Subgroups — C₁₉×#18403 #18024362

Order 4864.

998 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3502	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₉×#3208	C₂	1	2	yes	maximal
3	C₁₉×#2548	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1599	C₂	1	8	yes	maximal, complemented
5	C₁₉×#1503	C₂	1	2	yes	maximal, complemented
6	C₁₉×#1461	C₂	1	1	yes	maximal, characteristic, complemented
7	Q₈²×C₁₉	—	2	2	no	—
8	C₁₉×#557	—	2	2	no	—
9	C₁₉×#557	—	2	4	no	—
10	C₁₉×#544	—	2	1	no	—
11	C₁₉×#544	C₂²	1	4	yes	—
12	C₁₉×#533	—	2	2	no	—
13	C₁₉×#533	C₂²	1	2	yes	—
14	C₂×C₁₉×#127	C₂²	1	1	yes	characteristic
15	C₂×C₁₉×#124	—	2	1	no	—
16	C₁₉×#519	—	2	1	no	—
17	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#492	—	2	2	no	—
19	C₁₉×#486	—	2	2	no	—
20	C₁₉×#460	—	2	4	no	—
21	C₁₉×#450	C₂²	1	8	yes	—
22	C₁₉×#431	C₂²	1	2	yes	—
23	C₁₉×#394	C₂²	1	4	yes	—
24	C₁₉×#390	—	2	1	no	—
25	C₁₉×#390	—	2	2	no	—
26	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
27	C₁₉×#328	C₂²	1	8	yes	—
28	C₁₉×#324	C₂²	1	4	yes	—
29	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
30	C₂²×D₄×C₁₉	—	2	1	no	—
31	C₂²×D₄×C₁₉	—	2	1	no	—
32	C₂²×D₄×C₁₉	C₂³	1	2	yes	—
33	C₂×SD₁₆×C₁₉	—	2	8	no	—
34	C₁₉×C₄⋊Q₈	—	2	2	no	—
35	C₁₉×C₄⋊Q₈	—	2	2	no	—
36	C₁₉×C₄⋊Q₈	—	2	4	no	—
37	C₁₉×C₄⋊Q₈	—	2	8	no	—
38	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
39	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
40	C₁₉×#127	—	2	2	no	—
41	C₁₉×#127	—	2	2	no	—
42	C₁₉×#125	—	2	4	no	—
43	C₁₉×#125	—	2	4	no	—
44	C₁₉×#125	—	2	8	no	—
45	C₁₉×#125	—	2	2	no	—
46	C₁₉×#124	—	4	4	no	—
47	C₁₉×C₂²⋊Q₈	—	2	2	no	—
48	C₁₉×C₂²⋊Q₈	—	2	4	no	—
49	C₁₉×C₂²⋊Q₈	C₂³	1	4	yes	—
50	C₄×Q₈×C₁₉	—	2	4	no	—
51	C₄×Q₈×C₁₉	—	2	4	no	—
52	C₄×Q₈×C₁₉	—	2	4	no	—
53	C₄×Q₈×C₁₉	—	2	4	no	—
54	C₄×Q₈×C₁₉	—	2	2	no	—
55	C₄×Q₈×C₁₉	—	2	2	no	—
56	C₄×Q₈×C₁₉	—	2	2	no	—
57	C₄×Q₈×C₁₉	D₄	1	4	yes	—
58	C₁₉×#117	—	2	4	no	—
59	C₁₉×#117	—	2	2	no	—
60	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
61	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
63	C₂×C₂²⋊C₄×C₁₉	—	2	4	no	—
64	C₂×C₂²⋊C₄×C₁₉	—	2	4	no	—
65	C₂×C₄²×C₁₉	—	2	1	no	—
66	C₂×C₄²×C₁₉	D₄	1	2	yes	—
67	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
68	C₁₉×C₄⋊C₈	—	2	2	no	—
69	C₁₉×#102	—	2	8	no	—
70	C₁₉×C₂²⋊C₈	C₂³	1	4	yes	—
71	C₄×C₈×C₁₉	—	2	2	no	—
72	C₁₉×#95	C₂³	1	2	yes	—
73	C₂⁴×C₁₉	—	2	2	no	—
74	C₂×Q₈×C₁₉	—	2	4	no	—
75	C₂×Q₈×C₁₉	—	2	2	no	—
76	C₂×Q₈×C₁₉	—	2	2	no	—
77	C₂×Q₈×C₁₉	—	2	2	no	—
78	C₂×Q₈×C₁₉	SD₁₆	1	4	yes	complemented
79	C₂×Q₈×C₁₉	C₂×D₄	1	4	yes	—
80	C₂×D₄×C₁₉	—	2	4	no	—
81	C₂×D₄×C₁₉	—	4	4	no	—
82	C₂×D₄×C₁₉	—	4	1	no	—
83	C₂×D₄×C₁₉	—	4	2	no	—
84	C₂×D₄×C₁₉	—	4	1	no	—
85	C₂×D₄×C₁₉	—	4	2	no	—
86	C₂×D₄×C₁₉	—	4	2	no	—
87	C₂×D₄×C₁₉	—	8	2	no	—
88	C₂²×C₄×C₁₉	—	2	2	no	—
89	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
94	SD₁₆×C₁₉	—	8	8	no	—
95	C₂×C₈×C₁₉	—	2	4	no	—
96	C₄⋊C₄×C₁₉	—	2	4	no	—
97	C₄⋊C₄×C₁₉	—	2	4	no	—
98	C₄⋊C₄×C₁₉	—	2	4	no	—
99	C₄⋊C₄×C₁₉	—	2	8	no	—
100	C₄⋊C₄×C₁₉	—	2	2	no	—
101	C₄⋊C₄×C₁₉	—	2	2	no	—
102	C₄⋊C₄×C₁₉	—	2	2	no	—
103	C₄⋊C₄×C₁₉	—	2	2	no	—
104	C₄⋊C₄×C₁₉	—	2	4	no	—
105	C₄⋊C₄×C₁₉	—	2	4	no	—
106	C₄⋊C₄×C₁₉	—	2	4	no	—
107	C₄⋊C₄×C₁₉	—	4	2	no	—
108	C₄⋊C₄×C₁₉	C₂×D₄	1	4	yes	—
109	C₂²⋊C₄×C₁₉	—	2	4	no	—
110	C₂²⋊C₄×C₁₉	—	4	8	no	—
111	C₂²⋊C₄×C₁₉	—	8	4	no	—
112	C₄²×C₁₉	—	2	8	no	—
113	C₄²×C₁₉	—	2	2	no	—
114	C₄²×C₁₉	—	2	2	no	—
115	C₄²×C₁₉	—	2	4	no	—
116	C₄²×C₁₉	—	4	2	no	—
117	C₄²×C₁₉	C₂×D₄	1	2	yes	—
118	C₄²×C₁₉	C₂×D₄	1	2	yes	—
119	#18403	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂³×C₁₉	—	4	4	no	—
121	C₂³×C₁₉	—	4	4	no	—
122	C₂³×C₁₉	—	4	2	no	—
123	C₂³×C₁₉	—	4	2	no	—
124	C₂³×C₁₉	—	4	2	no	—
125	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
126	Q₈×C₁₉	—	2	8	no	—
127	Q₈×C₁₉	—	4	4	no	—
128	Q₈×C₁₉	—	4	4	no	—
129	Q₈×C₁₉	—	8	4	no	—
130	D₄×C₁₉	—	4	8	no	—
131	D₄×C₁₉	—	8	2	no	—
132	D₄×C₁₉	—	8	4	no	—
133	D₄×C₁₉	—	8	2	no	—
134	C₂×C₄×C₁₉	—	2	2	no	—
135	C₂×C₄×C₁₉	—	2	2	no	—
136	C₂×C₄×C₁₉	—	2	2	no	—
137	C₂×C₄×C₁₉	—	2	4	no	—
138	C₂×C₄×C₁₉	—	2	4	no	—
139	C₂×C₄×C₁₉	—	2	2	no	—
140	C₂×C₄×C₁₉	—	4	1	no	—
141	C₂×C₄×C₁₉	—	4	2	no	—
142	C₂×C₄×C₁₉	—	4	2	no	—
143	C₂×C₄×C₁₉	—	4	2	no	—
144	C₂×C₄×C₁₉	—	4	2	no	—
145	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
146	C₂×C₄×C₁₉	#120	1	2	yes	—
147	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
148	C₂×C₄×C₁₉	C₂×SD₁₆	1	4	yes	—
149	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
150	C₂×C₄×C₁₉	#137	1	2	yes	—
151	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
152	C₈×C₁₉	—	4	4	no	—
153	#3502	C₂×C₁₉	1	1	yes	characteristic, complemented
154	#3208	C₂×C₁₉	1	2	yes	—
155	#2548	C₂×C₁₉	1	1	yes	characteristic
156	#1599	C₂×C₁₉	1	8	yes	complemented
157	#1503	C₂×C₁₉	1	2	yes	complemented
158	#1461	C₂×C₁₉	1	1	yes	characteristic, complemented
159	C₂²×C₁₉	—	2	1	no	—
160	C₂²×C₁₉	—	2	1	no	—
161	C₂²×C₁₉	—	8	4	no	—
162	C₂²×C₁₉	—	8	4	no	—
163	C₂²×C₁₉	—	8	2	no	—
164	C₂²×C₁₉	—	8	2	no	—
165	C₂²×C₁₉	—	8	2	no	—
166	C₂²×C₁₉	C₂×#120	1	1	yes	characteristic
167	C₂²×C₁₉	C₂²×SD₁₆	1	1	yes	characteristic
168	C₂²×C₁₉	C₂×#137	1	1	yes	characteristic
169	C₄×C₁₉	—	2	4	no	—
170	C₄×C₁₉	—	2	1	no	—
171	C₄×C₁₉	—	2	2	no	—
172	C₄×C₁₉	—	4	4	no	—
173	C₄×C₁₉	—	4	2	no	—
174	C₄×C₁₉	—	4	4	no	—
175	C₄×C₁₉	—	8	2	no	—
176	C₄×C₁₉	#448	1	4	yes	—
177	C₄×C₁₉	#454	1	2	yes	—
178	Q₈²	—	2	2	no	—
179	#557	—	2	2	no	—
180	#557	—	2	4	no	—
181	#544	—	2	1	no	—
182	#544	C₂²×C₁₉	1	4	yes	—
183	#533	—	2	2	no	—
184	#533	C₂²×C₁₉	1	2	yes	—
185	C₂×#127	C₂²×C₁₉	1	1	yes	characteristic
186	C₂×#124	—	2	1	no	—
187	#519	—	2	1	no	—
188	C₂×C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
189	#492	—	2	2	no	—
190	#486	—	2	2	no	—
191	#460	—	2	4	no	—
192	#450	C₂²×C₁₉	1	8	yes	—
193	#431	C₂²×C₁₉	1	2	yes	—
194	#394	C₂²×C₁₉	1	4	yes	—
195	#390	—	2	1	no	—
196	#390	—	2	2	no	—
197	C₄×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
198	#328	C₂²×C₁₉	1	8	yes	—
199	#324	C₂²×C₁₉	1	4	yes	—
200	C₂×C₁₉	—	2	1	no	—
201	C₂×C₁₉	—	2	1	no	—
202	C₂×C₁₉	—	16	2	no	—
203	C₂×C₁₉	C₂×#448	1	1	yes	characteristic
204	C₂×C₁₉	C₂×#454	1	1	yes	characteristic
205	C₂×C₁₉	#3027	1	1	yes	characteristic
206	C₂²×Q₈	D₄×C₁₉	1	1	yes	characteristic
207	C₂²×D₄	—	2	1	no	—
208	C₂²×D₄	—	2	1	no	—
209	C₂²×D₄	C₂³×C₁₉	1	2	yes	—
210	C₂×SD₁₆	—	2	8	no	—
211	C₄⋊Q₈	—	2	2	no	—
212	C₄⋊Q₈	—	2	2	no	—
213	C₄⋊Q₈	—	2	4	no	—
214	C₄⋊Q₈	—	2	8	no	—
215	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
216	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
217	#127	—	2	2	no	—
218	#127	—	2	2	no	—
219	#125	—	2	4	no	—
220	#125	—	2	4	no	—
221	#125	—	2	8	no	—
222	#125	—	2	2	no	—
223	#124	—	4	4	no	—
224	C₂²⋊Q₈	—	2	2	no	—
225	C₂²⋊Q₈	—	2	4	no	—
226	C₂²⋊Q₈	C₂³×C₁₉	1	4	yes	—
227	C₄×Q₈	—	2	4	no	—
228	C₄×Q₈	—	2	4	no	—
229	C₄×Q₈	—	2	4	no	—
230	C₄×Q₈	—	2	4	no	—
231	C₄×Q₈	—	2	2	no	—
232	C₄×Q₈	—	2	2	no	—
233	C₄×Q₈	—	2	2	no	—
234	C₄×Q₈	D₄×C₁₉	1	4	yes	—
235	#117	—	2	4	no	—
236	#117	—	2	2	no	—
237	C₂×C₄⋊C₄	—	2	1	no	—
238	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
239	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
240	C₂×C₂²⋊C₄	—	2	4	no	—
241	C₂×C₂²⋊C₄	—	2	4	no	—
242	C₂×C₄²	—	2	1	no	—
243	C₂×C₄²	D₄×C₁₉	1	2	yes	—
244	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
245	C₄⋊C₈	—	2	2	no	—
246	#102	—	2	8	no	—
247	C₂²⋊C₈	C₂³×C₁₉	1	4	yes	—
248	C₄×C₈	—	2	2	no	—
249	#95	C₂³×C₁₉	1	2	yes	—
250	C₁₉	#18403	1	1	yes	characteristic, direct product factor, Sylow
251	C₂⁴	—	2	2	no	—
252	C₂×Q₈	—	2	4	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	—	2	2	no	—
256	C₂×Q₈	SD₁₆×C₁₉	1	4	yes	complemented
257	C₂×Q₈	C₂×D₄×C₁₉	1	4	yes	—
258	C₂×D₄	—	2	4	no	—
259	C₂×D₄	—	4	4	no	—
260	C₂×D₄	—	4	1	no	—
261	C₂×D₄	—	4	2	no	—
262	C₂×D₄	—	4	1	no	—
263	C₂×D₄	—	4	2	no	—
264	C₂×D₄	—	4	2	no	—
265	C₂×D₄	—	8	2	no	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
269	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
270	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
271	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
272	SD₁₆	—	8	8	no	—
273	C₂×C₈	—	2	4	no	—
274	C₄⋊C₄	—	2	4	no	—
275	C₄⋊C₄	—	2	4	no	—
276	C₄⋊C₄	—	2	4	no	—
277	C₄⋊C₄	—	2	8	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	4	no	—
283	C₄⋊C₄	—	2	4	no	—
284	C₄⋊C₄	—	2	4	no	—
285	C₄⋊C₄	—	4	2	no	—
286	C₄⋊C₄	C₂×D₄×C₁₉	1	4	yes	—
287	C₂²⋊C₄	—	2	4	no	—
288	C₂²⋊C₄	—	4	8	no	—
289	C₂²⋊C₄	—	8	4	no	—
290	C₄²	—	2	8	no	—
291	C₄²	—	2	2	no	—
292	C₄²	—	2	2	no	—
293	C₄²	—	2	4	no	—
294	C₄²	—	4	2	no	—
295	C₄²	C₂×D₄×C₁₉	1	2	yes	—
296	C₄²	C₂×D₄×C₁₉	1	2	yes	—
297	C₂³	—	4	4	no	—
298	C₂³	—	4	4	no	—
299	C₂³	—	4	2	no	—
300	C₂³	—	4	2	no	—
301	C₂³	—	4	2	no	—
302	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
303	Q₈	—	2	8	no	—
304	Q₈	—	4	4	no	—
305	Q₈	—	4	4	no	—
306	Q₈	—	8	4	no	—
307	D₄	—	4	8	no	—
308	D₄	—	8	2	no	—
309	D₄	—	8	4	no	—
310	D₄	—	8	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	4	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
323	C₂×C₄	C₁₉×#120	1	2	yes	—
324	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
325	C₂×C₄	C₂×SD₁₆×C₁₉	1	4	yes	—
326	C₂×C₄	C₂×SD₁₆×C₁₉	1	2	yes	—
327	C₂×C₄	C₁₉×#137	1	2	yes	—
328	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
329	C₈	—	4	4	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	8	4	no	—
333	C₂²	—	8	4	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	8	2	no	—
337	C₂²	C₂×C₁₉×#120	1	1	yes	characteristic
338	C₂²	C₂²×SD₁₆×C₁₉	1	1	yes	characteristic
339	C₂²	C₂×C₁₉×#137	1	1	yes	characteristic
340	C₄	—	2	4	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	2	no	—
343	C₄	—	4	4	no	—
344	C₄	—	4	2	no	—
345	C₄	—	4	4	no	—
346	C₄	—	8	2	no	—
347	C₄	C₁₉×#448	1	4	yes	—
348	C₄	C₁₉×#454	1	2	yes	—
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	—	16	2	no	—
352	C₂	C₂×C₁₉×#448	1	1	yes	characteristic
353	C₂	C₂×C₁₉×#454	1	1	yes	characteristic
354	C₂	C₁₉×#3027	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
prime radical	C₂×C₁₉×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ 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×#3502	C₂	semidirect
4. C₁₉×#1599	C₂	semidirect
5. C₁₉×#1503	C₂	semidirect
6. C₁₉×#1461	C₂	semidirect
78. C₂×Q₈×C₁₉	SD₁₆	semidirect
119. #18403	C₁₉	direct (N × H)
153. #3502	C₂×C₁₉	semidirect
156. #1599	C₂×C₁₉	semidirect
157. #1503	C₂×C₁₉	semidirect
158. #1461	C₂×C₁₉	semidirect
250. C₁₉	#18403	direct (N × H)
256. C₂×Q₈	SD₁₆×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₉×#3502	2
C₁₉×#3208	2
C₁₉×#2548	2
C₁₉×#1599	2
C₁₉×#1503	2
C₁₉×#1461	2
#18403	19

Subgroups — C19×#18403 #18024362

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#18403 #18024362