Subgroups — C₁₉×#9833 #18015792

Order 4864.

494 conjugacy classes of subgroups.

The subgroup lattice has 312 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×#1845	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×#107	—	2	2	no	—
13	C₂×C₁₉×#103	—	2	2	no	—
14	C₂×C₁₉×#103	—	2	2	no	—
15	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#384	C₂²	1	1	yes	characteristic
19	C₁₉×#384	C₂²	1	1	yes	characteristic
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×#107	—	4	4	no	—
49	C₁₉×#103	—	2	8	no	—
50	C₁₉×#103	—	4	4	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	—	2	1	no	—
55	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
56	C₄×C₈×C₁₉	D₄	1	2	yes	—
57	C₂×Q₈×C₁₉	—	2	2	no	—
58	C₂×Q₈×C₁₉	—	2	2	no	—
59	C₂×Q₈×C₁₉	—	4	1	no	—
60	C₂×Q₈×C₁₉	—	4	1	no	—
61	C₂×Q₈×C₁₉	—	4	2	no	—
62	C₂×Q₈×C₁₉	—	4	2	no	—
63	C₂×Q₈×C₁₉	—	4	2	no	—
64	C₂×Q₈×C₁₉	—	4	2	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	—	4	1	no	—
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	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	C₂×C₈×C₁₉	Q₁₆	1	4	yes	—
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	—	2	2	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	1	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	C₄⋊C₄×C₁₉	—	4	2	no	—
95	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₁₉	#41	1	1	yes	characteristic
98	C₄²×C₁₉	#41	1	1	yes	characteristic
99	#9833	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₁₉	#127	1	1	yes	characteristic
101	Q₈×C₁₉	—	4	4	no	—
102	Q₈×C₁₉	—	4	4	no	—
103	Q₈×C₁₉	—	8	2	no	—
104	Q₈×C₁₉	—	8	2	no	—
105	C₂×C₄×C₁₉	—	4	2	no	—
106	C₂×C₄×C₁₉	—	4	2	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	—	8	1	no	—
117	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
121	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
122	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
124	C₂×C₄×C₁₉	#135	1	2	yes	—
125	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
126	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
127	C₂×C₄×C₁₉	#137	1	2	yes	—
128	C₂×C₄×C₁₉	#137	1	2	yes	—
129	C₈×C₁₉	—	2	4	no	—
130	C₈×C₁₉	—	2	4	no	—
131	C₈×C₁₉	—	4	2	no	—
132	C₂×#487	C₂×C₁₉	1	1	yes	characteristic
133	#1990	C₂×C₁₉	1	1	yes	characteristic
134	#1990	C₂×C₁₉	1	1	yes	characteristic
135	#1990	C₂×C₁₉	1	1	yes	characteristic
136	#1956	C₂×C₁₉	1	1	yes	characteristic
137	#1845	C₂×C₁₉	1	1	yes	characteristic
138	C₂×#322	C₂×C₁₉	1	1	yes	characteristic
139	C₂²×C₁₉	#390	1	1	yes	characteristic
140	C₂²×C₁₉	#494	1	1	yes	characteristic
141	C₂²×C₁₉	#495	1	1	yes	characteristic
142	C₂²×C₁₉	#496	1	1	yes	characteristic
143	C₂²×C₁₉	#496	1	1	yes	characteristic
144	C₂²×C₁₉	#497	1	1	yes	characteristic
145	C₂²×C₁₉	#497	1	1	yes	characteristic
146	C₄×C₁₉	—	8	2	no	—
147	C₄×C₁₉	—	8	2	no	—
148	C₄×C₁₉	—	16	1	no	—
149	C₄×C₁₉	—	16	1	no	—
150	C₄×C₁₉	#417	1	2	yes	—
151	C₄×C₁₉	C₄×Q₁₆	1	2	yes	—
152	C₄×C₁₉	#440	1	2	yes	—
153	C₄×C₁₉	#467	1	2	yes	—
154	C₄×C₁₉	#468	1	2	yes	—
155	C₄×C₁₉	#489	1	2	yes	—
156	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
157	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
158	#487	—	2	2	no	—
159	#487	C₄×C₁₉	1	4	yes	complemented
160	C₂×#107	—	2	2	no	—
161	C₂×#103	—	2	2	no	—
162	C₂×#103	—	2	2	no	—
163	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
164	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
165	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
166	#384	C₂²×C₁₉	1	1	yes	characteristic
167	#384	C₂²×C₁₉	1	1	yes	characteristic
168	#340	—	2	2	no	—
169	#340	—	2	2	no	—
170	#340	—	2	2	no	—
171	#322	—	2	2	no	—
172	C₂×C₁₉	#1717	1	1	yes	characteristic
173	C₂×C₁₉	#1718	1	1	yes	characteristic
174	C₂×C₁₉	#1723	1	1	yes	characteristic
175	C₂×C₁₉	#1723	1	1	yes	characteristic
176	C₂×C₁₉	#1959	1	1	yes	characteristic
177	C₂×C₁₉	#1969	1	1	yes	characteristic
178	C₂×C₁₉	#1973	1	1	yes	characteristic
179	C₂²×Q₈	—	2	1	no	—
180	C₂²×Q₈	—	2	1	no	—
181	C₂²×C₈	D₄×C₁₉	1	2	yes	—
182	C₂²×C₈	D₄×C₁₉	1	2	yes	—
183	C₂²×C₈	D₄×C₁₉	1	2	yes	—
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	—	2	1	no	—
190	C₄⋊Q₈	C₂×C₄×C₁₉	1	2	yes	—
191	C₄⋊Q₈	C₂×C₄×C₁₉	1	2	yes	—
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×C₄⋊C₄	—	2	1	no	—
202	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
203	#107	—	4	4	no	—
204	#103	—	2	8	no	—
205	#103	—	4	4	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	2	1	no	—
208	C₄×C₈	—	2	1	no	—
209	C₄×C₈	—	2	1	no	—
210	C₄×C₈	C₂×C₄×C₁₉	1	2	yes	—
211	C₄×C₈	D₄×C₁₉	1	2	yes	—
212	C₁₉	#9833	1	1	yes	characteristic, direct product factor, Sylow
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	—	4	1	no	—
216	C₂×Q₈	—	4	1	no	—
217	C₂×Q₈	—	4	2	no	—
218	C₂×Q₈	—	4	2	no	—
219	C₂×Q₈	—	4	2	no	—
220	C₂×Q₈	—	4	2	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	—	4	1	no	—
224	C₂²×C₄	—	4	1	no	—
225	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
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	—	2	2	no	—
235	C₂×C₈	C₂²⋊C₄×C₁₉	1	4	yes	—
236	C₂×C₈	Q₁₆×C₁₉	1	4	yes	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	4	1	no	—
242	C₄⋊C₄	—	4	1	no	—
243	C₄⋊C₄	—	4	1	no	—
244	C₄⋊C₄	—	4	1	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	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
252	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
253	C₄²	#41×C₁₉	1	1	yes	characteristic
254	C₄²	#41×C₁₉	1	1	yes	characteristic
255	C₂³	C₁₉×#127	1	1	yes	characteristic
256	Q₈	—	4	4	no	—
257	Q₈	—	4	4	no	—
258	Q₈	—	8	2	no	—
259	Q₈	—	8	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	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	—	8	1	no	—
271	C₂×C₄	—	8	1	no	—
272	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
273	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
274	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
275	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
276	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
277	C₂×C₄	C₁₉×#124	1	1	yes	characteristic
278	C₂×C₄	C₂×Q₁₆×C₁₉	1	2	yes	—
279	C₂×C₄	C₁₉×#135	1	2	yes	—
280	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	2	yes	—
281	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	2	yes	—
282	C₂×C₄	C₁₉×#137	1	2	yes	—
283	C₂×C₄	C₁₉×#137	1	2	yes	—
284	C₈	—	2	4	no	—
285	C₈	—	2	4	no	—
286	C₈	—	4	2	no	—
287	C₂²	C₁₉×#390	1	1	yes	characteristic
288	C₂²	C₁₉×#494	1	1	yes	characteristic
289	C₂²	C₁₉×#495	1	1	yes	characteristic
290	C₂²	C₁₉×#496	1	1	yes	characteristic
291	C₂²	C₁₉×#496	1	1	yes	characteristic
292	C₂²	C₁₉×#497	1	1	yes	characteristic
293	C₂²	C₁₉×#497	1	1	yes	characteristic
294	C₄	—	8	2	no	—
295	C₄	—	8	2	no	—
296	C₄	—	16	1	no	—
297	C₄	—	16	1	no	—
298	C₄	C₁₉×#417	1	2	yes	—
299	C₄	C₄×Q₁₆×C₁₉	1	2	yes	—
300	C₄	C₁₉×#440	1	2	yes	—
301	C₄	C₁₉×#467	1	2	yes	—
302	C₄	C₁₉×#468	1	2	yes	—
303	C₄	C₁₉×#489	1	2	yes	—
304	C₂	C₁₉×#1717	1	1	yes	characteristic
305	C₂	C₁₉×#1718	1	1	yes	characteristic
306	C₂	C₁₉×#1723	1	1	yes	characteristic
307	C₂	C₁₉×#1723	1	1	yes	characteristic
308	C₂	C₁₉×#1959	1	1	yes	characteristic
309	C₂	C₁₉×#1969	1	1	yes	characteristic
310	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
99. #9833	C₁₉	direct (N × H)
159. #487	C₄×C₁₉	semidirect
212. C₁₉	#9833	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×#1845	2
C₂×C₁₉×#322	2
#9833	19

Subgroups — C19×#9833 #18015792

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#9833 #18015792