Subgroups — D₁₉×#559 #14577219

Order 2432.

1240 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927280	C₂	1	1	yes	maximal, characteristic, complemented
2	#11927279	C₂	1	2	yes	maximal, complemented
3	#11927278	C₂	1	4	yes	maximal, complemented
4	#11927277	C₂	1	4	yes	maximal, complemented
5	#11927276	C₂	1	4	yes	maximal, complemented
6	#127×D₁₉	C₂	1	1	yes	maximal, characteristic
7	#124×D₁₉	C₂	1	4	yes	maximal, complemented
8	#123×D₁₉	C₂	1	4	yes	maximal, complemented
9	C₄×Q₈×D₁₉	C₂	1	2	yes	maximal, complemented
10	C₄×D₄×D₁₉	C₂	1	4	yes	maximal, complemented
11	C₁₉×#559	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₁₉	C₂²	1	2	yes	—
13	C₂×D₄×D₁₉	C₂²	1	4	yes	—
14	C₂×D₄×D₁₉	C₂²	1	2	yes	—
15	C₂²×C₄×D₁₉	C₂²	1	4	yes	—
16	C₁₉×#127	C₂²	1	1	yes	characteristic
17	C₁₉×#124	C₂²	1	4	yes	complemented
18	C₁₉×#123	C₂²	1	4	yes	complemented
19	C₄×Q₈×C₁₉	C₂²	1	2	yes	complemented
20	C₄×D₄×C₁₉	C₂²	1	4	yes	complemented
21	#10597393	C₂²	1	4	yes	complemented
22	Q₈×Dic₁₉	C₂²	1	2	yes	complemented
23	#10597386	C₂²	1	4	yes	—
24	#10597386	C₂²	1	2	yes	—
25	#10597382	C₂²	1	4	yes	complemented
26	#10597381	C₂²	1	4	yes	complemented
27	D₄×Dic₁₉	C₂²	1	4	yes	complemented
28	#10597375	C₂²	1	4	yes	complemented
29	C₄×C₂²⋊D₁₉	C₂²	1	8	yes	complemented
30	#10597340	C₂²	1	8	yes	complemented
31	#10597339	C₂²	1	4	yes	complemented
32	C₄⋊C₄×D₁₉	C₂²	1	4	yes	—
33	C₄⋊C₄×D₁₉	C₂²	1	2	yes	—
34	#10597333	C₂²	1	2	yes	complemented
35	#10597333	C₂²	1	4	yes	complemented
36	#10597332	C₂²	1	4	yes	complemented
37	#10597331	C₂²	1	8	yes	complemented
38	#10597331	C₂²	1	8	yes	complemented
39	#10597329	C₂²	1	8	yes	complemented
40	(C₂²⋊D₁₉)⋊C₄	C₂²	1	8	yes	complemented
41	C₂²⋊C₄×D₁₉	C₂²	1	4	yes	—
42	C₂²⋊C₄×D₁₉	C₂²	1	8	yes	—
43	#10597321	C₂²	1	4	yes	complemented
44	#10597320	C₂²	1	1	yes	characteristic
45	C₄×D₇₆	C₂²	1	4	yes	complemented
46	C₄²×D₁₉	C₂²	1	4	yes	—
47	C₄²×D₁₉	C₂²	1	1	yes	characteristic
48	C₄×Dic₃₈	C₂²	1	4	yes	complemented
49	C₄×Dic₃₈	C₂²	1	2	yes	complemented
50	C₂³×D₁₉	C₂³	1	4	yes	—
51	C₂×Q₈×C₁₉	C₂³	1	2	yes	—
52	C₂×D₄×C₁₉	C₂³	1	4	yes	—
53	C₂×D₄×C₁₉	C₂³	1	2	yes	—
54	C₂²×C₄×C₁₉	C₂³	1	4	yes	—
55	C₂×C₂²⋊D₁₉	C₂³	1	8	yes	—
56	C₂×C₂²⋊D₁₉	C₂³	1	4	yes	—
57	C₂²×Dic₁₉	C₂³	1	4	yes	—
58	Q₈×D₁₉	—	2	8	no	—
59	D₄×D₁₉	—	2	16	no	—
60	D₄×D₁₉	—	4	4	no	—
61	C₂×D₇₆	C₂³	1	4	yes	—
62	C₂×D₇₆	C₂³	1	2	yes	—
63	C₂×C₄×D₁₉	—	4	8	no	—
64	C₂×C₄×D₁₉	C₂³	1	4	yes	—
65	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₁₉	C₂³	1	4	yes	—
67	C₂×C₄×D₁₉	C₂³	1	8	yes	—
68	C₂×C₄×D₁₉	C₂³	1	2	yes	—
69	C₂×Dic₃₈	C₂³	1	2	yes	—
70	C₂×Dic₃₈	C₂³	1	4	yes	—
71	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
72	C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
73	C₂²⋊C₄×C₁₉	C₂³	1	8	yes	—
74	C₂²⋊C₄×C₁₉	C₂³	1	4	yes	—
75	C₄²×C₁₉	C₂³	1	4	yes	—
76	C₄²×C₁₉	C₂³	1	1	yes	characteristic
77	C₂²⋊Dic₁₉	C₂³	1	8	yes	—
78	C₂²⋊Dic₁₉	C₂³	1	4	yes	—
79	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
80	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
81	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
82	C₄⋊Dic₁₉	C₂³	1	2	yes	—
83	C₄⋊Dic₁₉	C₂³	1	4	yes	—
84	Dic₁₉⋊C₄	C₂³	1	4	yes	—
85	Dic₁₉⋊C₄	C₂³	1	4	yes	—
86	Dic₁₉⋊C₄	C₂³	1	4	yes	—
87	C₄×Dic₁₉	C₂³	1	4	yes	—
88	C₄×Dic₁₉	C₂³	1	2	yes	—
89	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
90	C₄×Dic₁₉	C₂³	1	4	yes	—
91	C₄×Dic₁₉	C₂³	1	4	yes	—
92	C₂³×C₁₉	C₂⁴	1	4	yes	—
93	C₂²×D₁₉	—	2	16	no	—
94	C₂²×D₁₉	—	2	8	no	—
95	C₂²×D₁₉	C₂⁴	1	4	yes	—
96	C₂²×D₁₉	C₂⁴	1	1	yes	characteristic
97	Q₈×C₁₉	—	2	4	no	—
98	D₄×C₁₉	—	2	8	no	—
99	D₄×C₁₉	—	4	2	no	—
100	C₂×C₄×C₁₉	—	4	4	no	—
101	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
102	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
103	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
104	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
105	C₂²⋊D₁₉	—	2	16	no	—
106	C₂²⋊D₁₉	—	4	4	no	—
107	C₂×Dic₁₉	—	4	4	no	—
108	C₂×Dic₁₉	C₂⁴	1	1	yes	characteristic
109	C₂×Dic₁₉	C₂⁴	1	4	yes	—
110	C₂×Dic₁₉	C₂⁴	1	4	yes	—
111	C₂×Dic₁₉	C₂⁴	1	2	yes	—
112	D₇₆	—	2	8	no	—
113	D₇₆	—	4	2	no	—
114	C₄×D₁₉	—	2	8	no	—
115	C₄×D₁₉	—	2	8	no	—
116	C₄×D₁₉	—	2	2	no	—
117	C₄×D₁₉	—	4	8	no	—
118	C₄×D₁₉	#41	1	8	yes	—
119	Dic₃₈	—	2	4	no	—
120	Dic₃₈	—	2	8	no	—
121	C₂×#559	—	19	1	no	maximal, Sylow
122	C₂²×C₁₉	—	2	8	no	—
123	C₂²×C₁₉	—	2	4	no	—
124	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
125	C₂×D₁₉	—	2	8	no	—
126	C₂×D₁₉	—	2	4	no	—
127	C₂×D₁₉	—	4	16	no	—
128	C₂×D₁₉	C₂×#41	1	4	yes	—
129	C₂×D₁₉	H(5,𝔽₂)	1	2	yes	—
130	C₄×C₁₉	—	2	4	no	—
131	C₄×C₁₉	—	2	4	no	—
132	C₄×C₁₉	—	2	1	no	—
133	C₄×C₁₉	C₂×#41	1	4	yes	—
134	Dic₁₉	—	2	4	no	—
135	Dic₁₉	—	2	4	no	—
136	Dic₁₉	—	2	1	no	—
137	Dic₁₉	C₂×#41	1	4	yes	—
138	#559	—	19	1	no	—
139	#559	—	19	4	no	—
140	#559	—	19	4	no	—
141	#559	—	19	2	no	—
142	#559	—	19	4	no	—
143	#559	D₁₉	1	1	yes	characteristic, direct product factor
144	C₂×#127	—	19	1	no	—
145	C₂×#124	—	19	4	no	—
146	C₂×#123	—	19	4	no	—
147	C₂×C₄×Q₈	—	19	2	no	—
148	C₂×C₄×D₄	—	19	4	no	—
149	C₂×C₁₉	—	4	4	no	—
150	C₂×C₁₉	C₂²×#41	1	2	yes	—
151	C₂×C₁₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
152	D₁₉	—	4	4	no	—
153	D₁₉	#559	1	4	yes	direct product factor
154	C₂²×Q₈	—	19	2	no	—
155	C₂²×D₄	—	19	4	no	—
156	C₂²×D₄	—	19	2	no	—
157	C₂³×C₄	—	19	4	no	—
158	#127	—	19	2	no	—
159	#127	—	19	4	no	—
160	#127	—	19	1	no	—
161	#127	C₂×D₁₉	1	1	yes	characteristic
162	#124	—	19	4	no	—
163	#124	—	19	4	no	—
164	#124	—	19	8	no	—
165	#124	—	19	8	no	—
166	#124	—	19	4	no	—
167	#124	C₂×D₁₉	1	4	yes	complemented
168	#123	—	19	4	no	—
169	#123	—	19	4	no	—
170	#123	—	19	8	no	—
171	#123	—	19	8	no	—
172	#123	—	19	4	no	—
173	#123	C₂×D₁₉	1	4	yes	complemented
174	C₄×Q₈	—	19	2	no	—
175	C₄×Q₈	—	19	4	no	—
176	C₄×Q₈	—	19	2	no	—
177	C₄×Q₈	—	19	2	no	—
178	C₄×Q₈	—	19	4	no	—
179	C₄×Q₈	C₂×D₁₉	1	2	yes	complemented
180	C₄×D₄	—	19	4	no	—
181	C₄×D₄	—	19	8	no	—
182	C₄×D₄	—	19	4	no	—
183	C₄×D₄	—	19	8	no	—
184	C₄×D₄	—	19	4	no	—
185	C₄×D₄	C₂×D₁₉	1	4	yes	complemented
186	C₂×C₄⋊C₄	—	19	2	no	—
187	C₂×C₄⋊C₄	—	19	4	no	—
188	C₂×C₂²⋊C₄	—	19	8	no	—
189	C₂×C₂²⋊C₄	—	19	4	no	—
190	C₂×C₄²	—	19	4	no	—
191	C₂×C₄²	—	19	1	no	—
192	C₁₉	C₂×#559	1	1	yes	characteristic, complemented, Sylow
193	C₂⁴	—	19	4	no	—
194	C₂×Q₈	—	19	4	no	—
195	C₂×Q₈	—	19	2	no	—
196	C₂×Q₈	—	38	8	no	—
197	C₂×Q₈	C₂²×D₁₉	1	2	yes	—
198	C₂×D₄	—	19	8	no	—
199	C₂×D₄	—	19	4	no	—
200	C₂×D₄	—	19	4	no	—
201	C₂×D₄	—	19	2	no	—
202	C₂×D₄	—	38	16	no	—
203	C₂×D₄	—	76	4	no	—
204	C₂×D₄	C₂²×D₁₉	1	4	yes	—
205	C₂×D₄	C₂²×D₁₉	1	2	yes	—
206	C₂²×C₄	—	19	2	no	—
207	C₂²×C₄	—	19	4	no	—
208	C₂²×C₄	—	19	4	no	—
209	C₂²×C₄	—	19	4	no	—
210	C₂²×C₄	—	19	1	no	—
211	C₂²×C₄	—	19	8	no	—
212	C₂²×C₄	—	76	8	no	—
213	C₂²×C₄	C₂²×D₁₉	1	4	yes	—
214	C₄⋊C₄	—	19	4	no	—
215	C₄⋊C₄	—	19	4	no	—
216	C₄⋊C₄	—	19	4	no	—
217	C₄⋊C₄	—	19	2	no	—
218	C₄⋊C₄	—	19	4	no	—
219	C₄⋊C₄	C₂²×D₁₉	1	2	yes	—
220	C₄⋊C₄	C₂²×D₁₉	1	4	yes	—
221	C₂²⋊C₄	—	19	8	no	—
222	C₂²⋊C₄	—	19	4	no	—
223	C₂²⋊C₄	—	19	8	no	—
224	C₂²⋊C₄	—	19	8	no	—
225	C₂²⋊C₄	—	19	8	no	—
226	C₂²⋊C₄	C₂²×D₁₉	1	8	yes	—
227	C₂²⋊C₄	C₂²×D₁₉	1	4	yes	—
228	C₄²	—	19	4	no	—
229	C₄²	—	19	4	no	—
230	C₄²	—	19	4	no	—
231	C₄²	—	19	1	no	—
232	C₄²	—	19	2	no	—
233	C₄²	C₂²×D₁₉	1	4	yes	—
234	C₄²	C₂²×D₁₉	1	1	yes	characteristic
235	C₂³	—	19	1	no	—
236	C₂³	—	19	4	no	—
237	C₂³	—	38	8	no	—
238	C₂³	—	38	16	no	—
239	C₂³	C₂³×D₁₉	1	4	yes	—
240	Q₈	—	2	4	no	—
241	Q₈	—	38	8	no	—
242	Q₈	—	38	4	no	—
243	D₄	—	2	8	no	—
244	D₄	—	4	2	no	—
245	D₄	—	38	16	no	—
246	D₄	—	38	8	no	—
247	D₄	—	76	2	no	—
248	D₄	—	76	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	19	2	no	—
251	C₂×C₄	—	19	8	no	—
252	C₂×C₄	—	19	4	no	—
253	C₂×C₄	—	19	4	no	—
254	C₂×C₄	—	19	1	no	—
255	C₂×C₄	—	38	8	no	—
256	C₂×C₄	—	38	2	no	—
257	C₂×C₄	—	38	8	no	—
258	C₂×C₄	—	76	8	no	—
259	C₂×C₄	—	76	4	no	—
260	C₂×C₄	C₂³×D₁₉	1	4	yes	—
261	C₂×C₄	C₂³×D₁₉	1	4	yes	—
262	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
263	C₂×C₄	C₂³×D₁₉	1	2	yes	—
264	C₂²	—	2	8	no	—
265	C₂²	—	2	4	no	—
266	C₂²	—	19	4	no	—
267	C₂²	—	19	2	no	—
268	C₂²	—	38	8	no	—
269	C₂²	—	38	4	no	—
270	C₂²	—	76	16	no	—
271	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
272	C₄	—	2	4	no	—
273	C₄	—	2	4	no	—
274	C₄	—	2	1	no	—
275	C₄	—	19	4	no	—
276	C₄	—	38	4	no	—
277	C₄	—	38	4	no	—
278	C₄	—	38	1	no	—
279	C₄	#41×D₁₉	1	4	yes	—
280	C₂	—	4	4	no	—
281	C₂	—	19	4	no	—
282	C₂	—	76	4	no	—
283	C₂	C₂×#41×D₁₉	1	2	yes	—
284	C₂	H(5,𝔽₂)×D₁₉	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₉
commutator, socle, cosocle	C₂²×C₁₉	C₂⁵
Fitting	C₁₉×#559	C₂
prime radical	#127×D₁₉	C₂
hypercenter	#559	D₁₉
hypocenter	C₁₉	C₂×#559
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ #559

Complements

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

N	H ≅ G/N	type
1. #11927280	C₂	semidirect
2. #11927279	C₂	semidirect
3. #11927278	C₂	semidirect
4. #11927277	C₂	semidirect
5. #11927276	C₂	semidirect
7. #124×D₁₉	C₂	semidirect
8. #123×D₁₉	C₂	semidirect
9. C₄×Q₈×D₁₉	C₂	semidirect
10. C₄×D₄×D₁₉	C₂	semidirect
11. C₁₉×#559	C₂	semidirect
17. C₁₉×#124	C₂²	semidirect
18. C₁₉×#123	C₂²	semidirect
19. C₄×Q₈×C₁₉	C₂²	semidirect
20. C₄×D₄×C₁₉	C₂²	semidirect
21. #10597393	C₂²	semidirect
22. Q₈×Dic₁₉	C₂²	semidirect
25. #10597382	C₂²	semidirect
26. #10597381	C₂²	semidirect
27. D₄×Dic₁₉	C₂²	semidirect
28. #10597375	C₂²	semidirect
29. C₄×C₂²⋊D₁₉	C₂²	semidirect
30. #10597340	C₂²	semidirect
31. #10597339	C₂²	semidirect
34. #10597333	C₂²	semidirect
35. #10597333	C₂²	semidirect
36. #10597332	C₂²	semidirect
37. #10597331	C₂²	semidirect
38. #10597331	C₂²	semidirect
39. #10597329	C₂²	semidirect
40. (C₂²⋊D₁₉)⋊C₄	C₂²	semidirect
43. #10597321	C₂²	semidirect
45. C₄×D₇₆	C₂²	semidirect
48. C₄×Dic₃₈	C₂²	semidirect
49. C₄×Dic₃₈	C₂²	semidirect
143. #559	D₁₉	direct (N × H)
153. D₁₉	#559	direct (N × H)
167. #124	C₂×D₁₉	semidirect
173. #123	C₂×D₁₉	semidirect
179. C₄×Q₈	C₂×D₁₉	semidirect
185. C₄×D₄	C₂×D₁₉	semidirect
192. C₁₉	C₂×#559	semidirect

Maximal subgroups

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

subgroup	index
#11927280	2
#11927279	2
#11927278	2
#11927277	2
#11927276	2
#127×D₁₉	2
#124×D₁₉	2
#123×D₁₉	2
C₄×Q₈×D₁₉	2
C₄×D₄×D₁₉	2
C₁₉×#559	2
C₂×#559	19

Subgroups — D19×#559 #14577219

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₉×#559 #14577219