Subgroups — C₁₉×#10273 #18016232

Order 4864.

542 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#498	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#477	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#462	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1984	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₉×#333	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₉×#332	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1277	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₁₆×C₁₉	—	2	1	no	—
9	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
12	C₁₉×#498	—	2	2	no	—
13	C₁₉×#498	—	2	2	no	—
14	C₁₉×#477	—	2	2	no	—
15	C₁₉×#477	—	2	2	no	—
16	C₁₉×#462	—	2	2	no	—
17	C₁₉×#462	—	2	2	no	—
18	C₂×C₁₉×#107	—	2	1	no	—
19	C₂×C₁₉×#106	—	2	2	no	—
20	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₁₉×#103	—	2	1	no	—
24	C₂×C₁₉×#103	—	2	1	no	—
25	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
26	C₁₉×#336	—	2	2	no	—
27	C₁₉×#333	—	2	2	no	—
28	C₁₉×#332	—	2	2	no	—
29	C₂²×Q₈×C₁₉	—	2	1	no	—
30	C₂²×Q₈×C₁₉	—	2	1	no	—
31	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
32	C₂×Q₁₆×C₁₉	—	2	4	no	—
33	C₂×Q₁₆×C₁₉	—	4	2	no	—
34	C₂×Q₁₆×C₁₉	—	4	2	no	—
35	C₂²×C₈×C₁₉	—	2	1	no	—
36	C₂²×C₈×C₁₉	—	2	1	no	—
37	C₂²×C₈×C₁₉	—	2	1	no	—
38	C₂²×C₈×C₁₉	D₄	1	2	yes	—
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	—	2	1	no	—
43	C₁₉×C₄⋊Q₈	—	2	1	no	—
44	C₁₉×C₄⋊Q₈	—	2	1	no	—
45	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
46	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
47	C₄×Q₈×C₁₉	—	2	1	no	—
48	C₄×Q₈×C₁₉	—	2	1	no	—
49	C₄×Q₈×C₁₉	—	2	1	no	—
50	C₄×Q₈×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×C₄⋊C₄×C₁₉	—	2	1	no	—
55	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
56	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₁₉	—	2	1	no	—
58	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
59	C₁₉×#107	—	2	4	no	—
60	C₁₉×#106	—	2	8	no	—
61	C₁₉×C₄⋊C₈	—	2	1	no	—
62	C₁₉×C₄⋊C₈	—	2	1	no	—
63	C₁₉×C₄⋊C₈	—	2	1	no	—
64	C₁₉×C₄⋊C₈	—	2	1	no	—
65	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
66	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
67	C₁₉×#103	—	2	4	no	—
68	C₁₉×#103	—	2	4	no	—
69	C₄×C₈×C₁₉	—	2	1	no	—
70	C₄×C₈×C₁₉	—	2	1	no	—
71	C₂×Q₈×C₁₉	—	2	1	no	—
72	C₂×Q₈×C₁₉	—	2	1	no	—
73	C₂×Q₈×C₁₉	—	2	1	no	—
74	C₂×Q₈×C₁₉	—	2	2	no	—
75	C₂×Q₈×C₁₉	—	2	2	no	—
76	C₂×Q₈×C₁₉	—	2	1	no	—
77	C₂×Q₈×C₁₉	—	2	1	no	—
78	C₂×Q₈×C₁₉	—	4	1	no	—
79	C₂×Q₈×C₁₉	—	4	1	no	—
80	C₂×Q₈×C₁₉	—	4	1	no	—
81	C₂×Q₈×C₁₉	—	4	1	no	—
82	C₂×Q₈×C₁₉	—	4	1	no	—
83	C₂×Q₈×C₁₉	—	4	1	no	—
84	C₂×Q₈×C₁₉	—	8	2	no	—
85	C₂×Q₈×C₁₉	SD₁₆	1	2	yes	—
86	C₂²×C₄×C₁₉	—	2	1	no	—
87	C₂²×C₄×C₁₉	—	2	1	no	—
88	C₂²×C₄×C₁₉	—	4	1	no	—
89	C₂²×C₄×C₁₉	—	4	1	no	—
90	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
93	Q₁₆×C₁₉	—	4	8	no	—
94	C₂×C₈×C₁₉	—	2	2	no	—
95	C₂×C₈×C₁₉	—	2	2	no	—
96	C₂×C₈×C₁₉	—	2	2	no	—
97	C₂×C₈×C₁₉	—	2	2	no	—
98	C₂×C₈×C₁₉	—	4	1	no	—
99	C₂×C₈×C₁₉	—	4	1	no	—
100	C₂×C₈×C₁₉	—	4	1	no	—
101	C₂×C₈×C₁₉	—	4	1	no	—
102	C₂×C₈×C₁₉	—	4	1	no	—
103	C₂×C₈×C₁₉	—	4	2	no	—
104	C₂×C₈×C₁₉	Q₁₆	1	4	yes	complemented
105	C₄⋊C₄×C₁₉	—	2	2	no	—
106	C₄⋊C₄×C₁₉	—	2	2	no	—
107	C₄⋊C₄×C₁₉	—	2	2	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	—	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	C₄⋊C₄×C₁₉	—	4	1	no	—
117	C₄²×C₁₉	—	4	1	no	—
118	C₄²×C₁₉	—	4	1	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	C₂×D₄	1	1	yes	characteristic
123	#10273	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂³×C₁₉	#121	1	1	yes	characteristic
125	Q₈×C₁₉	—	2	2	no	—
126	Q₈×C₁₉	—	2	2	no	—
127	Q₈×C₁₉	—	4	1	no	—
128	Q₈×C₁₉	—	4	1	no	—
129	Q₈×C₁₉	—	4	2	no	—
130	Q₈×C₁₉	—	4	2	no	—
131	Q₈×C₁₉	—	8	2	no	—
132	Q₈×C₁₉	—	8	2	no	—
133	Q₈×C₁₉	—	8	1	no	—
134	Q₈×C₁₉	—	8	1	no	—
135	C₂×C₄×C₁₉	—	2	2	no	—
136	C₂×C₄×C₁₉	—	2	2	no	—
137	C₂×C₄×C₁₉	—	4	1	no	—
138	C₂×C₄×C₁₉	—	4	1	no	—
139	C₂×C₄×C₁₉	—	4	1	no	—
140	C₂×C₄×C₁₉	—	4	2	no	—
141	C₂×C₄×C₁₉	—	4	2	no	—
142	C₂×C₄×C₁₉	—	4	2	no	—
143	C₂×C₄×C₁₉	—	8	1	no	—
144	C₂×C₄×C₁₉	—	8	1	no	—
145	C₂×C₄×C₁₉	—	8	2	no	—
146	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
147	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
148	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
149	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
150	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
151	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
152	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
153	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
154	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
155	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
156	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
157	C₂×C₄×C₁₉	#137	1	2	yes	—
158	C₈×C₁₉	—	2	4	no	—
159	C₈×C₁₉	—	4	2	no	—
160	C₈×C₁₉	—	4	2	no	—
161	C₈×C₁₉	—	8	1	no	—
162	C₂×#498	C₂×C₁₉	1	1	yes	characteristic
163	C₂×#477	C₂×C₁₉	1	1	yes	characteristic
164	C₂×#462	C₂×C₁₉	1	1	yes	characteristic
165	#1984	C₂×C₁₉	1	1	yes	characteristic
166	C₂×#333	C₂×C₁₉	1	1	yes	characteristic
167	C₂×#332	C₂×C₁₉	1	1	yes	characteristic
168	#1277	C₂×C₁₉	1	1	yes	characteristic
169	C₂²×C₁₉	#392	1	1	yes	characteristic
170	C₂²×C₁₉	#462	1	2	yes	—
171	C₂²×C₁₉	#465	1	1	yes	characteristic
172	C₂²×C₁₉	#468	1	1	yes	characteristic
173	C₂²×C₁₉	#471	1	2	yes	—
174	C₄×C₁₉	—	2	1	no	—
175	C₄×C₁₉	—	2	1	no	—
176	C₄×C₁₉	—	4	2	no	—
177	C₄×C₁₉	—	8	2	no	—
178	C₄×C₁₉	—	8	1	no	—
179	C₄×C₁₉	—	8	2	no	—
180	C₄×C₁₉	#448	1	1	yes	characteristic
181	C₄×C₁₉	#449	1	1	yes	characteristic
182	C₄×C₁₉	#450	1	1	yes	characteristic
183	C₄×C₁₉	#451	1	2	yes	—
184	C₄×C₁₉	#452	1	1	yes	characteristic
185	C₄×C₁₉	#494	1	1	yes	characteristic
186	C₄×C₁₉	#497	1	1	yes	characteristic
187	C₂²×Q₁₆	—	2	1	no	—
188	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
189	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
190	C₂×C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
191	#498	—	2	2	no	—
192	#498	—	2	2	no	—
193	#477	—	2	2	no	—
194	#477	—	2	2	no	—
195	#462	—	2	2	no	—
196	#462	—	2	2	no	—
197	C₂×#107	—	2	1	no	—
198	C₂×#106	—	2	2	no	—
199	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
200	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
201	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
202	C₂×#103	—	2	1	no	—
203	C₂×#103	—	2	1	no	—
204	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
205	#336	—	2	2	no	—
206	#333	—	2	2	no	—
207	#332	—	2	2	no	—
208	C₂×C₁₉	#1672	1	2	yes	—
209	C₂×C₁₉	#1692	1	2	yes	—
210	C₂×C₁₉	#2000	1	1	yes	characteristic
211	C₂×C₁₉	#2016	1	1	yes	characteristic
212	C₂×C₁₉	#2019	1	1	yes	characteristic
213	C₂²×Q₈	—	2	1	no	—
214	C₂²×Q₈	—	2	1	no	—
215	C₂²×Q₈	D₄×C₁₉	1	1	yes	characteristic
216	C₂×Q₁₆	—	2	4	no	—
217	C₂×Q₁₆	—	4	2	no	—
218	C₂×Q₁₆	—	4	2	no	—
219	C₂²×C₈	—	2	1	no	—
220	C₂²×C₈	—	2	1	no	—
221	C₂²×C₈	—	2	1	no	—
222	C₂²×C₈	D₄×C₁₉	1	2	yes	—
223	C₄⋊Q₈	—	2	1	no	—
224	C₄⋊Q₈	—	2	1	no	—
225	C₄⋊Q₈	—	2	1	no	—
226	C₄⋊Q₈	—	2	1	no	—
227	C₄⋊Q₈	—	2	1	no	—
228	C₄⋊Q₈	—	2	1	no	—
229	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
230	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
231	C₄×Q₈	—	2	1	no	—
232	C₄×Q₈	—	2	1	no	—
233	C₄×Q₈	—	2	1	no	—
234	C₄×Q₈	—	2	1	no	—
235	C₂×C₄⋊C₄	—	2	1	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	—	2	1	no	—
238	C₂×C₄⋊C₄	—	2	1	no	—
239	C₂×C₄⋊C₄	—	2	1	no	—
240	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
241	C₂×C₄²	—	2	1	no	—
242	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
243	#107	—	2	4	no	—
244	#106	—	2	8	no	—
245	C₄⋊C₈	—	2	1	no	—
246	C₄⋊C₈	—	2	1	no	—
247	C₄⋊C₈	—	2	1	no	—
248	C₄⋊C₈	—	2	1	no	—
249	C₄⋊C₈	D₄×C₁₉	1	2	yes	—
250	C₄⋊C₈	D₄×C₁₉	1	2	yes	—
251	#103	—	2	4	no	—
252	#103	—	2	4	no	—
253	C₄×C₈	—	2	1	no	—
254	C₄×C₈	—	2	1	no	—
255	C₁₉	#10273	1	1	yes	characteristic, direct product factor, Sylow
256	C₂×Q₈	—	2	1	no	—
257	C₂×Q₈	—	2	1	no	—
258	C₂×Q₈	—	2	1	no	—
259	C₂×Q₈	—	2	2	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	2	1	no	—
262	C₂×Q₈	—	2	1	no	—
263	C₂×Q₈	—	4	1	no	—
264	C₂×Q₈	—	4	1	no	—
265	C₂×Q₈	—	4	1	no	—
266	C₂×Q₈	—	4	1	no	—
267	C₂×Q₈	—	4	1	no	—
268	C₂×Q₈	—	4	1	no	—
269	C₂×Q₈	—	8	2	no	—
270	C₂×Q₈	SD₁₆×C₁₉	1	2	yes	—
271	C₂²×C₄	—	2	1	no	—
272	C₂²×C₄	—	2	1	no	—
273	C₂²×C₄	—	4	1	no	—
274	C₂²×C₄	—	4	1	no	—
275	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
276	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
277	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
278	Q₁₆	—	4	8	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	4	1	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	4	1	no	—
286	C₂×C₈	—	4	1	no	—
287	C₂×C₈	—	4	1	no	—
288	C₂×C₈	—	4	2	no	—
289	C₂×C₈	Q₁₆×C₁₉	1	4	yes	complemented
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	1	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	4	1	no	—
297	C₄⋊C₄	—	4	1	no	—
298	C₄⋊C₄	—	4	1	no	—
299	C₄⋊C₄	—	4	1	no	—
300	C₄⋊C₄	—	4	1	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄²	—	4	1	no	—
303	C₄²	—	4	1	no	—
304	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
305	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
306	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
307	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
308	C₂³	C₁₉×#121	1	1	yes	characteristic
309	Q₈	—	2	2	no	—
310	Q₈	—	2	2	no	—
311	Q₈	—	4	1	no	—
312	Q₈	—	4	1	no	—
313	Q₈	—	4	2	no	—
314	Q₈	—	4	2	no	—
315	Q₈	—	8	2	no	—
316	Q₈	—	8	2	no	—
317	Q₈	—	8	1	no	—
318	Q₈	—	8	1	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	8	1	no	—
328	C₂×C₄	—	8	1	no	—
329	C₂×C₄	—	8	2	no	—
330	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
331	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
332	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
333	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
334	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
335	C₂×C₄	C₁₉×#127	1	1	yes	characteristic
336	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
337	C₂×C₄	C₂×Q₁₆×C₁₉	1	2	yes	—
338	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
339	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
340	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
341	C₂×C₄	C₁₉×#137	1	2	yes	—
342	C₈	—	2	4	no	—
343	C₈	—	4	2	no	—
344	C₈	—	4	2	no	—
345	C₈	—	8	1	no	—
346	C₂²	C₁₉×#392	1	1	yes	characteristic
347	C₂²	C₁₉×#462	1	2	yes	—
348	C₂²	C₁₉×#465	1	1	yes	characteristic
349	C₂²	C₁₉×#468	1	1	yes	characteristic
350	C₂²	C₁₉×#471	1	2	yes	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	4	2	no	—
354	C₄	—	8	2	no	—
355	C₄	—	8	1	no	—
356	C₄	—	8	2	no	—
357	C₄	C₁₉×#448	1	1	yes	characteristic
358	C₄	C₁₉×#449	1	1	yes	characteristic
359	C₄	C₁₉×#450	1	1	yes	characteristic
360	C₄	C₁₉×#451	1	2	yes	—
361	C₄	C₁₉×#452	1	1	yes	characteristic
362	C₄	C₁₉×#494	1	1	yes	characteristic
363	C₄	C₁₉×#497	1	1	yes	characteristic
364	C₂	C₁₉×#1672	1	2	yes	—
365	C₂	C₁₉×#1692	1	2	yes	—
366	C₂	C₁₉×#2000	1	1	yes	characteristic
367	C₂	C₁₉×#2016	1	1	yes	characteristic
368	C₂	C₁₉×#2019	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₉	#121
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
104. C₂×C₈×C₁₉	Q₁₆	semidirect
123. #10273	C₁₉	direct (N × H)
255. C₁₉	#10273	direct (N × H)
289. C₂×C₈	Q₁₆×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₉×#498	2
C₂×C₁₉×#477	2
C₂×C₁₉×#462	2
C₁₉×#1984	2
C₂×C₁₉×#333	2
C₂×C₁₉×#332	2
C₁₉×#1277	2
#10273	19

Subgroups — C19×#10273 #18016232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#10273 #18016232