Subgroups — D₄×SD₁₆×C₁₉ #14561138

Order 2432.

546 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×SD₁₆×C₁₉	C₂	1	2	yes	maximal, complemented
2	D₄×Q₈×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄²×C₁₉	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#465	C₂	1	2	yes	maximal, complemented
6	C₁₉×#461	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#460	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₉×#450	C₂	1	2	yes	maximal, complemented
9	C₁₉×#448	C₂	1	2	yes	maximal, complemented
10	C₄×SD₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₁₉	C₂²	1	2	yes	complemented
13	C₂²×D₄×C₁₉	—	2	1	no	—
14	C₂²×D₄×C₁₉	C₂²	1	2	yes	—
15	C₂×SD₁₆×C₁₉	—	2	8	no	—
16	C₂×SD₁₆×C₁₉	C₂²	1	2	yes	—
17	C₂×SD₁₆×C₁₉	C₂²	1	1	yes	characteristic
18	C₂×SD₁₆×C₁₉	C₂²	1	2	yes	complemented
19	C₂×SD₁₆×C₁₉	C₂²	1	2	yes	complemented
20	C₂×SD₁₆×C₁₉	C₂²	1	2	yes	complemented
21	C₂²×C₈×C₁₉	C₂²	1	2	yes	complemented
22	C₁₉×C₄⋊Q₈	—	2	1	no	—
23	C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
24	C₁₉×#127	C₂²	1	1	yes	characteristic
25	C₁₉×C₂²⋊Q₈	—	2	2	no	—
26	C₁₉×C₂²⋊Q₈	C₂²	1	2	yes	complemented
27	C₁₉×#121	—	2	1	no	—
28	C₁₉×#121	C₂²	1	2	yes	—
29	C₁₉×#120	—	2	2	no	—
30	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic, complemented
31	C₄×D₄×C₁₉	—	2	1	no	—
32	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
33	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
34	C₁₉×#106	C₂²	1	1	yes	characteristic, complemented
35	C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
36	C₁₉×#103	C₂²	1	1	yes	characteristic, complemented
37	C₁₉×#103	C₂²	1	2	yes	complemented
38	C₁₉×#102	C₂²	1	1	yes	characteristic
39	C₁₉×#102	C₂²	1	2	yes	complemented
40	C₁₉×C₂²⋊C₈	C₂²	1	2	yes	complemented
41	C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic, complemented
42	C₂⁴×C₁₉	—	2	2	no	—
43	C₂×Q₈×C₁₉	—	2	2	no	—
44	C₂×Q₈×C₁₉	—	2	4	no	—
45	C₂×Q₈×C₁₉	—	4	2	no	—
46	C₂×Q₈×C₁₉	C₂³	1	2	yes	—
47	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
48	C₂×D₄×C₁₉	—	2	4	no	—
49	C₂×D₄×C₁₉	—	2	2	no	—
50	C₂×D₄×C₁₉	—	2	2	no	—
51	C₂×D₄×C₁₉	—	2	1	no	—
52	C₂×D₄×C₁₉	—	2	1	no	—
53	C₂×D₄×C₁₉	—	4	2	no	—
54	C₂×D₄×C₁₉	—	4	4	no	—
55	C₂×D₄×C₁₉	D₄	1	1	yes	characteristic
56	C₂×D₄×C₁₉	C₂³	1	2	yes	—
57	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	—	2	2	no	—
59	C₂²×C₄×C₁₉	—	2	1	no	—
60	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
61	SD₁₆×C₁₉	—	2	4	no	—
62	SD₁₆×C₁₉	—	2	4	no	—
63	SD₁₆×C₁₉	—	2	4	no	—
64	SD₁₆×C₁₉	—	4	2	no	—
65	SD₁₆×C₁₉	D₄	1	4	yes	direct product factor
66	C₂×C₈×C₁₉	—	2	4	no	—
67	C₂×C₈×C₁₉	C₂³	1	2	yes	—
68	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₉	—	2	1	no	—
71	C₄⋊C₄×C₁₉	—	2	1	no	—
72	C₄⋊C₄×C₁₉	—	2	1	no	—
73	C₄⋊C₄×C₁₉	—	2	2	no	—
74	C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
75	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
77	C₂²⋊C₄×C₁₉	—	2	2	no	—
78	C₂²⋊C₄×C₁₉	—	2	1	no	—
79	C₂²⋊C₄×C₁₉	—	2	2	no	—
80	C₂²⋊C₄×C₁₉	D₄	1	2	yes	—
81	C₄²×C₁₉	—	2	1	no	—
82	C₄²×C₁₉	C₂³	1	1	yes	characteristic
83	C₂³×C₁₉	—	2	1	no	—
84	C₂³×C₁₉	—	2	2	no	—
85	C₂³×C₁₉	—	4	4	no	—
86	C₂³×C₁₉	—	4	4	no	—
87	C₂³×C₁₉	—	8	2	no	—
88	C₂³×C₁₉	C₂×D₄	1	2	yes	—
89	Q₈×C₁₉	—	2	1	no	—
90	Q₈×C₁₉	—	2	2	no	—
91	Q₈×C₁₉	—	2	2	no	—
92	Q₈×C₁₉	—	4	4	no	—
93	Q₈×C₁₉	C₂×D₄	1	2	yes	complemented
94	D₄×C₁₉	—	2	2	no	—
95	D₄×C₁₉	—	2	1	no	—
96	D₄×C₁₉	—	2	2	no	—
97	D₄×C₁₉	—	4	4	no	—
98	D₄×C₁₉	—	4	4	no	—
99	D₄×C₁₉	—	4	2	no	—
100	D₄×C₁₉	—	8	1	no	—
101	D₄×C₁₉	SD₁₆	1	4	yes	direct product factor
102	D₄×C₁₉	C₂×D₄	1	2	yes	—
103	C₂×C₄×C₁₉	—	2	1	no	—
104	C₂×C₄×C₁₉	—	2	2	no	—
105	C₂×C₄×C₁₉	—	2	2	no	—
106	C₂×C₄×C₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	—	2	1	no	—
108	C₂×C₄×C₁₉	—	2	2	no	—
109	C₂×C₄×C₁₉	—	4	4	no	—
110	C₂×C₄×C₁₉	—	4	2	no	—
111	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
113	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
115	C₈×C₁₉	—	2	2	no	—
116	C₈×C₁₉	—	2	1	no	—
117	C₈×C₁₉	C₂×D₄	1	2	yes	complemented
118	D₄×SD₁₆	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂²×C₁₉	—	2	2	no	—
120	C₂²×C₁₉	—	2	2	no	—
121	C₂²×C₁₉	—	2	2	no	—
122	C₂²×C₁₉	—	4	1	no	—
123	C₂²×C₁₉	—	4	1	no	—
124	C₂²×C₁₉	—	4	2	no	—
125	C₂²×C₁₉	—	4	2	no	—
126	C₂²×C₁₉	—	4	2	no	—
127	C₂²×C₁₉	—	8	8	no	—
128	C₂²×C₁₉	C₂×SD₁₆	1	4	yes	complemented
129	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
130	C₄×C₁₉	—	2	2	no	—
131	C₄×C₁₉	—	2	2	no	—
132	C₄×C₁₉	—	2	1	no	—
133	C₄×C₁₉	—	4	2	no	—
134	C₄×C₁₉	—	4	1	no	—
135	C₄×C₁₉	—	4	1	no	—
136	C₄×C₁₉	C₂×SD₁₆	1	2	yes	complemented
137	C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
138	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
139	C₂²×SD₁₆	C₂×C₁₉	1	2	yes	complemented
140	D₄×Q₈	C₂×C₁₉	1	1	yes	characteristic, complemented
141	D₄²	C₂×C₁₉	1	1	yes	characteristic
142	#492	C₂×C₁₉	1	1	yes	characteristic, complemented
143	#465	C₂×C₁₉	1	2	yes	complemented
144	#461	C₂×C₁₉	1	1	yes	characteristic, complemented
145	#460	C₂×C₁₉	1	1	yes	characteristic, complemented
146	#450	C₂×C₁₉	1	2	yes	complemented
147	#448	C₂×C₁₉	1	2	yes	complemented
148	C₄×SD₁₆	C₂×C₁₉	1	1	yes	characteristic, complemented
149	C₈×D₄	C₂×C₁₉	1	1	yes	characteristic, complemented
150	C₂×C₁₉	—	2	4	no	—
151	C₂×C₁₉	—	4	2	no	—
152	C₂×C₁₉	—	8	2	no	—
153	C₂×C₁₉	D₄²	1	1	yes	characteristic
154	C₂×C₁₉	C₂²×SD₁₆	1	1	yes	characteristic
155	C₂×C₁₉	#577	1	1	yes	characteristic
156	C₂²×Q₈	C₂²×C₁₉	1	2	yes	complemented
157	C₂²×D₄	—	2	1	no	—
158	C₂²×D₄	C₂²×C₁₉	1	2	yes	—
159	C₂×SD₁₆	—	2	8	no	—
160	C₂×SD₁₆	C₂²×C₁₉	1	2	yes	—
161	C₂×SD₁₆	C₂²×C₁₉	1	1	yes	characteristic
162	C₂×SD₁₆	C₂²×C₁₉	1	2	yes	complemented
163	C₂×SD₁₆	C₂²×C₁₉	1	2	yes	complemented
164	C₂×SD₁₆	C₂²×C₁₉	1	2	yes	complemented
165	C₂²×C₈	C₂²×C₁₉	1	2	yes	complemented
166	C₄⋊Q₈	—	2	1	no	—
167	C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic, complemented
168	#127	C₂²×C₁₉	1	1	yes	characteristic
169	C₂²⋊Q₈	—	2	2	no	—
170	C₂²⋊Q₈	C₂²×C₁₉	1	2	yes	complemented
171	#121	—	2	1	no	—
172	#121	C₂²×C₁₉	1	2	yes	—
173	#120	—	2	2	no	—
174	C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic, complemented
175	C₄×D₄	—	2	1	no	—
176	C₄×D₄	C₂²×C₁₉	1	1	yes	characteristic
177	C₄×D₄	C₂²×C₁₉	1	1	yes	characteristic
178	#106	C₂²×C₁₉	1	1	yes	characteristic, complemented
179	C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic, complemented
180	#103	C₂²×C₁₉	1	1	yes	characteristic, complemented
181	#103	C₂²×C₁₉	1	2	yes	complemented
182	#102	C₂²×C₁₉	1	1	yes	characteristic
183	#102	C₂²×C₁₉	1	2	yes	complemented
184	C₂²⋊C₈	C₂²×C₁₉	1	2	yes	complemented
185	C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic, complemented
186	C₁₉	D₄×SD₁₆	1	1	yes	characteristic, direct product factor, Sylow
187	C₂⁴	—	2	2	no	—
188	C₂×Q₈	—	2	2	no	—
189	C₂×Q₈	—	2	4	no	—
190	C₂×Q₈	—	4	2	no	—
191	C₂×Q₈	C₂³×C₁₉	1	2	yes	—
192	C₂×Q₈	C₂³×C₁₉	1	1	yes	characteristic
193	C₂×D₄	—	2	4	no	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	—	2	2	no	—
196	C₂×D₄	—	2	1	no	—
197	C₂×D₄	—	2	1	no	—
198	C₂×D₄	—	4	2	no	—
199	C₂×D₄	—	4	4	no	—
200	C₂×D₄	D₄×C₁₉	1	1	yes	characteristic
201	C₂×D₄	C₂³×C₁₉	1	2	yes	—
202	C₂×D₄	C₂³×C₁₉	1	1	yes	characteristic
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	C₂³×C₁₉	1	2	yes	—
206	SD₁₆	—	2	4	no	—
207	SD₁₆	—	2	4	no	—
208	SD₁₆	—	2	4	no	—
209	SD₁₆	—	4	2	no	—
210	SD₁₆	D₄×C₁₉	1	4	yes	direct product factor
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	C₂³×C₁₉	1	1	yes	characteristic
213	C₂×C₈	C₂³×C₁₉	1	2	yes	—
214	C₂×C₈	C₂³×C₁₉	1	1	yes	characteristic
215	C₄⋊C₄	—	2	1	no	—
216	C₄⋊C₄	—	2	1	no	—
217	C₄⋊C₄	—	2	1	no	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
220	C₄⋊C₄	C₂³×C₁₉	1	1	yes	characteristic
221	C₄⋊C₄	C₂³×C₁₉	1	1	yes	characteristic
222	C₂²⋊C₄	—	2	2	no	—
223	C₂²⋊C₄	—	2	1	no	—
224	C₂²⋊C₄	—	2	2	no	—
225	C₂²⋊C₄	D₄×C₁₉	1	2	yes	—
226	C₄²	—	2	1	no	—
227	C₄²	C₂³×C₁₉	1	1	yes	characteristic
228	C₂³	—	2	1	no	—
229	C₂³	—	2	2	no	—
230	C₂³	—	4	4	no	—
231	C₂³	—	4	4	no	—
232	C₂³	—	8	2	no	—
233	C₂³	C₂×D₄×C₁₉	1	2	yes	—
234	Q₈	—	2	1	no	—
235	Q₈	—	2	2	no	—
236	Q₈	—	2	2	no	—
237	Q₈	—	4	4	no	—
238	Q₈	C₂×D₄×C₁₉	1	2	yes	complemented
239	D₄	—	2	2	no	—
240	D₄	—	2	1	no	—
241	D₄	—	2	2	no	—
242	D₄	—	4	4	no	—
243	D₄	—	4	4	no	—
244	D₄	—	4	2	no	—
245	D₄	—	8	1	no	—
246	D₄	SD₁₆×C₁₉	1	4	yes	direct product factor
247	D₄	C₂×D₄×C₁₉	1	2	yes	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
257	C₂×C₄	C₂×D₄×C₁₉	1	2	yes	—
258	C₂×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
259	C₂×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₈	—	2	1	no	—
262	C₈	C₂×D₄×C₁₉	1	2	yes	complemented
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	2	no	—
271	C₂²	—	8	8	no	—
272	C₂²	C₂×SD₁₆×C₁₉	1	4	yes	complemented
273	C₂²	C₂²×D₄×C₁₉	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	1	no	—
277	C₄	—	4	2	no	—
278	C₄	—	4	1	no	—
279	C₄	—	4	1	no	—
280	C₄	C₂×SD₁₆×C₁₉	1	2	yes	complemented
281	C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
282	C₄	C₁₉×H(5,𝔽₂)	1	1	yes	characteristic
283	C₂	—	2	4	no	—
284	C₂	—	4	2	no	—
285	C₂	—	8	2	no	—
286	C₂	D₄²×C₁₉	1	1	yes	characteristic
287	C₂	C₂²×SD₁₆×C₁₉	1	1	yes	characteristic
288	C₂	C₁₉×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₉
prime radical	D₄²×C₁₉	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₄×D₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×SD₁₆×C₁₉	C₂	semidirect
2. D₄×Q₈×C₁₉	C₂	semidirect
4. C₁₉×#492	C₂	semidirect
5. C₁₉×#465	C₂	semidirect
6. C₁₉×#461	C₂	semidirect
7. C₁₉×#460	C₂	semidirect
8. C₁₉×#450	C₂	semidirect
9. C₁₉×#448	C₂	semidirect
10. C₄×SD₁₆×C₁₉	C₂	semidirect
11. C₈×D₄×C₁₉	C₂	semidirect
12. C₂²×Q₈×C₁₉	C₂²	semidirect
18. C₂×SD₁₆×C₁₉	C₂²	semidirect
19. C₂×SD₁₆×C₁₉	C₂²	semidirect
20. C₂×SD₁₆×C₁₉	C₂²	semidirect
21. C₂²×C₈×C₁₉	C₂²	semidirect
23. C₁₉×C₄⋊Q₈	C₂²	semidirect
26. C₁₉×C₂²⋊Q₈	C₂²	semidirect
30. C₄×Q₈×C₁₉	C₂²	semidirect
34. C₁₉×#106	C₂²	semidirect
35. C₁₉×C₄⋊C₈	C₂²	semidirect
36. C₁₉×#103	C₂²	semidirect
37. C₁₉×#103	C₂²	semidirect
39. C₁₉×#102	C₂²	semidirect
40. C₁₉×C₂²⋊C₈	C₂²	semidirect
41. C₄×C₈×C₁₉	C₂²	semidirect
65. SD₁₆×C₁₉	D₄	direct (N × H)
93. Q₈×C₁₉	C₂×D₄	semidirect
101. D₄×C₁₉	SD₁₆	direct (N × H)
117. C₈×C₁₉	C₂×D₄	semidirect
118. D₄×SD₁₆	C₁₉	direct (N × H)
128. C₂²×C₁₉	C₂×SD₁₆	semidirect
136. C₄×C₁₉	C₂×SD₁₆	semidirect
139. C₂²×SD₁₆	C₂×C₁₉	semidirect
140. D₄×Q₈	C₂×C₁₉	semidirect
142. #492	C₂×C₁₉	semidirect
143. #465	C₂×C₁₉	semidirect
144. #461	C₂×C₁₉	semidirect
145. #460	C₂×C₁₉	semidirect
146. #450	C₂×C₁₉	semidirect
147. #448	C₂×C₁₉	semidirect
148. C₄×SD₁₆	C₂×C₁₉	semidirect
149. C₈×D₄	C₂×C₁₉	semidirect
156. C₂²×Q₈	C₂²×C₁₉	semidirect
162. C₂×SD₁₆	C₂²×C₁₉	semidirect
163. C₂×SD₁₆	C₂²×C₁₉	semidirect
164. C₂×SD₁₆	C₂²×C₁₉	semidirect
165. C₂²×C₈	C₂²×C₁₉	semidirect
167. C₄⋊Q₈	C₂²×C₁₉	semidirect
170. C₂²⋊Q₈	C₂²×C₁₉	semidirect
174. C₄×Q₈	C₂²×C₁₉	semidirect
178. #106	C₂²×C₁₉	semidirect
179. C₄⋊C₈	C₂²×C₁₉	semidirect
180. #103	C₂²×C₁₉	semidirect
181. #103	C₂²×C₁₉	semidirect
183. #102	C₂²×C₁₉	semidirect
184. C₂²⋊C₈	C₂²×C₁₉	semidirect
185. C₄×C₈	C₂²×C₁₉	semidirect
186. C₁₉	D₄×SD₁₆	direct (N × H)
210. SD₁₆	D₄×C₁₉	direct (N × H)
238. Q₈	C₂×D₄×C₁₉	semidirect
246. D₄	SD₁₆×C₁₉	direct (N × H)
262. C₈	C₂×D₄×C₁₉	semidirect
272. C₂²	C₂×SD₁₆×C₁₉	semidirect
280. C₄	C₂×SD₁₆×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×SD₁₆×C₁₉	2
D₄×Q₈×C₁₉	2
D₄²×C₁₉	2
C₁₉×#492	2
C₁₉×#465	2
C₁₉×#461	2
C₁₉×#460	2
C₁₉×#450	2
C₁₉×#448	2
C₄×SD₁₆×C₁₉	2
C₈×D₄×C₁₉	2
D₄×SD₁₆	19

Subgroups — D4×SD16×C19 #14561138

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×SD₁₆×C₁₉ #14561138