Subgroups — C₁₉×#10206 #18016165

Order 4864.

560 conjugacy classes of subgroups.

The subgroup lattice has 386 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×#3229	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#3152	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1988	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1986	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#1502	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1283	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₉×#137	—	2	2	no	—
9	C₁₉×#533	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	2	no	—
14	C₁₉×#494	—	2	1	no	—
15	C₁₉×#492	—	2	1	no	—
16	C₁₉×#487	—	2	1	no	—
17	C₁₉×#483	—	2	2	no	—
18	C₁₉×#476	—	2	1	no	—
19	C₁₉×#475	—	2	2	no	—
20	C₁₉×#475	—	2	1	no	—
21	C₁₉×#465	—	2	2	no	—
22	C₁₉×#461	—	2	1	no	—
23	C₁₉×#460	—	2	1	no	—
24	C₁₉×#431	C₂²	1	1	yes	characteristic
25	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
28	C₁₉×#340	—	2	2	no	—
29	C₁₉×#339	—	2	2	no	—
30	C₁₉×#327	—	2	2	no	—
31	C₂×#41×C₁₉	—	2	1	no	—
32	C₂²×Q₈×C₁₉	—	2	1	no	—
33	C₁₉×#137	—	4	4	no	—
34	C₁₉×#137	—	4	4	no	—
35	C₂×Q₁₆×C₁₉	—	2	4	no	—
36	C₂×SD₁₆×C₁₉	—	2	4	no	—
37	C₂×SD₁₆×C₁₉	—	2	2	no	—
38	C₂×M₁₆×C₁₉	D₄	1	2	yes	—
39	C₂²×C₈×C₁₉	D₄	1	2	yes	—
40	C₁₉×C₄⋊Q₈	—	2	1	no	—
41	C₁₉×C₄⋊Q₈	—	2	1	no	—
42	C₁₉×C₄⋊Q₈	—	2	1	no	—
43	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
44	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
45	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
46	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
47	C₁₉×#127	D₄	1	1	yes	characteristic
48	C₁₉×#125	—	2	1	no	—
49	C₁₉×#124	—	2	1	no	—
50	C₁₉×C₂²⋊Q₈	—	2	1	no	—
51	C₁₉×C₂²⋊Q₈	—	2	1	no	—
52	C₁₉×#121	—	2	1	no	—
53	C₁₉×#121	—	2	1	no	—
54	C₄×Q₈×C₁₉	—	2	1	no	—
55	C₄×Q₈×C₁₉	—	2	1	no	—
56	C₄×D₄×C₁₉	—	2	1	no	—
57	C₄×D₄×C₁₉	—	2	1	no	—
58	C₁₉×#117	—	2	1	no	—
59	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
60	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
61	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
62	C₂×C₄²×C₁₉	—	2	1	no	—
63	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
64	C₁₉×#106	—	2	2	no	—
65	C₁₉×#106	—	2	4	no	—
66	C₁₉×C₄⋊C₈	—	2	1	no	—
67	C₁₉×C₄⋊C₈	—	2	1	no	—
68	C₁₉×C₄⋊C₈	—	2	1	no	—
69	C₁₉×#103	—	2	2	no	—
70	C₁₉×#103	—	2	4	no	—
71	C₁₉×#102	—	2	2	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	1	no	—
82	C₂×Q₈×C₁₉	—	2	2	no	—
83	C₂×Q₈×C₁₉	—	4	1	no	—
84	C₂×Q₈×C₁₉	—	4	2	no	—
85	C₂×Q₈×C₁₉	—	4	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	—	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	Q₁₆×C₁₉	—	4	8	no	—
97	SD₁₆×C₁₉	—	4	8	no	—
98	SD₁₆×C₁₉	—	8	2	no	—
99	M₁₆×C₁₉	—	2	2	no	—
100	M₁₆×C₁₉	SD₁₆	1	4	yes	complemented
101	C₂×C₈×C₁₉	—	2	2	no	—
102	C₂×C₈×C₁₉	—	2	2	no	—
103	C₂×C₈×C₁₉	—	2	2	no	—
104	C₂×C₈×C₁₉	—	4	2	no	—
105	C₄⋊C₄×C₁₉	—	2	2	no	—
106	C₄⋊C₄×C₁₉	—	2	2	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	2	no	—
111	C₄⋊C₄×C₁₉	—	2	1	no	—
112	C₄⋊C₄×C₁₉	—	2	1	no	—
113	C₄⋊C₄×C₁₉	—	2	1	no	—
114	C₄⋊C₄×C₁₉	—	4	1	no	—
115	C₄⋊C₄×C₁₉	—	4	1	no	—
116	C₄⋊C₄×C₁₉	—	4	1	no	—
117	C₄⋊C₄×C₁₉	—	4	1	no	—
118	C₄⋊C₄×C₁₉	—	4	1	no	—
119	C₂²⋊C₄×C₁₉	—	4	1	no	—
120	C₂²⋊C₄×C₁₉	—	4	1	no	—
121	C₂²⋊C₄×C₁₉	—	4	1	no	—
122	C₄²×C₁₉	—	4	1	no	—
123	C₄²×C₁₉	—	4	2	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	#10206	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂³×C₁₉	—	4	1	no	—
130	C₂³×C₁₉	#127	1	1	yes	characteristic
131	Q₈×C₁₉	—	4	2	no	—
132	Q₈×C₁₉	—	4	4	no	—
133	Q₈×C₁₉	—	4	2	no	—
134	Q₈×C₁₉	—	4	1	no	—
135	Q₈×C₁₉	—	4	1	no	—
136	Q₈×C₁₉	—	8	2	no	—
137	D₄×C₁₉	—	4	2	no	—
138	D₄×C₁₉	—	4	1	no	—
139	D₄×C₁₉	—	4	1	no	—
140	D₄×C₁₉	—	8	1	no	—
141	D₄×C₁₉	—	8	1	no	—
142	D₄×C₁₉	—	8	1	no	—
143	C₂×C₄×C₁₉	—	2	1	no	—
144	C₂×C₄×C₁₉	—	2	1	no	—
145	C₂×C₄×C₁₉	—	4	1	no	—
146	C₂×C₄×C₁₉	—	4	2	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	1	no	—
152	C₂×C₄×C₁₉	—	8	1	no	—
153	C₂×C₄×C₁₉	—	8	1	no	—
154	C₂×C₄×C₁₉	—	8	1	no	—
155	C₂×C₄×C₁₉	—	8	2	no	—
156	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
157	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
158	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
159	C₂×C₄×C₁₉	#121	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	C₂×SD₁₆	1	2	yes	—
163	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
164	C₈×C₁₉	—	2	4	no	—
165	C₈×C₁₉	—	4	2	no	—
166	C₈×C₁₉	—	4	2	no	—
167	#3246	C₂×C₁₉	1	1	yes	characteristic
168	#3229	C₂×C₁₉	1	1	yes	characteristic, complemented
169	#3152	C₂×C₁₉	1	1	yes	characteristic
170	#1988	C₂×C₁₉	1	1	yes	characteristic, complemented
171	#1986	C₂×C₁₉	1	1	yes	characteristic, complemented
172	#1502	C₂×C₁₉	1	1	yes	characteristic
173	#1283	C₂×C₁₉	1	1	yes	characteristic, complemented
174	C₂²×C₁₉	—	2	1	no	—
175	C₂²×C₁₉	—	2	1	no	—
176	C₂²×C₁₉	—	8	1	no	—
177	C₂²×C₁₉	—	8	1	no	—
178	C₂²×C₁₉	—	8	1	no	—
179	C₂²×C₁₉	#392	1	1	yes	characteristic
180	C₂²×C₁₉	#492	1	1	yes	characteristic
181	C₂²×C₁₉	#496	1	1	yes	characteristic
182	C₄×C₁₉	—	2	1	no	—
183	C₄×C₁₉	—	2	1	no	—
184	C₄×C₁₉	—	2	1	no	—
185	C₄×C₁₉	—	8	2	no	—
186	C₄×C₁₉	—	8	1	no	—
187	C₄×C₁₉	—	8	1	no	—
188	C₄×C₁₉	—	8	1	no	—
189	C₄×C₁₉	—	8	2	no	—
190	C₄×C₁₉	#450	1	2	yes	—
191	C₄×C₁₉	#455	1	1	yes	characteristic
192	C₄×C₁₉	#456	1	1	yes	characteristic
193	C₄×C₁₉	#460	1	2	yes	—
194	C₂×#137	—	2	2	no	—
195	#533	C₂²×C₁₉	1	1	yes	characteristic
196	#532	C₂²×C₁₉	1	1	yes	characteristic
197	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
198	#498	—	2	1	no	—
199	#497	—	2	2	no	—
200	#494	—	2	1	no	—
201	#492	—	2	1	no	—
202	#487	—	2	1	no	—
203	#483	—	2	2	no	—
204	#476	—	2	1	no	—
205	#475	—	2	2	no	—
206	#475	—	2	1	no	—
207	#465	—	2	2	no	—
208	#461	—	2	1	no	—
209	#460	—	2	1	no	—
210	#431	C₂²×C₁₉	1	1	yes	characteristic
211	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
212	C₄×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
213	C₄×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
214	#340	—	2	2	no	—
215	#339	—	2	2	no	—
216	#327	—	2	2	no	—
217	C₂×C₁₉	—	2	1	no	—
218	C₂×C₁₉	—	2	1	no	—
219	C₂×C₁₉	—	16	1	no	—
220	C₂×C₁₉	#2007	1	1	yes	characteristic
221	C₂×C₁₉	#2009	1	1	yes	characteristic
222	C₂×C₁₉	#2014	1	1	yes	characteristic
223	C₂×#41	—	2	1	no	—
224	C₂²×Q₈	—	2	1	no	—
225	#137	—	4	4	no	—
226	#137	—	4	4	no	—
227	C₂×Q₁₆	—	2	4	no	—
228	C₂×SD₁₆	—	2	4	no	—
229	C₂×SD₁₆	—	2	2	no	—
230	C₂×M₁₆	D₄×C₁₉	1	2	yes	—
231	C₂²×C₈	D₄×C₁₉	1	2	yes	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	2	1	no	—
234	C₄⋊Q₈	—	2	1	no	—
235	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
236	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
237	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
238	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
239	#127	D₄×C₁₉	1	1	yes	characteristic
240	#125	—	2	1	no	—
241	#124	—	2	1	no	—
242	C₂²⋊Q₈	—	2	1	no	—
243	C₂²⋊Q₈	—	2	1	no	—
244	#121	—	2	1	no	—
245	#121	—	2	1	no	—
246	C₄×Q₈	—	2	1	no	—
247	C₄×Q₈	—	2	1	no	—
248	C₄×D₄	—	2	1	no	—
249	C₄×D₄	—	2	1	no	—
250	#117	—	2	1	no	—
251	C₂×C₄⋊C₄	—	2	1	no	—
252	C₂×C₄⋊C₄	—	2	1	no	—
253	C₂×C₄⋊C₄	—	2	1	no	—
254	C₂×C₄²	—	2	1	no	—
255	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
256	#106	—	2	2	no	—
257	#106	—	2	4	no	—
258	C₄⋊C₈	—	2	1	no	—
259	C₄⋊C₈	—	2	1	no	—
260	C₄⋊C₈	—	2	1	no	—
261	#103	—	2	2	no	—
262	#103	—	2	4	no	—
263	#102	—	2	2	no	—
264	C₂²⋊C₈	D₄×C₁₉	1	2	yes	—
265	#97	—	2	1	no	—
266	C₄×C₈	—	2	1	no	—
267	C₄×C₈	—	2	1	no	—
268	#95	—	2	1	no	—
269	C₁₉	#10206	1	1	yes	characteristic, direct product factor, Sylow
270	#41	—	4	2	no	—
271	#41	—	4	2	no	—
272	C₂×Q₈	—	2	1	no	—
273	C₂×Q₈	—	2	1	no	—
274	C₂×Q₈	—	2	1	no	—
275	C₂×Q₈	—	2	2	no	—
276	C₂×Q₈	—	4	1	no	—
277	C₂×Q₈	—	4	2	no	—
278	C₂×Q₈	—	4	2	no	—
279	C₂×D₄	—	2	1	no	—
280	C₂×D₄	—	2	1	no	—
281	C₂×D₄	—	2	1	no	—
282	C₂×D₄	—	4	1	no	—
283	C₂²×C₄	—	4	1	no	—
284	C₂²×C₄	—	4	1	no	—
285	C₂²×C₄	—	4	1	no	—
286	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
287	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
288	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
289	Q₁₆	—	4	8	no	—
290	SD₁₆	—	4	8	no	—
291	SD₁₆	—	8	2	no	—
292	M₁₆	—	2	2	no	—
293	M₁₆	SD₁₆×C₁₉	1	4	yes	complemented
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	4	2	no	—
298	C₄⋊C₄	—	2	2	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	—	2	2	no	—
304	C₄⋊C₄	—	2	1	no	—
305	C₄⋊C₄	—	2	1	no	—
306	C₄⋊C₄	—	2	1	no	—
307	C₄⋊C₄	—	4	1	no	—
308	C₄⋊C₄	—	4	1	no	—
309	C₄⋊C₄	—	4	1	no	—
310	C₄⋊C₄	—	4	1	no	—
311	C₄⋊C₄	—	4	1	no	—
312	C₂²⋊C₄	—	4	1	no	—
313	C₂²⋊C₄	—	4	1	no	—
314	C₂²⋊C₄	—	4	1	no	—
315	C₄²	—	4	1	no	—
316	C₄²	—	4	2	no	—
317	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
318	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
319	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
320	C₄²	#41×C₁₉	1	1	yes	characteristic
321	C₂³	—	4	1	no	—
322	C₂³	C₁₉×#127	1	1	yes	characteristic
323	Q₈	—	4	2	no	—
324	Q₈	—	4	4	no	—
325	Q₈	—	4	2	no	—
326	Q₈	—	4	1	no	—
327	Q₈	—	4	1	no	—
328	Q₈	—	8	2	no	—
329	D₄	—	4	2	no	—
330	D₄	—	4	1	no	—
331	D₄	—	4	1	no	—
332	D₄	—	8	1	no	—
333	D₄	—	8	1	no	—
334	D₄	—	8	1	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	4	1	no	—
340	C₂×C₄	—	4	1	no	—
341	C₂×C₄	—	4	1	no	—
342	C₂×C₄	—	4	2	no	—
343	C₂×C₄	—	8	1	no	—
344	C₂×C₄	—	8	1	no	—
345	C₂×C₄	—	8	1	no	—
346	C₂×C₄	—	8	1	no	—
347	C₂×C₄	—	8	2	no	—
348	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
349	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
350	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
351	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
352	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
353	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
354	C₂×C₄	C₂×SD₁₆×C₁₉	1	2	yes	—
355	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	2	yes	—
356	C₈	—	2	4	no	—
357	C₈	—	4	2	no	—
358	C₈	—	4	2	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	2	1	no	—
361	C₂²	—	8	1	no	—
362	C₂²	—	8	1	no	—
363	C₂²	—	8	1	no	—
364	C₂²	C₁₉×#392	1	1	yes	characteristic
365	C₂²	C₁₉×#492	1	1	yes	characteristic
366	C₂²	C₁₉×#496	1	1	yes	characteristic
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	8	2	no	—
371	C₄	—	8	1	no	—
372	C₄	—	8	1	no	—
373	C₄	—	8	1	no	—
374	C₄	—	8	2	no	—
375	C₄	C₁₉×#450	1	2	yes	—
376	C₄	C₁₉×#455	1	1	yes	characteristic
377	C₄	C₁₉×#456	1	1	yes	characteristic
378	C₄	C₁₉×#460	1	2	yes	—
379	C₂	—	2	1	no	—
380	C₂	—	2	1	no	—
381	C₂	—	16	1	no	—
382	C₂	C₁₉×#2007	1	1	yes	characteristic
383	C₂	C₁₉×#2009	1	1	yes	characteristic
384	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×#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
2. C₁₉×#3229	C₂	semidirect
4. C₁₉×#1988	C₂	semidirect
5. C₁₉×#1986	C₂	semidirect
7. C₁₉×#1283	C₂	semidirect
100. M₁₆×C₁₉	SD₁₆	semidirect
128. #10206	C₁₉	direct (N × H)
168. #3229	C₂×C₁₉	semidirect
170. #1988	C₂×C₁₉	semidirect
171. #1986	C₂×C₁₉	semidirect
173. #1283	C₂×C₁₉	semidirect
269. C₁₉	#10206	direct (N × H)
293. 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×#3229	2
C₁₉×#3152	2
C₁₉×#1988	2
C₁₉×#1986	2
C₁₉×#1502	2
C₁₉×#1283	2
#10206	19

Subgroups — C19×#10206 #18016165

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#10206 #18016165