Subgroups — C₁₉×#2588 #14560445

Order 2432.

388 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#400	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#400	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#398	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#395	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#391	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#391	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#389	C₂	1	1	yes	maximal, characteristic
8	C₁₉×#387	C₂	1	1	yes	maximal, characteristic
9	C₁₉×#387	C₂	1	1	yes	maximal, characteristic
10	C₁₉×#387	C₂	1	1	yes	maximal, characteristic
11	C₁₉×#387	C₂	1	1	yes	maximal, characteristic
12	C₁₉×#387	C₂	1	1	yes	maximal, characteristic
13	C₁₉×#387	C₂	1	1	yes	maximal, characteristic
14	C₄×C₄⋊C₄×C₁₉	C₂	1	1	yes	maximal, characteristic
15	C₄×C₄⋊C₄×C₁₉	C₂	1	1	yes	maximal, characteristic
16	C₂²×Q₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
29	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
30	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
31	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
33	C₁₉×#95	C₂²	1	1	yes	characteristic
34	C₁₉×#95	C₂²	1	1	yes	characteristic
35	C₁₉×#95	C₂²	1	1	yes	characteristic
36	C₁₉×#95	C₂²	1	1	yes	characteristic
37	C₁₉×#95	C₂²	1	1	yes	characteristic
38	C₁₉×#95	C₂²	1	1	yes	characteristic
39	C₁₉×#95	C₂²	1	1	yes	characteristic
40	C₁₉×#95	C₂²	1	1	yes	characteristic
41	C₁₉×#95	C₂²	1	1	yes	characteristic
42	C₁₉×#95	C₂²	1	1	yes	characteristic
43	C₁₉×#95	C₂²	1	1	yes	characteristic
44	C₁₉×#95	C₂²	1	1	yes	characteristic
45	C₁₉×#95	C₂²	1	1	yes	characteristic
46	C₁₉×#95	C₂²	1	1	yes	characteristic
47	C₁₉×#95	C₂²	1	1	yes	characteristic
48	C₁₉×#95	C₂²	1	1	yes	characteristic
49	C₁₉×#95	C₂²	1	1	yes	characteristic
50	C₁₉×#95	C₂²	1	1	yes	characteristic
51	C₂×Q₈×C₁₉	—	2	2	no	—
52	C₂×Q₈×C₁₉	—	2	2	no	—
53	C₂×Q₈×C₁₉	—	4	1	no	—
54	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₁₉	—	2	2	no	—
70	C₄⋊C₄×C₁₉	—	2	2	no	—
71	C₄⋊C₄×C₁₉	—	2	2	no	—
72	C₄⋊C₄×C₁₉	—	2	2	no	—
73	C₄⋊C₄×C₁₉	—	2	2	no	—
74	C₄⋊C₄×C₁₉	—	2	2	no	—
75	C₄⋊C₄×C₁₉	—	2	2	no	—
76	C₄⋊C₄×C₁₉	—	2	2	no	—
77	C₄⋊C₄×C₁₉	—	4	1	no	—
78	C₄⋊C₄×C₁₉	—	4	1	no	—
79	C₄⋊C₄×C₁₉	Q₈	1	4	yes	complemented
80	C₄²×C₁₉	—	2	2	no	—
81	C₄²×C₁₉	—	2	2	no	—
82	C₄²×C₁₉	—	4	1	no	—
83	C₄²×C₁₉	—	4	1	no	—
84	C₄²×C₁₉	—	4	1	no	—
85	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
86	Q₈×C₁₉	—	4	4	no	—
87	C₂×C₄×C₁₉	—	2	1	no	—
88	C₂×C₄×C₁₉	—	2	1	no	—
89	C₂×C₄×C₁₉	—	2	1	no	—
90	C₂×C₄×C₁₉	—	2	1	no	—
91	C₂×C₄×C₁₉	—	2	1	no	—
92	C₂×C₄×C₁₉	—	2	1	no	—
93	C₂×C₄×C₁₉	—	2	1	no	—
94	C₂×C₄×C₁₉	—	2	1	no	—
95	C₂×C₄×C₁₉	—	2	1	no	—
96	C₂×C₄×C₁₉	—	2	1	no	—
97	C₂×C₄×C₁₉	—	2	1	no	—
98	C₂×C₄×C₁₉	—	2	1	no	—
99	C₂×C₄×C₁₉	—	2	1	no	—
100	C₂×C₄×C₁₉	—	2	1	no	—
101	C₂×C₄×C₁₉	—	2	1	no	—
102	C₂×C₄×C₁₉	—	2	1	no	—
103	C₂×C₄×C₁₉	—	2	1	no	—
104	C₂×C₄×C₁₉	—	2	1	no	—
105	C₂×C₄×C₁₉	—	2	1	no	—
106	C₂×C₄×C₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	—	2	1	no	—
108	C₂×C₄×C₁₉	—	2	1	no	—
109	C₂×C₄×C₁₉	—	2	1	no	—
110	C₂×C₄×C₁₉	—	2	1	no	—
111	C₂×C₄×C₁₉	—	2	1	no	—
112	C₂×C₄×C₁₉	—	2	1	no	—
113	C₂×C₄×C₁₉	—	2	1	no	—
114	C₂×C₄×C₁₉	—	2	1	no	—
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	2	1	no	—
118	C₂×C₄×C₁₉	—	2	1	no	—
119	C₂×C₄×C₁₉	—	2	1	no	—
120	C₂×C₄×C₁₉	—	2	1	no	—
121	C₂×C₄×C₁₉	—	2	1	no	—
122	C₂×C₄×C₁₉	—	2	1	no	—
123	C₂×C₄×C₁₉	—	2	1	no	—
124	C₂×C₄×C₁₉	—	2	1	no	—
125	C₂×C₄×C₁₉	C₂×Q₈	1	2	yes	—
126	C₂×C₄×C₁₉	C₂×Q₈	1	2	yes	—
127	C₂×C₄×C₁₉	C₂×Q₈	1	2	yes	—
128	C₂×C₄×C₁₉	#41	1	2	yes	—
129	C₂×C₄×C₁₉	#41	1	2	yes	—
130	C₂×C₄×C₁₉	#41	1	2	yes	—
131	C₂×C₄×C₁₉	#41	1	2	yes	—
132	#2588	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂²×C₁₉	C₂²×Q₈	1	1	yes	characteristic
134	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
135	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
136	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
137	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
138	C₂²×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
139	C₂²×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
140	C₄×C₁₉	—	2	2	no	—
141	C₄×C₁₉	—	2	2	no	—
142	C₄×C₁₉	—	2	2	no	—
143	C₄×C₁₉	—	2	2	no	—
144	C₄×C₁₉	—	2	2	no	—
145	C₄×C₁₉	—	2	2	no	—
146	C₄×C₁₉	—	2	2	no	—
147	C₄×C₁₉	—	4	1	no	—
148	C₄×C₁₉	—	4	1	no	—
149	C₄×C₁₉	—	4	1	no	—
150	C₄×C₁₉	—	4	1	no	—
151	C₄×C₁₉	—	4	1	no	—
152	C₄×C₁₉	—	4	1	no	—
153	C₄×C₁₉	—	4	1	no	—
154	C₄×C₁₉	—	4	1	no	—
155	#400	C₂×C₁₉	1	1	yes	characteristic
156	#400	C₂×C₁₉	1	1	yes	characteristic
157	#398	C₂×C₁₉	1	1	yes	characteristic
158	#395	C₂×C₁₉	1	1	yes	characteristic
159	#391	C₂×C₁₉	1	1	yes	characteristic
160	#391	C₂×C₁₉	1	1	yes	characteristic
161	#389	C₂×C₁₉	1	1	yes	characteristic
162	#387	C₂×C₁₉	1	1	yes	characteristic
163	#387	C₂×C₁₉	1	1	yes	characteristic
164	#387	C₂×C₁₉	1	1	yes	characteristic
165	#387	C₂×C₁₉	1	1	yes	characteristic
166	#387	C₂×C₁₉	1	1	yes	characteristic
167	#387	C₂×C₁₉	1	1	yes	characteristic
168	C₄×C₄⋊C₄	C₂×C₁₉	1	1	yes	characteristic
169	C₄×C₄⋊C₄	C₂×C₁₉	1	1	yes	characteristic
170	C₂×C₁₉	#529	1	1	yes	characteristic
171	C₂×C₁₉	#533	1	1	yes	characteristic
172	C₂×C₁₉	#539	1	1	yes	characteristic
173	C₂×C₁₉	#551	1	1	yes	characteristic
174	C₂×C₁₉	#554	1	1	yes	characteristic
175	C₂×C₁₉	#556	1	1	yes	characteristic
176	C₂×C₁₉	#557	1	1	yes	characteristic
177	C₂²×Q₈	C₂²×C₁₉	1	1	yes	characteristic
178	C₂×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
179	C₂×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
180	C₂×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
181	C₂×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
182	C₂×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
183	C₂×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
184	C₂×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
185	C₂×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
186	C₂×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
187	C₂×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
188	C₂×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
189	C₂×C₄²	C₂²×C₁₉	1	1	yes	characteristic
190	C₂×C₄²	C₂²×C₁₉	1	1	yes	characteristic
191	C₂×C₄²	C₂²×C₁₉	1	1	yes	characteristic
192	C₂×C₄²	C₂²×C₁₉	1	1	yes	characteristic
193	C₂×C₄²	C₂²×C₁₉	1	1	yes	characteristic
194	#95	C₂²×C₁₉	1	1	yes	characteristic
195	#95	C₂²×C₁₉	1	1	yes	characteristic
196	#95	C₂²×C₁₉	1	1	yes	characteristic
197	#95	C₂²×C₁₉	1	1	yes	characteristic
198	#95	C₂²×C₁₉	1	1	yes	characteristic
199	#95	C₂²×C₁₉	1	1	yes	characteristic
200	#95	C₂²×C₁₉	1	1	yes	characteristic
201	#95	C₂²×C₁₉	1	1	yes	characteristic
202	#95	C₂²×C₁₉	1	1	yes	characteristic
203	#95	C₂²×C₁₉	1	1	yes	characteristic
204	#95	C₂²×C₁₉	1	1	yes	characteristic
205	#95	C₂²×C₁₉	1	1	yes	characteristic
206	#95	C₂²×C₁₉	1	1	yes	characteristic
207	#95	C₂²×C₁₉	1	1	yes	characteristic
208	#95	C₂²×C₁₉	1	1	yes	characteristic
209	#95	C₂²×C₁₉	1	1	yes	characteristic
210	#95	C₂²×C₁₉	1	1	yes	characteristic
211	#95	C₂²×C₁₉	1	1	yes	characteristic
212	C₁₉	#2588	1	1	yes	characteristic, direct product factor, Sylow
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	—	4	1	no	—
216	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
217	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
218	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
219	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
220	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
221	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
222	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
223	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
224	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
225	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
226	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
227	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
228	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
229	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
230	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	4	1	no	—
240	C₄⋊C₄	—	4	1	no	—
241	C₄⋊C₄	Q₈×C₁₉	1	4	yes	complemented
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	4	1	no	—
246	C₄²	—	4	1	no	—
247	C₂³	C₂⁴×C₁₉	1	1	yes	characteristic
248	Q₈	—	4	4	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	C₂×Q₈×C₁₉	1	2	yes	—
288	C₂×C₄	C₂×Q₈×C₁₉	1	2	yes	—
289	C₂×C₄	C₂×Q₈×C₁₉	1	2	yes	—
290	C₂×C₄	#41×C₁₉	1	2	yes	—
291	C₂×C₄	#41×C₁₉	1	2	yes	—
292	C₂×C₄	#41×C₁₉	1	2	yes	—
293	C₂×C₄	#41×C₁₉	1	2	yes	—
294	C₂²	C₂²×Q₈×C₁₉	1	1	yes	characteristic
295	C₂²	C₂×#41×C₁₉	1	1	yes	characteristic
296	C₂²	C₂×#41×C₁₉	1	1	yes	characteristic
297	C₂²	C₂×#41×C₁₉	1	1	yes	characteristic
298	C₂²	C₂×#41×C₁₉	1	1	yes	characteristic
299	C₂²	C₁₉×H(5,𝔽₂)	1	1	yes	characteristic
300	C₂²	C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
301	C₄	—	2	2	no	—
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	4	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	1	no	—
316	C₂	C₁₉×#529	1	1	yes	characteristic
317	C₂	C₁₉×#533	1	1	yes	characteristic
318	C₂	C₁₉×#539	1	1	yes	characteristic
319	C₂	C₁₉×#551	1	1	yes	characteristic
320	C₂	C₁₉×#554	1	1	yes	characteristic
321	C₂	C₁₉×#556	1	1	yes	characteristic
322	C₂	C₁₉×#557	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₉
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
79. C₄⋊C₄×C₁₉	Q₈	semidirect
132. #2588	C₁₉	direct (N × H)
212. C₁₉	#2588	direct (N × H)
241. C₄⋊C₄	Q₈×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₉×#400	2
C₁₉×#398	2
C₁₉×#395	2
C₁₉×#391	2
C₁₉×#389	2
C₁₉×#387	2
C₄×C₄⋊C₄×C₁₉	2
#2588	19

Subgroups — C19×#2588 #14560445

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#2588 #14560445