Subgroups — #18075522

Order 4864.

456 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	#14572329	C₂	1	1	yes	maximal, characteristic
2	#14563664	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562179	C₂	1	2	yes	maximal
4	#14561542	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1810	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10597347	C₄	1	1	yes	characteristic, complemented
7	#11927055	C₂²	1	1	yes	characteristic
8	#11926615	—	2	1	no	—
9	#11926615	—	2	1	no	—
10	C₈×C₂²⋊D₁₉	—	2	1	no	—
11	C₈×C₂²⋊D₁₉	—	2	1	no	—
12	C₂×#10597275	—	2	1	no	—
13	#11926570	C₄	1	1	yes	characteristic, complemented
14	#11926510	C₄	1	1	yes	characteristic, complemented
15	#11926508	C₂²	1	1	yes	characteristic
16	#11926499	C₄	1	1	yes	characteristic, complemented
17	#11926404	C₂²	1	2	yes	—
18	#11926307	—	2	1	no	—
19	#11926282	—	2	2	no	—
20	#11926279	—	2	2	no	—
21	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
22	C₂×C₁₉×#108	—	2	1	no	—
23	C₁₉×#407	C₂²	1	2	yes	—
24	C₁₉×#341	—	2	1	no	—
25	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
26	C₂×#64541	C₂×C₄	1	1	yes	characteristic
27	C₂×M₁₆×C₁₉	—	2	2	no	—
28	C₂²×C₈×C₁₉	—	4	2	no	—
29	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
35	#10597399	—	2	1	no	—
36	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
37	#10597376	D₄	1	1	yes	characteristic
38	#10597376	Q₈	1	1	yes	characteristic
39	#10597375	—	2	1	no	—
40	C₄×C₂²⋊D₁₉	—	2	1	no	—
41	C₄×C₂²⋊D₁₉	—	2	1	no	—
42	#10597372	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄⋊Dic₁₉	—	2	2	no	—
44	#10597370	D₄	1	1	yes	characteristic
45	#10597370	Q₈	1	1	yes	characteristic
46	C₂×#64520	C₂×C₄	1	1	yes	characteristic
47	C₂×#64520	C₂×C₄	1	1	yes	characteristic
48	#10597347	—	4	2	no	—
49	C₂×#64515	—	2	1	no	—
50	C₂×C₈×D₁₉	—	2	1	no	—
51	C₁₉×#108	—	4	2	no	—
52	C₁₉×C₂²⋊C₈	—	2	2	no	—
53	C₁₉×C₂²⋊C₈	—	2	2	no	—
54	C₁₉⋊#95	—	2	2	no	—
55	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
56	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
57	(C₂×D₁₉)⋊C₈	—	2	1	no	—
58	(C₂×D₁₉)⋊C₈	—	2	1	no	—
59	#10597275	—	4	2	no	—
60	#10597272	—	2	1	no	—
61	Dic₁₉⋊C₈	—	2	1	no	—
62	Dic₁₉⋊C₈	—	2	1	no	—
63	C₈×Dic₁₉	—	2	1	no	—
64	C₂⁴×C₁₉	#41	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	—	4	1	no	—
66	C₂²×C₄×C₁₉	—	4	1	no	—
67	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
73	C₂×C₂²⋊D₁₉	—	2	1	no	—
74	C₂×C₂²⋊D₁₉	—	2	1	no	—
75	C₂×C₂²⋊D₁₉	—	2	1	no	—
76	C₂²×Dic₁₉	—	4	2	no	—
77	#64541	—	4	1	no	—
78	#64541	—	4	1	no	—
79	C₂×D₇₆	C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×D₁₉	—	2	1	no	—
81	C₂×Dic₃₈	C₄⋊C₄	1	1	yes	characteristic
82	M₁₆×C₁₉	—	4	2	no	—
83	M₁₆×C₁₉	—	4	2	no	—
84	C₂×C₈×C₁₉	—	2	2	no	—
85	C₂×C₈×C₁₉	—	2	2	no	—
86	C₂×C₈×C₁₉	—	2	2	no	—
87	C₂×C₈×C₁₉	—	4	1	no	—
88	C₂×C₈×C₁₉	—	4	1	no	—
89	C₂×C₈×C₁₉	—	4	1	no	—
90	C₂×C₈×C₁₉	—	4	2	no	—
91	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
95	C₂²⋊Dic₁₉	—	2	1	no	—
96	C₂²⋊Dic₁₉	—	2	1	no	—
97	C₂²⋊Dic₁₉	—	2	1	no	—
98	C₂²⋊Dic₁₉	—	8	2	no	—
99	(C₂×D₁₉)⋊C₄	—	2	1	no	—
100	(C₂×D₁₉)⋊C₄	—	2	1	no	—
101	C₄⋊Dic₁₉	—	4	2	no	—
102	C₄⋊Dic₁₉	—	4	2	no	—
103	C₄⋊Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
104	C₄⋊Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
105	Dic₁₉⋊C₄	—	2	1	no	—
106	Dic₁₉⋊C₄	—	2	1	no	—
107	C₄×Dic₁₉	—	2	1	no	—
108	#64520	—	4	1	no	—
109	#64520	—	4	1	no	—
110	C₂×C₁₉⋊C₈	—	2	1	no	—
111	C₂×C₁₉⋊C₈	—	2	1	no	—
112	#64515	—	4	2	no	—
113	C₈×D₁₉	—	4	2	no	—
114	#8628	—	19	1	no	maximal, Sylow
115	C₂³×C₁₉	—	2	1	no	—
116	C₂³×C₁₉	—	2	1	no	—
117	C₂³×C₁₉	—	4	1	no	—
118	C₂³×C₁₉	—	4	1	no	—
119	C₂³×C₁₉	#117	1	1	yes	characteristic
120	C₂³×C₁₉	#121	1	1	yes	characteristic
121	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂²×D₁₉	—	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	—	2	1	no	—
127	C₂×C₄×C₁₉	—	2	1	no	—
128	C₂×C₄×C₁₉	—	2	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	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
135	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
136	C₂²⋊D₁₉	—	4	2	no	—
137	C₂²⋊D₁₉	—	4	2	no	—
138	C₂²⋊D₁₉	—	4	1	no	—
139	C₂²⋊D₁₉	—	4	1	no	—
140	C₂×Dic₁₉	—	2	1	no	—
141	C₂×Dic₁₉	—	2	1	no	—
142	C₂×Dic₁₉	—	2	1	no	—
143	C₂×Dic₁₉	—	8	2	no	—
144	C₂×Dic₁₉	—	8	2	no	—
145	C₂×Dic₁₉	—	8	2	no	—
146	D₇₆	—	4	1	no	—
147	C₄×D₁₉	—	4	1	no	—
148	C₄×D₁₉	—	4	1	no	—
149	Dic₃₈	—	4	1	no	—
150	C₈×C₁₉	—	2	1	no	—
151	C₈×C₁₉	—	2	1	no	—
152	C₈×C₁₉	—	2	1	no	—
153	C₈×C₁₉	—	8	2	no	—
154	C₈×C₁₉	#99	1	2	yes	—
155	C₁₉⋊C₈	—	4	1	no	—
156	C₁₉⋊C₈	—	4	1	no	—
157	#2929	—	19	1	no	—
158	#1810	D₁₉	1	1	yes	characteristic, complemented
159	#1808	—	19	1	no	—
160	#1513	—	19	2	no	—
161	#1305	—	19	2	no	—
162	C₂²×C₁₉	—	2	1	no	—
163	C₂²×C₁₉	—	2	1	no	—
164	C₂²×C₁₉	—	2	1	no	—
165	C₂²×C₁₉	—	2	1	no	—
166	C₂²×C₁₉	—	2	1	no	—
167	C₂²×C₁₉	—	2	1	no	—
168	C₂²×C₁₉	—	2	1	no	—
169	C₂²×C₁₉	—	2	1	no	—
170	C₂²×C₁₉	—	4	2	no	—
171	C₂²×C₁₉	—	4	2	no	—
172	C₂²×C₁₉	#380	1	1	yes	characteristic
173	C₂²×C₁₉	#472	1	1	yes	characteristic
174	C₂²×C₁₉	#473	1	1	yes	characteristic
175	C₂×D₁₉	—	4	1	no	—
176	C₂×D₁₉	—	4	1	no	—
177	C₂×D₁₉	—	4	1	no	—
178	C₄×C₁₉	—	2	1	no	—
179	C₄×C₁₉	—	2	1	no	—
180	C₄×C₁₉	—	2	1	no	—
181	C₄×C₁₉	C₂×#99	1	1	yes	characteristic
182	C₄×C₁₉	#410	1	1	yes	characteristic
183	Dic₁₉	—	4	1	no	—
184	Dic₁₉	—	4	1	no	—
185	Dic₁₉	—	4	1	no	—
186	Dic₁₉	—	16	2	no	—
187	C₂×#131	—	19	1	no	—
188	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
189	#525	—	19	1	no	—
190	#435	—	38	1	no	—
191	#435	—	38	1	no	—
192	C₈×D₄	—	38	1	no	—
193	C₈×D₄	—	38	1	no	—
194	C₂×#108	—	2	1	no	—
195	C₂×#107	—	38	1	no	—
196	#424	—	19	1	no	—
197	#408	—	19	1	no	—
198	#407	—	19	1	no	—
199	#407	C₂×D₁₉	1	2	yes	—
200	#405	—	19	1	no	—
201	#380	—	19	2	no	—
202	#341	—	2	1	no	—
203	#340	—	38	1	no	—
204	#328	—	38	2	no	—
205	#327	—	38	2	no	—
206	C₂×C₁₉	—	2	1	no	—
207	C₂×C₁₉	—	2	1	no	—
208	C₂×C₁₉	—	4	1	no	—
209	C₂×C₁₉	—	4	1	no	—
210	C₂×C₁₉	#1799	1	1	yes	characteristic
211	C₂×C₁₉	#1814	1	1	yes	characteristic
212	C₂×C₁₉	#1816	1	1	yes	characteristic
213	D₁₉	—	8	1	no	—
214	C₂×#41	—	19	1	no	—
215	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
216	#131	—	76	2	no	—
217	C₂×M₁₆	—	2	2	no	—
218	C₂×M₁₆	—	19	1	no	—
219	C₂×M₁₆	—	19	1	no	—
220	C₂×M₁₆	—	38	1	no	—
221	C₂²×C₈	—	4	2	no	—
222	C₂²×C₈	—	38	1	no	—
223	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
224	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
225	C₂²×C₈	D₇₆	1	1	yes	characteristic
226	C₂²×C₈	D₇₆	1	1	yes	characteristic
227	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
228	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
229	#123	—	38	1	no	—
230	C₂²⋊Q₈	—	19	1	no	—
231	C₂²⋊Q₈	—	19	1	no	—
232	#121	—	19	1	no	—
233	#121	—	19	1	no	—
234	#120	—	38	1	no	—
235	C₄×D₄	—	38	1	no	—
236	C₄×D₄	—	38	1	no	—
237	#117	—	19	1	no	—
238	C₂×C₄⋊C₄	—	38	2	no	—
239	C₂×C₂²⋊C₄	—	38	2	no	—
240	#108	—	4	2	no	—
241	#107	—	76	2	no	—
242	C₄⋊C₈	—	38	1	no	—
243	C₄⋊C₈	—	38	1	no	—
244	C₂²⋊C₈	—	2	2	no	—
245	C₂²⋊C₈	—	2	2	no	—
246	C₂²⋊C₈	—	38	1	no	—
247	C₂²⋊C₈	—	38	1	no	—
248	C₂²⋊C₈	—	38	1	no	—
249	C₂²⋊C₈	—	38	1	no	—
250	#97	—	38	1	no	—
251	C₄×C₈	—	38	1	no	—
252	#95	—	38	2	no	—
253	C₁₉	#8628	1	1	yes	characteristic, complemented, Sylow
254	C₂⁴	#64545	1	1	yes	characteristic
255	#41	—	76	1	no	—
256	#41	—	76	1	no	—
257	C₂×Q₈	—	19	1	no	—
258	C₂×D₄	—	19	1	no	—
259	C₂×D₄	—	38	1	no	—
260	C₂×D₄	—	38	1	no	—
261	C₂×D₄	—	38	1	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	4	1	no	—
264	C₂²×C₄	—	38	1	no	—
265	C₂²×C₄	—	76	2	no	—
266	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
268	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
269	C₂²×C₄	#64543	1	1	yes	characteristic
270	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
271	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
272	M₁₆	—	4	2	no	—
273	M₁₆	—	4	2	no	—
274	M₁₆	—	76	2	no	—
275	M₁₆	—	76	1	no	—
276	M₁₆	—	76	1	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	4	1	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	4	1	no	—
283	C₂×C₈	—	4	2	no	—
284	C₂×C₈	—	38	1	no	—
285	C₂×C₈	—	38	1	no	—
286	C₂×C₈	—	76	2	no	—
287	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
288	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
289	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
290	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
291	C₄⋊C₄	—	19	1	no	—
292	C₄⋊C₄	—	19	1	no	—
293	C₄⋊C₄	—	38	1	no	—
294	C₄⋊C₄	—	38	1	no	—
295	C₄⋊C₄	—	76	2	no	—
296	C₄⋊C₄	—	76	2	no	—
297	C₂²⋊C₄	—	38	1	no	—
298	C₂²⋊C₄	—	38	1	no	—
299	C₂²⋊C₄	—	38	1	no	—
300	C₂²⋊C₄	—	38	1	no	—
301	C₂²⋊C₄	—	38	1	no	—
302	C₂²⋊C₄	—	152	2	no	—
303	C₄²	—	38	1	no	—
304	C₂³	—	2	1	no	—
305	C₂³	—	2	1	no	—
306	C₂³	—	4	1	no	—
307	C₂³	—	4	1	no	—
308	C₂³	—	38	1	no	—
309	C₂³	#10597338	1	1	yes	characteristic
310	C₂³	#10597341	1	1	yes	characteristic
311	C₂³	#10597392	1	1	yes	characteristic
312	Q₈	—	76	1	no	—
313	D₄	—	76	1	no	—
314	D₄	—	76	2	no	—
315	D₄	—	76	2	no	—
316	D₄	—	76	1	no	—
317	D₄	—	76	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	38	1	no	—
328	C₂×C₄	—	38	1	no	—
329	C₂×C₄	—	38	1	no	—
330	C₂×C₄	—	76	1	no	—
331	C₂×C₄	—	76	1	no	—
332	C₂×C₄	—	152	2	no	—
333	C₂×C₄	—	152	2	no	—
334	C₂×C₄	—	152	2	no	—
335	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
336	C₂×C₄	#10597343	1	1	yes	characteristic
337	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
338	C₂×C₄	#10597384	1	1	yes	characteristic
339	C₈	—	2	1	no	—
340	C₈	—	2	1	no	—
341	C₈	—	2	1	no	—
342	C₈	—	8	2	no	—
343	C₈	—	76	1	no	—
344	C₈	—	76	1	no	—
345	C₈	#10597263	1	2	yes	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	4	2	no	—
355	C₂²	—	4	2	no	—
356	C₂²	—	76	1	no	—
357	C₂²	—	76	1	no	—
358	C₂²	—	76	1	no	—
359	C₂²	#11926401	1	1	yes	characteristic
360	C₂²	#11926815	1	1	yes	characteristic
361	C₂²	#11926820	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	304	2	no	—
366	C₄	—	76	1	no	—
367	C₄	—	76	1	no	—
368	C₄	—	76	1	no	—
369	C₄	C₂×#10597263	1	1	yes	characteristic
370	C₄	#11926517	1	1	yes	characteristic
371	C₂	—	2	1	no	—
372	C₂	—	2	1	no	—
373	C₂	—	4	1	no	—
374	C₂	—	4	1	no	—
375	C₂	—	152	1	no	—
376	C₂	#14563620	1	1	yes	characteristic
377	C₂	#14563690	1	1	yes	characteristic
378	C₂	#14563696	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926401
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₁₉×#1810	C₂
socle	C₂²×C₁₉	#380
prime radical	#11927055	C₂²
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#1810	D₁₉
hypocenter	C₁₉	#8628
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1810

Complements

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

N	H ≅ G/N	type
2. #14563664	C₂	semidirect
4. #14561542	C₂	semidirect
5. C₁₉×#1810	C₂	semidirect
6. C₂×#10597347	C₄	semidirect
13. #11926570	C₄	semidirect
14. #11926510	C₄	semidirect
16. #11926499	C₄	semidirect
158. #1810	D₁₉	semidirect
253. C₁₉	#8628	semidirect

Maximal subgroups

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

subgroup	index
#14572329	2
#14563664	2
#14562179	2
#14561542	2
C₁₉×#1810	2
#8628	19