Subgroups — C₁₉×#10521 #18016480

Order 4864.

420 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3246	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1994	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#1962	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1831	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1550	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#1547	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1292	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×#476	—	2	1	no	—
11	C₁₉×#475	—	2	1	no	—
12	C₁₉×#465	—	2	2	no	—
13	C₁₉×#461	—	2	1	no	—
14	C₁₉×#460	—	2	1	no	—
15	C₁₉×#423	C₂²	1	1	yes	characteristic
16	C₁₉×#423	C₂²	1	1	yes	characteristic
17	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₁₉×#104	—	2	1	no	—
20	C₂×C₁₉×#104	—	2	1	no	—
21	C₄³×C₁₉	C₂²	1	1	yes	characteristic
22	C₁₉×#339	—	2	1	no	—
23	C₁₉×#339	—	2	1	no	—
24	C₁₉×#336	—	2	2	no	—
25	C₁₉×#333	—	2	1	no	—
26	C₁₉×#332	—	2	1	no	—
27	C₁₉×#332	—	2	1	no	—
28	C₁₉×#331	—	2	1	no	—
29	C₂×#41×C₁₉	—	2	1	no	—
30	C₂×SD₁₆×C₁₉	—	2	2	no	—
31	C₂×M₁₆×C₁₉	—	2	1	no	—
32	C₂×M₁₆×C₁₉	—	2	1	no	—
33	C₂²×C₈×C₁₉	—	2	1	no	—
34	C₂²×C₈×C₁₉	D₄	1	2	yes	—
35	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
36	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
37	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
38	C₁₉×#127	D₄	1	1	yes	characteristic
39	C₁₉×#125	—	2	1	no	—
40	C₁₉×#124	—	2	1	no	—
41	C₁₉×C₂²⋊Q₈	—	2	1	no	—
42	C₁₉×C₂²⋊Q₈	—	2	1	no	—
43	C₁₉×#121	—	2	1	no	—
44	C₁₉×#121	—	2	1	no	—
45	C₄×Q₈×C₁₉	—	2	1	no	—
46	C₄×Q₈×C₁₉	—	2	1	no	—
47	C₄×D₄×C₁₉	—	2	1	no	—
48	C₄×D₄×C₁₉	—	2	1	no	—
49	C₁₉×#117	—	2	1	no	—
50	C₂×C₄²×C₁₉	—	2	1	no	—
51	C₂×C₄²×C₁₉	—	2	1	no	—
52	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
55	C₁₉×#106	—	2	2	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	—	2	1	no	—
63	C₁₉×C₄⋊C₈	—	2	1	no	—
64	C₁₉×#104	—	4	2	no	—
65	C₁₉×#104	—	4	2	no	—
66	C₁₉×#103	—	2	2	no	—
67	C₁₉×#102	—	2	2	no	—
68	#41×C₁₉	—	4	1	no	—
69	#41×C₁₉	—	4	1	no	—
70	#41×C₁₉	—	4	1	no	—
71	#41×C₁₉	—	4	1	no	—
72	C₂×Q₈×C₁₉	—	2	1	no	—
73	C₂×Q₈×C₁₉	—	2	1	no	—
74	C₂×Q₈×C₁₉	—	2	1	no	—
75	C₂×D₄×C₁₉	—	2	1	no	—
76	C₂×D₄×C₁₉	—	2	1	no	—
77	C₂×D₄×C₁₉	—	2	1	no	—
78	C₂×D₄×C₁₉	—	4	1	no	—
79	C₂²×C₄×C₁₉	—	2	1	no	—
80	C₂²×C₄×C₁₉	—	2	1	no	—
81	C₂²×C₄×C₁₉	—	4	1	no	—
82	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
85	SD₁₆×C₁₉	—	8	2	no	—
86	M₁₆×C₁₉	—	4	1	no	—
87	M₁₆×C₁₉	—	4	1	no	—
88	M₁₆×C₁₉	—	4	1	no	—
89	M₁₆×C₁₉	—	4	1	no	—
90	C₂×C₈×C₁₉	—	2	2	no	—
91	C₂×C₈×C₁₉	—	4	2	no	—
92	C₂×C₈×C₁₉	—	4	2	no	—
93	C₂×C₈×C₁₉	—	4	1	no	—
94	C₂×C₈×C₁₉	—	4	1	no	—
95	C₂×C₈×C₁₉	—	4	1	no	—
96	C₄⋊C₄×C₁₉	—	2	1	no	—
97	C₄⋊C₄×C₁₉	—	2	1	no	—
98	C₄⋊C₄×C₁₉	—	2	1	no	—
99	C₄⋊C₄×C₁₉	—	2	1	no	—
100	C₄⋊C₄×C₁₉	—	2	1	no	—
101	C₄⋊C₄×C₁₉	—	2	1	no	—
102	C₄⋊C₄×C₁₉	—	4	1	no	—
103	C₄⋊C₄×C₁₉	—	4	1	no	—
104	C₂²⋊C₄×C₁₉	—	4	1	no	—
105	C₂²⋊C₄×C₁₉	—	4	1	no	—
106	C₂²⋊C₄×C₁₉	—	4	1	no	—
107	C₄²×C₁₉	—	2	1	no	—
108	C₄²×C₁₉	—	2	1	no	—
109	C₄²×C₁₉	—	2	1	no	—
110	C₄²×C₁₉	—	4	1	no	—
111	C₄²×C₁₉	—	4	1	no	—
112	C₄²×C₁₉	—	4	2	no	—
113	C₄²×C₁₉	—	4	1	no	—
114	C₄²×C₁₉	SD₁₆	1	2	yes	complemented
115	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₁₉	#41	1	1	yes	characteristic
118	C₄²×C₁₉	#41	1	1	yes	characteristic
119	#10521	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂³×C₁₉	—	4	1	no	—
121	C₂³×C₁₉	#121	1	1	yes	characteristic
122	Q₈×C₁₉	—	4	1	no	—
123	Q₈×C₁₉	—	4	1	no	—
124	Q₈×C₁₉	—	4	1	no	—
125	Q₈×C₁₉	—	4	1	no	—
126	Q₈×C₁₉	—	8	1	no	—
127	D₄×C₁₉	—	4	1	no	—
128	D₄×C₁₉	—	4	1	no	—
129	D₄×C₁₉	—	4	1	no	—
130	D₄×C₁₉	—	4	1	no	—
131	D₄×C₁₉	—	8	1	no	—
132	D₄×C₁₉	—	8	1	no	—
133	D₄×C₁₉	—	8	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	—	2	2	no	—
138	C₂×C₄×C₁₉	—	4	1	no	—
139	C₂×C₄×C₁₉	—	4	1	no	—
140	C₂×C₄×C₁₉	—	4	1	no	—
141	C₂×C₄×C₁₉	—	4	1	no	—
142	C₂×C₄×C₁₉	—	4	1	no	—
143	C₂×C₄×C₁₉	—	4	1	no	—
144	C₂×C₄×C₁₉	—	4	1	no	—
145	C₂×C₄×C₁₉	—	4	1	no	—
146	C₂×C₄×C₁₉	—	8	1	no	—
147	C₂×C₄×C₁₉	—	8	1	no	—
148	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
149	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
150	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
151	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
152	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
153	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
154	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
155	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
156	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
157	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
158	C₈×C₁₉	—	4	2	no	—
159	C₈×C₁₉	—	8	1	no	—
160	C₈×C₁₉	—	8	1	no	—
161	C₈×C₁₉	—	8	1	no	—
162	#3246	C₂×C₁₉	1	1	yes	characteristic
163	#1994	C₂×C₁₉	1	1	yes	characteristic, complemented
164	#1962	C₂×C₁₉	1	1	yes	characteristic
165	#1831	C₂×C₁₉	1	1	yes	characteristic, complemented
166	#1550	C₂×C₁₉	1	1	yes	characteristic, complemented
167	#1547	C₂×C₁₉	1	1	yes	characteristic
168	#1292	C₂×C₁₉	1	1	yes	characteristic, complemented
169	C₂²×C₁₉	—	2	1	no	—
170	C₂²×C₁₉	—	2	1	no	—
171	C₂²×C₁₉	—	8	1	no	—
172	C₂²×C₁₉	—	8	1	no	—
173	C₂²×C₁₉	—	8	1	no	—
174	C₂²×C₁₉	#394	1	1	yes	characteristic
175	C₂²×C₁₉	#465	1	1	yes	characteristic
176	C₂²×C₁₉	#468	1	1	yes	characteristic
177	C₄×C₁₉	—	2	1	no	—
178	C₄×C₁₉	—	2	1	no	—
179	C₄×C₁₉	—	2	1	no	—
180	C₄×C₁₉	—	4	2	no	—
181	C₄×C₁₉	—	4	2	no	—
182	C₄×C₁₉	—	8	1	no	—
183	C₄×C₁₉	—	8	1	no	—
184	C₄×C₁₉	—	8	1	no	—
185	C₄×C₁₉	#454	1	1	yes	characteristic
186	C₄×C₁₉	#455	1	1	yes	characteristic
187	C₄×C₁₉	#460	1	1	yes	characteristic
188	C₄×C₁₉	#461	1	1	yes	characteristic
189	C₄×C₁₉	#463	1	1	yes	characteristic
190	C₄×C₁₉	#464	1	1	yes	characteristic
191	#533	C₂²×C₁₉	1	1	yes	characteristic
192	#532	C₂²×C₁₉	1	1	yes	characteristic
193	#476	—	2	1	no	—
194	#475	—	2	1	no	—
195	#465	—	2	2	no	—
196	#461	—	2	1	no	—
197	#460	—	2	1	no	—
198	#423	C₂²×C₁₉	1	1	yes	characteristic
199	#423	C₂²×C₁₉	1	1	yes	characteristic
200	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
201	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
202	C₂×#104	—	2	1	no	—
203	C₂×#104	—	2	1	no	—
204	C₄³	C₂²×C₁₉	1	1	yes	characteristic
205	#339	—	2	1	no	—
206	#339	—	2	1	no	—
207	#336	—	2	2	no	—
208	#333	—	2	1	no	—
209	#332	—	2	1	no	—
210	#332	—	2	1	no	—
211	#331	—	2	1	no	—
212	C₂×C₁₉	—	2	1	no	—
213	C₂×C₁₉	—	2	1	no	—
214	C₂×C₁₉	—	16	1	no	—
215	C₂×C₁₉	#2040	1	1	yes	characteristic
216	C₂×C₁₉	#2050	1	1	yes	characteristic
217	C₂×C₁₉	#2055	1	1	yes	characteristic
218	C₂×#41	—	2	1	no	—
219	C₂×SD₁₆	—	2	2	no	—
220	C₂×M₁₆	—	2	1	no	—
221	C₂×M₁₆	—	2	1	no	—
222	C₂²×C₈	—	2	1	no	—
223	C₂²×C₈	D₄×C₁₉	1	2	yes	—
224	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
225	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
226	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
227	#127	D₄×C₁₉	1	1	yes	characteristic
228	#125	—	2	1	no	—
229	#124	—	2	1	no	—
230	C₂²⋊Q₈	—	2	1	no	—
231	C₂²⋊Q₈	—	2	1	no	—
232	#121	—	2	1	no	—
233	#121	—	2	1	no	—
234	C₄×Q₈	—	2	1	no	—
235	C₄×Q₈	—	2	1	no	—
236	C₄×D₄	—	2	1	no	—
237	C₄×D₄	—	2	1	no	—
238	#117	—	2	1	no	—
239	C₂×C₄²	—	2	1	no	—
240	C₂×C₄²	—	2	1	no	—
241	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
242	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
243	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
244	#106	—	2	2	no	—
245	C₄⋊C₈	—	2	1	no	—
246	C₄⋊C₈	—	2	1	no	—
247	C₄⋊C₈	—	2	1	no	—
248	C₄⋊C₈	—	2	1	no	—
249	C₄⋊C₈	—	2	1	no	—
250	C₄⋊C₈	—	2	1	no	—
251	C₄⋊C₈	—	2	1	no	—
252	C₄⋊C₈	—	2	1	no	—
253	#104	—	4	2	no	—
254	#104	—	4	2	no	—
255	#103	—	2	2	no	—
256	#102	—	2	2	no	—
257	C₁₉	#10521	1	1	yes	characteristic, direct product factor, Sylow
258	#41	—	4	1	no	—
259	#41	—	4	1	no	—
260	#41	—	4	1	no	—
261	#41	—	4	1	no	—
262	C₂×Q₈	—	2	1	no	—
263	C₂×Q₈	—	2	1	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×D₄	—	2	1	no	—
266	C₂×D₄	—	2	1	no	—
267	C₂×D₄	—	2	1	no	—
268	C₂×D₄	—	4	1	no	—
269	C₂²×C₄	—	2	1	no	—
270	C₂²×C₄	—	2	1	no	—
271	C₂²×C₄	—	4	1	no	—
272	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
273	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
274	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
275	SD₁₆	—	8	2	no	—
276	M₁₆	—	4	1	no	—
277	M₁₆	—	4	1	no	—
278	M₁₆	—	4	1	no	—
279	M₁₆	—	4	1	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	4	2	no	—
282	C₂×C₈	—	4	2	no	—
283	C₂×C₈	—	4	1	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	4	1	no	—
286	C₄⋊C₄	—	2	1	no	—
287	C₄⋊C₄	—	2	1	no	—
288	C₄⋊C₄	—	2	1	no	—
289	C₄⋊C₄	—	2	1	no	—
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₂²⋊C₄	—	4	1	no	—
295	C₂²⋊C₄	—	4	1	no	—
296	C₂²⋊C₄	—	4	1	no	—
297	C₄²	—	2	1	no	—
298	C₄²	—	2	1	no	—
299	C₄²	—	2	1	no	—
300	C₄²	—	4	1	no	—
301	C₄²	—	4	1	no	—
302	C₄²	—	4	2	no	—
303	C₄²	—	4	1	no	—
304	C₄²	SD₁₆×C₁₉	1	2	yes	complemented
305	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
306	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
307	C₄²	#41×C₁₉	1	1	yes	characteristic
308	C₄²	#41×C₁₉	1	1	yes	characteristic
309	C₂³	—	4	1	no	—
310	C₂³	C₁₉×#121	1	1	yes	characteristic
311	Q₈	—	4	1	no	—
312	Q₈	—	4	1	no	—
313	Q₈	—	4	1	no	—
314	Q₈	—	4	1	no	—
315	Q₈	—	8	1	no	—
316	D₄	—	4	1	no	—
317	D₄	—	4	1	no	—
318	D₄	—	4	1	no	—
319	D₄	—	4	1	no	—
320	D₄	—	8	1	no	—
321	D₄	—	8	1	no	—
322	D₄	—	8	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	8	1	no	—
336	C₂×C₄	—	8	1	no	—
337	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
338	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
339	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
340	C₂×C₄	C₁₉×#123	1	1	yes	characteristic
341	C₂×C₄	C₁₉×#123	1	1	yes	characteristic
342	C₂×C₄	C₁₉×#124	1	1	yes	characteristic
343	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
344	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
345	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
346	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
347	C₈	—	4	2	no	—
348	C₈	—	8	1	no	—
349	C₈	—	8	1	no	—
350	C₈	—	8	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	8	1	no	—
354	C₂²	—	8	1	no	—
355	C₂²	—	8	1	no	—
356	C₂²	C₁₉×#394	1	1	yes	characteristic
357	C₂²	C₁₉×#465	1	1	yes	characteristic
358	C₂²	C₁₉×#468	1	1	yes	characteristic
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	4	2	no	—
363	C₄	—	4	2	no	—
364	C₄	—	8	1	no	—
365	C₄	—	8	1	no	—
366	C₄	—	8	1	no	—
367	C₄	C₁₉×#454	1	1	yes	characteristic
368	C₄	C₁₉×#455	1	1	yes	characteristic
369	C₄	C₁₉×#460	1	1	yes	characteristic
370	C₄	C₁₉×#461	1	1	yes	characteristic
371	C₄	C₁₉×#463	1	1	yes	characteristic
372	C₄	C₁₉×#464	1	1	yes	characteristic
373	C₂	—	2	1	no	—
374	C₂	—	2	1	no	—
375	C₂	—	16	1	no	—
376	C₂	C₁₉×#2040	1	1	yes	characteristic
377	C₂	C₁₉×#2050	1	1	yes	characteristic
378	C₂	C₁₉×#2055	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
prime radical	C₁₉×#532	C₂²
LCS	G ▷ C₂×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
2. C₁₉×#1994	C₂	semidirect
4. C₁₉×#1831	C₂	semidirect
5. C₁₉×#1550	C₂	semidirect
7. C₁₉×#1292	C₂	semidirect
114. C₄²×C₁₉	SD₁₆	semidirect
119. #10521	C₁₉	direct (N × H)
163. #1994	C₂×C₁₉	semidirect
165. #1831	C₂×C₁₉	semidirect
166. #1550	C₂×C₁₉	semidirect
168. #1292	C₂×C₁₉	semidirect
257. C₁₉	#10521	direct (N × H)
304. C₄²	SD₁₆×C₁₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₁₉×#3246	2
C₁₉×#1994	2
C₁₉×#1962	2
C₁₉×#1831	2
C₁₉×#1550	2
C₁₉×#1547	2
C₁₉×#1292	2
#10521	19

Subgroups — C19×#10521 #18016480

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#10521 #18016480