Subgroups — C₁₉×#10502 #18016461

Order 4864.

412 conjugacy classes of subgroups.

The subgroup lattice has 364 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×#1997	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#1987	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1980	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1565	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#1547	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1303	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₉×#532	C₂²	1	1	yes	characteristic
9	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₁₉×#498	—	2	1	no	—
11	C₁₉×#486	—	2	1	no	—
12	C₁₉×#481	—	2	2	no	—
13	C₁₉×#476	—	2	1	no	—
14	C₁₉×#476	—	2	1	no	—
15	C₁₉×#431	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×#104	—	2	1	no	—
19	C₂×C₁₉×#104	—	2	1	no	—
20	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
22	C₁₉×#343	—	2	1	no	—
23	C₁₉×#343	—	2	1	no	—
24	C₁₉×#340	—	2	2	no	—
25	C₁₉×#335	—	2	1	no	—
26	C₁₉×#334	—	2	1	no	—
27	C₁₉×#332	—	2	1	no	—
28	C₁₉×#331	—	2	1	no	—
29	C₂×#41×C₁₉	—	2	1	no	—
30	C₂²×Q₈×C₁₉	—	2	1	no	—
31	C₂×M₁₆×C₁₉	—	2	1	no	—
32	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
33	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₉	—	2	1	no	—
35	C₁₉×C₄⋊Q₈	—	2	1	no	—
36	C₁₉×C₄⋊Q₈	—	2	1	no	—
37	C₁₉×C₄⋊Q₈	—	2	1	no	—
38	C₁₉×C₄⋊Q₈	—	2	1	no	—
39	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
40	C₁₉×#127	D₄	1	1	yes	characteristic
41	C₁₉×#124	—	2	1	no	—
42	C₁₉×#121	—	2	1	no	—
43	C₁₉×#121	—	2	1	no	—
44	C₄×D₄×C₁₉	—	2	1	no	—
45	C₄×D₄×C₁₉	—	2	1	no	—
46	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
47	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
48	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
49	C₂×C₄²×C₁₉	—	2	1	no	—
50	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
51	C₁₉×#106	—	2	2	no	—
52	C₁₉×#106	—	2	2	no	—
53	C₁₉×C₄⋊C₈	—	2	1	no	—
54	C₁₉×C₄⋊C₈	—	2	1	no	—
55	C₁₉×C₄⋊C₈	—	2	1	no	—
56	C₁₉×C₄⋊C₈	—	2	1	no	—
57	C₁₉×C₄⋊C₈	D₄	1	1	yes	characteristic
58	C₁₉×C₄⋊C₈	D₄	1	1	yes	characteristic
59	C₁₉×#104	—	4	2	no	—
60	C₁₉×#104	—	4	2	no	—
61	C₁₉×#102	—	2	2	no	—
62	C₁₉×#102	—	2	2	no	—
63	C₁₉×C₂²⋊C₈	—	2	1	no	—
64	C₁₉×#97	—	2	1	no	—
65	C₄×C₈×C₁₉	—	2	1	no	—
66	C₄×C₈×C₁₉	—	2	1	no	—
67	C₁₉×#95	—	2	1	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	—	4	1	no	—
74	C₂×Q₈×C₁₉	—	4	1	no	—
75	C₂×Q₈×C₁₉	—	8	2	no	—
76	C₂×D₄×C₁₉	—	2	1	no	—
77	C₂×D₄×C₁₉	—	2	1	no	—
78	C₂×D₄×C₁₉	—	2	1	no	—
79	C₂×D₄×C₁₉	—	4	1	no	—
80	C₂²×C₄×C₁₉	—	2	2	no	—
81	C₂²×C₄×C₁₉	—	4	1	no	—
82	C₂²×C₄×C₁₉	—	4	1	no	—
83	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
86	M₁₆×C₁₉	—	2	1	no	—
87	M₁₆×C₁₉	—	2	1	no	—
88	M₁₆×C₁₉	—	2	1	no	—
89	M₁₆×C₁₉	—	4	1	no	—
90	M₁₆×C₁₉	—	4	1	no	—
91	M₁₆×C₁₉	SD₁₆	1	2	yes	—
92	C₂×C₈×C₁₉	—	2	1	no	—
93	C₂×C₈×C₁₉	—	2	1	no	—
94	C₂×C₈×C₁₉	—	2	1	no	—
95	C₂×C₈×C₁₉	—	2	1	no	—
96	C₂×C₈×C₁₉	—	4	2	no	—
97	C₂×C₈×C₁₉	—	4	1	no	—
98	C₂×C₈×C₁₉	—	4	1	no	—
99	C₄⋊C₄×C₁₉	—	2	2	no	—
100	C₄⋊C₄×C₁₉	—	2	2	no	—
101	C₄⋊C₄×C₁₉	—	2	1	no	—
102	C₄⋊C₄×C₁₉	—	2	2	no	—
103	C₄⋊C₄×C₁₉	—	2	1	no	—
104	C₄⋊C₄×C₁₉	—	4	1	no	—
105	C₄⋊C₄×C₁₉	—	4	1	no	—
106	C₄⋊C₄×C₁₉	—	4	1	no	—
107	C₂²⋊C₄×C₁₉	—	4	1	no	—
108	C₂²⋊C₄×C₁₉	—	4	1	no	—
109	C₄²×C₁₉	—	4	1	no	—
110	C₄²×C₁₉	—	4	1	no	—
111	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₁₉	#41	1	1	yes	characteristic
114	C₄²×C₁₉	#41	1	1	yes	characteristic
115	#10502	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂³×C₁₉	—	4	1	no	—
117	C₂³×C₁₉	#121	1	1	yes	characteristic
118	Q₈×C₁₉	—	4	1	no	—
119	Q₈×C₁₉	—	4	1	no	—
120	Q₈×C₁₉	—	8	2	no	—
121	Q₈×C₁₉	—	8	2	no	—
122	D₄×C₁₉	—	4	1	no	—
123	D₄×C₁₉	—	4	1	no	—
124	D₄×C₁₉	—	4	1	no	—
125	D₄×C₁₉	—	4	1	no	—
126	D₄×C₁₉	—	8	1	no	—
127	D₄×C₁₉	—	8	1	no	—
128	D₄×C₁₉	—	8	1	no	—
129	C₂×C₄×C₁₉	—	2	1	no	—
130	C₂×C₄×C₁₉	—	2	1	no	—
131	C₂×C₄×C₁₉	—	4	1	no	—
132	C₂×C₄×C₁₉	—	4	2	no	—
133	C₂×C₄×C₁₉	—	4	1	no	—
134	C₂×C₄×C₁₉	—	4	2	no	—
135	C₂×C₄×C₁₉	—	8	1	no	—
136	C₂×C₄×C₁₉	—	8	1	no	—
137	C₂×C₄×C₁₉	—	8	1	no	—
138	C₂×C₄×C₁₉	—	8	1	no	—
139	C₂×C₄×C₁₉	—	8	2	no	—
140	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
141	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
142	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
143	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
144	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
145	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
146	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
147	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
148	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
150	C₈×C₁₉	—	2	2	no	—
151	C₈×C₁₉	—	4	1	no	—
152	C₈×C₁₉	—	4	1	no	—
153	C₈×C₁₉	—	4	1	no	—
154	C₈×C₁₉	—	8	1	no	—
155	C₈×C₁₉	—	8	1	no	—
156	#3227	C₂×C₁₉	1	1	yes	characteristic
157	#1997	C₂×C₁₉	1	1	yes	characteristic, complemented
158	#1987	C₂×C₁₉	1	1	yes	characteristic, complemented
159	#1980	C₂×C₁₉	1	1	yes	characteristic
160	#1565	C₂×C₁₉	1	1	yes	characteristic, complemented
161	#1547	C₂×C₁₉	1	1	yes	characteristic
162	#1303	C₂×C₁₉	1	1	yes	characteristic, complemented
163	C₂²×C₁₉	—	2	1	no	—
164	C₂²×C₁₉	—	2	1	no	—
165	C₂²×C₁₉	—	8	1	no	—
166	C₂²×C₁₉	—	8	1	no	—
167	C₂²×C₁₉	—	8	1	no	—
168	C₂²×C₁₉	#394	1	1	yes	characteristic
169	C₂²×C₁₉	#460	1	1	yes	characteristic
170	C₂²×C₁₉	#464	1	1	yes	characteristic
171	C₄×C₁₉	—	2	1	no	—
172	C₄×C₁₉	—	2	1	no	—
173	C₄×C₁₉	—	2	1	no	—
174	C₄×C₁₉	—	8	2	no	—
175	C₄×C₁₉	—	8	2	no	—
176	C₄×C₁₉	—	8	1	no	—
177	C₄×C₁₉	#454	1	1	yes	characteristic
178	C₄×C₁₉	#456	1	1	yes	characteristic
179	C₄×C₁₉	#461	1	1	yes	characteristic
180	C₄×C₁₉	#463	1	1	yes	characteristic
181	C₄×C₁₉	#465	1	1	yes	characteristic
182	C₄×C₁₉	#468	1	1	yes	characteristic
183	#532	C₂²×C₁₉	1	1	yes	characteristic
184	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
185	#498	—	2	1	no	—
186	#486	—	2	1	no	—
187	#481	—	2	2	no	—
188	#476	—	2	1	no	—
189	#476	—	2	1	no	—
190	#431	C₂²×C₁₉	1	1	yes	characteristic
191	#423	C₂²×C₁₉	1	1	yes	characteristic
192	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
193	C₂×#104	—	2	1	no	—
194	C₂×#104	—	2	1	no	—
195	C₄×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
196	C₄×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
197	#343	—	2	1	no	—
198	#343	—	2	1	no	—
199	#340	—	2	2	no	—
200	#335	—	2	1	no	—
201	#334	—	2	1	no	—
202	#332	—	2	1	no	—
203	#331	—	2	1	no	—
204	C₂×C₁₉	—	2	1	no	—
205	C₂×C₁₉	—	2	1	no	—
206	C₂×C₁₉	—	16	1	no	—
207	C₂×C₁₉	#2038	1	1	yes	characteristic
208	C₂×C₁₉	#2048	1	1	yes	characteristic
209	C₂×C₁₉	#2055	1	1	yes	characteristic
210	C₂×#41	—	2	1	no	—
211	C₂²×Q₈	—	2	1	no	—
212	C₂×M₁₆	—	2	1	no	—
213	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
214	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
215	C₂²×C₈	—	2	1	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	C₄⋊Q₈	—	2	1	no	—
218	C₄⋊Q₈	—	2	1	no	—
219	C₄⋊Q₈	—	2	1	no	—
220	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
221	#127	D₄×C₁₉	1	1	yes	characteristic
222	#124	—	2	1	no	—
223	#121	—	2	1	no	—
224	#121	—	2	1	no	—
225	C₄×D₄	—	2	1	no	—
226	C₄×D₄	—	2	1	no	—
227	C₂×C₄⋊C₄	—	2	1	no	—
228	C₂×C₄⋊C₄	—	2	1	no	—
229	C₂×C₄⋊C₄	D₄×C₁₉	1	2	yes	—
230	C₂×C₄²	—	2	1	no	—
231	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
232	#106	—	2	2	no	—
233	#106	—	2	2	no	—
234	C₄⋊C₈	—	2	1	no	—
235	C₄⋊C₈	—	2	1	no	—
236	C₄⋊C₈	—	2	1	no	—
237	C₄⋊C₈	—	2	1	no	—
238	C₄⋊C₈	D₄×C₁₉	1	1	yes	characteristic
239	C₄⋊C₈	D₄×C₁₉	1	1	yes	characteristic
240	#104	—	4	2	no	—
241	#104	—	4	2	no	—
242	#102	—	2	2	no	—
243	#102	—	2	2	no	—
244	C₂²⋊C₈	—	2	1	no	—
245	#97	—	2	1	no	—
246	C₄×C₈	—	2	1	no	—
247	C₄×C₈	—	2	1	no	—
248	#95	—	2	1	no	—
249	C₁₉	#10502	1	1	yes	characteristic, direct product factor, Sylow
250	#41	—	4	1	no	—
251	#41	—	4	1	no	—
252	#41	—	4	1	no	—
253	#41	—	4	1	no	—
254	C₂×Q₈	—	2	1	no	—
255	C₂×Q₈	—	4	1	no	—
256	C₂×Q₈	—	4	1	no	—
257	C₂×Q₈	—	8	2	no	—
258	C₂×D₄	—	2	1	no	—
259	C₂×D₄	—	2	1	no	—
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	—	4	1	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	4	1	no	—
264	C₂²×C₄	—	4	1	no	—
265	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
266	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
267	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
268	M₁₆	—	2	1	no	—
269	M₁₆	—	2	1	no	—
270	M₁₆	—	2	1	no	—
271	M₁₆	—	4	1	no	—
272	M₁₆	—	4	1	no	—
273	M₁₆	SD₁₆×C₁₉	1	2	yes	—
274	C₂×C₈	—	2	1	no	—
275	C₂×C₈	—	2	1	no	—
276	C₂×C₈	—	2	1	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	4	2	no	—
279	C₂×C₈	—	4	1	no	—
280	C₂×C₈	—	4	1	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	1	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	1	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	—	4	1	no	—
288	C₄⋊C₄	—	4	1	no	—
289	C₂²⋊C₄	—	4	1	no	—
290	C₂²⋊C₄	—	4	1	no	—
291	C₄²	—	4	1	no	—
292	C₄²	—	4	1	no	—
293	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
294	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
295	C₄²	#41×C₁₉	1	1	yes	characteristic
296	C₄²	#41×C₁₉	1	1	yes	characteristic
297	C₂³	—	4	1	no	—
298	C₂³	C₁₉×#121	1	1	yes	characteristic
299	Q₈	—	4	1	no	—
300	Q₈	—	4	1	no	—
301	Q₈	—	8	2	no	—
302	Q₈	—	8	2	no	—
303	D₄	—	4	1	no	—
304	D₄	—	4	1	no	—
305	D₄	—	4	1	no	—
306	D₄	—	4	1	no	—
307	D₄	—	8	1	no	—
308	D₄	—	8	1	no	—
309	D₄	—	8	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	8	1	no	—
317	C₂×C₄	—	8	1	no	—
318	C₂×C₄	—	8	1	no	—
319	C₂×C₄	—	8	1	no	—
320	C₂×C₄	—	8	2	no	—
321	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
322	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
323	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
324	C₂×C₄	C₁₉×#123	1	1	yes	characteristic
325	C₂×C₄	C₁₉×#123	1	1	yes	characteristic
326	C₂×C₄	C₁₉×#124	1	1	yes	characteristic
327	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
328	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
329	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
330	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
331	C₈	—	2	2	no	—
332	C₈	—	4	1	no	—
333	C₈	—	4	1	no	—
334	C₈	—	4	1	no	—
335	C₈	—	8	1	no	—
336	C₈	—	8	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	8	1	no	—
340	C₂²	—	8	1	no	—
341	C₂²	—	8	1	no	—
342	C₂²	C₁₉×#394	1	1	yes	characteristic
343	C₂²	C₁₉×#460	1	1	yes	characteristic
344	C₂²	C₁₉×#464	1	1	yes	characteristic
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	8	2	no	—
349	C₄	—	8	2	no	—
350	C₄	—	8	1	no	—
351	C₄	C₁₉×#454	1	1	yes	characteristic
352	C₄	C₁₉×#456	1	1	yes	characteristic
353	C₄	C₁₉×#461	1	1	yes	characteristic
354	C₄	C₁₉×#463	1	1	yes	characteristic
355	C₄	C₁₉×#465	1	1	yes	characteristic
356	C₄	C₁₉×#468	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	16	1	no	—
360	C₂	C₁₉×#2038	1	1	yes	characteristic
361	C₂	C₁₉×#2048	1	1	yes	characteristic
362	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×#1997	C₂	semidirect
3. C₁₉×#1987	C₂	semidirect
5. C₁₉×#1565	C₂	semidirect
7. C₁₉×#1303	C₂	semidirect
115. #10502	C₁₉	direct (N × H)
157. #1997	C₂×C₁₉	semidirect
158. #1987	C₂×C₁₉	semidirect
160. #1565	C₂×C₁₉	semidirect
162. #1303	C₂×C₁₉	semidirect
249. C₁₉	#10502	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×#1997	2
C₁₉×#1987	2
C₁₉×#1980	2
C₁₉×#1565	2
C₁₉×#1547	2
C₁₉×#1303	2
#10502	19

Subgroups — C19×#10502 #18016461

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#10502 #18016461