Subgroups — #18071152

Order 4864.

464 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563615	C₂	1	1	yes	maximal, characteristic
2	#14563615	C₂	1	1	yes	maximal, characteristic
3	#14561461	C₂	1	2	yes	maximal, complemented
4	#14561458	C₂	1	2	yes	maximal, complemented
5	C₁₉×#2855	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
7	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
8	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
9	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
10	#11926299	C₂²	1	2	yes	—
11	#11926299	C₂²	1	2	yes	—
12	C₂×C₁₉×C₂²⋊Q₈	C₄	1	1	yes	characteristic
13	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
14	C₁₉×#391	C₄	1	1	yes	characteristic
15	C₁₉×#391	C₂²	1	1	yes	characteristic
16	C₁₉×#389	C₄	1	1	yes	characteristic
17	C₁₉×#389	C₂²	1	1	yes	characteristic
18	C₁₉×#387	—	2	1	no	—
19	C₁₉×#387	—	2	1	no	—
20	C₁₉×#385	—	2	1	no	—
21	C₁₉×#381	—	2	1	no	—
22	C₁₉×#380	C₄	1	1	yes	characteristic
23	C₂²×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
27	C₁₉×C₂²⋊Q₈	—	4	2	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×C₂²⋊C₄×C₁₉	—	2	1	no	—
35	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
36	C₂×C₄²×C₁₉	—	4	2	no	—
37	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
41	C₂²×C₁₉⋊C₈	—	2	2	no	—
42	C₂²×C₁₉⋊C₈	—	2	2	no	—
43	C₁₉×#95	—	2	1	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	C₄⋊(C₁₉⋊C₈)	—	4	2	no	—
54	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Q₈×C₁₉	—	2	2	no	—
56	C₂×Q₈×C₁₉	—	4	1	no	—
57	C₂×Q₈×C₁₉	—	4	1	no	—
58	C₂²×C₄×C₁₉	—	2	2	no	—
59	C₂²×C₄×C₁₉	—	2	2	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	—	2	1	no	—
65	C₂²×C₄×C₁₉	—	2	1	no	—
66	C₂²×C₄×C₁₉	—	4	1	no	—
67	C₂²×C₄×C₁₉	—	4	1	no	—
68	C₂²×C₄×C₁₉	—	4	1	no	—
69	C₂²×C₄×C₁₉	—	4	1	no	—
70	C₂²×C₄×C₁₉	—	4	1	no	—
71	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
75	C₂²×C₄×C₁₉	Q₁₆	1	2	yes	—
76	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
79	C₄⋊C₄×C₁₉	—	2	2	no	—
80	C₄⋊C₄×C₁₉	—	4	1	no	—
81	C₄⋊C₄×C₁₉	—	4	1	no	—
82	C₄⋊C₄×C₁₉	—	4	2	no	—
83	C₄⋊C₄×C₁₉	—	4	2	no	—
84	C₄⋊C₄×C₁₉	—	4	1	no	—
85	C₄⋊C₄×C₁₉	—	4	1	no	—
86	C₂²⋊C₄×C₁₉	—	4	2	no	—
87	C₂²⋊C₄×C₁₉	—	4	2	no	—
88	C₄²×C₁₉	—	2	2	no	—
89	C₄²×C₁₉	—	2	2	no	—
90	C₄²×C₁₉	—	4	1	no	—
91	C₄²×C₁₉	—	4	1	no	—
92	C₂×C₁₉⋊C₈	—	4	2	no	—
93	C₂×C₁₉⋊C₈	—	4	2	no	—
94	C₂×C₁₉⋊C₈	—	4	2	no	—
95	C₂×C₁₉⋊C₈	—	4	2	no	—
96	C₂×C₁₉⋊C₈	—	4	2	no	—
97	C₂×C₁₉⋊C₈	—	4	2	no	—
98	#7715	—	19	1	no	maximal, Sylow
99	C₂³×C₁₉	—	2	1	no	—
100	C₂³×C₁₉	—	2	1	no	—
101	C₂³×C₁₉	—	2	1	no	—
102	C₂³×C₁₉	—	2	1	no	—
103	C₂³×C₁₉	—	2	1	no	—
104	C₂³×C₁₉	—	2	1	no	—
105	C₂³×C₁₉	#103	1	2	yes	—
106	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	Q₈×C₁₉	—	8	2	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	—	2	1	no	—
119	C₂×C₄×C₁₉	—	2	1	no	—
120	C₂×C₄×C₁₉	—	2	1	no	—
121	C₂×C₄×C₁₉	—	4	2	no	—
122	C₂×C₄×C₁₉	—	4	2	no	—
123	C₂×C₄×C₁₉	—	4	4	no	—
124	C₂×C₄×C₁₉	—	4	2	no	—
125	C₂×C₄×C₁₉	—	4	2	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	—	4	1	no	—
135	C₂×C₄×C₁₉	—	4	1	no	—
136	C₂×C₄×C₁₉	—	4	1	no	—
137	C₂×C₄×C₁₉	—	4	1	no	—
138	C₂×C₄×C₁₉	#99	1	2	yes	—
139	C₂×C₄×C₁₉	#103	1	2	yes	—
140	C₂×C₄×C₁₉	#131	1	2	yes	—
141	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
142	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
143	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
145	C₁₉⋊C₈	—	8	2	no	—
146	C₁₉⋊C₈	—	8	2	no	—
147	#2855	D₁₉	1	1	yes	characteristic
148	#1797	—	19	1	no	—
149	#1797	—	19	1	no	—
150	#1272	—	19	2	no	—
151	#1271	—	19	2	no	—
152	C₂²×C₁₉	—	2	2	no	—
153	C₂²×C₁₉	—	2	2	no	—
154	C₂²×C₁₉	—	2	2	no	—
155	C₂²×C₁₉	—	4	1	no	—
156	C₂²×C₁₉	—	4	1	no	—
157	C₂²×C₁₉	—	4	1	no	—
158	C₂²×C₁₉	—	4	1	no	—
159	C₂²×C₁₉	#408	1	1	yes	characteristic
160	C₂²×C₁₉	C₂×#99	1	1	yes	characteristic
161	C₂²×C₁₉	#413	1	1	yes	characteristic
162	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
163	C₂²×C₁₉	#417	1	1	yes	characteristic
164	C₂²×C₁₉	#421	1	1	yes	characteristic
165	C₂²×C₁₉	#421	1	1	yes	characteristic
166	C₄×C₁₉	—	2	2	no	—
167	C₄×C₁₉	—	2	2	no	—
168	C₄×C₁₉	—	2	1	no	—
169	C₄×C₁₉	—	2	1	no	—
170	C₄×C₁₉	—	2	1	no	—
171	C₄×C₁₉	—	2	2	no	—
172	C₄×C₁₉	—	2	1	no	—
173	C₄×C₁₉	—	4	1	no	—
174	C₄×C₁₉	—	8	1	no	—
175	C₄×C₁₉	—	8	1	no	—
176	C₄×C₁₉	—	8	1	no	—
177	C₄×C₁₉	—	8	1	no	—
178	C₂×C₂²⋊Q₈	Dic₁₉	1	1	yes	characteristic
179	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
180	C₂×C₄⋊C₈	—	38	1	no	—
181	C₂×C₄⋊C₈	—	38	1	no	—
182	C₂×C₂²⋊C₈	—	38	1	no	—
183	C₂×C₂²⋊C₈	—	38	1	no	—
184	#391	Dic₁₉	1	1	yes	characteristic
185	#391	C₂×D₁₉	1	1	yes	characteristic
186	#389	Dic₁₉	1	1	yes	characteristic
187	#389	C₂×D₁₉	1	1	yes	characteristic
188	#387	—	2	1	no	—
189	#387	—	2	1	no	—
190	#385	—	2	1	no	—
191	#381	—	2	1	no	—
192	#380	Dic₁₉	1	1	yes	characteristic
193	#336	—	19	2	no	—
194	#336	—	19	2	no	—
195	C₂×C₁₉	—	4	2	no	—
196	C₂×C₁₉	#1468	1	1	yes	characteristic
197	C₂×C₁₉	#1481	1	1	yes	characteristic
198	C₂×C₁₉	#1495	1	1	yes	characteristic
199	C₂×C₁₉	#1513	1	1	yes	characteristic
200	C₂×C₁₉	#1520	1	1	yes	characteristic
201	C₂×C₁₉	#1535	1	1	yes	characteristic
202	C₂×C₁₉	#1550	1	1	yes	characteristic
203	C₂²×Q₈	C₂×Dic₁₉	1	1	yes	characteristic
204	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
205	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
206	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
207	C₂²×C₈	—	38	2	no	—
208	C₂²×C₈	—	38	2	no	—
209	C₂²⋊Q₈	—	4	2	no	—
210	C₂×C₄⋊C₄	—	2	1	no	—
211	C₂×C₄⋊C₄	—	2	1	no	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
214	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
215	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
216	C₂×C₂²⋊C₄	—	2	1	no	—
217	C₂×C₂²⋊C₄	—	2	1	no	—
218	C₂×C₄²	—	4	2	no	—
219	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
220	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
221	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
222	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
223	C₄⋊C₈	—	76	2	no	—
224	C₄⋊C₈	—	76	2	no	—
225	C₂²⋊C₈	—	76	2	no	—
226	C₂²⋊C₈	—	76	2	no	—
227	#95	—	2	1	no	—
228	#95	—	2	1	no	—
229	#95	—	2	1	no	—
230	#95	—	2	1	no	—
231	#95	—	2	1	no	—
232	#95	—	2	1	no	—
233	#95	—	2	1	no	—
234	C₁₉	#7715	1	1	yes	characteristic, complemented, Sylow
235	C₂⁴	C₂²⋊Dic₁₉	1	1	yes	characteristic
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	4	1	no	—
238	C₂×Q₈	—	4	1	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	4	1	no	—
248	C₂²×C₄	—	4	1	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	#64527	1	2	yes	—
253	C₂²×C₄	#64528	1	2	yes	—
254	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
255	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
256	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
257	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
258	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
259	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
260	C₂×C₈	—	76	2	no	—
261	C₂×C₈	—	76	2	no	—
262	C₂×C₈	—	76	2	no	—
263	C₂×C₈	—	76	2	no	—
264	C₂×C₈	—	76	2	no	—
265	C₂×C₈	—	76	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	4	1	no	—
278	C₄²	—	4	1	no	—
279	C₂³	—	2	1	no	—
280	C₂³	—	2	1	no	—
281	C₂³	—	2	1	no	—
282	C₂³	—	2	1	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	2	1	no	—
285	C₂³	#10597292	1	2	yes	—
286	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
287	Q₈	—	8	2	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	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	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	#10597291	1	2	yes	—
319	C₂×C₄	#10597292	1	2	yes	—
320	C₂×C₄	C₂×#64527	1	1	yes	characteristic
321	C₂×C₄	C₂×#64528	1	1	yes	characteristic
322	C₂×C₄	#10597394	1	2	yes	—
323	C₂×C₄	#10597395	1	1	yes	characteristic
324	C₂×C₄	#10597397	1	1	yes	characteristic
325	C₈	—	152	2	no	—
326	C₈	—	152	2	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	2	2	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	#11926511	1	1	yes	characteristic
335	C₂²	C₂×#10597291	1	1	yes	characteristic
336	C₂²	#11926528	1	1	yes	characteristic
337	C₂²	C₂×#10597292	1	1	yes	characteristic
338	C₂²	#11926545	1	1	yes	characteristic
339	C₂²	#11926561	1	1	yes	characteristic
340	C₂²	#11926561	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	2	2	no	—
346	C₄	—	2	2	no	—
347	C₄	—	2	2	no	—
348	C₄	—	4	1	no	—
349	C₄	—	8	1	no	—
350	C₄	—	8	1	no	—
351	C₄	—	8	1	no	—
352	C₄	—	8	1	no	—
353	C₂	—	4	2	no	—
354	C₂	#14561983	1	1	yes	characteristic
355	C₂	#14562038	1	1	yes	characteristic
356	C₂	#14562100	1	1	yes	characteristic
357	C₂	#14562178	1	1	yes	characteristic
358	C₂	#14562207	1	1	yes	characteristic
359	C₂	#14562266	1	1	yes	characteristic
360	C₂	#14562325	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×#2855	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2855	D₁₉
hypocenter	C₁₉	#7715
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #2855

Complements

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

N	H ≅ G/N	type
3. #14561461	C₂	semidirect
4. #14561458	C₂	semidirect
234. C₁₉	#7715	semidirect

Maximal subgroups

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

subgroup	index
#14563615	2
#14561461	2
#14561458	2
C₁₉×#2855	2
#7715	19