Subgroups — C₁₉×#12921 #18018880

Order 4864.

442 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#427	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#367	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₉×#366	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₉×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1808	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1380	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1380	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
9	C₂×C₁₉×#117	—	2	1	no	—
10	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
12	C₁₉×#427	—	2	2	no	—
13	C₁₉×#427	—	2	2	no	—
14	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
15	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
16	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
17	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
18	C₂×C₁₉×#106	—	2	1	no	—
19	C₁₉×#380	—	2	1	no	—
20	C₁₉×#367	—	2	2	no	—
21	C₁₉×#366	—	2	2	no	—
22	C₁₉×C₂²⋊C₁₆	C₄	1	4	yes	complemented
23	C₁₉×#340	—	2	1	no	—
24	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₉	—	2	1	no	—
26	C₂²×C₈×C₁₉	—	2	1	no	—
27	C₂²×C₈×C₁₉	—	2	1	no	—
28	C₂²×C₈×C₁₉	—	2	1	no	—
29	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₉	Q₈	1	2	yes	—
32	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
33	C₁₉×#117	—	2	4	no	—
34	C₁₉×#117	—	4	2	no	—
35	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
36	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
37	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
38	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
39	C₂×C₂²⋊C₄×C₁₉	—	4	1	no	—
40	C₂×C₂²⋊C₄×C₁₉	—	4	1	no	—
41	C₂×C₄²×C₁₉	—	4	1	no	—
42	C₂×C₁₆×C₁₉	—	2	1	no	—
43	C₂×C₁₆×C₁₉	—	2	1	no	—
44	C₂×C₁₆×C₁₉	—	2	1	no	—
45	C₂×C₁₆×C₁₉	—	2	1	no	—
46	C₂×C₁₆×C₁₉	C₂×C₄	1	2	yes	—
47	C₂×C₁₆×C₁₉	C₂×C₄	1	2	yes	—
48	C₁₉×#107	—	2	1	no	—
49	C₁₉×#107	—	2	1	no	—
50	C₁₉×#107	—	2	1	no	—
51	C₁₉×#107	—	2	1	no	—
52	C₁₉×#107	—	2	1	no	—
53	C₁₉×#107	—	2	1	no	—
54	C₁₉×#107	D₄	1	2	yes	—
55	C₁₉×#107	D₄	1	2	yes	—
56	C₁₉×#106	—	2	4	no	—
57	C₁₉×#95	—	4	1	no	—
58	C₂⁴×C₁₉	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	—	2	1	no	—
60	C₂²×C₄×C₁₉	—	2	1	no	—
61	C₂²×C₄×C₁₉	—	2	1	no	—
62	C₂²×C₄×C₁₉	—	2	1	no	—
63	C₂²×C₄×C₁₉	—	4	1	no	—
64	C₂²×C₄×C₁₉	—	4	1	no	—
65	C₂²×C₄×C₁₉	—	4	1	no	—
66	C₂²×C₄×C₁₉	—	4	1	no	—
67	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	Q₁₆	1	2	yes	—
70	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₁₉	—	2	1	no	—
72	C₂×C₈×C₁₉	—	2	1	no	—
73	C₂×C₈×C₁₉	—	2	2	no	—
74	C₂×C₈×C₁₉	—	2	2	no	—
75	C₂×C₈×C₁₉	—	2	2	no	—
76	C₂×C₈×C₁₉	—	2	2	no	—
77	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
78	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
82	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
83	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
84	C₄⋊C₄×C₁₉	—	2	2	no	—
85	C₄⋊C₄×C₁₉	—	2	2	no	—
86	C₄⋊C₄×C₁₉	—	4	1	no	—
87	C₄⋊C₄×C₁₉	—	4	1	no	—
88	C₄⋊C₄×C₁₉	—	4	1	no	—
89	C₄⋊C₄×C₁₉	—	4	1	no	—
90	C₄⋊C₄×C₁₉	—	4	1	no	—
91	C₄⋊C₄×C₁₉	—	4	1	no	—
92	C₂²⋊C₄×C₁₉	—	4	4	no	—
93	C₂²⋊C₄×C₁₉	—	8	2	no	—
94	C₄²×C₁₉	—	4	4	no	—
95	C₁₆×C₁₉	—	2	2	no	—
96	C₁₆×C₁₉	—	2	2	no	—
97	#12921	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂³×C₁₉	—	2	1	no	—
99	C₂³×C₁₉	—	2	1	no	—
100	C₂³×C₁₉	—	2	1	no	—
101	C₂³×C₁₉	—	2	1	no	—
102	C₂³×C₁₉	—	2	1	no	—
103	C₂³×C₁₉	—	2	1	no	—
104	C₂³×C₁₉	#107	1	2	yes	—
105	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	—	2	1	no	—
108	C₂×C₄×C₁₉	—	2	2	no	—
109	C₂×C₄×C₁₉	—	2	2	no	—
110	C₂×C₄×C₁₉	—	2	2	no	—
111	C₂×C₄×C₁₉	—	2	2	no	—
112	C₂×C₄×C₁₉	—	4	2	no	—
113	C₂×C₄×C₁₉	—	4	2	no	—
114	C₂×C₄×C₁₉	—	8	1	no	—
115	C₂×C₄×C₁₉	—	8	1	no	—
116	C₂×C₄×C₁₉	—	8	1	no	—
117	C₂×C₄×C₁₉	—	8	1	no	—
118	C₂×C₄×C₁₉	—	8	1	no	—
119	C₂×C₄×C₁₉	—	8	1	no	—
120	C₂×C₄×C₁₉	—	8	1	no	—
121	C₂×C₄×C₁₉	—	8	1	no	—
122	C₂×C₄×C₁₉	—	8	1	no	—
123	C₂×C₄×C₁₉	—	8	1	no	—
124	C₂×C₄×C₁₉	#107	1	2	yes	—
125	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
127	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
128	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
129	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
131	C₈×C₁₉	—	2	2	no	—
132	C₈×C₁₉	C₄×D₄	1	1	yes	characteristic
133	C₈×C₁₉	C₄×D₄	1	1	yes	characteristic
134	C₈×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
135	C₈×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
136	C₂×#427	C₂×C₁₉	1	1	yes	characteristic
137	C₂×#367	C₂×C₁₉	1	1	yes	characteristic, complemented
138	C₂×#366	C₂×C₁₉	1	1	yes	characteristic, complemented
139	C₂×C₂²⋊C₁₆	C₂×C₁₉	1	1	yes	characteristic
140	#1808	C₂×C₁₉	1	1	yes	characteristic
141	#1380	C₂×C₁₉	1	1	yes	characteristic, complemented
142	#1380	C₂×C₁₉	1	1	yes	characteristic, complemented
143	C₂²×C₁₉	—	2	2	no	—
144	C₂²×C₁₉	—	2	2	no	—
145	C₂²×C₁₉	—	2	2	no	—
146	C₂²×C₁₉	—	4	1	no	—
147	C₂²×C₁₉	—	4	1	no	—
148	C₂²×C₁₉	—	4	1	no	—
149	C₂²×C₁₉	—	4	1	no	—
150	C₂²×C₁₉	C₂×#107	1	1	yes	characteristic
151	C₂²×C₁₉	#447	1	1	yes	characteristic
152	C₂²×C₁₉	#480	1	1	yes	characteristic
153	C₂²×C₁₉	#508	1	1	yes	characteristic
154	C₂²×C₁₉	#508	1	1	yes	characteristic
155	C₂²×C₁₉	#509	1	1	yes	characteristic
156	C₂²×C₁₉	#510	1	1	yes	characteristic
157	C₄×C₁₉	—	2	2	no	—
158	C₄×C₁₉	—	8	2	no	—
159	C₄×C₁₉	—	8	2	no	—
160	C₄×C₁₉	—	16	1	no	—
161	C₄×C₁₉	—	16	1	no	—
162	C₄×C₁₉	#385	1	1	yes	characteristic
163	C₄×C₁₉	#428	1	1	yes	characteristic
164	C₄×C₁₉	#451	1	1	yes	characteristic
165	C₄×C₁₉	#484	1	1	yes	characteristic
166	C₂³×C₈	C₂²×C₁₉	1	1	yes	characteristic
167	C₂×#117	—	2	1	no	—
168	C₂²×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
169	C₂²×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
170	#427	—	2	2	no	—
171	#427	—	2	2	no	—
172	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
173	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
174	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
175	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
176	C₂×#106	—	2	1	no	—
177	#380	—	2	1	no	—
178	#367	—	2	2	no	—
179	#366	—	2	2	no	—
180	C₂²⋊C₁₆	C₄×C₁₉	1	4	yes	complemented
181	#340	—	2	1	no	—
182	C₂×C₁₉	—	4	2	no	—
183	C₂×C₁₉	#1852	1	1	yes	characteristic
184	C₂×C₁₉	#2158	1	1	yes	characteristic
185	C₂×C₁₉	#2159	1	1	yes	characteristic
186	C₂×C₁₉	#2186	1	1	yes	characteristic
187	C₂×C₁₉	#2189	1	1	yes	characteristic
188	C₂×C₁₉	#2234	1	1	yes	characteristic
189	C₂×C₁₉	#2237	1	1	yes	characteristic
190	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
191	C₂²×C₈	—	2	1	no	—
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	—	2	1	no	—
194	C₂²×C₈	—	2	1	no	—
195	C₂²×C₈	C₂×C₄×C₁₉	1	2	yes	—
196	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
197	C₂²×C₈	Q₈×C₁₉	1	2	yes	—
198	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
199	#117	—	2	4	no	—
200	#117	—	4	2	no	—
201	C₂×C₄⋊C₄	—	2	1	no	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₄⋊C₄	—	2	1	no	—
205	C₂×C₂²⋊C₄	—	4	1	no	—
206	C₂×C₂²⋊C₄	—	4	1	no	—
207	C₂×C₄²	—	4	1	no	—
208	C₂×C₁₆	—	2	1	no	—
209	C₂×C₁₆	—	2	1	no	—
210	C₂×C₁₆	—	2	1	no	—
211	C₂×C₁₆	—	2	1	no	—
212	C₂×C₁₆	C₂×C₄×C₁₉	1	2	yes	—
213	C₂×C₁₆	C₂×C₄×C₁₉	1	2	yes	—
214	#107	—	2	1	no	—
215	#107	—	2	1	no	—
216	#107	—	2	1	no	—
217	#107	—	2	1	no	—
218	#107	—	2	1	no	—
219	#107	—	2	1	no	—
220	#107	D₄×C₁₉	1	2	yes	—
221	#107	D₄×C₁₉	1	2	yes	—
222	#106	—	2	4	no	—
223	#95	—	4	1	no	—
224	C₁₉	#12921	1	1	yes	characteristic, direct product factor, Sylow
225	C₂⁴	D₈×C₁₉	1	1	yes	characteristic
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	4	1	no	—
231	C₂²×C₄	—	4	1	no	—
232	C₂²×C₄	—	4	1	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	C₄⋊C₄×C₁₉	1	2	yes	—
235	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
236	C₂²×C₄	Q₁₆×C₁₉	1	2	yes	—
237	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	C₄⋊C₄×C₁₉	1	2	yes	—
245	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
246	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
247	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
248	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
249	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
250	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	4	1	no	—
254	C₄⋊C₄	—	4	1	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₄⋊C₄	—	4	1	no	—
258	C₄⋊C₄	—	4	1	no	—
259	C₂²⋊C₄	—	4	4	no	—
260	C₂²⋊C₄	—	8	2	no	—
261	C₄²	—	4	4	no	—
262	C₁₆	—	2	2	no	—
263	C₁₆	—	2	2	no	—
264	C₂³	—	2	1	no	—
265	C₂³	—	2	1	no	—
266	C₂³	—	2	1	no	—
267	C₂³	—	2	1	no	—
268	C₂³	—	2	1	no	—
269	C₂³	—	2	1	no	—
270	C₂³	C₁₉×#107	1	2	yes	—
271	C₂³	C₂×D₈×C₁₉	1	1	yes	characteristic
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	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	8	1	no	—
281	C₂×C₄	—	8	1	no	—
282	C₂×C₄	—	8	1	no	—
283	C₂×C₄	—	8	1	no	—
284	C₂×C₄	—	8	1	no	—
285	C₂×C₄	—	8	1	no	—
286	C₂×C₄	—	8	1	no	—
287	C₂×C₄	—	8	1	no	—
288	C₂×C₄	—	8	1	no	—
289	C₂×C₄	—	8	1	no	—
290	C₂×C₄	C₁₉×#107	1	2	yes	—
291	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
292	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
293	C₂×C₄	C₁₉×#123	1	1	yes	characteristic
294	C₂×C₄	C₂×Q₁₆×C₁₉	1	1	yes	characteristic
295	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
296	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
297	C₈	—	2	2	no	—
298	C₈	C₄×D₄×C₁₉	1	1	yes	characteristic
299	C₈	C₄×D₄×C₁₉	1	1	yes	characteristic
300	C₈	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
301	C₈	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
302	C₂²	—	2	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	C₂×C₁₉×#107	1	1	yes	characteristic
310	C₂²	C₁₉×#447	1	1	yes	characteristic
311	C₂²	C₁₉×#480	1	1	yes	characteristic
312	C₂²	C₁₉×#508	1	1	yes	characteristic
313	C₂²	C₁₉×#508	1	1	yes	characteristic
314	C₂²	C₁₉×#509	1	1	yes	characteristic
315	C₂²	C₁₉×#510	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	8	2	no	—
318	C₄	—	8	2	no	—
319	C₄	—	16	1	no	—
320	C₄	—	16	1	no	—
321	C₄	C₁₉×#385	1	1	yes	characteristic
322	C₄	C₁₉×#428	1	1	yes	characteristic
323	C₄	C₁₉×#451	1	1	yes	characteristic
324	C₄	C₁₉×#484	1	1	yes	characteristic
325	C₂	—	4	2	no	—
326	C₂	C₁₉×#1852	1	1	yes	characteristic
327	C₂	C₁₉×#2158	1	1	yes	characteristic
328	C₂	C₁₉×#2159	1	1	yes	characteristic
329	C₂	C₁₉×#2186	1	1	yes	characteristic
330	C₂	C₁₉×#2189	1	1	yes	characteristic
331	C₂	C₁₉×#2234	1	1	yes	characteristic
332	C₂	C₁₉×#2237	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₂⁴×C₁₉	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂³×C₄×C₁₉ ◁ C₂³×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₉×#367	C₂	semidirect
3. C₂×C₁₉×#366	C₂	semidirect
6. C₁₉×#1380	C₂	semidirect
7. C₁₉×#1380	C₂	semidirect
22. C₁₉×C₂²⋊C₁₆	C₄	semidirect
97. #12921	C₁₉	direct (N × H)
137. C₂×#367	C₂×C₁₉	semidirect
138. C₂×#366	C₂×C₁₉	semidirect
141. #1380	C₂×C₁₉	semidirect
142. #1380	C₂×C₁₉	semidirect
180. C₂²⋊C₁₆	C₄×C₁₉	semidirect
224. C₁₉	#12921	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×#427	2
C₂×C₁₉×#367	2
C₂×C₁₉×#366	2
C₂×C₁₉×C₂²⋊C₁₆	2
C₁₉×#1808	2
C₁₉×#1380	2
#12921	19

Subgroups — C19×#12921 #18018880

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12921 #18018880