Subgroups — C₁₉×#9269 #18015228

Order 4864.

480 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3227	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2935	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1997	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1547	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₉×#332	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#1301	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1283	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₉×#131	—	2	1	no	—
9	C₁₉×#532	C₂²	1	1	yes	characteristic
10	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₁₉×#498	—	2	1	no	—
12	C₁₉×#486	—	2	1	no	—
13	C₁₉×#481	C₄	1	4	yes	—
14	C₁₉×#476	—	2	2	no	—
15	C₁₉×#436	—	2	1	no	—
16	C₁₉×#435	—	2	2	no	—
17	C₈×D₄×C₁₉	—	2	1	no	—
18	C₁₉×#431	C₂²	1	1	yes	characteristic
19	C₁₉×#423	C₂²	1	1	yes	characteristic
20	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₁₉×#408	—	2	1	no	—
23	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
24	C₁₉×#343	—	2	1	no	—
25	C₁₉×#343	—	2	1	no	—
26	C₁₉×#332	—	2	1	no	—
27	C₁₉×#332	—	2	1	no	—
28	C₁₉×#332	—	2	1	no	—
29	C₁₉×#331	—	2	1	no	—
30	C₂×#41×C₁₉	—	2	1	no	—
31	C₂²×Q₈×C₁₉	—	2	1	no	—
32	C₁₉×#131	—	4	2	no	—
33	C₁₉×#131	—	4	2	no	—
34	C₂×M₁₆×C₁₉	—	2	2	no	—
35	C₂×M₁₆×C₁₉	—	2	1	no	—
36	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
37	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₉	—	2	2	no	—
39	C₂²×C₈×C₁₉	—	2	1	no	—
40	C₂²×C₈×C₁₉	—	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	1	yes	characteristic
46	C₁₉×#127	D₄	1	1	yes	characteristic
47	C₁₉×#124	—	2	1	no	—
48	C₁₉×#121	—	2	1	no	—
49	C₁₉×#121	C₂×C₄	1	2	yes	—
50	C₄×D₄×C₁₉	—	2	1	no	—
51	C₄×D₄×C₁₉	D₄	1	2	yes	—
52	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
53	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
55	C₁₉×#106	—	2	4	no	—
56	C₁₉×C₄⋊C₈	—	2	1	no	—
57	C₁₉×C₄⋊C₈	—	2	1	no	—
58	C₁₉×C₄⋊C₈	—	2	1	no	—
59	C₁₉×C₄⋊C₈	—	2	1	no	—
60	C₁₉×C₄⋊C₈	—	2	1	no	—
61	C₁₉×C₄⋊C₈	—	2	1	no	—
62	C₁₉×C₄⋊C₈	D₄	1	1	yes	characteristic
63	C₁₉×C₄⋊C₈	D₄	1	1	yes	characteristic
64	C₁₉×#102	—	2	4	no	—
65	C₁₉×C₂²⋊C₈	—	2	2	no	—
66	C₁₉×C₂²⋊C₈	—	2	2	no	—
67	C₁₉×C₂²⋊C₈	C₂×C₄	1	2	yes	—
68	C₁₉×#97	—	2	1	no	—
69	C₄×C₈×C₁₉	—	2	1	no	—
70	C₄×C₈×C₁₉	—	2	1	no	—
71	#41×C₁₉	—	4	1	no	—
72	#41×C₁₉	—	4	1	no	—
73	#41×C₁₉	—	4	1	no	—
74	#41×C₁₉	—	4	1	no	—
75	C₂×Q₈×C₁₉	—	2	1	no	—
76	C₂×Q₈×C₁₉	—	4	1	no	—
77	C₂×Q₈×C₁₉	—	4	1	no	—
78	C₂×Q₈×C₁₉	—	8	1	no	—
79	C₂×Q₈×C₁₉	—	8	1	no	—
80	C₂×D₄×C₁₉	—	2	2	no	—
81	C₂×D₄×C₁₉	—	2	1	no	—
82	C₂×D₄×C₁₉	—	2	1	no	—
83	C₂×D₄×C₁₉	C₂²⋊C₄	1	2	yes	—
84	C₂²×C₄×C₁₉	—	2	2	no	—
85	C₂²×C₄×C₁₉	—	2	2	no	—
86	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
89	M₁₆×C₁₉	—	2	1	no	—
90	M₁₆×C₁₉	—	2	1	no	—
91	M₁₆×C₁₉	—	2	1	no	—
92	M₁₆×C₁₉	—	4	4	no	—
93	M₁₆×C₁₉	—	4	1	no	—
94	M₁₆×C₁₉	—	4	1	no	—
95	M₁₆×C₁₉	SD₁₆	1	2	yes	—
96	C₂×C₈×C₁₉	—	2	1	no	—
97	C₂×C₈×C₁₉	—	2	2	no	—
98	C₂×C₈×C₁₉	—	2	1	no	—
99	C₂×C₈×C₁₉	—	2	1	no	—
100	C₂×C₈×C₁₉	—	2	1	no	—
101	C₂×C₈×C₁₉	—	4	4	no	—
102	C₂×C₈×C₁₉	—	4	1	no	—
103	C₂×C₈×C₁₉	—	4	1	no	—
104	C₂×C₈×C₁₉	—	4	1	no	—
105	C₂×C₈×C₁₉	—	4	1	no	—
106	C₂×C₈×C₁₉	—	8	1	no	—
107	C₄⋊C₄×C₁₉	—	2	1	no	—
108	C₄⋊C₄×C₁₉	—	2	2	no	—
109	C₄⋊C₄×C₁₉	—	2	2	no	—
110	C₄⋊C₄×C₁₉	—	4	1	no	—
111	C₄⋊C₄×C₁₉	—	4	1	no	—
112	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
113	C₂²⋊C₄×C₁₉	—	2	2	no	—
114	C₂²⋊C₄×C₁₉	—	2	2	no	—
115	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₁₉	#41	1	1	yes	characteristic
119	#9269	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂³×C₁₉	—	2	2	no	—
121	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
122	Q₈×C₁₉	—	4	1	no	—
123	Q₈×C₁₉	—	4	1	no	—
124	Q₈×C₁₉	—	8	2	no	—
125	Q₈×C₁₉	—	8	1	no	—
126	Q₈×C₁₉	—	8	1	no	—
127	D₄×C₁₉	—	4	2	no	—
128	D₄×C₁₉	—	4	2	no	—
129	D₄×C₁₉	—	4	2	no	—
130	D₄×C₁₉	—	4	1	no	—
131	D₄×C₁₉	—	4	1	no	—
132	D₄×C₁₉	—	4	1	no	—
133	D₄×C₁₉	—	4	1	no	—
134	C₂×C₄×C₁₉	—	2	1	no	—
135	C₂×C₄×C₁₉	—	2	1	no	—
136	C₂×C₄×C₁₉	—	2	2	no	—
137	C₂×C₄×C₁₉	—	4	1	no	—
138	C₂×C₄×C₁₉	—	4	2	no	—
139	C₂×C₄×C₁₉	—	4	2	no	—
140	C₂×C₄×C₁₉	—	4	2	no	—
141	C₂×C₄×C₁₉	—	8	2	no	—
142	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
145	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
146	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
147	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
148	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
149	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
150	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
152	C₈×C₁₉	—	2	2	no	—
153	C₈×C₁₉	—	4	2	no	—
154	C₈×C₁₉	—	4	1	no	—
155	C₈×C₁₉	—	4	1	no	—
156	C₈×C₁₉	—	4	1	no	—
157	C₈×C₁₉	—	8	1	no	—
158	C₈×C₁₉	—	8	1	no	—
159	#3227	C₂×C₁₉	1	1	yes	characteristic
160	#2935	C₂×C₁₉	1	1	yes	characteristic
161	#1997	C₂×C₁₉	1	1	yes	characteristic, complemented
162	#1547	C₂×C₁₉	1	1	yes	characteristic
163	C₂×#332	C₂×C₁₉	1	1	yes	characteristic, complemented
164	#1301	C₂×C₁₉	1	1	yes	characteristic, complemented
165	#1283	C₂×C₁₉	1	1	yes	characteristic, complemented
166	C₂²×C₁₉	—	2	1	no	—
167	C₂²×C₁₉	—	2	1	no	—
168	C₂²×C₁₉	—	4	2	no	—
169	C₂²×C₁₉	—	4	2	no	—
170	C₂²×C₁₉	—	4	2	no	—
171	C₂²×C₁₉	#386	1	1	yes	characteristic
172	C₂²×C₁₉	C₄×SD₁₆	1	1	yes	characteristic
173	C₂²×C₁₉	#440	1	1	yes	characteristic
174	C₄×C₁₉	—	2	1	no	—
175	C₄×C₁₉	—	2	1	no	—
176	C₄×C₁₉	—	2	1	no	—
177	C₄×C₁₉	—	4	2	no	—
178	C₄×C₁₉	—	8	2	no	—
179	C₄×C₁₉	#450	1	1	yes	characteristic
180	C₄×C₁₉	#452	1	1	yes	characteristic
181	C₄×C₁₉	#461	1	1	yes	characteristic
182	C₄×C₁₉	#463	1	1	yes	characteristic
183	C₄×C₁₉	#471	1	1	yes	characteristic
184	C₄×C₁₉	#472	1	1	yes	characteristic
185	C₂×#131	—	2	1	no	—
186	#532	C₂²×C₁₉	1	1	yes	characteristic
187	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
188	#498	—	2	1	no	—
189	#486	—	2	1	no	—
190	#481	C₄×C₁₉	1	4	yes	—
191	#476	—	2	2	no	—
192	#436	—	2	1	no	—
193	#435	—	2	2	no	—
194	C₈×D₄	—	2	1	no	—
195	#431	C₂²×C₁₉	1	1	yes	characteristic
196	#423	C₂²×C₁₉	1	1	yes	characteristic
197	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
198	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
199	#408	—	2	1	no	—
200	C₄×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
201	#343	—	2	1	no	—
202	#343	—	2	1	no	—
203	#332	—	2	1	no	—
204	#332	—	2	1	no	—
205	#332	—	2	1	no	—
206	#331	—	2	1	no	—
207	C₂×C₁₉	—	2	1	no	—
208	C₂×C₁₉	—	2	1	no	—
209	C₂×C₁₉	—	8	2	no	—
210	C₂×C₁₉	#1877	1	1	yes	characteristic
211	C₂×C₁₉	#1907	1	1	yes	characteristic
212	C₂×C₁₉	#1915	1	1	yes	characteristic
213	C₂×#41	—	2	1	no	—
214	C₂²×Q₈	—	2	1	no	—
215	#131	—	4	2	no	—
216	#131	—	4	2	no	—
217	C₂×M₁₆	—	2	2	no	—
218	C₂×M₁₆	—	2	1	no	—
219	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
220	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
221	C₂²×C₈	—	2	2	no	—
222	C₂²×C₈	—	2	1	no	—
223	C₂²×C₈	—	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	D₄×C₁₉	1	1	yes	characteristic
229	#127	D₄×C₁₉	1	1	yes	characteristic
230	#124	—	2	1	no	—
231	#121	—	2	1	no	—
232	#121	C₂×C₄×C₁₉	1	2	yes	—
233	C₄×D₄	—	2	1	no	—
234	C₄×D₄	D₄×C₁₉	1	2	yes	—
235	C₂×C₄⋊C₄	—	2	1	no	—
236	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
237	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
238	#106	—	2	4	no	—
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	—	2	1	no	—
241	C₄⋊C₈	—	2	1	no	—
242	C₄⋊C₈	—	2	1	no	—
243	C₄⋊C₈	—	2	1	no	—
244	C₄⋊C₈	—	2	1	no	—
245	C₄⋊C₈	D₄×C₁₉	1	1	yes	characteristic
246	C₄⋊C₈	D₄×C₁₉	1	1	yes	characteristic
247	#102	—	2	4	no	—
248	C₂²⋊C₈	—	2	2	no	—
249	C₂²⋊C₈	—	2	2	no	—
250	C₂²⋊C₈	C₂×C₄×C₁₉	1	2	yes	—
251	#97	—	2	1	no	—
252	C₄×C₈	—	2	1	no	—
253	C₄×C₈	—	2	1	no	—
254	C₁₉	#9269	1	1	yes	characteristic, direct product factor, Sylow
255	#41	—	4	1	no	—
256	#41	—	4	1	no	—
257	#41	—	4	1	no	—
258	#41	—	4	1	no	—
259	C₂×Q₈	—	2	1	no	—
260	C₂×Q₈	—	4	1	no	—
261	C₂×Q₈	—	4	1	no	—
262	C₂×Q₈	—	8	1	no	—
263	C₂×Q₈	—	8	1	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	2	1	no	—
266	C₂×D₄	—	2	1	no	—
267	C₂×D₄	C₂²⋊C₄×C₁₉	1	2	yes	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
271	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
272	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
273	M₁₆	—	2	1	no	—
274	M₁₆	—	2	1	no	—
275	M₁₆	—	2	1	no	—
276	M₁₆	—	4	4	no	—
277	M₁₆	—	4	1	no	—
278	M₁₆	—	4	1	no	—
279	M₁₆	SD₁₆×C₁₉	1	2	yes	—
280	C₂×C₈	—	2	1	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	4	4	no	—
286	C₂×C₈	—	4	1	no	—
287	C₂×C₈	—	4	1	no	—
288	C₂×C₈	—	4	1	no	—
289	C₂×C₈	—	4	1	no	—
290	C₂×C₈	—	8	1	no	—
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	4	1	no	—
295	C₄⋊C₄	—	4	1	no	—
296	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
297	C₂²⋊C₄	—	2	2	no	—
298	C₂²⋊C₄	—	2	2	no	—
299	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
300	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
301	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
302	C₄²	#41×C₁₉	1	1	yes	characteristic
303	C₂³	—	2	2	no	—
304	C₂³	C₄×D₄×C₁₉	1	1	yes	characteristic
305	Q₈	—	4	1	no	—
306	Q₈	—	4	1	no	—
307	Q₈	—	8	2	no	—
308	Q₈	—	8	1	no	—
309	Q₈	—	8	1	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	2	no	—
312	D₄	—	4	2	no	—
313	D₄	—	4	1	no	—
314	D₄	—	4	1	no	—
315	D₄	—	4	1	no	—
316	D₄	—	4	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	8	2	no	—
325	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
326	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
327	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
328	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
329	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
330	C₂×C₄	C₁₉×#123	1	1	yes	characteristic
331	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
332	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
333	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
334	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
335	C₈	—	2	2	no	—
336	C₈	—	4	2	no	—
337	C₈	—	4	1	no	—
338	C₈	—	4	1	no	—
339	C₈	—	4	1	no	—
340	C₈	—	8	1	no	—
341	C₈	—	8	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	4	2	no	—
346	C₂²	—	4	2	no	—
347	C₂²	C₁₉×#386	1	1	yes	characteristic
348	C₂²	C₄×SD₁₆×C₁₉	1	1	yes	characteristic
349	C₂²	C₁₉×#440	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	4	2	no	—
354	C₄	—	8	2	no	—
355	C₄	C₁₉×#450	1	1	yes	characteristic
356	C₄	C₁₉×#452	1	1	yes	characteristic
357	C₄	C₁₉×#461	1	1	yes	characteristic
358	C₄	C₁₉×#463	1	1	yes	characteristic
359	C₄	C₁₉×#471	1	1	yes	characteristic
360	C₄	C₁₉×#472	1	1	yes	characteristic
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	—	8	2	no	—
364	C₂	C₁₉×#1877	1	1	yes	characteristic
365	C₂	C₁₉×#1907	1	1	yes	characteristic
366	C₂	C₁₉×#1915	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#386
commutator	C₂²×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
prime radical	C₁₉×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ 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×#1997	C₂	semidirect
5. C₂×C₁₉×#332	C₂	semidirect
6. C₁₉×#1301	C₂	semidirect
7. C₁₉×#1283	C₂	semidirect
119. #9269	C₁₉	direct (N × H)
161. #1997	C₂×C₁₉	semidirect
163. C₂×#332	C₂×C₁₉	semidirect
164. #1301	C₂×C₁₉	semidirect
165. #1283	C₂×C₁₉	semidirect
254. C₁₉	#9269	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×#3227	2
C₁₉×#2935	2
C₁₉×#1997	2
C₁₉×#1547	2
C₂×C₁₉×#332	2
C₁₉×#1301	2
C₁₉×#1283	2
#9269	19

Subgroups — C19×#9269 #18015228

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#9269 #18015228