Subgroups — C₁₉×#12815 #18018774

Order 4864.

422 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₁₉×#135	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₉×#359	C₂	1	2	yes	maximal
4	C₁₉×#1848	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1382	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₉×#135	C₂²	1	1	yes	characteristic
7	C₄×#41×C₁₉	—	2	1	no	—
8	C₂²×C₁₆×C₁₉	C₂²	1	2	yes	—
9	C₄×Q₁₆×C₁₉	C₄	1	1	yes	characteristic
10	C₄×Q₁₆×C₁₉	C₄	1	1	yes	characteristic
11	C₄×SD₁₆×C₁₉	—	2	1	no	—
12	C₄×SD₁₆×C₁₉	—	2	1	no	—
13	C₄×D₈×C₁₉	C₄	1	1	yes	characteristic
14	C₄×D₈×C₁₉	C₄	1	1	yes	characteristic
15	C₂×C₁₉×#108	—	2	1	no	—
16	C₂×C₁₉×#108	C₂²	1	2	yes	—
17	C₁₉×#427	C₂²	1	1	yes	characteristic
18	C₁₉×#423	—	2	1	no	—
19	C₁₉×#416	—	2	1	no	—
20	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
21	C₁₉×C₄⋊C₁₆	—	2	1	no	—
22	C₁₉×C₄⋊C₁₆	—	2	1	no	—
23	C₁₉×#359	—	2	4	no	—
24	C₂×#41×C₁₉	—	2	1	no	—
25	C₁₉×#135	—	2	1	no	—
26	C₁₉×#135	—	2	1	no	—
27	C₁₉×#135	D₄	1	2	yes	—
28	C₁₉×#135	Q₈	1	2	yes	—
29	C₂×Q₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₂×SD₁₆×C₁₉	—	2	1	no	—
31	C₂×D₈×C₁₉	C₂×C₄	1	1	yes	characteristic
32	C₂×M₁₆×C₁₉	—	2	2	no	—
33	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
35	C₄×Q₈×C₁₉	—	2	1	no	—
36	C₄×Q₈×C₁₉	—	2	1	no	—
37	C₄×D₄×C₁₉	—	2	1	no	—
38	C₄×D₄×C₁₉	—	2	1	no	—
39	C₄×D₄×C₁₉	—	4	1	no	—
40	C₄×D₄×C₁₉	—	4	1	no	—
41	C₁₉×#117	—	2	1	no	—
42	C₁₉×#117	—	4	1	no	—
43	C₂×C₄²×C₁₉	—	4	1	no	—
44	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
45	C₂×C₁₆×C₁₉	—	2	2	no	—
46	C₂×C₁₆×C₁₉	—	2	2	no	—
47	C₂×C₁₆×C₁₉	—	2	2	no	—
48	C₁₉×#108	—	2	2	no	—
49	C₁₉×#108	—	2	2	no	—
50	C₁₉×#108	—	4	2	no	—
51	C₁₉×#107	C₂×C₄	1	1	yes	characteristic
52	C₁₉×#107	C₂×C₄	1	1	yes	characteristic
53	C₁₉×#106	—	2	1	no	—
54	C₁₉×C₄⋊C₈	—	4	1	no	—
55	C₁₉×C₄⋊C₈	—	4	1	no	—
56	C₁₉×#103	—	2	1	no	—
57	C₁₉×#103	—	2	1	no	—
58	C₁₉×#102	—	2	1	no	—
59	C₁₉×#102	—	2	1	no	—
60	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
63	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
64	#41×C₁₉	—	2	2	no	—
65	#41×C₁₉	—	2	2	no	—
66	#41×C₁₉	—	4	1	no	—
67	#41×C₁₉	—	4	1	no	—
68	C₂×Q₈×C₁₉	—	2	1	no	—
69	C₂×D₄×C₁₉	—	2	1	no	—
70	C₂×D₄×C₁₉	—	4	1	no	—
71	C₂²×C₄×C₁₉	—	4	1	no	—
72	C₂²×C₄×C₁₉	—	4	1	no	—
73	C₂²×C₄×C₁₉	—	4	1	no	—
74	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
76	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
77	Q₁₆×C₁₉	—	2	1	no	—
78	Q₁₆×C₁₉	C₄⋊C₄	1	2	yes	—
79	SD₁₆×C₁₉	—	2	2	no	—
80	SD₁₆×C₁₉	—	2	2	no	—
81	D₈×C₁₉	—	2	1	no	—
82	D₈×C₁₉	C₄⋊C₄	1	2	yes	—
83	M₁₆×C₁₉	—	4	2	no	—
84	M₁₆×C₁₉	—	4	2	no	—
85	C₂×C₈×C₁₉	—	2	1	no	—
86	C₂×C₈×C₁₉	—	2	1	no	—
87	C₂×C₈×C₁₉	—	4	2	no	—
88	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
94	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
95	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
96	C₄⋊C₄×C₁₉	—	2	1	no	—
97	C₄⋊C₄×C₁₉	—	2	1	no	—
98	C₄⋊C₄×C₁₉	—	4	1	no	—
99	C₄⋊C₄×C₁₉	—	4	1	no	—
100	C₂²⋊C₄×C₁₉	—	4	1	no	—
101	C₂²⋊C₄×C₁₉	—	4	1	no	—
102	C₂²⋊C₄×C₁₉	—	4	1	no	—
103	C₄²×C₁₉	—	4	1	no	—
104	C₄²×C₁₉	—	4	1	no	—
105	C₄²×C₁₉	—	4	1	no	—
106	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
107	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
108	C₄²×C₁₉	D₈	1	1	yes	characteristic
109	C₄²×C₁₉	D₈	1	1	yes	characteristic
110	C₁₆×C₁₉	—	4	2	no	—
111	#12815	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂³×C₁₉	—	4	1	no	—
113	C₂³×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
114	Q₈×C₁₉	—	2	2	no	—
115	Q₈×C₁₉	—	4	1	no	—
116	D₄×C₁₉	—	2	2	no	—
117	D₄×C₁₉	—	4	2	no	—
118	D₄×C₁₉	—	4	2	no	—
119	D₄×C₁₉	—	4	1	no	—
120	C₂×C₄×C₁₉	—	2	2	no	—
121	C₂×C₄×C₁₉	—	4	1	no	—
122	C₂×C₄×C₁₉	—	4	1	no	—
123	C₂×C₄×C₁₉	—	4	2	no	—
124	C₂×C₄×C₁₉	—	4	2	no	—
125	C₂×C₄×C₁₉	—	4	1	no	—
126	C₂×C₄×C₁₉	—	4	1	no	—
127	C₂×C₄×C₁₉	—	8	2	no	—
128	C₂×C₄×C₁₉	—	8	2	no	—
129	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
130	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
131	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
132	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
133	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
135	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
136	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
139	C₈×C₁₉	—	2	1	no	—
140	C₈×C₁₉	—	2	1	no	—
141	C₈×C₁₉	—	8	2	no	—
142	C₈×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₈×C₁₉	#117	1	1	yes	characteristic
144	C₈×C₁₉	#121	1	1	yes	characteristic
145	C₈×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
146	C₄×#135	C₂×C₁₉	1	1	yes	characteristic
147	C₂×C₄⋊C₁₆	C₂×C₁₉	1	1	yes	characteristic, complemented
148	C₂×#359	C₂×C₁₉	1	2	yes	—
149	#1848	C₂×C₁₉	1	1	yes	characteristic, complemented
150	#1382	C₂×C₁₉	1	2	yes	complemented
151	C₂²×C₁₉	—	2	1	no	—
152	C₂²×C₁₉	—	2	1	no	—
153	C₂²×C₁₉	—	4	2	no	—
154	C₂²×C₁₉	—	8	1	no	—
155	C₂²×C₁₉	—	8	1	no	—
156	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
157	C₂²×C₁₉	#461	1	1	yes	characteristic
158	C₂²×C₁₉	#476	1	1	yes	characteristic
159	C₄×C₁₉	—	2	1	no	—
160	C₄×C₁₉	—	2	1	no	—
161	C₄×C₁₉	—	2	1	no	—
162	C₄×C₁₉	—	4	2	no	—
163	C₄×C₁₉	—	8	1	no	—
164	C₄×C₁₉	—	8	1	no	—
165	C₄×C₁₉	#359	1	2	yes	complemented
166	C₄×C₁₉	#380	1	1	yes	characteristic
167	C₄×C₁₉	#417	1	1	yes	characteristic
168	C₄×C₁₉	#459	1	1	yes	characteristic
169	C₄×C₁₉	#474	1	1	yes	characteristic
170	C₂×#135	C₂²×C₁₉	1	1	yes	characteristic
171	C₄×#41	—	2	1	no	—
172	C₂²×C₁₆	C₂²×C₁₉	1	2	yes	—
173	C₄×Q₁₆	C₄×C₁₉	1	1	yes	characteristic
174	C₄×Q₁₆	C₄×C₁₉	1	1	yes	characteristic
175	C₄×SD₁₆	—	2	1	no	—
176	C₄×SD₁₆	—	2	1	no	—
177	C₄×D₈	C₄×C₁₉	1	1	yes	characteristic
178	C₄×D₈	C₄×C₁₉	1	1	yes	characteristic
179	C₂×#108	—	2	1	no	—
180	C₂×#108	C₂²×C₁₉	1	2	yes	—
181	#427	C₂²×C₁₉	1	1	yes	characteristic
182	#423	—	2	1	no	—
183	#416	—	2	1	no	—
184	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
185	C₄⋊C₁₆	—	2	1	no	—
186	C₄⋊C₁₆	—	2	1	no	—
187	#359	—	2	4	no	—
188	C₂×C₁₉	—	2	1	no	—
189	C₂×C₁₉	—	2	1	no	—
190	C₂×C₁₉	—	8	2	no	—
191	C₂×C₁₉	#1802	1	1	yes	characteristic
192	C₂×C₁₉	C₂×#359	1	1	yes	characteristic
193	C₂×C₁₉	#2143	1	1	yes	characteristic
194	C₂×#41	—	2	1	no	—
195	#135	—	2	1	no	—
196	#135	—	2	1	no	—
197	#135	D₄×C₁₉	1	2	yes	—
198	#135	Q₈×C₁₉	1	2	yes	—
199	C₂×Q₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
200	C₂×SD₁₆	—	2	1	no	—
201	C₂×D₈	C₂×C₄×C₁₉	1	1	yes	characteristic
202	C₂×M₁₆	—	2	2	no	—
203	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
204	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
205	C₄×Q₈	—	2	1	no	—
206	C₄×Q₈	—	2	1	no	—
207	C₄×D₄	—	2	1	no	—
208	C₄×D₄	—	2	1	no	—
209	C₄×D₄	—	4	1	no	—
210	C₄×D₄	—	4	1	no	—
211	#117	—	2	1	no	—
212	#117	—	4	1	no	—
213	C₂×C₄²	—	4	1	no	—
214	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
215	C₂×C₁₆	—	2	2	no	—
216	C₂×C₁₆	—	2	2	no	—
217	C₂×C₁₆	—	2	2	no	—
218	#108	—	2	2	no	—
219	#108	—	2	2	no	—
220	#108	—	4	2	no	—
221	#107	C₂×C₄×C₁₉	1	1	yes	characteristic
222	#107	C₂×C₄×C₁₉	1	1	yes	characteristic
223	#106	—	2	1	no	—
224	C₄⋊C₈	—	4	1	no	—
225	C₄⋊C₈	—	4	1	no	—
226	#103	—	2	1	no	—
227	#103	—	2	1	no	—
228	#102	—	2	1	no	—
229	#102	—	2	1	no	—
230	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
231	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
232	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
233	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
234	C₁₉	#12815	1	1	yes	characteristic, direct product factor, Sylow
235	#41	—	2	2	no	—
236	#41	—	2	2	no	—
237	#41	—	4	1	no	—
238	#41	—	4	1	no	—
239	C₂×Q₈	—	2	1	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	4	1	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	—	4	1	no	—
245	C₂²×C₄	SD₁₆×C₁₉	1	1	yes	characteristic
246	C₂²×C₄	SD₁₆×C₁₉	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
248	Q₁₆	—	2	1	no	—
249	Q₁₆	C₄⋊C₄×C₁₉	1	2	yes	—
250	SD₁₆	—	2	2	no	—
251	SD₁₆	—	2	2	no	—
252	D₈	—	2	1	no	—
253	D₈	C₄⋊C₄×C₁₉	1	2	yes	—
254	M₁₆	—	4	2	no	—
255	M₁₆	—	4	2	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	4	2	no	—
259	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
260	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
261	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
262	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
263	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
264	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
265	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
266	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₂²⋊C₄	—	4	1	no	—
272	C₂²⋊C₄	—	4	1	no	—
273	C₂²⋊C₄	—	4	1	no	—
274	C₄²	—	4	1	no	—
275	C₄²	—	4	1	no	—
276	C₄²	—	4	1	no	—
277	C₄²	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
278	C₄²	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
279	C₄²	D₈×C₁₉	1	1	yes	characteristic
280	C₄²	D₈×C₁₉	1	1	yes	characteristic
281	C₁₆	—	4	2	no	—
282	C₂³	—	4	1	no	—
283	C₂³	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
284	Q₈	—	2	2	no	—
285	Q₈	—	4	1	no	—
286	D₄	—	2	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	4	2	no	—
289	D₄	—	4	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	8	2	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	C₁₉×#102	1	1	yes	characteristic
300	C₂×C₄	C₁₉×#102	1	1	yes	characteristic
301	C₂×C₄	C₁₉×#102	1	1	yes	characteristic
302	C₂×C₄	C₁₉×#102	1	1	yes	characteristic
303	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
304	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
305	C₂×C₄	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
306	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
307	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
308	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	8	2	no	—
312	C₈	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
313	C₈	C₁₉×#117	1	1	yes	characteristic
314	C₈	C₁₉×#121	1	1	yes	characteristic
315	C₈	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	4	2	no	—
319	C₂²	—	8	1	no	—
320	C₂²	—	8	1	no	—
321	C₂²	C₂×C₁₉×#102	1	1	yes	characteristic
322	C₂²	C₁₉×#461	1	1	yes	characteristic
323	C₂²	C₁₉×#476	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	4	2	no	—
328	C₄	—	8	1	no	—
329	C₄	—	8	1	no	—
330	C₄	C₁₉×#359	1	2	yes	complemented
331	C₄	C₁₉×#380	1	1	yes	characteristic
332	C₄	C₁₉×#417	1	1	yes	characteristic
333	C₄	C₁₉×#459	1	1	yes	characteristic
334	C₄	C₁₉×#474	1	1	yes	characteristic
335	C₂	—	2	1	no	—
336	C₂	—	2	1	no	—
337	C₂	—	8	2	no	—
338	C₂	C₁₉×#1802	1	1	yes	characteristic
339	C₂	C₂×C₁₉×#359	1	1	yes	characteristic
340	C₂	C₁₉×#2143	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₉	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
socle	C₂²×C₁₉	C₂×#102
prime radical	C₂×C₁₉×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₉×C₄⋊C₁₆	C₂	semidirect
4. C₁₉×#1848	C₂	semidirect
5. C₁₉×#1382	C₂	semidirect
111. #12815	C₁₉	direct (N × H)
147. C₂×C₄⋊C₁₆	C₂×C₁₉	semidirect
149. #1848	C₂×C₁₉	semidirect
150. #1382	C₂×C₁₉	semidirect
165. C₄×C₁₉	#359	semidirect
234. C₁₉	#12815	direct (N × H)
330. C₄	C₁₉×#359	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₁₉×#135	2
C₂×C₁₉×C₄⋊C₁₆	2
C₂×C₁₉×#359	2
C₁₉×#1848	2
C₁₉×#1382	2
#12815	19

Subgroups — C19×#12815 #18018774

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12815 #18018774