Subgroups — #18073596

Order 4864.

400 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572604	C₂	1	1	yes	maximal, characteristic
2	#14563387	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562224	C₂	1	1	yes	maximal, characteristic
4	#14562119	C₂	1	1	yes	maximal, characteristic
5	M₁₆×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561533	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1300	C₂	1	1	yes	maximal, characteristic, complemented
8	#11927078	C₂²	1	1	yes	characteristic
9	#11926671	C₄	1	1	yes	characteristic, complemented
10	#11926668	C₄	1	1	yes	characteristic, complemented
11	#11926621	C₄	1	1	yes	characteristic, complemented
12	#11926612	—	2	1	no	—
13	#11926609	C₄	1	1	yes	characteristic, complemented
14	(C₂²⋊D₁₉)⋊C₈	—	2	1	no	—
15	#11926606	—	2	1	no	—
16	#11926597	C₂²	1	1	yes	characteristic
17	#11926501	—	2	1	no	—
18	C₂×#10597260	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
20	#11926299	—	2	1	no	—
21	#11926299	—	2	1	no	—
22	#11926285	C₄	1	2	yes	complemented
23	#11926283	C₄	1	2	yes	complemented
24	#11926280	C₄	1	2	yes	—
25	#11926277	C₄	1	2	yes	—
26	#11926265	—	2	1	no	—
27	C₈×C₁₉⋊C₈	—	2	1	no	—
28	C₁₉×#432	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
30	C₁₉×#376	C₂²	1	1	yes	characteristic
31	C₂×M₁₆×C₁₉	—	2	1	no	—
32	C₂×C₄²×C₁₉	—	2	1	no	—
33	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
34	#10597376	C₂×C₄	1	1	yes	characteristic
35	#10597375	—	2	1	no	—
36	C₄×C₂²⋊D₁₉	—	2	1	no	—
37	#10597370	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₁₉⋊C₈	—	2	1	no	—
39	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
40	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
41	C₂×#64515	—	2	1	no	—
42	C₂×C₈×D₁₉	—	2	1	no	—
43	#10597322	—	2	1	no	—
44	#10597321	C₂×C₄	1	1	yes	characteristic
45	C₄×D₇₆	C₂×C₄	1	1	yes	characteristic
46	#10597318	—	2	1	no	—
47	#10597316	C₂×C₄	1	1	yes	characteristic
48	C₄×Dic₃₈	C₂×C₄	1	1	yes	characteristic
49	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
50	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
51	C₁₉×C₂²⋊C₈	—	2	1	no	—
52	C₁₉×#97	—	2	1	no	—
53	C₁₉×#97	C₂×C₄	1	2	yes	—
54	C₄×C₈×C₁₉	—	2	1	no	—
55	C₁₉×#95	—	2	1	no	—
56	C₁₉×#95	C₂×C₄	1	2	yes	—
57	C₂²⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
58	(C₂×D₁₉)⋊C₈	—	2	1	no	—
59	(C₂×D₁₉)⋊C₈	—	2	1	no	—
60	#10597272	—	2	1	no	—
61	Dic₁₉⋊C₈	—	2	1	no	—
62	Dic₁₉⋊C₈	—	2	1	no	—
63	C₈×Dic₁₉	—	2	1	no	—
64	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
65	#10597260	—	2	1	no	—
66	#10597260	C₂×C₄	1	2	yes	—
67	C₄×C₁₉⋊C₈	—	2	1	no	—
68	C₄×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
69	C₄×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
70	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
71	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	—	2	2	no	—
73	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
76	C₂×C₂²⋊D₁₉	—	2	1	no	—
77	C₂×D₇₆	C₄²	1	1	yes	characteristic
78	C₂×C₄×D₁₉	—	2	1	no	—
79	C₂×Dic₃₈	C₄²	1	1	yes	characteristic
80	M₁₆×C₁₉	—	2	4	no	—
81	C₂×C₈×C₁₉	—	2	2	no	—
82	C₂×C₈×C₁₉	—	2	1	no	—
83	C₂×C₈×C₁₉	—	2	1	no	—
84	C₄²×C₁₉	—	4	1	no	—
85	C₄²×C₁₉	—	4	1	no	—
86	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₁₉	#41	1	1	yes	characteristic
90	C₂²⋊Dic₁₉	—	2	1	no	—
91	(C₂×D₁₉)⋊C₄	—	2	1	no	—
92	(C₂×D₁₉)⋊C₄	—	2	1	no	—
93	(C₂×D₁₉)⋊C₄	—	2	1	no	—
94	(C₂×D₁₉)⋊C₄	—	2	1	no	—
95	C₄⋊Dic₁₉	C₄²	1	1	yes	characteristic
96	C₄⋊Dic₁₉	C₄²	1	1	yes	characteristic
97	Dic₁₉⋊C₄	—	2	1	no	—
98	Dic₁₉⋊C₄	—	2	1	no	—
99	Dic₁₉⋊C₄	—	2	1	no	—
100	Dic₁₉⋊C₄	—	2	1	no	—
101	C₄×Dic₁₉	—	2	1	no	—
102	C₂×C₁₉⋊C₈	—	2	2	no	—
103	C₂×C₁₉⋊C₈	—	2	2	no	—
104	C₂×C₁₉⋊C₈	—	4	1	no	—
105	C₂×C₁₉⋊C₈	—	4	1	no	—
106	C₂×C₁₉⋊C₈	—	4	1	no	—
107	C₂×C₁₉⋊C₈	—	4	1	no	—
108	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
112	#64515	—	4	2	no	—
113	C₈×D₁₉	—	4	2	no	—
114	#8250	—	19	1	no	maximal, Sylow
115	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
116	C₂²×D₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	2	2	no	—
118	C₂×C₄×C₁₉	—	2	1	no	—
119	C₂×C₄×C₁₉	—	2	1	no	—
120	C₂×C₄×C₁₉	—	4	1	no	—
121	C₂×C₄×C₁₉	—	4	2	no	—
122	C₂×C₄×C₁₉	—	4	2	no	—
123	C₂×C₄×C₁₉	—	4	2	no	—
124	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
127	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
130	C₂²⋊D₁₉	—	4	2	no	—
131	C₂×Dic₁₉	—	2	1	no	—
132	C₂×Dic₁₉	—	2	1	no	—
133	C₂×Dic₁₉	—	2	1	no	—
134	D₇₆	—	4	1	no	—
135	C₄×D₁₉	—	4	1	no	—
136	C₄×D₁₉	—	4	1	no	—
137	Dic₃₈	—	4	1	no	—
138	C₈×C₁₉	—	2	4	no	—
139	C₈×C₁₉	—	4	1	no	—
140	C₈×C₁₉	—	4	1	no	—
141	C₁₉⋊C₈	—	2	1	no	—
142	C₁₉⋊C₈	—	2	1	no	—
143	C₁₉⋊C₈	—	2	1	no	—
144	C₁₉⋊C₈	—	2	1	no	—
145	C₁₉⋊C₈	—	4	2	no	—
146	C₁₉⋊C₈	—	4	2	no	—
147	#2966	—	19	1	no	—
148	#1746	—	19	1	no	—
149	#1524	—	19	1	no	—
150	#1500	—	19	1	no	—
151	C₈×M₁₆	—	19	1	no	—
152	#1302	—	19	1	no	—
153	#1300	D₁₉	1	1	yes	characteristic, complemented
154	C₂²×C₁₉	—	2	1	no	—
155	C₂²×C₁₉	—	2	1	no	—
156	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
157	C₂²×C₁₉	#443	1	2	yes	—
158	C₂×D₁₉	—	4	1	no	—
159	C₂×D₁₉	—	4	1	no	—
160	C₂×D₁₉	—	4	1	no	—
161	C₄×C₁₉	—	2	1	no	—
162	C₄×C₁₉	—	2	1	no	—
163	C₄×C₁₉	—	2	1	no	—
164	C₄×C₁₉	—	2	1	no	—
165	C₄×C₁₉	—	2	1	no	—
166	C₄×C₁₉	—	8	2	no	—
167	C₄×C₁₉	#421	1	1	yes	characteristic
168	C₄×C₁₉	#421	1	1	yes	characteristic
169	Dic₁₉	—	4	1	no	—
170	Dic₁₉	—	4	1	no	—
171	Dic₁₉	—	4	1	no	—
172	#529	—	19	1	no	—
173	#446	—	19	1	no	—
174	C₈×Q₈	—	19	1	no	—
175	#436	—	19	1	no	—
176	#435	—	38	1	no	—
177	C₈×D₄	—	19	1	no	—
178	C₈×D₄	—	38	1	no	—
179	#433	—	38	1	no	—
180	#432	C₂×D₁₉	1	1	yes	characteristic
181	#431	—	19	1	no	—
182	#405	—	38	1	no	—
183	C₄×M₁₆	C₂×D₁₉	1	1	yes	characteristic
184	C₂×#97	—	19	1	no	—
185	C₂×C₄×C₈	—	19	1	no	—
186	#376	C₂×D₁₉	1	1	yes	characteristic
187	#336	—	38	1	no	—
188	#336	—	38	1	no	—
189	#330	—	19	2	no	—
190	#329	—	19	2	no	—
191	#328	—	19	2	no	—
192	#327	—	19	2	no	—
193	#322	—	38	1	no	—
194	C₈²	—	38	1	no	—
195	C₂×C₁₉	—	2	2	no	—
196	C₂×C₁₉	#1757	1	1	yes	characteristic
197	C₂×C₁₉	#1759	1	1	yes	characteristic
198	C₂×C₁₉	#1764	1	1	yes	characteristic
199	D₁₉	—	8	1	no	—
200	C₂×M₁₆	—	2	1	no	—
201	C₂×M₁₆	—	38	1	no	—
202	C₂²×C₈	—	19	1	no	—
203	C₂²×C₈	—	19	1	no	—
204	C₂²×C₈	—	38	1	no	—
205	C₂²×C₈	—	38	1	no	—
206	#126	—	38	1	no	—
207	#125	—	19	1	no	—
208	#124	—	19	1	no	—
209	#123	—	38	1	no	—
210	C₂²⋊Q₈	—	19	1	no	—
211	#121	—	19	1	no	—
212	C₄×Q₈	—	19	1	no	—
213	C₄×D₄	—	19	1	no	—
214	C₄×D₄	—	38	1	no	—
215	#117	—	38	1	no	—
216	C₂×C₄²	—	2	1	no	—
217	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
218	C₄⋊C₈	—	38	1	no	—
219	C₄⋊C₈	—	38	1	no	—
220	C₄⋊C₈	—	38	1	no	—
221	C₄⋊C₈	C₄×D₁₉	1	1	yes	characteristic
222	C₄⋊C₈	C₄×D₁₉	1	1	yes	characteristic
223	C₂²⋊C₈	—	2	1	no	—
224	C₂²⋊C₈	—	19	2	no	—
225	C₂²⋊C₈	—	38	1	no	—
226	C₂²⋊C₈	—	38	1	no	—
227	#97	—	2	1	no	—
228	#97	—	19	2	no	—
229	#97	—	38	1	no	—
230	#97	—	38	1	no	—
231	#97	C₄×D₁₉	1	2	yes	—
232	C₄×C₈	—	2	1	no	—
233	C₄×C₈	—	19	1	no	—
234	C₄×C₈	—	19	1	no	—
235	C₄×C₈	—	19	1	no	—
236	C₄×C₈	—	19	1	no	—
237	C₄×C₈	—	38	1	no	—
238	C₄×C₈	—	38	1	no	—
239	#95	—	2	1	no	—
240	#95	C₄×D₁₉	1	2	yes	—
241	C₁₉	#8250	1	1	yes	characteristic, complemented, Sylow
242	C₂×Q₈	—	19	1	no	—
243	C₂×D₄	—	19	1	no	—
244	C₂×D₄	—	38	1	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	38	1	no	—
247	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
248	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
249	C₂²×C₄	#64543	1	1	yes	characteristic
250	M₁₆	—	2	4	no	—
251	M₁₆	—	76	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	19	1	no	—
256	C₂×C₈	—	19	1	no	—
257	C₂×C₈	—	19	1	no	—
258	C₂×C₈	—	19	1	no	—
259	C₂×C₈	—	38	2	no	—
260	C₂×C₈	—	38	2	no	—
261	C₂×C₈	—	76	2	no	—
262	C₂×C₈	—	76	1	no	—
263	C₂×C₈	—	76	1	no	—
264	C₂×C₈	—	76	1	no	—
265	C₂×C₈	—	76	1	no	—
266	C₄⋊C₄	—	19	1	no	—
267	C₄⋊C₄	—	19	1	no	—
268	C₄⋊C₄	—	38	1	no	—
269	C₄⋊C₄	—	38	1	no	—
270	C₄⋊C₄	—	38	1	no	—
271	C₄⋊C₄	—	38	1	no	—
272	C₂²⋊C₄	—	38	1	no	—
273	C₂²⋊C₄	—	38	1	no	—
274	C₂²⋊C₄	—	38	1	no	—
275	C₂²⋊C₄	—	38	1	no	—
276	C₂²⋊C₄	—	38	1	no	—
277	C₄²	—	4	1	no	—
278	C₄²	—	4	1	no	—
279	C₄²	—	38	1	no	—
280	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
281	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
282	C₄²	D₄×D₁₉	1	1	yes	characteristic
283	C₄²	#64545	1	1	yes	characteristic
284	C₂³	—	38	1	no	—
285	C₂³	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
286	Q₈	—	76	1	no	—
287	D₄	—	76	1	no	—
288	D₄	—	76	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	38	1	no	—
297	C₂×C₄	—	38	1	no	—
298	C₂×C₄	—	38	1	no	—
299	C₂×C₄	—	76	1	no	—
300	C₂×C₄	—	76	1	no	—
301	C₂×C₄	C₄²×D₁₉	1	1	yes	characteristic
302	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
303	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
304	C₂×C₄	#10597338	1	1	yes	characteristic
305	C₂×C₄	#10597339	1	1	yes	characteristic
306	C₂×C₄	#10597339	1	1	yes	characteristic
307	C₈	—	2	4	no	—
308	C₈	—	4	1	no	—
309	C₈	—	4	1	no	—
310	C₈	—	38	1	no	—
311	C₈	—	38	1	no	—
312	C₈	—	38	1	no	—
313	C₈	—	38	1	no	—
314	C₈	—	76	2	no	—
315	C₈	—	76	2	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	76	1	no	—
319	C₂²	—	76	1	no	—
320	C₂²	—	76	1	no	—
321	C₂²	#11926391	1	1	yes	characteristic
322	C₂²	#11926658	1	2	yes	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	8	2	no	—
329	C₄	—	76	1	no	—
330	C₄	—	76	1	no	—
331	C₄	—	76	1	no	—
332	C₄	#11926559	1	1	yes	characteristic
333	C₄	#11926559	1	1	yes	characteristic
334	C₂	—	2	2	no	—
335	C₂	—	152	1	no	—
336	C₂	#14563434	1	1	yes	characteristic
337	C₂	#14563444	1	1	yes	characteristic
338	C₂	#14563469	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1300	C₂
socle	C₂²×C₁₉	C₄×C₂²⋊C₄
prime radical	#10597376	C₂×C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1300	D₁₉
hypocenter	C₁₉	#8250
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1300

Complements

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

N	H ≅ G/N	type
2. #14563387	C₂	semidirect
5. M₁₆×C₁₉⋊C₈	C₂	semidirect
6. #14561533	C₂	semidirect
7. C₁₉×#1300	C₂	semidirect
9. #11926671	C₄	semidirect
10. #11926668	C₄	semidirect
11. #11926621	C₄	semidirect
13. #11926609	C₄	semidirect
22. #11926285	C₄	semidirect
23. #11926283	C₄	semidirect
153. #1300	D₁₉	semidirect
241. C₁₉	#8250	semidirect

Maximal subgroups

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

subgroup	index
#14572604	2
#14563387	2
#14562224	2
#14562119	2
M₁₆×C₁₉⋊C₈	2
#14561533	2
C₁₉×#1300	2
#8250	19