Subgroups — C₁₉×#13024 #18018983

Order 4864.

466 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×#3414	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#444	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#2148	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#2143	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1975	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#1388	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	C₂×C₁₉×#135	C₂²	1	1	yes	characteristic
10	C₂×C₁₉×#131	—	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	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
18	C₁₉×#444	—	2	2	no	—
19	C₁₉×#444	—	2	2	no	—
20	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
21	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
22	C₂×C₁₉×#104	—	2	1	no	—
23	C₂×C₁₉×#104	—	2	1	no	—
24	C₁₉×#408	—	2	1	no	—
25	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
26	C₁₉×#362	—	2	1	no	—
27	C₁₉×#361	—	2	1	no	—
28	C₁₉×#359	—	2	2	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	—	2	1	no	—
49	C₂×M₁₆×C₁₉	—	4	1	no	—
50	C₂²×C₈×C₁₉	—	4	1	no	—
51	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
54	C₁₉×Q₃₂	—	2	2	no	—
55	C₁₉×SD₃₂	—	2	2	no	—
56	C₁₉×SD₃₂	—	2	2	no	—
57	C₁₉×D₁₆	—	2	2	no	—
58	C₁₉×M₃₂	—	2	1	no	—
59	C₁₉×M₃₂	—	2	1	no	—
60	C₁₉×M₃₂	—	2	1	no	—
61	C₁₉×M₃₂	C₂×C₄	1	2	yes	—
62	C₂×C₁₆×C₁₉	—	2	1	no	—
63	C₂×C₁₆×C₁₉	—	2	1	no	—
64	C₁₉×#108	—	2	1	no	—
65	C₁₉×#108	—	2	1	no	—
66	C₁₉×#108	—	2	1	no	—
67	C₁₉×#108	D₄	1	2	yes	—
68	C₁₉×#104	—	2	4	no	—
69	C₁₉×#104	—	4	2	no	—
70	C₁₉×C₂²⋊C₈	—	4	2	no	—
71	C₁₉×#97	—	2	1	no	—
72	C₁₉×#97	—	2	1	no	—
73	#41×C₁₉	—	2	2	no	—
74	#41×C₁₉	—	2	2	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	#41×C₁₉	—	4	1	no	—
81	C₂×Q₈×C₁₉	—	2	1	no	—
82	C₂×Q₈×C₁₉	—	2	1	no	—
83	C₂×D₄×C₁₉	—	2	1	no	—
84	C₂×D₄×C₁₉	—	2	1	no	—
85	C₂×D₄×C₁₉	—	4	1	no	—
86	C₂×D₄×C₁₉	—	4	1	no	—
87	C₂²×C₄×C₁₉	—	2	1	no	—
88	C₂²×C₄×C₁₉	—	4	1	no	—
89	C₂²×C₄×C₁₉	—	4	1	no	—
90	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
91	Q₁₆×C₁₉	—	2	1	no	—
92	Q₁₆×C₁₉	—	2	1	no	—
93	Q₁₆×C₁₉	—	2	1	no	—
94	Q₁₆×C₁₉	C₂²⋊C₄	1	2	yes	—
95	SD₁₆×C₁₉	—	2	2	no	—
96	SD₁₆×C₁₉	—	2	2	no	—
97	SD₁₆×C₁₉	—	4	1	no	—
98	SD₁₆×C₁₉	—	4	1	no	—
99	D₈×C₁₉	—	2	1	no	—
100	D₈×C₁₉	—	2	1	no	—
101	D₈×C₁₉	—	2	1	no	—
102	D₈×C₁₉	C₂²⋊C₄	1	2	yes	—
103	M₁₆×C₁₉	—	2	2	no	—
104	M₁₆×C₁₉	—	4	4	no	—
105	M₁₆×C₁₉	—	4	1	no	—
106	M₁₆×C₁₉	—	4	1	no	—
107	M₁₆×C₁₉	—	4	1	no	—
108	C₂×C₈×C₁₉	—	2	1	no	—
109	C₂×C₈×C₁₉	—	2	1	no	—
110	C₂×C₈×C₁₉	—	4	4	no	—
111	C₂×C₈×C₁₉	—	4	1	no	—
112	C₂×C₈×C₁₉	—	4	1	no	—
113	C₂×C₈×C₁₉	D₈	1	2	yes	—
114	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
119	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
120	C₄²×C₁₉	—	2	1	no	—
121	C₄²×C₁₉	—	2	1	no	—
122	C₁₆×C₁₉	—	2	2	no	—
123	C₁₆×C₁₉	—	4	1	no	—
124	#13024	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂³×C₁₉	—	4	1	no	—
126	C₂³×C₁₉	—	4	1	no	—
127	C₂³×C₁₉	#135	1	1	yes	characteristic
128	Q₈×C₁₉	—	2	2	no	—
129	Q₈×C₁₉	—	4	1	no	—
130	Q₈×C₁₉	—	4	1	no	—
131	Q₈×C₁₉	—	4	1	no	—
132	D₄×C₁₉	—	2	2	no	—
133	D₄×C₁₉	—	4	2	no	—
134	D₄×C₁₉	—	4	2	no	—
135	D₄×C₁₉	—	4	1	no	—
136	D₄×C₁₉	—	4	1	no	—
137	D₄×C₁₉	—	4	1	no	—
138	D₄×C₁₉	—	8	1	no	—
139	D₄×C₁₉	—	8	1	no	—
140	C₂×C₄×C₁₉	—	4	1	no	—
141	C₂×C₄×C₁₉	—	4	2	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	—	8	1	no	—
148	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
149	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
150	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
151	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
154	C₈×C₁₉	—	2	2	no	—
155	C₈×C₁₉	—	4	2	no	—
156	C₈×C₁₉	—	8	1	no	—
157	C₈×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
158	C₈×C₁₉	C₄×D₄	1	1	yes	characteristic
159	C₈×C₁₉	#121	1	1	yes	characteristic
160	C₈×C₁₉	#123	1	1	yes	characteristic
161	#3414	C₂×C₁₉	1	1	yes	characteristic
162	C₂×#444	C₂×C₁₉	1	1	yes	characteristic, complemented
163	#2148	C₂×C₁₉	1	1	yes	characteristic, complemented
164	#2143	C₂×C₁₉	1	1	yes	characteristic, complemented
165	#1975	C₂×C₁₉	1	1	yes	characteristic, complemented
166	#1388	C₂×C₁₉	1	1	yes	characteristic, complemented
167	#1377	C₂×C₁₉	1	1	yes	characteristic, complemented
168	C₂²×C₁₉	—	4	1	no	—
169	C₂²×C₁₉	—	4	2	no	—
170	C₂²×C₁₉	—	8	2	no	—
171	C₂²×C₁₉	—	8	2	no	—
172	C₂²×C₁₉	—	8	1	no	—
173	C₂²×C₁₉	C₄×D₈	1	1	yes	characteristic
174	C₂²×C₁₉	#449	1	1	yes	characteristic
175	C₂²×C₁₉	#463	1	1	yes	characteristic
176	C₄×C₁₉	—	4	2	no	—
177	C₄×C₁₉	—	8	1	no	—
178	C₄×C₁₉	—	8	1	no	—
179	C₄×C₁₉	#386	1	1	yes	characteristic
180	C₄×C₁₉	#442	1	1	yes	characteristic
181	C₄×C₁₉	#447	1	1	yes	characteristic
182	C₄×C₁₉	#459	1	1	yes	characteristic
183	C₂×#135	C₂²×C₁₉	1	1	yes	characteristic
184	C₂×#135	C₂²×C₁₉	1	1	yes	characteristic
185	C₂×#131	—	2	1	no	—
186	#510	—	2	1	no	—
187	#510	C₄×C₁₉	1	2	yes	complemented
188	#509	—	2	1	no	—
189	#509	C₄×C₁₉	1	2	yes	complemented
190	#508	—	2	2	no	—
191	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
192	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
193	#444	—	2	2	no	—
194	#444	—	2	2	no	—
195	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
196	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
197	C₂×#104	—	2	1	no	—
198	C₂×#104	—	2	1	no	—
199	#408	—	2	1	no	—
200	C₂×#97	C₂²×C₁₉	1	1	yes	characteristic
201	#362	—	2	1	no	—
202	#361	—	2	1	no	—
203	#359	—	2	2	no	—
204	C₂×C₁₉	—	2	1	no	—
205	C₂×C₁₉	—	2	1	no	—
206	C₂×C₁₉	—	2	1	no	—
207	C₂×C₁₉	—	8	2	no	—
208	C₂×C₁₉	—	16	1	no	—
209	C₂×C₁₉	#1879	1	1	yes	characteristic
210	C₂×#41	—	2	1	no	—
211	C₂×#41	—	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	—	2	1	no	—
218	#135	C₂×C₄×C₁₉	1	2	yes	—
219	#135	D₄×C₁₉	1	2	yes	—
220	C₂×Q₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
221	C₂×Q₁₆	D₄×C₁₉	1	1	yes	characteristic
222	C₂×SD₁₆	—	2	1	no	—
223	C₂×SD₁₆	—	2	1	no	—
224	C₂×D₈	C₂×C₄×C₁₉	1	1	yes	characteristic
225	C₂×D₈	D₄×C₁₉	1	1	yes	characteristic
226	#131	—	2	4	no	—
227	#131	—	4	2	no	—
228	C₂×M₁₆	—	2	1	no	—
229	C₂×M₁₆	—	2	1	no	—
230	C₂×M₁₆	—	4	1	no	—
231	C₂²×C₈	—	4	1	no	—
232	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
233	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
234	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
235	Q₃₂	—	2	2	no	—
236	SD₃₂	—	2	2	no	—
237	SD₃₂	—	2	2	no	—
238	D₁₆	—	2	2	no	—
239	M₃₂	—	2	1	no	—
240	M₃₂	—	2	1	no	—
241	M₃₂	—	2	1	no	—
242	M₃₂	C₂×C₄×C₁₉	1	2	yes	—
243	C₂×C₁₆	—	2	1	no	—
244	C₂×C₁₆	—	2	1	no	—
245	#108	—	2	1	no	—
246	#108	—	2	1	no	—
247	#108	—	2	1	no	—
248	#108	D₄×C₁₉	1	2	yes	—
249	#104	—	2	4	no	—
250	#104	—	4	2	no	—
251	C₂²⋊C₈	—	4	2	no	—
252	#97	—	2	1	no	—
253	#97	—	2	1	no	—
254	C₁₉	#13024	1	1	yes	characteristic, direct product factor, Sylow
255	#41	—	2	2	no	—
256	#41	—	2	2	no	—
257	#41	—	4	1	no	—
258	#41	—	4	1	no	—
259	#41	—	4	1	no	—
260	#41	—	4	1	no	—
261	#41	—	4	1	no	—
262	#41	—	4	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	—	4	1	no	—
268	C₂×D₄	—	4	1	no	—
269	C₂²×C₄	—	2	1	no	—
270	C₂²×C₄	—	4	1	no	—
271	C₂²×C₄	—	4	1	no	—
272	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
273	Q₁₆	—	2	1	no	—
274	Q₁₆	—	2	1	no	—
275	Q₁₆	—	2	1	no	—
276	Q₁₆	C₂²⋊C₄×C₁₉	1	2	yes	—
277	SD₁₆	—	2	2	no	—
278	SD₁₆	—	2	2	no	—
279	SD₁₆	—	4	1	no	—
280	SD₁₆	—	4	1	no	—
281	D₈	—	2	1	no	—
282	D₈	—	2	1	no	—
283	D₈	—	2	1	no	—
284	D₈	C₂²⋊C₄×C₁₉	1	2	yes	—
285	M₁₆	—	2	2	no	—
286	M₁₆	—	4	4	no	—
287	M₁₆	—	4	1	no	—
288	M₁₆	—	4	1	no	—
289	M₁₆	—	4	1	no	—
290	C₂×C₈	—	2	1	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	4	4	no	—
293	C₂×C₈	—	4	1	no	—
294	C₂×C₈	—	4	1	no	—
295	C₂×C₈	D₈×C₁₉	1	2	yes	—
296	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
297	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
298	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
299	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
300	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
301	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
302	C₄²	—	2	1	no	—
303	C₄²	—	2	1	no	—
304	C₁₆	—	2	2	no	—
305	C₁₆	—	4	1	no	—
306	C₂³	—	4	1	no	—
307	C₂³	—	4	1	no	—
308	C₂³	C₁₉×#135	1	1	yes	characteristic
309	Q₈	—	2	2	no	—
310	Q₈	—	4	1	no	—
311	Q₈	—	4	1	no	—
312	Q₈	—	4	1	no	—
313	D₄	—	2	2	no	—
314	D₄	—	4	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	1	no	—
317	D₄	—	4	1	no	—
318	D₄	—	4	1	no	—
319	D₄	—	8	1	no	—
320	D₄	—	8	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	8	1	no	—
327	C₂×C₄	—	8	1	no	—
328	C₂×C₄	—	8	1	no	—
329	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
330	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
331	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
332	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
333	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
334	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
335	C₈	—	2	2	no	—
336	C₈	—	4	2	no	—
337	C₈	—	8	1	no	—
338	C₈	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
339	C₈	C₄×D₄×C₁₉	1	1	yes	characteristic
340	C₈	C₁₉×#121	1	1	yes	characteristic
341	C₈	C₁₉×#123	1	1	yes	characteristic
342	C₂²	—	4	1	no	—
343	C₂²	—	4	2	no	—
344	C₂²	—	8	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	—	8	1	no	—
347	C₂²	C₄×D₈×C₁₉	1	1	yes	characteristic
348	C₂²	C₁₉×#449	1	1	yes	characteristic
349	C₂²	C₁₉×#463	1	1	yes	characteristic
350	C₄	—	4	2	no	—
351	C₄	—	8	1	no	—
352	C₄	—	8	1	no	—
353	C₄	C₁₉×#386	1	1	yes	characteristic
354	C₄	C₁₉×#442	1	1	yes	characteristic
355	C₄	C₁₉×#447	1	1	yes	characteristic
356	C₄	C₁₉×#459	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	8	2	no	—
361	C₂	—	16	1	no	—
362	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 ◁ C₂×C₁₉×#97 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₉×#444	C₂	semidirect
3. C₁₉×#2148	C₂	semidirect
4. C₁₉×#2143	C₂	semidirect
5. C₁₉×#1975	C₂	semidirect
6. C₁₉×#1388	C₂	semidirect
7. C₁₉×#1377	C₂	semidirect
12. C₁₉×#510	C₄	semidirect
14. C₁₉×#509	C₄	semidirect
124. #13024	C₁₉	direct (N × H)
162. C₂×#444	C₂×C₁₉	semidirect
163. #2148	C₂×C₁₉	semidirect
164. #2143	C₂×C₁₉	semidirect
165. #1975	C₂×C₁₉	semidirect
166. #1388	C₂×C₁₉	semidirect
167. #1377	C₂×C₁₉	semidirect
187. #510	C₄×C₁₉	semidirect
189. #509	C₄×C₁₉	semidirect
254. C₁₉	#13024	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
C₂×C₁₉×#444	2
C₁₉×#2148	2
C₁₉×#2143	2
C₁₉×#1975	2
C₁₉×#1388	2
C₁₉×#1377	2
#13024	19

Subgroups — C19×#13024 #18018983

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13024 #18018983