Subgroups — D₁₉×#550 #14577111

Order 2432.

1736 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×D₁₉	C₂	1	2	yes	maximal, complemented
2	#11927216	C₂	1	1	yes	maximal, characteristic, complemented
3	#11927215	C₂	1	3	yes	maximal, complemented
4	#11927214	C₂	1	3	yes	maximal, complemented
5	#11927213	C₂	1	2	yes	maximal, complemented
6	#11927212	C₂	1	6	yes	maximal, complemented
7	#127×D₁₉	C₂	1	3	yes	maximal, complemented
8	#121×D₁₉	C₂	1	6	yes	maximal, complemented
9	C₄×Q₈×D₁₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₄×D₁₉	C₂	1	3	yes	maximal, complemented
11	C₁₉×#550	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₁₉	C₂²	1	2	yes	complemented
13	#41×D₁₉	—	2	16	no	—
14	C₂×#64545	C₂²	1	2	yes	complemented
15	C₂×Q₈×D₁₉	C₂²	1	1	yes	characteristic
16	C₂×#64543	C₂²	1	6	yes	complemented
17	C₂×D₄×D₁₉	C₂²	1	3	yes	—
18	C₂×D₄×D₁₉	C₂²	1	6	yes	—
19	C₂×D₄×D₁₉	C₂²	1	6	yes	—
20	C₂×#64541	C₂²	1	6	yes	complemented
21	C₂²×C₄×D₁₉	C₂²	1	6	yes	—
22	C₁₉×#127	C₂²	1	3	yes	complemented
23	C₁₉×#121	C₂²	1	6	yes	complemented
24	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic, complemented
25	C₄×D₄×C₁₉	C₂²	1	3	yes	complemented
26	Q₈×Dic₁₉	C₂²	1	1	yes	characteristic, complemented
27	#10597386	C₂²	1	6	yes	complemented
28	#10597386	C₂²	1	6	yes	complemented
29	#10597386	C₂²	1	6	yes	complemented
30	#10597385	C₂²	1	12	yes	complemented
31	#10597384	C₂²	1	6	yes	complemented
32	D₄×Dic₁₉	C₂²	1	3	yes	complemented
33	#10597376	C₂²	1	6	yes	complemented
34	C₄×C₂²⋊D₁₉	C₂²	1	6	yes	complemented
35	#10597341	C₂²	1	6	yes	complemented
36	#10597339	C₂²	1	3	yes	complemented
37	C₄⋊C₄×D₁₉	C₂²	1	1	yes	characteristic
38	C₄⋊C₄×D₁₉	C₂²	1	3	yes	—
39	#10597333	C₂²	1	3	yes	complemented
40	#10597330	C₂²	1	12	yes	complemented
41	(C₂²⋊D₁₉)⋊C₄	C₂²	1	6	yes	complemented
42	C₂²⋊C₄×D₁₉	C₂²	1	6	yes	—
43	#10597320	C₂²	1	3	yes	complemented
44	C₄×D₇₆	C₂²	1	3	yes	complemented
45	C₄²×D₁₉	C₂²	1	3	yes	—
46	C₄×Dic₃₈	C₂²	1	3	yes	complemented
47	C₂³×D₁₉	C₂³	1	6	yes	—
48	#41×C₁₉	—	2	8	no	—
49	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₁₉	C₂³	1	3	yes	—
51	C₂×D₄×C₁₉	C₂³	1	6	yes	—
52	C₂×D₄×C₁₉	C₂³	1	6	yes	—
53	C₂²×C₄×C₁₉	C₂³	1	6	yes	—
54	C₂×C₂²⋊D₁₉	C₂³	1	12	yes	—
55	C₂×C₂²⋊D₁₉	C₂³	1	6	yes	—
56	C₂×C₂²⋊D₁₉	C₂³	1	12	yes	—
57	C₂²×Dic₁₉	C₂³	1	6	yes	—
58	#64545	—	2	8	no	—
59	Q₈×D₁₉	D₄	1	8	yes	complemented
60	#64543	—	2	24	no	—
61	D₄×D₁₉	—	2	24	no	—
62	D₄×D₁₉	—	2	12	no	—
63	D₄×D₁₉	—	4	12	no	—
64	#64541	—	2	24	no	—
65	C₂×D₇₆	C₂³	1	6	yes	—
66	C₂×D₇₆	C₂³	1	3	yes	—
67	C₂×D₇₆	C₂³	1	6	yes	—
68	C₂×C₄×D₁₉	—	2	24	no	—
69	C₂×C₄×D₁₉	C₂³	1	6	yes	—
70	C₂×C₄×D₁₉	C₂³	1	2	yes	—
71	C₂×C₄×D₁₉	C₂³	1	6	yes	—
72	C₂×C₄×D₁₉	C₂³	1	3	yes	—
73	C₂×C₄×D₁₉	C₂³	1	3	yes	—
74	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
75	C₂×Dic₃₈	C₂³	1	3	yes	—
76	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₁₉	C₂³	1	3	yes	—
78	C₂²⋊C₄×C₁₉	C₂³	1	6	yes	—
79	C₄²×C₁₉	C₂³	1	3	yes	—
80	C₂²⋊Dic₁₉	C₂³	1	6	yes	—
81	(C₂×D₁₉)⋊C₄	C₂³	1	6	yes	—
82	(C₂×D₁₉)⋊C₄	C₂³	1	6	yes	—
83	C₄⋊Dic₁₉	C₂³	1	3	yes	—
84	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
85	Dic₁₉⋊C₄	C₂³	1	6	yes	—
86	Dic₁₉⋊C₄	C₂³	1	2	yes	—
87	C₄×Dic₁₉	C₂³	1	3	yes	—
88	C₄×Dic₁₉	C₂³	1	3	yes	—
89	C₄×Dic₁₉	C₂³	1	3	yes	—
90	C₂³×C₁₉	C₂⁴	1	6	yes	—
91	C₂²×D₁₉	—	2	12	no	—
92	C₂²×D₁₉	—	2	12	no	—
93	C₂²×D₁₉	—	2	12	no	—
94	C₂²×D₁₉	C₂⁴	1	6	yes	—
95	C₂²×D₁₉	C₂⁴	1	1	yes	characteristic
96	Q₈×C₁₉	C₂×D₄	1	4	yes	complemented
97	D₄×C₁₉	—	2	12	no	—
98	D₄×C₁₉	—	2	6	no	—
99	D₄×C₁₉	—	4	6	no	—
100	C₂×C₄×C₁₉	—	2	12	no	—
101	C₂×C₄×C₁₉	C₂⁴	1	3	yes	—
102	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
103	C₂×C₄×C₁₉	C₂⁴	1	3	yes	—
104	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
105	C₂²⋊D₁₉	—	2	24	no	—
106	C₂²⋊D₁₉	—	2	12	no	—
107	C₂²⋊D₁₉	—	4	12	no	—
108	C₂×Dic₁₉	—	2	12	no	—
109	C₂×Dic₁₉	C₂⁴	1	3	yes	—
110	C₂×Dic₁₉	C₂⁴	1	2	yes	—
111	C₂×Dic₁₉	C₂⁴	1	3	yes	—
112	C₂×Dic₁₉	C₂⁴	1	1	yes	characteristic
113	D₇₆	—	2	12	no	—
114	D₇₆	—	2	6	no	—
115	D₇₆	—	4	6	no	—
116	C₄×D₁₉	—	2	4	no	—
117	C₄×D₁₉	—	2	6	no	—
118	C₄×D₁₉	—	2	12	no	—
119	C₄×D₁₉	—	2	12	no	—
120	C₄×D₁₉	C₂×D₄	1	12	yes	—
121	C₄×D₁₉	#41	1	4	yes	complemented
122	Dic₃₈	C₂×D₄	1	12	yes	complemented
123	C₂×#550	—	19	1	no	maximal, Sylow
124	C₂²×C₁₉	—	2	6	no	—
125	C₂²×C₁₉	—	2	6	no	—
126	C₂²×C₁₉	—	2	6	no	—
127	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
128	C₂×D₁₉	—	2	6	no	—
129	C₂×D₁₉	—	2	6	no	—
130	C₂×D₁₉	—	2	6	no	—
131	C₂×D₁₉	—	4	24	no	—
132	C₂×D₁₉	C₂²×D₄	1	2	yes	—
133	C₂×D₁₉	C₂×#41	1	2	yes	—
134	C₂×D₁₉	H(5,𝔽₂)	1	2	yes	—
135	C₄×C₁₉	—	2	2	no	—
136	C₄×C₁₉	—	2	3	no	—
137	C₄×C₁₉	C₂²×D₄	1	6	yes	—
138	C₄×C₁₉	C₂×#41	1	2	yes	complemented
139	Dic₁₉	—	2	2	no	—
140	Dic₁₉	—	2	3	no	—
141	Dic₁₉	C₂²×D₄	1	6	yes	—
142	Dic₁₉	C₂×#41	1	2	yes	complemented
143	C₂²×#41	—	19	2	no	—
144	#550	—	19	2	no	—
145	#550	—	19	3	no	—
146	#550	—	19	6	no	—
147	#550	—	19	1	no	—
148	#550	—	19	3	no	—
149	#550	D₁₉	1	1	yes	characteristic, direct product factor
150	C₂×#127	—	19	3	no	—
151	C₂×#121	—	19	6	no	—
152	C₂×C₄×Q₈	—	19	1	no	—
153	C₂×C₄×D₄	—	19	3	no	—
154	C₂×C₁₉	—	4	6	no	—
155	C₂×C₁₉	C₂³×D₄	1	1	yes	characteristic
156	C₂×C₁₉	C₂²×#41	1	1	yes	characteristic
157	C₂×C₁₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
158	D₁₉	—	4	6	no	—
159	D₁₉	#550	1	4	yes	direct product factor
160	C₂×#41	—	19	2	no	—
161	C₂×#41	—	19	6	no	—
162	C₂×#41	—	19	6	no	—
163	C₂×#41	—	38	16	no	—
164	C₂×#41	C₂×D₁₉	1	2	yes	complemented
165	C₂²×Q₈	—	19	1	no	—
166	C₂²×D₄	—	19	3	no	—
167	C₂²×D₄	—	19	6	no	—
168	C₂²×D₄	—	19	6	no	—
169	C₂³×C₄	—	19	6	no	—
170	#127	—	19	6	no	—
171	#127	—	19	6	no	—
172	#127	—	19	6	no	—
173	#127	—	19	3	no	—
174	#127	C₂×D₁₉	1	3	yes	complemented
175	#121	—	19	6	no	—
176	#121	—	19	12	no	—
177	#121	—	19	6	no	—
178	#121	—	19	6	no	—
179	#121	—	19	12	no	—
180	#121	C₂×D₁₉	1	6	yes	complemented
181	C₄×Q₈	—	19	1	no	—
182	C₄×Q₈	—	19	3	no	—
183	C₄×Q₈	—	19	3	no	—
184	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic, complemented
185	C₄×D₄	—	19	3	no	—
186	C₄×D₄	—	19	6	no	—
187	C₄×D₄	—	19	3	no	—
188	C₄×D₄	—	19	6	no	—
189	C₄×D₄	—	19	3	no	—
190	C₄×D₄	C₂×D₁₉	1	3	yes	complemented
191	C₂×C₄⋊C₄	—	19	1	no	—
192	C₂×C₄⋊C₄	—	19	3	no	—
193	C₂×C₂²⋊C₄	—	19	6	no	—
194	C₂×C₄²	—	19	3	no	—
195	C₁₉	C₂×#550	1	1	yes	characteristic, complemented, Sylow
196	C₂⁴	—	19	6	no	—
197	#41	—	2	8	no	—
198	#41	—	38	24	no	—
199	#41	—	38	24	no	—
200	#41	—	38	8	no	—
201	C₂×Q₈	—	19	8	no	—
202	C₂×Q₈	—	19	3	no	—
203	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
204	C₂×D₄	—	19	6	no	—
205	C₂×D₄	—	19	12	no	—
206	C₂×D₄	—	19	12	no	—
207	C₂×D₄	—	19	3	no	—
208	C₂×D₄	—	19	6	no	—
209	C₂×D₄	—	19	6	no	—
210	C₂×D₄	—	38	12	no	—
211	C₂×D₄	—	38	24	no	—
212	C₂×D₄	—	76	12	no	—
213	C₂×D₄	C₂²×D₁₉	1	3	yes	—
214	C₂×D₄	C₂²×D₁₉	1	6	yes	—
215	C₂×D₄	C₂²×D₁₉	1	6	yes	—
216	C₂²×C₄	—	19	1	no	—
217	C₂²×C₄	—	19	3	no	—
218	C₂²×C₄	—	19	6	no	—
219	C₂²×C₄	—	19	3	no	—
220	C₂²×C₄	—	19	2	no	—
221	C₂²×C₄	—	19	6	no	—
222	C₂²×C₄	—	19	6	no	—
223	C₂²×C₄	—	38	24	no	—
224	C₂²×C₄	C₂²×D₁₉	1	6	yes	—
225	C₄⋊C₄	—	19	2	no	—
226	C₄⋊C₄	—	19	6	no	—
227	C₄⋊C₄	—	19	1	no	—
228	C₄⋊C₄	—	19	3	no	—
229	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
230	C₄⋊C₄	C₂²×D₁₉	1	3	yes	—
231	C₂²⋊C₄	—	19	6	no	—
232	C₂²⋊C₄	—	19	6	no	—
233	C₂²⋊C₄	—	19	6	no	—
234	C₂²⋊C₄	C₂²×D₁₉	1	6	yes	—
235	C₄²	—	19	3	no	—
236	C₄²	—	19	3	no	—
237	C₄²	—	19	3	no	—
238	C₄²	C₂²×D₁₉	1	3	yes	—
239	C₂³	—	19	1	no	—
240	C₂³	—	19	6	no	—
241	C₂³	—	38	12	no	—
242	C₂³	—	38	12	no	—
243	C₂³	—	38	12	no	—
244	C₂³	C₂³×D₁₉	1	6	yes	—
245	Q₈	—	19	12	no	—
246	Q₈	D₄×D₁₉	1	4	yes	complemented
247	D₄	—	2	12	no	—
248	D₄	—	2	6	no	—
249	D₄	—	4	6	no	—
250	D₄	—	38	24	no	—
251	D₄	—	38	12	no	—
252	D₄	—	38	12	no	—
253	D₄	—	38	6	no	—
254	D₄	—	76	6	no	—
255	D₄	—	76	12	no	—
256	C₂×C₄	—	2	12	no	—
257	C₂×C₄	—	19	1	no	—
258	C₂×C₄	—	19	3	no	—
259	C₂×C₄	—	19	12	no	—
260	C₂×C₄	—	19	4	no	—
261	C₂×C₄	—	19	3	no	—
262	C₂×C₄	—	19	2	no	—
263	C₂×C₄	—	38	12	no	—
264	C₂×C₄	—	38	4	no	—
265	C₂×C₄	—	38	6	no	—
266	C₂×C₄	—	38	12	no	—
267	C₂×C₄	—	38	12	no	—
268	C₂×C₄	C₂³×D₁₉	1	3	yes	—
269	C₂×C₄	C₂³×D₁₉	1	2	yes	—
270	C₂×C₄	C₂³×D₁₉	1	3	yes	—
271	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
272	C₂²	—	2	6	no	—
273	C₂²	—	2	6	no	—
274	C₂²	—	2	6	no	—
275	C₂²	—	19	2	no	—
276	C₂²	—	19	2	no	—
277	C₂²	—	19	2	no	—
278	C₂²	—	38	6	no	—
279	C₂²	—	38	6	no	—
280	C₂²	—	38	6	no	—
281	C₂²	—	76	24	no	—
282	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
283	C₄	—	2	2	no	—
284	C₄	—	2	3	no	—
285	C₄	—	19	6	no	—
286	C₄	—	19	2	no	—
287	C₄	—	38	2	no	—
288	C₄	—	38	3	no	—
289	C₄	C₂×D₄×D₁₉	1	6	yes	—
290	C₄	#41×D₁₉	1	2	yes	complemented
291	C₂	—	4	6	no	—
292	C₂	—	19	4	no	—
293	C₂	—	76	6	no	—
294	C₂	C₂²×D₄×D₁₉	1	1	yes	characteristic
295	C₂	C₂×#41×D₁₉	1	1	yes	characteristic
296	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×#550	C₂
hypercenter	#550	D₁₉
hypocenter	C₁₉	C₂×#550
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ #550

Complements

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

N	H ≅ G/N	type
1. C₂×#41×D₁₉	C₂	semidirect
2. #11927216	C₂	semidirect
3. #11927215	C₂	semidirect
4. #11927214	C₂	semidirect
5. #11927213	C₂	semidirect
6. #11927212	C₂	semidirect
7. #127×D₁₉	C₂	semidirect
8. #121×D₁₉	C₂	semidirect
9. C₄×Q₈×D₁₉	C₂	semidirect
10. C₄×D₄×D₁₉	C₂	semidirect
11. C₁₉×#550	C₂	semidirect
12. C₂×#41×C₁₉	C₂²	semidirect
14. C₂×#64545	C₂²	semidirect
16. C₂×#64543	C₂²	semidirect
20. C₂×#64541	C₂²	semidirect
22. C₁₉×#127	C₂²	semidirect
23. C₁₉×#121	C₂²	semidirect
24. C₄×Q₈×C₁₉	C₂²	semidirect
25. C₄×D₄×C₁₉	C₂²	semidirect
26. Q₈×Dic₁₉	C₂²	semidirect
27. #10597386	C₂²	semidirect
28. #10597386	C₂²	semidirect
29. #10597386	C₂²	semidirect
30. #10597385	C₂²	semidirect
31. #10597384	C₂²	semidirect
32. D₄×Dic₁₉	C₂²	semidirect
33. #10597376	C₂²	semidirect
34. C₄×C₂²⋊D₁₉	C₂²	semidirect
35. #10597341	C₂²	semidirect
36. #10597339	C₂²	semidirect
39. #10597333	C₂²	semidirect
40. #10597330	C₂²	semidirect
41. (C₂²⋊D₁₉)⋊C₄	C₂²	semidirect
43. #10597320	C₂²	semidirect
44. C₄×D₇₆	C₂²	semidirect
46. C₄×Dic₃₈	C₂²	semidirect
59. Q₈×D₁₉	D₄	semidirect
96. Q₈×C₁₉	C₂×D₄	semidirect
121. C₄×D₁₉	#41	semidirect
122. Dic₃₈	C₂×D₄	semidirect
138. C₄×C₁₉	C₂×#41	semidirect
142. Dic₁₉	C₂×#41	semidirect
149. #550	D₁₉	direct (N × H)
159. D₁₉	#550	direct (N × H)
164. C₂×#41	C₂×D₁₉	semidirect
174. #127	C₂×D₁₉	semidirect
180. #121	C₂×D₁₉	semidirect
184. C₄×Q₈	C₂×D₁₉	semidirect
190. C₄×D₄	C₂×D₁₉	semidirect
195. C₁₉	C₂×#550	semidirect
246. Q₈	D₄×D₁₉	semidirect
290. C₄	#41×D₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#41×D₁₉	2
#11927216	2
#11927215	2
#11927214	2
#11927213	2
#11927212	2
#127×D₁₉	2
#121×D₁₉	2
C₄×Q₈×D₁₉	2
C₄×D₄×D₁₉	2
C₁₉×#550	2
C₂×#550	19

Subgroups — D19×#550 #14577111

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₉×#550 #14577111