Subgroups — C₁₉×#9817 #18015776

Order 4864.

484 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#467	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1810	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1752	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1508	C₂	1	2	yes	maximal
5	C₁₉×#1472	C₂	1	2	yes	maximal
6	C₂²×Q₁₆×C₁₉	C₄	1	1	yes	characteristic, complemented
7	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
8	C₂×C₁₉×C₂²⋊Q₈	C₂²	1	2	yes	—
9	C₁₉×#467	—	2	1	no	—
10	C₁₉×#467	—	2	1	no	—
11	C₁₉×#467	—	2	1	no	—
12	C₁₉×#467	—	2	1	no	—
13	C₂×C₁₉×#108	—	2	1	no	—
14	C₂×C₁₉×#107	C₄	1	1	yes	characteristic, complemented
15	C₂×C₁₉×#103	C₄	1	2	yes	complemented
16	C₁₉×#407	C₂²	1	2	yes	—
17	C₂×C₁₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₁₉×#97	—	2	1	no	—
19	C₄×C₂²⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
20	C₁₉×#341	—	2	1	no	—
21	C₁₉×#336	—	2	1	no	—
22	C₁₉×#328	—	2	4	no	—
23	C₁₉×#324	—	2	4	no	—
24	C₂²×Q₈×C₁₉	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
26	C₂×Q₁₆×C₁₉	—	2	1	no	—
27	C₂×Q₁₆×C₁₉	—	2	1	no	—
28	C₂×Q₁₆×C₁₉	—	4	2	no	—
29	C₂×M₁₆×C₁₉	—	2	2	no	—
30	C₂²×C₈×C₁₉	—	2	1	no	—
31	C₂²×C₈×C₁₉	—	4	2	no	—
32	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
38	C₁₉×C₂²⋊Q₈	—	2	2	no	—
39	C₁₉×C₂²⋊Q₈	—	2	2	no	—
40	C₁₉×C₂²⋊Q₈	—	2	2	no	—
41	C₁₉×C₂²⋊Q₈	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
44	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
45	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
46	C₂×C₄²×C₁₉	—	2	1	no	—
47	C₁₉×#108	—	4	2	no	—
48	C₁₉×#107	—	2	1	no	—
49	C₁₉×#107	—	2	1	no	—
50	C₁₉×#103	—	2	2	no	—
51	C₁₉×#103	—	2	2	no	—
52	C₁₉×C₂²⋊C₈	—	2	1	no	—
53	C₁₉×C₂²⋊C₈	—	2	2	no	—
54	C₁₉×C₂²⋊C₈	—	2	2	no	—
55	C₁₉×C₂²⋊C₈	D₄	1	2	yes	—
56	C₁₉×#97	—	4	2	no	—
57	C₁₉×#95	D₄	1	2	yes	—
58	C₂⁴×C₁₉	#41	1	1	yes	characteristic
59	C₂×Q₈×C₁₉	—	2	2	no	—
60	C₂×Q₈×C₁₉	—	2	2	no	—
61	C₂×Q₈×C₁₉	—	4	2	no	—
62	C₂×Q₈×C₁₉	—	4	2	no	—
63	C₂²×C₄×C₁₉	—	2	2	no	—
64	C₂²×C₄×C₁₉	—	2	2	no	—
65	C₂²×C₄×C₁₉	—	2	2	no	—
66	C₂²×C₄×C₁₉	—	4	1	no	—
67	C₂²×C₄×C₁₉	—	4	1	no	—
68	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
74	Q₁₆×C₁₉	—	4	2	no	—
75	Q₁₆×C₁₉	—	4	2	no	—
76	M₁₆×C₁₉	—	4	2	no	—
77	M₁₆×C₁₉	—	4	2	no	—
78	C₂×C₈×C₁₉	—	2	2	no	—
79	C₂×C₈×C₁₉	—	2	2	no	—
80	C₂×C₈×C₁₉	—	2	2	no	—
81	C₂×C₈×C₁₉	—	2	2	no	—
82	C₂×C₈×C₁₉	—	4	1	no	—
83	C₂×C₈×C₁₉	—	4	1	no	—
84	C₂×C₈×C₁₉	—	4	1	no	—
85	C₂×C₈×C₁₉	—	4	1	no	—
86	C₂×C₈×C₁₉	—	4	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	C₂²⋊C₄	1	1	yes	characteristic
92	C₄⋊C₄×C₁₉	—	2	2	no	—
93	C₄⋊C₄×C₁₉	—	2	2	no	—
94	C₄⋊C₄×C₁₉	—	4	2	no	—
95	C₄⋊C₄×C₁₉	—	4	2	no	—
96	C₂²⋊C₄×C₁₉	—	4	2	no	—
97	C₂²⋊C₄×C₁₉	—	4	2	no	—
98	C₂²⋊C₄×C₁₉	—	4	2	no	—
99	C₄²×C₁₉	—	4	1	no	—
100	C₄²×C₁₉	—	4	1	no	—
101	#9817	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂³×C₁₉	—	2	2	no	—
103	C₂³×C₁₉	—	2	2	no	—
104	C₂³×C₁₉	—	4	1	no	—
105	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
106	C₂³×C₁₉	#121	1	1	yes	characteristic
107	C₂³×C₁₉	#124	1	1	yes	characteristic
108	Q₈×C₁₉	—	4	2	no	—
109	Q₈×C₁₉	—	4	2	no	—
110	Q₈×C₁₉	—	8	2	no	—
111	C₂×C₄×C₁₉	—	2	1	no	—
112	C₂×C₄×C₁₉	—	2	1	no	—
113	C₂×C₄×C₁₉	—	2	1	no	—
114	C₂×C₄×C₁₉	—	2	1	no	—
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	4	1	no	—
118	C₂×C₄×C₁₉	—	4	1	no	—
119	C₂×C₄×C₁₉	—	4	1	no	—
120	C₂×C₄×C₁₉	—	4	2	no	—
121	C₂×C₄×C₁₉	—	4	2	no	—
122	C₂×C₄×C₁₉	—	4	2	no	—
123	C₂×C₄×C₁₉	—	4	2	no	—
124	C₂×C₄×C₁₉	—	4	4	no	—
125	C₂×C₄×C₁₉	—	4	2	no	—
126	C₂×C₄×C₁₉	—	8	2	no	—
127	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
130	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
131	C₈×C₁₉	—	2	1	no	—
132	C₈×C₁₉	—	2	1	no	—
133	C₈×C₁₉	—	2	1	no	—
134	C₈×C₁₉	—	4	2	no	—
135	C₈×C₁₉	—	8	2	no	—
136	C₈×C₁₉	#99	1	2	yes	—
137	C₂×#467	C₂×C₁₉	1	1	yes	characteristic
138	#1810	C₂×C₁₉	1	1	yes	characteristic
139	#1752	C₂×C₁₉	1	1	yes	characteristic
140	#1508	C₂×C₁₉	1	2	yes	—
141	#1472	C₂×C₁₉	1	2	yes	—
142	C₂²×C₁₉	—	2	1	no	—
143	C₂²×C₁₉	—	2	1	no	—
144	C₂²×C₁₉	—	2	1	no	—
145	C₂²×C₁₉	—	2	1	no	—
146	C₂²×C₁₉	—	2	1	no	—
147	C₂²×C₁₉	—	2	1	no	—
148	C₂²×C₁₉	—	2	1	no	—
149	C₂²×C₁₉	—	4	4	no	—
150	C₂²×C₁₉	#390	1	1	yes	characteristic
151	C₂²×C₁₉	#444	1	2	yes	—
152	C₂²×C₁₉	#473	1	2	yes	—
153	C₄×C₁₉	—	2	1	no	—
154	C₄×C₁₉	—	2	1	no	—
155	C₄×C₁₉	—	2	1	no	—
156	C₄×C₁₉	—	8	2	no	—
157	C₄×C₁₉	—	8	2	no	—
158	C₄×C₁₉	—	8	2	no	—
159	C₄×C₁₉	C₂×#99	1	1	yes	characteristic
160	C₄×C₁₉	#410	1	1	yes	characteristic
161	C₂²×Q₁₆	C₄×C₁₉	1	1	yes	characteristic, complemented
162	C₂³×C₈	C₂²×C₁₉	1	1	yes	characteristic
163	C₂×C₂²⋊Q₈	C₂²×C₁₉	1	2	yes	—
164	#467	—	2	1	no	—
165	#467	—	2	1	no	—
166	#467	—	2	1	no	—
167	#467	—	2	1	no	—
168	C₂×#108	—	2	1	no	—
169	C₂×#107	C₄×C₁₉	1	1	yes	characteristic, complemented
170	C₂×#103	C₄×C₁₉	1	2	yes	complemented
171	#407	C₂²×C₁₉	1	2	yes	—
172	C₂×C₂²⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
173	C₂×#97	—	2	1	no	—
174	C₄×C₂²⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
175	#341	—	2	1	no	—
176	#336	—	2	1	no	—
177	#328	—	2	4	no	—
178	#324	—	2	4	no	—
179	C₂×C₁₉	—	2	2	no	—
180	C₂×C₁₉	—	2	2	no	—
181	C₂×C₁₉	—	4	1	no	—
182	C₂×C₁₉	#1964	1	1	yes	characteristic
183	C₂×C₁₉	#1975	1	1	yes	characteristic
184	C₂×C₁₉	#1979	1	1	yes	characteristic
185	C₂²×Q₈	C₂×C₄×C₁₉	1	2	yes	—
186	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
187	C₂×Q₁₆	—	2	1	no	—
188	C₂×Q₁₆	—	2	1	no	—
189	C₂×Q₁₆	—	4	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂²×C₈	—	2	1	no	—
192	C₂²×C₈	—	4	2	no	—
193	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
194	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
195	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
196	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
197	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
198	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
199	C₂²⋊Q₈	—	2	2	no	—
200	C₂²⋊Q₈	—	2	2	no	—
201	C₂²⋊Q₈	—	2	2	no	—
202	C₂²⋊Q₈	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
205	C₂×C₂²⋊C₄	—	2	2	no	—
206	C₂×C₂²⋊C₄	—	2	1	no	—
207	C₂×C₄²	—	2	1	no	—
208	#108	—	4	2	no	—
209	#107	—	2	1	no	—
210	#107	—	2	1	no	—
211	#103	—	2	2	no	—
212	#103	—	2	2	no	—
213	C₂²⋊C₈	—	2	1	no	—
214	C₂²⋊C₈	—	2	2	no	—
215	C₂²⋊C₈	—	2	2	no	—
216	C₂²⋊C₈	D₄×C₁₉	1	2	yes	—
217	#97	—	4	2	no	—
218	#95	D₄×C₁₉	1	2	yes	—
219	C₁₉	#9817	1	1	yes	characteristic, direct product factor, Sylow
220	C₂⁴	#41×C₁₉	1	1	yes	characteristic
221	C₂×Q₈	—	2	2	no	—
222	C₂×Q₈	—	2	2	no	—
223	C₂×Q₈	—	4	2	no	—
224	C₂×Q₈	—	4	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	4	1	no	—
229	C₂²×C₄	—	4	1	no	—
230	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
235	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
236	Q₁₆	—	4	2	no	—
237	Q₁₆	—	4	2	no	—
238	M₁₆	—	4	2	no	—
239	M₁₆	—	4	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	4	2	no	—
250	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
251	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
252	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
253	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₂²⋊C₄	—	4	2	no	—
259	C₂²⋊C₄	—	4	2	no	—
260	C₂²⋊C₄	—	4	2	no	—
261	C₄²	—	4	1	no	—
262	C₄²	—	4	1	no	—
263	C₂³	—	2	2	no	—
264	C₂³	—	2	2	no	—
265	C₂³	—	4	1	no	—
266	C₂³	C₄×D₄×C₁₉	1	1	yes	characteristic
267	C₂³	C₁₉×#121	1	1	yes	characteristic
268	C₂³	C₁₉×#124	1	1	yes	characteristic
269	Q₈	—	4	2	no	—
270	Q₈	—	4	2	no	—
271	Q₈	—	8	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	8	2	no	—
288	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
289	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
290	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
291	C₂×C₄	C₁₉×#127	1	1	yes	characteristic
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	—	4	2	no	—
296	C₈	—	8	2	no	—
297	C₈	C₁₉×#99	1	2	yes	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	4	4	no	—
306	C₂²	C₁₉×#390	1	1	yes	characteristic
307	C₂²	C₁₉×#444	1	2	yes	—
308	C₂²	C₁₉×#473	1	2	yes	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	8	2	no	—
313	C₄	—	8	2	no	—
314	C₄	—	8	2	no	—
315	C₄	C₂×C₁₉×#99	1	1	yes	characteristic
316	C₄	C₁₉×#410	1	1	yes	characteristic
317	C₂	—	2	2	no	—
318	C₂	—	2	2	no	—
319	C₂	—	4	1	no	—
320	C₂	C₁₉×#1964	1	1	yes	characteristic
321	C₂	C₁₉×#1975	1	1	yes	characteristic
322	C₂	C₁₉×#1979	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#390
commutator	C₂²×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂³×C₄	C₂³×C₁₉
prime radical	C₂⁴×C₁₉	#41
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂³×C₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₂²×Q₁₆×C₁₉	C₄	semidirect
14. C₂×C₁₉×#107	C₄	semidirect
15. C₂×C₁₉×#103	C₄	semidirect
101. #9817	C₁₉	direct (N × H)
161. C₂²×Q₁₆	C₄×C₁₉	semidirect
169. C₂×#107	C₄×C₁₉	semidirect
170. C₂×#103	C₄×C₁₉	semidirect
219. C₁₉	#9817	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×#467	2
C₁₉×#1810	2
C₁₉×#1752	2
C₁₉×#1508	2
C₁₉×#1472	2
#9817	19

Subgroups — C19×#9817 #18015776

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#9817 #18015776