Subgroups — C₁₉×#10504 #18016463

Order 4864.

470 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3208	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1986	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#1982	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1965	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1565	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#1550	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1303	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₉×#533	C₂²	1	1	yes	characteristic
9	C₂×C₁₉×#127	C₂²	1	1	yes	characteristic
10	C₁₉×#492	—	2	1	no	—
11	C₁₉×#486	—	2	1	no	—
12	C₁₉×#460	—	2	1	no	—
13	C₁₉×#460	—	2	1	no	—
14	C₁₉×#450	—	2	2	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×#102	—	2	2	no	—
19	C₂×C₁₉×#100	—	2	1	no	—
20	C₂×C₁₉×#100	—	2	1	no	—
21	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
23	C₁₉×#339	—	2	1	no	—
24	C₁₉×#339	—	2	1	no	—
25	C₁₉×#335	—	2	1	no	—
26	C₁₉×#334	—	2	1	no	—
27	C₁₉×#333	—	2	1	no	—
28	C₁₉×#332	—	2	1	no	—
29	C₂²×D₄×C₁₉	—	2	1	no	—
30	C₂²×D₄×C₁₉	—	2	1	no	—
31	C₂²×D₄×C₁₉	D₄	1	2	yes	—
32	C₂×SD₁₆×C₁₉	—	2	2	no	—
33	C₂×SD₁₆×C₁₉	—	2	2	no	—
34	C₂×M₁₆×C₁₉	—	2	1	no	—
35	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
36	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₉	—	2	1	no	—
38	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
39	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
40	C₁₉×#127	—	2	1	no	—
41	C₁₉×#127	—	2	1	no	—
42	C₁₉×#127	—	2	1	no	—
43	C₁₉×#127	—	2	1	no	—
44	C₁₉×#125	—	2	1	no	—
45	C₁₉×C₂²⋊Q₈	—	2	1	no	—
46	C₁₉×C₂²⋊Q₈	—	2	1	no	—
47	C₄×Q₈×C₁₉	—	2	1	no	—
48	C₄×Q₈×C₁₉	—	2	1	no	—
49	C₁₉×#117	—	2	1	no	—
50	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
51	C₂×C₄²×C₁₉	—	2	1	no	—
52	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
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×#102	—	2	2	no	—
60	C₁₉×#102	—	2	2	no	—
61	C₁₉×#102	—	4	4	no	—
62	C₁₉×#100	—	4	2	no	—
63	C₁₉×#100	—	4	2	no	—
64	C₁₉×C₂²⋊C₈	—	2	1	no	—
65	C₁₉×#97	—	2	1	no	—
66	C₄×C₈×C₁₉	—	2	1	no	—
67	C₄×C₈×C₁₉	—	2	1	no	—
68	C₁₉×#95	—	2	1	no	—
69	C₂⁴×C₁₉	—	2	2	no	—
70	C₂×Q₈×C₁₉	—	2	1	no	—
71	C₂×Q₈×C₁₉	—	2	1	no	—
72	C₂×D₄×C₁₉	—	2	2	no	—
73	C₂×D₄×C₁₉	—	2	2	no	—
74	C₂×D₄×C₁₉	—	4	2	no	—
75	C₂×D₄×C₁₉	—	4	2	no	—
76	C₂×D₄×C₁₉	—	4	1	no	—
77	C₂×D₄×C₁₉	—	4	1	no	—
78	C₂×D₄×C₁₉	—	4	1	no	—
79	C₂×D₄×C₁₉	—	4	1	no	—
80	C₂×D₄×C₁₉	—	4	1	no	—
81	C₂×D₄×C₁₉	—	4	1	no	—
82	C₂×D₄×C₁₉	—	4	1	no	—
83	C₂×D₄×C₁₉	—	4	1	no	—
84	C₂×D₄×C₁₉	—	8	2	no	—
85	C₂²×C₄×C₁₉	—	4	1	no	—
86	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
89	SD₁₆×C₁₉	—	8	2	no	—
90	SD₁₆×C₁₉	—	8	2	no	—
91	M₁₆×C₁₉	—	2	1	no	—
92	M₁₆×C₁₉	—	2	1	no	—
93	M₁₆×C₁₉	—	2	1	no	—
94	M₁₆×C₁₉	—	4	1	no	—
95	M₁₆×C₁₉	—	4	1	no	—
96	M₁₆×C₁₉	SD₁₆	1	2	yes	complemented
97	C₂×C₈×C₁₉	—	2	1	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	2	no	—
102	C₂×C₈×C₁₉	—	4	1	no	—
103	C₂×C₈×C₁₉	—	4	1	no	—
104	C₄⋊C₄×C₁₉	—	2	2	no	—
105	C₄⋊C₄×C₁₉	—	2	1	no	—
106	C₄⋊C₄×C₁₉	—	2	1	no	—
107	C₄⋊C₄×C₁₉	—	2	1	no	—
108	C₄⋊C₄×C₁₉	—	2	1	no	—
109	C₄⋊C₄×C₁₉	—	4	1	no	—
110	C₄⋊C₄×C₁₉	—	4	1	no	—
111	C₄⋊C₄×C₁₉	—	4	1	no	—
112	C₂²⋊C₄×C₁₉	—	4	1	no	—
113	C₄²×C₁₉	—	4	1	no	—
114	C₄²×C₁₉	—	4	1	no	—
115	C₄²×C₁₉	—	4	1	no	—
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	C₄²×C₁₉	#41	1	1	yes	characteristic
120	#10504	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂³×C₁₉	—	4	2	no	—
122	C₂³×C₁₉	—	4	2	no	—
123	C₂³×C₁₉	—	4	2	no	—
124	C₂³×C₁₉	—	8	2	no	—
125	C₂³×C₁₉	—	8	2	no	—
126	C₂³×C₁₉	#121	1	1	yes	characteristic
127	Q₈×C₁₉	—	4	1	no	—
128	Q₈×C₁₉	—	4	1	no	—
129	Q₈×C₁₉	—	8	1	no	—
130	D₄×C₁₉	—	4	4	no	—
131	D₄×C₁₉	—	4	2	no	—
132	D₄×C₁₉	—	4	2	no	—
133	D₄×C₁₉	—	8	2	no	—
134	D₄×C₁₉	—	8	2	no	—
135	D₄×C₁₉	—	8	1	no	—
136	D₄×C₁₉	—	8	1	no	—
137	D₄×C₁₉	—	8	1	no	—
138	D₄×C₁₉	—	8	1	no	—
139	C₂×C₄×C₁₉	—	2	1	no	—
140	C₂×C₄×C₁₉	—	2	1	no	—
141	C₂×C₄×C₁₉	—	4	1	no	—
142	C₂×C₄×C₁₉	—	4	2	no	—
143	C₂×C₄×C₁₉	—	4	1	no	—
144	C₂×C₄×C₁₉	—	4	1	no	—
145	C₂×C₄×C₁₉	—	8	1	no	—
146	C₂×C₄×C₁₉	—	8	1	no	—
147	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
148	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
149	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
150	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
151	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
152	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
153	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
154	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
155	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
156	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
157	C₈×C₁₉	—	2	2	no	—
158	C₈×C₁₉	—	4	1	no	—
159	C₈×C₁₉	—	4	1	no	—
160	C₈×C₁₉	—	4	1	no	—
161	C₈×C₁₉	—	8	1	no	—
162	C₈×C₁₉	—	8	1	no	—
163	#3208	C₂×C₁₉	1	1	yes	characteristic
164	#1986	C₂×C₁₉	1	1	yes	characteristic, complemented
165	#1982	C₂×C₁₉	1	1	yes	characteristic
166	#1965	C₂×C₁₉	1	1	yes	characteristic
167	#1565	C₂×C₁₉	1	1	yes	characteristic, complemented
168	#1550	C₂×C₁₉	1	1	yes	characteristic, complemented
169	#1303	C₂×C₁₉	1	1	yes	characteristic, complemented
170	C₂²×C₁₉	—	2	1	no	—
171	C₂²×C₁₉	—	2	1	no	—
172	C₂²×C₁₉	—	8	2	no	—
173	C₂²×C₁₉	—	8	2	no	—
174	C₂²×C₁₉	—	8	2	no	—
175	C₂²×C₁₉	—	16	2	no	—
176	C₂²×C₁₉	—	16	2	no	—
177	C₂²×C₁₉	#394	1	1	yes	characteristic
178	C₂²×C₁₉	#461	1	1	yes	characteristic
179	C₂²×C₁₉	#463	1	1	yes	characteristic
180	C₄×C₁₉	—	2	1	no	—
181	C₄×C₁₉	—	2	1	no	—
182	C₄×C₁₉	—	2	1	no	—
183	C₄×C₁₉	—	8	2	no	—
184	C₄×C₁₉	—	8	1	no	—
185	C₄×C₁₉	—	8	1	no	—
186	C₄×C₁₉	#453	1	1	yes	characteristic
187	C₄×C₁₉	#455	1	1	yes	characteristic
188	C₄×C₁₉	#460	1	1	yes	characteristic
189	C₄×C₁₉	#464	1	1	yes	characteristic
190	C₄×C₁₉	#465	1	1	yes	characteristic
191	C₄×C₁₉	#468	1	1	yes	characteristic
192	#533	C₂²×C₁₉	1	1	yes	characteristic
193	C₂×#127	C₂²×C₁₉	1	1	yes	characteristic
194	#492	—	2	1	no	—
195	#486	—	2	1	no	—
196	#460	—	2	1	no	—
197	#460	—	2	1	no	—
198	#450	—	2	2	no	—
199	#431	C₂²×C₁₉	1	1	yes	characteristic
200	#423	C₂²×C₁₉	1	1	yes	characteristic
201	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
202	C₂×#102	—	2	2	no	—
203	C₂×#100	—	2	1	no	—
204	C₂×#100	—	2	1	no	—
205	C₄×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
206	C₄×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
207	#339	—	2	1	no	—
208	#339	—	2	1	no	—
209	#335	—	2	1	no	—
210	#334	—	2	1	no	—
211	#333	—	2	1	no	—
212	#332	—	2	1	no	—
213	C₂×C₁₉	—	2	1	no	—
214	C₂×C₁₉	—	2	1	no	—
215	C₂×C₁₉	—	16	2	no	—
216	C₂×C₁₉	#2037	1	1	yes	characteristic
217	C₂×C₁₉	#2048	1	1	yes	characteristic
218	C₂×C₁₉	#2055	1	1	yes	characteristic
219	C₂²×D₄	—	2	1	no	—
220	C₂²×D₄	—	2	1	no	—
221	C₂²×D₄	D₄×C₁₉	1	2	yes	—
222	C₂×SD₁₆	—	2	2	no	—
223	C₂×SD₁₆	—	2	2	no	—
224	C₂×M₁₆	—	2	1	no	—
225	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
226	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
227	C₂²×C₈	—	2	1	no	—
228	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
229	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
230	#127	—	2	1	no	—
231	#127	—	2	1	no	—
232	#127	—	2	1	no	—
233	#127	—	2	1	no	—
234	#125	—	2	1	no	—
235	C₂²⋊Q₈	—	2	1	no	—
236	C₂²⋊Q₈	—	2	1	no	—
237	C₄×Q₈	—	2	1	no	—
238	C₄×Q₈	—	2	1	no	—
239	#117	—	2	1	no	—
240	C₂×C₄⋊C₄	—	2	1	no	—
241	C₂×C₄²	—	2	1	no	—
242	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
243	C₄⋊C₈	—	2	1	no	—
244	C₄⋊C₈	—	2	1	no	—
245	C₄⋊C₈	—	2	1	no	—
246	C₄⋊C₈	—	2	1	no	—
247	C₄⋊C₈	D₄×C₁₉	1	1	yes	characteristic
248	C₄⋊C₈	D₄×C₁₉	1	1	yes	characteristic
249	#102	—	2	2	no	—
250	#102	—	2	2	no	—
251	#102	—	4	4	no	—
252	#100	—	4	2	no	—
253	#100	—	4	2	no	—
254	C₂²⋊C₈	—	2	1	no	—
255	#97	—	2	1	no	—
256	C₄×C₈	—	2	1	no	—
257	C₄×C₈	—	2	1	no	—
258	#95	—	2	1	no	—
259	C₁₉	#10504	1	1	yes	characteristic, direct product factor, Sylow
260	C₂⁴	—	2	2	no	—
261	C₂×Q₈	—	2	1	no	—
262	C₂×Q₈	—	2	1	no	—
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	4	2	no	—
266	C₂×D₄	—	4	2	no	—
267	C₂×D₄	—	4	1	no	—
268	C₂×D₄	—	4	1	no	—
269	C₂×D₄	—	4	1	no	—
270	C₂×D₄	—	4	1	no	—
271	C₂×D₄	—	4	1	no	—
272	C₂×D₄	—	4	1	no	—
273	C₂×D₄	—	4	1	no	—
274	C₂×D₄	—	4	1	no	—
275	C₂×D₄	—	8	2	no	—
276	C₂²×C₄	—	4	1	no	—
277	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
278	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
279	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
280	SD₁₆	—	8	2	no	—
281	SD₁₆	—	8	2	no	—
282	M₁₆	—	2	1	no	—
283	M₁₆	—	2	1	no	—
284	M₁₆	—	2	1	no	—
285	M₁₆	—	4	1	no	—
286	M₁₆	—	4	1	no	—
287	M₁₆	SD₁₆×C₁₉	1	2	yes	complemented
288	C₂×C₈	—	2	1	no	—
289	C₂×C₈	—	2	1	no	—
290	C₂×C₈	—	2	1	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	4	2	no	—
293	C₂×C₈	—	4	1	no	—
294	C₂×C₈	—	4	1	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	2	1	no	—
297	C₄⋊C₄	—	2	1	no	—
298	C₄⋊C₄	—	2	1	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	4	1	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	4	1	no	—
303	C₂²⋊C₄	—	4	1	no	—
304	C₄²	—	4	1	no	—
305	C₄²	—	4	1	no	—
306	C₄²	—	4	1	no	—
307	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
308	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
309	C₄²	#41×C₁₉	1	1	yes	characteristic
310	C₄²	#41×C₁₉	1	1	yes	characteristic
311	C₂³	—	4	2	no	—
312	C₂³	—	4	2	no	—
313	C₂³	—	4	2	no	—
314	C₂³	—	8	2	no	—
315	C₂³	—	8	2	no	—
316	C₂³	C₁₉×#121	1	1	yes	characteristic
317	Q₈	—	4	1	no	—
318	Q₈	—	4	1	no	—
319	Q₈	—	8	1	no	—
320	D₄	—	4	4	no	—
321	D₄	—	4	2	no	—
322	D₄	—	4	2	no	—
323	D₄	—	8	2	no	—
324	D₄	—	8	2	no	—
325	D₄	—	8	1	no	—
326	D₄	—	8	1	no	—
327	D₄	—	8	1	no	—
328	D₄	—	8	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	2	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	—	2	2	no	—
348	C₈	—	4	1	no	—
349	C₈	—	4	1	no	—
350	C₈	—	4	1	no	—
351	C₈	—	8	1	no	—
352	C₈	—	8	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	8	2	no	—
356	C₂²	—	8	2	no	—
357	C₂²	—	8	2	no	—
358	C₂²	—	16	2	no	—
359	C₂²	—	16	2	no	—
360	C₂²	C₁₉×#394	1	1	yes	characteristic
361	C₂²	C₁₉×#461	1	1	yes	characteristic
362	C₂²	C₁₉×#463	1	1	yes	characteristic
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	8	2	no	—
367	C₄	—	8	1	no	—
368	C₄	—	8	1	no	—
369	C₄	C₁₉×#453	1	1	yes	characteristic
370	C₄	C₁₉×#455	1	1	yes	characteristic
371	C₄	C₁₉×#460	1	1	yes	characteristic
372	C₄	C₁₉×#464	1	1	yes	characteristic
373	C₄	C₁₉×#465	1	1	yes	characteristic
374	C₄	C₁₉×#468	1	1	yes	characteristic
375	C₂	—	2	1	no	—
376	C₂	—	2	1	no	—
377	C₂	—	16	2	no	—
378	C₂	C₁₉×#2037	1	1	yes	characteristic
379	C₂	C₁₉×#2048	1	1	yes	characteristic
380	C₂	C₁₉×#2055	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
prime radical	C₂×C₁₉×#127	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×#1986	C₂	semidirect
5. C₁₉×#1565	C₂	semidirect
6. C₁₉×#1550	C₂	semidirect
7. C₁₉×#1303	C₂	semidirect
96. M₁₆×C₁₉	SD₁₆	semidirect
120. #10504	C₁₉	direct (N × H)
164. #1986	C₂×C₁₉	semidirect
167. #1565	C₂×C₁₉	semidirect
168. #1550	C₂×C₁₉	semidirect
169. #1303	C₂×C₁₉	semidirect
259. C₁₉	#10504	direct (N × H)
287. M₁₆	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×#3208	2
C₁₉×#1986	2
C₁₉×#1982	2
C₁₉×#1965	2
C₁₉×#1565	2
C₁₉×#1550	2
C₁₉×#1303	2
#10504	19

Subgroups — C19×#10504 #18016463

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#10504 #18016463