Subgroups — C₁₉×#18994 #18024953

Order 4864.

936 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#563	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₉×#3234	C₂	1	2	yes	maximal, complemented
3	C₁₉×#3153	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1660	C₂	1	4	yes	maximal, complemented
5	C₁₉×#1657	C₂	1	4	yes	maximal, complemented
6	C₁₉×#1533	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1528	C₂	1	2	yes	maximal, complemented
8	C₂×C₁₉×#137	—	2	1	no	—
9	C₂×C₁₉×AGL(1,ℤ/8)	—	2	1	no	—
10	C₁₉×#563	—	2	4	no	—
11	C₁₉×#563	—	2	2	no	—
12	C₁₉×#563	C₂²	1	4	yes	—
13	C₁₉×#532	C₂²	1	1	yes	characteristic
14	C₂×C₁₉×C₄⋊Q₈	—	2	1	no	—
15	C₂×C₁₉×#126	—	2	2	no	—
16	C₂×C₁₉×#125	C₂²	1	2	yes	—
17	C₂×C₁₉×#124	C₂²	1	1	yes	characteristic
18	C₂×C₁₉×#123	—	2	1	no	—
19	C₂×C₁₉×C₂²⋊Q₈	—	2	2	no	—
20	C₁₉×#501	C₂²	1	4	yes	complemented
21	C₁₉×#497	C₂²	1	2	yes	—
22	C₁₉×#496	C₂²	1	2	yes	—
23	C₁₉×#495	—	2	2	no	—
24	C₁₉×#490	C₂²	1	4	yes	complemented
25	C₁₉×#481	—	2	2	no	—
26	C₁₉×#480	—	2	2	no	—
27	C₁₉×#478	—	2	4	no	—
28	C₁₉×#432	C₂²	1	2	yes	—
29	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
30	C₁₉×#330	C₂²	1	8	yes	complemented
31	C₁₉×#329	C₂²	1	4	yes	—
32	C₂×#41×C₁₉	—	2	1	no	—
33	C₂²×Q₈×C₁₉	—	2	1	no	—
34	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
35	C₂²×D₄×C₁₉	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₁₉	—	2	1	no	—
37	C₁₉×#137	—	4	4	no	—
38	C₁₉×AGL(1,ℤ/8)	—	4	4	no	—
39	C₂×Q₁₆×C₁₉	—	2	2	no	—
40	C₂×SD₁₆×C₁₉	—	2	2	no	—
41	C₂×SD₁₆×C₁₉	—	2	2	no	—
42	C₂×D₈×C₁₉	—	2	2	no	—
43	C₂×M₁₆×C₁₉	—	2	1	no	—
44	C₁₉×C₄⋊Q₈	—	2	4	no	—
45	C₁₉×C₄⋊Q₈	—	2	4	no	—
46	C₁₉×C₄⋊Q₈	C₂³	1	1	yes	characteristic
47	C₁₉×#127	C₂³	1	1	yes	characteristic
48	C₁₉×#126	—	2	8	no	—
49	C₁₉×#126	—	2	8	no	—
50	C₁₉×#125	—	2	2	no	—
51	C₁₉×#125	—	2	2	no	—
52	C₁₉×#125	D₄	1	4	yes	—
53	C₁₉×#125	C₂³	1	4	yes	—
54	C₁₉×#124	—	2	2	no	—
55	C₁₉×#124	—	2	1	no	—
56	C₁₉×#124	D₄	1	2	yes	—
57	C₁₉×#124	C₂³	1	2	yes	—
58	C₁₉×#123	—	2	4	no	—
59	C₁₉×#123	—	2	4	no	—
60	C₁₉×C₂²⋊Q₈	—	2	8	no	—
61	C₁₉×C₂²⋊Q₈	—	2	8	no	—
62	C₁₉×#121	—	2	2	no	—
63	C₄×D₄×C₁₉	—	2	2	no	—
64	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
65	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
66	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
67	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
68	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
69	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
70	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
71	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
72	C₂×C₄²×C₁₉	—	2	1	no	—
73	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
74	C₁₉×#107	—	2	4	no	—
75	C₁₉×#106	—	2	4	no	—
76	C₁₉×C₄⋊C₈	—	2	2	no	—
77	C₁₉×#102	—	2	4	no	—
78	C₁₉×#102	—	2	4	no	—
79	C₁₉×C₂²⋊C₈	—	2	2	no	—
80	C₁₉×#97	C₂³	1	2	yes	—
81	C₁₉×#97	C₂³	1	4	yes	—
82	C₄×C₈×C₁₉	—	2	1	no	—
83	C₂⁴×C₁₉	—	2	1	no	—
84	#41×C₁₉	—	4	2	no	—
85	#41×C₁₉	—	4	2	no	—
86	C₂×Q₈×C₁₉	—	2	4	no	—
87	C₂×Q₈×C₁₉	—	2	1	no	—
88	C₂×Q₈×C₁₉	—	2	1	no	—
89	C₂×Q₈×C₁₉	—	4	2	no	—
90	C₂×Q₈×C₁₉	—	8	2	no	—
91	C₂×Q₈×C₁₉	C₂×D₄	1	2	yes	—
92	C₂×D₄×C₁₉	—	2	1	no	—
93	C₂×D₄×C₁₉	—	2	2	no	—
94	C₂×D₄×C₁₉	—	2	1	no	—
95	C₂×D₄×C₁₉	—	4	2	no	—
96	C₂×D₄×C₁₉	—	4	1	no	—
97	C₂×D₄×C₁₉	C₂×D₄	1	2	yes	—
98	C₂²×C₄×C₁₉	—	2	2	no	—
99	C₂²×C₄×C₁₉	—	2	1	no	—
100	C₂²×C₄×C₁₉	—	2	2	no	—
101	C₂²×C₄×C₁₉	—	2	4	no	—
102	C₂²×C₄×C₁₉	—	4	1	no	—
103	C₂²×C₄×C₁₉	—	8	2	no	—
104	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
106	Q₁₆×C₁₉	—	4	4	no	—
107	SD₁₆×C₁₉	—	4	4	no	—
108	SD₁₆×C₁₉	—	4	4	no	—
109	D₈×C₁₉	—	4	4	no	—
110	M₁₆×C₁₉	—	2	4	no	—
111	C₂×C₈×C₁₉	—	2	2	no	—
112	C₂×C₈×C₁₉	—	2	4	no	—
113	C₄⋊C₄×C₁₉	—	2	8	no	—
114	C₄⋊C₄×C₁₉	—	2	4	no	—
115	C₄⋊C₄×C₁₉	—	2	4	no	—
116	C₄⋊C₄×C₁₉	—	2	2	no	—
117	C₄⋊C₄×C₁₉	—	2	2	no	—
118	C₄⋊C₄×C₁₉	—	2	4	no	—
119	C₄⋊C₄×C₁₉	—	2	4	no	—
120	C₄⋊C₄×C₁₉	—	2	2	no	—
121	C₄⋊C₄×C₁₉	C₂×D₄	1	4	yes	—
122	C₄⋊C₄×C₁₉	C₂×D₄	1	4	yes	—
123	C₂²⋊C₄×C₁₉	—	2	8	no	—
124	C₂²⋊C₄×C₁₉	—	2	4	no	—
125	C₂²⋊C₄×C₁₉	—	2	4	no	—
126	C₂²⋊C₄×C₁₉	—	2	4	no	—
127	C₂²⋊C₄×C₁₉	—	4	2	no	—
128	C₄²×C₁₉	—	2	4	no	—
129	C₄²×C₁₉	C₂×D₄	1	2	yes	—
130	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
131	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
132	#18994	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂³×C₁₉	—	2	2	no	—
134	C₂³×C₁₉	—	4	4	no	—
135	C₂³×C₁₉	—	4	2	no	—
136	C₂³×C₁₉	—	4	1	no	—
137	C₂³×C₁₉	#120	1	1	yes	characteristic
138	Q₈×C₁₉	—	4	2	no	—
139	Q₈×C₁₉	—	4	1	no	—
140	Q₈×C₁₉	—	4	1	no	—
141	Q₈×C₁₉	—	8	1	no	—
142	Q₈×C₁₉	—	8	4	no	—
143	D₄×C₁₉	—	4	2	no	—
144	D₄×C₁₉	—	4	1	no	—
145	D₄×C₁₉	—	4	1	no	—
146	D₄×C₁₉	—	4	4	no	—
147	D₄×C₁₉	—	8	1	no	—
148	D₄×C₁₉	—	8	2	no	—
149	C₂×C₄×C₁₉	—	2	2	no	—
150	C₂×C₄×C₁₉	—	2	4	no	—
151	C₂×C₄×C₁₉	—	2	2	no	—
152	C₂×C₄×C₁₉	—	2	4	no	—
153	C₂×C₄×C₁₉	—	4	2	no	—
154	C₂×C₄×C₁₉	—	4	4	no	—
155	C₂×C₄×C₁₉	—	4	4	no	—
156	C₂×C₄×C₁₉	—	4	2	no	—
157	C₂×C₄×C₁₉	—	4	1	no	—
158	C₂×C₄×C₁₉	—	8	2	no	—
159	C₂×C₄×C₁₉	—	8	4	no	—
160	C₂×C₄×C₁₉	#120	1	2	yes	—
161	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
162	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
163	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
164	C₈×C₁₉	—	2	4	no	—
165	C₈×C₁₉	—	4	4	no	—
166	C₂×#563	C₂×C₁₉	1	1	yes	characteristic, complemented
167	#3234	C₂×C₁₉	1	2	yes	complemented
168	#3153	C₂×C₁₉	1	1	yes	characteristic
169	#1660	C₂×C₁₉	1	4	yes	complemented
170	#1657	C₂×C₁₉	1	4	yes	complemented
171	#1533	C₂×C₁₉	1	1	yes	characteristic, complemented
172	#1528	C₂×C₁₉	1	2	yes	complemented
173	C₂²×C₁₉	—	2	2	no	—
174	C₂²×C₁₉	—	4	2	no	—
175	C₂²×C₁₉	—	4	4	no	—
176	C₂²×C₁₉	—	8	4	no	—
177	C₂²×C₁₉	—	8	1	no	—
178	C₂²×C₁₉	—	8	2	no	—
179	C₂²×C₁₉	#453	1	2	yes	—
180	C₂²×C₁₉	C₂×#120	1	1	yes	characteristic
181	C₄×C₁₉	—	2	1	no	—
182	C₄×C₁₉	—	2	1	no	—
183	C₄×C₁₉	—	2	2	no	—
184	C₄×C₁₉	—	2	2	no	—
185	C₄×C₁₉	—	4	4	no	—
186	C₄×C₁₉	—	4	2	no	—
187	C₄×C₁₉	—	4	4	no	—
188	C₄×C₁₉	—	8	1	no	—
189	C₂×#137	—	2	1	no	—
190	C₂×AGL(1,ℤ/8)	—	2	1	no	—
191	#563	—	2	4	no	—
192	#563	—	2	2	no	—
193	#563	C₂²×C₁₉	1	4	yes	—
194	#532	C₂²×C₁₉	1	1	yes	characteristic
195	C₂×C₄⋊Q₈	—	2	1	no	—
196	C₂×#126	—	2	2	no	—
197	C₂×#125	C₂²×C₁₉	1	2	yes	—
198	C₂×#124	C₂²×C₁₉	1	1	yes	characteristic
199	C₂×#123	—	2	1	no	—
200	C₂×C₂²⋊Q₈	—	2	2	no	—
201	#501	C₂²×C₁₉	1	4	yes	complemented
202	#497	C₂²×C₁₉	1	2	yes	—
203	#496	C₂²×C₁₉	1	2	yes	—
204	#495	—	2	2	no	—
205	#490	C₂²×C₁₉	1	4	yes	complemented
206	#481	—	2	2	no	—
207	#480	—	2	2	no	—
208	#478	—	2	4	no	—
209	#432	C₂²×C₁₉	1	2	yes	—
210	C₄×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
211	#330	C₂²×C₁₉	1	8	yes	complemented
212	#329	C₂²×C₁₉	1	4	yes	—
213	C₂×C₁₉	—	2	2	no	—
214	C₂×C₁₉	—	8	2	no	—
215	C₂×C₁₉	—	16	1	no	—
216	C₂×C₁₉	C₂×#453	1	1	yes	characteristic
217	C₂×C₁₉	#3020	1	2	yes	—
218	C₂×#41	—	2	1	no	—
219	C₂²×Q₈	—	2	1	no	—
220	C₂²×Q₈	D₄×C₁₉	1	1	yes	characteristic
221	C₂²×D₄	D₄×C₁₉	1	1	yes	characteristic
222	C₂³×C₄	—	2	1	no	—
223	#137	—	4	4	no	—
224	AGL(1,ℤ/8)	—	4	4	no	—
225	C₂×Q₁₆	—	2	2	no	—
226	C₂×SD₁₆	—	2	2	no	—
227	C₂×SD₁₆	—	2	2	no	—
228	C₂×D₈	—	2	2	no	—
229	C₂×M₁₆	—	2	1	no	—
230	C₄⋊Q₈	—	2	4	no	—
231	C₄⋊Q₈	—	2	4	no	—
232	C₄⋊Q₈	C₂³×C₁₉	1	1	yes	characteristic
233	#127	C₂³×C₁₉	1	1	yes	characteristic
234	#126	—	2	8	no	—
235	#126	—	2	8	no	—
236	#125	—	2	2	no	—
237	#125	—	2	2	no	—
238	#125	D₄×C₁₉	1	4	yes	—
239	#125	C₂³×C₁₉	1	4	yes	—
240	#124	—	2	2	no	—
241	#124	—	2	1	no	—
242	#124	D₄×C₁₉	1	2	yes	—
243	#124	C₂³×C₁₉	1	2	yes	—
244	#123	—	2	4	no	—
245	#123	—	2	4	no	—
246	C₂²⋊Q₈	—	2	8	no	—
247	C₂²⋊Q₈	—	2	8	no	—
248	#121	—	2	2	no	—
249	C₄×D₄	—	2	2	no	—
250	C₂×C₄⋊C₄	—	2	2	no	—
251	C₂×C₄⋊C₄	—	2	2	no	—
252	C₂×C₄⋊C₄	—	2	2	no	—
253	C₂×C₄⋊C₄	D₄×C₁₉	1	2	yes	—
254	C₂×C₄⋊C₄	D₄×C₁₉	1	2	yes	—
255	C₂×C₂²⋊C₄	—	2	2	no	—
256	C₂×C₂²⋊C₄	—	2	1	no	—
257	C₂×C₂²⋊C₄	—	2	2	no	—
258	C₂×C₄²	—	2	1	no	—
259	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
260	#107	—	2	4	no	—
261	#106	—	2	4	no	—
262	C₄⋊C₈	—	2	2	no	—
263	#102	—	2	4	no	—
264	#102	—	2	4	no	—
265	C₂²⋊C₈	—	2	2	no	—
266	#97	C₂³×C₁₉	1	4	yes	—
267	#97	C₂³×C₁₉	1	2	yes	—
268	C₄×C₈	—	2	1	no	—
269	C₁₉	#18994	1	1	yes	characteristic, direct product factor, Sylow
270	C₂⁴	—	2	1	no	—
271	#41	—	4	2	no	—
272	#41	—	4	2	no	—
273	C₂×Q₈	—	2	4	no	—
274	C₂×Q₈	—	2	1	no	—
275	C₂×Q₈	—	2	1	no	—
276	C₂×Q₈	—	4	2	no	—
277	C₂×Q₈	—	8	2	no	—
278	C₂×Q₈	C₂×D₄×C₁₉	1	2	yes	—
279	C₂×D₄	—	2	1	no	—
280	C₂×D₄	—	2	2	no	—
281	C₂×D₄	—	2	1	no	—
282	C₂×D₄	—	4	2	no	—
283	C₂×D₄	—	4	1	no	—
284	C₂×D₄	C₂×D₄×C₁₉	1	2	yes	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	—	2	1	no	—
287	C₂²×C₄	—	2	2	no	—
288	C₂²×C₄	—	2	4	no	—
289	C₂²×C₄	—	4	1	no	—
290	C₂²×C₄	—	8	2	no	—
291	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
292	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
293	Q₁₆	—	4	4	no	—
294	SD₁₆	—	4	4	no	—
295	SD₁₆	—	4	4	no	—
296	D₈	—	4	4	no	—
297	M₁₆	—	2	4	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	4	no	—
300	C₄⋊C₄	—	2	8	no	—
301	C₄⋊C₄	—	2	4	no	—
302	C₄⋊C₄	—	2	4	no	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	2	2	no	—
305	C₄⋊C₄	—	2	4	no	—
306	C₄⋊C₄	—	2	4	no	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	C₂×D₄×C₁₉	1	4	yes	—
309	C₄⋊C₄	C₂×D₄×C₁₉	1	4	yes	—
310	C₂²⋊C₄	—	2	8	no	—
311	C₂²⋊C₄	—	2	4	no	—
312	C₂²⋊C₄	—	2	4	no	—
313	C₂²⋊C₄	—	2	4	no	—
314	C₂²⋊C₄	—	4	2	no	—
315	C₄²	—	2	4	no	—
316	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
317	C₄²	C₂×D₄×C₁₉	1	2	yes	—
318	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
319	C₂³	—	2	2	no	—
320	C₂³	—	4	4	no	—
321	C₂³	—	4	2	no	—
322	C₂³	—	4	1	no	—
323	C₂³	C₁₉×#120	1	1	yes	characteristic
324	Q₈	—	4	2	no	—
325	Q₈	—	4	1	no	—
326	Q₈	—	4	1	no	—
327	Q₈	—	8	1	no	—
328	Q₈	—	8	4	no	—
329	D₄	—	4	2	no	—
330	D₄	—	4	1	no	—
331	D₄	—	4	1	no	—
332	D₄	—	4	4	no	—
333	D₄	—	8	1	no	—
334	D₄	—	8	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	4	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	4	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	4	4	no	—
341	C₂×C₄	—	4	4	no	—
342	C₂×C₄	—	4	2	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	8	2	no	—
345	C₂×C₄	—	8	4	no	—
346	C₂×C₄	C₁₉×#120	1	2	yes	—
347	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
348	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
349	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
350	C₈	—	2	4	no	—
351	C₈	—	4	4	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	4	2	no	—
354	C₂²	—	4	4	no	—
355	C₂²	—	8	4	no	—
356	C₂²	—	8	1	no	—
357	C₂²	—	8	2	no	—
358	C₂²	C₁₉×#453	1	2	yes	—
359	C₂²	C₂×C₁₉×#120	1	1	yes	characteristic
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	2	no	—
363	C₄	—	2	2	no	—
364	C₄	—	4	4	no	—
365	C₄	—	4	2	no	—
366	C₄	—	4	4	no	—
367	C₄	—	8	1	no	—
368	C₂	—	2	2	no	—
369	C₂	—	8	2	no	—
370	C₂	—	16	1	no	—
371	C₂	C₂×C₁₉×#453	1	1	yes	characteristic
372	C₂	C₁₉×#3020	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₉
prime radical	C₁₉×#3153	C₂
LCS	G ▷ 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
1. C₂×C₁₉×#563	C₂	semidirect
2. C₁₉×#3234	C₂	semidirect
4. C₁₉×#1660	C₂	semidirect
5. C₁₉×#1657	C₂	semidirect
6. C₁₉×#1533	C₂	semidirect
7. C₁₉×#1528	C₂	semidirect
20. C₁₉×#501	C₂²	semidirect
24. C₁₉×#490	C₂²	semidirect
30. C₁₉×#330	C₂²	semidirect
132. #18994	C₁₉	direct (N × H)
166. C₂×#563	C₂×C₁₉	semidirect
167. #3234	C₂×C₁₉	semidirect
169. #1660	C₂×C₁₉	semidirect
170. #1657	C₂×C₁₉	semidirect
171. #1533	C₂×C₁₉	semidirect
172. #1528	C₂×C₁₉	semidirect
201. #501	C₂²×C₁₉	semidirect
205. #490	C₂²×C₁₉	semidirect
211. #330	C₂²×C₁₉	semidirect
269. C₁₉	#18994	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×#563	2
C₁₉×#3234	2
C₁₉×#3153	2
C₁₉×#1660	2
C₁₉×#1657	2
C₁₉×#1533	2
C₁₉×#1528	2
#18994	19

Subgroups — C19×#18994 #18024953

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#18994 #18024953