Subgroups — C₁₉×#9415 #18015374

Order 4864.

420 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3182	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2980	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1537	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1515	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1514	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1291	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1288	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₉×#533	C₂²	1	1	yes	characteristic
9	C₁₉×#532	C₂²	1	1	yes	characteristic
10	C₁₉×#500	C₄	1	1	yes	characteristic, complemented
11	C₁₉×#492	C₄	1	1	yes	characteristic, complemented
12	C₁₉×#487	C₄	1	1	yes	characteristic, complemented
13	C₁₉×#486	C₄	1	1	yes	characteristic, complemented
14	C₁₉×#468	—	2	2	no	—
15	C₁₉×#440	—	2	2	no	—
16	C₁₉×#435	—	2	2	no	—
17	C₁₉×#435	—	2	2	no	—
18	C₁₉×#432	C₂²	1	1	yes	characteristic
19	C₁₉×#432	C₂²	1	1	yes	characteristic
20	C₁₉×#432	C₂²	1	1	yes	characteristic
21	C₁₉×#408	—	2	2	no	—
22	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
23	C₁₉×#383	C₂²	1	1	yes	characteristic
24	C₁₉×#330	—	2	2	no	—
25	C₁₉×#328	—	2	2	no	—
26	C₁₉×#327	—	2	2	no	—
27	C₂×#41×C₁₉	—	2	1	no	—
28	C₂×SD₁₆×C₁₉	—	2	2	no	—
29	C₂×M₁₆×C₁₉	—	2	1	no	—
30	C₂×M₁₆×C₁₉	—	2	2	no	—
31	C₂²×C₈×C₁₉	—	2	2	no	—
32	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
33	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
34	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
35	C₁₉×#127	C₂×C₄	1	1	yes	characteristic
36	C₁₉×#125	—	2	1	no	—
37	C₁₉×#124	—	2	1	no	—
38	C₁₉×C₂²⋊Q₈	—	2	1	no	—
39	C₁₉×C₂²⋊Q₈	—	2	1	no	—
40	C₁₉×#121	—	2	1	no	—
41	C₁₉×#121	D₄	1	2	yes	—
42	C₄×Q₈×C₁₉	—	2	1	no	—
43	C₄×Q₈×C₁₉	—	2	1	no	—
44	C₄×D₄×C₁₉	—	2	1	no	—
45	C₄×D₄×C₁₉	D₄	1	2	yes	—
46	C₁₉×#117	—	2	1	no	—
47	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
48	C₁₉×#107	—	2	2	no	—
49	C₁₉×C₄⋊C₈	—	2	1	no	—
50	C₁₉×C₄⋊C₈	—	2	1	no	—
51	C₁₉×C₄⋊C₈	—	2	1	no	—
52	C₁₉×#103	—	2	2	no	—
53	C₁₉×#102	—	2	2	no	—
54	C₁₉×C₂²⋊C₈	—	2	4	no	—
55	C₁₉×C₂²⋊C₈	—	2	1	no	—
56	C₁₉×C₂²⋊C₈	—	2	1	no	—
57	C₁₉×C₂²⋊C₈	D₄	1	2	yes	—
58	C₁₉×#97	—	2	1	no	—
59	C₁₉×#97	—	2	1	no	—
60	C₁₉×#97	—	2	1	no	—
61	C₁₉×#97	D₄	1	2	yes	—
62	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
64	C₁₉×#95	—	2	1	no	—
65	C₁₉×#95	—	2	1	no	—
66	C₁₉×#95	—	2	1	no	—
67	#41×C₁₉	—	8	1	no	—
68	#41×C₁₉	—	8	1	no	—
69	C₂×Q₈×C₁₉	—	2	1	no	—
70	C₂×Q₈×C₁₉	—	2	1	no	—
71	C₂×Q₈×C₁₉	—	2	1	no	—
72	C₂×D₄×C₁₉	—	2	2	no	—
73	C₂×D₄×C₁₉	—	2	1	no	—
74	C₂×D₄×C₁₉	—	2	1	no	—
75	C₂×D₄×C₁₉	C₂²⋊C₄	1	2	yes	—
76	C₂²×C₄×C₁₉	—	2	2	no	—
77	C₂²×C₄×C₁₉	—	4	1	no	—
78	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
81	SD₁₆×C₁₉	—	4	4	no	—
82	M₁₆×C₁₉	—	4	2	no	—
83	M₁₆×C₁₉	—	8	2	no	—
84	C₂×C₈×C₁₉	—	2	1	no	—
85	C₂×C₈×C₁₉	—	2	1	no	—
86	C₂×C₈×C₁₉	—	2	2	no	—
87	C₂×C₈×C₁₉	—	2	2	no	—
88	C₂×C₈×C₁₉	—	4	1	no	—
89	C₂×C₈×C₁₉	—	8	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	—	2	1	no	—
95	C₄⋊C₄×C₁₉	—	4	1	no	—
96	C₄⋊C₄×C₁₉	—	4	1	no	—
97	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
98	C₂²⋊C₄×C₁₉	—	2	2	no	—
99	C₂²⋊C₄×C₁₉	—	2	2	no	—
100	C₂²⋊C₄×C₁₉	—	4	1	no	—
101	C₄²×C₁₉	—	4	1	no	—
102	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₁₉	#41	1	1	yes	characteristic
106	#9415	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂³×C₁₉	—	2	2	no	—
108	C₂³×C₁₉	#121	1	1	yes	characteristic
109	Q₈×C₁₉	—	4	2	no	—
110	Q₈×C₁₉	—	8	1	no	—
111	Q₈×C₁₉	—	8	1	no	—
112	D₄×C₁₉	—	4	2	no	—
113	D₄×C₁₉	—	8	1	no	—
114	D₄×C₁₉	—	8	1	no	—
115	D₄×C₁₉	—	8	2	no	—
116	C₂×C₄×C₁₉	—	2	2	no	—
117	C₂×C₄×C₁₉	—	4	1	no	—
118	C₂×C₄×C₁₉	—	4	1	no	—
119	C₂×C₄×C₁₉	—	4	1	no	—
120	C₂×C₄×C₁₉	—	4	2	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	—	8	1	no	—
127	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
131	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
132	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
133	C₈×C₁₉	—	2	2	no	—
134	C₈×C₁₉	—	4	2	no	—
135	C₈×C₁₉	—	4	2	no	—
136	C₈×C₁₉	—	4	1	no	—
137	C₈×C₁₉	—	8	1	no	—
138	#3182	C₂×C₁₉	1	1	yes	characteristic
139	#2980	C₂×C₁₉	1	1	yes	characteristic
140	#1537	C₂×C₁₉	1	1	yes	characteristic, complemented
141	#1515	C₂×C₁₉	1	1	yes	characteristic, complemented
142	#1514	C₂×C₁₉	1	1	yes	characteristic
143	#1291	C₂×C₁₉	1	1	yes	characteristic, complemented
144	#1288	C₂×C₁₉	1	1	yes	characteristic, complemented
145	C₂²×C₁₉	—	2	1	no	—
146	C₂²×C₁₉	—	2	1	no	—
147	C₂²×C₁₉	—	2	1	no	—
148	C₂²×C₁₉	—	4	2	no	—
149	C₂²×C₁₉	—	4	2	no	—
150	C₂²×C₁₉	—	4	2	no	—
151	C₂²×C₁₉	#386	1	1	yes	characteristic
152	C₄×C₁₉	—	2	1	no	—
153	C₄×C₁₉	—	2	1	no	—
154	C₄×C₁₉	—	2	1	no	—
155	C₄×C₁₉	—	2	1	no	—
156	C₄×C₁₉	—	2	1	no	—
157	C₄×C₁₉	—	2	1	no	—
158	C₄×C₁₉	—	4	2	no	—
159	C₄×C₁₉	—	8	1	no	—
160	C₄×C₁₉	—	8	1	no	—
161	C₄×C₁₉	—	16	1	no	—
162	#533	C₂²×C₁₉	1	1	yes	characteristic
163	#532	C₂²×C₁₉	1	1	yes	characteristic
164	#500	C₄×C₁₉	1	1	yes	characteristic, complemented
165	#492	C₄×C₁₉	1	1	yes	characteristic, complemented
166	#487	C₄×C₁₉	1	1	yes	characteristic, complemented
167	#486	C₄×C₁₉	1	1	yes	characteristic, complemented
168	#468	—	2	2	no	—
169	#440	—	2	2	no	—
170	#435	—	2	2	no	—
171	#435	—	2	2	no	—
172	#432	C₂²×C₁₉	1	1	yes	characteristic
173	#432	C₂²×C₁₉	1	1	yes	characteristic
174	#432	C₂²×C₁₉	1	1	yes	characteristic
175	#408	—	2	2	no	—
176	C₄×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
177	#383	C₂²×C₁₉	1	1	yes	characteristic
178	#330	—	2	2	no	—
179	#328	—	2	2	no	—
180	#327	—	2	2	no	—
181	C₂×C₁₉	—	4	1	no	—
182	C₂×C₁₉	—	8	2	no	—
183	C₂×C₁₉	#1884	1	1	yes	characteristic
184	C₂×C₁₉	#1902	1	1	yes	characteristic
185	C₂×C₁₉	#1904	1	1	yes	characteristic
186	C₂×#41	—	2	1	no	—
187	C₂×SD₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	1	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂²×C₈	—	2	2	no	—
191	C₄⋊Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
192	C₄⋊Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
193	C₄⋊Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
194	#127	C₂×C₄×C₁₉	1	1	yes	characteristic
195	#125	—	2	1	no	—
196	#124	—	2	1	no	—
197	C₂²⋊Q₈	—	2	1	no	—
198	C₂²⋊Q₈	—	2	1	no	—
199	#121	—	2	1	no	—
200	#121	D₄×C₁₉	1	2	yes	—
201	C₄×Q₈	—	2	1	no	—
202	C₄×Q₈	—	2	1	no	—
203	C₄×D₄	—	2	1	no	—
204	C₄×D₄	D₄×C₁₉	1	2	yes	—
205	#117	—	2	1	no	—
206	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
207	#107	—	2	2	no	—
208	C₄⋊C₈	—	2	1	no	—
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	—	2	1	no	—
211	#103	—	2	2	no	—
212	#102	—	2	2	no	—
213	C₂²⋊C₈	—	2	4	no	—
214	C₂²⋊C₈	—	2	1	no	—
215	C₂²⋊C₈	—	2	1	no	—
216	C₂²⋊C₈	D₄×C₁₉	1	2	yes	—
217	#97	—	2	1	no	—
218	#97	—	2	1	no	—
219	#97	—	2	1	no	—
220	#97	D₄×C₁₉	1	2	yes	—
221	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
222	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
223	#95	—	2	1	no	—
224	#95	—	2	1	no	—
225	#95	—	2	1	no	—
226	C₁₉	#9415	1	1	yes	characteristic, direct product factor, Sylow
227	#41	—	8	1	no	—
228	#41	—	8	1	no	—
229	C₂×Q₈	—	2	1	no	—
230	C₂×Q₈	—	2	1	no	—
231	C₂×Q₈	—	2	1	no	—
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	—	2	1	no	—
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	C₂²⋊C₄×C₁₉	1	2	yes	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
240	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
241	SD₁₆	—	4	4	no	—
242	M₁₆	—	4	2	no	—
243	M₁₆	—	8	2	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	8	2	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
258	C₂²⋊C₄	—	2	2	no	—
259	C₂²⋊C₄	—	2	2	no	—
260	C₂²⋊C₄	—	4	1	no	—
261	C₄²	—	4	1	no	—
262	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
263	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
264	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
265	C₄²	#41×C₁₉	1	1	yes	characteristic
266	C₂³	—	2	2	no	—
267	C₂³	C₁₉×#121	1	1	yes	characteristic
268	Q₈	—	4	2	no	—
269	Q₈	—	8	1	no	—
270	Q₈	—	8	1	no	—
271	D₄	—	4	2	no	—
272	D₄	—	8	1	no	—
273	D₄	—	8	1	no	—
274	D₄	—	8	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	8	1	no	—
284	C₂×C₄	—	8	1	no	—
285	C₂×C₄	—	8	1	no	—
286	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
287	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
288	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
289	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
290	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
291	C₂×C₄	C₁₉×#123	1	1	yes	characteristic
292	C₈	—	2	2	no	—
293	C₈	—	4	2	no	—
294	C₈	—	4	2	no	—
295	C₈	—	4	1	no	—
296	C₈	—	8	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	4	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	C₁₉×#386	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	4	2	no	—
311	C₄	—	8	1	no	—
312	C₄	—	8	1	no	—
313	C₄	—	16	1	no	—
314	C₂	—	4	1	no	—
315	C₂	—	8	2	no	—
316	C₂	C₁₉×#1884	1	1	yes	characteristic
317	C₂	C₁₉×#1902	1	1	yes	characteristic
318	C₂	C₁₉×#1904	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	C₁₉×#532	C₂²
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
3. C₁₉×#1537	C₂	semidirect
4. C₁₉×#1515	C₂	semidirect
6. C₁₉×#1291	C₂	semidirect
7. C₁₉×#1288	C₂	semidirect
10. C₁₉×#500	C₄	semidirect
11. C₁₉×#492	C₄	semidirect
12. C₁₉×#487	C₄	semidirect
13. C₁₉×#486	C₄	semidirect
106. #9415	C₁₉	direct (N × H)
140. #1537	C₂×C₁₉	semidirect
141. #1515	C₂×C₁₉	semidirect
143. #1291	C₂×C₁₉	semidirect
144. #1288	C₂×C₁₉	semidirect
164. #500	C₄×C₁₉	semidirect
165. #492	C₄×C₁₉	semidirect
166. #487	C₄×C₁₉	semidirect
167. #486	C₄×C₁₉	semidirect
226. C₁₉	#9415	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×#3182	2
C₁₉×#2980	2
C₁₉×#1537	2
C₁₉×#1515	2
C₁₉×#1514	2
C₁₉×#1291	2
C₁₉×#1288	2
#9415	19

Subgroups — C19×#9415 #18015374

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#9415 #18015374