—
359	C₂²	—	8	1	no	—
360	C₂²	C₁₉×#392	1	1	yes	characteristic
361	C₂²	C₁₉×#492	1	1	yes	characteristic
362	C₂²	C₁₉×#497	1	1	yes	characteristic
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	8	2	no	—
367	C₄	—	8	2	no	—
368	C₄	—	8	1	no	—
369	C₄	—	8	1	no	—
370	C₄	—	8	1	no	—
371	C₄	C₁₉×#448	1	2	yes	—
372	C₄	C₁₉×#453	1	1	yes	characteristic
373	C₄	C₁₉×#454	1	1	yes	characteristic
374	C₄	C₁₉×#461	1	2	yes	—
375	C₂	—	2	1	no	—
376	C₂	—	2	1	no	—
377	C₂	—	16	1	no	—
378	C₂	C₁₉×#2006	1	1	yes	characteristic
379	C₂	C₁₉×#2009	1	1	yes	characteristic
380	C₂	C₁₉×#2013	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
prime radical	C₁₉×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₉×#3165	C₂	semidirect
4. C₁₉×#1981	C₂	semidirect
6. C₁₉×#1503	C₂	semidirect
7. C₁₉×#1283	C₂	semidirect
98. M₁₆×C₁₉	SD₁₆	semidirect
126. #10201	C₁₉	direct (N × H)
167. #3165	C₂×C₁₉	semidirect
168. #1981	C₂×C₁₉	semidirect
170. #1503	C₂×C₁₉	semidirect
171. #1283	C₂×C₁₉	semidirect
267. C₁₉	#10201	direct (N × H)
289. M₁₆	SD₁₆×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₉×#3246	2
C₁₉×#3227	2
C₁₉×#3165	2
C₁₉×#1981	2
C₁₉×#1980	2
C₁₉×#1503	2
C₁₉×#1283	2
#10201	19

Subgroups — C19×#10201 #18016160

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#10201 #18016160