Subgroups — C₁₉×#10221 #18016180

Order 4864.

570 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3227	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₉×#3206	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#3150	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1993	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1517	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#1517	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1281	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₉×#135	—	2	2	no	—
9	C₁₉×#532	C₂²	1	1	yes	characteristic
10	C₁₉×#532	C₂²	1	1	yes	characteristic
11	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₁₉×#498	—	2	1	no	—
13	C₁₉×#497	—	2	1	no	—
14	C₁₉×#497	—	2	1	no	—
15	C₁₉×#496	—	2	1	no	—
16	C₁₉×#496	—	2	1	no	—
17	C₁₉×#492	—	2	1	no	—
18	C₁₉×#487	—	2	1	no	—
19	C₁₉×#486	—	2	1	no	—
20	C₁₉×#481	—	2	2	no	—
21	C₁₉×#476	—	2	2	no	—
22	C₁₉×#461	—	2	2	no	—
23	C₁₉×#448	—	2	2	no	—
24	C₁₉×#431	C₂²	1	1	yes	characteristic
25	C₁₉×#431	C₂²	1	1	yes	characteristic
26	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
27	C₁₉×#382	C₂²	1	1	yes	characteristic
28	C₁₉×#340	—	2	2	no	—
29	C₁₉×#327	—	2	2	no	—
30	C₁₉×#327	—	2	2	no	—
31	C₂×#41×C₁₉	—	2	1	no	—
32	C₂×#41×C₁₉	—	2	1	no	—
33	C₂²×Q₈×C₁₉	—	2	1	no	—
34	C₁₉×#135	—	4	4	no	—
35	C₁₉×#135	—	4	4	no	—
36	C₂×Q₁₆×C₁₉	—	2	2	no	—
37	C₂×SD₁₆×C₁₉	—	2	2	no	—
38	C₂×SD₁₆×C₁₉	—	2	2	no	—
39	C₂×SD₁₆×C₁₉	—	2	4	no	—
40	C₂×D₈×C₁₉	—	2	2	no	—
41	C₂²×C₈×C₁₉	D₄	1	2	yes	—
42	C₁₉×C₄⋊Q₈	—	2	1	no	—
43	C₁₉×C₄⋊Q₈	—	2	1	no	—
44	C₁₉×C₄⋊Q₈	—	2	1	no	—
45	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
46	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
47	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
48	C₁₉×#127	D₄	1	1	yes	characteristic
49	C₁₉×#127	D₄	1	1	yes	characteristic
50	C₁₉×#124	—	2	1	no	—
51	C₁₉×#124	—	2	1	no	—
52	C₁₉×#121	—	2	1	no	—
53	C₁₉×#121	—	2	1	no	—
54	C₁₉×#121	—	2	1	no	—
55	C₁₉×#121	—	2	1	no	—
56	C₄×D₄×C₁₉	—	2	1	no	—
57	C₄×D₄×C₁₉	—	2	1	no	—
58	C₄×D₄×C₁₉	—	2	1	no	—
59	C₄×D₄×C₁₉	—	2	1	no	—
60	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
61	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
62	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
63	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
64	C₁₉×#106	—	2	4	no	—
65	C₁₉×C₄⋊C₈	—	2	1	no	—
66	C₁₉×C₄⋊C₈	—	2	1	no	—
67	C₁₉×#103	—	2	4	no	—
68	C₁₉×#102	—	2	4	no	—
69	C₁₉×C₂²⋊C₈	D₄	1	2	yes	—
70	C₁₉×C₂²⋊C₈	D₄	1	2	yes	—
71	C₁₉×#97	—	2	1	no	—
72	C₁₉×#97	—	2	1	no	—
73	C₄×C₈×C₁₉	—	2	1	no	—
74	C₄×C₈×C₁₉	—	2	1	no	—
75	C₁₉×#95	—	2	1	no	—
76	C₁₉×#95	—	2	1	no	—
77	#41×C₁₉	—	4	2	no	—
78	#41×C₁₉	—	4	2	no	—
79	#41×C₁₉	—	4	2	no	—
80	#41×C₁₉	—	4	2	no	—
81	C₂×Q₈×C₁₉	—	2	1	no	—
82	C₂×Q₈×C₁₉	—	2	1	no	—
83	C₂×Q₈×C₁₉	—	2	2	no	—
84	C₂×Q₈×C₁₉	—	4	1	no	—
85	C₂×Q₈×C₁₉	—	8	2	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	—	2	1	no	—
90	C₂×D₄×C₁₉	—	2	1	no	—
91	C₂×D₄×C₁₉	—	2	1	no	—
92	C₂×D₄×C₁₉	—	4	1	no	—
93	C₂×D₄×C₁₉	—	4	1	no	—
94	C₂²×C₄×C₁₉	—	4	1	no	—
95	C₂²×C₄×C₁₉	—	4	1	no	—
96	C₂²×C₄×C₁₉	—	4	1	no	—
97	C₂²×C₄×C₁₉	—	4	1	no	—
98	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
101	Q₁₆×C₁₉	—	4	4	no	—
102	SD₁₆×C₁₉	—	4	4	no	—
103	SD₁₆×C₁₉	—	4	4	no	—
104	SD₁₆×C₁₉	—	8	4	no	—
105	D₈×C₁₉	—	4	4	no	—
106	C₂×C₈×C₁₉	—	2	2	no	—
107	C₂×C₈×C₁₉	—	2	2	no	—
108	C₂×C₈×C₁₉	—	2	2	no	—
109	C₂×C₈×C₁₉	—	2	2	no	—
110	C₂×C₈×C₁₉	SD₁₆	1	4	yes	complemented
111	C₄⋊C₄×C₁₉	—	2	2	no	—
112	C₄⋊C₄×C₁₉	—	2	1	no	—
113	C₄⋊C₄×C₁₉	—	2	1	no	—
114	C₄⋊C₄×C₁₉	—	2	2	no	—
115	C₄⋊C₄×C₁₉	—	2	1	no	—
116	C₄⋊C₄×C₁₉	—	2	1	no	—
117	C₄⋊C₄×C₁₉	—	4	1	no	—
118	C₄⋊C₄×C₁₉	—	4	1	no	—
119	C₄⋊C₄×C₁₉	—	4	2	no	—
120	C₂²⋊C₄×C₁₉	—	4	1	no	—
121	C₂²⋊C₄×C₁₉	—	4	1	no	—
122	C₂²⋊C₄×C₁₉	—	4	1	no	—
123	C₂²⋊C₄×C₁₉	—	4	1	no	—
124	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₁₉	#41	1	1	yes	characteristic
128	#10221	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂³×C₁₉	—	4	1	no	—
130	C₂³×C₁₉	—	4	1	no	—
131	C₂³×C₁₉	#127	1	1	yes	characteristic
132	Q₈×C₁₉	—	4	4	no	—
133	Q₈×C₁₉	—	4	2	no	—
134	Q₈×C₁₉	—	4	2	no	—
135	Q₈×C₁₉	—	8	2	no	—
136	D₄×C₁₉	—	4	2	no	—
137	D₄×C₁₉	—	4	2	no	—
138	D₄×C₁₉	—	4	2	no	—
139	D₄×C₁₉	—	4	2	no	—
140	D₄×C₁₉	—	8	1	no	—
141	D₄×C₁₉	—	8	1	no	—
142	D₄×C₁₉	—	8	1	no	—
143	D₄×C₁₉	—	8	1	no	—
144	D₄×C₁₉	—	8	1	no	—
145	D₄×C₁₉	—	8	1	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	1	no	—
150	C₂×C₄×C₁₉	—	4	2	no	—
151	C₂×C₄×C₁₉	—	8	2	no	—
152	C₂×C₄×C₁₉	—	8	2	no	—
153	C₂×C₄×C₁₉	—	8	2	no	—
154	C₂×C₄×C₁₉	—	8	1	no	—
155	C₂×C₄×C₁₉	—	8	1	no	—
156	C₂×C₄×C₁₉	—	8	1	no	—
157	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
158	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
159	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
160	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
161	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
162	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
163	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
164	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
165	C₈×C₁₉	—	2	4	no	—
166	C₈×C₁₉	—	4	2	no	—
167	C₈×C₁₉	—	4	2	no	—
168	#3227	C₂×C₁₉	1	1	yes	characteristic, complemented
169	#3206	C₂×C₁₉	1	1	yes	characteristic, complemented
170	#3150	C₂×C₁₉	1	1	yes	characteristic
171	#1993	C₂×C₁₉	1	1	yes	characteristic, complemented
172	#1517	C₂×C₁₉	1	1	yes	characteristic, complemented
173	#1517	C₂×C₁₉	1	1	yes	characteristic, complemented
174	#1281	C₂×C₁₉	1	1	yes	characteristic, complemented
175	C₂²×C₁₉	—	2	1	no	—
176	C₂²×C₁₉	—	2	1	no	—
177	C₂²×C₁₉	—	8	1	no	—
178	C₂²×C₁₉	—	8	1	no	—
179	C₂²×C₁₉	—	8	1	no	—
180	C₂²×C₁₉	—	8	1	no	—
181	C₂²×C₁₉	—	8	1	no	—
182	C₂²×C₁₉	—	8	1	no	—
183	C₂²×C₁₉	#392	1	1	yes	characteristic
184	C₂²×C₁₉	#492	1	1	yes	characteristic
185	C₂²×C₁₉	#496	1	1	yes	characteristic
186	C₄×C₁₉	—	2	1	no	—
187	C₄×C₁₉	—	2	1	no	—
188	C₄×C₁₉	—	2	1	no	—
189	C₄×C₁₉	—	2	1	no	—
190	C₄×C₁₉	—	8	2	no	—
191	C₄×C₁₉	—	8	1	no	—
192	C₄×C₁₉	—	8	1	no	—
193	C₄×C₁₉	—	16	1	no	—
194	C₄×C₁₉	#450	1	2	yes	—
195	C₄×C₁₉	#460	1	2	yes	—
196	C₂×#135	—	2	2	no	—
197	#532	C₂²×C₁₉	1	1	yes	characteristic
198	#532	C₂²×C₁₉	1	1	yes	characteristic
199	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
200	#498	—	2	1	no	—
201	#497	—	2	1	no	—
202	#497	—	2	1	no	—
203	#496	—	2	1	no	—
204	#496	—	2	1	no	—
205	#492	—	2	1	no	—
206	#487	—	2	1	no	—
207	#486	—	2	1	no	—
208	#481	—	2	2	no	—
209	#476	—	2	2	no	—
210	#461	—	2	2	no	—
211	#448	—	2	2	no	—
212	#431	C₂²×C₁₉	1	1	yes	characteristic
213	#431	C₂²×C₁₉	1	1	yes	characteristic
214	C₂×#97	C₂²×C₁₉	1	1	yes	characteristic
215	#382	C₂²×C₁₉	1	1	yes	characteristic
216	#340	—	2	2	no	—
217	#327	—	2	2	no	—
218	#327	—	2	2	no	—
219	C₂×C₁₉	—	2	1	no	—
220	C₂×C₁₉	—	2	1	no	—
221	C₂×C₁₉	—	16	1	no	—
222	C₂×C₁₉	—	16	1	no	—
223	C₂×C₁₉	#2008	1	1	yes	characteristic
224	C₂×C₁₉	#2009	1	1	yes	characteristic
225	C₂×C₁₉	#2014	1	1	yes	characteristic
226	C₂×#41	—	2	1	no	—
227	C₂×#41	—	2	1	no	—
228	C₂²×Q₈	—	2	1	no	—
229	#135	—	4	4	no	—
230	#135	—	4	4	no	—
231	C₂×Q₁₆	—	2	2	no	—
232	C₂×SD₁₆	—	2	2	no	—
233	C₂×SD₁₆	—	2	2	no	—
234	C₂×SD₁₆	—	2	4	no	—
235	C₂×D₈	—	2	2	no	—
236	C₂²×C₈	D₄×C₁₉	1	2	yes	—
237	C₄⋊Q₈	—	2	1	no	—
238	C₄⋊Q₈	—	2	1	no	—
239	C₄⋊Q₈	—	2	1	no	—
240	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
241	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
242	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
243	#127	D₄×C₁₉	1	1	yes	characteristic
244	#127	D₄×C₁₉	1	1	yes	characteristic
245	#124	—	2	1	no	—
246	#124	—	2	1	no	—
247	#121	—	2	1	no	—
248	#121	—	2	1	no	—
249	#121	—	2	1	no	—
250	#121	—	2	1	no	—
251	C₄×D₄	—	2	1	no	—
252	C₄×D₄	—	2	1	no	—
253	C₄×D₄	—	2	1	no	—
254	C₄×D₄	—	2	1	no	—
255	C₂×C₄⋊C₄	—	2	1	no	—
256	C₂×C₄⋊C₄	—	2	1	no	—
257	C₂×C₄⋊C₄	—	2	1	no	—
258	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
259	#106	—	2	4	no	—
260	C₄⋊C₈	—	2	1	no	—
261	C₄⋊C₈	—	2	1	no	—
262	#103	—	2	4	no	—
263	#102	—	2	4	no	—
264	C₂²⋊C₈	D₄×C₁₉	1	2	yes	—
265	C₂²⋊C₈	D₄×C₁₉	1	2	yes	—
266	#97	—	2	1	no	—
267	#97	—	2	1	no	—
268	C₄×C₈	—	2	1	no	—
269	C₄×C₈	—	2	1	no	—
270	#95	—	2	1	no	—
271	#95	—	2	1	no	—
272	C₁₉	#10221	1	1	yes	characteristic, direct product factor, Sylow
273	#41	—	4	2	no	—
274	#41	—	4	2	no	—
275	#41	—	4	2	no	—
276	#41	—	4	2	no	—
277	C₂×Q₈	—	2	2	no	—
278	C₂×Q₈	—	2	1	no	—
279	C₂×Q₈	—	2	1	no	—
280	C₂×Q₈	—	4	1	no	—
281	C₂×Q₈	—	8	2	no	—
282	C₂×D₄	—	2	1	no	—
283	C₂×D₄	—	2	1	no	—
284	C₂×D₄	—	2	1	no	—
285	C₂×D₄	—	2	1	no	—
286	C₂×D₄	—	2	1	no	—
287	C₂×D₄	—	2	1	no	—
288	C₂×D₄	—	4	1	no	—
289	C₂×D₄	—	4	1	no	—
290	C₂²×C₄	—	4	1	no	—
291	C₂²×C₄	—	4	1	no	—
292	C₂²×C₄	—	4	1	no	—
293	C₂²×C₄	—	4	1	no	—
294	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
295	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
296	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
297	Q₁₆	—	4	4	no	—
298	SD₁₆	—	4	4	no	—
299	SD₁₆	—	4	4	no	—
300	SD₁₆	—	8	4	no	—
301	D₈	—	4	4	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	SD₁₆×C₁₉	1	4	yes	complemented
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	2	1	no	—
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	2	1	no	—
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	4	1	no	—
314	C₄⋊C₄	—	4	1	no	—
315	C₄⋊C₄	—	4	2	no	—
316	C₂²⋊C₄	—	4	1	no	—
317	C₂²⋊C₄	—	4	1	no	—
318	C₂²⋊C₄	—	4	1	no	—
319	C₂²⋊C₄	—	4	1	no	—
320	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
321	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
322	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
323	C₄²	#41×C₁₉	1	1	yes	characteristic
324	C₂³	—	4	1	no	—
325	C₂³	—	4	1	no	—
326	C₂³	C₁₉×#127	1	1	yes	characteristic
327	Q₈	—	4	4	no	—
328	Q₈	—	4	2	no	—
329	Q₈	—	4	2	no	—
330	Q₈	—	8	2	no	—
331	D₄	—	4	2	no	—
332	D₄	—	4	2	no	—
333	D₄	—	4	2	no	—
334	D₄	—	4	2	no	—
335	D₄	—	8	1	no	—
336	D₄	—	8	1	no	—
337	D₄	—	8	1	no	—
338	D₄	—	8	1	no	—
339	D₄	—	8	1	no	—
340	D₄	—	8	1	no	—
341	C₂×C₄	—	4	1	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	1	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	8	2	no	—
347	C₂×C₄	—	8	2	no	—
348	C₂×C₄	—	8	2	no	—
349	C₂×C₄	—	8	1	no	—
350	C₂×C₄	—	8	1	no	—
351	C₂×C₄	—	8	1	no	—
352	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
353	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
354	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
355	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
356	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
357	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
358	C₂×C₄	C₂×SD₁₆×C₁₉	1	2	yes	—
359	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	2	yes	—
360	C₈	—	2	4	no	—
361	C₈	—	4	2	no	—
362	C₈	—	4	2	no	—
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	8	1	no	—
366	C₂²	—	8	1	no	—
367	C₂²	—	8	1	no	—
368	C₂²	—	8	1	no	—
369	C₂²	—	8	1	no	—
370	C₂²	—	8	1	no	—
371	C₂²	C₁₉×#392	1	1	yes	characteristic
372	C₂²	C₁₉×#492	1	1	yes	characteristic
373	C₂²	C₁₉×#496	1	1	yes	characteristic
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	8	2	no	—
379	C₄	—	8	1	no	—
380	C₄	—	8	1	no	—
381	C₄	—	16	1	no	—
382	C₄	C₁₉×#450	1	2	yes	—
383	C₄	C₁₉×#460	1	2	yes	—
384	C₂	—	2	1	no	—
385	C₂	—	2	1	no	—
386	C₂	—	16	1	no	—
387	C₂	—	16	1	no	—
388	C₂	C₁₉×#2008	1	1	yes	characteristic
389	C₂	C₁₉×#2009	1	1	yes	characteristic
390	C₂	C₁₉×#2014	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
prime radical	C₁₉×#3150	C₂
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
1. C₁₉×#3227	C₂	semidirect
2. C₁₉×#3206	C₂	semidirect
4. C₁₉×#1993	C₂	semidirect
5. C₁₉×#1517	C₂	semidirect
6. C₁₉×#1517	C₂	semidirect
7. C₁₉×#1281	C₂	semidirect
110. C₂×C₈×C₁₉	SD₁₆	semidirect
128. #10221	C₁₉	direct (N × H)
168. #3227	C₂×C₁₉	semidirect
169. #3206	C₂×C₁₉	semidirect
171. #1993	C₂×C₁₉	semidirect
172. #1517	C₂×C₁₉	semidirect
173. #1517	C₂×C₁₉	semidirect
174. #1281	C₂×C₁₉	semidirect
272. C₁₉	#10221	direct (N × H)
306. C₂×C₈	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×#3227	2
C₁₉×#3206	2
C₁₉×#3150	2
C₁₉×#1993	2
C₁₉×#1517	2
C₁₉×#1281	2
#10221	19

Subgroups — C19×#10221 #18016180

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#10221 #18016180