Subgroups — C₁₉×#19689 #18025648

Order 4864.

832 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3207	C₂	1	2	yes	maximal, complemented
2	C₁₉×#3184	C₂	1	2	yes	maximal, complemented
3	C₁₉×#3169	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#3147	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1721	C₂	1	2	yes	maximal, complemented
6	C₁₉×#1721	C₂	1	2	yes	maximal, complemented
7	C₁₉×#1720	C₂	1	2	yes	maximal, complemented
8	C₁₉×#1714	C₂	1	2	yes	maximal, complemented
9	C₁₉×#1451	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₉×#135	—	2	2	no	—
11	C₁₉×#532	C₂²	1	2	yes	—
12	C₁₉×#532	C₂²	1	2	yes	—
13	C₁₉×#500	C₂²	1	1	yes	characteristic
14	C₁₉×#498	C₂²	1	2	yes	—
15	C₁₉×#495	—	2	2	no	—
16	C₁₉×#494	C₂²	1	1	yes	characteristic
17	C₁₉×#493	C₂²	1	1	yes	characteristic
18	C₁₉×#493	C₂²	1	2	yes	—
19	C₁₉×#493	C₂²	1	2	yes	—
20	C₁₉×#493	C₂²	1	1	yes	characteristic
21	C₁₉×#492	C₂²	1	2	yes	—
22	C₁₉×#492	C₂²	1	2	yes	—
23	C₁₉×#487	C₂²	1	2	yes	—
24	C₁₉×#486	C₂²	1	2	yes	—
25	C₁₉×#486	C₂²	1	2	yes	—
26	C₁₉×#486	C₂²	1	2	yes	—
27	C₁₉×#486	C₂²	1	2	yes	—
28	C₁₉×#469	—	2	4	no	—
29	C₁₉×#466	—	2	2	no	—
30	C₁₉×#463	—	2	4	no	—
31	C₁₉×#449	—	2	4	no	—
32	C₁₉×#442	—	2	4	no	—
33	C₄×D₈×C₁₉	—	2	2	no	—
34	C₁₉×#431	C₂²	1	1	yes	characteristic
35	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
37	C₁₉×#322	C₂²	1	2	yes	—
38	C₁₉×#322	C₂²	1	2	yes	—
39	C₂×#41×C₁₉	—	2	2	no	—
40	C₂×#41×C₁₉	—	2	2	no	—
41	C₁₉×#135	—	4	8	no	—
42	C₂×Q₁₆×C₁₉	—	2	2	no	—
43	C₂×SD₁₆×C₁₉	—	2	4	no	—
44	C₂×SD₁₆×C₁₉	—	2	4	no	—
45	C₂×D₈×C₁₉	—	2	2	no	—
46	C₂×D₈×C₁₉	—	2	4	no	—
47	C₂×D₈×C₁₉	—	2	2	no	—
48	C₂×D₈×C₁₉	—	2	4	no	—
49	C₂×M₁₆×C₁₉	D₄	1	2	yes	—
50	C₂²×C₈×C₁₉	D₄	1	2	yes	—
51	C₁₉×C₄⋊Q₈	C₂³	1	2	yes	—
52	C₁₉×C₄⋊Q₈	C₂³	1	2	yes	—
53	C₁₉×#127	C₂³	1	2	yes	—
54	C₁₉×#127	C₂³	1	2	yes	—
55	C₁₉×#124	—	2	2	no	—
56	C₁₉×#124	—	2	2	no	—
57	C₁₉×#121	—	2	2	no	—
58	C₁₉×#121	—	2	2	no	—
59	C₁₉×#121	—	2	2	no	—
60	C₁₉×#121	—	2	2	no	—
61	C₄×D₄×C₁₉	—	2	2	no	—
62	C₄×D₄×C₁₉	—	2	2	no	—
63	C₄×D₄×C₁₉	—	2	2	no	—
64	C₄×D₄×C₁₉	—	2	2	no	—
65	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
66	C₁₉×#107	—	2	2	no	—
67	C₁₉×#106	—	2	4	no	—
68	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
69	C₁₉×#103	—	2	4	no	—
70	C₁₉×#102	—	2	4	no	—
71	C₁₉×#102	—	2	4	no	—
72	C₁₉×#102	—	2	4	no	—
73	C₁₉×#102	—	2	4	no	—
74	C₁₉×C₂²⋊C₈	D₄	1	2	yes	—
75	C₁₉×#97	D₄	1	2	yes	—
76	C₄×C₈×C₁₉	D₄	1	2	yes	—
77	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₁₉	C₂³	1	2	yes	—
80	C₄×C₈×C₁₉	C₂³	1	2	yes	—
81	#41×C₁₉	—	4	8	no	—
82	#41×C₁₉	—	8	4	no	—
83	C₂×Q₈×C₁₉	—	2	2	no	—
84	C₂×Q₈×C₁₉	—	2	2	no	—
85	C₂×D₄×C₁₉	—	2	2	no	—
86	C₂×D₄×C₁₉	—	2	2	no	—
87	C₂×D₄×C₁₉	—	2	2	no	—
88	C₂×D₄×C₁₉	—	2	2	no	—
89	C₂×D₄×C₁₉	—	2	2	no	—
90	C₂×D₄×C₁₉	—	2	2	no	—
91	C₂×D₄×C₁₉	—	4	2	no	—
92	C₂×D₄×C₁₉	—	4	2	no	—
93	C₂²×C₄×C₁₉	—	4	2	no	—
94	C₂²×C₄×C₁₉	—	4	2	no	—
95	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
98	Q₁₆×C₁₉	—	4	4	no	—
99	SD₁₆×C₁₉	—	4	8	no	—
100	SD₁₆×C₁₉	—	8	4	no	—
101	D₈×C₁₉	—	4	8	no	—
102	D₈×C₁₉	—	4	4	no	—
103	D₈×C₁₉	—	4	4	no	—
104	D₈×C₁₉	—	8	4	no	—
105	M₁₆×C₁₉	—	4	2	no	—
106	C₂×C₈×C₁₉	—	2	4	no	—
107	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₁₉	C₂×D₄	1	4	yes	—
109	C₂×C₈×C₁₉	C₂×D₄	1	4	yes	—
110	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
111	C₄⋊C₄×C₁₉	—	2	2	no	—
112	C₄⋊C₄×C₁₉	—	2	2	no	—
113	C₄⋊C₄×C₁₉	—	2	2	no	—
114	C₄⋊C₄×C₁₉	—	2	2	no	—
115	C₂²⋊C₄×C₁₉	—	4	2	no	—
116	C₂²⋊C₄×C₁₉	—	4	2	no	—
117	C₂²⋊C₄×C₁₉	—	4	2	no	—
118	C₂²⋊C₄×C₁₉	—	4	2	no	—
119	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
123	#19689	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂³×C₁₉	—	4	2	no	—
125	C₂³×C₁₉	—	4	2	no	—
126	C₂³×C₁₉	#127	1	1	yes	characteristic
127	Q₈×C₁₉	—	4	4	no	—
128	Q₈×C₁₉	—	4	4	no	—
129	D₄×C₁₉	—	4	4	no	—
130	D₄×C₁₉	—	4	4	no	—
131	D₄×C₁₉	—	4	4	no	—
132	D₄×C₁₉	—	4	4	no	—
133	D₄×C₁₉	—	4	4	no	—
134	D₄×C₁₉	—	4	4	no	—
135	D₄×C₁₉	—	8	4	no	—
136	D₄×C₁₉	—	8	4	no	—
137	C₂×C₄×C₁₉	—	2	2	no	—
138	C₂×C₄×C₁₉	—	2	2	no	—
139	C₂×C₄×C₁₉	—	2	2	no	—
140	C₂×C₄×C₁₉	—	4	2	no	—
141	C₂×C₄×C₁₉	—	4	2	no	—
142	C₂×C₄×C₁₉	—	8	4	no	—
143	C₂×C₄×C₁₉	—	8	4	no	—
144	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
145	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
146	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
147	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
150	C₈×C₁₉	—	2	2	no	—
151	C₈×C₁₉	—	2	4	no	—
152	C₈×C₁₉	—	2	4	no	—
153	C₈×C₁₉	AGL(1,ℤ/8)	1	4	yes	—
154	#3207	C₂×C₁₉	1	2	yes	complemented
155	#3184	C₂×C₁₉	1	2	yes	complemented
156	#3169	C₂×C₁₉	1	1	yes	characteristic, complemented
157	#3147	C₂×C₁₉	1	1	yes	characteristic, complemented
158	#1721	C₂×C₁₉	1	2	yes	complemented
159	#1721	C₂×C₁₉	1	2	yes	complemented
160	#1720	C₂×C₁₉	1	2	yes	complemented
161	#1714	C₂×C₁₉	1	2	yes	complemented
162	#1451	C₂×C₁₉	1	1	yes	characteristic, complemented
163	C₂²×C₁₉	—	2	1	no	—
164	C₂²×C₁₉	—	2	1	no	—
165	C₂²×C₁₉	—	2	1	no	—
166	C₂²×C₁₉	—	8	2	no	—
167	C₂²×C₁₉	—	8	2	no	—
168	C₂²×C₁₉	—	8	2	no	—
169	C₂²×C₁₉	—	8	2	no	—
170	C₂²×C₁₉	—	8	2	no	—
171	C₂²×C₁₉	—	8	2	no	—
172	C₂²×C₁₉	C₂×#127	1	1	yes	characteristic
173	C₄×C₁₉	—	2	1	no	—
174	C₄×C₁₉	—	2	1	no	—
175	C₄×C₁₉	—	2	1	no	—
176	C₄×C₁₉	—	8	2	no	—
177	C₄×C₁₉	—	8	2	no	—
178	C₄×C₁₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
179	C₄×C₁₉	#575	1	2	yes	—
180	C₄×C₁₉	#576	1	2	yes	—
181	C₂×#135	—	2	2	no	—
182	#532	C₂²×C₁₉	1	2	yes	—
183	#532	C₂²×C₁₉	1	2	yes	—
184	#500	C₂²×C₁₉	1	1	yes	characteristic
185	#498	C₂²×C₁₉	1	2	yes	—
186	#495	—	2	2	no	—
187	#494	C₂²×C₁₉	1	1	yes	characteristic
188	#493	C₂²×C₁₉	1	1	yes	characteristic
189	#493	C₂²×C₁₉	1	1	yes	characteristic
190	#493	C₂²×C₁₉	1	2	yes	—
191	#493	C₂²×C₁₉	1	2	yes	—
192	#492	C₂²×C₁₉	1	2	yes	—
193	#492	C₂²×C₁₉	1	2	yes	—
194	#487	C₂²×C₁₉	1	2	yes	—
195	#486	C₂²×C₁₉	1	2	yes	—
196	#486	C₂²×C₁₉	1	2	yes	—
197	#486	C₂²×C₁₉	1	2	yes	—
198	#486	C₂²×C₁₉	1	2	yes	—
199	#469	—	2	4	no	—
200	#466	—	2	2	no	—
201	#463	—	2	4	no	—
202	#449	—	2	4	no	—
203	#442	—	2	4	no	—
204	C₄×D₈	—	2	2	no	—
205	#431	C₂²×C₁₉	1	1	yes	characteristic
206	C₄×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
207	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
208	#322	C₂²×C₁₉	1	2	yes	—
209	#322	C₂²×C₁₉	1	2	yes	—
210	C₂×C₁₉	—	4	1	no	—
211	C₂×C₁₉	—	16	2	no	—
212	C₂×C₁₉	—	16	2	no	—
213	C₂×C₁₉	#3150	1	1	yes	characteristic
214	C₂×C₁₉	#3151	1	1	yes	characteristic
215	C₂×C₁₉	#3155	1	1	yes	characteristic
216	C₂×#41	—	2	2	no	—
217	C₂×#41	—	2	2	no	—
218	#135	—	4	8	no	—
219	C₂×Q₁₆	—	2	2	no	—
220	C₂×SD₁₆	—	2	4	no	—
221	C₂×SD₁₆	—	2	4	no	—
222	C₂×D₈	—	2	2	no	—
223	C₂×D₈	—	2	2	no	—
224	C₂×D₈	—	2	4	no	—
225	C₂×D₈	—	2	4	no	—
226	C₂×M₁₆	D₄×C₁₉	1	2	yes	—
227	C₂²×C₈	D₄×C₁₉	1	2	yes	—
228	C₄⋊Q₈	C₂³×C₁₉	1	2	yes	—
229	C₄⋊Q₈	C₂³×C₁₉	1	2	yes	—
230	#127	C₂³×C₁₉	1	2	yes	—
231	#127	C₂³×C₁₉	1	2	yes	—
232	#124	—	2	2	no	—
233	#124	—	2	2	no	—
234	#121	—	2	2	no	—
235	#121	—	2	2	no	—
236	#121	—	2	2	no	—
237	#121	—	2	2	no	—
238	C₄×D₄	—	2	2	no	—
239	C₄×D₄	—	2	2	no	—
240	C₄×D₄	—	2	2	no	—
241	C₄×D₄	—	2	2	no	—
242	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
243	#107	—	2	2	no	—
244	#106	—	2	4	no	—
245	C₄⋊C₈	D₄×C₁₉	1	2	yes	—
246	#103	—	2	4	no	—
247	#102	—	2	4	no	—
248	#102	—	2	4	no	—
249	#102	—	2	4	no	—
250	#102	—	2	4	no	—
251	C₂²⋊C₈	D₄×C₁₉	1	2	yes	—
252	#97	D₄×C₁₉	1	2	yes	—
253	C₄×C₈	D₄×C₁₉	1	2	yes	—
254	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
255	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
256	C₄×C₈	C₂³×C₁₉	1	2	yes	—
257	C₄×C₈	C₂³×C₁₉	1	2	yes	—
258	C₁₉	#19689	1	1	yes	characteristic, direct product factor, Sylow
259	#41	—	4	8	no	—
260	#41	—	8	4	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	2	2	no	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	—	2	2	no	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	—	4	2	no	—
270	C₂×D₄	—	4	2	no	—
271	C₂²×C₄	—	4	2	no	—
272	C₂²×C₄	—	4	2	no	—
273	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
274	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
275	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
276	Q₁₆	—	4	4	no	—
277	SD₁₆	—	4	8	no	—
278	SD₁₆	—	8	4	no	—
279	D₈	—	4	8	no	—
280	D₈	—	4	4	no	—
281	D₈	—	4	4	no	—
282	D₈	—	8	4	no	—
283	M₁₆	—	4	2	no	—
284	C₂×C₈	—	2	4	no	—
285	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
286	C₂×C₈	C₂×D₄×C₁₉	1	4	yes	—
287	C₂×C₈	C₂×D₄×C₁₉	1	4	yes	—
288	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₂²⋊C₄	—	4	2	no	—
294	C₂²⋊C₄	—	4	2	no	—
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	4	2	no	—
297	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
298	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
299	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
300	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
301	C₂³	—	4	2	no	—
302	C₂³	—	4	2	no	—
303	C₂³	C₁₉×#127	1	1	yes	characteristic
304	Q₈	—	4	4	no	—
305	Q₈	—	4	4	no	—
306	D₄	—	4	4	no	—
307	D₄	—	4	4	no	—
308	D₄	—	4	4	no	—
309	D₄	—	4	4	no	—
310	D₄	—	4	4	no	—
311	D₄	—	4	4	no	—
312	D₄	—	8	4	no	—
313	D₄	—	8	4	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	8	4	no	—
320	C₂×C₄	—	8	4	no	—
321	C₂×C₄	C₁₉×#127	1	1	yes	characteristic
322	C₂×C₄	C₁₉×#127	1	1	yes	characteristic
323	C₂×C₄	C₁₉×#127	1	1	yes	characteristic
324	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
325	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
326	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
327	C₈	—	2	2	no	—
328	C₈	—	2	4	no	—
329	C₈	—	2	4	no	—
330	C₈	C₁₉×AGL(1,ℤ/8)	1	4	yes	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	8	2	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	8	2	no	—
339	C₂²	—	8	2	no	—
340	C₂²	C₂×C₁₉×#127	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	8	2	no	—
345	C₄	—	8	2	no	—
346	C₄	C₂×C₁₉×AGL(1,ℤ/8)	1	2	yes	—
347	C₄	C₁₉×#575	1	2	yes	—
348	C₄	C₁₉×#576	1	2	yes	—
349	C₂	—	4	1	no	—
350	C₂	—	16	2	no	—
351	C₂	—	16	2	no	—
352	C₂	C₁₉×#3150	1	1	yes	characteristic
353	C₂	C₁₉×#3151	1	1	yes	characteristic
354	C₂	C₁₉×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₉
LCS	G ▷ 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×#3207	C₂	semidirect
2. C₁₉×#3184	C₂	semidirect
3. C₁₉×#3169	C₂	semidirect
4. C₁₉×#3147	C₂	semidirect
5. C₁₉×#1721	C₂	semidirect
6. C₁₉×#1721	C₂	semidirect
7. C₁₉×#1720	C₂	semidirect
8. C₁₉×#1714	C₂	semidirect
9. C₁₉×#1451	C₂	semidirect
123. #19689	C₁₉	direct (N × H)
154. #3207	C₂×C₁₉	semidirect
155. #3184	C₂×C₁₉	semidirect
156. #3169	C₂×C₁₉	semidirect
157. #3147	C₂×C₁₉	semidirect
158. #1721	C₂×C₁₉	semidirect
159. #1721	C₂×C₁₉	semidirect
160. #1720	C₂×C₁₉	semidirect
161. #1714	C₂×C₁₉	semidirect
162. #1451	C₂×C₁₉	semidirect
258. C₁₉	#19689	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×#3207	2
C₁₉×#3184	2
C₁₉×#3169	2
C₁₉×#3147	2
C₁₉×#1721	2
C₁₉×#1720	2
C₁₉×#1714	2
C₁₉×#1451	2
#19689	19

Subgroups — C19×#19689 #18025648

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#19689 #18025648