Subgroups — C₁₉×#13033 #18018992

Order 4864.

498 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#509	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2948	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#2145	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#2140	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1976	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#1384	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1378	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₉×#135	C₂²	1	1	yes	characteristic
9	C₂²×D₈×C₁₉	C₂²	1	1	yes	characteristic
10	C₁₉×#518	—	2	1	no	—
11	C₁₉×#509	—	2	2	no	—
12	C₁₉×#509	—	2	2	no	—
13	C₂×C₁₉×SD₃₂	C₄	1	2	yes	complemented
14	C₂×C₁₉×D₁₆	C₄	1	2	yes	complemented
15	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
17	C₁₉×#442	—	2	2	no	—
18	C₄×D₈×C₁₉	—	2	2	no	—
19	C₁₉×#427	C₂²	1	1	yes	characteristic
20	C₁₉×#427	C₂²	1	1	yes	characteristic
21	C₁₉×#421	—	2	1	no	—
22	C₁₉×#418	—	2	1	no	—
23	C₁₉×#410	—	2	1	no	—
24	C₁₉×#405	C₂²	1	1	yes	characteristic
25	C₁₉×#360	—	2	2	no	—
26	C₁₉×#357	—	2	2	no	—
27	C₂×#41×C₁₉	—	2	1	no	—
28	C₂²×D₄×C₁₉	—	2	1	no	—
29	C₁₉×#135	—	2	1	no	—
30	C₁₉×#135	—	2	1	no	—
31	C₁₉×#135	—	2	1	no	—
32	C₁₉×#135	—	2	1	no	—
33	C₂×Q₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic, complemented
34	C₂×SD₁₆×C₁₉	—	2	1	no	—
35	C₂×D₈×C₁₉	—	2	1	no	—
36	C₂×D₈×C₁₉	—	2	1	no	—
37	C₂×D₈×C₁₉	—	2	1	no	—
38	C₂×D₈×C₁₉	C₂×C₄	1	2	yes	—
39	C₂×D₈×C₁₉	C₂×C₄	1	1	yes	characteristic, complemented
40	C₂×D₈×C₁₉	D₄	1	2	yes	—
41	C₂×D₈×C₁₉	D₄	1	2	yes	—
42	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
44	C₄×D₄×C₁₉	—	2	4	no	—
45	C₄×D₄×C₁₉	—	4	2	no	—
46	C₁₉×#117	—	2	1	no	—
47	C₁₉×#117	—	2	1	no	—
48	C₁₉×#117	D₄	1	1	yes	characteristic
49	C₂×C₂²⋊C₄×C₁₉	—	4	2	no	—
50	C₁₉×SD₃₂	—	2	4	no	—
51	C₁₉×D₁₆	—	2	4	no	—
52	C₁₉×M₃₂	—	2	1	no	—
53	C₁₉×M₃₂	—	2	1	no	—
54	C₁₉×M₃₂	—	2	1	no	—
55	C₁₉×M₃₂	—	2	1	no	—
56	C₂×C₁₆×C₁₉	—	2	1	no	—
57	C₂×C₁₆×C₁₉	C₂×C₄	1	2	yes	complemented
58	C₁₉×#107	—	2	1	no	—
59	C₁₉×#107	—	2	1	no	—
60	C₁₉×#106	—	2	1	no	—
61	C₁₉×#106	D₄	1	2	yes	—
62	C₁₉×#104	—	4	2	no	—
63	C₁₉×#102	—	2	4	no	—
64	C₁₉×#99	—	4	2	no	—
65	C₁₉×#97	—	2	1	no	—
66	C₄×C₈×C₁₉	—	2	1	no	—
67	C₂⁴×C₁₉	—	4	1	no	—
68	#41×C₁₉	—	4	1	no	—
69	#41×C₁₉	—	4	1	no	—
70	#41×C₁₉	—	4	1	no	—
71	#41×C₁₉	—	4	1	no	—
72	C₂×Q₈×C₁₉	—	2	1	no	—
73	C₂×D₄×C₁₉	—	2	2	no	—
74	C₂×D₄×C₁₉	—	2	1	no	—
75	C₂×D₄×C₁₉	—	2	2	no	—
76	C₂×D₄×C₁₉	—	2	2	no	—
77	C₂×D₄×C₁₉	—	4	1	no	—
78	C₂×D₄×C₁₉	—	4	1	no	—
79	C₂×D₄×C₁₉	—	4	1	no	—
80	C₂×D₄×C₁₉	—	4	1	no	—
81	C₂²×C₄×C₁₉	—	4	4	no	—
82	C₂²×C₄×C₁₉	—	4	1	no	—
83	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
84	Q₁₆×C₁₉	—	2	1	no	—
85	Q₁₆×C₁₉	—	2	1	no	—
86	SD₁₆×C₁₉	—	4	1	no	—
87	SD₁₆×C₁₉	—	4	1	no	—
88	D₈×C₁₉	—	2	1	no	—
89	D₈×C₁₉	—	2	2	no	—
90	D₈×C₁₉	—	2	1	no	—
91	D₈×C₁₉	—	2	2	no	—
92	D₈×C₁₉	—	2	2	no	—
93	D₈×C₁₉	C₂²⋊C₄	1	4	yes	—
94	M₁₆×C₁₉	—	2	1	no	—
95	M₁₆×C₁₉	—	2	1	no	—
96	C₂×C₈×C₁₉	—	2	1	no	—
97	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
102	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
103	C₄⋊C₄×C₁₉	—	2	1	no	—
104	C₄⋊C₄×C₁₉	—	2	2	no	—
105	C₄⋊C₄×C₁₉	—	4	1	no	—
106	C₂²⋊C₄×C₁₉	—	4	4	no	—
107	C₂²⋊C₄×C₁₉	—	4	1	no	—
108	C₂²⋊C₄×C₁₉	—	4	1	no	—
109	C₂²⋊C₄×C₁₉	D₈	1	2	yes	—
110	C₄²×C₁₉	—	2	1	no	—
111	C₄²×C₁₉	—	4	1	no	—
112	C₄²×C₁₉	—	4	1	no	—
113	C₁₆×C₁₉	—	2	2	no	—
114	C₁₆×C₁₉	—	4	1	no	—
115	#13033	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂³×C₁₉	—	4	2	no	—
117	C₂³×C₁₉	—	4	2	no	—
118	C₂³×C₁₉	—	4	2	no	—
119	C₂³×C₁₉	—	4	1	no	—
120	C₂³×C₁₉	—	16	2	no	—
121	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
122	Q₈×C₁₉	—	4	1	no	—
123	Q₈×C₁₉	—	4	1	no	—
124	D₄×C₁₉	—	2	4	no	—
125	D₄×C₁₉	—	4	1	no	—
126	D₄×C₁₉	—	4	2	no	—
127	D₄×C₁₉	—	4	1	no	—
128	D₄×C₁₉	—	4	2	no	—
129	D₄×C₁₉	—	4	2	no	—
130	D₄×C₁₉	—	8	1	no	—
131	D₄×C₁₉	—	8	1	no	—
132	C₂×C₄×C₁₉	—	2	2	no	—
133	C₂×C₄×C₁₉	—	4	1	no	—
134	C₂×C₄×C₁₉	—	4	2	no	—
135	C₂×C₄×C₁₉	—	8	1	no	—
136	C₂×C₄×C₁₉	—	8	1	no	—
137	C₂×C₄×C₁₉	—	8	1	no	—
138	C₂×C₄×C₁₉	—	16	4	no	—
139	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
141	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
142	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
143	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₈×C₁₉	—	4	1	no	—
146	C₈×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₈×C₁₉	C₄×D₄	1	1	yes	characteristic
148	C₈×C₁₉	#121	1	1	yes	characteristic
149	C₈×C₁₉	#123	1	1	yes	characteristic
150	C₂×#509	C₂×C₁₉	1	1	yes	characteristic
151	#2948	C₂×C₁₉	1	1	yes	characteristic, complemented
152	#2145	C₂×C₁₉	1	1	yes	characteristic, complemented
153	#2140	C₂×C₁₉	1	1	yes	characteristic, complemented
154	#1976	C₂×C₁₉	1	1	yes	characteristic, complemented
155	#1384	C₂×C₁₉	1	1	yes	characteristic, complemented
156	#1378	C₂×C₁₉	1	1	yes	characteristic, complemented
157	C₂²×C₁₉	—	4	1	no	—
158	C₂²×C₁₉	—	4	4	no	—
159	C₂²×C₁₉	—	8	4	no	—
160	C₂²×C₁₉	—	8	1	no	—
161	C₂²×C₁₉	—	16	2	no	—
162	C₂²×C₁₉	—	16	2	no	—
163	C₂²×C₁₉	—	16	1	no	—
164	C₂²×C₁₉	C₄×D₈	1	1	yes	characteristic
165	C₂²×C₁₉	#447	1	1	yes	characteristic
166	C₂²×C₁₉	#459	1	1	yes	characteristic
167	C₄×C₁₉	—	4	2	no	—
168	C₄×C₁₉	—	8	2	no	—
169	C₄×C₁₉	—	8	1	no	—
170	C₄×C₁₉	—	16	1	no	—
171	C₄×C₁₉	#386	1	1	yes	characteristic
172	C₄×C₁₉	#442	1	1	yes	characteristic
173	C₄×C₁₉	#449	1	1	yes	characteristic
174	C₄×C₁₉	#463	1	1	yes	characteristic
175	C₂×#135	C₂²×C₁₉	1	1	yes	characteristic
176	C₂²×D₈	C₂²×C₁₉	1	1	yes	characteristic
177	#518	—	2	1	no	—
178	#509	—	2	2	no	—
179	#509	—	2	2	no	—
180	C₂×SD₃₂	C₄×C₁₉	1	2	yes	complemented
181	C₂×D₁₆	C₄×C₁₉	1	2	yes	complemented
182	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
183	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
184	#442	—	2	2	no	—
185	C₄×D₈	—	2	2	no	—
186	#427	C₂²×C₁₉	1	1	yes	characteristic
187	#427	C₂²×C₁₉	1	1	yes	characteristic
188	#421	—	2	1	no	—
189	#418	—	2	1	no	—
190	#410	—	2	1	no	—
191	#405	C₂²×C₁₉	1	1	yes	characteristic
192	#360	—	2	2	no	—
193	#357	—	2	2	no	—
194	C₂×C₁₉	—	2	1	no	—
195	C₂×C₁₉	—	2	1	no	—
196	C₂×C₁₉	—	2	1	no	—
197	C₂×C₁₉	—	8	4	no	—
198	C₂×C₁₉	—	16	1	no	—
199	C₂×C₁₉	#1879	1	1	yes	characteristic
200	C₂×#41	—	2	1	no	—
201	C₂²×D₄	—	2	1	no	—
202	#135	—	2	1	no	—
203	#135	—	2	1	no	—
204	#135	—	2	1	no	—
205	#135	—	2	1	no	—
206	C₂×Q₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic, complemented
207	C₂×SD₁₆	—	2	1	no	—
208	C₂×D₈	—	2	1	no	—
209	C₂×D₈	—	2	1	no	—
210	C₂×D₈	—	2	1	no	—
211	C₂×D₈	C₂×C₄×C₁₉	1	2	yes	—
212	C₂×D₈	C₂×C₄×C₁₉	1	1	yes	characteristic, complemented
213	C₂×D₈	D₄×C₁₉	1	2	yes	—
214	C₂×D₈	D₄×C₁₉	1	2	yes	—
215	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
216	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
217	C₄×D₄	—	2	4	no	—
218	C₄×D₄	—	4	2	no	—
219	#117	—	2	1	no	—
220	#117	—	2	1	no	—
221	#117	D₄×C₁₉	1	1	yes	characteristic
222	C₂×C₂²⋊C₄	—	4	2	no	—
223	SD₃₂	—	2	4	no	—
224	D₁₆	—	2	4	no	—
225	M₃₂	—	2	1	no	—
226	M₃₂	—	2	1	no	—
227	M₃₂	—	2	1	no	—
228	M₃₂	—	2	1	no	—
229	C₂×C₁₆	—	2	1	no	—
230	C₂×C₁₆	C₂×C₄×C₁₉	1	2	yes	complemented
231	#107	—	2	1	no	—
232	#107	—	2	1	no	—
233	#106	—	2	1	no	—
234	#106	D₄×C₁₉	1	2	yes	—
235	#104	—	4	2	no	—
236	#102	—	2	4	no	—
237	#99	—	4	2	no	—
238	#97	—	2	1	no	—
239	C₄×C₈	—	2	1	no	—
240	C₁₉	#13033	1	1	yes	characteristic, direct product factor, Sylow
241	C₂⁴	—	4	1	no	—
242	#41	—	4	1	no	—
243	#41	—	4	1	no	—
244	#41	—	4	1	no	—
245	#41	—	4	1	no	—
246	C₂×Q₈	—	2	1	no	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	—	2	1	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	4	1	no	—
254	C₂×D₄	—	4	1	no	—
255	C₂²×C₄	—	4	4	no	—
256	C₂²×C₄	—	4	1	no	—
257	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
258	Q₁₆	—	2	1	no	—
259	Q₁₆	—	2	1	no	—
260	SD₁₆	—	4	1	no	—
261	SD₁₆	—	4	1	no	—
262	D₈	—	2	1	no	—
263	D₈	—	2	2	no	—
264	D₈	—	2	1	no	—
265	D₈	—	2	2	no	—
266	D₈	—	2	2	no	—
267	D₈	C₂²⋊C₄×C₁₉	1	4	yes	—
268	M₁₆	—	2	1	no	—
269	M₁₆	—	2	1	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
272	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
273	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
274	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
275	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
276	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	4	4	no	—
281	C₂²⋊C₄	—	4	1	no	—
282	C₂²⋊C₄	—	4	1	no	—
283	C₂²⋊C₄	D₈×C₁₉	1	2	yes	—
284	C₄²	—	2	1	no	—
285	C₄²	—	4	1	no	—
286	C₄²	—	4	1	no	—
287	C₁₆	—	2	2	no	—
288	C₁₆	—	4	1	no	—
289	C₂³	—	4	2	no	—
290	C₂³	—	4	2	no	—
291	C₂³	—	4	2	no	—
292	C₂³	—	4	1	no	—
293	C₂³	—	16	2	no	—
294	C₂³	C₂×D₈×C₁₉	1	1	yes	characteristic
295	Q₈	—	4	1	no	—
296	Q₈	—	4	1	no	—
297	D₄	—	2	4	no	—
298	D₄	—	4	1	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	1	no	—
301	D₄	—	4	2	no	—
302	D₄	—	4	2	no	—
303	D₄	—	8	1	no	—
304	D₄	—	8	1	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	8	1	no	—
309	C₂×C₄	—	8	1	no	—
310	C₂×C₄	—	8	1	no	—
311	C₂×C₄	—	16	4	no	—
312	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
313	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
314	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
315	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
316	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
317	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
318	C₈	—	4	1	no	—
319	C₈	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
320	C₈	C₄×D₄×C₁₉	1	1	yes	characteristic
321	C₈	C₁₉×#121	1	1	yes	characteristic
322	C₈	C₁₉×#123	1	1	yes	characteristic
323	C₂²	—	4	1	no	—
324	C₂²	—	4	4	no	—
325	C₂²	—	8	4	no	—
326	C₂²	—	8	1	no	—
327	C₂²	—	16	2	no	—
328	C₂²	—	16	2	no	—
329	C₂²	—	16	1	no	—
330	C₂²	C₄×D₈×C₁₉	1	1	yes	characteristic
331	C₂²	C₁₉×#447	1	1	yes	characteristic
332	C₂²	C₁₉×#459	1	1	yes	characteristic
333	C₄	—	4	2	no	—
334	C₄	—	8	2	no	—
335	C₄	—	8	1	no	—
336	C₄	—	16	1	no	—
337	C₄	C₁₉×#386	1	1	yes	characteristic
338	C₄	C₁₉×#442	1	1	yes	characteristic
339	C₄	C₁₉×#449	1	1	yes	characteristic
340	C₄	C₁₉×#463	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	2	1	no	—
344	C₂	—	8	4	no	—
345	C₂	—	16	1	no	—
346	C₂	C₁₉×#1879	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₉	#1879
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₂×C₁₉×#509	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#2948	C₂	semidirect
3. C₁₉×#2145	C₂	semidirect
4. C₁₉×#2140	C₂	semidirect
5. C₁₉×#1976	C₂	semidirect
6. C₁₉×#1384	C₂	semidirect
7. C₁₉×#1378	C₂	semidirect
13. C₂×C₁₉×SD₃₂	C₄	semidirect
14. C₂×C₁₉×D₁₆	C₄	semidirect
33. C₂×Q₁₆×C₁₉	C₂×C₄	semidirect
39. C₂×D₈×C₁₉	C₂×C₄	semidirect
57. C₂×C₁₆×C₁₉	C₂×C₄	semidirect
115. #13033	C₁₉	direct (N × H)
151. #2948	C₂×C₁₉	semidirect
152. #2145	C₂×C₁₉	semidirect
153. #2140	C₂×C₁₉	semidirect
154. #1976	C₂×C₁₉	semidirect
155. #1384	C₂×C₁₉	semidirect
156. #1378	C₂×C₁₉	semidirect
180. C₂×SD₃₂	C₄×C₁₉	semidirect
181. C₂×D₁₆	C₄×C₁₉	semidirect
206. C₂×Q₁₆	C₂×C₄×C₁₉	semidirect
212. C₂×D₈	C₂×C₄×C₁₉	semidirect
230. C₂×C₁₆	C₂×C₄×C₁₉	semidirect
240. C₁₉	#13033	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×#509	2
C₁₉×#2948	2
C₁₉×#2145	2
C₁₉×#2140	2
C₁₉×#1976	2
C₁₉×#1384	2
C₁₉×#1378	2
#13033	19

Subgroups — C19×#13033 #18018992

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13033 #18018992