Subgroups — C₁₉×#9527 #18015486

Order 4864.

436 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3112	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2981	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1557	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1487	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1467	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#1307	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1287	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₉×#532	C₂²	1	1	yes	characteristic
9	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
10	C₁₉×#480	C₄	1	2	yes	—
11	C₁₉×#477	—	2	1	no	—
12	C₁₉×#460	—	2	1	no	—
13	C₁₉×#448	C₄	1	2	yes	—
14	C₁₉×#440	—	2	2	no	—
15	C₁₉×#436	—	2	1	no	—
16	C₁₉×#435	—	2	2	no	—
17	C₈×D₄×C₁₉	—	2	1	no	—
18	C₁₉×#432	C₂²	1	1	yes	characteristic
19	C₁₉×#432	C₂²	1	1	yes	characteristic
20	C₁₉×#432	C₂²	1	1	yes	characteristic
21	C₁₉×#431	C₂²	1	1	yes	characteristic
22	C₁₉×#423	C₂²	1	1	yes	characteristic
23	C₁₉×#408	—	2	2	no	—
24	C₁₉×#332	—	2	1	no	—
25	C₁₉×#331	—	2	1	no	—
26	C₁₉×Q₈⋊C₈	—	2	1	no	—
27	C₁₉×Q₈⋊C₈	—	2	1	no	—
28	C₁₉×#324	—	2	2	no	—
29	C₂×#41×C₁₉	—	2	1	no	—
30	C₂²×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂×SD₁₆×C₁₉	—	2	2	no	—
32	C₂×M₁₆×C₁₉	—	2	2	no	—
33	C₂×M₁₆×C₁₉	—	2	1	no	—
34	C₂²×C₈×C₁₉	—	2	2	no	—
35	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
36	C₁₉×#127	D₄	1	1	yes	characteristic
37	C₁₉×#124	—	2	1	no	—
38	C₁₉×#121	—	2	1	no	—
39	C₁₉×#121	C₂×C₄	1	2	yes	—
40	C₄×Q₈×C₁₉	—	2	1	no	—
41	C₄×Q₈×C₁₉	—	2	1	no	—
42	C₄×Q₈×C₁₉	—	2	1	no	—
43	C₄×Q₈×C₁₉	—	2	1	no	—
44	C₄×D₄×C₁₉	—	2	1	no	—
45	C₄×D₄×C₁₉	D₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
47	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₁₉	—	2	1	no	—
49	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
50	C₁₉×#107	—	2	2	no	—
51	C₁₉×C₄⋊C₈	—	2	1	no	—
52	C₁₉×C₄⋊C₈	—	2	1	no	—
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×#103	—	2	2	no	—
58	C₁₉×#102	—	2	2	no	—
59	C₁₉×C₂²⋊C₈	—	2	2	no	—
60	C₁₉×C₂²⋊C₈	—	2	2	no	—
61	C₁₉×C₂²⋊C₈	—	2	1	no	—
62	C₁₉×C₂²⋊C₈	—	2	1	no	—
63	C₁₉×C₂²⋊C₈	C₂×C₄	1	2	yes	—
64	C₁₉×C₂²⋊C₈	D₄	1	2	yes	—
65	C₁₉×#97	—	2	1	no	—
66	C₁₉×#97	—	2	1	no	—
67	C₁₉×#97	—	2	1	no	—
68	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
69	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
70	#41×C₁₉	—	8	2	no	—
71	C₂×Q₈×C₁₉	—	2	1	no	—
72	C₂×Q₈×C₁₉	—	2	1	no	—
73	C₂×Q₈×C₁₉	—	2	1	no	—
74	C₂×Q₈×C₁₉	—	4	2	no	—
75	C₂×D₄×C₁₉	—	2	2	no	—
76	C₂×D₄×C₁₉	—	2	1	no	—
77	C₂×D₄×C₁₉	—	2	1	no	—
78	C₂×D₄×C₁₉	C₂²⋊C₄	1	2	yes	—
79	C₂²×C₄×C₁₉	—	2	1	no	—
80	C₂²×C₄×C₁₉	—	2	2	no	—
81	C₂²×C₄×C₁₉	—	2	1	no	—
82	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
85	SD₁₆×C₁₉	—	4	4	no	—
86	M₁₆×C₁₉	—	8	2	no	—
87	M₁₆×C₁₉	—	8	1	no	—
88	C₂×C₈×C₁₉	—	2	2	no	—
89	C₂×C₈×C₁₉	—	2	1	no	—
90	C₂×C₈×C₁₉	—	2	2	no	—
91	C₂×C₈×C₁₉	—	2	1	no	—
92	C₂×C₈×C₁₉	—	4	1	no	—
93	C₂×C₈×C₁₉	—	4	1	no	—
94	C₂×C₈×C₁₉	—	8	2	no	—
95	C₄⋊C₄×C₁₉	—	2	1	no	—
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	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
101	C₂²⋊C₄×C₁₉	—	2	2	no	—
102	C₂²⋊C₄×C₁₉	—	2	2	no	—
103	C₄²×C₁₉	—	4	1	no	—
104	C₄²×C₁₉	—	4	1	no	—
105	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₉	#41	1	1	yes	characteristic
109	#9527	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂³×C₁₉	—	2	2	no	—
111	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
112	Q₈×C₁₉	—	2	2	no	—
113	Q₈×C₁₉	—	2	2	no	—
114	Q₈×C₁₉	—	4	1	no	—
115	Q₈×C₁₉	—	4	1	no	—
116	Q₈×C₁₉	—	8	1	no	—
117	D₄×C₁₉	—	4	2	no	—
118	D₄×C₁₉	—	4	1	no	—
119	D₄×C₁₉	—	4	1	no	—
120	D₄×C₁₉	—	8	1	no	—
121	D₄×C₁₉	—	8	2	no	—
122	C₂×C₄×C₁₉	—	2	1	no	—
123	C₂×C₄×C₁₉	—	2	1	no	—
124	C₂×C₄×C₁₉	—	2	2	no	—
125	C₂×C₄×C₁₉	—	2	2	no	—
126	C₂×C₄×C₁₉	—	4	1	no	—
127	C₂×C₄×C₁₉	—	4	1	no	—
128	C₂×C₄×C₁₉	—	4	2	no	—
129	C₂×C₄×C₁₉	—	4	2	no	—
130	C₂×C₄×C₁₉	—	4	1	no	—
131	C₂×C₄×C₁₉	—	4	1	no	—
132	C₂×C₄×C₁₉	—	4	1	no	—
133	C₂×C₄×C₁₉	—	8	1	no	—
134	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
137	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
138	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
139	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
140	C₈×C₁₉	—	4	1	no	—
141	C₈×C₁₉	—	4	2	no	—
142	C₈×C₁₉	—	4	1	no	—
143	C₈×C₁₉	—	4	2	no	—
144	C₈×C₁₉	—	8	1	no	—
145	C₈×C₁₉	—	8	1	no	—
146	#3112	C₂×C₁₉	1	1	yes	characteristic
147	#2981	C₂×C₁₉	1	1	yes	characteristic
148	#1557	C₂×C₁₉	1	1	yes	characteristic
149	#1487	C₂×C₁₉	1	1	yes	characteristic, complemented
150	#1467	C₂×C₁₉	1	1	yes	characteristic, complemented
151	#1307	C₂×C₁₉	1	1	yes	characteristic, complemented
152	#1287	C₂×C₁₉	1	1	yes	characteristic, complemented
153	C₂²×C₁₉	—	2	1	no	—
154	C₂²×C₁₉	—	2	1	no	—
155	C₂²×C₁₉	—	2	1	no	—
156	C₂²×C₁₉	—	4	2	no	—
157	C₂²×C₁₉	—	4	2	no	—
158	C₂²×C₁₉	—	4	2	no	—
159	C₂²×C₁₉	#386	1	1	yes	characteristic
160	C₄×C₁₉	—	2	1	no	—
161	C₄×C₁₉	—	2	1	no	—
162	C₄×C₁₉	—	2	1	no	—
163	C₄×C₁₉	—	2	1	no	—
164	C₄×C₁₉	—	2	1	no	—
165	C₄×C₁₉	—	4	2	no	—
166	C₄×C₁₉	—	4	2	no	—
167	C₄×C₁₉	—	8	1	no	—
168	C₄×C₁₉	#471	1	1	yes	characteristic
169	C₄×C₁₉	#473	1	1	yes	characteristic
170	#532	C₂²×C₁₉	1	1	yes	characteristic
171	C₂×C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
172	#480	C₄×C₁₉	1	2	yes	—
173	#477	—	2	1	no	—
174	#460	—	2	1	no	—
175	#448	C₄×C₁₉	1	2	yes	—
176	#440	—	2	2	no	—
177	#436	—	2	1	no	—
178	#435	—	2	2	no	—
179	C₈×D₄	—	2	1	no	—
180	#432	C₂²×C₁₉	1	1	yes	characteristic
181	#432	C₂²×C₁₉	1	1	yes	characteristic
182	#432	C₂²×C₁₉	1	1	yes	characteristic
183	#431	C₂²×C₁₉	1	1	yes	characteristic
184	#423	C₂²×C₁₉	1	1	yes	characteristic
185	#408	—	2	2	no	—
186	#332	—	2	1	no	—
187	#331	—	2	1	no	—
188	Q₈⋊C₈	—	2	1	no	—
189	Q₈⋊C₈	—	2	1	no	—
190	#324	—	2	2	no	—
191	C₂×C₁₉	—	4	1	no	—
192	C₂×C₁₉	—	8	2	no	—
193	C₂×C₁₉	#1891	1	1	yes	characteristic
194	C₂×C₁₉	#1904	1	1	yes	characteristic
195	C₂×C₁₉	#1908	1	1	yes	characteristic
196	C₂×#41	—	2	1	no	—
197	C₂²×Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
198	C₂×SD₁₆	—	2	2	no	—
199	C₂×M₁₆	—	2	2	no	—
200	C₂×M₁₆	—	2	1	no	—
201	C₂²×C₈	—	2	2	no	—
202	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
203	#127	D₄×C₁₉	1	1	yes	characteristic
204	#124	—	2	1	no	—
205	#121	—	2	1	no	—
206	#121	C₂×C₄×C₁₉	1	2	yes	—
207	C₄×Q₈	—	2	1	no	—
208	C₄×Q₈	—	2	1	no	—
209	C₄×Q₈	—	2	1	no	—
210	C₄×Q₈	—	2	1	no	—
211	C₄×D₄	—	2	1	no	—
212	C₄×D₄	D₄×C₁₉	1	2	yes	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
215	C₂×C₄²	—	2	1	no	—
216	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
217	#107	—	2	2	no	—
218	C₄⋊C₈	—	2	1	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	2	1	no	—
221	C₄⋊C₈	—	2	1	no	—
222	C₄⋊C₈	—	2	1	no	—
223	C₄⋊C₈	—	2	1	no	—
224	#103	—	2	2	no	—
225	#102	—	2	2	no	—
226	C₂²⋊C₈	—	2	2	no	—
227	C₂²⋊C₈	—	2	2	no	—
228	C₂²⋊C₈	—	2	1	no	—
229	C₂²⋊C₈	—	2	1	no	—
230	C₂²⋊C₈	C₂×C₄×C₁₉	1	2	yes	—
231	C₂²⋊C₈	D₄×C₁₉	1	2	yes	—
232	#97	—	2	1	no	—
233	#97	—	2	1	no	—
234	#97	—	2	1	no	—
235	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
236	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
237	C₁₉	#9527	1	1	yes	characteristic, direct product factor, Sylow
238	#41	—	8	2	no	—
239	C₂×Q₈	—	2	1	no	—
240	C₂×Q₈	—	2	1	no	—
241	C₂×Q₈	—	2	1	no	—
242	C₂×Q₈	—	4	2	no	—
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	2	1	no	—
245	C₂×D₄	—	2	1	no	—
246	C₂×D₄	C₂²⋊C₄×C₁₉	1	2	yes	—
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
252	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
253	SD₁₆	—	4	4	no	—
254	M₁₆	—	8	2	no	—
255	M₁₆	—	8	1	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	4	1	no	—
262	C₂×C₈	—	8	2	no	—
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
269	C₂²⋊C₄	—	2	2	no	—
270	C₂²⋊C₄	—	2	2	no	—
271	C₄²	—	4	1	no	—
272	C₄²	—	4	1	no	—
273	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
274	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
275	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
276	C₄²	#41×C₁₉	1	1	yes	characteristic
277	C₂³	—	2	2	no	—
278	C₂³	C₄×D₄×C₁₉	1	1	yes	characteristic
279	Q₈	—	2	2	no	—
280	Q₈	—	2	2	no	—
281	Q₈	—	4	1	no	—
282	Q₈	—	4	1	no	—
283	Q₈	—	8	1	no	—
284	D₄	—	4	2	no	—
285	D₄	—	4	1	no	—
286	D₄	—	4	1	no	—
287	D₄	—	8	1	no	—
288	D₄	—	8	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	8	1	no	—
301	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
302	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
303	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
304	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
305	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
306	C₂×C₄	C₁₉×#123	1	1	yes	characteristic
307	C₈	—	4	1	no	—
308	C₈	—	4	2	no	—
309	C₈	—	4	1	no	—
310	C₈	—	4	2	no	—
311	C₈	—	8	1	no	—
312	C₈	—	8	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	2	no	—
318	C₂²	—	4	2	no	—
319	C₂²	C₁₉×#386	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	2	no	—
327	C₄	—	8	1	no	—
328	C₄	C₁₉×#471	1	1	yes	characteristic
329	C₄	C₁₉×#473	1	1	yes	characteristic
330	C₂	—	4	1	no	—
331	C₂	—	8	2	no	—
332	C₂	C₁₉×#1891	1	1	yes	characteristic
333	C₂	C₁₉×#1904	1	1	yes	characteristic
334	C₂	C₁₉×#1908	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#386
commutator	C₂²×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
prime radical	C₁₉×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ 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
4. C₁₉×#1487	C₂	semidirect
5. C₁₉×#1467	C₂	semidirect
6. C₁₉×#1307	C₂	semidirect
7. C₁₉×#1287	C₂	semidirect
109. #9527	C₁₉	direct (N × H)
149. #1487	C₂×C₁₉	semidirect
150. #1467	C₂×C₁₉	semidirect
151. #1307	C₂×C₁₉	semidirect
152. #1287	C₂×C₁₉	semidirect
237. C₁₉	#9527	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×#3112	2
C₁₉×#2981	2
C₁₉×#1557	2
C₁₉×#1487	2
C₁₉×#1467	2
C₁₉×#1307	2
C₁₉×#1287	2
#9527	19

Subgroups — C19×#9527 #18015486

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#9527 #18015486