Subgroups — C₁₉×#11709 #18017668

Order 4864.

450 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3050	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₉×#341	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1350	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1349	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₉×#135	C₄	1	1	yes	characteristic
7	C₂²×M₁₆×C₁₉	—	2	1	no	—
8	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
9	C₁₉×#525	—	2	1	no	—
10	C₂²×C₁₆×C₁₉	—	2	1	no	—
11	C₁₉×#467	C₄	1	1	yes	characteristic
12	C₁₉×#467	C₂²	1	1	yes	characteristic
13	C₁₉×#466	C₄	1	1	yes	characteristic
14	C₁₉×#466	C₂²	1	1	yes	characteristic
15	C₁₉×#465	—	2	1	no	—
16	C₁₉×#465	—	2	1	no	—
17	C₁₉×#427	C₄	1	1	yes	characteristic
18	C₁₉×#416	—	2	1	no	—
19	C₁₉×C₂²⋊C₁₆	—	2	1	no	—
20	C₁₉×C₂²⋊C₁₆	C₂²	1	2	yes	—
21	C₁₉×#341	—	2	1	no	—
22	C₁₉×#341	C₂²	1	2	yes	—
23	C₂×#41×C₁₉	—	2	1	no	—
24	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
25	C₁₉×#135	—	4	2	no	—
26	C₂×Q₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
27	C₂×SD₁₆×C₁₉	—	2	1	no	—
28	C₂×D₈×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₂×M₁₆×C₁₉	—	2	4	no	—
30	C₂×M₁₆×C₁₉	—	2	2	no	—
31	C₂×M₁₆×C₁₉	—	4	2	no	—
32	C₂×M₁₆×C₁₉	—	4	1	no	—
33	C₂²×C₈×C₁₉	—	4	2	no	—
34	C₂²×C₈×C₁₉	—	4	1	no	—
35	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
41	C₁₉×#123	—	4	1	no	—
42	C₁₉×C₂²⋊Q₈	—	2	1	no	—
43	C₁₉×C₂²⋊Q₈	—	2	1	no	—
44	C₁₉×#121	—	2	1	no	—
45	C₁₉×#121	—	2	1	no	—
46	C₁₉×#120	—	4	1	no	—
47	C₄×D₄×C₁₉	—	4	1	no	—
48	C₄×D₄×C₁₉	—	4	1	no	—
49	C₁₉×#117	—	2	1	no	—
50	C₂×C₁₆×C₁₉	—	2	2	no	—
51	C₂×C₁₆×C₁₉	—	4	2	no	—
52	C₁₉×#107	C₂×C₄	1	1	yes	characteristic
53	C₁₉×#107	C₂×C₄	1	1	yes	characteristic
54	C₁₉×#106	—	2	1	no	—
55	C₁₉×#103	—	2	1	no	—
56	C₁₉×#103	—	2	1	no	—
57	C₁₉×#102	—	2	1	no	—
58	C₁₉×#102	—	2	1	no	—
59	C₂⁴×C₁₉	SD₁₆	1	1	yes	characteristic
60	#41×C₁₉	—	4	2	no	—
61	#41×C₁₉	—	4	2	no	—
62	C₂×Q₈×C₁₉	—	2	1	no	—
63	C₂×D₄×C₁₉	—	2	1	no	—
64	C₂×D₄×C₁₉	—	4	1	no	—
65	C₂×D₄×C₁₉	—	4	1	no	—
66	C₂×D₄×C₁₉	—	4	1	no	—
67	C₂²×C₄×C₁₉	—	2	2	no	—
68	C₂²×C₄×C₁₉	—	2	2	no	—
69	C₂²×C₄×C₁₉	—	4	1	no	—
70	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
76	Q₁₆×C₁₉	—	4	1	no	—
77	SD₁₆×C₁₉	—	4	2	no	—
78	D₈×C₁₉	—	4	1	no	—
79	M₁₆×C₁₉	—	2	4	no	—
80	M₁₆×C₁₉	—	4	4	no	—
81	M₁₆×C₁₉	—	4	2	no	—
82	C₂×C₈×C₁₉	—	2	2	no	—
83	C₂×C₈×C₁₉	—	2	2	no	—
84	C₂×C₈×C₁₉	—	4	1	no	—
85	C₂×C₈×C₁₉	—	4	1	no	—
86	C₂×C₈×C₁₉	—	4	1	no	—
87	C₂×C₈×C₁₉	—	4	1	no	—
88	C₂×C₈×C₁₉	—	4	4	no	—
89	C₂×C₈×C₁₉	—	4	2	no	—
90	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
94	C₄⋊C₄×C₁₉	—	2	1	no	—
95	C₄⋊C₄×C₁₉	—	2	1	no	—
96	C₄⋊C₄×C₁₉	—	4	1	no	—
97	C₄⋊C₄×C₁₉	—	4	1	no	—
98	C₂²⋊C₄×C₁₉	—	4	1	no	—
99	C₂²⋊C₄×C₁₉	—	4	1	no	—
100	C₂²⋊C₄×C₁₉	—	4	1	no	—
101	C₂²⋊C₄×C₁₉	—	4	1	no	—
102	C₂²⋊C₄×C₁₉	—	4	1	no	—
103	C₄²×C₁₉	—	4	1	no	—
104	C₁₆×C₁₉	—	4	2	no	—
105	#11709	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂³×C₁₉	—	2	2	no	—
107	C₂³×C₁₉	—	2	2	no	—
108	C₂³×C₁₉	—	4	1	no	—
109	C₂³×C₁₉	—	4	1	no	—
110	C₂³×C₁₉	#102	1	1	yes	characteristic
111	C₂³×C₁₉	#102	1	1	yes	characteristic
112	C₂³×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
113	Q₈×C₁₉	—	4	1	no	—
114	Q₈×C₁₉	—	4	1	no	—
115	D₄×C₁₉	—	4	1	no	—
116	D₄×C₁₉	—	4	1	no	—
117	D₄×C₁₉	—	8	2	no	—
118	D₄×C₁₉	—	8	2	no	—
119	D₄×C₁₉	—	8	2	no	—
120	C₂×C₄×C₁₉	—	2	2	no	—
121	C₂×C₄×C₁₉	—	2	2	no	—
122	C₂×C₄×C₁₉	—	2	2	no	—
123	C₂×C₄×C₁₉	—	2	1	no	—
124	C₂×C₄×C₁₉	—	2	1	no	—
125	C₂×C₄×C₁₉	—	2	1	no	—
126	C₂×C₄×C₁₉	—	2	1	no	—
127	C₂×C₄×C₁₉	—	4	1	no	—
128	C₂×C₄×C₁₉	—	4	1	no	—
129	C₂×C₄×C₁₉	—	4	1	no	—
130	C₂×C₄×C₁₉	—	8	2	no	—
131	C₂×C₄×C₁₉	#99	1	2	yes	—
132	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
133	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
134	C₂×C₄×C₁₉	#104	1	2	yes	—
135	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
137	C₈×C₁₉	—	2	1	no	—
138	C₈×C₁₉	—	2	1	no	—
139	C₈×C₁₉	—	2	1	no	—
140	C₈×C₁₉	—	2	1	no	—
141	C₈×C₁₉	—	4	4	no	—
142	#3050	C₂×C₁₉	1	1	yes	characteristic
143	C₂×C₂²⋊C₁₆	C₂×C₁₉	1	1	yes	characteristic, complemented
144	C₂×#341	C₂×C₁₉	1	1	yes	characteristic, complemented
145	#1350	C₂×C₁₉	1	2	yes	complemented
146	#1349	C₂×C₁₉	1	2	yes	complemented
147	C₂²×C₁₉	—	2	1	no	—
148	C₂²×C₁₉	—	2	1	no	—
149	C₂²×C₁₉	—	2	1	no	—
150	C₂²×C₁₉	—	2	1	no	—
151	C₂²×C₁₉	—	2	1	no	—
152	C₂²×C₁₉	—	2	1	no	—
153	C₂²×C₁₉	—	2	1	no	—
154	C₂²×C₁₉	—	4	4	no	—
155	C₂²×C₁₉	—	8	1	no	—
156	C₂²×C₁₉	—	8	1	no	—
157	C₂²×C₁₉	—	8	1	no	—
158	C₂²×C₁₉	#327	1	2	yes	—
159	C₂²×C₁₉	#359	1	2	yes	—
160	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
161	C₄×C₁₉	—	2	1	no	—
162	C₄×C₁₉	—	2	1	no	—
163	C₄×C₁₉	—	8	1	no	—
164	C₄×C₁₉	—	8	1	no	—
165	C₄×C₁₉	—	8	1	no	—
166	C₄×C₁₉	#327	1	2	yes	—
167	C₄×C₁₉	C₂×#99	1	1	yes	characteristic
168	C₄×C₁₉	C₂×#104	1	1	yes	characteristic
169	C₂×#135	C₄×C₁₉	1	1	yes	characteristic
170	C₂²×M₁₆	—	2	1	no	—
171	C₂³×C₈	C₂²×C₁₉	1	1	yes	characteristic
172	#525	—	2	1	no	—
173	C₂²×C₁₆	—	2	1	no	—
174	#467	C₄×C₁₉	1	1	yes	characteristic
175	#467	C₂²×C₁₉	1	1	yes	characteristic
176	#466	C₄×C₁₉	1	1	yes	characteristic
177	#466	C₂²×C₁₉	1	1	yes	characteristic
178	#465	—	2	1	no	—
179	#465	—	2	1	no	—
180	#427	C₄×C₁₉	1	1	yes	characteristic
181	#416	—	2	1	no	—
182	C₂²⋊C₁₆	—	2	1	no	—
183	C₂²⋊C₁₆	C₂²×C₁₉	1	2	yes	—
184	#341	—	2	1	no	—
185	#341	C₂²×C₁₉	1	2	yes	—
186	C₂×C₁₉	—	2	2	no	—
187	C₂×C₁₉	—	2	2	no	—
188	C₂×C₁₉	—	4	1	no	—
189	C₂×C₁₉	—	16	1	no	—
190	C₂×C₁₉	C₂×#327	1	1	yes	characteristic
191	C₂×C₁₉	C₂×#359	1	1	yes	characteristic
192	C₂×C₁₉	#2148	1	1	yes	characteristic
193	C₂×#41	—	2	1	no	—
194	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
195	#135	—	4	2	no	—
196	C₂×Q₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
197	C₂×SD₁₆	—	2	1	no	—
198	C₂×D₈	C₂×C₄×C₁₉	1	1	yes	characteristic
199	C₂×M₁₆	—	2	4	no	—
200	C₂×M₁₆	—	2	2	no	—
201	C₂×M₁₆	—	4	2	no	—
202	C₂×M₁₆	—	4	1	no	—
203	C₂²×C₈	—	4	2	no	—
204	C₂²×C₈	—	4	1	no	—
205	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
206	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
207	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
208	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
209	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
210	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
211	#123	—	4	1	no	—
212	C₂²⋊Q₈	—	2	1	no	—
213	C₂²⋊Q₈	—	2	1	no	—
214	#121	—	2	1	no	—
215	#121	—	2	1	no	—
216	#120	—	4	1	no	—
217	C₄×D₄	—	4	1	no	—
218	C₄×D₄	—	4	1	no	—
219	#117	—	2	1	no	—
220	C₂×C₁₆	—	2	2	no	—
221	C₂×C₁₆	—	4	2	no	—
222	#107	C₂×C₄×C₁₉	1	1	yes	characteristic
223	#107	C₂×C₄×C₁₉	1	1	yes	characteristic
224	#106	—	2	1	no	—
225	#103	—	2	1	no	—
226	#103	—	2	1	no	—
227	#102	—	2	1	no	—
228	#102	—	2	1	no	—
229	C₁₉	#11709	1	1	yes	characteristic, direct product factor, Sylow
230	C₂⁴	SD₁₆×C₁₉	1	1	yes	characteristic
231	#41	—	4	2	no	—
232	#41	—	4	2	no	—
233	C₂×Q₈	—	2	1	no	—
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	—	4	1	no	—
236	C₂×D₄	—	4	1	no	—
237	C₂×D₄	—	4	1	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
242	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
243	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
244	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
245	C₂²×C₄	SD₁₆×C₁₉	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
247	Q₁₆	—	4	1	no	—
248	SD₁₆	—	4	2	no	—
249	D₈	—	4	1	no	—
250	M₁₆	—	2	4	no	—
251	M₁₆	—	4	4	no	—
252	M₁₆	—	4	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	4	4	no	—
260	C₂×C₈	—	4	2	no	—
261	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
262	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
263	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
264	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₂²⋊C₄	—	4	1	no	—
270	C₂²⋊C₄	—	4	1	no	—
271	C₂²⋊C₄	—	4	1	no	—
272	C₂²⋊C₄	—	4	1	no	—
273	C₂²⋊C₄	—	4	1	no	—
274	C₄²	—	4	1	no	—
275	C₁₆	—	4	2	no	—
276	C₂³	—	2	2	no	—
277	C₂³	—	2	2	no	—
278	C₂³	—	4	1	no	—
279	C₂³	—	4	1	no	—
280	C₂³	C₁₉×#102	1	1	yes	characteristic
281	C₂³	C₁₉×#102	1	1	yes	characteristic
282	C₂³	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
283	Q₈	—	4	1	no	—
284	Q₈	—	4	1	no	—
285	D₄	—	4	1	no	—
286	D₄	—	4	1	no	—
287	D₄	—	8	2	no	—
288	D₄	—	8	2	no	—
289	D₄	—	8	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	8	2	no	—
301	C₂×C₄	C₁₉×#99	1	2	yes	—
302	C₂×C₄	C₁₉×#102	1	1	yes	characteristic
303	C₂×C₄	C₁₉×#102	1	1	yes	characteristic
304	C₂×C₄	C₁₉×#104	1	2	yes	—
305	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
306	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	4	4	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	4	4	no	—
320	C₂²	—	8	1	no	—
321	C₂²	—	8	1	no	—
322	C₂²	—	8	1	no	—
323	C₂²	C₁₉×#327	1	2	yes	—
324	C₂²	C₁₉×#359	1	2	yes	—
325	C₂²	C₂×C₁₉×#102	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	8	1	no	—
329	C₄	—	8	1	no	—
330	C₄	—	8	1	no	—
331	C₄	C₁₉×#327	1	2	yes	—
332	C₄	C₂×C₁₉×#99	1	1	yes	characteristic
333	C₄	C₂×C₁₉×#104	1	1	yes	characteristic
334	C₂	—	2	2	no	—
335	C₂	—	2	2	no	—
336	C₂	—	4	1	no	—
337	C₂	—	16	1	no	—
338	C₂	C₂×C₁₉×#327	1	1	yes	characteristic
339	C₂	C₂×C₁₉×#359	1	1	yes	characteristic
340	C₂	C₁₉×#2148	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₉	#102
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
socle	C₂²×C₁₉	C₂×#102
prime radical	C₁₉×#3050	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂³×C₄×C₁₉ ◁ C₂³×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₉×C₂²⋊C₁₆	C₂	semidirect
3. C₂×C₁₉×#341	C₂	semidirect
4. C₁₉×#1350	C₂	semidirect
5. C₁₉×#1349	C₂	semidirect
105. #11709	C₁₉	direct (N × H)
143. C₂×C₂²⋊C₁₆	C₂×C₁₉	semidirect
144. C₂×#341	C₂×C₁₉	semidirect
145. #1350	C₂×C₁₉	semidirect
146. #1349	C₂×C₁₉	semidirect
229. C₁₉	#11709	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×#3050	2
C₂×C₁₉×C₂²⋊C₁₆	2
C₂×C₁₉×#341	2
C₁₉×#1350	2
C₁₉×#1349	2
#11709	19

Subgroups — C19×#11709 #18017668

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#11709 #18017668