Subgroups — C₁₉×#13021 #18018980

Order 4864.

462 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3414	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2957	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₉×#359	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#2138	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1976	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#1389	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1379	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₉×#135	C₂²	1	1	yes	characteristic
9	C₂×C₁₉×#135	C₂²	1	1	yes	characteristic
10	C₁₉×#568	—	2	1	no	—
11	C₁₉×#510	—	2	1	no	—
12	C₁₉×#510	C₄	1	2	yes	complemented
13	C₁₉×#509	—	2	1	no	—
14	C₁₉×#509	C₄	1	2	yes	complemented
15	C₁₉×#508	—	2	2	no	—
16	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#444	—	2	2	no	—
19	C₁₉×#443	—	2	2	no	—
20	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
21	C₁₉×#427	C₂²	1	1	yes	characteristic
22	C₁₉×#421	—	2	1	no	—
23	C₁₉×#421	—	2	1	no	—
24	C₁₉×#410	—	2	1	no	—
25	C₁₉×#405	C₂²	1	1	yes	characteristic
26	C₁₉×#359	—	2	2	no	—
27	C₁₉×#358	—	2	1	no	—
28	C₁₉×#357	—	2	1	no	—
29	C₂×#41×C₁₉	—	2	1	no	—
30	C₂×#41×C₁₉	—	2	1	no	—
31	C₁₉×#135	—	2	1	no	—
32	C₁₉×#135	—	2	1	no	—
33	C₁₉×#135	—	2	1	no	—
34	C₁₉×#135	—	2	1	no	—
35	C₁₉×#135	—	2	1	no	—
36	C₁₉×#135	—	2	1	no	—
37	C₁₉×#135	C₂×C₄	1	2	yes	—
38	C₁₉×#135	D₄	1	2	yes	—
39	C₂×Q₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
40	C₂×Q₁₆×C₁₉	D₄	1	1	yes	characteristic
41	C₂×SD₁₆×C₁₉	—	2	1	no	—
42	C₂×SD₁₆×C₁₉	—	2	1	no	—
43	C₂×D₈×C₁₉	C₂×C₄	1	1	yes	characteristic
44	C₂×D₈×C₁₉	D₄	1	1	yes	characteristic
45	C₁₉×#131	—	2	4	no	—
46	C₁₉×#131	—	4	2	no	—
47	C₂×M₁₆×C₁₉	—	2	1	no	—
48	C₂×M₁₆×C₁₉	—	4	2	no	—
49	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
51	C₁₉×#117	—	2	1	no	—
52	C₁₉×#117	D₄	1	1	yes	characteristic
53	C₁₉×Q₃₂	—	2	2	no	—
54	C₁₉×SD₃₂	—	2	2	no	—
55	C₁₉×SD₃₂	—	2	2	no	—
56	C₁₉×D₁₆	—	2	2	no	—
57	C₁₉×M₃₂	—	2	1	no	—
58	C₁₉×M₃₂	C₂×C₄	1	2	yes	—
59	C₂×C₁₆×C₁₉	—	2	1	no	—
60	C₂×C₁₆×C₁₉	—	2	1	no	—
61	C₂×C₁₆×C₁₉	—	2	1	no	—
62	C₂×C₁₆×C₁₉	—	2	1	no	—
63	C₁₉×#108	—	2	1	no	—
64	C₁₉×#108	D₄	1	2	yes	—
65	C₁₉×#107	—	2	1	no	—
66	C₁₉×#106	—	2	1	no	—
67	C₁₉×#104	—	2	4	no	—
68	C₁₉×#104	—	4	2	no	—
69	C₁₉×#99	—	4	2	no	—
70	C₁₉×#97	—	2	1	no	—
71	C₄×C₈×C₁₉	—	2	1	no	—
72	#41×C₁₉	—	2	2	no	—
73	#41×C₁₉	—	2	2	no	—
74	#41×C₁₉	—	4	1	no	—
75	#41×C₁₉	—	4	1	no	—
76	#41×C₁₉	—	4	1	no	—
77	#41×C₁₉	—	4	1	no	—
78	#41×C₁₉	—	4	1	no	—
79	#41×C₁₉	—	4	1	no	—
80	C₂×Q₈×C₁₉	—	2	1	no	—
81	C₂×Q₈×C₁₉	—	2	1	no	—
82	C₂×D₄×C₁₉	—	2	1	no	—
83	C₂×D₄×C₁₉	—	2	1	no	—
84	C₂×D₄×C₁₉	—	4	1	no	—
85	C₂×D₄×C₁₉	—	4	1	no	—
86	C₂²×C₄×C₁₉	—	4	1	no	—
87	C₂²×C₄×C₁₉	—	4	1	no	—
88	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
89	Q₁₆×C₁₉	—	2	1	no	—
90	Q₁₆×C₁₉	—	2	1	no	—
91	Q₁₆×C₁₉	—	2	1	no	—
92	Q₁₆×C₁₉	C₂²⋊C₄	1	2	yes	—
93	SD₁₆×C₁₉	—	2	2	no	—
94	SD₁₆×C₁₉	—	2	2	no	—
95	SD₁₆×C₁₉	—	4	1	no	—
96	SD₁₆×C₁₉	—	4	1	no	—
97	D₈×C₁₉	—	2	1	no	—
98	D₈×C₁₉	—	2	1	no	—
99	D₈×C₁₉	—	2	1	no	—
100	D₈×C₁₉	C₂²⋊C₄	1	2	yes	—
101	M₁₆×C₁₉	—	2	1	no	—
102	M₁₆×C₁₉	—	2	2	no	—
103	M₁₆×C₁₉	—	4	4	no	—
104	M₁₆×C₁₉	—	4	1	no	—
105	M₁₆×C₁₉	D₈	1	2	yes	—
106	C₂×C₈×C₁₉	—	2	1	no	—
107	C₂×C₈×C₁₉	—	4	4	no	—
108	C₂×C₈×C₁₉	—	4	1	no	—
109	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
114	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
115	C₄⋊C₄×C₁₉	—	2	1	no	—
116	C₄⋊C₄×C₁₉	—	4	1	no	—
117	C₂²⋊C₄×C₁₉	—	2	1	no	—
118	C₂²⋊C₄×C₁₉	—	4	1	no	—
119	C₄²×C₁₉	—	2	1	no	—
120	C₄²×C₁₉	—	4	1	no	—
121	C₁₆×C₁₉	—	2	2	no	—
122	C₁₆×C₁₉	—	4	1	no	—
123	#13021	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂³×C₁₉	—	4	1	no	—
125	C₂³×C₁₉	—	4	1	no	—
126	C₂³×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
127	Q₈×C₁₉	—	2	2	no	—
128	Q₈×C₁₉	—	4	1	no	—
129	Q₈×C₁₉	—	4	1	no	—
130	Q₈×C₁₉	—	4	1	no	—
131	D₄×C₁₉	—	2	2	no	—
132	D₄×C₁₉	—	4	2	no	—
133	D₄×C₁₉	—	4	2	no	—
134	D₄×C₁₉	—	4	1	no	—
135	D₄×C₁₉	—	4	1	no	—
136	D₄×C₁₉	—	4	1	no	—
137	D₄×C₁₉	—	8	1	no	—
138	D₄×C₁₉	—	8	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	1	no	—
143	C₂×C₄×C₁₉	—	4	1	no	—
144	C₂×C₄×C₁₉	—	8	1	no	—
145	C₂×C₄×C₁₉	—	8	1	no	—
146	C₂×C₄×C₁₉	—	8	1	no	—
147	C₂×C₄×C₁₉	C₄×D₄	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	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
152	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₈×C₁₉	—	2	2	no	—
154	C₈×C₁₉	—	4	2	no	—
155	C₈×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
156	C₈×C₁₉	C₄×D₄	1	1	yes	characteristic
157	C₈×C₁₉	#121	1	1	yes	characteristic
158	C₈×C₁₉	#123	1	1	yes	characteristic
159	#3414	C₂×C₁₉	1	1	yes	characteristic
160	#2957	C₂×C₁₉	1	1	yes	characteristic, complemented
161	C₂×#359	C₂×C₁₉	1	1	yes	characteristic, complemented
162	#2138	C₂×C₁₉	1	1	yes	characteristic, complemented
163	#1976	C₂×C₁₉	1	1	yes	characteristic, complemented
164	#1389	C₂×C₁₉	1	1	yes	characteristic, complemented
165	#1379	C₂×C₁₉	1	1	yes	characteristic, complemented
166	C₂²×C₁₉	—	4	1	no	—
167	C₂²×C₁₉	—	4	2	no	—
168	C₂²×C₁₉	—	8	1	no	—
169	C₂²×C₁₉	—	16	1	no	—
170	C₂²×C₁₉	—	16	1	no	—
171	C₂²×C₁₉	#442	1	1	yes	characteristic
172	C₂²×C₁₉	#449	1	1	yes	characteristic
173	C₂²×C₁₉	#459	1	1	yes	characteristic
174	C₄×C₁₉	—	4	2	no	—
175	C₄×C₁₉	—	8	1	no	—
176	C₄×C₁₉	—	8	1	no	—
177	C₄×C₁₉	—	16	1	no	—
178	C₄×C₁₉	#386	1	1	yes	characteristic
179	C₄×C₁₉	C₄×D₈	1	1	yes	characteristic
180	C₄×C₁₉	#447	1	1	yes	characteristic
181	C₄×C₁₉	#463	1	1	yes	characteristic
182	C₂×#135	C₂²×C₁₉	1	1	yes	characteristic
183	C₂×#135	C₂²×C₁₉	1	1	yes	characteristic
184	#568	—	2	1	no	—
185	#510	—	2	1	no	—
186	#510	C₄×C₁₉	1	2	yes	complemented
187	#509	—	2	1	no	—
188	#509	C₄×C₁₉	1	2	yes	complemented
189	#508	—	2	2	no	—
190	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
191	C₂²×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
192	#444	—	2	2	no	—
193	#443	—	2	2	no	—
194	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
195	#427	C₂²×C₁₉	1	1	yes	characteristic
196	#421	—	2	1	no	—
197	#421	—	2	1	no	—
198	#410	—	2	1	no	—
199	#405	C₂²×C₁₉	1	1	yes	characteristic
200	#359	—	2	2	no	—
201	#358	—	2	1	no	—
202	#357	—	2	1	no	—
203	C₂×C₁₉	—	2	1	no	—
204	C₂×C₁₉	—	2	1	no	—
205	C₂×C₁₉	—	2	1	no	—
206	C₂×C₁₉	—	8	2	no	—
207	C₂×C₁₉	—	16	1	no	—
208	C₂×C₁₉	#1879	1	1	yes	characteristic
209	C₂×#41	—	2	1	no	—
210	C₂×#41	—	2	1	no	—
211	#135	—	2	1	no	—
212	#135	—	2	1	no	—
213	#135	—	2	1	no	—
214	#135	—	2	1	no	—
215	#135	—	2	1	no	—
216	#135	—	2	1	no	—
217	#135	C₂×C₄×C₁₉	1	2	yes	—
218	#135	D₄×C₁₉	1	2	yes	—
219	C₂×Q₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
220	C₂×Q₁₆	D₄×C₁₉	1	1	yes	characteristic
221	C₂×SD₁₆	—	2	1	no	—
222	C₂×SD₁₆	—	2	1	no	—
223	C₂×D₈	C₂×C₄×C₁₉	1	1	yes	characteristic
224	C₂×D₈	D₄×C₁₉	1	1	yes	characteristic
225	#131	—	2	4	no	—
226	#131	—	4	2	no	—
227	C₂×M₁₆	—	2	1	no	—
228	C₂×M₁₆	—	4	2	no	—
229	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
230	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
231	#117	—	2	1	no	—
232	#117	D₄×C₁₉	1	1	yes	characteristic
233	Q₃₂	—	2	2	no	—
234	SD₃₂	—	2	2	no	—
235	SD₃₂	—	2	2	no	—
236	D₁₆	—	2	2	no	—
237	M₃₂	—	2	1	no	—
238	M₃₂	C₂×C₄×C₁₉	1	2	yes	—
239	C₂×C₁₆	—	2	1	no	—
240	C₂×C₁₆	—	2	1	no	—
241	C₂×C₁₆	—	2	1	no	—
242	C₂×C₁₆	—	2	1	no	—
243	#108	—	2	1	no	—
244	#108	D₄×C₁₉	1	2	yes	—
245	#107	—	2	1	no	—
246	#106	—	2	1	no	—
247	#104	—	2	4	no	—
248	#104	—	4	2	no	—
249	#99	—	4	2	no	—
250	#97	—	2	1	no	—
251	C₄×C₈	—	2	1	no	—
252	C₁₉	#13021	1	1	yes	characteristic, direct product factor, Sylow
253	#41	—	2	2	no	—
254	#41	—	2	2	no	—
255	#41	—	4	1	no	—
256	#41	—	4	1	no	—
257	#41	—	4	1	no	—
258	#41	—	4	1	no	—
259	#41	—	4	1	no	—
260	#41	—	4	1	no	—
261	C₂×Q₈	—	2	1	no	—
262	C₂×Q₈	—	2	1	no	—
263	C₂×D₄	—	2	1	no	—
264	C₂×D₄	—	2	1	no	—
265	C₂×D₄	—	4	1	no	—
266	C₂×D₄	—	4	1	no	—
267	C₂²×C₄	—	4	1	no	—
268	C₂²×C₄	—	4	1	no	—
269	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
270	Q₁₆	—	2	1	no	—
271	Q₁₆	—	2	1	no	—
272	Q₁₆	—	2	1	no	—
273	Q₁₆	C₂²⋊C₄×C₁₉	1	2	yes	—
274	SD₁₆	—	2	2	no	—
275	SD₁₆	—	2	2	no	—
276	SD₁₆	—	4	1	no	—
277	SD₁₆	—	4	1	no	—
278	D₈	—	2	1	no	—
279	D₈	—	2	1	no	—
280	D₈	—	2	1	no	—
281	D₈	C₂²⋊C₄×C₁₉	1	2	yes	—
282	M₁₆	—	2	1	no	—
283	M₁₆	—	2	2	no	—
284	M₁₆	—	4	4	no	—
285	M₁₆	—	4	1	no	—
286	M₁₆	D₈×C₁₉	1	2	yes	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	4	4	no	—
289	C₂×C₈	—	4	1	no	—
290	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
291	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
292	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
293	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
294	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
295	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
296	C₄⋊C₄	—	2	1	no	—
297	C₄⋊C₄	—	4	1	no	—
298	C₂²⋊C₄	—	2	1	no	—
299	C₂²⋊C₄	—	4	1	no	—
300	C₄²	—	2	1	no	—
301	C₄²	—	4	1	no	—
302	C₁₆	—	2	2	no	—
303	C₁₆	—	4	1	no	—
304	C₂³	—	4	1	no	—
305	C₂³	—	4	1	no	—
306	C₂³	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
307	Q₈	—	2	2	no	—
308	Q₈	—	4	1	no	—
309	Q₈	—	4	1	no	—
310	Q₈	—	4	1	no	—
311	D₄	—	2	2	no	—
312	D₄	—	4	2	no	—
313	D₄	—	4	2	no	—
314	D₄	—	4	1	no	—
315	D₄	—	4	1	no	—
316	D₄	—	4	1	no	—
317	D₄	—	8	1	no	—
318	D₄	—	8	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	8	1	no	—
325	C₂×C₄	—	8	1	no	—
326	C₂×C₄	—	8	1	no	—
327	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
328	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
329	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
330	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
331	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
332	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
333	C₈	—	2	2	no	—
334	C₈	—	4	2	no	—
335	C₈	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
336	C₈	C₄×D₄×C₁₉	1	1	yes	characteristic
337	C₈	C₁₉×#121	1	1	yes	characteristic
338	C₈	C₁₉×#123	1	1	yes	characteristic
339	C₂²	—	4	1	no	—
340	C₂²	—	4	2	no	—
341	C₂²	—	8	1	no	—
342	C₂²	—	16	1	no	—
343	C₂²	—	16	1	no	—
344	C₂²	C₁₉×#442	1	1	yes	characteristic
345	C₂²	C₁₉×#449	1	1	yes	characteristic
346	C₂²	C₁₉×#459	1	1	yes	characteristic
347	C₄	—	4	2	no	—
348	C₄	—	8	1	no	—
349	C₄	—	8	1	no	—
350	C₄	—	16	1	no	—
351	C₄	C₁₉×#386	1	1	yes	characteristic
352	C₄	C₄×D₈×C₁₉	1	1	yes	characteristic
353	C₄	C₁₉×#447	1	1	yes	characteristic
354	C₄	C₁₉×#463	1	1	yes	characteristic
355	C₂	—	2	1	no	—
356	C₂	—	2	1	no	—
357	C₂	—	2	1	no	—
358	C₂	—	8	2	no	—
359	C₂	—	16	1	no	—
360	C₂	C₁₉×#1879	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₉	#447
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
socle	C₂×C₁₉	#1879
prime radical	C₁₉×#3414	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#2957	C₂	semidirect
3. C₂×C₁₉×#359	C₂	semidirect
4. C₁₉×#2138	C₂	semidirect
5. C₁₉×#1976	C₂	semidirect
6. C₁₉×#1389	C₂	semidirect
7. C₁₉×#1379	C₂	semidirect
12. C₁₉×#510	C₄	semidirect
14. C₁₉×#509	C₄	semidirect
123. #13021	C₁₉	direct (N × H)
160. #2957	C₂×C₁₉	semidirect
161. C₂×#359	C₂×C₁₉	semidirect
162. #2138	C₂×C₁₉	semidirect
163. #1976	C₂×C₁₉	semidirect
164. #1389	C₂×C₁₉	semidirect
165. #1379	C₂×C₁₉	semidirect
186. #510	C₄×C₁₉	semidirect
188. #509	C₄×C₁₉	semidirect
252. C₁₉	#13021	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×#3414	2
C₁₉×#2957	2
C₂×C₁₉×#359	2
C₁₉×#2138	2
C₁₉×#1976	2
C₁₉×#1389	2
C₁₉×#1379	2
#13021	19

Subgroups — C19×#13021 #18018980

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13021 #18018980