Subgroups — C₁₉×#10197 #18016156

Order 4864.

524 conjugacy classes of subgroups.

The subgroup lattice has 378 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×#3206	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#3165	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1987	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×#492	—	2	1	no	—
15	C₁₉×#487	—	2	1	no	—
16	C₁₉×#476	—	2	1	no	—
17	C₁₉×#475	—	2	1	no	—
18	C₁₉×#465	—	2	2	no	—
19	C₁₉×#461	—	2	2	no	—
20	C₁₉×#461	—	2	1	no	—
21	C₁₉×#460	—	2	1	no	—
22	C₁₉×#448	—	2	2	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	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
29	C₁₉×#340	—	2	2	no	—
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	4	no	—
34	C₂×SD₁₆×C₁₉	—	2	2	no	—
35	C₂×M₁₆×C₁₉	D₄	1	2	yes	—
36	C₂²×C₈×C₁₉	D₄	1	2	yes	—
37	C₁₉×C₄⋊Q₈	—	2	1	no	—
38	C₁₉×C₄⋊Q₈	—	2	1	no	—
39	C₁₉×C₄⋊Q₈	—	2	1	no	—
40	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
41	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
42	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
43	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
44	C₁₉×#127	D₄	1	1	yes	characteristic
45	C₁₉×#125	—	2	1	no	—
46	C₁₉×#124	—	2	1	no	—
47	C₁₉×C₂²⋊Q₈	—	2	1	no	—
48	C₁₉×C₂²⋊Q₈	—	2	1	no	—
49	C₁₉×#121	—	2	1	no	—
50	C₁₉×#121	—	2	1	no	—
51	C₄×Q₈×C₁₉	—	2	1	no	—
52	C₄×Q₈×C₁₉	—	2	1	no	—
53	C₄×D₄×C₁₉	—	2	1	no	—
54	C₄×D₄×C₁₉	—	2	1	no	—
55	C₁₉×#117	—	2	1	no	—
56	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
57	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
58	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
59	C₂×C₄²×C₁₉	—	2	1	no	—
60	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
61	C₁₉×#107	—	2	4	no	—
62	C₁₉×#106	—	2	4	no	—
63	C₁₉×#106	—	2	2	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	4	no	—
69	C₁₉×#103	—	2	2	no	—
70	C₁₉×#102	—	2	2	no	—
71	C₁₉×C₂²⋊C₈	D₄	1	2	yes	—
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	#41×C₁₉	—	4	2	no	—
77	#41×C₁₉	—	4	2	no	—
78	C₂×Q₈×C₁₉	—	2	1	no	—
79	C₂×Q₈×C₁₉	—	2	2	no	—
80	C₂×Q₈×C₁₉	—	2	1	no	—
81	C₂×Q₈×C₁₉	—	2	1	no	—
82	C₂×Q₈×C₁₉	—	4	1	no	—
83	C₂×Q₈×C₁₉	—	8	2	no	—
84	C₂×D₄×C₁₉	—	2	1	no	—
85	C₂×D₄×C₁₉	—	2	1	no	—
86	C₂×D₄×C₁₉	—	2	1	no	—
87	C₂×D₄×C₁₉	—	4	1	no	—
88	C₂²×C₄×C₁₉	—	4	1	no	—
89	C₂²×C₄×C₁₉	—	4	1	no	—
90	C₂²×C₄×C₁₉	—	4	1	no	—
91	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
94	SD₁₆×C₁₉	—	8	4	no	—
95	SD₁₆×C₁₉	—	8	2	no	—
96	M₁₆×C₁₉	—	2	2	no	—
97	M₁₆×C₁₉	SD₁₆	1	4	yes	complemented
98	C₂×C₈×C₁₉	—	2	2	no	—
99	C₂×C₈×C₁₉	—	2	2	no	—
100	C₂×C₈×C₁₉	—	2	2	no	—
101	C₂×C₈×C₁₉	—	4	2	no	—
102	C₄⋊C₄×C₁₉	—	2	2	no	—
103	C₄⋊C₄×C₁₉	—	2	2	no	—
104	C₄⋊C₄×C₁₉	—	2	1	no	—
105	C₄⋊C₄×C₁₉	—	2	1	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	2	no	—
110	C₄⋊C₄×C₁₉	—	2	1	no	—
111	C₄⋊C₄×C₁₉	—	4	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	C₂²⋊C₄×C₁₉	—	4	1	no	—
117	C₂²⋊C₄×C₁₉	—	4	1	no	—
118	C₂²⋊C₄×C₁₉	—	4	1	no	—
119	C₄²×C₁₉	—	4	1	no	—
120	C₄²×C₁₉	—	4	2	no	—
121	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₁₉	#41	1	1	yes	characteristic
125	#10197	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂³×C₁₉	—	4	1	no	—
127	C₂³×C₁₉	#127	1	1	yes	characteristic
128	Q₈×C₁₉	—	4	4	no	—
129	Q₈×C₁₉	—	4	1	no	—
130	Q₈×C₁₉	—	4	1	no	—
131	Q₈×C₁₉	—	4	2	no	—
132	Q₈×C₁₉	—	8	1	no	—
133	Q₈×C₁₉	—	8	2	no	—
134	D₄×C₁₉	—	4	2	no	—
135	D₄×C₁₉	—	4	1	no	—
136	D₄×C₁₉	—	4	1	no	—
137	D₄×C₁₉	—	4	2	no	—
138	D₄×C₁₉	—	8	1	no	—
139	D₄×C₁₉	—	8	1	no	—
140	C₂×C₄×C₁₉	—	2	1	no	—
141	C₂×C₄×C₁₉	—	2	1	no	—
142	C₂×C₄×C₁₉	—	4	1	no	—
143	C₂×C₄×C₁₉	—	4	2	no	—
144	C₂×C₄×C₁₉	—	4	1	no	—
145	C₂×C₄×C₁₉	—	4	1	no	—
146	C₂×C₄×C₁₉	—	4	1	no	—
147	C₂×C₄×C₁₉	—	4	2	no	—
148	C₂×C₄×C₁₉	—	8	1	no	—
149	C₂×C₄×C₁₉	—	8	2	no	—
150	C₂×C₄×C₁₉	—	8	1	no	—
151	C₂×C₄×C₁₉	—	8	2	no	—
152	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
153	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
154	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
155	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
156	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
157	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
158	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
159	C₂×C₄×C₁₉	#137	1	2	yes	—
160	C₈×C₁₉	—	2	4	no	—
161	C₈×C₁₉	—	4	2	no	—
162	C₈×C₁₉	—	4	2	no	—
163	#3246	C₂×C₁₉	1	1	yes	characteristic
164	#3206	C₂×C₁₉	1	1	yes	characteristic
165	#3165	C₂×C₁₉	1	1	yes	characteristic, complemented
166	#1987	C₂×C₁₉	1	1	yes	characteristic, complemented
167	#1980	C₂×C₁₉	1	1	yes	characteristic
168	#1503	C₂×C₁₉	1	1	yes	characteristic, complemented
169	#1283	C₂×C₁₉	1	1	yes	characteristic, complemented
170	C₂²×C₁₉	—	2	1	no	—
171	C₂²×C₁₉	—	2	1	no	—
172	C₂²×C₁₉	—	8	1	no	—
173	C₂²×C₁₉	—	8	1	no	—
174	C₂²×C₁₉	—	8	1	no	—
175	C₂²×C₁₉	#392	1	1	yes	characteristic
176	C₂²×C₁₉	#492	1	1	yes	characteristic
177	C₂²×C₁₉	#497	1	1	yes	characteristic
178	C₄×C₁₉	—	2	1	no	—
179	C₄×C₁₉	—	2	1	no	—
180	C₄×C₁₉	—	2	1	no	—
181	C₄×C₁₉	—	8	2	no	—
182	C₄×C₁₉	—	8	1	no	—
183	C₄×C₁₉	—	8	2	no	—
184	C₄×C₁₉	—	8	1	no	—
185	C₄×C₁₉	—	8	1	no	—
186	C₄×C₁₉	#448	1	2	yes	—
187	C₄×C₁₉	#455	1	1	yes	characteristic
188	C₄×C₁₉	#456	1	1	yes	characteristic
189	C₄×C₁₉	#461	1	2	yes	—
190	#533	C₂²×C₁₉	1	1	yes	characteristic
191	#532	C₂²×C₁₉	1	1	yes	characteristic
192	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
193	#501	—	2	2	no	—
194	#500	—	2	1	no	—
195	#498	—	2	1	no	—
196	#492	—	2	1	no	—
197	#487	—	2	1	no	—
198	#476	—	2	1	no	—
199	#475	—	2	1	no	—
200	#465	—	2	2	no	—
201	#461	—	2	2	no	—
202	#461	—	2	1	no	—
203	#460	—	2	1	no	—
204	#448	—	2	2	no	—
205	#431	C₂²×C₁₉	1	1	yes	characteristic
206	#428	—	2	2	no	—
207	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
208	C₂×#104	—	2	2	no	—
209	C₄×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
210	C₄×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
211	#340	—	2	2	no	—
212	#328	—	2	2	no	—
213	C₂×C₁₉	—	2	1	no	—
214	C₂×C₁₉	—	2	1	no	—
215	C₂×C₁₉	—	16	1	no	—
216	C₂×C₁₉	#2007	1	1	yes	characteristic
217	C₂×C₁₉	#2008	1	1	yes	characteristic
218	C₂×C₁₉	#2013	1	1	yes	characteristic
219	C₂×#41	—	2	1	no	—
220	C₂²×Q₈	—	2	1	no	—
221	C₂×SD₁₆	—	2	4	no	—
222	C₂×SD₁₆	—	2	2	no	—
223	C₂×M₁₆	D₄×C₁₉	1	2	yes	—
224	C₂²×C₈	D₄×C₁₉	1	2	yes	—
225	C₄⋊Q₈	—	2	1	no	—
226	C₄⋊Q₈	—	2	1	no	—
227	C₄⋊Q₈	—	2	1	no	—
228	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
229	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
230	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
231	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
232	#127	D₄×C₁₉	1	1	yes	characteristic
233	#125	—	2	1	no	—
234	#124	—	2	1	no	—
235	C₂²⋊Q₈	—	2	1	no	—
236	C₂²⋊Q₈	—	2	1	no	—
237	#121	—	2	1	no	—
238	#121	—	2	1	no	—
239	C₄×Q₈	—	2	1	no	—
240	C₄×Q₈	—	2	1	no	—
241	C₄×D₄	—	2	1	no	—
242	C₄×D₄	—	2	1	no	—
243	#117	—	2	1	no	—
244	C₂×C₄⋊C₄	—	2	1	no	—
245	C₂×C₄⋊C₄	—	2	1	no	—
246	C₂×C₄⋊C₄	—	2	1	no	—
247	C₂×C₄²	—	2	1	no	—
248	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
249	#107	—	2	4	no	—
250	#106	—	2	4	no	—
251	#106	—	2	2	no	—
252	C₄⋊C₈	—	2	1	no	—
253	C₄⋊C₈	—	2	1	no	—
254	C₄⋊C₈	—	2	1	no	—
255	#104	—	4	4	no	—
256	#103	—	2	4	no	—
257	#103	—	2	2	no	—
258	#102	—	2	2	no	—
259	C₂²⋊C₈	D₄×C₁₉	1	2	yes	—
260	#97	—	2	1	no	—
261	C₄×C₈	—	2	1	no	—
262	C₄×C₈	—	2	1	no	—
263	#95	—	2	1	no	—
264	C₁₉	#10197	1	1	yes	characteristic, direct product factor, Sylow
265	#41	—	4	2	no	—
266	#41	—	4	2	no	—
267	C₂×Q₈	—	2	1	no	—
268	C₂×Q₈	—	2	2	no	—
269	C₂×Q₈	—	2	1	no	—
270	C₂×Q₈	—	2	1	no	—
271	C₂×Q₈	—	4	1	no	—
272	C₂×Q₈	—	8	2	no	—
273	C₂×D₄	—	2	1	no	—
274	C₂×D₄	—	2	1	no	—
275	C₂×D₄	—	2	1	no	—
276	C₂×D₄	—	4	1	no	—
277	C₂²×C₄	—	4	1	no	—
278	C₂²×C₄	—	4	1	no	—
279	C₂²×C₄	—	4	1	no	—
280	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
281	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
282	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
283	SD₁₆	—	8	4	no	—
284	SD₁₆	—	8	2	no	—
285	M₁₆	—	2	2	no	—
286	M₁₆	SD₁₆×C₁₉	1	4	yes	complemented
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	4	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	1	no	—
293	C₄⋊C₄	—	2	1	no	—
294	C₄⋊C₄	—	2	1	no	—
295	C₄⋊C₄	—	2	1	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	1	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	4	1	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	4	1	no	—
303	C₄⋊C₄	—	4	1	no	—
304	C₄⋊C₄	—	4	1	no	—
305	C₂²⋊C₄	—	4	1	no	—
306	C₂²⋊C₄	—	4	1	no	—
307	C₂²⋊C₄	—	4	1	no	—
308	C₄²	—	4	1	no	—
309	C₄²	—	4	2	no	—
310	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
311	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
312	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
313	C₄²	#41×C₁₉	1	1	yes	characteristic
314	C₂³	—	4	1	no	—
315	C₂³	C₁₉×#127	1	1	yes	characteristic
316	Q₈	—	4	4	no	—
317	Q₈	—	4	1	no	—
318	Q₈	—	4	1	no	—
319	Q₈	—	4	2	no	—
320	Q₈	—	8	1	no	—
321	Q₈	—	8	2	no	—
322	D₄	—	4	2	no	—
323	D₄	—	4	1	no	—
324	D₄	—	4	1	no	—
325	D₄	—	4	2	no	—
326	D₄	—	8	1	no	—
327	D₄	—	8	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	8	2	no	—
337	C₂×C₄	—	8	1	no	—
338	C₂×C₄	—	8	1	no	—
339	C₂×C₄	—	8	2	no	—
340	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
341	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
342	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
343	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
344	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
345	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
346	C₂×C₄	C₂×SD₁₆×C₁₉	1	2	yes	—
347	C₂×C₄	C₁₉×#137	1	2	yes	—
348	C₈	—	2	4	no	—
349	C₈	—	4	2	no	—
350	C₈	—	4	2	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	8	1	no	—
354	C₂²	—	8	1	no	—
355	C₂²	—	8	1	no	—
356	C₂²	C₁₉×#392	1	1	yes	characteristic
357	C₂²	C₁₉×#492	1	1	yes	characteristic
358	C₂²	C₁₉×#497	1	1	yes	characteristic
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	8	2	no	—
363	C₄	—	8	1	no	—
364	C₄	—	8	2	no	—
365	C₄	—	8	1	no	—
366	C₄	—	8	1	no	—
367	C₄	C₁₉×#448	1	2	yes	—
368	C₄	C₁₉×#455	1	1	yes	characteristic
369	C₄	C₁₉×#456	1	1	yes	characteristic
370	C₄	C₁₉×#461	1	2	yes	—
371	C₂	—	2	1	no	—
372	C₂	—	2	1	no	—
373	C₂	—	16	1	no	—
374	C₂	C₁₉×#2007	1	1	yes	characteristic
375	C₂	C₁₉×#2008	1	1	yes	characteristic
376	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×#1987	C₂	semidirect
6. C₁₉×#1503	C₂	semidirect
7. C₁₉×#1283	C₂	semidirect
97. M₁₆×C₁₉	SD₁₆	semidirect
125. #10197	C₁₉	direct (N × H)
165. #3165	C₂×C₁₉	semidirect
166. #1987	C₂×C₁₉	semidirect
168. #1503	C₂×C₁₉	semidirect
169. #1283	C₂×C₁₉	semidirect
264. C₁₉	#10197	direct (N × H)
286. 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×#3206	2
C₁₉×#3165	2
C₁₉×#1987	2
C₁₉×#1980	2
C₁₉×#1503	2
C₁₉×#1283	2
#10197	19

Subgroups — C19×#10197 #18016156

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#10197 #18016156