Subgroups — #18074381

Order 4864.

692 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563619	C₂	1	1	yes	maximal, characteristic
2	#14562292	C₂	1	2	yes	maximal, complemented
3	#14562038	C₂	1	2	yes	maximal, complemented
4	#14561949	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#3458	C₂	1	1	yes	maximal, characteristic
6	#11926597	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
10	#11926299	—	2	1	no	—
11	#11926293	—	2	2	no	—
12	#11926290	—	2	2	no	—
13	Q₈⋊(C₁₉⋊C₈)	—	2	4	no	—
14	#11926270	—	2	1	no	—
15	#11926268	—	2	1	no	—
16	C₁₉×#544	—	2	2	no	—
17	C₁₉×#544	—	2	2	no	—
18	C₁₉×#544	—	2	1	no	—
19	C₁₉×#544	C₄	1	2	yes	—
20	C₁₉×#543	—	2	1	no	—
21	C₁₉×#543	C₄	1	2	yes	—
22	C₁₉×#533	—	2	4	no	—
23	C₁₉×#533	—	2	1	no	—
24	C₁₉×#533	C₂²	1	2	yes	—
25	C₂×C₁₉×#117	—	2	1	no	—
26	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
27	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
28	C₁₉×C₄⋊Q₈	—	2	4	no	—
29	C₁₉×C₄⋊Q₈	—	2	1	no	—
30	C₁₉×C₄⋊Q₈	—	2	1	no	—
31	C₁₉×C₄⋊Q₈	—	2	4	no	—
32	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
33	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
34	C₁₉×#125	—	2	4	no	—
35	C₁₉×#125	—	2	2	no	—
36	C₁₉×#125	—	2	4	no	—
37	C₁₉×#125	—	2	2	no	—
38	C₁₉×C₂²⋊Q₈	—	2	4	no	—
39	C₁₉×C₂²⋊Q₈	—	2	4	no	—
40	C₁₉×C₂²⋊Q₈	—	2	4	no	—
41	C₁₉×C₂²⋊Q₈	—	2	2	no	—
42	C₁₉×C₂²⋊Q₈	—	2	2	no	—
43	C₁₉×C₂²⋊Q₈	—	2	2	no	—
44	C₁₉×C₂²⋊Q₈	—	2	4	no	—
45	C₁₉×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
46	C₄×Q₈×C₁₉	—	2	2	no	—
47	C₄×Q₈×C₁₉	—	2	2	no	—
48	C₄×Q₈×C₁₉	—	2	2	no	—
49	C₄×Q₈×C₁₉	D₄	1	4	yes	—
50	C₁₉×#117	—	2	2	no	—
51	C₁₉×#117	—	2	2	no	—
52	C₁₉×#117	—	2	2	no	—
53	C₁₉×#117	—	2	2	no	—
54	C₁₉×#117	—	2	2	no	—
55	C₁₉×#117	—	2	2	no	—
56	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
57	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
58	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
60	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
61	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
63	C₂×C₄²×C₁₉	—	2	2	no	—
64	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
65	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
66	C₂²×C₁₉⋊C₈	—	2	1	no	—
67	C₂²×C₁₉⋊C₈	—	2	1	no	—
68	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
69	C₂²⋊(C₁₉⋊C₈)	—	4	2	no	—
70	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
71	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
72	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
73	C₄×C₁₉⋊C₈	—	2	2	no	—
74	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×Q₈×C₁₉	—	2	2	no	—
76	C₂×Q₈×C₁₉	—	2	2	no	—
77	C₂×Q₈×C₁₉	—	2	2	no	—
78	C₂×Q₈×C₁₉	C₂²⋊C₄	1	4	yes	—
79	C₂²×C₄×C₁₉	—	2	1	no	—
80	C₂²×C₄×C₁₉	—	2	1	no	—
81	C₂²×C₄×C₁₉	—	2	1	no	—
82	C₂²×C₄×C₁₉	—	2	1	no	—
83	C₂²×C₄×C₁₉	—	2	2	no	—
84	C₂²×C₄×C₁₉	—	2	2	no	—
85	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
89	C₂²×C₄×C₁₉	Q₁₆	1	2	yes	—
90	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
93	C₄⋊C₄×C₁₉	—	2	2	no	—
94	C₄⋊C₄×C₁₉	—	2	2	no	—
95	C₄⋊C₄×C₁₉	—	2	2	no	—
96	C₄⋊C₄×C₁₉	—	2	4	no	—
97	C₄⋊C₄×C₁₉	—	2	4	no	—
98	C₄⋊C₄×C₁₉	—	2	2	no	—
99	C₄⋊C₄×C₁₉	—	2	2	no	—
100	C₄⋊C₄×C₁₉	—	2	4	no	—
101	C₄⋊C₄×C₁₉	—	2	2	no	—
102	C₄⋊C₄×C₁₉	—	2	2	no	—
103	C₄⋊C₄×C₁₉	—	2	2	no	—
104	C₄⋊C₄×C₁₉	—	2	2	no	—
105	C₄⋊C₄×C₁₉	—	2	4	no	—
106	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
107	C₂²⋊C₄×C₁₉	—	2	4	no	—
108	C₂²⋊C₄×C₁₉	—	2	4	no	—
109	C₂²⋊C₄×C₁₉	—	2	2	no	—
110	C₂²⋊C₄×C₁₉	—	2	2	no	—
111	C₄²×C₁₉	—	2	4	no	—
112	C₄²×C₁₉	—	2	4	no	—
113	C₄²×C₁₉	—	2	1	no	—
114	C₄²×C₁₉	—	2	1	no	—
115	C₄²×C₁₉	C₂×D₄	1	2	yes	—
116	C₄²×C₁₉	C₂×D₄	1	2	yes	—
117	C₂×C₁₉⋊C₈	—	4	2	no	—
118	C₂×C₁₉⋊C₈	—	4	1	no	—
119	C₂×C₁₉⋊C₈	—	4	1	no	—
120	C₂×C₁₉⋊C₈	—	4	1	no	—
121	C₂×C₁₉⋊C₈	—	4	1	no	—
122	C₂×C₁₉⋊C₈	—	4	1	no	—
123	C₂×C₁₉⋊C₈	—	4	1	no	—
124	#8410	—	19	1	no	maximal, Sylow
125	C₂³×C₁₉	—	2	1	no	—
126	C₂³×C₁₉	—	2	1	no	—
127	C₂³×C₁₉	—	4	1	no	—
128	C₂³×C₁₉	—	4	1	no	—
129	C₂³×C₁₉	#103	1	2	yes	—
130	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	Q₈×C₁₉	—	4	4	no	—
132	Q₈×C₁₉	—	8	2	no	—
133	Q₈×C₁₉	—	8	2	no	—
134	Q₈×C₁₉	—	8	2	no	—
135	C₂×C₄×C₁₉	—	2	1	no	—
136	C₂×C₄×C₁₉	—	2	1	no	—
137	C₂×C₄×C₁₉	—	2	2	no	—
138	C₂×C₄×C₁₉	—	2	2	no	—
139	C₂×C₄×C₁₉	—	2	2	no	—
140	C₂×C₄×C₁₉	—	2	2	no	—
141	C₂×C₄×C₁₉	—	2	4	no	—
142	C₂×C₄×C₁₉	—	2	1	no	—
143	C₂×C₄×C₁₉	—	2	1	no	—
144	C₂×C₄×C₁₉	—	2	4	no	—
145	C₂×C₄×C₁₉	—	4	1	no	—
146	C₂×C₄×C₁₉	—	4	1	no	—
147	C₂×C₄×C₁₉	—	4	1	no	—
148	C₂×C₄×C₁₉	—	4	1	no	—
149	C₂×C₄×C₁₉	—	8	2	no	—
150	C₂×C₄×C₁₉	—	8	2	no	—
151	C₂×C₄×C₁₉	#103	1	2	yes	—
152	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
153	C₂×C₄×C₁₉	#120	1	2	yes	—
154	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
155	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
156	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
157	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
158	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
159	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
160	C₁₉⋊C₈	—	8	2	no	—
161	C₁₉⋊C₈	—	8	1	no	—
162	C₁₉⋊C₈	—	8	1	no	—
163	#3458	D₁₉	1	1	yes	characteristic
164	#1798	—	19	1	no	—
165	#1542	—	19	2	no	—
166	#1481	—	19	2	no	—
167	#1458	—	19	1	no	—
168	C₂²×C₁₉	—	2	1	no	—
169	C₂²×C₁₉	—	2	1	no	—
170	C₂²×C₁₉	—	2	2	no	—
171	C₂²×C₁₉	—	4	2	no	—
172	C₂²×C₁₉	—	4	1	no	—
173	C₂²×C₁₉	—	4	1	no	—
174	C₂²×C₁₉	—	8	1	no	—
175	C₂²×C₁₉	#379	1	1	yes	characteristic
176	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
177	C₂²×C₁₉	#419	1	1	yes	characteristic
178	C₄×C₁₉	—	2	2	no	—
179	C₄×C₁₉	—	2	2	no	—
180	C₄×C₁₉	—	2	2	no	—
181	C₄×C₁₉	—	4	1	no	—
182	C₄×C₁₉	—	4	4	no	—
183	C₄×C₁₉	—	4	1	no	—
184	C₄×C₁₉	—	4	4	no	—
185	C₄×C₁₉	#448	1	1	yes	characteristic
186	C₄×C₁₉	#448	1	1	yes	characteristic
187	C₄×C₁₉	#451	1	1	yes	characteristic
188	C₄×C₁₉	#451	1	1	yes	characteristic
189	#544	—	2	2	no	—
190	#544	—	2	2	no	—
191	#544	—	2	1	no	—
192	#544	Dic₁₉	1	2	yes	—
193	#543	—	2	1	no	—
194	#543	Dic₁₉	1	2	yes	—
195	#533	—	2	4	no	—
196	#533	—	2	1	no	—
197	#533	C₂×D₁₉	1	2	yes	—
198	C₂×#117	—	2	1	no	—
199	C₂×#117	C₂×D₁₉	1	1	yes	characteristic
200	#431	—	19	2	no	—
201	C₂×C₄⋊C₈	—	19	1	no	—
202	C₂×C₄⋊C₈	—	19	1	no	—
203	C₂×C₂²⋊C₈	—	38	1	no	—
204	#336	—	38	1	no	—
205	#333	—	38	2	no	—
206	#332	—	38	2	no	—
207	Q₈⋊C₈	—	38	4	no	—
208	#324	—	38	1	no	—
209	#323	—	38	1	no	—
210	C₂×C₁₉	—	2	1	no	—
211	C₂×C₁₉	—	2	1	no	—
212	C₂×C₁₉	—	4	2	no	—
213	C₂×C₁₉	#1787	1	1	yes	characteristic
214	C₂×C₁₉	#1791	1	1	yes	characteristic
215	C₂×C₁₉	#1794	1	1	yes	characteristic
216	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
217	C₂²×C₈	—	38	1	no	—
218	C₂²×C₈	—	38	1	no	—
219	C₄⋊Q₈	—	2	4	no	—
220	C₄⋊Q₈	—	2	1	no	—
221	C₄⋊Q₈	—	2	1	no	—
222	C₄⋊Q₈	—	2	4	no	—
223	C₄⋊Q₈	C₂²⋊D₁₉	1	2	yes	—
224	C₄⋊Q₈	C₂²⋊D₁₉	1	2	yes	—
225	#125	—	2	4	no	—
226	#125	—	2	2	no	—
227	#125	—	2	4	no	—
228	#125	—	2	2	no	—
229	C₂²⋊Q₈	—	2	4	no	—
230	C₂²⋊Q₈	—	2	4	no	—
231	C₂²⋊Q₈	—	2	4	no	—
232	C₂²⋊Q₈	—	2	2	no	—
233	C₂²⋊Q₈	—	2	2	no	—
234	C₂²⋊Q₈	—	2	2	no	—
235	C₂²⋊Q₈	—	2	4	no	—
236	C₂²⋊Q₈	C₂×Dic₁₉	1	4	yes	—
237	C₄×Q₈	—	2	2	no	—
238	C₄×Q₈	—	2	2	no	—
239	C₄×Q₈	—	2	2	no	—
240	C₄×Q₈	C₂²⋊D₁₉	1	4	yes	—
241	#117	—	2	2	no	—
242	#117	—	2	2	no	—
243	#117	—	2	2	no	—
244	#117	—	2	2	no	—
245	#117	—	2	2	no	—
246	#117	—	2	2	no	—
247	C₂×C₄⋊C₄	—	2	1	no	—
248	C₂×C₄⋊C₄	—	2	1	no	—
249	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
250	C₂×C₄⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
251	C₂×C₂²⋊C₄	—	2	2	no	—
252	C₂×C₂²⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
253	C₂×C₂²⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
254	C₂×C₄²	—	2	2	no	—
255	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
256	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
257	C₄⋊C₈	—	38	2	no	—
258	C₄⋊C₈	—	38	2	no	—
259	C₄⋊C₈	—	38	2	no	—
260	C₂²⋊C₈	—	38	2	no	—
261	C₂²⋊C₈	—	76	2	no	—
262	C₄×C₈	—	38	2	no	—
263	C₁₉	#8410	1	1	yes	characteristic, complemented, Sylow
264	C₂⁴	C₂²⋊Dic₁₉	1	1	yes	characteristic
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	—	2	2	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	C₂²⋊Dic₁₉	1	4	yes	—
269	C₂²×C₄	—	2	1	no	—
270	C₂²×C₄	—	2	1	no	—
271	C₂²×C₄	—	2	1	no	—
272	C₂²×C₄	—	2	1	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	#64527	1	2	yes	—
276	C₂²×C₄	#64528	1	2	yes	—
277	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
278	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
279	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
280	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
281	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
282	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
283	C₂×C₈	—	76	1	no	—
284	C₂×C₈	—	76	1	no	—
285	C₂×C₈	—	76	1	no	—
286	C₂×C₈	—	76	1	no	—
287	C₂×C₈	—	76	2	no	—
288	C₂×C₈	—	76	1	no	—
289	C₂×C₈	—	76	1	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	4	no	—
294	C₄⋊C₄	—	2	4	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	4	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	2	4	no	—
303	C₄⋊C₄	C₂²⋊Dic₁₉	1	4	yes	—
304	C₂²⋊C₄	—	2	4	no	—
305	C₂²⋊C₄	—	2	4	no	—
306	C₂²⋊C₄	—	2	2	no	—
307	C₂²⋊C₄	—	2	2	no	—
308	C₄²	—	2	4	no	—
309	C₄²	—	2	1	no	—
310	C₄²	—	2	4	no	—
311	C₄²	—	2	1	no	—
312	C₄²	C₂×C₂²⋊D₁₉	1	2	yes	—
313	C₄²	C₂×C₂²⋊D₁₉	1	2	yes	—
314	C₂³	—	2	1	no	—
315	C₂³	—	2	1	no	—
316	C₂³	—	4	1	no	—
317	C₂³	—	4	1	no	—
318	C₂³	#10597292	1	2	yes	—
319	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
320	Q₈	—	4	4	no	—
321	Q₈	—	8	2	no	—
322	Q₈	—	8	2	no	—
323	Q₈	—	8	2	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	4	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	4	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	8	2	no	—
339	C₂×C₄	—	8	2	no	—
340	C₂×C₄	#10597292	1	2	yes	—
341	C₂×C₄	C₂×#64527	1	1	yes	characteristic
342	C₂×C₄	#10597388	1	1	yes	characteristic
343	C₂×C₄	#10597388	1	1	yes	characteristic
344	C₂×C₄	C₂×#64528	1	1	yes	characteristic
345	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	2	yes	—
346	C₂×C₄	#10597399	1	2	yes	—
347	C₂×C₄	#10597399	1	1	yes	characteristic
348	C₂×C₄	#10597399	1	1	yes	characteristic
349	C₈	—	152	2	no	—
350	C₈	—	152	1	no	—
351	C₈	—	152	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	2	no	—
355	C₂²	—	4	2	no	—
356	C₂²	—	4	1	no	—
357	C₂²	—	4	1	no	—
358	C₂²	—	8	1	no	—
359	C₂²	#11926400	1	1	yes	characteristic
360	C₂²	C₂×#10597292	1	1	yes	characteristic
361	C₂²	#11926553	1	1	yes	characteristic
362	C₄	—	2	2	no	—
363	C₄	—	2	2	no	—
364	C₄	—	2	2	no	—
365	C₄	—	4	1	no	—
366	C₄	—	4	4	no	—
367	C₄	—	4	1	no	—
368	C₄	—	4	4	no	—
369	C₄	#11926683	1	1	yes	characteristic
370	C₄	#11926683	1	1	yes	characteristic
371	C₄	#11926700	1	1	yes	characteristic
372	C₄	#11926700	1	1	yes	characteristic
373	C₂	—	2	1	no	—
374	C₂	—	2	1	no	—
375	C₂	—	4	2	no	—
376	C₂	#14563570	1	1	yes	characteristic
377	C₂	#14563586	1	1	yes	characteristic
378	C₂	#14563599	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597399
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#3458	C₂
socle	C₂²×C₁₉	#379
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#3458	D₁₉
hypocenter	C₁₉	#8410
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3458

Complements

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

N	H ≅ G/N	type
2. #14562292	C₂	semidirect
3. #14562038	C₂	semidirect
263. C₁₉	#8410	semidirect

Maximal subgroups

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

subgroup	index
#14563619	2
#14562292	2
#14562038	2
#14561949	2
C₁₉×#3458	2
#8410	19