Subgroups — #18078190

Order 4864.

588 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926597	C₂	1	1	yes	maximal, characteristic
2	#14563615	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926290	C₂	1	2	yes	maximal, complemented
4	#14561518	C₂	1	2	yes	maximal, complemented
5	C₁₉×#2857	C₂	1	1	yes	maximal, characteristic
6	#11926597	—	2	4	no	—
7	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
10	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
11	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
12	#11926299	—	2	1	no	—
13	#11926290	—	2	4	no	—
14	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₁₉×C₂²⋊Q₈	—	2	1	no	—
16	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#391	—	2	1	no	—
18	C₁₉×#389	—	2	1	no	—
19	C₁₉×#389	C₄	1	2	yes	—
20	C₁₉×#384	C₂²	1	1	yes	characteristic
21	C₁₉×#380	—	2	1	no	—
22	C₁₉×#380	C₄	1	2	yes	—
23	C₂²×Q₈×C₁₉	—	2	1	no	—
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×C₄⋊Q₈	—	2	2	no	—
28	C₁₉×C₄⋊Q₈	—	2	2	no	—
29	C₁₉×C₂²⋊Q₈	—	4	2	no	—
30	C₁₉×C₂²⋊Q₈	—	4	2	no	—
31	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
32	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
33	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
34	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
35	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
37	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
38	C₂×C₄²×C₁₉	—	2	2	no	—
39	C₂×C₄²×C₁₉	—	2	2	no	—
40	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
44	C₂²×C₁₉⋊C₈	—	2	2	no	—
45	C₂²×C₁₉⋊C₈	—	2	2	no	—
46	C₁₉×#95	—	2	2	no	—
47	C₁₉×#95	—	2	2	no	—
48	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
49	C₂²⋊(C₁₉⋊C₈)	—	4	2	no	—
50	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
51	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
52	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
53	C₄⋊(C₁₉⋊C₈)	D₄	1	4	yes	—
54	C₄×C₁₉⋊C₈	—	2	4	no	—
55	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Q₈×C₁₉	—	4	2	no	—
57	C₂×Q₈×C₁₉	—	4	1	no	—
58	C₂×Q₈×C₁₉	—	4	1	no	—
59	C₂×Q₈×C₁₉	—	4	2	no	—
60	C₂²×C₄×C₁₉	—	2	2	no	—
61	C₂²×C₄×C₁₉	—	2	2	no	—
62	C₂²×C₄×C₁₉	—	2	2	no	—
63	C₂²×C₄×C₁₉	—	2	2	no	—
64	C₂²×C₄×C₁₉	—	2	1	no	—
65	C₂²×C₄×C₁₉	—	2	1	no	—
66	C₂²×C₄×C₁₉	—	2	2	no	—
67	C₂²×C₄×C₁₉	—	2	2	no	—
68	C₂²×C₄×C₁₉	—	2	2	no	—
69	C₂²×C₄×C₁₉	—	2	2	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	D₈	1	2	yes	—
75	C₂²×C₄×C₁₉	SD₁₆	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	4	no	—
80	C₄⋊C₄×C₁₉	—	4	2	no	—
81	C₄⋊C₄×C₁₉	—	4	4	no	—
82	C₄⋊C₄×C₁₉	—	4	1	no	—
83	C₄⋊C₄×C₁₉	—	4	1	no	—
84	C₄⋊C₄×C₁₉	—	4	1	no	—
85	C₄⋊C₄×C₁₉	—	4	1	no	—
86	C₄⋊C₄×C₁₉	—	4	2	no	—
87	C₂²⋊C₄×C₁₉	—	4	4	no	—
88	C₄²×C₁₉	—	2	2	no	—
89	C₄²×C₁₉	—	2	1	no	—
90	C₄²×C₁₉	—	2	2	no	—
91	C₄²×C₁₉	—	2	1	no	—
92	C₄²×C₁₉	C₂×D₄	1	2	yes	—
93	C₄²×C₁₉	#41	1	2	yes	—
94	C₂×C₁₉⋊C₈	—	2	4	no	—
95	C₂×C₁₉⋊C₈	—	4	2	no	—
96	C₂×C₁₉⋊C₈	—	4	2	no	—
97	C₂×C₁₉⋊C₈	—	4	2	no	—
98	C₂×C₁₉⋊C₈	—	4	2	no	—
99	C₂×C₁₉⋊C₈	—	4	2	no	—
100	#9130	—	19	1	no	maximal, Sylow
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	—	2	1	no	—
106	C₂³×C₁₉	—	2	1	no	—
107	C₂³×C₁₉	#102	1	2	yes	—
108	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	Q₈×C₁₉	—	8	2	no	—
110	Q₈×C₁₉	—	8	2	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	4	no	—
118	C₂×C₄×C₁₉	—	2	4	no	—
119	C₂×C₄×C₁₉	—	2	1	no	—
120	C₂×C₄×C₁₉	—	2	1	no	—
121	C₂×C₄×C₁₉	—	2	1	no	—
122	C₂×C₄×C₁₉	—	4	4	no	—
123	C₂×C₄×C₁₉	—	4	2	no	—
124	C₂×C₄×C₁₉	—	4	2	no	—
125	C₂×C₄×C₁₉	—	4	2	no	—
126	C₂×C₄×C₁₉	—	4	2	no	—
127	C₂×C₄×C₁₉	—	4	2	no	—
128	C₂×C₄×C₁₉	—	4	2	no	—
129	C₂×C₄×C₁₉	—	4	2	no	—
130	C₂×C₄×C₁₉	—	4	2	no	—
131	C₂×C₄×C₁₉	#101	1	2	yes	—
132	C₂×C₄×C₁₉	#102	1	2	yes	—
133	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
135	C₂×C₄×C₁₉	#121	1	2	yes	—
136	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
137	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
139	C₂×C₄×C₁₉	#135	1	2	yes	—
140	C₂×C₄×C₁₉	#135	1	2	yes	—
141	C₂×C₄×C₁₉	#137	1	2	yes	—
142	C₁₉⋊C₈	—	4	4	no	—
143	C₁₉⋊C₈	—	8	2	no	—
144	C₂×#431	—	19	1	no	—
145	#2857	D₁₉	1	1	yes	characteristic
146	#1797	—	19	1	no	—
147	C₂×#332	—	19	2	no	—
148	#1296	—	19	2	no	—
149	C₂²×C₁₉	—	2	2	no	—
150	C₂²×C₁₉	—	2	2	no	—
151	C₂²×C₁₉	—	2	2	no	—
152	C₂²×C₁₉	—	4	1	no	—
153	C₂²×C₁₉	—	4	1	no	—
154	C₂²×C₁₉	—	4	1	no	—
155	C₂²×C₁₉	—	4	1	no	—
156	C₂²×C₁₉	#386	1	1	yes	characteristic
157	C₂²×C₁₉	C₂×#101	1	1	yes	characteristic
158	C₂²×C₁₉	#413	1	1	yes	characteristic
159	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
160	C₂²×C₁₉	#416	1	1	yes	characteristic
161	C₂²×C₁₉	#416	1	1	yes	characteristic
162	C₂²×C₁₉	#419	1	1	yes	characteristic
163	C₄×C₁₉	—	2	2	no	—
164	C₄×C₁₉	—	2	2	no	—
165	C₄×C₁₉	—	2	2	no	—
166	C₄×C₁₉	—	4	1	no	—
167	C₄×C₁₉	—	8	2	no	—
168	C₄×C₁₉	—	8	2	no	—
169	C₄×C₁₉	#452	1	2	yes	—
170	C₄×C₁₉	#465	1	2	yes	—
171	C₄×C₁₉	#466	1	2	yes	—
172	C₄×C₁₉	#485	1	2	yes	—
173	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
174	C₂×C₂²⋊Q₈	—	2	1	no	—
175	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
176	#431	—	38	4	no	—
177	C₂×C₄⋊C₈	—	19	2	no	—
178	C₂×C₄⋊C₈	—	19	2	no	—
179	C₂×C₂²⋊C₈	—	38	1	no	—
180	C₂×C₂²⋊C₈	—	38	1	no	—
181	C₂×C₄×C₈	—	38	1	no	—
182	#391	—	2	1	no	—
183	#389	—	2	1	no	—
184	#389	Dic₁₉	1	2	yes	—
185	#384	C₂×D₁₉	1	1	yes	characteristic
186	#380	—	2	1	no	—
187	#380	Dic₁₉	1	2	yes	—
188	#336	—	38	1	no	—
189	#332	—	38	4	no	—
190	C₂×C₁₉	—	4	2	no	—
191	C₂×C₁₉	#1540	1	1	yes	characteristic
192	C₂×C₁₉	#1544	1	1	yes	characteristic
193	C₂×C₁₉	#1546	1	1	yes	characteristic
194	C₂×C₁₉	#1554	1	1	yes	characteristic
195	C₂×C₁₉	#1873	1	1	yes	characteristic
196	C₂×C₁₉	#1893	1	1	yes	characteristic
197	C₂×C₁₉	#1901	1	1	yes	characteristic
198	C₂²×Q₈	—	2	1	no	—
199	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
200	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
201	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
202	C₂²×C₈	—	38	2	no	—
203	C₂²×C₈	—	38	2	no	—
204	C₄⋊Q₈	—	2	2	no	—
205	C₄⋊Q₈	—	2	2	no	—
206	C₂²⋊Q₈	—	4	2	no	—
207	C₂²⋊Q₈	—	4	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	1	no	—
210	C₂×C₄⋊C₄	—	2	1	no	—
211	C₂×C₄⋊C₄	—	2	1	no	—
212	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
213	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
214	C₂×C₂²⋊C₄	—	2	2	no	—
215	C₂×C₄²	—	2	2	no	—
216	C₂×C₄²	—	2	2	no	—
217	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
218	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
219	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
220	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
221	C₄⋊C₈	—	19	4	no	—
222	C₄⋊C₈	—	38	2	no	—
223	C₄⋊C₈	—	38	2	no	—
224	C₄⋊C₈	—	38	2	no	—
225	C₂²⋊C₈	—	38	4	no	—
226	C₂²⋊C₈	—	76	2	no	—
227	C₄×C₈	—	38	4	no	—
228	#95	—	2	2	no	—
229	#95	—	2	2	no	—
230	C₁₉	#9130	1	1	yes	characteristic, complemented, Sylow
231	C₂⁴	C₂²⋊Dic₁₉	1	1	yes	characteristic
232	C₂×Q₈	—	4	1	no	—
233	C₂×Q₈	—	4	1	no	—
234	C₂×Q₈	—	4	2	no	—
235	C₂×Q₈	—	4	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	4	1	no	—
247	C₂²×C₄	#64525	1	2	yes	—
248	C₂²×C₄	#64526	1	2	yes	—
249	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
250	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
251	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
252	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
254	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
255	C₂×C₈	—	38	4	no	—
256	C₂×C₈	—	76	2	no	—
257	C₂×C₈	—	76	2	no	—
258	C₂×C₈	—	76	2	no	—
259	C₂×C₈	—	76	2	no	—
260	C₂×C₈	—	76	2	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	4	no	—
269	C₂²⋊C₄	—	4	4	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	2	1	no	—
273	C₄²	—	2	2	no	—
274	C₄²	D₄×D₁₉	1	2	yes	—
275	C₄²	#64543	1	2	yes	—
276	C₂³	—	2	1	no	—
277	C₂³	—	2	1	no	—
278	C₂³	—	2	1	no	—
279	C₂³	—	2	1	no	—
280	C₂³	—	2	1	no	—
281	C₂³	—	2	1	no	—
282	C₂³	#10597289	1	2	yes	—
283	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
284	Q₈	—	8	2	no	—
285	Q₈	—	8	2	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	—	2	1	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	#10597289	1	2	yes	—
307	C₂×C₄	#10597293	1	2	yes	—
308	C₂×C₄	C₂×#64525	1	1	yes	characteristic
309	C₂×C₄	C₂×#64526	1	1	yes	characteristic
310	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
311	C₂×C₄	#10597381	1	1	yes	characteristic
312	C₂×C₄	#10597383	1	1	yes	characteristic
313	C₂×C₄	#10597385	1	2	yes	—
314	C₂×C₄	#10597388	1	2	yes	—
315	C₂×C₄	#10597396	1	2	yes	—
316	C₂×C₄	#10597396	1	2	yes	—
317	C₈	—	76	4	no	—
318	C₈	—	152	2	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	4	1	no	—
326	C₂²	#11926424	1	1	yes	characteristic
327	C₂²	C₂×#10597293	1	1	yes	characteristic
328	C₂²	#11926528	1	1	yes	characteristic
329	C₂²	C₂×#10597289	1	1	yes	characteristic
330	C₂²	#11926540	1	1	yes	characteristic
331	C₂²	#11926540	1	1	yes	characteristic
332	C₂²	#11926553	1	1	yes	characteristic
333	C₄	—	2	2	no	—
334	C₄	—	2	2	no	—
335	C₄	—	2	2	no	—
336	C₄	—	4	1	no	—
337	C₄	—	8	2	no	—
338	C₄	—	8	2	no	—
339	C₄	#11926704	1	2	yes	—
340	C₄	#11926771	1	2	yes	—
341	C₄	#11926778	1	2	yes	—
342	C₄	#11926882	1	2	yes	—
343	C₂	—	4	2	no	—
344	C₂	#14562284	1	1	yes	characteristic
345	C₂	#14562300	1	1	yes	characteristic
346	C₂	#14562307	1	1	yes	characteristic
347	C₂	#14562345	1	1	yes	characteristic
348	C₂	#14563948	1	1	yes	characteristic
349	C₂	#14564064	1	1	yes	characteristic
350	C₂	#14564104	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×#2857	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2857	D₁₉
hypocenter	C₁₉	#9130
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #2857

Complements

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

N	H ≅ G/N	type
3. C₂×#11926290	C₂	semidirect
4. #14561518	C₂	semidirect
230. C₁₉	#9130	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926597	2
#14563615	2
C₂×#11926290	2
#14561518	2
C₁₉×#2857	2
#9130	19