Subgroups — C₁₉×#13026 #18018985

Order 4864.

486 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#510	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2943	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#360	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#2141	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1971	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#1386	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1377	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₉×#135	C₂²	1	1	yes	characteristic
9	C₂²×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
10	C₄×#41×C₁₉	—	2	1	no	—
11	C₁₉×#510	—	2	2	no	—
12	C₁₉×#510	C₄	1	4	yes	complemented
13	C₂×C₁₉×Q₃₂	—	2	1	no	—
14	C₂×C₁₉×SD₃₂	—	2	1	no	—
15	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
17	C₁₉×#442	—	2	1	no	—
18	C₁₉×#441	—	2	1	no	—
19	C₁₉×#440	—	2	2	no	—
20	C₁₉×#427	C₂²	1	1	yes	characteristic
21	C₂×C₁₉×#106	C₂²	1	1	yes	characteristic
22	C₁₉×#416	—	2	1	no	—
23	C₂×C₁₉×#103	—	2	1	no	—
24	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
25	C₁₉×#391	—	2	1	no	—
26	C₁₉×#360	—	2	2	no	—
27	C₁₉×#358	—	2	2	no	—
28	C₂×#41×C₁₉	—	2	1	no	—
29	C₂²×Q₈×C₁₉	—	2	1	no	—
30	C₁₉×#135	—	2	1	no	—
31	C₁₉×#135	—	2	1	no	—
32	C₁₉×#135	C₂×C₄	1	2	yes	—
33	C₁₉×#135	D₄	1	2	yes	—
34	C₂×Q₁₆×C₁₉	—	2	1	no	—
35	C₂×Q₁₆×C₁₉	—	2	1	no	—
36	C₂×Q₁₆×C₁₉	—	2	1	no	—
37	C₂×Q₁₆×C₁₉	—	2	1	no	—
38	C₂×Q₁₆×C₁₉	—	2	1	no	—
39	C₂×Q₁₆×C₁₉	C₂×C₄	1	2	yes	complemented
40	C₂×Q₁₆×C₁₉	D₄	1	1	yes	characteristic
41	C₂×SD₁₆×C₁₉	—	2	1	no	—
42	C₂×D₈×C₁₉	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
45	C₄×Q₈×C₁₉	—	2	2	no	—
46	C₄×D₄×C₁₉	—	2	2	no	—
47	C₄×D₄×C₁₉	—	4	2	no	—
48	C₁₉×#117	—	2	1	no	—
49	C₁₉×#117	—	4	1	no	—
50	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
51	C₂×C₄²×C₁₉	—	4	1	no	—
52	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
53	C₁₉×Q₃₂	—	2	4	no	—
54	C₁₉×SD₃₂	—	2	4	no	—
55	C₁₉×M₃₂	—	2	1	no	—
56	C₁₉×M₃₂	—	2	1	no	—
57	C₁₉×M₃₂	—	2	1	no	—
58	C₁₉×M₃₂	C₂×C₄	1	2	yes	complemented
59	C₂×C₁₆×C₁₉	—	2	1	no	—
60	C₂×C₁₆×C₁₉	—	2	1	no	—
61	C₁₉×#107	D₄	1	2	yes	—
62	C₁₉×#106	—	2	1	no	—
63	C₁₉×#106	—	2	1	no	—
64	C₁₉×#106	—	2	1	no	—
65	C₁₉×#103	—	2	2	no	—
66	C₁₉×#103	—	4	2	no	—
67	C₁₉×#102	—	2	2	no	—
68	C₁₉×#97	—	2	1	no	—
69	C₁₉×#97	—	2	1	no	—
70	C₁₉×#95	—	4	2	no	—
71	#41×C₁₉	—	2	2	no	—
72	#41×C₁₉	—	2	2	no	—
73	#41×C₁₉	—	4	1	no	—
74	#41×C₁₉	—	4	1	no	—
75	C₂×Q₈×C₁₉	—	2	2	no	—
76	C₂×Q₈×C₁₉	—	2	1	no	—
77	C₂×Q₈×C₁₉	—	4	1	no	—
78	C₂×Q₈×C₁₉	—	4	1	no	—
79	C₂×Q₈×C₁₉	—	4	1	no	—
80	C₂×Q₈×C₁₉	—	4	1	no	—
81	C₂×Q₈×C₁₉	—	4	1	no	—
82	C₂×D₄×C₁₉	—	2	1	no	—
83	C₂×D₄×C₁₉	—	4	1	no	—
84	C₂²×C₄×C₁₉	—	4	2	no	—
85	C₂²×C₄×C₁₉	—	4	1	no	—
86	C₂²×C₄×C₁₉	—	4	1	no	—
87	C₂²×C₄×C₁₉	—	4	1	no	—
88	C₂²×C₄×C₁₉	D₈	1	2	yes	—
89	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
90	Q₁₆×C₁₉	—	2	2	no	—
91	Q₁₆×C₁₉	—	2	2	no	—
92	Q₁₆×C₁₉	—	2	1	no	—
93	Q₁₆×C₁₉	—	4	1	no	—
94	Q₁₆×C₁₉	—	4	1	no	—
95	Q₁₆×C₁₉	C₂²⋊C₄	1	2	yes	—
96	SD₁₆×C₁₉	—	2	2	no	—
97	SD₁₆×C₁₉	—	2	2	no	—
98	D₈×C₁₉	—	2	1	no	—
99	D₈×C₁₉	C₂²⋊C₄	1	2	yes	—
100	C₂×C₈×C₁₉	—	2	1	no	—
101	C₂×C₈×C₁₉	—	2	1	no	—
102	C₂×C₈×C₁₉	—	2	1	no	—
103	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
108	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
109	C₄⋊C₄×C₁₉	—	2	2	no	—
110	C₄⋊C₄×C₁₉	—	4	2	no	—
111	C₄⋊C₄×C₁₉	—	4	1	no	—
112	C₄⋊C₄×C₁₉	—	4	1	no	—
113	C₂²⋊C₄×C₁₉	—	4	2	no	—
114	C₂²⋊C₄×C₁₉	—	4	1	no	—
115	C₄²×C₁₉	—	2	1	no	—
116	C₄²×C₁₉	—	2	1	no	—
117	C₄²×C₁₉	—	4	2	no	—
118	C₄²×C₁₉	—	4	1	no	—
119	C₁₆×C₁₉	—	2	2	no	—
120	C₁₆×C₁₉	—	4	1	no	—
121	#13026	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂³×C₁₉	—	4	1	no	—
123	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
124	Q₈×C₁₉	—	2	2	no	—
125	Q₈×C₁₉	—	4	1	no	—
126	Q₈×C₁₉	—	4	2	no	—
127	Q₈×C₁₉	—	4	2	no	—
128	Q₈×C₁₉	—	8	1	no	—
129	Q₈×C₁₉	—	8	1	no	—
130	D₄×C₁₉	—	2	2	no	—
131	D₄×C₁₉	—	4	2	no	—
132	D₄×C₁₉	—	4	2	no	—
133	D₄×C₁₉	—	4	1	no	—
134	C₂×C₄×C₁₉	—	2	2	no	—
135	C₂×C₄×C₁₉	—	4	1	no	—
136	C₂×C₄×C₁₉	—	4	2	no	—
137	C₂×C₄×C₁₉	—	4	2	no	—
138	C₂×C₄×C₁₉	—	4	2	no	—
139	C₂×C₄×C₁₉	—	4	2	no	—
140	C₂×C₄×C₁₉	—	4	2	no	—
141	C₂×C₄×C₁₉	—	8	4	no	—
142	C₂×C₄×C₁₉	—	8	1	no	—
143	C₂×C₄×C₁₉	—	8	1	no	—
144	C₂×C₄×C₁₉	—	8	2	no	—
145	C₂×C₄×C₁₉	—	8	1	no	—
146	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
148	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
149	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
150	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₈×C₁₉	—	4	1	no	—
153	C₈×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₈×C₁₉	C₄×D₄	1	1	yes	characteristic
155	C₈×C₁₉	#121	1	1	yes	characteristic
156	C₈×C₁₉	#123	1	1	yes	characteristic
157	C₂×#510	C₂×C₁₉	1	1	yes	characteristic
158	#2943	C₂×C₁₉	1	1	yes	characteristic
159	C₂×#360	C₂×C₁₉	1	1	yes	characteristic
160	#2141	C₂×C₁₉	1	1	yes	characteristic, complemented
161	#1971	C₂×C₁₉	1	1	yes	characteristic, complemented
162	#1386	C₂×C₁₉	1	1	yes	characteristic, complemented
163	#1377	C₂×C₁₉	1	1	yes	characteristic, complemented
164	C₂²×C₁₉	—	2	2	no	—
165	C₂²×C₁₉	—	4	2	no	—
166	C₂²×C₁₉	—	8	2	no	—
167	C₂²×C₁₉	C₄×D₈	1	1	yes	characteristic
168	C₂²×C₁₉	#447	1	1	yes	characteristic
169	C₂²×C₁₉	#459	1	1	yes	characteristic
170	C₄×C₁₉	—	4	2	no	—
171	C₄×C₁₉	—	4	2	no	—
172	C₄×C₁₉	—	8	2	no	—
173	C₄×C₁₉	—	8	2	no	—
174	C₄×C₁₉	—	16	1	no	—
175	C₄×C₁₉	#386	1	1	yes	characteristic
176	C₄×C₁₉	#442	1	1	yes	characteristic
177	C₄×C₁₉	#449	1	1	yes	characteristic
178	C₄×C₁₉	#463	1	1	yes	characteristic
179	C₂×#135	C₂²×C₁₉	1	1	yes	characteristic
180	C₂²×Q₁₆	C₂²×C₁₉	1	1	yes	characteristic
181	C₄×#41	—	2	1	no	—
182	#510	—	2	2	no	—
183	#510	C₄×C₁₉	1	4	yes	complemented
184	C₂×Q₃₂	—	2	1	no	—
185	C₂×SD₃₂	—	2	1	no	—
186	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
187	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
188	#442	—	2	1	no	—
189	#441	—	2	1	no	—
190	#440	—	2	2	no	—
191	#427	C₂²×C₁₉	1	1	yes	characteristic
192	C₂×#106	C₂²×C₁₉	1	1	yes	characteristic
193	#416	—	2	1	no	—
194	C₂×#103	—	2	1	no	—
195	C₂×#97	C₂²×C₁₉	1	1	yes	characteristic
196	#391	—	2	1	no	—
197	#360	—	2	2	no	—
198	#358	—	2	2	no	—
199	C₂×C₁₉	—	2	1	no	—
200	C₂×C₁₉	—	2	1	no	—
201	C₂×C₁₉	—	8	2	no	—
202	C₂×C₁₉	#1879	1	1	yes	characteristic
203	C₂×C₁₉	#2191	1	2	yes	—
204	C₂×#41	—	2	1	no	—
205	C₂²×Q₈	—	2	1	no	—
206	#135	—	2	1	no	—
207	#135	—	2	1	no	—
208	#135	C₂×C₄×C₁₉	1	2	yes	—
209	#135	D₄×C₁₉	1	2	yes	—
210	C₂×Q₁₆	—	2	1	no	—
211	C₂×Q₁₆	—	2	1	no	—
212	C₂×Q₁₆	—	2	1	no	—
213	C₂×Q₁₆	—	2	1	no	—
214	C₂×Q₁₆	—	2	1	no	—
215	C₂×Q₁₆	C₂×C₄×C₁₉	1	2	yes	complemented
216	C₂×Q₁₆	D₄×C₁₉	1	1	yes	characteristic
217	C₂×SD₁₆	—	2	1	no	—
218	C₂×D₈	D₄×C₁₉	1	1	yes	characteristic
219	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
220	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
221	C₄×Q₈	—	2	2	no	—
222	C₄×D₄	—	2	2	no	—
223	C₄×D₄	—	4	2	no	—
224	#117	—	2	1	no	—
225	#117	—	4	1	no	—
226	C₂×C₄⋊C₄	—	2	1	no	—
227	C₂×C₄²	—	4	1	no	—
228	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
229	Q₃₂	—	2	4	no	—
230	SD₃₂	—	2	4	no	—
231	M₃₂	—	2	1	no	—
232	M₃₂	—	2	1	no	—
233	M₃₂	—	2	1	no	—
234	M₃₂	C₂×C₄×C₁₉	1	2	yes	complemented
235	C₂×C₁₆	—	2	1	no	—
236	C₂×C₁₆	—	2	1	no	—
237	#107	D₄×C₁₉	1	2	yes	—
238	#106	—	2	1	no	—
239	#106	—	2	1	no	—
240	#106	—	2	1	no	—
241	#103	—	2	2	no	—
242	#103	—	4	2	no	—
243	#102	—	2	2	no	—
244	#97	—	2	1	no	—
245	#97	—	2	1	no	—
246	#95	—	4	2	no	—
247	C₁₉	#13026	1	1	yes	characteristic, direct product factor, Sylow
248	#41	—	2	2	no	—
249	#41	—	2	2	no	—
250	#41	—	4	1	no	—
251	#41	—	4	1	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	—	2	1	no	—
254	C₂×Q₈	—	4	1	no	—
255	C₂×Q₈	—	4	1	no	—
256	C₂×Q₈	—	4	1	no	—
257	C₂×Q₈	—	4	1	no	—
258	C₂×Q₈	—	4	1	no	—
259	C₂×D₄	—	2	1	no	—
260	C₂×D₄	—	4	1	no	—
261	C₂²×C₄	—	4	2	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	4	1	no	—
264	C₂²×C₄	—	4	1	no	—
265	C₂²×C₄	D₈×C₁₉	1	2	yes	—
266	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
267	Q₁₆	—	2	2	no	—
268	Q₁₆	—	2	2	no	—
269	Q₁₆	—	2	1	no	—
270	Q₁₆	—	4	1	no	—
271	Q₁₆	—	4	1	no	—
272	Q₁₆	C₂²⋊C₄×C₁₉	1	2	yes	—
273	SD₁₆	—	2	2	no	—
274	SD₁₆	—	2	2	no	—
275	D₈	—	2	1	no	—
276	D₈	C₂²⋊C₄×C₁₉	1	2	yes	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	2	1	no	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
281	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
282	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
283	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
284	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
285	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	4	2	no	—
288	C₄⋊C₄	—	4	1	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₂²⋊C₄	—	4	2	no	—
291	C₂²⋊C₄	—	4	1	no	—
292	C₄²	—	2	1	no	—
293	C₄²	—	2	1	no	—
294	C₄²	—	4	2	no	—
295	C₄²	—	4	1	no	—
296	C₁₆	—	2	2	no	—
297	C₁₆	—	4	1	no	—
298	C₂³	—	4	1	no	—
299	C₂³	C₂×D₈×C₁₉	1	1	yes	characteristic
300	Q₈	—	2	2	no	—
301	Q₈	—	4	1	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	8	1	no	—
305	Q₈	—	8	1	no	—
306	D₄	—	2	2	no	—
307	D₄	—	4	2	no	—
308	D₄	—	4	2	no	—
309	D₄	—	4	1	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	8	4	no	—
318	C₂×C₄	—	8	1	no	—
319	C₂×C₄	—	8	1	no	—
320	C₂×C₄	—	8	2	no	—
321	C₂×C₄	—	8	1	no	—
322	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
323	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
324	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
325	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
326	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
327	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
328	C₈	—	4	1	no	—
329	C₈	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
330	C₈	C₄×D₄×C₁₉	1	1	yes	characteristic
331	C₈	C₁₉×#121	1	1	yes	characteristic
332	C₈	C₁₉×#123	1	1	yes	characteristic
333	C₂²	—	2	2	no	—
334	C₂²	—	4	2	no	—
335	C₂²	—	8	2	no	—
336	C₂²	C₄×D₈×C₁₉	1	1	yes	characteristic
337	C₂²	C₁₉×#447	1	1	yes	characteristic
338	C₂²	C₁₉×#459	1	1	yes	characteristic
339	C₄	—	4	2	no	—
340	C₄	—	4	2	no	—
341	C₄	—	8	2	no	—
342	C₄	—	8	2	no	—
343	C₄	—	16	1	no	—
344	C₄	C₁₉×#386	1	1	yes	characteristic
345	C₄	C₁₉×#442	1	1	yes	characteristic
346	C₄	C₁₉×#449	1	1	yes	characteristic
347	C₄	C₁₉×#463	1	1	yes	characteristic
348	C₂	—	2	1	no	—
349	C₂	—	2	1	no	—
350	C₂	—	8	2	no	—
351	C₂	C₁₉×#1879	1	1	yes	characteristic
352	C₂	C₁₉×#2191	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉	#447
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₂×C₁₉×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₁₉×#97 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₉×#2141	C₂	semidirect
5. C₁₉×#1971	C₂	semidirect
6. C₁₉×#1386	C₂	semidirect
7. C₁₉×#1377	C₂	semidirect
12. C₁₉×#510	C₄	semidirect
39. C₂×Q₁₆×C₁₉	C₂×C₄	semidirect
58. C₁₉×M₃₂	C₂×C₄	semidirect
121. #13026	C₁₉	direct (N × H)
160. #2141	C₂×C₁₉	semidirect
161. #1971	C₂×C₁₉	semidirect
162. #1386	C₂×C₁₉	semidirect
163. #1377	C₂×C₁₉	semidirect
183. #510	C₄×C₁₉	semidirect
215. C₂×Q₁₆	C₂×C₄×C₁₉	semidirect
234. M₃₂	C₂×C₄×C₁₉	semidirect
247. C₁₉	#13026	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×#510	2
C₁₉×#2943	2
C₂×C₁₉×#360	2
C₁₉×#2141	2
C₁₉×#1971	2
C₁₉×#1386	2
C₁₉×#1377	2
#13026	19

Subgroups — C19×#13026 #18018985

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13026 #18018985