Subgroups — C₁₉×#9840 #18015799

Order 4864.

478 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#487	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1990	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1990	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1990	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1956	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1834	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₉×#322	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₁₉×#487	—	2	2	no	—
11	C₁₉×#487	C₄	1	4	yes	complemented
12	C₂×C₁₉×#103	—	2	2	no	—
13	C₂×C₁₉×#103	—	2	2	no	—
14	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#384	C₂²	1	1	yes	characteristic
18	C₁₉×#384	C₂²	1	1	yes	characteristic
19	C₁₉×#340	—	2	2	no	—
20	C₁₉×#340	—	2	2	no	—
21	C₁₉×#340	—	2	2	no	—
22	C₁₉×#340	—	2	2	no	—
23	C₁₉×#322	—	2	2	no	—
24	C₂²×Q₈×C₁₉	—	2	1	no	—
25	C₂²×Q₈×C₁₉	—	2	1	no	—
26	C₂²×C₈×C₁₉	D₄	1	2	yes	—
27	C₂²×C₈×C₁₉	D₄	1	2	yes	—
28	C₂²×C₈×C₁₉	D₄	1	2	yes	—
29	C₁₉×C₄⋊Q₈	—	2	1	no	—
30	C₁₉×C₄⋊Q₈	—	2	1	no	—
31	C₁₉×C₄⋊Q₈	—	2	1	no	—
32	C₁₉×C₄⋊Q₈	—	2	1	no	—
33	C₁₉×C₄⋊Q₈	—	2	1	no	—
34	C₁₉×C₄⋊Q₈	—	2	1	no	—
35	C₁₉×C₄⋊Q₈	C₂×C₄	1	2	yes	—
36	C₁₉×C₄⋊Q₈	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
38	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
39	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
40	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
41	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
42	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
43	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
44	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
45	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
46	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
47	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
48	C₁₉×#103	—	2	8	no	—
49	C₁₉×#103	—	4	4	no	—
50	C₄×C₈×C₁₉	—	2	1	no	—
51	C₄×C₈×C₁₉	—	2	1	no	—
52	C₄×C₈×C₁₉	—	2	1	no	—
53	C₄×C₈×C₁₉	—	2	1	no	—
54	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₁₉	D₄	1	2	yes	—
56	C₂×Q₈×C₁₉	—	2	2	no	—
57	C₂×Q₈×C₁₉	—	2	2	no	—
58	C₂×Q₈×C₁₉	—	4	1	no	—
59	C₂×Q₈×C₁₉	—	4	1	no	—
60	C₂×Q₈×C₁₉	—	4	2	no	—
61	C₂×Q₈×C₁₉	—	4	2	no	—
62	C₂×Q₈×C₁₉	—	4	2	no	—
63	C₂×Q₈×C₁₉	—	4	2	no	—
64	C₂²×C₄×C₁₉	—	4	1	no	—
65	C₂²×C₄×C₁₉	—	4	1	no	—
66	C₂²×C₄×C₁₉	—	4	1	no	—
67	C₂²×C₄×C₁₉	—	4	1	no	—
68	C₂²×C₄×C₁₉	C₂×D₄	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	C₂×C₈×C₁₉	—	2	2	no	—
72	C₂×C₈×C₁₉	—	2	2	no	—
73	C₂×C₈×C₁₉	—	2	2	no	—
74	C₂×C₈×C₁₉	—	2	2	no	—
75	C₂×C₈×C₁₉	—	2	2	no	—
76	C₂×C₈×C₁₉	—	2	2	no	—
77	C₂×C₈×C₁₉	—	2	2	no	—
78	C₂×C₈×C₁₉	—	2	2	no	—
79	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
80	C₄⋊C₄×C₁₉	—	2	2	no	—
81	C₄⋊C₄×C₁₉	—	2	2	no	—
82	C₄⋊C₄×C₁₉	—	2	2	no	—
83	C₄⋊C₄×C₁₉	—	2	2	no	—
84	C₄⋊C₄×C₁₉	—	4	1	no	—
85	C₄⋊C₄×C₁₉	—	4	1	no	—
86	C₄⋊C₄×C₁₉	—	4	1	no	—
87	C₄⋊C₄×C₁₉	—	4	1	no	—
88	C₄⋊C₄×C₁₉	—	4	2	no	—
89	C₄⋊C₄×C₁₉	—	4	2	no	—
90	C₄⋊C₄×C₁₉	—	4	2	no	—
91	C₄⋊C₄×C₁₉	—	4	2	no	—
92	C₄⋊C₄×C₁₉	—	4	2	no	—
93	C₄⋊C₄×C₁₉	—	4	2	no	—
94	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₁₉	#41	1	1	yes	characteristic
97	C₄²×C₁₉	#41	1	1	yes	characteristic
98	#9840	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂³×C₁₉	#127	1	1	yes	characteristic
100	Q₈×C₁₉	—	4	4	no	—
101	Q₈×C₁₉	—	4	4	no	—
102	Q₈×C₁₉	—	8	2	no	—
103	Q₈×C₁₉	—	8	2	no	—
104	C₂×C₄×C₁₉	—	4	2	no	—
105	C₂×C₄×C₁₉	—	4	2	no	—
106	C₂×C₄×C₁₉	—	8	1	no	—
107	C₂×C₄×C₁₉	—	8	1	no	—
108	C₂×C₄×C₁₉	—	8	1	no	—
109	C₂×C₄×C₁₉	—	8	1	no	—
110	C₂×C₄×C₁₉	—	8	1	no	—
111	C₂×C₄×C₁₉	—	8	1	no	—
112	C₂×C₄×C₁₉	—	8	1	no	—
113	C₂×C₄×C₁₉	—	8	1	no	—
114	C₂×C₄×C₁₉	—	8	1	no	—
115	C₂×C₄×C₁₉	—	8	1	no	—
116	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
120	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
121	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
122	C₂×C₄×C₁₉	#135	1	2	yes	—
123	C₂×C₄×C₁₉	#135	1	2	yes	—
124	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
125	C₂×C₄×C₁₉	#137	1	2	yes	—
126	C₂×C₄×C₁₉	#137	1	2	yes	—
127	C₂×C₄×C₁₉	#137	1	2	yes	—
128	C₈×C₁₉	—	2	4	no	—
129	C₈×C₁₉	—	4	2	no	—
130	C₈×C₁₉	—	4	2	no	—
131	C₂×#487	C₂×C₁₉	1	1	yes	characteristic
132	#1990	C₂×C₁₉	1	1	yes	characteristic
133	#1990	C₂×C₁₉	1	1	yes	characteristic
134	#1990	C₂×C₁₉	1	1	yes	characteristic
135	#1956	C₂×C₁₉	1	1	yes	characteristic
136	#1834	C₂×C₁₉	1	1	yes	characteristic
137	C₂×#322	C₂×C₁₉	1	1	yes	characteristic
138	C₂²×C₁₉	#390	1	1	yes	characteristic
139	C₂²×C₁₉	#495	1	1	yes	characteristic
140	C₂²×C₁₉	#495	1	1	yes	characteristic
141	C₂²×C₁₉	#496	1	1	yes	characteristic
142	C₂²×C₁₉	#497	1	1	yes	characteristic
143	C₂²×C₁₉	#497	1	1	yes	characteristic
144	C₂²×C₁₉	#497	1	1	yes	characteristic
145	C₄×C₁₉	—	8	2	no	—
146	C₄×C₁₉	—	8	2	no	—
147	C₄×C₁₉	—	16	1	no	—
148	C₄×C₁₉	—	16	1	no	—
149	C₄×C₁₉	#416	1	2	yes	—
150	C₄×C₁₉	#440	1	2	yes	—
151	C₄×C₁₉	#441	1	2	yes	—
152	C₄×C₁₉	#468	1	2	yes	—
153	C₄×C₁₉	#470	1	2	yes	—
154	C₄×C₁₉	#488	1	2	yes	—
155	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
156	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
157	#487	—	2	2	no	—
158	#487	C₄×C₁₉	1	4	yes	complemented
159	C₂×#103	—	2	2	no	—
160	C₂×#103	—	2	2	no	—
161	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
162	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
163	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
164	#384	C₂²×C₁₉	1	1	yes	characteristic
165	#384	C₂²×C₁₉	1	1	yes	characteristic
166	#340	—	2	2	no	—
167	#340	—	2	2	no	—
168	#340	—	2	2	no	—
169	#340	—	2	2	no	—
170	#322	—	2	2	no	—
171	C₂×C₁₉	#1718	1	1	yes	characteristic
172	C₂×C₁₉	#1718	1	1	yes	characteristic
173	C₂×C₁₉	#1719	1	1	yes	characteristic
174	C₂×C₁₉	#1720	1	1	yes	characteristic
175	C₂×C₁₉	#1958	1	1	yes	characteristic
176	C₂×C₁₉	#1971	1	1	yes	characteristic
177	C₂×C₁₉	#1973	1	1	yes	characteristic
178	C₂²×Q₈	—	2	1	no	—
179	C₂²×Q₈	—	2	1	no	—
180	C₂²×C₈	D₄×C₁₉	1	2	yes	—
181	C₂²×C₈	D₄×C₁₉	1	2	yes	—
182	C₂²×C₈	D₄×C₁₉	1	2	yes	—
183	C₄⋊Q₈	—	2	1	no	—
184	C₄⋊Q₈	—	2	1	no	—
185	C₄⋊Q₈	—	2	1	no	—
186	C₄⋊Q₈	—	2	1	no	—
187	C₄⋊Q₈	—	2	1	no	—
188	C₄⋊Q₈	—	2	1	no	—
189	C₄⋊Q₈	C₂×C₄×C₁₉	1	2	yes	—
190	C₄⋊Q₈	C₂×C₄×C₁₉	1	2	yes	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₄⋊C₄	—	2	1	no	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄⋊C₄	—	2	1	no	—
199	C₂×C₄⋊C₄	—	2	1	no	—
200	C₂×C₄⋊C₄	—	2	1	no	—
201	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
202	#103	—	2	8	no	—
203	#103	—	4	4	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	2	1	no	—
208	C₄×C₈	C₂×C₄×C₁₉	1	2	yes	—
209	C₄×C₈	D₄×C₁₉	1	2	yes	—
210	C₁₉	#9840	1	1	yes	characteristic, direct product factor, Sylow
211	C₂×Q₈	—	2	2	no	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	4	1	no	—
214	C₂×Q₈	—	4	1	no	—
215	C₂×Q₈	—	4	2	no	—
216	C₂×Q₈	—	4	2	no	—
217	C₂×Q₈	—	4	2	no	—
218	C₂×Q₈	—	4	2	no	—
219	C₂²×C₄	—	4	1	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
225	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	C₂²⋊C₄×C₁₉	1	4	yes	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	4	1	no	—
240	C₄⋊C₄	—	4	1	no	—
241	C₄⋊C₄	—	4	1	no	—
242	C₄⋊C₄	—	4	1	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
250	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
251	C₄²	#41×C₁₉	1	1	yes	characteristic
252	C₄²	#41×C₁₉	1	1	yes	characteristic
253	C₂³	C₁₉×#127	1	1	yes	characteristic
254	Q₈	—	4	4	no	—
255	Q₈	—	4	4	no	—
256	Q₈	—	8	2	no	—
257	Q₈	—	8	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	8	1	no	—
261	C₂×C₄	—	8	1	no	—
262	C₂×C₄	—	8	1	no	—
263	C₂×C₄	—	8	1	no	—
264	C₂×C₄	—	8	1	no	—
265	C₂×C₄	—	8	1	no	—
266	C₂×C₄	—	8	1	no	—
267	C₂×C₄	—	8	1	no	—
268	C₂×C₄	—	8	1	no	—
269	C₂×C₄	—	8	1	no	—
270	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
271	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
272	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
273	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
274	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
275	C₂×C₄	C₁₉×#124	1	1	yes	characteristic
276	C₂×C₄	C₁₉×#135	1	2	yes	—
277	C₂×C₄	C₁₉×#135	1	2	yes	—
278	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	2	yes	—
279	C₂×C₄	C₁₉×#137	1	2	yes	—
280	C₂×C₄	C₁₉×#137	1	2	yes	—
281	C₂×C₄	C₁₉×#137	1	2	yes	—
282	C₈	—	2	4	no	—
283	C₈	—	4	2	no	—
284	C₈	—	4	2	no	—
285	C₂²	C₁₉×#390	1	1	yes	characteristic
286	C₂²	C₁₉×#495	1	1	yes	characteristic
287	C₂²	C₁₉×#495	1	1	yes	characteristic
288	C₂²	C₁₉×#496	1	1	yes	characteristic
289	C₂²	C₁₉×#497	1	1	yes	characteristic
290	C₂²	C₁₉×#497	1	1	yes	characteristic
291	C₂²	C₁₉×#497	1	1	yes	characteristic
292	C₄	—	8	2	no	—
293	C₄	—	8	2	no	—
294	C₄	—	16	1	no	—
295	C₄	—	16	1	no	—
296	C₄	C₁₉×#416	1	2	yes	—
297	C₄	C₁₉×#440	1	2	yes	—
298	C₄	C₁₉×#441	1	2	yes	—
299	C₄	C₁₉×#468	1	2	yes	—
300	C₄	C₁₉×#470	1	2	yes	—
301	C₄	C₁₉×#488	1	2	yes	—
302	C₂	C₁₉×#1718	1	1	yes	characteristic
303	C₂	C₁₉×#1718	1	1	yes	characteristic
304	C₂	C₁₉×#1719	1	1	yes	characteristic
305	C₂	C₁₉×#1720	1	1	yes	characteristic
306	C₂	C₁₉×#1958	1	1	yes	characteristic
307	C₂	C₁₉×#1971	1	1	yes	characteristic
308	C₂	C₁₉×#1973	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₉	#127
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₁₉×#487	C₄	semidirect
98. #9840	C₁₉	direct (N × H)
158. #487	C₄×C₁₉	semidirect
210. C₁₉	#9840	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×#487	2
C₁₉×#1990	2
C₁₉×#1956	2
C₁₉×#1834	2
C₂×C₁₉×#322	2
#9840	19

Subgroups — C19×#9840 #18015799

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#9840 #18015799