Subgroups — C₁₉×#9373 #18015332

Order 4864.

420 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3216	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2981	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1551	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1537	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1481	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#1290	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1287	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₉×#533	C₂²	1	1	yes	characteristic
9	C₁₉×#532	C₂²	1	1	yes	characteristic
10	C₁₉×#497	C₄	1	2	yes	—
11	C₁₉×#489	C₄	1	2	yes	—
12	C₁₉×#462	—	2	1	no	—
13	C₁₉×#460	—	2	1	no	—
14	C₁₉×#452	—	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	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₁₉×#408	—	2	2	no	—
24	C₁₉×#333	—	2	1	no	—
25	C₁₉×#331	—	2	1	no	—
26	C₁₉×#330	—	2	2	no	—
27	C₁₉×Q₈⋊C₈	—	2	1	no	—
28	C₁₉×Q₈⋊C₈	—	2	1	no	—
29	C₂×#41×C₁₉	D₄	1	2	yes	—
30	C₂×Q₁₆×C₁₉	—	2	2	no	—
31	C₂×SD₁₆×C₁₉	—	2	2	no	—
32	C₂×M₁₆×C₁₉	—	2	2	no	—
33	C₂²×C₈×C₁₉	—	2	2	no	—
34	C₂²×C₈×C₁₉	—	2	1	no	—
35	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
36	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
37	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
38	C₁₉×#127	D₄	1	1	yes	characteristic
39	C₁₉×#125	—	2	1	no	—
40	C₁₉×#124	C₂×C₄	1	2	yes	—
41	C₁₉×C₂²⋊Q₈	—	2	1	no	—
42	C₁₉×C₂²⋊Q₈	—	2	1	no	—
43	C₁₉×#121	—	2	1	no	—
44	C₁₉×#121	—	2	1	no	—
45	C₄×Q₈×C₁₉	—	2	1	no	—
46	C₄×Q₈×C₁₉	—	2	1	no	—
47	C₄×D₄×C₁₉	—	2	1	no	—
48	C₄×D₄×C₁₉	—	2	1	no	—
49	C₁₉×#117	—	2	1	no	—
50	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
64	C₁₉×C₂²⋊C₈	—	2	1	no	—
65	C₁₉×#97	—	2	1	no	—
66	C₁₉×#97	C₂×C₄	1	2	yes	—
67	C₁₉×#97	D₄	1	2	yes	—
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	C₂²⋊C₄	1	2	yes	—
74	C₂×D₄×C₁₉	—	2	2	no	—
75	C₂×D₄×C₁₉	—	2	1	no	—
76	C₂×D₄×C₁₉	—	2	1	no	—
77	C₂×D₄×C₁₉	C₂²⋊C₄	1	2	yes	—
78	C₂²×C₄×C₁₉	—	2	2	no	—
79	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
81	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
82	Q₁₆×C₁₉	—	8	2	no	—
83	SD₁₆×C₁₉	—	8	2	no	—
84	M₁₆×C₁₉	—	8	2	no	—
85	C₂×C₈×C₁₉	—	2	2	no	—
86	C₂×C₈×C₁₉	—	2	1	no	—
87	C₂×C₈×C₁₉	—	2	2	no	—
88	C₂×C₈×C₁₉	—	2	1	no	—
89	C₂×C₈×C₁₉	—	4	1	no	—
90	C₂×C₈×C₁₉	—	4	1	no	—
91	C₂×C₈×C₁₉	—	8	2	no	—
92	C₂×C₈×C₁₉	—	8	1	no	—
93	C₄⋊C₄×C₁₉	—	2	1	no	—
94	C₄⋊C₄×C₁₉	—	2	1	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	—	2	1	no	—
99	C₄⋊C₄×C₁₉	—	4	1	no	—
100	C₄⋊C₄×C₁₉	—	4	1	no	—
101	C₂²⋊C₄×C₁₉	—	2	2	no	—
102	C₂²⋊C₄×C₁₉	—	2	2	no	—
103	C₂²⋊C₄×C₁₉	—	4	1	no	—
104	C₄²×C₁₉	—	4	1	no	—
105	C₄²×C₁₉	C₂²×C₄	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	C₂×D₄	1	1	yes	characteristic
109	#9373	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂³×C₁₉	—	2	2	no	—
111	C₂³×C₁₉	#123	1	1	yes	characteristic
112	Q₈×C₁₉	—	4	2	no	—
113	Q₈×C₁₉	—	4	1	no	—
114	Q₈×C₁₉	—	4	1	no	—
115	Q₈×C₁₉	—	4	2	no	—
116	D₄×C₁₉	—	4	2	no	—
117	D₄×C₁₉	—	4	1	no	—
118	D₄×C₁₉	—	4	1	no	—
119	D₄×C₁₉	—	8	1	no	—
120	D₄×C₁₉	—	8	2	no	—
121	C₂×C₄×C₁₉	—	2	1	no	—
122	C₂×C₄×C₁₉	—	2	1	no	—
123	C₂×C₄×C₁₉	—	2	2	no	—
124	C₂×C₄×C₁₉	—	4	1	no	—
125	C₂×C₄×C₁₉	—	4	1	no	—
126	C₂×C₄×C₁₉	—	4	2	no	—
127	C₂×C₄×C₁₉	—	4	2	no	—
128	C₂×C₄×C₁₉	—	4	1	no	—
129	C₂×C₄×C₁₉	—	4	1	no	—
130	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
134	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
135	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
136	C₈×C₁₉	—	4	1	no	—
137	C₈×C₁₉	—	4	2	no	—
138	C₈×C₁₉	—	4	1	no	—
139	C₈×C₁₉	—	4	2	no	—
140	C₈×C₁₉	—	8	1	no	—
141	C₈×C₁₉	—	8	1	no	—
142	#3216	C₂×C₁₉	1	1	yes	characteristic
143	#2981	C₂×C₁₉	1	1	yes	characteristic
144	#1551	C₂×C₁₉	1	1	yes	characteristic
145	#1537	C₂×C₁₉	1	1	yes	characteristic, complemented
146	#1481	C₂×C₁₉	1	1	yes	characteristic, complemented
147	#1290	C₂×C₁₉	1	1	yes	characteristic, complemented
148	#1287	C₂×C₁₉	1	1	yes	characteristic, complemented
149	C₂²×C₁₉	—	2	1	no	—
150	C₂²×C₁₉	—	2	1	no	—
151	C₂²×C₁₉	—	2	1	no	—
152	C₂²×C₁₉	—	4	2	no	—
153	C₂²×C₁₉	—	4	2	no	—
154	C₂²×C₁₉	—	4	2	no	—
155	C₂²×C₁₉	#386	1	1	yes	characteristic
156	C₄×C₁₉	—	2	1	no	—
157	C₄×C₁₉	—	2	1	no	—
158	C₄×C₁₉	—	2	1	no	—
159	C₄×C₁₉	—	2	1	no	—
160	C₄×C₁₉	—	2	1	no	—
161	C₄×C₁₉	—	4	2	no	—
162	C₄×C₁₉	—	8	1	no	—
163	C₄×C₁₉	—	8	1	no	—
164	C₄×C₁₉	#471	1	1	yes	characteristic
165	C₄×C₁₉	#473	1	1	yes	characteristic
166	#533	C₂²×C₁₉	1	1	yes	characteristic
167	#532	C₂²×C₁₉	1	1	yes	characteristic
168	#497	C₄×C₁₉	1	2	yes	—
169	#489	C₄×C₁₉	1	2	yes	—
170	#462	—	2	1	no	—
171	#460	—	2	1	no	—
172	#452	—	2	2	no	—
173	#436	—	2	1	no	—
174	#435	—	2	2	no	—
175	C₈×D₄	—	2	1	no	—
176	#432	C₂²×C₁₉	1	1	yes	characteristic
177	#432	C₂²×C₁₉	1	1	yes	characteristic
178	#432	C₂²×C₁₉	1	1	yes	characteristic
179	#431	C₂²×C₁₉	1	1	yes	characteristic
180	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
181	#408	—	2	2	no	—
182	#333	—	2	1	no	—
183	#331	—	2	1	no	—
184	#330	—	2	2	no	—
185	Q₈⋊C₈	—	2	1	no	—
186	Q₈⋊C₈	—	2	1	no	—
187	C₂×C₁₉	—	4	1	no	—
188	C₂×C₁₉	—	8	2	no	—
189	C₂×C₁₉	#1881	1	1	yes	characteristic
190	C₂×C₁₉	#1908	1	1	yes	characteristic
191	C₂×C₁₉	#1911	1	1	yes	characteristic
192	C₂×#41	D₄×C₁₉	1	2	yes	—
193	C₂×Q₁₆	—	2	2	no	—
194	C₂×SD₁₆	—	2	2	no	—
195	C₂×M₁₆	—	2	2	no	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	2	1	no	—
198	C₄⋊Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
199	C₄⋊Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
200	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
201	#127	D₄×C₁₉	1	1	yes	characteristic
202	#125	—	2	1	no	—
203	#124	C₂×C₄×C₁₉	1	2	yes	—
204	C₂²⋊Q₈	—	2	1	no	—
205	C₂²⋊Q₈	—	2	1	no	—
206	#121	—	2	1	no	—
207	#121	—	2	1	no	—
208	C₄×Q₈	—	2	1	no	—
209	C₄×Q₈	—	2	1	no	—
210	C₄×D₄	—	2	1	no	—
211	C₄×D₄	—	2	1	no	—
212	#117	—	2	1	no	—
213	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
214	C₄⋊C₈	—	2	1	no	—
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	—	2	1	no	—
217	C₄⋊C₈	—	2	1	no	—
218	C₄⋊C₈	—	2	1	no	—
219	C₄⋊C₈	—	2	1	no	—
220	#103	—	2	2	no	—
221	#102	—	2	2	no	—
222	C₂²⋊C₈	—	2	2	no	—
223	C₂²⋊C₈	—	2	2	no	—
224	C₂²⋊C₈	—	2	1	no	—
225	C₂²⋊C₈	—	2	1	no	—
226	C₂²⋊C₈	—	2	1	no	—
227	C₂²⋊C₈	—	2	1	no	—
228	#97	—	2	1	no	—
229	#97	C₂×C₄×C₁₉	1	2	yes	—
230	#97	D₄×C₁₉	1	2	yes	—
231	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
232	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
233	C₁₉	#9373	1	1	yes	characteristic, direct product factor, Sylow
234	#41	—	8	2	no	—
235	C₂×Q₈	—	2	1	no	—
236	C₂×Q₈	—	2	1	no	—
237	C₂×Q₈	C₂²⋊C₄×C₁₉	1	2	yes	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	C₂²⋊C₄×C₁₉	1	2	yes	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
244	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
245	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
246	Q₁₆	—	8	2	no	—
247	SD₁₆	—	8	2	no	—
248	M₁₆	—	8	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	8	2	no	—
256	C₂×C₈	—	8	1	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₂²⋊C₄	—	2	2	no	—
266	C₂²⋊C₄	—	2	2	no	—
267	C₂²⋊C₄	—	4	1	no	—
268	C₄²	—	4	1	no	—
269	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
270	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
271	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
272	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
273	C₂³	—	2	2	no	—
274	C₂³	C₁₉×#123	1	1	yes	characteristic
275	Q₈	—	4	2	no	—
276	Q₈	—	4	1	no	—
277	Q₈	—	4	1	no	—
278	Q₈	—	4	2	no	—
279	D₄	—	4	2	no	—
280	D₄	—	4	1	no	—
281	D₄	—	4	1	no	—
282	D₄	—	8	1	no	—
283	D₄	—	8	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
294	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
295	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
296	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
297	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
298	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
299	C₈	—	4	1	no	—
300	C₈	—	4	2	no	—
301	C₈	—	4	1	no	—
302	C₈	—	4	2	no	—
303	C₈	—	8	1	no	—
304	C₈	—	8	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	4	2	no	—
309	C₂²	—	4	2	no	—
310	C₂²	—	4	2	no	—
311	C₂²	C₁₉×#386	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	4	2	no	—
318	C₄	—	8	1	no	—
319	C₄	—	8	1	no	—
320	C₄	C₁₉×#471	1	1	yes	characteristic
321	C₄	C₁₉×#473	1	1	yes	characteristic
322	C₂	—	4	1	no	—
323	C₂	—	8	2	no	—
324	C₂	C₁₉×#1881	1	1	yes	characteristic
325	C₂	C₁₉×#1908	1	1	yes	characteristic
326	C₂	C₁₉×#1911	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#386
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×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
4. C₁₉×#1537	C₂	semidirect
5. C₁₉×#1481	C₂	semidirect
6. C₁₉×#1290	C₂	semidirect
7. C₁₉×#1287	C₂	semidirect
109. #9373	C₁₉	direct (N × H)
145. #1537	C₂×C₁₉	semidirect
146. #1481	C₂×C₁₉	semidirect
147. #1290	C₂×C₁₉	semidirect
148. #1287	C₂×C₁₉	semidirect
233. C₁₉	#9373	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×#3216	2
C₁₉×#2981	2
C₁₉×#1551	2
C₁₉×#1537	2
C₁₉×#1481	2
C₁₉×#1290	2
C₁₉×#1287	2
#9373	19

Subgroups — C19×#9373 #18015332

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#9373 #18015332