Subgroups — #18071224

Order 4864.

450 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563703	C₂	1	1	yes	maximal, characteristic
2	#14563615	C₂	1	1	yes	maximal, characteristic
3	#14561458	C₂	1	2	yes	maximal, complemented
4	#14561458	C₂	1	2	yes	maximal, complemented
5	C₁₉×#2851	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
7	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
8	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
9	#11926299	—	2	1	no	—
10	#11926299	C₂²	1	2	yes	—
11	#11926299	C₂²	1	2	yes	—
12	C₂×C₁₉×#125	C₄	1	1	yes	characteristic
13	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
14	C₁₉×#389	C₂²	1	1	yes	characteristic
15	C₁₉×#389	C₂²	1	1	yes	characteristic
16	C₁₉×#387	—	2	1	no	—
17	C₁₉×#387	—	2	1	no	—
18	C₁₉×#385	—	2	1	no	—
19	C₁₉×#385	—	2	1	no	—
20	C₁₉×#382	C₄	1	1	yes	characteristic
21	C₁₉×#380	C₄	1	1	yes	characteristic
22	C₁₉×#380	C₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
26	C₁₉×#125	—	4	2	no	—
27	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
28	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
31	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
36	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
37	C₂×C₄²×C₁₉	—	4	2	no	—
38	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
42	C₂²×C₁₉⋊C₈	—	2	2	no	—
43	C₂²×C₁₉⋊C₈	—	2	2	no	—
44	C₁₉×#95	—	2	1	no	—
45	C₁₉×#95	—	2	1	no	—
46	C₁₉×#95	—	2	1	no	—
47	C₁₉×#95	—	2	1	no	—
48	C₁₉×#95	—	2	1	no	—
49	C₁₉×#95	—	2	1	no	—
50	C₂²⋊(C₁₉⋊C₈)	—	4	2	no	—
51	C₂²⋊(C₁₉⋊C₈)	—	4	2	no	—
52	C₄⋊(C₁₉⋊C₈)	—	4	2	no	—
53	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	—	2	2	no	—
55	C₂²×C₄×C₁₉	—	2	2	no	—
56	C₂²×C₄×C₁₉	—	2	2	no	—
57	C₂²×C₄×C₁₉	—	2	2	no	—
58	C₂²×C₄×C₁₉	—	2	1	no	—
59	C₂²×C₄×C₁₉	—	2	1	no	—
60	C₂²×C₄×C₁₉	—	2	1	no	—
61	C₂²×C₄×C₁₉	—	2	1	no	—
62	C₂²×C₄×C₁₉	—	2	1	no	—
63	C₂²×C₄×C₁₉	—	2	1	no	—
64	C₂²×C₄×C₁₉	—	4	1	no	—
65	C₂²×C₄×C₁₉	—	4	1	no	—
66	C₂²×C₄×C₁₉	—	4	1	no	—
67	C₂²×C₄×C₁₉	—	4	1	no	—
68	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
74	C₄⋊C₄×C₁₉	—	2	2	no	—
75	C₄⋊C₄×C₁₉	—	2	2	no	—
76	C₄⋊C₄×C₁₉	—	4	1	no	—
77	C₄⋊C₄×C₁₉	—	4	1	no	—
78	C₄⋊C₄×C₁₉	—	4	2	no	—
79	C₄⋊C₄×C₁₉	—	4	2	no	—
80	C₄⋊C₄×C₁₉	—	4	2	no	—
81	C₄⋊C₄×C₁₉	—	4	1	no	—
82	C₄⋊C₄×C₁₉	—	4	1	no	—
83	C₂²⋊C₄×C₁₉	—	4	2	no	—
84	C₂²⋊C₄×C₁₉	—	4	2	no	—
85	C₄²×C₁₉	—	2	2	no	—
86	C₄²×C₁₉	—	2	2	no	—
87	C₄²×C₁₉	—	4	1	no	—
88	C₄²×C₁₉	—	4	1	no	—
89	C₂×C₁₉⋊C₈	—	4	2	no	—
90	C₂×C₁₉⋊C₈	—	4	2	no	—
91	C₂×C₁₉⋊C₈	—	4	2	no	—
92	C₂×C₁₉⋊C₈	—	4	2	no	—
93	C₂×C₁₉⋊C₈	—	4	2	no	—
94	C₂×C₁₉⋊C₈	—	4	2	no	—
95	#7731	—	19	1	no	maximal, Sylow
96	C₂³×C₁₉	—	2	1	no	—
97	C₂³×C₁₉	—	2	1	no	—
98	C₂³×C₁₉	—	2	1	no	—
99	C₂³×C₁₉	—	2	1	no	—
100	C₂³×C₁₉	—	2	1	no	—
101	C₂³×C₁₉	—	2	1	no	—
102	C₂³×C₁₉	#101	1	2	yes	—
103	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₉	—	2	1	no	—
105	C₂×C₄×C₁₉	—	2	1	no	—
106	C₂×C₄×C₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	—	2	1	no	—
108	C₂×C₄×C₁₉	—	2	1	no	—
109	C₂×C₄×C₁₉	—	2	1	no	—
110	C₂×C₄×C₁₉	—	2	1	no	—
111	C₂×C₄×C₁₉	—	2	1	no	—
112	C₂×C₄×C₁₉	—	2	1	no	—
113	C₂×C₄×C₁₉	—	2	1	no	—
114	C₂×C₄×C₁₉	—	2	1	no	—
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	2	1	no	—
118	C₂×C₄×C₁₉	—	4	4	no	—
119	C₂×C₄×C₁₉	—	4	4	no	—
120	C₂×C₄×C₁₉	—	4	2	no	—
121	C₂×C₄×C₁₉	—	4	2	no	—
122	C₂×C₄×C₁₉	—	4	1	no	—
123	C₂×C₄×C₁₉	—	4	1	no	—
124	C₂×C₄×C₁₉	—	4	1	no	—
125	C₂×C₄×C₁₉	—	4	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	—	4	1	no	—
130	C₂×C₄×C₁₉	—	4	1	no	—
131	C₂×C₄×C₁₉	—	4	1	no	—
132	C₂×C₄×C₁₉	—	4	1	no	—
133	C₂×C₄×C₁₉	—	4	1	no	—
134	C₂×C₄×C₁₉	#99	1	2	yes	—
135	C₂×C₄×C₁₉	#131	1	2	yes	—
136	C₂×C₄×C₁₉	#135	1	2	yes	—
137	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
139	C₁₉⋊C₈	—	8	2	no	—
140	C₁₉⋊C₈	—	8	2	no	—
141	#2851	D₁₉	1	1	yes	characteristic
142	#1817	—	19	1	no	—
143	#1797	—	19	1	no	—
144	#1271	—	19	2	no	—
145	#1271	—	19	2	no	—
146	C₂²×C₁₉	—	2	2	no	—
147	C₂²×C₁₉	—	2	2	no	—
148	C₂²×C₁₉	—	2	2	no	—
149	C₂²×C₁₉	—	4	1	no	—
150	C₂²×C₁₉	—	4	1	no	—
151	C₂²×C₁₉	—	4	1	no	—
152	C₂²×C₁₉	—	4	1	no	—
153	C₂²×C₁₉	#408	1	1	yes	characteristic
154	C₂²×C₁₉	C₂×#99	1	1	yes	characteristic
155	C₂²×C₁₉	C₂×#101	1	1	yes	characteristic
156	C₂²×C₁₉	#416	1	1	yes	characteristic
157	C₂²×C₁₉	#417	1	1	yes	characteristic
158	C₂²×C₁₉	#421	1	1	yes	characteristic
159	C₂²×C₁₉	#421	1	1	yes	characteristic
160	C₄×C₁₉	—	2	2	no	—
161	C₄×C₁₉	—	2	1	no	—
162	C₄×C₁₉	—	2	1	no	—
163	C₄×C₁₉	—	2	2	no	—
164	C₄×C₁₉	—	2	2	no	—
165	C₄×C₁₉	—	4	1	no	—
166	C₄×C₁₉	—	4	1	no	—
167	C₄×C₁₉	—	8	1	no	—
168	C₄×C₁₉	—	8	1	no	—
169	C₄×C₁₉	—	8	1	no	—
170	C₄×C₁₉	—	8	1	no	—
171	C₂×#125	Dic₁₉	1	1	yes	characteristic
172	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
173	C₂×C₄⋊C₈	—	38	1	no	—
174	C₂×C₂²⋊C₈	—	38	1	no	—
175	C₂×C₂²⋊C₈	—	38	1	no	—
176	#389	C₂×D₁₉	1	1	yes	characteristic
177	#389	C₂×D₁₉	1	1	yes	characteristic
178	#387	—	2	1	no	—
179	#387	—	2	1	no	—
180	#385	—	2	1	no	—
181	#385	—	2	1	no	—
182	#382	Dic₁₉	1	1	yes	characteristic
183	#380	Dic₁₉	1	1	yes	characteristic
184	#380	Dic₁₉	1	1	yes	characteristic
185	#336	—	19	2	no	—
186	#336	—	19	2	no	—
187	#336	—	38	1	no	—
188	C₂×C₁₉	—	4	2	no	—
189	C₂×C₁₉	#1472	1	1	yes	characteristic
190	C₂×C₁₉	#1482	1	1	yes	characteristic
191	C₂×C₁₉	#1495	1	1	yes	characteristic
192	C₂×C₁₉	#1513	1	1	yes	characteristic
193	C₂×C₁₉	#1516	1	1	yes	characteristic
194	C₂×C₁₉	#1534	1	1	yes	characteristic
195	C₂×C₁₉	#1552	1	1	yes	characteristic
196	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
197	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
198	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
199	C₂²×C₈	—	38	2	no	—
200	C₂²×C₈	—	38	2	no	—
201	#125	—	4	2	no	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₄⋊C₄	—	2	1	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
207	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
208	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
209	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
210	C₂×C₂²⋊C₄	—	2	1	no	—
211	C₂×C₂²⋊C₄	—	2	1	no	—
212	C₂×C₄²	—	4	2	no	—
213	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
214	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
215	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
216	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
217	C₄⋊C₈	—	76	2	no	—
218	C₂²⋊C₈	—	76	2	no	—
219	C₂²⋊C₈	—	76	2	no	—
220	#95	—	2	1	no	—
221	#95	—	2	1	no	—
222	#95	—	2	1	no	—
223	#95	—	2	1	no	—
224	#95	—	2	1	no	—
225	#95	—	2	1	no	—
226	C₁₉	#7731	1	1	yes	characteristic, complemented, Sylow
227	C₂⁴	C₂²⋊Dic₁₉	1	1	yes	characteristic
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
243	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
244	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
245	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
248	C₂×C₈	—	76	2	no	—
249	C₂×C₈	—	76	2	no	—
250	C₂×C₈	—	76	2	no	—
251	C₂×C₈	—	76	2	no	—
252	C₂×C₈	—	76	2	no	—
253	C₂×C₈	—	76	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₄⋊C₄	—	4	1	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	4	1	no	—
268	C₄²	—	4	1	no	—
269	C₂³	—	2	1	no	—
270	C₂³	—	2	1	no	—
271	C₂³	—	2	1	no	—
272	C₂³	—	2	1	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	#10597293	1	2	yes	—
276	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	#10597291	1	2	yes	—
308	C₂×C₄	#10597394	1	2	yes	—
309	C₂×C₄	#10597395	1	1	yes	characteristic
310	C₂×C₄	#10597396	1	2	yes	—
311	C₂×C₄	#10597397	1	1	yes	characteristic
312	C₈	—	152	2	no	—
313	C₈	—	152	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	#11926511	1	1	yes	characteristic
322	C₂²	C₂×#10597291	1	1	yes	characteristic
323	C₂²	C₂×#10597293	1	1	yes	characteristic
324	C₂²	#11926540	1	1	yes	characteristic
325	C₂²	#11926545	1	1	yes	characteristic
326	C₂²	#11926561	1	1	yes	characteristic
327	C₂²	#11926561	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	8	1	no	—
336	C₄	—	8	1	no	—
337	C₄	—	8	1	no	—
338	C₄	—	8	1	no	—
339	C₂	—	4	2	no	—
340	C₂	#14561999	1	1	yes	characteristic
341	C₂	#14562043	1	1	yes	characteristic
342	C₂	#14562100	1	1	yes	characteristic
343	C₂	#14562178	1	1	yes	characteristic
344	C₂	#14562191	1	1	yes	characteristic
345	C₂	#14562262	1	1	yes	characteristic
346	C₂	#14562335	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2851	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2851	D₁₉
hypocenter	C₁₉	#7731
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #2851

Complements

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

N	H ≅ G/N	type
3. #14561458	C₂	semidirect
4. #14561458	C₂	semidirect
226. C₁₉	#7731	semidirect

Maximal subgroups

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

subgroup	index
#14563703	2
#14563615	2
#14561458	2
C₁₉×#2851	2
#7731	19