Subgroups — C₁₉×#13019 #18018978

Order 4864.

496 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₉×SD₃₂	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2948	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#361	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#2141	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1979	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#1382	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1379	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
9	C₂²×D₈×C₁₉	C₂²	1	1	yes	characteristic
10	C₁₉×#518	—	2	1	no	—
11	C₂×C₁₉×SD₃₂	—	2	2	no	—
12	C₂×C₁₉×SD₃₂	—	2	2	no	—
13	C₂×C₁₉×SD₃₂	C₄	1	4	yes	complemented
14	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
16	C₁₉×#442	—	2	2	no	—
17	C₄×D₈×C₁₉	—	2	2	no	—
18	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
19	C₁₉×#427	C₂²	1	1	yes	characteristic
20	C₁₉×#419	—	2	1	no	—
21	C₁₉×#418	—	2	1	no	—
22	C₂×C₁₉×#101	—	2	1	no	—
23	C₁₉×#405	C₂²	1	1	yes	characteristic
24	C₁₉×#361	—	2	2	no	—
25	C₁₉×#358	—	2	2	no	—
26	C₂²×Q₈×C₁₉	—	2	1	no	—
27	C₂²×D₄×C₁₉	—	2	1	no	—
28	C₂×Q₁₆×C₁₉	—	2	1	no	—
29	C₂×Q₁₆×C₁₉	—	2	1	no	—
30	C₂×Q₁₆×C₁₉	—	2	1	no	—
31	C₂×Q₁₆×C₁₉	—	2	1	no	—
32	C₂×Q₁₆×C₁₉	—	2	1	no	—
33	C₂×Q₁₆×C₁₉	C₂×C₄	1	2	yes	complemented
34	C₂×D₈×C₁₉	—	2	1	no	—
35	C₂×D₈×C₁₉	—	2	1	no	—
36	C₂×D₈×C₁₉	—	2	1	no	—
37	C₂×D₈×C₁₉	C₂×C₄	1	2	yes	—
38	C₂×D₈×C₁₉	D₄	1	2	yes	—
39	C₂×D₈×C₁₉	D₄	1	2	yes	—
40	C₂×M₁₆×C₁₉	—	2	1	no	—
41	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
43	C₄×D₄×C₁₉	—	2	4	no	—
44	C₄×D₄×C₁₉	—	4	2	no	—
45	C₁₉×#117	—	2	1	no	—
46	C₁₉×#117	D₄	1	1	yes	characteristic
47	C₂×C₂²⋊C₄×C₁₉	—	4	2	no	—
48	C₁₉×SD₃₂	—	2	8	no	—
49	C₁₉×M₃₂	—	2	1	no	—
50	C₁₉×M₃₂	—	2	1	no	—
51	C₂×C₁₆×C₁₉	—	2	1	no	—
52	C₂×C₁₆×C₁₉	—	2	1	no	—
53	C₂×C₁₆×C₁₉	—	2	1	no	—
54	C₂×C₁₆×C₁₉	C₂×C₄	1	2	yes	complemented
55	C₁₉×#108	—	2	1	no	—
56	C₁₉×#108	—	2	1	no	—
57	C₁₉×#107	D₄	1	2	yes	—
58	C₁₉×#106	—	2	1	no	—
59	C₁₉×#103	—	4	2	no	—
60	C₁₉×#102	—	2	4	no	—
61	C₁₉×#101	—	4	2	no	—
62	C₁₉×#97	—	2	1	no	—
63	C₄×C₈×C₁₉	—	2	1	no	—
64	C₂⁴×C₁₉	—	4	1	no	—
65	C₂×Q₈×C₁₉	—	2	2	no	—
66	C₂×Q₈×C₁₉	—	4	1	no	—
67	C₂×Q₈×C₁₉	—	4	1	no	—
68	C₂×Q₈×C₁₉	—	4	1	no	—
69	C₂×Q₈×C₁₉	—	4	1	no	—
70	C₂×Q₈×C₁₉	—	4	1	no	—
71	C₂×D₄×C₁₉	—	2	2	no	—
72	C₂×D₄×C₁₉	—	2	2	no	—
73	C₂×D₄×C₁₉	—	2	2	no	—
74	C₂×D₄×C₁₉	—	4	1	no	—
75	C₂×D₄×C₁₉	—	4	1	no	—
76	C₂×D₄×C₁₉	—	4	1	no	—
77	C₂²×C₄×C₁₉	—	4	4	no	—
78	C₂²×C₄×C₁₉	—	4	1	no	—
79	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
80	Q₁₆×C₁₉	—	2	2	no	—
81	Q₁₆×C₁₉	—	2	2	no	—
82	Q₁₆×C₁₉	—	4	1	no	—
83	Q₁₆×C₁₉	—	4	1	no	—
84	D₈×C₁₉	—	2	2	no	—
85	D₈×C₁₉	—	2	2	no	—
86	D₈×C₁₉	—	2	2	no	—
87	D₈×C₁₉	C₂²⋊C₄	1	4	yes	—
88	M₁₆×C₁₉	—	2	1	no	—
89	M₁₆×C₁₉	—	2	1	no	—
90	M₁₆×C₁₉	—	4	1	no	—
91	M₁₆×C₁₉	—	4	1	no	—
92	C₂×C₈×C₁₉	—	2	1	no	—
93	C₂×C₈×C₁₉	—	4	1	no	—
94	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
99	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
100	C₄⋊C₄×C₁₉	—	2	2	no	—
101	C₄⋊C₄×C₁₉	D₈	1	2	yes	—
102	C₂²⋊C₄×C₁₉	—	2	1	no	—
103	C₂²⋊C₄×C₁₉	—	4	4	no	—
104	C₂²⋊C₄×C₁₉	—	4	1	no	—
105	C₄²×C₁₉	—	2	1	no	—
106	C₄²×C₁₉	—	4	1	no	—
107	C₁₆×C₁₉	—	2	2	no	—
108	C₁₆×C₁₉	—	4	1	no	—
109	#13019	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂³×C₁₉	—	4	2	no	—
111	C₂³×C₁₉	—	4	2	no	—
112	C₂³×C₁₉	—	4	2	no	—
113	C₂³×C₁₉	—	16	2	no	—
114	C₂³×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
115	Q₈×C₁₉	—	4	2	no	—
116	Q₈×C₁₉	—	4	2	no	—
117	Q₈×C₁₉	—	8	1	no	—
118	Q₈×C₁₉	—	8	1	no	—
119	D₄×C₁₉	—	2	4	no	—
120	D₄×C₁₉	—	4	2	no	—
121	D₄×C₁₉	—	4	2	no	—
122	D₄×C₁₉	—	4	2	no	—
123	C₂×C₄×C₁₉	—	2	2	no	—
124	C₂×C₄×C₁₉	—	4	2	no	—
125	C₂×C₄×C₁₉	—	4	2	no	—
126	C₂×C₄×C₁₉	—	8	1	no	—
127	C₂×C₄×C₁₉	—	8	1	no	—
128	C₂×C₄×C₁₉	—	16	4	no	—
129	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
131	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
132	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
134	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₈×C₁₉	—	4	1	no	—
136	C₈×C₁₉	—	8	1	no	—
137	C₈×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₈×C₁₉	C₄×D₄	1	1	yes	characteristic
139	C₈×C₁₉	#121	1	1	yes	characteristic
140	C₈×C₁₉	#123	1	1	yes	characteristic
141	C₂²×SD₃₂	C₂×C₁₉	1	1	yes	characteristic
142	#2948	C₂×C₁₉	1	1	yes	characteristic
143	C₂×#361	C₂×C₁₉	1	1	yes	characteristic
144	#2141	C₂×C₁₉	1	1	yes	characteristic, complemented
145	#1979	C₂×C₁₉	1	1	yes	characteristic, complemented
146	#1382	C₂×C₁₉	1	1	yes	characteristic, complemented
147	#1379	C₂×C₁₉	1	1	yes	characteristic, complemented
148	C₂²×C₁₉	—	2	2	no	—
149	C₂²×C₁₉	—	4	4	no	—
150	C₂²×C₁₉	—	8	4	no	—
151	C₂²×C₁₉	—	16	2	no	—
152	C₂²×C₁₉	—	16	2	no	—
153	C₂²×C₁₉	#442	1	1	yes	characteristic
154	C₂²×C₁₉	#447	1	1	yes	characteristic
155	C₂²×C₁₉	#463	1	1	yes	characteristic
156	C₄×C₁₉	—	4	2	no	—
157	C₄×C₁₉	—	8	2	no	—
158	C₄×C₁₉	—	8	2	no	—
159	C₄×C₁₉	#386	1	1	yes	characteristic
160	C₄×C₁₉	C₄×D₈	1	1	yes	characteristic
161	C₄×C₁₉	#449	1	1	yes	characteristic
162	C₄×C₁₉	#459	1	1	yes	characteristic
163	C₂²×Q₁₆	C₂²×C₁₉	1	1	yes	characteristic
164	C₂²×D₈	C₂²×C₁₉	1	1	yes	characteristic
165	#518	—	2	1	no	—
166	C₂×SD₃₂	—	2	2	no	—
167	C₂×SD₃₂	—	2	2	no	—
168	C₂×SD₃₂	C₄×C₁₉	1	4	yes	complemented
169	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
170	C₂²×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
171	#442	—	2	2	no	—
172	C₄×D₈	—	2	2	no	—
173	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
174	#427	C₂²×C₁₉	1	1	yes	characteristic
175	#419	—	2	1	no	—
176	#418	—	2	1	no	—
177	C₂×#101	—	2	1	no	—
178	#405	C₂²×C₁₉	1	1	yes	characteristic
179	#361	—	2	2	no	—
180	#358	—	2	2	no	—
181	C₂×C₁₉	—	2	1	no	—
182	C₂×C₁₉	—	2	1	no	—
183	C₂×C₁₉	—	8	4	no	—
184	C₂×C₁₉	#1879	1	1	yes	characteristic
185	C₂×C₁₉	#2194	1	2	yes	—
186	C₂²×Q₈	—	2	1	no	—
187	C₂²×D₄	—	2	1	no	—
188	C₂×Q₁₆	—	2	1	no	—
189	C₂×Q₁₆	—	2	1	no	—
190	C₂×Q₁₆	—	2	1	no	—
191	C₂×Q₁₆	—	2	1	no	—
192	C₂×Q₁₆	—	2	1	no	—
193	C₂×Q₁₆	C₂×C₄×C₁₉	1	2	yes	complemented
194	C₂×D₈	—	2	1	no	—
195	C₂×D₈	—	2	1	no	—
196	C₂×D₈	—	2	1	no	—
197	C₂×D₈	C₂×C₄×C₁₉	1	2	yes	—
198	C₂×D₈	D₄×C₁₉	1	2	yes	—
199	C₂×D₈	D₄×C₁₉	1	2	yes	—
200	C₂×M₁₆	—	2	1	no	—
201	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
202	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
203	C₄×D₄	—	2	4	no	—
204	C₄×D₄	—	4	2	no	—
205	#117	—	2	1	no	—
206	#117	D₄×C₁₉	1	1	yes	characteristic
207	C₂×C₂²⋊C₄	—	4	2	no	—
208	SD₃₂	—	2	8	no	—
209	M₃₂	—	2	1	no	—
210	M₃₂	—	2	1	no	—
211	C₂×C₁₆	—	2	1	no	—
212	C₂×C₁₆	—	2	1	no	—
213	C₂×C₁₆	—	2	1	no	—
214	C₂×C₁₆	C₂×C₄×C₁₉	1	2	yes	complemented
215	#108	—	2	1	no	—
216	#108	—	2	1	no	—
217	#107	D₄×C₁₉	1	2	yes	—
218	#106	—	2	1	no	—
219	#103	—	4	2	no	—
220	#102	—	2	4	no	—
221	#101	—	4	2	no	—
222	#97	—	2	1	no	—
223	C₄×C₈	—	2	1	no	—
224	C₁₉	#13019	1	1	yes	characteristic, direct product factor, Sylow
225	C₂⁴	—	4	1	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂×Q₈	—	4	1	no	—
229	C₂×Q₈	—	4	1	no	—
230	C₂×Q₈	—	4	1	no	—
231	C₂×Q₈	—	4	1	no	—
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	—	2	2	no	—
234	C₂×D₄	—	2	2	no	—
235	C₂×D₄	—	4	1	no	—
236	C₂×D₄	—	4	1	no	—
237	C₂×D₄	—	4	1	no	—
238	C₂²×C₄	—	4	4	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
241	Q₁₆	—	2	2	no	—
242	Q₁₆	—	2	2	no	—
243	Q₁₆	—	4	1	no	—
244	Q₁₆	—	4	1	no	—
245	D₈	—	2	2	no	—
246	D₈	—	2	2	no	—
247	D₈	—	2	2	no	—
248	D₈	C₂²⋊C₄×C₁₉	1	4	yes	—
249	M₁₆	—	2	1	no	—
250	M₁₆	—	2	1	no	—
251	M₁₆	—	4	1	no	—
252	M₁₆	—	4	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
256	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
257	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
258	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
259	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
260	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	D₈×C₁₉	1	2	yes	—
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	4	4	no	—
265	C₂²⋊C₄	—	4	1	no	—
266	C₄²	—	2	1	no	—
267	C₄²	—	4	1	no	—
268	C₁₆	—	2	2	no	—
269	C₁₆	—	4	1	no	—
270	C₂³	—	4	2	no	—
271	C₂³	—	4	2	no	—
272	C₂³	—	4	2	no	—
273	C₂³	—	16	2	no	—
274	C₂³	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
275	Q₈	—	4	2	no	—
276	Q₈	—	4	2	no	—
277	Q₈	—	8	1	no	—
278	Q₈	—	8	1	no	—
279	D₄	—	2	4	no	—
280	D₄	—	4	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	8	1	no	—
287	C₂×C₄	—	8	1	no	—
288	C₂×C₄	—	16	4	no	—
289	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
290	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
291	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
292	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
293	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
294	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
295	C₈	—	4	1	no	—
296	C₈	—	8	1	no	—
297	C₈	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
298	C₈	C₄×D₄×C₁₉	1	1	yes	characteristic
299	C₈	C₁₉×#121	1	1	yes	characteristic
300	C₈	C₁₉×#123	1	1	yes	characteristic
301	C₂²	—	2	2	no	—
302	C₂²	—	4	4	no	—
303	C₂²	—	8	4	no	—
304	C₂²	—	16	2	no	—
305	C₂²	—	16	2	no	—
306	C₂²	C₁₉×#442	1	1	yes	characteristic
307	C₂²	C₁₉×#447	1	1	yes	characteristic
308	C₂²	C₁₉×#463	1	1	yes	characteristic
309	C₄	—	4	2	no	—
310	C₄	—	8	2	no	—
311	C₄	—	8	2	no	—
312	C₄	C₁₉×#386	1	1	yes	characteristic
313	C₄	C₄×D₈×C₁₉	1	1	yes	characteristic
314	C₄	C₁₉×#449	1	1	yes	characteristic
315	C₄	C₁₉×#459	1	1	yes	characteristic
316	C₂	—	2	1	no	—
317	C₂	—	2	1	no	—
318	C₂	—	8	4	no	—
319	C₂	C₁₉×#1879	1	1	yes	characteristic
320	C₂	C₁₉×#2194	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉	#447
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₂²×D₈×C₁₉	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#405 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₉×#2141	C₂	semidirect
5. C₁₉×#1979	C₂	semidirect
6. C₁₉×#1382	C₂	semidirect
7. C₁₉×#1379	C₂	semidirect
13. C₂×C₁₉×SD₃₂	C₄	semidirect
33. C₂×Q₁₆×C₁₉	C₂×C₄	semidirect
54. C₂×C₁₆×C₁₉	C₂×C₄	semidirect
109. #13019	C₁₉	direct (N × H)
144. #2141	C₂×C₁₉	semidirect
145. #1979	C₂×C₁₉	semidirect
146. #1382	C₂×C₁₉	semidirect
147. #1379	C₂×C₁₉	semidirect
168. C₂×SD₃₂	C₄×C₁₉	semidirect
193. C₂×Q₁₆	C₂×C₄×C₁₉	semidirect
214. C₂×C₁₆	C₂×C₄×C₁₉	semidirect
224. C₁₉	#13019	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₁₉×SD₃₂	2
C₁₉×#2948	2
C₂×C₁₉×#361	2
C₁₉×#2141	2
C₁₉×#1979	2
C₁₉×#1382	2
C₁₉×#1379	2
#13019	19

Subgroups — C19×#13019 #18018978

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13019 #18018978