Subgroups — #18073597

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	#14563386	C₂	1	1	yes	maximal, characteristic, complemented
2	#14562223	C₂	1	1	yes	maximal, characteristic
3	#14562121	C₂	1	1	yes	maximal, characteristic
4	C₈×#64520	C₂	1	1	yes	maximal, characteristic, complemented
5	#14561533	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561527	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#2966	C₂	1	1	yes	maximal, characteristic
8	#11926597	C₂²	1	1	yes	characteristic
9	C₄×#64520	C₂²	1	1	yes	characteristic
10	C₂×#10597260	C₂²	1	1	yes	characteristic
11	#11926390	C₂²	1	1	yes	characteristic
12	#11926298	—	2	1	no	—
13	#11926298	—	2	1	no	—
14	#11926286	C₄	1	2	yes	complemented
15	#11926284	C₄	1	2	yes	complemented
16	#11926281	C₄	1	2	yes	—
17	#11926278	C₄	1	2	yes	—
18	#11926266	—	2	1	no	—
19	C₈×C₁₉⋊C₈	—	2	1	no	—
20	C₁₉×#529	C₂²	1	1	yes	characteristic
21	C₁₉×#446	C₄	1	1	yes	characteristic, complemented
22	C₈×Q₈×C₁₉	C₄	1	1	yes	characteristic, complemented
23	C₁₉×#436	C₄	1	1	yes	characteristic, complemented
24	C₁₉×#435	—	2	1	no	—
25	C₈×D₄×C₁₉	—	2	1	no	—
26	C₈×D₄×C₁₉	C₄	1	1	yes	characteristic, complemented
27	C₁₉×#433	—	2	1	no	—
28	C₁₉×#432	C₂²	1	1	yes	characteristic
29	C₁₉×#405	—	2	1	no	—
30	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
31	C₂×M₁₆×C₁₉	—	2	1	no	—
32	C₂²×C₈×C₁₉	—	2	1	no	—
33	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
35	C₁₉×#126	—	2	1	no	—
36	C₁₉×#125	C₂×C₄	1	1	yes	characteristic
37	C₁₉×#124	C₂×C₄	1	1	yes	characteristic
38	C₁₉×#123	—	2	1	no	—
39	C₁₉×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
40	C₁₉×#121	C₂×C₄	1	1	yes	characteristic
41	C₄×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
42	C₄×D₄×C₁₉	—	2	1	no	—
43	C₄×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
44	C₁₉×#117	—	2	1	no	—
45	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
46	C₂×C₄×Dic₁₉	—	2	1	no	—
47	C₂×#64520	—	2	1	no	—
48	C₂²×C₁₉⋊C₈	—	2	1	no	—
49	C₁₉×C₄⋊C₈	—	2	1	no	—
50	C₁₉×C₄⋊C₈	—	2	1	no	—
51	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
52	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₁₉×C₂²⋊C₈	—	2	1	no	—
54	C₁₉×C₂²⋊C₈	—	2	1	no	—
55	C₁₉×C₂²⋊C₈	—	2	1	no	—
56	C₁₉×#97	—	2	1	no	—
57	C₁₉×#97	—	2	1	no	—
58	C₄×C₈×C₁₉	—	2	1	no	—
59	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
62	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
63	C₁₉⋊#95	—	2	1	no	—
64	C₁₉⋊#95	C₂×C₄	1	2	yes	—
65	C₂²⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
66	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
67	#10597260	—	2	1	no	—
68	#10597260	C₂×C₄	1	2	yes	—
69	#10597260	C₂×C₄	1	2	yes	—
70	C₄×C₁₉⋊C₈	—	2	1	no	—
71	C₄×C₁₉⋊C₈	—	2	1	no	—
72	C₂×Q₈×C₁₉	C₄²	1	1	yes	characteristic
73	C₂×D₄×C₁₉	—	2	1	no	—
74	C₂×D₄×C₁₉	C₄²	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	—	2	1	no	—
76	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
79	C₂²×Dic₁₉	—	2	2	no	—
80	M₁₆×C₁₉	—	4	2	no	—
81	C₂×C₈×C₁₉	—	2	1	no	—
82	C₂×C₈×C₁₉	—	2	1	no	—
83	C₂×C₈×C₁₉	—	4	1	no	—
84	C₂×C₈×C₁₉	—	4	1	no	—
85	C₂×C₈×C₁₉	—	4	2	no	—
86	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
90	C₄⋊C₄×C₁₉	—	2	1	no	—
91	C₄⋊C₄×C₁₉	—	2	1	no	—
92	C₄⋊C₄×C₁₉	—	2	1	no	—
93	C₄⋊C₄×C₁₉	—	2	1	no	—
94	C₄⋊C₄×C₁₉	C₄²	1	1	yes	characteristic
95	C₄⋊C₄×C₁₉	C₄²	1	1	yes	characteristic
96	C₂²⋊C₄×C₁₉	—	2	1	no	—
97	C₂²⋊C₄×C₁₉	—	2	1	no	—
98	C₂²⋊C₄×C₁₉	—	2	1	no	—
99	C₂²⋊C₄×C₁₉	—	2	1	no	—
100	C₂²⋊C₄×C₁₉	—	2	1	no	—
101	C₄²×C₁₉	—	2	1	no	—
102	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₁₉	#41	1	1	yes	characteristic
106	C₄×Dic₁₉	—	4	1	no	—
107	C₄×Dic₁₉	—	4	1	no	—
108	#64520	—	2	4	no	—
109	C₂×C₁₉⋊C₈	—	2	2	no	—
110	C₂×C₁₉⋊C₈	—	2	2	no	—
111	C₂×C₁₉⋊C₈	—	2	2	no	—
112	C₂×C₁₉⋊C₈	—	4	1	no	—
113	C₂×C₁₉⋊C₈	—	4	1	no	—
114	#8250	—	19	1	no	maximal, Sylow
115	C₂³×C₁₉	—	2	1	no	—
116	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
117	Q₈×C₁₉	—	4	1	no	—
118	D₄×C₁₉	—	4	1	no	—
119	D₄×C₁₉	—	4	2	no	—
120	C₂×C₄×C₁₉	—	2	2	no	—
121	C₂×C₄×C₁₉	—	2	1	no	—
122	C₂×C₄×C₁₉	—	2	1	no	—
123	C₂×C₄×C₁₉	—	2	1	no	—
124	C₂×C₄×C₁₉	—	2	1	no	—
125	C₂×C₄×C₁₉	—	2	1	no	—
126	C₂×C₄×C₁₉	—	4	1	no	—
127	C₂×C₄×C₁₉	—	4	1	no	—
128	C₂×C₄×C₁₉	—	4	1	no	—
129	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
130	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
132	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
135	C₂×Dic₁₉	—	4	2	no	—
136	C₂×Dic₁₉	—	4	2	no	—
137	C₂×Dic₁₉	—	4	2	no	—
138	C₈×C₁₉	—	2	1	no	—
139	C₈×C₁₉	—	2	1	no	—
140	C₈×C₁₉	—	2	1	no	—
141	C₈×C₁₉	—	2	1	no	—
142	C₈×C₁₉	—	4	1	no	—
143	C₈×C₁₉	—	4	1	no	—
144	C₁₉⋊C₈	—	2	4	no	—
145	C₁₉⋊C₈	—	4	2	no	—
146	C₁₉⋊C₈	—	4	2	no	—
147	#2966	D₁₉	1	1	yes	characteristic
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	—	19	1	no	—
154	C₂²×C₁₉	—	2	1	no	—
155	C₂²×C₁₉	—	2	1	no	—
156	C₂²×C₁₉	—	4	1	no	—
157	C₂²×C₁₉	—	4	1	no	—
158	C₂²×C₁₉	—	4	1	no	—
159	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
160	C₂²×C₁₉	#443	1	2	yes	—
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	—	4	1	no	—
167	C₄×C₁₉	—	4	1	no	—
168	C₄×C₁₉	—	4	1	no	—
169	C₄×C₁₉	#421	1	1	yes	characteristic
170	C₄×C₁₉	#421	1	1	yes	characteristic
171	Dic₁₉	—	8	2	no	—
172	#529	C₂×D₁₉	1	1	yes	characteristic
173	#446	Dic₁₉	1	1	yes	characteristic, complemented
174	C₈×Q₈	Dic₁₉	1	1	yes	characteristic, complemented
175	#436	Dic₁₉	1	1	yes	characteristic, complemented
176	#435	—	2	1	no	—
177	C₈×D₄	—	2	1	no	—
178	C₈×D₄	Dic₁₉	1	1	yes	characteristic, complemented
179	#433	—	2	1	no	—
180	#432	C₂×D₁₉	1	1	yes	characteristic
181	#431	—	19	1	no	—
182	#405	—	2	1	no	—
183	C₄×M₁₆	—	19	1	no	—
184	C₂×#97	—	19	1	no	—
185	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
186	#376	—	19	1	no	—
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	—	8	1	no	—
197	C₂×C₁₉	#1757	1	1	yes	characteristic
198	C₂×C₁₉	#1759	1	1	yes	characteristic
199	C₂×C₁₉	#1764	1	1	yes	characteristic
200	C₂×M₁₆	—	2	1	no	—
201	C₂×M₁₆	—	38	1	no	—
202	C₂²×C₈	—	2	1	no	—
203	C₂²×C₈	—	38	1	no	—
204	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
205	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
206	#126	—	2	1	no	—
207	#125	C₄×D₁₉	1	1	yes	characteristic
208	#124	C₄×D₁₉	1	1	yes	characteristic
209	#123	—	2	1	no	—
210	C₂²⋊Q₈	C₄×D₁₉	1	1	yes	characteristic
211	#121	C₄×D₁₉	1	1	yes	characteristic
212	C₄×Q₈	C₂×Dic₁₉	1	1	yes	characteristic
213	C₄×D₄	—	2	1	no	—
214	C₄×D₄	C₂×Dic₁₉	1	1	yes	characteristic
215	#117	—	2	1	no	—
216	C₂×C₄²	—	38	1	no	—
217	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
218	C₄⋊C₈	—	2	1	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	38	1	no	—
221	C₄⋊C₈	C₂×Dic₁₉	1	1	yes	characteristic
222	C₄⋊C₈	C₂×Dic₁₉	1	1	yes	characteristic
223	C₂²⋊C₈	—	2	1	no	—
224	C₂²⋊C₈	—	2	1	no	—
225	C₂²⋊C₈	—	2	1	no	—
226	C₂²⋊C₈	—	19	2	no	—
227	#97	—	2	1	no	—
228	#97	—	2	1	no	—
229	#97	—	19	2	no	—
230	#97	—	19	2	no	—
231	#97	—	38	1	no	—
232	C₄×C₈	—	2	1	no	—
233	C₄×C₈	—	38	1	no	—
234	C₄×C₈	—	38	1	no	—
235	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
236	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
237	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
238	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
239	#95	—	19	2	no	—
240	#95	—	38	1	no	—
241	C₁₉	#8250	1	1	yes	characteristic, complemented, Sylow
242	C₂×Q₈	C₄×Dic₁₉	1	1	yes	characteristic
243	C₂×D₄	—	2	1	no	—
244	C₂×D₄	C₄×Dic₁₉	1	1	yes	characteristic
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	38	2	no	—
247	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
248	C₂²×C₄	#64541	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
250	M₁₆	—	4	2	no	—
251	M₁₆	—	38	4	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	—	38	2	no	—
258	C₂×C₈	—	38	2	no	—
259	C₂×C₈	—	38	2	no	—
260	C₂×C₈	—	76	1	no	—
261	C₂×C₈	—	76	1	no	—
262	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
263	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
264	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
265	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	C₄×Dic₁₉	1	1	yes	characteristic
271	C₄⋊C₄	C₄×Dic₁₉	1	1	yes	characteristic
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	2	1	no	—
274	C₂²⋊C₄	—	2	1	no	—
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₄²	—	2	1	no	—
278	C₄²	—	76	1	no	—
279	C₄²	—	76	1	no	—
280	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
281	C₄²	#64541	1	1	yes	characteristic
282	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
283	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
284	C₂³	—	2	1	no	—
285	C₂³	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
286	Q₈	—	4	1	no	—
287	D₄	—	4	1	no	—
288	D₄	—	4	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	76	2	no	—
299	C₂×C₄	—	76	2	no	—
300	C₂×C₄	—	76	2	no	—
301	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
302	C₂×C₄	#10597372	1	1	yes	characteristic
303	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
304	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
305	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
306	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	4	1	no	—
312	C₈	—	4	1	no	—
313	C₈	—	38	4	no	—
314	C₈	—	76	2	no	—
315	C₈	—	76	2	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	C₄×C₂²⋊Dic₁₉	1	1	yes	characteristic
322	C₂²	#11926659	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	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	152	2	no	—
332	C₄	#11926561	1	1	yes	characteristic
333	C₄	#11926561	1	1	yes	characteristic
334	C₂	—	2	2	no	—
335	C₂	—	8	1	no	—
336	C₂	#14563435	1	1	yes	characteristic
337	C₂	#14563445	1	1	yes	characteristic
338	C₂	#14563470	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2966	C₂
socle	C₂²×C₁₉	C₄×C₂²⋊C₄
prime radical	C₁₉×#121	C₂×C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2966	D₁₉
hypocenter	C₁₉	#8250
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2966

Complements

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

N	H ≅ G/N	type
1. #14563386	C₂	semidirect
4. C₈×#64520	C₂	semidirect
5. #14561533	C₂	semidirect
6. #14561527	C₂	semidirect
14. #11926286	C₄	semidirect
15. #11926284	C₄	semidirect
21. C₁₉×#446	C₄	semidirect
22. C₈×Q₈×C₁₉	C₄	semidirect
23. C₁₉×#436	C₄	semidirect
26. C₈×D₄×C₁₉	C₄	semidirect
173. #446	Dic₁₉	semidirect
174. C₈×Q₈	Dic₁₉	semidirect
175. #436	Dic₁₉	semidirect
178. C₈×D₄	Dic₁₉	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
#14563386	2
#14562223	2
#14562121	2
C₈×#64520	2
#14561533	2
#14561527	2
C₁₉×#2966	2
#8250	19