Subgroups — C₁₉×#10195 #18016154

Order 4864.

554 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3249	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₉×#3208	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#3138	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1986	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1982	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1503	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1283	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₉×#533	C₂²	1	1	yes	characteristic
9	C₁₉×#533	C₂²	1	1	yes	characteristic
10	C₂×C₁₉×#127	C₂²	1	1	yes	characteristic
11	C₁₉×#492	—	2	1	no	—
12	C₁₉×#490	—	2	2	no	—
13	C₁₉×#486	—	2	1	no	—
14	C₁₉×#486	—	2	1	no	—
15	C₁₉×#486	—	2	1	no	—
16	C₁₉×#477	—	2	1	no	—
17	C₁₉×#477	—	2	1	no	—
18	C₁₉×#467	—	2	2	no	—
19	C₁₉×#462	—	2	1	no	—
20	C₁₉×#462	—	2	1	no	—
21	C₁₉×#460	—	2	2	no	—
22	C₁₉×#450	—	2	2	no	—
23	C₁₉×#431	C₂²	1	1	yes	characteristic
24	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₁₉×#418	—	2	2	no	—
26	C₂×C₁₉×#102	—	2	2	no	—
27	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
29	C₁₉×#339	—	2	2	no	—
30	C₁₉×#328	—	2	2	no	—
31	C₂²×D₄×C₁₉	—	2	1	no	—
32	C₂²×D₄×C₁₉	—	2	1	no	—
33	C₂²×D₄×C₁₉	—	2	1	no	—
34	C₂×Q₁₆×C₁₉	—	2	2	no	—
35	C₂×SD₁₆×C₁₉	—	2	4	no	—
36	C₂×M₁₆×C₁₉	D₄	1	2	yes	—
37	C₂²×C₈×C₁₉	D₄	1	2	yes	—
38	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
39	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
40	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
41	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
42	C₁₉×#127	—	2	1	no	—
43	C₁₉×#127	—	2	1	no	—
44	C₁₉×#127	—	2	1	no	—
45	C₁₉×#127	D₄	1	2	yes	—
46	C₁₉×#125	—	2	1	no	—
47	C₁₉×#125	—	2	1	no	—
48	C₁₉×C₂²⋊Q₈	—	2	1	no	—
49	C₁₉×C₂²⋊Q₈	—	2	1	no	—
50	C₁₉×C₂²⋊Q₈	—	2	1	no	—
51	C₁₉×C₂²⋊Q₈	—	2	1	no	—
52	C₄×Q₈×C₁₉	—	2	1	no	—
53	C₄×Q₈×C₁₉	—	2	1	no	—
54	C₄×Q₈×C₁₉	—	2	1	no	—
55	C₄×Q₈×C₁₉	—	2	1	no	—
56	C₁₉×#117	—	2	1	no	—
57	C₁₉×#117	—	2	1	no	—
58	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
59	C₂×C₄²×C₁₉	—	2	1	no	—
60	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
61	C₁₉×#107	—	2	2	no	—
62	C₁₉×C₄⋊C₈	—	2	1	no	—
63	C₁₉×C₄⋊C₈	—	2	1	no	—
64	C₁₉×C₄⋊C₈	—	2	1	no	—
65	C₁₉×#103	—	2	2	no	—
66	C₁₉×#103	—	2	2	no	—
67	C₁₉×#102	—	2	4	no	—
68	C₁₉×#102	—	2	4	no	—
69	C₁₉×#102	—	2	4	no	—
70	C₁₉×#102	—	4	4	no	—
71	C₁₉×C₂²⋊C₈	D₄	1	2	yes	—
72	C₁₉×#97	—	2	1	no	—
73	C₄×C₈×C₁₉	—	2	1	no	—
74	C₄×C₈×C₁₉	—	2	1	no	—
75	C₁₉×#95	—	2	1	no	—
76	C₂⁴×C₁₉	—	4	1	no	—
77	C₂×Q₈×C₁₉	—	2	1	no	—
78	C₂×Q₈×C₁₉	—	2	1	no	—
79	C₂×Q₈×C₁₉	—	2	1	no	—
80	C₂×Q₈×C₁₉	—	2	1	no	—
81	C₂×D₄×C₁₉	—	2	2	no	—
82	C₂×D₄×C₁₉	—	2	2	no	—
83	C₂×D₄×C₁₉	—	2	2	no	—
84	C₂×D₄×C₁₉	—	4	2	no	—
85	C₂×D₄×C₁₉	—	4	2	no	—
86	C₂×D₄×C₁₉	—	4	1	no	—
87	C₂×D₄×C₁₉	—	4	1	no	—
88	C₂×D₄×C₁₉	—	4	1	no	—
89	C₂×D₄×C₁₉	—	4	2	no	—
90	C₂×D₄×C₁₉	—	4	2	no	—
91	C₂×D₄×C₁₉	—	8	2	no	—
92	C₂²×C₄×C₁₉	—	4	1	no	—
93	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
96	Q₁₆×C₁₉	—	8	2	no	—
97	SD₁₆×C₁₉	—	8	4	no	—
98	M₁₆×C₁₉	—	2	2	no	—
99	M₁₆×C₁₉	—	2	2	no	—
100	C₂×C₈×C₁₉	—	2	2	no	—
101	C₂×C₈×C₁₉	—	2	2	no	—
102	C₂×C₈×C₁₉	—	2	2	no	—
103	C₂×C₈×C₁₉	—	4	2	no	—
104	C₄⋊C₄×C₁₉	—	2	2	no	—
105	C₄⋊C₄×C₁₉	—	2	1	no	—
106	C₄⋊C₄×C₁₉	—	2	1	no	—
107	C₄⋊C₄×C₁₉	—	2	1	no	—
108	C₄⋊C₄×C₁₉	—	2	1	no	—
109	C₄⋊C₄×C₁₉	—	2	1	no	—
110	C₄⋊C₄×C₁₉	—	2	1	no	—
111	C₄⋊C₄×C₁₉	—	2	1	no	—
112	C₄⋊C₄×C₁₉	—	2	1	no	—
113	C₄⋊C₄×C₁₉	—	4	1	no	—
114	C₄⋊C₄×C₁₉	—	4	1	no	—
115	C₄⋊C₄×C₁₉	—	4	1	no	—
116	C₄⋊C₄×C₁₉	—	4	1	no	—
117	C₄⋊C₄×C₁₉	—	4	1	no	—
118	C₂²⋊C₄×C₁₉	—	4	1	no	—
119	C₂²⋊C₄×C₁₉	—	4	1	no	—
120	C₄²×C₁₉	—	4	1	no	—
121	C₄²×C₁₉	—	4	1	no	—
122	C₄²×C₁₉	—	4	2	no	—
123	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₁₉	#41	1	1	yes	characteristic
127	#10195	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂³×C₁₉	—	4	2	no	—
129	C₂³×C₁₉	—	8	2	no	—
130	C₂³×C₁₉	—	8	2	no	—
131	C₂³×C₁₉	—	8	1	no	—
132	C₂³×C₁₉	—	8	1	no	—
133	C₂³×C₁₉	#127	1	1	yes	characteristic
134	Q₈×C₁₉	—	4	2	no	—
135	Q₈×C₁₉	—	4	1	no	—
136	Q₈×C₁₉	—	4	1	no	—
137	Q₈×C₁₉	—	4	1	no	—
138	Q₈×C₁₉	—	4	1	no	—
139	Q₈×C₁₉	—	8	1	no	—
140	D₄×C₁₉	—	4	4	no	—
141	D₄×C₁₉	—	4	4	no	—
142	D₄×C₁₉	—	4	4	no	—
143	D₄×C₁₉	—	8	2	no	—
144	D₄×C₁₉	—	8	1	no	—
145	D₄×C₁₉	—	8	1	no	—
146	D₄×C₁₉	—	8	2	no	—
147	C₂×C₄×C₁₉	—	2	1	no	—
148	C₂×C₄×C₁₉	—	2	1	no	—
149	C₂×C₄×C₁₉	—	4	1	no	—
150	C₂×C₄×C₁₉	—	4	2	no	—
151	C₂×C₄×C₁₉	—	4	1	no	—
152	C₂×C₄×C₁₉	—	4	1	no	—
153	C₂×C₄×C₁₉	—	4	1	no	—
154	C₂×C₄×C₁₉	—	4	1	no	—
155	C₂×C₄×C₁₉	—	8	1	no	—
156	C₂×C₄×C₁₉	—	8	1	no	—
157	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
158	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
159	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
160	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
161	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
162	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
163	C₂×C₄×C₁₉	#135	1	2	yes	—
164	C₂×C₄×C₁₉	#137	1	2	yes	—
165	C₈×C₁₉	—	4	2	no	—
166	C₈×C₁₉	—	4	2	no	—
167	C₈×C₁₉	—	4	2	no	—
168	#3249	C₂×C₁₉	1	1	yes	characteristic, complemented
169	#3208	C₂×C₁₉	1	1	yes	characteristic
170	#3138	C₂×C₁₉	1	1	yes	characteristic
171	#1986	C₂×C₁₉	1	1	yes	characteristic, complemented
172	#1982	C₂×C₁₉	1	1	yes	characteristic
173	#1503	C₂×C₁₉	1	1	yes	characteristic, complemented
174	#1283	C₂×C₁₉	1	1	yes	characteristic, complemented
175	C₂²×C₁₉	—	2	1	no	—
176	C₂²×C₁₉	—	2	1	no	—
177	C₂²×C₁₉	—	8	2	no	—
178	C₂²×C₁₉	—	8	2	no	—
179	C₂²×C₁₉	—	8	2	no	—
180	C₂²×C₁₉	—	16	2	no	—
181	C₂²×C₁₉	—	16	2	no	—
182	C₂²×C₁₉	#392	1	1	yes	characteristic
183	C₂²×C₁₉	#495	1	1	yes	characteristic
184	C₂²×C₁₉	#497	1	1	yes	characteristic
185	C₄×C₁₉	—	2	1	no	—
186	C₄×C₁₉	—	2	1	no	—
187	C₄×C₁₉	—	2	1	no	—
188	C₄×C₁₉	—	8	2	no	—
189	C₄×C₁₉	—	8	1	no	—
190	C₄×C₁₉	—	8	1	no	—
191	C₄×C₁₉	—	8	1	no	—
192	C₄×C₁₉	—	8	1	no	—
193	C₄×C₁₉	#452	1	2	yes	—
194	C₄×C₁₉	#453	1	1	yes	characteristic
195	C₄×C₁₉	#454	1	1	yes	characteristic
196	C₄×C₁₉	#464	1	2	yes	—
197	#533	C₂²×C₁₉	1	1	yes	characteristic
198	#533	C₂²×C₁₉	1	1	yes	characteristic
199	C₂×#127	C₂²×C₁₉	1	1	yes	characteristic
200	#492	—	2	1	no	—
201	#490	—	2	2	no	—
202	#486	—	2	1	no	—
203	#486	—	2	1	no	—
204	#486	—	2	1	no	—
205	#477	—	2	1	no	—
206	#477	—	2	1	no	—
207	#467	—	2	2	no	—
208	#462	—	2	1	no	—
209	#462	—	2	1	no	—
210	#460	—	2	2	no	—
211	#450	—	2	2	no	—
212	#431	C₂²×C₁₉	1	1	yes	characteristic
213	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
214	#418	—	2	2	no	—
215	C₂×#102	—	2	2	no	—
216	C₄×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
217	C₄×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
218	#339	—	2	2	no	—
219	#328	—	2	2	no	—
220	C₂×C₁₉	—	2	1	no	—
221	C₂×C₁₉	—	2	1	no	—
222	C₂×C₁₉	—	16	2	no	—
223	C₂×C₁₉	#2006	1	1	yes	characteristic
224	C₂×C₁₉	#2008	1	1	yes	characteristic
225	C₂×C₁₉	#2015	1	1	yes	characteristic
226	C₂²×D₄	—	2	1	no	—
227	C₂²×D₄	—	2	1	no	—
228	C₂²×D₄	—	2	1	no	—
229	C₂×Q₁₆	—	2	2	no	—
230	C₂×SD₁₆	—	2	4	no	—
231	C₂×M₁₆	D₄×C₁₉	1	2	yes	—
232	C₂²×C₈	D₄×C₁₉	1	2	yes	—
233	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
234	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
235	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
236	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
237	#127	—	2	1	no	—
238	#127	—	2	1	no	—
239	#127	—	2	1	no	—
240	#127	D₄×C₁₉	1	2	yes	—
241	#125	—	2	1	no	—
242	#125	—	2	1	no	—
243	C₂²⋊Q₈	—	2	1	no	—
244	C₂²⋊Q₈	—	2	1	no	—
245	C₂²⋊Q₈	—	2	1	no	—
246	C₂²⋊Q₈	—	2	1	no	—
247	C₄×Q₈	—	2	1	no	—
248	C₄×Q₈	—	2	1	no	—
249	C₄×Q₈	—	2	1	no	—
250	C₄×Q₈	—	2	1	no	—
251	#117	—	2	1	no	—
252	#117	—	2	1	no	—
253	C₂×C₄⋊C₄	—	2	1	no	—
254	C₂×C₄²	—	2	1	no	—
255	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
256	#107	—	2	2	no	—
257	C₄⋊C₈	—	2	1	no	—
258	C₄⋊C₈	—	2	1	no	—
259	C₄⋊C₈	—	2	1	no	—
260	#103	—	2	2	no	—
261	#103	—	2	2	no	—
262	#102	—	2	4	no	—
263	#102	—	2	4	no	—
264	#102	—	2	4	no	—
265	#102	—	4	4	no	—
266	C₂²⋊C₈	D₄×C₁₉	1	2	yes	—
267	#97	—	2	1	no	—
268	C₄×C₈	—	2	1	no	—
269	C₄×C₈	—	2	1	no	—
270	#95	—	2	1	no	—
271	C₁₉	#10195	1	1	yes	characteristic, direct product factor, Sylow
272	C₂⁴	—	4	1	no	—
273	C₂×Q₈	—	2	1	no	—
274	C₂×Q₈	—	2	1	no	—
275	C₂×Q₈	—	2	1	no	—
276	C₂×Q₈	—	2	1	no	—
277	C₂×D₄	—	2	2	no	—
278	C₂×D₄	—	2	2	no	—
279	C₂×D₄	—	2	2	no	—
280	C₂×D₄	—	4	2	no	—
281	C₂×D₄	—	4	2	no	—
282	C₂×D₄	—	4	1	no	—
283	C₂×D₄	—	4	1	no	—
284	C₂×D₄	—	4	1	no	—
285	C₂×D₄	—	4	2	no	—
286	C₂×D₄	—	4	2	no	—
287	C₂×D₄	—	8	2	no	—
288	C₂²×C₄	—	4	1	no	—
289	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
290	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
291	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
292	Q₁₆	—	8	2	no	—
293	SD₁₆	—	8	4	no	—
294	M₁₆	—	2	2	no	—
295	M₁₆	—	2	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	4	2	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	2	1	no	—
302	C₄⋊C₄	—	2	1	no	—
303	C₄⋊C₄	—	2	1	no	—
304	C₄⋊C₄	—	2	1	no	—
305	C₄⋊C₄	—	2	1	no	—
306	C₄⋊C₄	—	2	1	no	—
307	C₄⋊C₄	—	2	1	no	—
308	C₄⋊C₄	—	2	1	no	—
309	C₄⋊C₄	—	4	1	no	—
310	C₄⋊C₄	—	4	1	no	—
311	C₄⋊C₄	—	4	1	no	—
312	C₄⋊C₄	—	4	1	no	—
313	C₄⋊C₄	—	4	1	no	—
314	C₂²⋊C₄	—	4	1	no	—
315	C₂²⋊C₄	—	4	1	no	—
316	C₄²	—	4	1	no	—
317	C₄²	—	4	1	no	—
318	C₄²	—	4	2	no	—
319	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
320	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
321	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
322	C₄²	#41×C₁₉	1	1	yes	characteristic
323	C₂³	—	4	2	no	—
324	C₂³	—	8	2	no	—
325	C₂³	—	8	2	no	—
326	C₂³	—	8	1	no	—
327	C₂³	—	8	1	no	—
328	C₂³	C₁₉×#127	1	1	yes	characteristic
329	Q₈	—	4	2	no	—
330	Q₈	—	4	1	no	—
331	Q₈	—	4	1	no	—
332	Q₈	—	4	1	no	—
333	Q₈	—	4	1	no	—
334	Q₈	—	8	1	no	—
335	D₄	—	4	4	no	—
336	D₄	—	4	4	no	—
337	D₄	—	4	4	no	—
338	D₄	—	8	2	no	—
339	D₄	—	8	1	no	—
340	D₄	—	8	1	no	—
341	D₄	—	8	2	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	4	1	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	1	no	—
350	C₂×C₄	—	8	1	no	—
351	C₂×C₄	—	8	1	no	—
352	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
353	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
354	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
355	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
356	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
357	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
358	C₂×C₄	C₁₉×#135	1	2	yes	—
359	C₂×C₄	C₁₉×#137	1	2	yes	—
360	C₈	—	4	2	no	—
361	C₈	—	4	2	no	—
362	C₈	—	4	2	no	—
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	8	2	no	—
366	C₂²	—	8	2	no	—
367	C₂²	—	8	2	no	—
368	C₂²	—	16	2	no	—
369	C₂²	—	16	2	no	—
370	C₂²	C₁₉×#392	1	1	yes	characteristic
371	C₂²	C₁₉×#495	1	1	yes	characteristic
372	C₂²	C₁₉×#497	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	8	2	no	—
377	C₄	—	8	1	no	—
378	C₄	—	8	1	no	—
379	C₄	—	8	1	no	—
380	C₄	—	8	1	no	—
381	C₄	C₁₉×#452	1	2	yes	—
382	C₄	C₁₉×#453	1	1	yes	characteristic
383	C₄	C₁₉×#454	1	1	yes	characteristic
384	C₄	C₁₉×#464	1	2	yes	—
385	C₂	—	2	1	no	—
386	C₂	—	2	1	no	—
387	C₂	—	16	2	no	—
388	C₂	C₁₉×#2006	1	1	yes	characteristic
389	C₂	C₁₉×#2008	1	1	yes	characteristic
390	C₂	C₁₉×#2015	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
prime radical	C₂×C₁₉×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₉×#3249	C₂	semidirect
4. C₁₉×#1986	C₂	semidirect
6. C₁₉×#1503	C₂	semidirect
7. C₁₉×#1283	C₂	semidirect
127. #10195	C₁₉	direct (N × H)
168. #3249	C₂×C₁₉	semidirect
171. #1986	C₂×C₁₉	semidirect
173. #1503	C₂×C₁₉	semidirect
174. #1283	C₂×C₁₉	semidirect
271. C₁₉	#10195	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₉×#3249	2
C₁₉×#3208	2
C₁₉×#3138	2
C₁₉×#1986	2
C₁₉×#1982	2
C₁₉×#1503	2
C₁₉×#1283	2
#10195	19

Subgroups — C19×#10195 #18016154

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#10195 #18016154