Subgroups — C₁₉×#10313 #18016272

Order 4864.

554 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	C₂×C₁₉×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#487	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#487	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1990	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1990	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1956	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1273	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₁₉×#500	—	2	2	no	—
12	C₁₉×#500	—	2	2	no	—
13	C₁₉×#487	—	2	2	no	—
14	C₁₉×#487	—	2	2	no	—
15	C₁₉×#487	—	2	2	no	—
16	C₁₉×#487	—	2	2	no	—
17	C₂×C₁₉×#107	—	2	2	no	—
18	C₂×C₁₉×#103	—	2	2	no	—
19	C₂×C₁₉×#103	—	2	2	no	—
20	C₂×C₁₉×#103	—	2	2	no	—
21	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
24	C₁₉×#384	C₂²	1	1	yes	characteristic
25	C₁₉×#340	—	2	2	no	—
26	C₁₉×#340	—	2	2	no	—
27	C₂²×Q₈×C₁₉	—	2	1	no	—
28	C₂²×Q₈×C₁₉	—	2	1	no	—
29	C₂²×Q₈×C₁₉	—	2	1	no	—
30	C₂²×C₈×C₁₉	D₄	1	2	yes	—
31	C₂²×C₈×C₁₉	D₄	1	2	yes	—
32	C₂²×C₈×C₁₉	D₄	1	2	yes	—
33	C₁₉×C₄⋊Q₈	—	2	1	no	—
34	C₁₉×C₄⋊Q₈	—	2	1	no	—
35	C₁₉×C₄⋊Q₈	—	2	1	no	—
36	C₁₉×C₄⋊Q₈	—	2	1	no	—
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	—	2	1	no	—
41	C₁₉×C₄⋊Q₈	—	2	1	no	—
42	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
43	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
44	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
46	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
47	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
48	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
49	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
50	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
51	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
52	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
53	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
54	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
55	C₁₉×#107	—	2	8	no	—
56	C₁₉×#103	—	2	8	no	—
57	C₁₉×#103	—	2	8	no	—
58	C₁₉×#103	—	4	4	no	—
59	C₄×C₈×C₁₉	—	2	1	no	—
60	C₄×C₈×C₁₉	—	2	1	no	—
61	C₄×C₈×C₁₉	—	2	1	no	—
62	C₄×C₈×C₁₉	—	2	1	no	—
63	C₄×C₈×C₁₉	—	2	1	no	—
64	C₄×C₈×C₁₉	—	2	1	no	—
65	C₂×Q₈×C₁₉	—	2	2	no	—
66	C₂×Q₈×C₁₉	—	2	2	no	—
67	C₂×Q₈×C₁₉	—	2	2	no	—
68	C₂×Q₈×C₁₉	—	4	1	no	—
69	C₂×Q₈×C₁₉	—	4	1	no	—
70	C₂×Q₈×C₁₉	—	4	1	no	—
71	C₂×Q₈×C₁₉	—	4	2	no	—
72	C₂×Q₈×C₁₉	—	4	2	no	—
73	C₂×Q₈×C₁₉	—	4	2	no	—
74	C₂×Q₈×C₁₉	—	4	2	no	—
75	C₂×Q₈×C₁₉	—	4	2	no	—
76	C₂×Q₈×C₁₉	—	4	2	no	—
77	C₂²×C₄×C₁₉	—	4	1	no	—
78	C₂²×C₄×C₁₉	—	4	1	no	—
79	C₂²×C₄×C₁₉	—	4	1	no	—
80	C₂²×C₄×C₁₉	—	4	1	no	—
81	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₉	—	2	2	no	—
85	C₂×C₈×C₁₉	—	2	2	no	—
86	C₂×C₈×C₁₉	—	2	2	no	—
87	C₂×C₈×C₁₉	—	2	2	no	—
88	C₂×C₈×C₁₉	—	2	2	no	—
89	C₂×C₈×C₁₉	—	2	2	no	—
90	C₂×C₈×C₁₉	—	2	2	no	—
91	C₂×C₈×C₁₉	—	2	2	no	—
92	C₂×C₈×C₁₉	Q₁₆	1	4	yes	—
93	C₄⋊C₄×C₁₉	—	2	2	no	—
94	C₄⋊C₄×C₁₉	—	2	2	no	—
95	C₄⋊C₄×C₁₉	—	2	2	no	—
96	C₄⋊C₄×C₁₉	—	2	2	no	—
97	C₄⋊C₄×C₁₉	—	2	2	no	—
98	C₄⋊C₄×C₁₉	—	2	2	no	—
99	C₄⋊C₄×C₁₉	—	4	1	no	—
100	C₄⋊C₄×C₁₉	—	4	1	no	—
101	C₄⋊C₄×C₁₉	—	4	1	no	—
102	C₄⋊C₄×C₁₉	—	4	1	no	—
103	C₄⋊C₄×C₁₉	—	4	1	no	—
104	C₄⋊C₄×C₁₉	—	4	1	no	—
105	C₄⋊C₄×C₁₉	—	4	2	no	—
106	C₄⋊C₄×C₁₉	—	4	2	no	—
107	C₄⋊C₄×C₁₉	—	4	2	no	—
108	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₁₉	#41	1	1	yes	characteristic
112	#10313	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂³×C₁₉	#127	1	1	yes	characteristic
114	Q₈×C₁₉	—	4	4	no	—
115	Q₈×C₁₉	—	4	4	no	—
116	Q₈×C₁₉	—	4	4	no	—
117	Q₈×C₁₉	—	8	2	no	—
118	Q₈×C₁₉	—	8	2	no	—
119	Q₈×C₁₉	—	8	2	no	—
120	C₂×C₄×C₁₉	—	4	2	no	—
121	C₂×C₄×C₁₉	—	4	2	no	—
122	C₂×C₄×C₁₉	—	4	2	no	—
123	C₂×C₄×C₁₉	—	8	1	no	—
124	C₂×C₄×C₁₉	—	8	1	no	—
125	C₂×C₄×C₁₉	—	8	1	no	—
126	C₂×C₄×C₁₉	—	8	1	no	—
127	C₂×C₄×C₁₉	—	8	1	no	—
128	C₂×C₄×C₁₉	—	8	1	no	—
129	C₂×C₄×C₁₉	—	8	1	no	—
130	C₂×C₄×C₁₉	—	8	1	no	—
131	C₂×C₄×C₁₉	—	8	1	no	—
132	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
133	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
134	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
135	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
136	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
137	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
138	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
139	C₂×C₄×C₁₉	#135	1	2	yes	—
140	C₂×C₄×C₁₉	#135	1	2	yes	—
141	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
142	C₂×C₄×C₁₉	#137	1	2	yes	—
143	C₂×C₄×C₁₉	#137	1	2	yes	—
144	C₈×C₁₉	—	2	4	no	—
145	C₈×C₁₉	—	4	2	no	—
146	C₈×C₁₉	—	4	2	no	—
147	C₂×#500	C₂×C₁₉	1	1	yes	characteristic
148	C₂×#487	C₂×C₁₉	1	1	yes	characteristic
149	C₂×#487	C₂×C₁₉	1	1	yes	characteristic
150	#1990	C₂×C₁₉	1	1	yes	characteristic
151	#1990	C₂×C₁₉	1	1	yes	characteristic
152	#1956	C₂×C₁₉	1	1	yes	characteristic
153	#1273	C₂×C₁₉	1	1	yes	characteristic
154	C₂²×C₁₉	#392	1	1	yes	characteristic
155	C₂²×C₁₉	#494	1	1	yes	characteristic
156	C₂²×C₁₉	#495	1	1	yes	characteristic
157	C₂²×C₁₉	#495	1	1	yes	characteristic
158	C₂²×C₁₉	#496	1	1	yes	characteristic
159	C₂²×C₁₉	#497	1	1	yes	characteristic
160	C₂²×C₁₉	#497	1	1	yes	characteristic
161	C₄×C₁₉	—	8	2	no	—
162	C₄×C₁₉	—	8	2	no	—
163	C₄×C₁₉	—	8	2	no	—
164	C₄×C₁₉	—	16	1	no	—
165	C₄×C₁₉	#449	1	2	yes	—
166	C₄×C₁₉	#451	1	2	yes	—
167	C₄×C₁₉	#452	1	2	yes	—
168	C₄×C₁₉	#462	1	2	yes	—
169	C₄×C₁₉	#463	1	2	yes	—
170	C₄×C₁₉	#464	1	2	yes	—
171	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
172	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
173	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
174	#500	—	2	2	no	—
175	#500	—	2	2	no	—
176	#487	—	2	2	no	—
177	#487	—	2	2	no	—
178	#487	—	2	2	no	—
179	#487	—	2	2	no	—
180	C₂×#107	—	2	2	no	—
181	C₂×#103	—	2	2	no	—
182	C₂×#103	—	2	2	no	—
183	C₂×#103	—	2	2	no	—
184	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
185	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
186	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
187	#384	C₂²×C₁₉	1	1	yes	characteristic
188	#340	—	2	2	no	—
189	#340	—	2	2	no	—
190	C₂×C₁₉	#1710	1	1	yes	characteristic
191	C₂×C₁₉	#1718	1	1	yes	characteristic
192	C₂×C₁₉	#1719	1	1	yes	characteristic
193	C₂×C₁₉	#1723	1	1	yes	characteristic
194	C₂×C₁₉	#2012	1	1	yes	characteristic
195	C₂×C₁₉	#2015	1	1	yes	characteristic
196	C₂×C₁₉	#2016	1	1	yes	characteristic
197	C₂²×Q₈	—	2	1	no	—
198	C₂²×Q₈	—	2	1	no	—
199	C₂²×Q₈	—	2	1	no	—
200	C₂²×C₈	D₄×C₁₉	1	2	yes	—
201	C₂²×C₈	D₄×C₁₉	1	2	yes	—
202	C₂²×C₈	D₄×C₁₉	1	2	yes	—
203	C₄⋊Q₈	—	2	1	no	—
204	C₄⋊Q₈	—	2	1	no	—
205	C₄⋊Q₈	—	2	1	no	—
206	C₄⋊Q₈	—	2	1	no	—
207	C₄⋊Q₈	—	2	1	no	—
208	C₄⋊Q₈	—	2	1	no	—
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
213	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
214	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄⋊C₄	—	2	1	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
225	#107	—	2	8	no	—
226	#103	—	2	8	no	—
227	#103	—	2	8	no	—
228	#103	—	4	4	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	—	2	1	no	—
232	C₄×C₈	—	2	1	no	—
233	C₄×C₈	—	2	1	no	—
234	C₄×C₈	—	2	1	no	—
235	C₁₉	#10313	1	1	yes	characteristic, direct product factor, Sylow
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	—	4	2	no	—
243	C₂×Q₈	—	4	2	no	—
244	C₂×Q₈	—	4	2	no	—
245	C₂×Q₈	—	4	2	no	—
246	C₂×Q₈	—	4	2	no	—
247	C₂×Q₈	—	4	2	no	—
248	C₂²×C₄	—	4	1	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
254	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	Q₁₆×C₁₉	1	4	yes	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
280	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
281	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
282	C₄²	#41×C₁₉	1	1	yes	characteristic
283	C₂³	C₁₉×#127	1	1	yes	characteristic
284	Q₈	—	4	4	no	—
285	Q₈	—	4	4	no	—
286	Q₈	—	4	4	no	—
287	Q₈	—	8	2	no	—
288	Q₈	—	8	2	no	—
289	Q₈	—	8	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	8	1	no	—
294	C₂×C₄	—	8	1	no	—
295	C₂×C₄	—	8	1	no	—
296	C₂×C₄	—	8	1	no	—
297	C₂×C₄	—	8	1	no	—
298	C₂×C₄	—	8	1	no	—
299	C₂×C₄	—	8	1	no	—
300	C₂×C₄	—	8	1	no	—
301	C₂×C₄	—	8	1	no	—
302	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
303	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
304	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
305	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
306	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
307	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
308	C₂×C₄	C₂×Q₁₆×C₁₉	1	2	yes	—
309	C₂×C₄	C₁₉×#135	1	2	yes	—
310	C₂×C₄	C₁₉×#135	1	2	yes	—
311	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	2	yes	—
312	C₂×C₄	C₁₉×#137	1	2	yes	—
313	C₂×C₄	C₁₉×#137	1	2	yes	—
314	C₈	—	2	4	no	—
315	C₈	—	4	2	no	—
316	C₈	—	4	2	no	—
317	C₂²	C₁₉×#392	1	1	yes	characteristic
318	C₂²	C₁₉×#494	1	1	yes	characteristic
319	C₂²	C₁₉×#495	1	1	yes	characteristic
320	C₂²	C₁₉×#495	1	1	yes	characteristic
321	C₂²	C₁₉×#496	1	1	yes	characteristic
322	C₂²	C₁₉×#497	1	1	yes	characteristic
323	C₂²	C₁₉×#497	1	1	yes	characteristic
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	—	8	2	no	—
327	C₄	—	16	1	no	—
328	C₄	C₁₉×#449	1	2	yes	—
329	C₄	C₁₉×#451	1	2	yes	—
330	C₄	C₁₉×#452	1	2	yes	—
331	C₄	C₁₉×#462	1	2	yes	—
332	C₄	C₁₉×#463	1	2	yes	—
333	C₄	C₁₉×#464	1	2	yes	—
334	C₂	C₁₉×#1710	1	1	yes	characteristic
335	C₂	C₁₉×#1718	1	1	yes	characteristic
336	C₂	C₁₉×#1719	1	1	yes	characteristic
337	C₂	C₁₉×#1723	1	1	yes	characteristic
338	C₂	C₁₉×#2012	1	1	yes	characteristic
339	C₂	C₁₉×#2015	1	1	yes	characteristic
340	C₂	C₁₉×#2016	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₉	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
112. #10313	C₁₉	direct (N × H)
235. C₁₉	#10313	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₁₉×#500	2
C₂×C₁₉×#487	2
C₁₉×#1990	2
C₁₉×#1956	2
C₁₉×#1273	2
#10313	19

Subgroups — C19×#10313 #18016272

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#10313 #18016272