Subgroups — C₁₉×#12806 #18018765

Order 4864.

420 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2930	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2120	C₂	1	2	yes	maximal
3	C₁₉×#1813	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1812	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1395	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₉×#131	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₁₉	C₄	1	1	yes	characteristic, complemented
8	C₁₉×#525	C₂²	1	1	yes	characteristic
9	C₁₉×#436	—	2	1	no	—
10	C₁₉×#436	—	2	1	no	—
11	C₁₉×#435	—	2	1	no	—
12	C₁₉×#435	—	2	1	no	—
13	C₂×C₁₉×#108	—	2	1	no	—
14	C₁₉×#427	—	2	1	no	—
15	C₁₉×#424	C₄	1	1	yes	characteristic, complemented
16	C₁₉×#413	C₂²	1	2	yes	—
17	C₁₉×#410	C₂²	1	2	yes	—
18	C₁₉×#408	C₂²	1	1	yes	characteristic
19	C₁₉×#407	C₄	1	1	yes	characteristic, complemented
20	C₁₉×#405	C₄	1	1	yes	characteristic, complemented
21	C₁₉×#352	—	2	2	no	—
22	C₁₉×AΣL(1,𝔽₁₆)	—	2	2	no	—
23	C₁₉×#338	—	2	1	no	—
24	C₁₉×H(3,ℤ/4)	—	2	1	no	—
25	C₂×#41×C₁₉	C₂³	1	1	yes	characteristic
26	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
27	C₁₉×#131	—	4	1	no	—
28	C₁₉×#131	—	4	1	no	—
29	C₂×M₁₆×C₁₉	—	2	1	no	—
30	C₂×M₁₆×C₁₉	—	2	1	no	—
31	C₂×M₁₆×C₁₉	—	2	2	no	—
32	C₂×M₁₆×C₁₉	—	4	1	no	—
33	C₂×M₁₆×C₁₉	—	4	1	no	—
34	C₂×M₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
35	C₂×M₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
36	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
37	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
42	C₁₉×#123	D₄	1	1	yes	characteristic
43	C₁₉×#123	Q₈	1	1	yes	characteristic
44	C₁₉×C₂²⋊Q₈	—	2	1	no	—
45	C₁₉×#121	—	2	1	no	—
46	C₁₉×#120	D₄	1	1	yes	characteristic
47	C₁₉×#120	Q₈	1	1	yes	characteristic
48	C₄×D₄×C₁₉	—	2	1	no	—
49	C₄×D₄×C₁₉	—	2	1	no	—
50	C₁₉×#117	—	2	2	no	—
51	C₁₉×#117	C₂×C₄	1	1	yes	characteristic
52	C₁₉×#108	—	4	1	no	—
53	C₁₉×#108	—	4	1	no	—
54	C₁₉×#107	—	4	1	no	—
55	C₁₉×#106	—	4	1	no	—
56	C₁₉×C₄⋊C₈	—	2	1	no	—
57	C₁₉×C₄⋊C₈	—	2	1	no	—
58	C₁₉×#101	—	2	2	no	—
59	C₁₉×#100	—	2	2	no	—
60	C₁₉×#99	—	2	2	no	—
61	C₁₉×#99	—	2	2	no	—
62	C₁₉×C₂²⋊C₈	—	2	1	no	—
63	C₁₉×C₂²⋊C₈	—	2	1	no	—
64	C₁₉×C₂²⋊C₈	—	2	1	no	—
65	C₁₉×C₂²⋊C₈	—	2	1	no	—
66	C₁₉×#97	—	2	1	no	—
67	C₄×C₈×C₁₉	—	2	1	no	—
68	C₂⁴×C₁₉	C₄⋊C₄	1	1	yes	characteristic
69	#41×C₁₉	—	4	1	no	—
70	#41×C₁₉	—	4	1	no	—
71	C₂×Q₈×C₁₉	#41	1	1	yes	characteristic
72	C₂×D₄×C₁₉	—	2	1	no	—
73	C₂×D₄×C₁₉	—	2	1	no	—
74	C₂×D₄×C₁₉	C₂×D₄	1	1	yes	characteristic
75	C₂×D₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
76	C₂×D₄×C₁₉	#41	1	1	yes	characteristic
77	C₂²×C₄×C₁₉	—	2	1	no	—
78	C₂²×C₄×C₁₉	—	2	1	no	—
79	C₂²×C₄×C₁₉	—	4	1	no	—
80	C₂²×C₄×C₁₉	—	4	1	no	—
81	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
85	M₁₆×C₁₉	—	4	1	no	—
86	M₁₆×C₁₉	—	4	1	no	—
87	M₁₆×C₁₉	—	4	1	no	—
88	M₁₆×C₁₉	—	4	2	no	—
89	M₁₆×C₁₉	—	4	1	no	—
90	M₁₆×C₁₉	—	4	1	no	—
91	M₁₆×C₁₉	—	4	1	no	—
92	M₁₆×C₁₉	—	4	2	no	—
93	M₁₆×C₁₉	—	4	2	no	—
94	C₂×C₈×C₁₉	—	2	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	—	2	1	no	—
99	C₂×C₈×C₁₉	—	2	1	no	—
100	C₂×C₈×C₁₉	—	2	1	no	—
101	C₂×C₈×C₁₉	—	2	1	no	—
102	C₂×C₈×C₁₉	—	4	1	no	—
103	C₂×C₈×C₁₉	—	4	2	no	—
104	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
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	—	4	2	no	—
112	C₂²⋊C₄×C₁₉	—	2	1	no	—
113	C₂²⋊C₄×C₁₉	—	2	1	no	—
114	C₂²⋊C₄×C₁₉	—	2	1	no	—
115	C₂²⋊C₄×C₁₉	—	2	1	no	—
116	C₂²⋊C₄×C₁₉	—	4	2	no	—
117	C₂²⋊C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
118	C₂²⋊C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
119	C₄²×C₁₉	—	2	1	no	—
120	C₄²×C₁₉	—	4	2	no	—
121	#12806	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂³×C₁₉	—	2	1	no	—
123	C₂³×C₁₉	—	4	1	no	—
124	C₂³×C₁₉	—	8	1	no	—
125	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂³×C₁₉	#121	1	1	yes	characteristic
127	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
128	Q₈×C₁₉	—	4	1	no	—
129	D₄×C₁₉	—	4	1	no	—
130	D₄×C₁₉	—	4	1	no	—
131	D₄×C₁₉	—	4	1	no	—
132	D₄×C₁₉	—	8	1	no	—
133	D₄×C₁₉	—	8	1	no	—
134	C₂×C₄×C₁₉	—	2	1	no	—
135	C₂×C₄×C₁₉	—	2	1	no	—
136	C₂×C₄×C₁₉	—	2	1	no	—
137	C₂×C₄×C₁₉	—	2	1	no	—
138	C₂×C₄×C₁₉	—	2	1	no	—
139	C₂×C₄×C₁₉	—	2	1	no	—
140	C₂×C₄×C₁₉	—	2	1	no	—
141	C₂×C₄×C₁₉	—	2	1	no	—
142	C₂×C₄×C₁₉	—	2	1	no	—
143	C₂×C₄×C₁₉	—	4	2	no	—
144	C₂×C₄×C₁₉	—	4	1	no	—
145	C₂×C₄×C₁₉	—	4	1	no	—
146	C₂×C₄×C₁₉	—	4	1	no	—
147	C₂×C₄×C₁₉	—	8	2	no	—
148	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
150	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
151	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
152	C₈×C₁₉	—	2	1	no	—
153	C₈×C₁₉	—	2	1	no	—
154	C₈×C₁₉	—	2	1	no	—
155	C₈×C₁₉	—	4	1	no	—
156	C₈×C₁₉	—	4	1	no	—
157	C₈×C₁₉	—	8	2	no	—
158	C₈×C₁₉	#99	1	1	yes	characteristic
159	C₈×C₁₉	#99	1	1	yes	characteristic
160	#2930	C₂×C₁₉	1	1	yes	characteristic
161	#2120	C₂×C₁₉	1	2	yes	—
162	#1813	C₂×C₁₉	1	1	yes	characteristic, complemented
163	#1812	C₂×C₁₉	1	1	yes	characteristic, complemented
164	#1395	C₂×C₁₉	1	2	yes	complemented
165	C₂²×C₁₉	—	2	1	no	—
166	C₂²×C₁₉	—	2	1	no	—
167	C₂²×C₁₉	—	2	1	no	—
168	C₂²×C₁₉	—	4	1	no	—
169	C₂²×C₁₉	—	4	1	no	—
170	C₂²×C₁₉	—	4	1	no	—
171	C₂²×C₁₉	—	4	1	no	—
172	C₂²×C₁₉	—	4	1	no	—
173	C₂²×C₁₉	—	4	1	no	—
174	C₂²×C₁₉	—	8	2	no	—
175	C₂²×C₁₉	#380	1	1	yes	characteristic
176	C₄×C₁₉	—	2	1	no	—
177	C₄×C₁₉	—	2	1	no	—
178	C₄×C₁₉	—	2	1	no	—
179	C₄×C₁₉	—	4	1	no	—
180	C₄×C₁₉	—	4	1	no	—
181	C₄×C₁₉	—	4	1	no	—
182	C₄×C₁₉	—	16	2	no	—
183	C₄×C₁₉	C₂×#99	1	1	yes	characteristic
184	C₄×C₁₉	#410	1	1	yes	characteristic
185	C₂×#131	C₂²×C₁₉	1	1	yes	characteristic
186	C₂²×M₁₆	C₄×C₁₉	1	1	yes	characteristic, complemented
187	#525	C₂²×C₁₉	1	1	yes	characteristic
188	#436	—	2	1	no	—
189	#436	—	2	1	no	—
190	#435	—	2	1	no	—
191	#435	—	2	1	no	—
192	C₂×#108	—	2	1	no	—
193	#427	—	2	1	no	—
194	#424	C₄×C₁₉	1	1	yes	characteristic, complemented
195	#413	C₂²×C₁₉	1	2	yes	—
196	#410	C₂²×C₁₉	1	2	yes	—
197	#408	C₂²×C₁₉	1	1	yes	characteristic
198	#407	C₄×C₁₉	1	1	yes	characteristic, complemented
199	#405	C₄×C₁₉	1	1	yes	characteristic, complemented
200	#352	—	2	2	no	—
201	AΣL(1,𝔽₁₆)	—	2	2	no	—
202	#338	—	2	1	no	—
203	H(3,ℤ/4)	—	2	1	no	—
204	C₂×C₁₉	—	2	1	no	—
205	C₂×C₁₉	—	4	1	no	—
206	C₂×C₁₉	—	4	1	no	—
207	C₂×C₁₉	—	4	1	no	—
208	C₂×C₁₉	—	8	1	no	—
209	C₂×C₁₉	#1799	1	1	yes	characteristic
210	C₂×#41	C₂³×C₁₉	1	1	yes	characteristic
211	C₂³×C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
212	#131	—	4	1	no	—
213	#131	—	4	1	no	—
214	C₂×M₁₆	—	2	1	no	—
215	C₂×M₁₆	—	2	1	no	—
216	C₂×M₁₆	—	2	2	no	—
217	C₂×M₁₆	—	4	1	no	—
218	C₂×M₁₆	—	4	1	no	—
219	C₂×M₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
220	C₂×M₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
221	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
222	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
223	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
224	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
225	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
226	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
227	#123	D₄×C₁₉	1	1	yes	characteristic
228	#123	Q₈×C₁₉	1	1	yes	characteristic
229	C₂²⋊Q₈	—	2	1	no	—
230	#121	—	2	1	no	—
231	#120	D₄×C₁₉	1	1	yes	characteristic
232	#120	Q₈×C₁₉	1	1	yes	characteristic
233	C₄×D₄	—	2	1	no	—
234	C₄×D₄	—	2	1	no	—
235	#117	—	2	2	no	—
236	#117	C₂×C₄×C₁₉	1	1	yes	characteristic
237	#108	—	4	1	no	—
238	#108	—	4	1	no	—
239	#107	—	4	1	no	—
240	#106	—	4	1	no	—
241	C₄⋊C₈	—	2	1	no	—
242	C₄⋊C₈	—	2	1	no	—
243	#101	—	2	2	no	—
244	#100	—	2	2	no	—
245	#99	—	2	2	no	—
246	#99	—	2	2	no	—
247	C₂²⋊C₈	—	2	1	no	—
248	C₂²⋊C₈	—	2	1	no	—
249	C₂²⋊C₈	—	2	1	no	—
250	C₂²⋊C₈	—	2	1	no	—
251	#97	—	2	1	no	—
252	C₄×C₈	—	2	1	no	—
253	C₁₉	#12806	1	1	yes	characteristic, direct product factor, Sylow
254	C₂⁴	C₄⋊C₄×C₁₉	1	1	yes	characteristic
255	#41	—	4	1	no	—
256	#41	—	4	1	no	—
257	C₂×Q₈	#41×C₁₉	1	1	yes	characteristic
258	C₂×D₄	—	2	1	no	—
259	C₂×D₄	—	2	1	no	—
260	C₂×D₄	C₂×D₄×C₁₉	1	1	yes	characteristic
261	C₂×D₄	C₂×Q₈×C₁₉	1	1	yes	characteristic
262	C₂×D₄	#41×C₁₉	1	1	yes	characteristic
263	C₂²×C₄	—	2	1	no	—
264	C₂²×C₄	—	2	1	no	—
265	C₂²×C₄	—	4	1	no	—
266	C₂²×C₄	—	4	1	no	—
267	C₂²×C₄	C₄⋊C₄×C₁₉	1	1	yes	characteristic
268	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
270	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
271	M₁₆	—	4	1	no	—
272	M₁₆	—	4	1	no	—
273	M₁₆	—	4	1	no	—
274	M₁₆	—	4	2	no	—
275	M₁₆	—	4	1	no	—
276	M₁₆	—	4	1	no	—
277	M₁₆	—	4	1	no	—
278	M₁₆	—	4	2	no	—
279	M₁₆	—	4	2	no	—
280	C₂×C₈	—	2	1	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	4	1	no	—
289	C₂×C₈	—	4	2	no	—
290	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
291	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
292	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
293	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
294	C₄⋊C₄	—	2	1	no	—
295	C₄⋊C₄	—	2	1	no	—
296	C₄⋊C₄	—	2	1	no	—
297	C₄⋊C₄	—	4	2	no	—
298	C₂²⋊C₄	—	2	1	no	—
299	C₂²⋊C₄	—	2	1	no	—
300	C₂²⋊C₄	—	2	1	no	—
301	C₂²⋊C₄	—	2	1	no	—
302	C₂²⋊C₄	—	4	2	no	—
303	C₂²⋊C₄	C₄⋊C₄×C₁₉	1	1	yes	characteristic
304	C₂²⋊C₄	C₄⋊C₄×C₁₉	1	1	yes	characteristic
305	C₄²	—	2	1	no	—
306	C₄²	—	4	2	no	—
307	C₂³	—	2	1	no	—
308	C₂³	—	4	1	no	—
309	C₂³	—	8	1	no	—
310	C₂³	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
311	C₂³	C₁₉×#121	1	1	yes	characteristic
312	C₂³	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
313	Q₈	—	4	1	no	—
314	D₄	—	4	1	no	—
315	D₄	—	4	1	no	—
316	D₄	—	4	1	no	—
317	D₄	—	8	1	no	—
318	D₄	—	8	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	8	2	no	—
333	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
334	C₂×C₄	C₁₉×#117	1	1	yes	characteristic
335	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
336	C₂×C₄	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
337	C₈	—	2	1	no	—
338	C₈	—	2	1	no	—
339	C₈	—	2	1	no	—
340	C₈	—	4	1	no	—
341	C₈	—	4	1	no	—
342	C₈	—	8	2	no	—
343	C₈	C₁₉×#99	1	1	yes	characteristic
344	C₈	C₁₉×#99	1	1	yes	characteristic
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	4	1	no	—
349	C₂²	—	4	1	no	—
350	C₂²	—	4	1	no	—
351	C₂²	—	4	1	no	—
352	C₂²	—	4	1	no	—
353	C₂²	—	4	1	no	—
354	C₂²	—	8	2	no	—
355	C₂²	C₁₉×#380	1	1	yes	characteristic
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	4	1	no	—
360	C₄	—	4	1	no	—
361	C₄	—	4	1	no	—
362	C₄	—	16	2	no	—
363	C₄	C₂×C₁₉×#99	1	1	yes	characteristic
364	C₄	C₁₉×#410	1	1	yes	characteristic
365	C₂	—	2	1	no	—
366	C₂	—	4	1	no	—
367	C₂	—	4	1	no	—
368	C₂	—	4	1	no	—
369	C₂	—	8	1	no	—
370	C₂	C₁₉×#1799	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₉	#410
commutator	C₂²×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×#41	C₂³×C₁₉
socle	C₂×C₁₉	#1799
prime radical	C₁₉×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂×C₈×C₁₉ ◁ C₂×C₁₉×#131 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₉×#1813	C₂	semidirect
4. C₁₉×#1812	C₂	semidirect
5. C₁₉×#1395	C₂	semidirect
7. C₂²×M₁₆×C₁₉	C₄	semidirect
15. C₁₉×#424	C₄	semidirect
19. C₁₉×#407	C₄	semidirect
20. C₁₉×#405	C₄	semidirect
121. #12806	C₁₉	direct (N × H)
162. #1813	C₂×C₁₉	semidirect
163. #1812	C₂×C₁₉	semidirect
164. #1395	C₂×C₁₉	semidirect
186. C₂²×M₁₆	C₄×C₁₉	semidirect
194. #424	C₄×C₁₉	semidirect
198. #407	C₄×C₁₉	semidirect
199. #405	C₄×C₁₉	semidirect
253. C₁₉	#12806	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×#2930	2
C₁₉×#2120	2
C₁₉×#1813	2
C₁₉×#1812	2
C₁₉×#1395	2
#12806	19

Subgroups — C19×#12806 #18018765

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12806 #18018765