Subgroups — C₁₉×#18719 #18024678

Order 4864.

928 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3550	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#3230	C₂	1	2	yes	maximal, complemented
3	C₁₉×#3152	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1633	C₂	1	4	yes	maximal, complemented
5	C₁₉×#1630	C₂	1	4	yes	maximal, complemented
6	C₁₉×#1543	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1485	C₂	1	2	yes	maximal, complemented
8	C₂×C₁₉×#137	—	2	2	no	—
9	C₁₉×#563	—	2	2	no	—
10	C₁₉×#559	C₂²	1	2	yes	—
11	Q₈²×C₁₉	C₂²	1	2	yes	—
12	C₁₉×#556	—	2	2	no	—
13	D₄×Q₈×C₁₉	—	2	2	no	—
14	C₁₉×#542	—	2	4	no	—
15	C₁₉×#536	—	2	2	no	—
16	C₁₉×#533	C₂²	1	2	yes	—
17	C₁₉×#532	C₂²	1	1	yes	characteristic
18	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₁₉×#500	—	2	2	no	—
20	C₁₉×#497	—	2	2	no	—
21	C₁₉×#494	C₂²	1	2	yes	—
22	C₁₉×#492	C₂²	1	2	yes	—
23	C₁₉×#487	—	2	2	no	—
24	C₁₉×#484	—	2	4	no	—
25	C₁₉×#477	C₂²	1	8	yes	—
26	C₁₉×#431	C₂²	1	2	yes	—
27	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
28	C₁₉×#333	C₂²	1	2	yes	—
29	C₁₉×#331	C₂²	1	2	yes	—
30	C₁₉×Q₈⋊C₈	C₂²	1	8	yes	—
31	C₂×#41×C₁₉	D₄	1	2	yes	—
32	C₂²×Q₈×C₁₉	—	2	2	no	—
33	C₂²×Q₈×C₁₉	—	2	1	no	—
34	C₁₉×#137	—	4	8	no	—
35	C₂×Q₁₆×C₁₉	—	2	4	no	—
36	C₂×SD₁₆×C₁₉	—	2	4	no	—
37	C₂×M₁₆×C₁₉	—	2	1	no	—
38	C₁₉×C₄⋊Q₈	—	2	4	no	—
39	C₁₉×C₄⋊Q₈	—	2	4	no	—
40	C₁₉×C₄⋊Q₈	—	2	2	no	—
41	C₁₉×C₄⋊Q₈	—	2	1	no	—
42	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
43	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
44	C₁₉×C₄⋊Q₈	C₂³	1	2	yes	—
45	C₁₉×C₄⋊Q₈	C₂³	1	1	yes	characteristic
46	C₁₉×#127	C₂³	1	1	yes	characteristic
47	C₁₉×#126	—	2	8	no	—
48	C₁₉×#125	—	2	2	no	—
49	C₁₉×#124	—	2	4	no	—
50	C₁₉×#124	—	2	4	no	—
51	C₁₉×#124	D₄	1	2	yes	—
52	C₁₉×#123	—	2	4	no	—
53	C₁₉×#123	—	2	4	no	—
54	C₁₉×C₂²⋊Q₈	—	2	4	no	—
55	C₁₉×C₂²⋊Q₈	—	2	8	no	—
56	C₁₉×C₂²⋊Q₈	—	2	2	no	—
57	C₁₉×C₂²⋊Q₈	D₄	1	4	yes	—
58	C₁₉×#121	—	2	2	no	—
59	C₄×Q₈×C₁₉	—	2	4	no	—
60	C₄×Q₈×C₁₉	—	2	2	no	—
61	C₄×Q₈×C₁₉	C₂³	1	4	yes	—
62	C₄×D₄×C₁₉	—	2	4	no	—
63	C₄×D₄×C₁₉	—	2	2	no	—
64	C₁₉×#117	—	2	2	no	—
65	C₁₉×#117	—	2	2	no	—
66	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
67	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
68	C₁₉×#107	—	2	8	no	—
69	C₁₉×C₄⋊C₈	C₂³	1	4	yes	—
70	C₁₉×#103	—	2	8	no	—
71	C₁₉×C₂²⋊C₈	—	2	2	no	—
72	C₁₉×#97	—	2	1	no	—
73	C₄×C₈×C₁₉	—	2	2	no	—
74	C₄×C₈×C₁₉	C₂³	1	2	yes	—
75	#41×C₁₉	—	4	4	no	—
76	C₂×Q₈×C₁₉	—	2	4	no	—
77	C₂×Q₈×C₁₉	—	2	2	no	—
78	C₂×Q₈×C₁₉	—	2	2	no	—
79	C₂×Q₈×C₁₉	—	4	4	no	—
80	C₂×Q₈×C₁₉	—	4	1	no	—
81	C₂×Q₈×C₁₉	—	8	4	no	—
82	C₂×Q₈×C₁₉	C₂×D₄	1	4	yes	—
83	C₂×Q₈×C₁₉	C₂×D₄	1	2	yes	—
84	C₂×D₄×C₁₉	—	2	2	no	—
85	C₂×D₄×C₁₉	—	2	2	no	—
86	C₂×D₄×C₁₉	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₁₉	—	2	4	no	—
88	C₂²×C₄×C₁₉	—	2	2	no	—
89	C₂²×C₄×C₁₉	—	4	1	no	—
90	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
92	Q₁₆×C₁₉	—	4	8	no	—
93	SD₁₆×C₁₉	—	4	8	no	—
94	M₁₆×C₁₉	—	2	4	no	—
95	C₂×C₈×C₁₉	—	2	2	no	—
96	C₂×C₈×C₁₉	—	2	4	no	—
97	C₄⋊C₄×C₁₉	—	2	8	no	—
98	C₄⋊C₄×C₁₉	—	2	2	no	—
99	C₄⋊C₄×C₁₉	—	2	2	no	—
100	C₄⋊C₄×C₁₉	—	2	2	no	—
101	C₄⋊C₄×C₁₉	—	2	4	no	—
102	C₄⋊C₄×C₁₉	—	2	2	no	—
103	C₄⋊C₄×C₁₉	—	2	4	no	—
104	C₄⋊C₄×C₁₉	—	2	4	no	—
105	C₄⋊C₄×C₁₉	—	4	2	no	—
106	C₄⋊C₄×C₁₉	C₂×D₄	1	4	yes	—
107	C₂²⋊C₄×C₁₉	—	2	4	no	—
108	C₂²⋊C₄×C₁₉	—	2	8	no	—
109	C₂²⋊C₄×C₁₉	—	2	4	no	—
110	C₂²⋊C₄×C₁₉	—	2	4	no	—
111	C₄²×C₁₉	—	2	4	no	—
112	C₄²×C₁₉	—	2	4	no	—
113	C₄²×C₁₉	C₂×D₄	1	2	yes	—
114	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
116	#18719	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂³×C₁₉	—	2	2	no	—
118	C₂³×C₁₉	#120	1	1	yes	characteristic
119	Q₈×C₁₉	—	2	8	no	—
120	Q₈×C₁₉	—	4	2	no	—
121	Q₈×C₁₉	—	4	4	no	—
122	Q₈×C₁₉	—	4	4	no	—
123	Q₈×C₁₉	—	8	4	no	—
124	Q₈×C₁₉	—	8	2	no	—
125	D₄×C₁₉	—	4	2	no	—
126	D₄×C₁₉	—	4	4	no	—
127	D₄×C₁₉	—	8	2	no	—
128	C₂×C₄×C₁₉	—	2	4	no	—
129	C₂×C₄×C₁₉	—	2	2	no	—
130	C₂×C₄×C₁₉	—	2	4	no	—
131	C₂×C₄×C₁₉	—	2	2	no	—
132	C₂×C₄×C₁₉	—	2	4	no	—
133	C₂×C₄×C₁₉	—	4	4	no	—
134	C₂×C₄×C₁₉	—	4	2	no	—
135	C₂×C₄×C₁₉	—	8	4	no	—
136	C₂×C₄×C₁₉	—	8	2	no	—
137	C₂×C₄×C₁₉	#120	1	2	yes	—
138	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
139	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
141	C₈×C₁₉	—	2	4	no	—
142	C₈×C₁₉	—	4	4	no	—
143	#3550	C₂×C₁₉	1	1	yes	characteristic
144	#3230	C₂×C₁₉	1	2	yes	complemented
145	#3152	C₂×C₁₉	1	1	yes	characteristic
146	#1633	C₂×C₁₉	1	4	yes	complemented
147	#1630	C₂×C₁₉	1	4	yes	complemented
148	#1543	C₂×C₁₉	1	1	yes	characteristic
149	#1485	C₂×C₁₉	1	2	yes	complemented
150	C₂²×C₁₉	—	2	2	no	—
151	C₂²×C₁₉	—	2	1	no	—
152	C₂²×C₁₉	—	4	4	no	—
153	C₂²×C₁₉	—	4	2	no	—
154	C₂²×C₁₉	C₂×#120	1	1	yes	characteristic
155	C₄×C₁₉	—	2	2	no	—
156	C₄×C₁₉	—	2	1	no	—
157	C₄×C₁₉	—	4	4	no	—
158	C₄×C₁₉	—	4	2	no	—
159	C₄×C₁₉	—	4	4	no	—
160	C₄×C₁₉	—	8	2	no	—
161	C₄×C₁₉	#453	1	2	yes	—
162	C₄×C₁₉	#578	1	4	yes	—
163	C₂×#137	—	2	2	no	—
164	#563	—	2	2	no	—
165	#559	C₂²×C₁₉	1	2	yes	—
166	Q₈²	C₂²×C₁₉	1	2	yes	—
167	#556	—	2	2	no	—
168	D₄×Q₈	—	2	2	no	—
169	#542	—	2	4	no	—
170	#536	—	2	2	no	—
171	#533	C₂²×C₁₉	1	2	yes	—
172	#532	C₂²×C₁₉	1	1	yes	characteristic
173	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
174	#500	—	2	2	no	—
175	#497	—	2	2	no	—
176	#494	C₂²×C₁₉	1	2	yes	—
177	#492	C₂²×C₁₉	1	2	yes	—
178	#487	—	2	2	no	—
179	#484	—	2	4	no	—
180	#477	C₂²×C₁₉	1	8	yes	—
181	#431	C₂²×C₁₉	1	2	yes	—
182	C₄×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
183	#333	C₂²×C₁₉	1	2	yes	—
184	#331	C₂²×C₁₉	1	2	yes	—
185	Q₈⋊C₈	C₂²×C₁₉	1	8	yes	—
186	C₂×C₁₉	—	4	1	no	—
187	C₂×C₁₉	—	8	2	no	—
188	C₂×C₁₉	#3011	1	2	yes	—
189	C₂×C₁₉	C₂×#453	1	1	yes	characteristic
190	C₂×#41	D₄×C₁₉	1	2	yes	—
191	C₂²×Q₈	—	2	2	no	—
192	C₂²×Q₈	—	2	1	no	—
193	#137	—	4	8	no	—
194	C₂×Q₁₆	—	2	4	no	—
195	C₂×SD₁₆	—	2	4	no	—
196	C₂×M₁₆	—	2	1	no	—
197	C₄⋊Q₈	—	2	4	no	—
198	C₄⋊Q₈	—	2	4	no	—
199	C₄⋊Q₈	—	2	2	no	—
200	C₄⋊Q₈	—	2	1	no	—
201	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
202	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
203	C₄⋊Q₈	C₂³×C₁₉	1	2	yes	—
204	C₄⋊Q₈	C₂³×C₁₉	1	1	yes	characteristic
205	#127	C₂³×C₁₉	1	1	yes	characteristic
206	#126	—	2	8	no	—
207	#125	—	2	2	no	—
208	#124	—	2	4	no	—
209	#124	—	2	4	no	—
210	#124	D₄×C₁₉	1	2	yes	—
211	#123	—	2	4	no	—
212	#123	—	2	4	no	—
213	C₂²⋊Q₈	—	2	4	no	—
214	C₂²⋊Q₈	—	2	8	no	—
215	C₂²⋊Q₈	—	2	2	no	—
216	C₂²⋊Q₈	D₄×C₁₉	1	4	yes	—
217	#121	—	2	2	no	—
218	C₄×Q₈	—	2	4	no	—
219	C₄×Q₈	—	2	2	no	—
220	C₄×Q₈	C₂³×C₁₉	1	4	yes	—
221	C₄×D₄	—	2	4	no	—
222	C₄×D₄	—	2	2	no	—
223	#117	—	2	2	no	—
224	#117	—	2	2	no	—
225	C₂×C₄⋊C₄	—	2	2	no	—
226	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
227	#107	—	2	8	no	—
228	C₄⋊C₈	C₂³×C₁₉	1	4	yes	—
229	#103	—	2	8	no	—
230	C₂²⋊C₈	—	2	2	no	—
231	#97	—	2	1	no	—
232	C₄×C₈	—	2	2	no	—
233	C₄×C₈	C₂³×C₁₉	1	2	yes	—
234	C₁₉	#18719	1	1	yes	characteristic, direct product factor, Sylow
235	#41	—	4	4	no	—
236	C₂×Q₈	—	2	4	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	4	4	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×Q₈	—	8	4	no	—
242	C₂×Q₈	C₂×D₄×C₁₉	1	4	yes	—
243	C₂×Q₈	C₂×D₄×C₁₉	1	2	yes	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	C₂×D₄×C₁₉	1	2	yes	—
247	C₂²×C₄	—	2	4	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
251	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
252	Q₁₆	—	4	8	no	—
253	SD₁₆	—	4	8	no	—
254	M₁₆	—	2	4	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	4	no	—
257	C₄⋊C₄	—	2	8	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	C₂×D₄×C₁₉	1	4	yes	—
267	C₂²⋊C₄	—	2	4	no	—
268	C₂²⋊C₄	—	2	8	no	—
269	C₂²⋊C₄	—	2	4	no	—
270	C₂²⋊C₄	—	2	4	no	—
271	C₄²	—	2	4	no	—
272	C₄²	—	2	4	no	—
273	C₄²	C₂×D₄×C₁₉	1	2	yes	—
274	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
275	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
276	C₂³	—	2	2	no	—
277	C₂³	C₁₉×#120	1	1	yes	characteristic
278	Q₈	—	2	8	no	—
279	Q₈	—	4	2	no	—
280	Q₈	—	4	4	no	—
281	Q₈	—	4	4	no	—
282	Q₈	—	8	4	no	—
283	Q₈	—	8	2	no	—
284	D₄	—	4	2	no	—
285	D₄	—	4	4	no	—
286	D₄	—	8	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	8	4	no	—
295	C₂×C₄	—	8	2	no	—
296	C₂×C₄	C₁₉×#120	1	2	yes	—
297	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
298	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
299	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
300	C₈	—	2	4	no	—
301	C₈	—	4	4	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	4	4	no	—
305	C₂²	—	4	2	no	—
306	C₂²	C₂×C₁₉×#120	1	1	yes	characteristic
307	C₄	—	2	2	no	—
308	C₄	—	2	1	no	—
309	C₄	—	4	4	no	—
310	C₄	—	4	2	no	—
311	C₄	—	4	4	no	—
312	C₄	—	8	2	no	—
313	C₄	C₁₉×#453	1	2	yes	—
314	C₄	C₁₉×#578	1	4	yes	—
315	C₂	—	4	1	no	—
316	C₂	—	8	2	no	—
317	C₂	C₁₉×#3011	1	2	yes	—
318	C₂	C₂×C₁₉×#453	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₉
prime radical	C₁₉×#532	C₂²
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
2. C₁₉×#3230	C₂	semidirect
4. C₁₉×#1633	C₂	semidirect
5. C₁₉×#1630	C₂	semidirect
7. C₁₉×#1485	C₂	semidirect
116. #18719	C₁₉	direct (N × H)
144. #3230	C₂×C₁₉	semidirect
146. #1633	C₂×C₁₉	semidirect
147. #1630	C₂×C₁₉	semidirect
149. #1485	C₂×C₁₉	semidirect
234. C₁₉	#18719	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×#3550	2
C₁₉×#3230	2
C₁₉×#3152	2
C₁₉×#1633	2
C₁₉×#1630	2
C₁₉×#1543	2
C₁₉×#1485	2
#18719	19

Subgroups — C19×#18719 #18024678

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#18719 #18024678