Subgroups — C₁₉×#13249 #18019208

Order 4864.

444 conjugacy classes of subgroups.

The subgroup lattice has 346 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	C₂×C₁₉×#365	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₉×#361	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₉×#360	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₉×#347	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#1975	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1972	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×#509	—	2	2	no	—
11	C₁₉×#509	—	2	2	no	—
12	C₂×C₁₉×SD₃₂	C₄	1	2	yes	complemented
13	C₂×C₁₉×D₁₆	C₄	1	2	yes	complemented
14	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
17	C₂×C₁₉×#106	C₂²	1	1	yes	characteristic
18	C₂×C₁₉×#104	—	2	1	no	—
19	C₂×C₁₉×#102	—	2	1	no	—
20	C₁₉×#408	—	2	1	no	—
21	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
22	C₁₉×#390	—	2	1	no	—
23	C₁₉×#365	—	2	2	no	—
24	C₁₉×#361	—	2	2	no	—
25	C₁₉×#360	—	2	2	no	—
26	C₁₉×#347	—	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
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	—	2	1	no	—
39	C₂×D₈×C₁₉	—	2	1	no	—
40	C₂×D₈×C₁₉	C₂×C₄	1	1	yes	characteristic
41	C₂×D₈×C₁₉	C₂×C₄	1	2	yes	—
42	C₂×M₁₆×C₁₉	—	2	1	no	—
43	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
46	C₂×C₂²⋊C₄×C₁₉	—	4	2	no	—
47	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
48	C₁₉×SD₃₂	—	2	4	no	—
49	C₁₉×D₁₆	—	2	4	no	—
50	C₁₉×M₃₂	—	2	1	no	—
51	C₁₉×M₃₂	D₄	1	2	yes	—
52	C₁₉×M₃₂	D₄	1	2	yes	—
53	C₁₉×M₃₂	D₄	1	2	yes	—
54	C₂×C₁₆×C₁₉	—	2	1	no	—
55	C₂×C₁₆×C₁₉	C₂×C₄	1	2	yes	—
56	C₁₉×#108	—	2	1	no	—
57	C₁₉×#108	—	2	1	no	—
58	C₁₉×#106	—	2	1	no	—
59	C₁₉×#106	—	2	1	no	—
60	C₁₉×#104	—	4	2	no	—
61	C₁₉×#102	—	4	2	no	—
62	C₁₉×C₂²⋊C₈	—	4	2	no	—
63	C₁₉×#97	—	2	1	no	—
64	C₁₉×#97	—	2	1	no	—
65	C₂⁴×C₁₉	—	4	1	no	—
66	#41×C₁₉	—	4	1	no	—
67	#41×C₁₉	—	4	1	no	—
68	#41×C₁₉	—	4	1	no	—
69	#41×C₁₉	—	4	1	no	—
70	C₂×Q₈×C₁₉	—	2	1	no	—
71	C₂×D₄×C₁₉	—	2	2	no	—
72	C₂×D₄×C₁₉	—	2	1	no	—
73	C₂×D₄×C₁₉	—	4	1	no	—
74	C₂×D₄×C₁₉	—	4	1	no	—
75	C₂×D₄×C₁₉	—	4	1	no	—
76	C₂×D₄×C₁₉	—	4	1	no	—
77	C₂×D₄×C₁₉	—	4	1	no	—
78	C₂×D₄×C₁₉	—	4	1	no	—
79	C₂²×C₄×C₁₉	—	2	1	no	—
80	C₂²×C₄×C₁₉	—	4	1	no	—
81	C₂²×C₄×C₁₉	—	4	1	no	—
82	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
83	Q₁₆×C₁₉	—	2	1	no	—
84	Q₁₆×C₁₉	—	2	1	no	—
85	SD₁₆×C₁₉	—	4	1	no	—
86	SD₁₆×C₁₉	—	4	1	no	—
87	D₈×C₁₉	—	2	2	no	—
88	D₈×C₁₉	—	2	1	no	—
89	D₈×C₁₉	—	2	2	no	—
90	D₈×C₁₉	—	2	1	no	—
91	D₈×C₁₉	—	4	1	no	—
92	D₈×C₁₉	—	4	1	no	—
93	M₁₆×C₁₉	—	4	1	no	—
94	M₁₆×C₁₉	—	4	1	no	—
95	C₂×C₈×C₁₉	—	2	1	no	—
96	C₂×C₈×C₁₉	—	2	1	no	—
97	C₂×C₈×C₁₉	—	2	1	no	—
98	C₂×C₈×C₁₉	—	4	1	no	—
99	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
104	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
105	C₄⋊C₄×C₁₉	—	4	1	no	—
106	C₄⋊C₄×C₁₉	—	4	1	no	—
107	C₂²⋊C₄×C₁₉	—	8	4	no	—
108	C₄²×C₁₉	—	2	1	no	—
109	C₄²×C₁₉	—	2	1	no	—
110	C₁₆×C₁₉	—	2	2	no	—
111	C₁₆×C₁₉	C₂²⋊C₄	1	4	yes	complemented
112	#13249	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂³×C₁₉	—	4	1	no	—
114	C₂³×C₁₉	—	4	2	no	—
115	C₂³×C₁₉	—	8	2	no	—
116	C₂³×C₁₉	—	8	2	no	—
117	C₂³×C₁₉	—	8	1	no	—
118	C₂³×C₁₉	—	8	1	no	—
119	C₂³×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
120	Q₈×C₁₉	—	4	1	no	—
121	Q₈×C₁₉	—	4	1	no	—
122	D₄×C₁₉	—	4	2	no	—
123	D₄×C₁₉	—	4	1	no	—
124	D₄×C₁₉	—	4	2	no	—
125	D₄×C₁₉	—	4	1	no	—
126	D₄×C₁₉	—	8	1	no	—
127	D₄×C₁₉	—	8	1	no	—
128	D₄×C₁₉	—	8	1	no	—
129	D₄×C₁₉	—	8	1	no	—
130	C₂×C₄×C₁₉	—	4	2	no	—
131	C₂×C₄×C₁₉	—	4	1	no	—
132	C₂×C₄×C₁₉	—	4	1	no	—
133	C₂×C₄×C₁₉	—	8	1	no	—
134	C₂×C₄×C₁₉	—	8	1	no	—
135	C₂×C₄×C₁₉	—	8	2	no	—
136	C₂×C₄×C₁₉	—	8	1	no	—
137	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
139	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
140	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₈×C₁₉	—	4	1	no	—
144	C₈×C₁₉	—	8	1	no	—
145	C₈×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₈×C₁₉	C₄×D₄	1	1	yes	characteristic
147	C₈×C₁₉	#121	1	1	yes	characteristic
148	C₈×C₁₉	#124	1	1	yes	characteristic
149	C₂×#509	C₂×C₁₉	1	1	yes	characteristic
150	C₂×#365	C₂×C₁₉	1	1	yes	characteristic, complemented
151	C₂×#361	C₂×C₁₉	1	1	yes	characteristic, complemented
152	C₂×#360	C₂×C₁₉	1	1	yes	characteristic, complemented
153	C₂×#347	C₂×C₁₉	1	1	yes	characteristic, complemented
154	#1975	C₂×C₁₉	1	1	yes	characteristic, complemented
155	#1972	C₂×C₁₉	1	1	yes	characteristic, complemented
156	C₂²×C₁₉	—	2	2	no	—
157	C₂²×C₁₉	—	8	2	no	—
158	C₂²×C₁₉	—	8	4	no	—
159	C₂²×C₁₉	—	8	1	no	—
160	C₂²×C₁₉	—	8	2	no	—
161	C₂²×C₁₉	—	16	2	no	—
162	C₂²×C₁₉	—	16	1	no	—
163	C₂²×C₁₉	—	16	1	no	—
164	C₂²×C₁₉	#442	1	1	yes	characteristic
165	C₂²×C₁₉	#469	1	1	yes	characteristic
166	C₂²×C₁₉	#496	1	1	yes	characteristic
167	C₄×C₁₉	—	8	1	no	—
168	C₄×C₁₉	—	8	1	no	—
169	C₄×C₁₉	—	16	1	no	—
170	C₄×C₁₉	#390	1	1	yes	characteristic
171	C₄×C₁₉	#442	1	1	yes	characteristic
172	C₄×C₁₉	#469	1	1	yes	characteristic
173	C₄×C₁₉	#496	1	1	yes	characteristic
174	C₂×#135	C₂²×C₁₉	1	1	yes	characteristic
175	C₂²×D₈	C₂²×C₁₉	1	1	yes	characteristic
176	#509	—	2	2	no	—
177	#509	—	2	2	no	—
178	C₂×SD₃₂	C₄×C₁₉	1	2	yes	complemented
179	C₂×D₁₆	C₄×C₁₉	1	2	yes	complemented
180	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
181	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
182	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
183	C₂×#106	C₂²×C₁₉	1	1	yes	characteristic
184	C₂×#104	—	2	1	no	—
185	C₂×#102	—	2	1	no	—
186	#408	—	2	1	no	—
187	C₂×#97	C₂²×C₁₉	1	1	yes	characteristic
188	#390	—	2	1	no	—
189	#365	—	2	2	no	—
190	#361	—	2	2	no	—
191	#360	—	2	2	no	—
192	#347	—	2	2	no	—
193	C₂×C₁₉	—	2	1	no	—
194	C₂×C₁₉	—	2	1	no	—
195	C₂×C₁₉	—	16	1	no	—
196	C₂×C₁₉	—	16	2	no	—
197	C₂×C₁₉	#1972	1	1	yes	characteristic
198	C₂×C₁₉	AGL(1,ℤ/16)	1	2	yes	—
199	C₂×#41	—	2	1	no	—
200	C₂²×D₄	—	2	1	no	—
201	#135	—	2	1	no	—
202	#135	—	2	1	no	—
203	#135	—	2	1	no	—
204	#135	—	2	1	no	—
205	C₂×Q₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
206	C₂×SD₁₆	—	2	1	no	—
207	C₂×D₈	—	2	1	no	—
208	C₂×D₈	—	2	1	no	—
209	C₂×D₈	—	2	1	no	—
210	C₂×D₈	—	2	1	no	—
211	C₂×D₈	—	2	1	no	—
212	C₂×D₈	C₂×C₄×C₁₉	1	2	yes	—
213	C₂×D₈	C₂×C₄×C₁₉	1	1	yes	characteristic
214	C₂×M₁₆	—	2	1	no	—
215	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
216	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₂²⋊C₄	—	4	2	no	—
219	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
220	SD₃₂	—	2	4	no	—
221	D₁₆	—	2	4	no	—
222	M₃₂	—	2	1	no	—
223	M₃₂	D₄×C₁₉	1	2	yes	—
224	M₃₂	D₄×C₁₉	1	2	yes	—
225	M₃₂	D₄×C₁₉	1	2	yes	—
226	C₂×C₁₆	—	2	1	no	—
227	C₂×C₁₆	C₂×C₄×C₁₉	1	2	yes	—
228	#108	—	2	1	no	—
229	#108	—	2	1	no	—
230	#106	—	2	1	no	—
231	#106	—	2	1	no	—
232	#104	—	4	2	no	—
233	#102	—	4	2	no	—
234	C₂²⋊C₈	—	4	2	no	—
235	#97	—	2	1	no	—
236	#97	—	2	1	no	—
237	C₁₉	#13249	1	1	yes	characteristic, direct product factor, Sylow
238	C₂⁴	—	4	1	no	—
239	#41	—	4	1	no	—
240	#41	—	4	1	no	—
241	#41	—	4	1	no	—
242	#41	—	4	1	no	—
243	C₂×Q₈	—	2	1	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	2	1	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂×D₄	—	4	1	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂×D₄	—	4	1	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
256	Q₁₆	—	2	1	no	—
257	Q₁₆	—	2	1	no	—
258	SD₁₆	—	4	1	no	—
259	SD₁₆	—	4	1	no	—
260	D₈	—	2	1	no	—
261	D₈	—	2	2	no	—
262	D₈	—	2	2	no	—
263	D₈	—	2	1	no	—
264	D₈	—	4	1	no	—
265	D₈	—	4	1	no	—
266	M₁₆	—	4	1	no	—
267	M₁₆	—	4	1	no	—
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	4	1	no	—
272	C₂×C₈	C₂²×C₄×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	C₂×D₄×C₁₉	1	1	yes	characteristic
276	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
277	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	8	4	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	2	1	no	—
283	C₁₆	—	2	2	no	—
284	C₁₆	C₂²⋊C₄×C₁₉	1	4	yes	complemented
285	C₂³	—	4	1	no	—
286	C₂³	—	4	2	no	—
287	C₂³	—	8	2	no	—
288	C₂³	—	8	2	no	—
289	C₂³	—	8	1	no	—
290	C₂³	—	8	1	no	—
291	C₂³	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
292	Q₈	—	4	1	no	—
293	Q₈	—	4	1	no	—
294	D₄	—	4	2	no	—
295	D₄	—	4	1	no	—
296	D₄	—	4	2	no	—
297	D₄	—	4	1	no	—
298	D₄	—	8	1	no	—
299	D₄	—	8	1	no	—
300	D₄	—	8	1	no	—
301	D₄	—	8	1	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	8	1	no	—
306	C₂×C₄	—	8	1	no	—
307	C₂×C₄	—	8	2	no	—
308	C₂×C₄	—	8	1	no	—
309	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
310	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
311	C₂×C₄	C₁₉×#127	1	1	yes	characteristic
312	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
313	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
314	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
315	C₈	—	4	1	no	—
316	C₈	—	8	1	no	—
317	C₈	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
318	C₈	C₄×D₄×C₁₉	1	1	yes	characteristic
319	C₈	C₁₉×#121	1	1	yes	characteristic
320	C₈	C₁₉×#124	1	1	yes	characteristic
321	C₂²	—	2	2	no	—
322	C₂²	—	8	4	no	—
323	C₂²	—	8	2	no	—
324	C₂²	—	8	1	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	16	2	no	—
327	C₂²	—	16	1	no	—
328	C₂²	—	16	1	no	—
329	C₂²	C₁₉×#442	1	1	yes	characteristic
330	C₂²	C₁₉×#469	1	1	yes	characteristic
331	C₂²	C₁₉×#496	1	1	yes	characteristic
332	C₄	—	8	1	no	—
333	C₄	—	8	1	no	—
334	C₄	—	16	1	no	—
335	C₄	C₁₉×#390	1	1	yes	characteristic
336	C₄	C₁₉×#442	1	1	yes	characteristic
337	C₄	C₁₉×#469	1	1	yes	characteristic
338	C₄	C₁₉×#496	1	1	yes	characteristic
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	—	16	2	no	—
342	C₂	—	16	1	no	—
343	C₂	C₁₉×#1972	1	1	yes	characteristic
344	C₂	C₁₉×AGL(1,ℤ/16)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉	#442
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 ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₁₉×#97 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₉×#365	C₂	semidirect
3. C₂×C₁₉×#361	C₂	semidirect
4. C₂×C₁₉×#360	C₂	semidirect
5. C₂×C₁₉×#347	C₂	semidirect
6. C₁₉×#1975	C₂	semidirect
7. C₁₉×#1972	C₂	semidirect
12. C₂×C₁₉×SD₃₂	C₄	semidirect
13. C₂×C₁₉×D₁₆	C₄	semidirect
111. C₁₆×C₁₉	C₂²⋊C₄	semidirect
112. #13249	C₁₉	direct (N × H)
150. C₂×#365	C₂×C₁₉	semidirect
151. C₂×#361	C₂×C₁₉	semidirect
152. C₂×#360	C₂×C₁₉	semidirect
153. C₂×#347	C₂×C₁₉	semidirect
154. #1975	C₂×C₁₉	semidirect
155. #1972	C₂×C₁₉	semidirect
178. C₂×SD₃₂	C₄×C₁₉	semidirect
179. C₂×D₁₆	C₄×C₁₉	semidirect
237. C₁₉	#13249	direct (N × H)
284. C₁₆	C₂²⋊C₄×C₁₉	semidirect

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
C₂×C₁₉×#365	2
C₂×C₁₉×#361	2
C₂×C₁₉×#360	2
C₂×C₁₉×#347	2
C₁₉×#1975	2
C₁₉×#1972	2
#13249	19

Subgroups — C19×#13249 #18019208

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13249 #18019208