Subgroups — C₁₉×#9284 #18015243

Order 4864.

396 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3228	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2985	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#1531	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1530	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1496	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1287	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1275	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×#500	C₄	1	1	yes	characteristic
11	C₁₉×#498	C₄	1	1	yes	characteristic
12	C₁₉×#486	C₄	1	1	yes	characteristic
13	C₁₉×#486	C₄	1	1	yes	characteristic
14	C₁₉×#482	—	2	2	no	—
15	C₁₉×#478	—	2	2	no	—
16	C₁₉×#446	—	2	2	no	—
17	C₁₉×#446	—	2	1	no	—
18	C₈×Q₈×C₁₉	—	2	1	no	—
19	C₁₉×#433	—	2	2	no	—
20	C₁₉×#432	C₂²	1	1	yes	characteristic
21	C₁₉×#432	C₂²	1	1	yes	characteristic
22	C₁₉×#431	C₂²	1	1	yes	characteristic
23	C₁₉×#431	C₂²	1	1	yes	characteristic
24	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
25	C₁₉×#330	—	2	2	no	—
26	C₁₉×#329	—	2	2	no	—
27	C₁₉×D₄⋊C₈	—	2	1	no	—
28	C₁₉×D₄⋊C₈	—	2	1	no	—
29	C₂×#41×C₁₉	—	2	1	no	—
30	C₂²×C₈×C₁₉	—	2	1	no	—
31	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
32	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
33	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
34	C₁₉×#127	C₂×C₄	1	1	yes	characteristic
35	C₁₉×#125	D₄	1	2	yes	—
36	C₁₉×#124	—	2	1	no	—
37	C₁₉×C₂²⋊Q₈	—	2	1	no	—
38	C₁₉×C₂²⋊Q₈	—	2	1	no	—
39	C₁₉×#121	—	2	1	no	—
40	C₁₉×#121	—	2	1	no	—
41	C₄×Q₈×C₁₉	—	2	1	no	—
42	C₄×Q₈×C₁₉	—	2	1	no	—
43	C₄×D₄×C₁₉	—	2	1	no	—
44	C₄×D₄×C₁₉	—	2	1	no	—
45	C₁₉×#117	D₄	1	2	yes	—
46	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
47	C₁₉×#107	—	2	2	no	—
48	C₁₉×#106	—	2	2	no	—
49	C₁₉×C₄⋊C₈	—	2	2	no	—
50	C₁₉×C₄⋊C₈	—	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×#102	—	2	2	no	—
56	C₁₉×#102	—	2	2	no	—
57	C₁₉×C₂²⋊C₈	—	2	1	no	—
58	C₁₉×C₂²⋊C₈	—	2	1	no	—
59	C₁₉×C₂²⋊C₈	—	2	1	no	—
60	C₁₉×C₂²⋊C₈	—	2	1	no	—
61	C₁₉×#97	—	2	2	no	—
62	C₁₉×#97	—	2	1	no	—
63	C₁₉×#97	—	2	1	no	—
64	C₁₉×#97	—	2	1	no	—
65	C₁₉×#97	D₄	1	2	yes	—
66	C₄×C₈×C₁₉	—	2	2	no	—
67	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
68	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
70	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
71	#41×C₁₉	—	8	2	no	—
72	C₂×Q₈×C₁₉	—	2	1	no	—
73	C₂×Q₈×C₁₉	—	2	1	no	—
74	C₂×Q₈×C₁₉	—	2	1	no	—
75	C₂×D₄×C₁₉	—	2	1	no	—
76	C₂×D₄×C₁₉	—	2	1	no	—
77	C₂×D₄×C₁₉	—	2	1	no	—
78	C₂×D₄×C₁₉	—	4	1	no	—
79	C₂²×C₄×C₁₉	—	4	1	no	—
80	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
82	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
83	C₂×C₈×C₁₉	—	2	2	no	—
84	C₂×C₈×C₁₉	—	2	1	no	—
85	C₂×C₈×C₁₉	—	2	1	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	—	4	1	no	—
92	C₄⋊C₄×C₁₉	—	2	1	no	—
93	C₄⋊C₄×C₁₉	—	2	1	no	—
94	C₄⋊C₄×C₁₉	—	2	2	no	—
95	C₄⋊C₄×C₁₉	—	2	2	no	—
96	C₄⋊C₄×C₁₉	—	2	1	no	—
97	C₄⋊C₄×C₁₉	—	2	1	no	—
98	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
99	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
100	C₂²⋊C₄×C₁₉	—	2	2	no	—
101	C₂²⋊C₄×C₁₉	—	4	1	no	—
102	C₂²⋊C₄×C₁₉	—	4	1	no	—
103	C₄²×C₁₉	—	2	2	no	—
104	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
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	#9284	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂³×C₁₉	—	4	1	no	—
110	C₂³×C₁₉	#123	1	1	yes	characteristic
111	Q₈×C₁₉	—	4	1	no	—
112	Q₈×C₁₉	—	4	1	no	—
113	Q₈×C₁₉	—	8	1	no	—
114	Q₈×C₁₉	—	8	1	no	—
115	D₄×C₁₉	—	4	2	no	—
116	D₄×C₁₉	—	4	1	no	—
117	D₄×C₁₉	—	4	1	no	—
118	D₄×C₁₉	—	8	1	no	—
119	D₄×C₁₉	—	8	2	no	—
120	C₂×C₄×C₁₉	—	2	1	no	—
121	C₂×C₄×C₁₉	—	2	1	no	—
122	C₂×C₄×C₁₉	—	2	2	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	1	no	—
127	C₂×C₄×C₁₉	—	8	2	no	—
128	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
132	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
133	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
134	C₈×C₁₉	—	4	1	no	—
135	C₈×C₁₉	—	4	2	no	—
136	C₈×C₁₉	—	4	1	no	—
137	C₈×C₁₉	—	4	1	no	—
138	C₈×C₁₉	—	4	1	no	—
139	C₈×C₁₉	—	4	2	no	—
140	C₈×C₁₉	—	8	1	no	—
141	#3228	C₂×C₁₉	1	1	yes	characteristic
142	#2985	C₂×C₁₉	1	1	yes	characteristic, complemented
143	#1531	C₂×C₁₉	1	1	yes	characteristic, complemented
144	#1530	C₂×C₁₉	1	1	yes	characteristic
145	#1496	C₂×C₁₉	1	1	yes	characteristic
146	#1287	C₂×C₁₉	1	1	yes	characteristic, complemented
147	#1275	C₂×C₁₉	1	1	yes	characteristic, complemented
148	C₂²×C₁₉	—	2	1	no	—
149	C₂²×C₁₉	—	2	1	no	—
150	C₂²×C₁₉	—	2	1	no	—
151	C₂²×C₁₉	—	8	1	no	—
152	C₂²×C₁₉	—	8	1	no	—
153	C₂²×C₁₉	—	8	1	no	—
154	C₂²×C₁₉	#386	1	1	yes	characteristic
155	C₄×C₁₉	—	2	1	no	—
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	—	4	2	no	—
161	C₄×C₁₉	—	4	2	no	—
162	C₄×C₁₉	—	8	1	no	—
163	C₄×C₁₉	#443	1	1	yes	characteristic
164	C₄×C₁₉	#444	1	1	yes	characteristic
165	#533	C₂²×C₁₉	1	1	yes	characteristic
166	#532	C₂²×C₁₉	1	1	yes	characteristic
167	#500	C₄×C₁₉	1	1	yes	characteristic
168	#498	C₄×C₁₉	1	1	yes	characteristic
169	#486	C₄×C₁₉	1	1	yes	characteristic
170	#486	C₄×C₁₉	1	1	yes	characteristic
171	#482	—	2	2	no	—
172	#478	—	2	2	no	—
173	#446	—	2	2	no	—
174	#446	—	2	1	no	—
175	C₈×Q₈	—	2	1	no	—
176	#433	—	2	2	no	—
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	#431	C₂²×C₁₉	1	1	yes	characteristic
181	C₂×#97	C₂²×C₁₉	1	1	yes	characteristic
182	#330	—	2	2	no	—
183	#329	—	2	2	no	—
184	D₄⋊C₈	—	2	1	no	—
185	D₄⋊C₈	—	2	1	no	—
186	C₂×C₁₉	—	4	1	no	—
187	C₂×C₁₉	—	16	1	no	—
188	C₂×C₁₉	#1876	1	1	yes	characteristic
189	C₂×C₁₉	#1903	1	1	yes	characteristic
190	C₂×C₁₉	#1912	1	1	yes	characteristic
191	C₂×#41	—	2	1	no	—
192	C₂²×C₈	—	2	1	no	—
193	C₄⋊Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
194	C₄⋊Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
195	C₄⋊Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
196	#127	C₂×C₄×C₁₉	1	1	yes	characteristic
197	#125	D₄×C₁₉	1	2	yes	—
198	#124	—	2	1	no	—
199	C₂²⋊Q₈	—	2	1	no	—
200	C₂²⋊Q₈	—	2	1	no	—
201	#121	—	2	1	no	—
202	#121	—	2	1	no	—
203	C₄×Q₈	—	2	1	no	—
204	C₄×Q₈	—	2	1	no	—
205	C₄×D₄	—	2	1	no	—
206	C₄×D₄	—	2	1	no	—
207	#117	D₄×C₁₉	1	2	yes	—
208	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
209	#107	—	2	2	no	—
210	#106	—	2	2	no	—
211	C₄⋊C₈	—	2	2	no	—
212	C₄⋊C₈	—	2	2	no	—
213	C₄⋊C₈	—	2	1	no	—
214	C₄⋊C₈	—	2	1	no	—
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	—	2	1	no	—
217	#102	—	2	2	no	—
218	#102	—	2	2	no	—
219	C₂²⋊C₈	—	2	1	no	—
220	C₂²⋊C₈	—	2	1	no	—
221	C₂²⋊C₈	—	2	1	no	—
222	C₂²⋊C₈	—	2	1	no	—
223	#97	—	2	2	no	—
224	#97	—	2	1	no	—
225	#97	—	2	1	no	—
226	#97	—	2	1	no	—
227	#97	D₄×C₁₉	1	2	yes	—
228	C₄×C₈	—	2	2	no	—
229	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
230	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
231	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
232	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
233	C₁₉	#9284	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	—	2	1	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	4	1	no	—
242	C₂²×C₄	—	4	1	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	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
262	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
263	C₂²⋊C₄	—	2	2	no	—
264	C₂²⋊C₄	—	4	1	no	—
265	C₂²⋊C₄	—	4	1	no	—
266	C₄²	—	2	2	no	—
267	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
268	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
269	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
270	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
271	C₂³	—	4	1	no	—
272	C₂³	C₁₉×#123	1	1	yes	characteristic
273	Q₈	—	4	1	no	—
274	Q₈	—	4	1	no	—
275	Q₈	—	8	1	no	—
276	Q₈	—	8	1	no	—
277	D₄	—	4	2	no	—
278	D₄	—	4	1	no	—
279	D₄	—	4	1	no	—
280	D₄	—	8	1	no	—
281	D₄	—	8	2	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	8	2	no	—
290	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
291	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
292	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
293	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
294	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
295	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
296	C₈	—	4	2	no	—
297	C₈	—	4	1	no	—
298	C₈	—	4	2	no	—
299	C₈	—	4	1	no	—
300	C₈	—	4	1	no	—
301	C₈	—	4	1	no	—
302	C₈	—	8	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	8	1	no	—
307	C₂²	—	8	1	no	—
308	C₂²	—	8	1	no	—
309	C₂²	C₁₉×#386	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	4	2	no	—
316	C₄	—	4	2	no	—
317	C₄	—	8	1	no	—
318	C₄	C₁₉×#443	1	1	yes	characteristic
319	C₄	C₁₉×#444	1	1	yes	characteristic
320	C₂	—	4	1	no	—
321	C₂	—	16	1	no	—
322	C₂	C₁₉×#1876	1	1	yes	characteristic
323	C₂	C₁₉×#1903	1	1	yes	characteristic
324	C₂	C₁₉×#1912	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
2. C₁₉×#2985	C₂	semidirect
3. C₁₉×#1531	C₂	semidirect
6. C₁₉×#1287	C₂	semidirect
7. C₁₉×#1275	C₂	semidirect
108. #9284	C₁₉	direct (N × H)
142. #2985	C₂×C₁₉	semidirect
143. #1531	C₂×C₁₉	semidirect
146. #1287	C₂×C₁₉	semidirect
147. #1275	C₂×C₁₉	semidirect
233. C₁₉	#9284	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×#3228	2
C₁₉×#2985	2
C₁₉×#1531	2
C₁₉×#1530	2
C₁₉×#1496	2
C₁₉×#1287	2
C₁₉×#1275	2
#9284	19

Subgroups — C19×#9284 #18015243

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#9284 #18015243