Subgroups — C₁₉×#9280 #18015239

Order 4864.

382 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3231	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2985	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1506	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1506	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1497	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1287	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1274	C₂	1	1	yes	maximal, characteristic
8	C₁₉×#533	C₂²	1	1	yes	characteristic
9	C₁₉×#533	C₂²	1	1	yes	characteristic
10	C₁₉×#500	C₄	1	1	yes	characteristic, complemented
11	C₁₉×#498	C₄	1	1	yes	characteristic, complemented
12	C₁₉×#487	C₄	1	1	yes	characteristic, complemented
13	C₁₉×#487	C₄	1	1	yes	characteristic, complemented
14	C₁₉×#485	—	2	2	no	—
15	C₁₉×#479	—	2	2	no	—
16	C₁₉×#446	—	2	1	no	—
17	C₁₉×#446	—	2	2	no	—
18	C₈×Q₈×C₁₉	—	2	1	no	—
19	C₁₉×#433	—	2	2	no	—
20	C₁₉×#432	C₂²	1	1	yes	characteristic
21	C₁₉×#432	C₂²	1	1	yes	characteristic
22	C₁₉×#431	C₂²	1	1	yes	characteristic
23	C₁₉×#431	C₂²	1	1	yes	characteristic
24	C₁₉×#376	C₂²	1	1	yes	characteristic
25	C₁₉×#328	—	2	2	no	—
26	C₁₉×#328	—	2	2	no	—
27	C₁₉×Q₈⋊C₈	—	2	1	no	—
28	C₁₉×Q₈⋊C₈	—	2	1	no	—
29	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
30	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
31	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
32	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
33	C₁₉×#125	—	2	1	no	—
34	C₁₉×#125	—	2	1	no	—
35	C₁₉×C₂²⋊Q₈	—	2	1	no	—
36	C₁₉×C₂²⋊Q₈	—	2	1	no	—
37	C₁₉×C₂²⋊Q₈	—	2	1	no	—
38	C₁₉×C₂²⋊Q₈	D₄	1	2	yes	—
39	C₄×Q₈×C₁₉	—	2	1	no	—
40	C₄×Q₈×C₁₉	—	2	1	no	—
41	C₄×Q₈×C₁₉	—	2	1	no	—
42	C₄×Q₈×C₁₉	D₄	1	2	yes	—
43	C₁₉×#117	—	2	1	no	—
44	C₁₉×#117	—	2	1	no	—
45	C₂×C₄²×C₁₉	—	2	1	no	—
46	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
47	C₁₉×#107	—	2	2	no	—
48	C₁₉×#106	—	2	2	no	—
49	C₁₉×C₄⋊C₈	—	2	2	no	—
50	C₁₉×C₄⋊C₈	—	2	2	no	—
51	C₁₉×C₄⋊C₈	—	2	1	no	—
52	C₁₉×C₄⋊C₈	—	2	1	no	—
53	C₁₉×C₄⋊C₈	—	2	1	no	—
54	C₁₉×C₄⋊C₈	—	2	1	no	—
55	C₁₉×#103	—	2	2	no	—
56	C₁₉×#103	—	2	2	no	—
57	C₁₉×C₂²⋊C₈	—	2	1	no	—
58	C₁₉×C₂²⋊C₈	—	2	1	no	—
59	C₁₉×C₂²⋊C₈	—	2	1	no	—
60	C₁₉×C₂²⋊C₈	D₄	1	2	yes	—
61	C₁₉×#97	—	2	2	no	—
62	C₁₉×#97	—	2	1	no	—
63	C₁₉×#97	—	2	1	no	—
64	C₄×C₈×C₁₉	—	2	2	no	—
65	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
68	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
69	C₁₉×#95	—	2	1	no	—
70	C₁₉×#95	—	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	C₂²⋊C₄	1	2	yes	—
75	C₂²×C₄×C₁₉	—	4	1	no	—
76	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
79	C₂×C₈×C₁₉	—	2	1	no	—
80	C₂×C₈×C₁₉	—	2	1	no	—
81	C₂×C₈×C₁₉	—	2	1	no	—
82	C₂×C₈×C₁₉	—	2	2	no	—
83	C₂×C₈×C₁₉	—	2	1	no	—
84	C₂×C₈×C₁₉	—	4	1	no	—
85	C₄⋊C₄×C₁₉	—	2	1	no	—
86	C₄⋊C₄×C₁₉	—	2	1	no	—
87	C₄⋊C₄×C₁₉	—	2	1	no	—
88	C₄⋊C₄×C₁₉	—	2	2	no	—
89	C₄⋊C₄×C₁₉	—	2	2	no	—
90	C₄⋊C₄×C₁₉	—	2	1	no	—
91	C₄⋊C₄×C₁₉	—	2	1	no	—
92	C₄⋊C₄×C₁₉	—	2	1	no	—
93	C₄⋊C₄×C₁₉	—	2	1	no	—
94	C₄⋊C₄×C₁₉	—	4	1	no	—
95	C₄⋊C₄×C₁₉	—	4	1	no	—
96	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
97	C₂²⋊C₄×C₁₉	—	2	2	no	—
98	C₂²⋊C₄×C₁₉	—	4	1	no	—
99	C₄²×C₁₉	—	2	2	no	—
100	C₄²×C₁₉	—	2	2	no	—
101	C₄²×C₁₉	—	4	1	no	—
102	C₄²×C₁₉	—	4	1	no	—
103	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₁₉	#41	1	1	yes	characteristic
107	#9280	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂³×C₁₉	#121	1	1	yes	characteristic
109	Q₈×C₁₉	—	4	2	no	—
110	Q₈×C₁₉	—	4	1	no	—
111	Q₈×C₁₉	—	4	1	no	—
112	Q₈×C₁₉	—	4	1	no	—
113	Q₈×C₁₉	—	4	1	no	—
114	Q₈×C₁₉	—	8	1	no	—
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	2	2	no	—
118	C₂×C₄×C₁₉	—	2	2	no	—
119	C₂×C₄×C₁₉	—	4	1	no	—
120	C₂×C₄×C₁₉	—	4	1	no	—
121	C₂×C₄×C₁₉	—	4	2	no	—
122	C₂×C₄×C₁₉	—	4	1	no	—
123	C₂×C₄×C₁₉	—	4	1	no	—
124	C₂×C₄×C₁₉	—	8	1	no	—
125	C₂×C₄×C₁₉	—	8	1	no	—
126	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
130	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
131	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
132	C₈×C₁₉	—	4	1	no	—
133	C₈×C₁₉	—	4	2	no	—
134	C₈×C₁₉	—	4	1	no	—
135	C₈×C₁₉	—	4	1	no	—
136	C₈×C₁₉	—	4	1	no	—
137	C₈×C₁₉	—	8	1	no	—
138	#3231	C₂×C₁₉	1	1	yes	characteristic
139	#2985	C₂×C₁₉	1	1	yes	characteristic
140	#1506	C₂×C₁₉	1	1	yes	characteristic
141	#1506	C₂×C₁₉	1	1	yes	characteristic
142	#1497	C₂×C₁₉	1	1	yes	characteristic
143	#1287	C₂×C₁₉	1	1	yes	characteristic
144	#1274	C₂×C₁₉	1	1	yes	characteristic
145	C₂²×C₁₉	—	2	1	no	—
146	C₂²×C₁₉	—	2	1	no	—
147	C₂²×C₁₉	—	2	1	no	—
148	C₂²×C₁₉	#386	1	1	yes	characteristic
149	C₄×C₁₉	—	2	1	no	—
150	C₄×C₁₉	—	2	1	no	—
151	C₄×C₁₉	—	2	1	no	—
152	C₄×C₁₉	—	2	1	no	—
153	C₄×C₁₉	—	2	1	no	—
154	C₄×C₁₉	—	4	2	no	—
155	C₄×C₁₉	—	4	2	no	—
156	C₄×C₁₉	—	8	2	no	—
157	C₄×C₁₉	—	8	1	no	—
158	C₄×C₁₉	—	8	1	no	—
159	C₄×C₁₉	#443	1	1	yes	characteristic
160	C₄×C₁₉	#444	1	1	yes	characteristic
161	#533	C₂²×C₁₉	1	1	yes	characteristic
162	#533	C₂²×C₁₉	1	1	yes	characteristic
163	#500	C₄×C₁₉	1	1	yes	characteristic, complemented
164	#498	C₄×C₁₉	1	1	yes	characteristic, complemented
165	#487	C₄×C₁₉	1	1	yes	characteristic, complemented
166	#487	C₄×C₁₉	1	1	yes	characteristic, complemented
167	#485	—	2	2	no	—
168	#479	—	2	2	no	—
169	#446	—	2	2	no	—
170	#446	—	2	1	no	—
171	C₈×Q₈	—	2	1	no	—
172	#433	—	2	2	no	—
173	#432	C₂²×C₁₉	1	1	yes	characteristic
174	#432	C₂²×C₁₉	1	1	yes	characteristic
175	#431	C₂²×C₁₉	1	1	yes	characteristic
176	#431	C₂²×C₁₉	1	1	yes	characteristic
177	#376	C₂²×C₁₉	1	1	yes	characteristic
178	#328	—	2	2	no	—
179	#328	—	2	2	no	—
180	Q₈⋊C₈	—	2	1	no	—
181	Q₈⋊C₈	—	2	1	no	—
182	C₂×C₁₉	—	4	1	no	—
183	C₂×C₁₉	#1876	1	1	yes	characteristic
184	C₂×C₁₉	#1905	1	1	yes	characteristic
185	C₂×C₁₉	#1909	1	1	yes	characteristic
186	C₄⋊Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
187	C₄⋊Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
188	C₄⋊Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
189	C₄⋊Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
190	#125	—	2	1	no	—
191	#125	—	2	1	no	—
192	C₂²⋊Q₈	—	2	1	no	—
193	C₂²⋊Q₈	—	2	1	no	—
194	C₂²⋊Q₈	—	2	1	no	—
195	C₂²⋊Q₈	D₄×C₁₉	1	2	yes	—
196	C₄×Q₈	—	2	1	no	—
197	C₄×Q₈	—	2	1	no	—
198	C₄×Q₈	—	2	1	no	—
199	C₄×Q₈	D₄×C₁₉	1	2	yes	—
200	#117	—	2	1	no	—
201	#117	—	2	1	no	—
202	C₂×C₄²	—	2	1	no	—
203	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
204	#107	—	2	2	no	—
205	#106	—	2	2	no	—
206	C₄⋊C₈	—	2	2	no	—
207	C₄⋊C₈	—	2	2	no	—
208	C₄⋊C₈	—	2	1	no	—
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	—	2	1	no	—
211	C₄⋊C₈	—	2	1	no	—
212	#103	—	2	2	no	—
213	#103	—	2	2	no	—
214	C₂²⋊C₈	—	2	1	no	—
215	C₂²⋊C₈	—	2	1	no	—
216	C₂²⋊C₈	—	2	1	no	—
217	C₂²⋊C₈	D₄×C₁₉	1	2	yes	—
218	#97	—	2	2	no	—
219	#97	—	2	1	no	—
220	#97	—	2	1	no	—
221	C₄×C₈	—	2	2	no	—
222	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
223	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
224	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
225	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
226	#95	—	2	1	no	—
227	#95	—	2	1	no	—
228	C₁₉	#9280	1	1	yes	characteristic, direct product factor, Sylow
229	C₂×Q₈	—	2	1	no	—
230	C₂×Q₈	—	2	1	no	—
231	C₂×Q₈	—	2	1	no	—
232	C₂×Q₈	C₂²⋊C₄×C₁₉	1	2	yes	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
235	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
236	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	4	1	no	—
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	4	1	no	—
253	C₄⋊C₄	—	4	1	no	—
254	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
255	C₂²⋊C₄	—	2	2	no	—
256	C₂²⋊C₄	—	4	1	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	4	1	no	—
260	C₄²	—	4	1	no	—
261	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
262	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
263	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
264	C₄²	#41×C₁₉	1	1	yes	characteristic
265	C₂³	C₁₉×#121	1	1	yes	characteristic
266	Q₈	—	4	2	no	—
267	Q₈	—	4	1	no	—
268	Q₈	—	4	1	no	—
269	Q₈	—	4	1	no	—
270	Q₈	—	4	1	no	—
271	Q₈	—	8	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	8	1	no	—
282	C₂×C₄	—	8	1	no	—
283	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
284	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
285	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
286	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
287	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
288	C₂×C₄	C₁₉×#123	1	1	yes	characteristic
289	C₈	—	4	1	no	—
290	C₈	—	4	2	no	—
291	C₈	—	4	1	no	—
292	C₈	—	4	1	no	—
293	C₈	—	4	1	no	—
294	C₈	—	8	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	C₁₉×#386	1	1	yes	characteristic
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	4	2	no	—
305	C₄	—	4	2	no	—
306	C₄	—	8	2	no	—
307	C₄	—	8	1	no	—
308	C₄	—	8	1	no	—
309	C₄	C₁₉×#443	1	1	yes	characteristic
310	C₄	C₁₉×#444	1	1	yes	characteristic
311	C₂	—	4	1	no	—
312	C₂	C₁₉×#1876	1	1	yes	characteristic
313	C₂	C₁₉×#1905	1	1	yes	characteristic
314	C₂	C₁₉×#1909	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#386
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
prime radical	C₂³×C₁₉	#121
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
10. C₁₉×#500	C₄	semidirect
11. C₁₉×#498	C₄	semidirect
12. C₁₉×#487	C₄	semidirect
13. C₁₉×#487	C₄	semidirect
107. #9280	C₁₉	direct (N × H)
163. #500	C₄×C₁₉	semidirect
164. #498	C₄×C₁₉	semidirect
165. #487	C₄×C₁₉	semidirect
166. #487	C₄×C₁₉	semidirect
228. C₁₉	#9280	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×#3231	2
C₁₉×#2985	2
C₁₉×#1506	2
C₁₉×#1497	2
C₁₉×#1287	2
C₁₉×#1274	2
#9280	19

Subgroups — C19×#9280 #18015239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#9280 #18015239