Subgroups — D₁₉×#578 #14578397

Order 2432.

1408 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	2₋¹⁺⁴×D₁₉	C₂	1	2	yes	maximal, complemented
2	#11927361	C₂	1	1	yes	maximal, characteristic, complemented
3	#11927360	C₂	1	6	yes	maximal, complemented
4	#11927359	C₂	1	2	yes	maximal, complemented
5	#11927358	C₂	1	3	yes	maximal, complemented
6	#11927357	C₂	1	3	yes	maximal, complemented
7	#137×D₁₉	C₂	1	6	yes	maximal, complemented
8	#135×D₁₉	C₂	1	3	yes	maximal, complemented
9	C₂×Q₁₆×D₁₉	C₂	1	3	yes	maximal, complemented
10	#131×D₁₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₉×#578	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₉×2₋¹⁺⁴	C₂²	1	2	yes	complemented
13	#10597436	—	2	6	no	—
14	#10597436	C₂²	1	6	yes	complemented
15	#10597436	C₂²	1	2	yes	complemented
16	#41×D₁₉	—	2	6	no	—
17	#41×D₁₉	C₂²	1	6	yes	—
18	#41×D₁₉	C₂²	1	1	yes	characteristic
19	#10597433	—	2	2	no	—
20	#10597433	C₂²	1	6	yes	complemented
21	C₂×Q₈×D₁₉	—	2	2	no	—
22	C₂×Q₈×D₁₉	C₂²	1	6	yes	—
23	C₁₉×#137	C₂²	1	6	yes	complemented
24	C₁₉×#135	C₂²	1	3	yes	complemented
25	C₂×Q₁₆×C₁₉	C₂²	1	3	yes	complemented
26	C₁₉×#131	C₂²	1	1	yes	characteristic, complemented
27	#10597397	C₂²	1	6	yes	complemented
28	#10597396	C₂²	1	6	yes	complemented
29	#10597394	C₂²	1	1	yes	characteristic, complemented
30	C₂×#64528	C₂²	1	6	yes	complemented
31	#10597388	C₂²	1	6	yes	complemented
32	#10597365	C₂²	1	3	yes	complemented
33	#10597364	C₂²	1	12	yes	complemented
34	Q₁₆×D₁₉	C₂²	1	3	yes	—
35	Q₁₆×D₁₉	C₂²	1	6	yes	—
36	Q₁₆×D₁₉	C₂²	1	6	yes	complemented
37	Q₁₆×D₁₉	C₂²	1	6	yes	complemented
38	#10597362	C₂²	1	6	yes	complemented
39	#10597361	C₂²	1	12	yes	complemented
40	SD₁₆×D₁₉	C₂²	1	6	yes	—
41	#10597358	C₂²	1	3	yes	complemented
42	D₈×D₁₉	C₂²	1	1	yes	characteristic
43	#10597355	C₂²	1	6	yes	complemented
44	#10597353	C₂²	1	3	yes	complemented
45	M₁₆×D₁₉	C₂²	1	3	yes	—
46	C₂×Dic₇₆	C₂²	1	3	yes	complemented
47	#10597350	C₂²	1	3	yes	complemented
48	#10597347	C₂²	1	3	yes	complemented
49	C₂×C₈×D₁₉	C₂²	1	3	yes	—
50	#41×C₁₉	—	2	6	no	—
51	#41×C₁₉	C₂³	1	6	yes	—
52	#41×C₁₉	C₂³	1	1	yes	characteristic
53	C₂×Q₈×C₁₉	—	2	2	no	—
54	C₂×Q₈×C₁₉	C₂³	1	6	yes	—
55	#64545	—	2	6	no	—
56	#64545	D₄	1	1	yes	characteristic
57	#64545	C₂³	1	6	yes	—
58	Q₈×D₁₉	—	2	2	no	—
59	Q₈×D₁₉	—	2	6	no	—
60	Q₈×D₁₉	—	2	6	no	—
61	Q₈×D₁₉	—	2	6	no	—
62	Q₈×D₁₉	D₄	1	1	yes	characteristic
63	Q₈×D₁₉	C₂³	1	6	yes	—
64	Q₈×D₁₉	C₂³	1	12	yes	—
65	#64543	—	2	6	no	—
66	#64543	—	2	12	no	—
67	#64543	—	2	6	no	—
68	#64543	D₄	1	3	yes	—
69	#64543	C₂³	1	6	yes	—
70	D₄×D₁₉	—	2	6	no	—
71	D₄×D₁₉	D₄	1	3	yes	—
72	D₄×D₁₉	C₂³	1	2	yes	—
73	#64541	—	2	12	no	—
74	#64541	—	2	6	no	—
75	#64541	—	2	6	no	—
76	#64541	C₂³	1	3	yes	—
77	#64541	C₂³	1	6	yes	—
78	C₂×C₄×D₁₉	—	2	6	no	—
79	C₂×C₄×D₁₉	—	2	6	no	—
80	C₂×C₄×D₁₉	C₂³	1	3	yes	—
81	C₂×Dic₃₈	—	2	6	no	—
82	C₂×Dic₃₈	—	2	6	no	—
83	C₂×Dic₃₈	C₂³	1	6	yes	—
84	Q₁₆×C₁₉	C₂³	1	3	yes	—
85	Q₁₆×C₁₉	C₂³	1	6	yes	—
86	SD₁₆×C₁₉	C₂³	1	6	yes	—
87	D₈×C₁₉	C₂³	1	1	yes	characteristic
88	M₁₆×C₁₉	C₂³	1	3	yes	—
89	C₂×C₈×C₁₉	C₂³	1	3	yes	—
90	#64528	C₂³	1	12	yes	—
91	#64528	C₂³	1	6	yes	—
92	#64527	C₂³	1	6	yes	—
93	#64526	C₂³	1	6	yes	—
94	#64525	C₂³	1	2	yes	—
95	#64520	C₂³	1	3	yes	—
96	C₂×C₁₉⋊C₈	C₂³	1	3	yes	—
97	Dic₇₆	C₂³	1	3	yes	—
98	Dic₇₆	C₂³	1	6	yes	—
99	D₁₅₂	C₂³	1	1	yes	characteristic
100	#64516	C₂³	1	6	yes	—
101	#64515	C₂³	1	6	yes	—
102	C₈×D₁₉	C₂³	1	1	yes	characteristic
103	C₈×D₁₉	C₂³	1	3	yes	—
104	C₈×D₁₉	C₂³	1	3	yes	—
105	C₈×D₁₉	C₂³	1	3	yes	—
106	C₂²×D₁₉	—	2	2	no	—
107	C₂²×D₁₉	C₂×D₄	1	3	yes	—
108	Q₈×C₁₉	—	2	6	no	—
109	Q₈×C₁₉	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₁₉	C₂⁴	1	6	yes	—
111	D₄×C₁₉	—	2	6	no	—
112	D₄×C₁₉	C₂×D₄	1	3	yes	—
113	D₄×C₁₉	C₂⁴	1	2	yes	—
114	C₂×C₄×C₁₉	—	2	6	no	—
115	C₂×C₄×C₁₉	—	2	6	no	—
116	C₂×C₄×C₁₉	C₂⁴	1	3	yes	—
117	C₂²⋊D₁₉	—	2	6	no	—
118	C₂²⋊D₁₉	—	2	6	no	—
119	C₂²⋊D₁₉	—	2	2	no	—
120	C₂²⋊D₁₉	C₂×D₄	1	6	yes	—
121	C₂×Dic₁₉	—	2	6	no	—
122	C₂×Dic₁₉	—	2	6	no	—
123	C₂×Dic₁₉	C₂×D₄	1	3	yes	—
124	D₇₆	—	2	6	no	—
125	D₇₆	C₂×D₄	1	3	yes	—
126	D₇₆	C₂⁴	1	2	yes	—
127	C₄×D₁₉	—	2	6	no	—
128	C₄×D₁₉	—	2	12	no	—
129	C₄×D₁₉	—	2	6	no	—
130	C₄×D₁₉	—	2	6	no	—
131	C₄×D₁₉	C₂×D₄	1	3	yes	—
132	C₄×D₁₉	C₂×D₄	1	3	yes	—
133	C₄×D₁₉	C₂⁴	1	1	yes	characteristic
134	C₄×D₁₉	C₂⁴	1	3	yes	—
135	Dic₃₈	—	2	6	no	—
136	Dic₃₈	—	2	6	no	—
137	Dic₃₈	—	2	12	no	—
138	Dic₃₈	C₂×D₄	1	3	yes	—
139	Dic₃₈	C₂⁴	1	6	yes	—
140	C₈×C₁₉	C₂⁴	1	1	yes	characteristic
141	C₈×C₁₉	C₂⁴	1	3	yes	—
142	C₁₉⋊C₈	C₂⁴	1	1	yes	characteristic
143	C₁₉⋊C₈	C₂⁴	1	3	yes	—
144	C₂×#578	—	19	1	no	maximal, Sylow
145	C₂²×C₁₉	—	2	2	no	—
146	C₂²×C₁₉	C₂²×D₄	1	3	yes	—
147	C₂×D₁₉	—	2	6	no	—
148	C₂×D₁₉	—	2	2	no	—
149	C₂×D₁₉	—	4	4	no	—
150	C₂×D₁₉	C₂²×D₄	1	3	yes	—
151	C₂×D₁₉	C₂²×D₄	1	1	yes	characteristic
152	C₄×C₁₉	—	2	6	no	—
153	C₄×C₁₉	C₂²×D₄	1	3	yes	—
154	C₄×C₁₉	C₂⁵	1	1	yes	characteristic
155	Dic₁₉	—	2	6	no	—
156	Dic₁₉	C₂²×D₄	1	1	yes	characteristic
157	Dic₁₉	C₂²×D₄	1	3	yes	—
158	C₂×2₋¹⁺⁴	—	19	2	no	—
159	#578	—	19	2	no	—
160	#578	—	19	6	no	—
161	#578	—	19	3	no	—
162	#578	—	19	3	no	—
163	#578	—	19	1	no	—
164	#578	D₁₉	1	1	yes	characteristic, direct product factor
165	C₂×#137	—	19	6	no	—
166	C₂×#135	—	19	3	no	—
167	C₂²×Q₁₆	—	19	3	no	—
168	C₂×#131	—	19	1	no	—
169	C₂×C₁₉	—	2	3	no	—
170	C₂×C₁₉	—	4	2	no	—
171	C₂×C₁₉	C₂³×D₄	1	1	yes	characteristic
172	D₁₉	—	2	3	no	—
173	D₁₉	—	4	2	no	—
174	D₁₉	#578	1	2	yes	direct product factor
175	2₋¹⁺⁴	—	19	6	no	—
176	2₋¹⁺⁴	—	19	2	no	—
177	2₋¹⁺⁴	—	19	6	no	—
178	2₋¹⁺⁴	—	38	2	no	—
179	2₋¹⁺⁴	—	38	6	no	—
180	2₋¹⁺⁴	C₂×D₁₉	1	2	yes	complemented
181	C₂×#41	—	19	1	no	—
182	C₂×#41	—	19	6	no	—
183	C₂×#41	—	38	6	no	—
184	C₂²×Q₈	—	19	6	no	—
185	C₂²×Q₈	—	38	2	no	—
186	#137	—	19	6	no	—
187	#137	—	19	6	no	—
188	#137	—	19	12	no	—
189	#137	—	19	12	no	—
190	#137	—	19	6	no	—
191	#137	C₂×D₁₉	1	6	yes	complemented
192	#135	—	19	6	no	—
193	#135	—	19	3	no	—
194	#135	—	19	6	no	—
195	#135	—	19	3	no	—
196	#135	—	19	3	no	—
197	#135	C₂×D₁₉	1	3	yes	complemented
198	C₂×Q₁₆	—	19	6	no	—
199	C₂×Q₁₆	—	19	3	no	—
200	C₂×Q₁₆	—	19	6	no	—
201	C₂×Q₁₆	—	19	6	no	—
202	C₂×Q₁₆	—	19	6	no	—
203	C₂×Q₁₆	—	19	3	no	—
204	C₂×Q₁₆	C₂×D₁₉	1	3	yes	complemented
205	C₂×SD₁₆	—	19	6	no	—
206	C₂×D₈	—	19	1	no	—
207	#131	—	19	1	no	—
208	#131	—	19	3	no	—
209	#131	—	19	3	no	—
210	#131	C₂×D₁₉	1	1	yes	characteristic, complemented
211	C₂×M₁₆	—	19	3	no	—
212	C₂²×C₈	—	19	3	no	—
213	C₁₉	C₂×#578	1	1	yes	characteristic, complemented, Sylow
214	#41	—	2	6	no	—
215	#41	—	19	1	no	—
216	#41	—	19	3	no	—
217	#41	—	19	3	no	—
218	#41	—	19	6	no	—
219	#41	—	19	6	no	—
220	#41	—	19	6	no	—
221	#41	—	38	6	no	—
222	#41	—	38	12	no	—
223	#41	—	38	6	no	—
224	#41	—	38	6	no	—
225	#41	—	38	12	no	—
226	#41	—	38	6	no	—
227	#41	—	38	6	no	—
228	#41	C₂²×D₁₉	1	6	yes	—
229	#41	C₂²×D₁₉	1	1	yes	characteristic
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	19	1	no	—
232	C₂×Q₈	—	19	6	no	—
233	C₂×Q₈	—	19	12	no	—
234	C₂×Q₈	—	19	6	no	—
235	C₂×Q₈	—	38	6	no	—
236	C₂×Q₈	—	38	6	no	—
237	C₂×Q₈	—	38	6	no	—
238	C₂×Q₈	—	38	6	no	—
239	C₂×Q₈	—	38	6	no	—
240	C₂×Q₈	—	38	2	no	—
241	C₂×Q₈	C₂²×D₁₉	1	6	yes	—
242	C₂×D₄	—	19	3	no	—
243	C₂×D₄	—	19	2	no	—
244	C₂×D₄	—	38	6	no	—
245	C₂²×C₄	—	19	3	no	—
246	C₂²×C₄	—	38	6	no	—
247	C₂²×C₄	—	38	6	no	—
248	Q₁₆	—	19	6	no	—
249	Q₁₆	—	19	12	no	—
250	Q₁₆	—	19	3	no	—
251	Q₁₆	—	19	6	no	—
252	Q₁₆	C₂²×D₁₉	1	3	yes	—
253	Q₁₆	C₂²×D₁₉	1	6	yes	—
254	SD₁₆	—	19	6	no	—
255	SD₁₆	—	19	6	no	—
256	SD₁₆	—	19	6	no	—
257	SD₁₆	C₂²×D₁₉	1	6	yes	—
258	D₈	—	19	2	no	—
259	D₈	—	19	1	no	—
260	D₈	C₂²×D₁₉	1	1	yes	characteristic
261	M₁₆	—	19	3	no	—
262	M₁₆	—	19	6	no	—
263	M₁₆	C₂²×D₁₉	1	3	yes	—
264	C₂×C₈	—	19	3	no	—
265	C₂×C₈	—	19	1	no	—
266	C₂×C₈	—	19	3	no	—
267	C₂×C₈	—	19	3	no	—
268	C₂×C₈	—	19	3	no	—
269	C₂×C₈	C₂²×D₁₉	1	3	yes	—
270	C₂³	—	19	3	no	—
271	C₂³	—	38	2	no	—
272	Q₈	—	2	6	no	—
273	Q₈	—	19	3	no	—
274	Q₈	—	19	6	no	—
275	Q₈	—	38	6	no	—
276	Q₈	—	38	12	no	—
277	Q₈	—	38	6	no	—
278	Q₈	D₄×D₁₉	1	1	yes	characteristic
279	Q₈	C₂³×D₁₉	1	6	yes	—
280	D₄	—	2	6	no	—
281	D₄	—	19	6	no	—
282	D₄	—	19	3	no	—
283	D₄	—	19	2	no	—
284	D₄	—	38	6	no	—
285	D₄	—	38	2	no	—
286	D₄	—	38	6	no	—
287	D₄	—	38	6	no	—
288	D₄	D₄×D₁₉	1	3	yes	—
289	D₄	C₂³×D₁₉	1	2	yes	—
290	C₂×C₄	—	2	6	no	—
291	C₂×C₄	—	2	6	no	—
292	C₂×C₄	—	19	3	no	—
293	C₂×C₄	—	19	3	no	—
294	C₂×C₄	—	19	3	no	—
295	C₂×C₄	—	19	1	no	—
296	C₂×C₄	—	19	3	no	—
297	C₂×C₄	—	38	6	no	—
298	C₂×C₄	—	38	6	no	—
299	C₂×C₄	—	38	12	no	—
300	C₂×C₄	—	38	6	no	—
301	C₂×C₄	—	38	6	no	—
302	C₂×C₄	—	38	6	no	—
303	C₂×C₄	C₂³×D₁₉	1	3	yes	—
304	C₈	—	19	1	no	—
305	C₈	—	19	3	no	—
306	C₈	C₂³×D₁₉	1	1	yes	characteristic
307	C₈	C₂³×D₁₉	1	3	yes	—
308	C₂²	—	2	2	no	—
309	C₂²	—	19	1	no	—
310	C₂²	—	19	3	no	—
311	C₂²	—	38	6	no	—
312	C₂²	—	38	2	no	—
313	C₂²	—	76	4	no	—
314	C₂²	C₂×D₄×D₁₉	1	3	yes	—
315	C₄	—	2	6	no	—
316	C₄	—	19	1	no	—
317	C₄	—	19	3	no	—
318	C₄	—	38	6	no	—
319	C₄	C₂×D₄×D₁₉	1	3	yes	—
320	C₄	C₂⁴×D₁₉	1	1	yes	characteristic
321	C₂	—	2	3	no	—
322	C₂	—	4	2	no	—
323	C₂	—	19	2	no	—
324	C₂	—	38	3	no	—
325	C₂	—	76	2	no	—
326	C₂	C₂²×D₄×D₁₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₄×D₁₉
commutator, cosocle	C₄×C₁₉	C₂⁵
Frattini	C₄	C₂⁴×D₁₉
Fitting	C₁₉×#578	C₂
socle	C₂×C₁₉	C₂³×D₄
hypercenter	#578	D₁₉
hypocenter	C₁₉	C₂×#578
LCS	G ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂ ◁ #41 ◁ #578

Complements

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

N	H ≅ G/N	type
1. 2₋¹⁺⁴×D₁₉	C₂	semidirect
2. #11927361	C₂	semidirect
3. #11927360	C₂	semidirect
4. #11927359	C₂	semidirect
5. #11927358	C₂	semidirect
6. #11927357	C₂	semidirect
7. #137×D₁₉	C₂	semidirect
8. #135×D₁₉	C₂	semidirect
9. C₂×Q₁₆×D₁₉	C₂	semidirect
10. #131×D₁₉	C₂	semidirect
11. C₁₉×#578	C₂	semidirect
12. C₁₉×2₋¹⁺⁴	C₂²	semidirect
14. #10597436	C₂²	semidirect
15. #10597436	C₂²	semidirect
20. #10597433	C₂²	semidirect
23. C₁₉×#137	C₂²	semidirect
24. C₁₉×#135	C₂²	semidirect
25. C₂×Q₁₆×C₁₉	C₂²	semidirect
26. C₁₉×#131	C₂²	semidirect
27. #10597397	C₂²	semidirect
28. #10597396	C₂²	semidirect
29. #10597394	C₂²	semidirect
30. C₂×#64528	C₂²	semidirect
31. #10597388	C₂²	semidirect
32. #10597365	C₂²	semidirect
33. #10597364	C₂²	semidirect
36. Q₁₆×D₁₉	C₂²	semidirect
37. Q₁₆×D₁₉	C₂²	semidirect
38. #10597362	C₂²	semidirect
39. #10597361	C₂²	semidirect
41. #10597358	C₂²	semidirect
43. #10597355	C₂²	semidirect
44. #10597353	C₂²	semidirect
46. C₂×Dic₇₆	C₂²	semidirect
47. #10597350	C₂²	semidirect
48. #10597347	C₂²	semidirect
164. #578	D₁₉	direct (N × H)
174. D₁₉	#578	direct (N × H)
180. 2₋¹⁺⁴	C₂×D₁₉	semidirect
191. #137	C₂×D₁₉	semidirect
197. #135	C₂×D₁₉	semidirect
204. C₂×Q₁₆	C₂×D₁₉	semidirect
210. #131	C₂×D₁₉	semidirect
213. C₁₉	C₂×#578	semidirect

Maximal subgroups

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

subgroup	index
2₋¹⁺⁴×D₁₉	2
#11927361	2
#11927360	2
#11927359	2
#11927358	2
#11927357	2
#137×D₁₉	2
#135×D₁₉	2
C₂×Q₁₆×D₁₉	2
#131×D₁₉	2
C₁₉×#578	2
C₂×#578	19

Subgroups — D19×#578 #14578397

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₉×#578 #14578397