Subgroups — #18076409

Order 4864.

454 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572675	C₂	1	1	yes	maximal, characteristic
2	#14564428	C₂	1	1	yes	maximal, characteristic
3	C₄×#10597280	C₂	1	1	yes	maximal, characteristic
4	#14562385	C₂	1	1	yes	maximal, characteristic, complemented
5	#14561473	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561471	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1565	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₇₆	C₂²	1	1	yes	characteristic
9	#11926623	—	2	1	no	—
10	#11926623	—	2	1	no	—
11	#11926621	—	2	1	no	—
12	#11926621	—	2	1	no	—
13	#11926597	C₂²	1	1	yes	characteristic
14	C₂×#10597280	C₄	1	1	yes	characteristic
15	C₂×#10597280	C₄	1	1	yes	characteristic
16	#11926506	—	2	2	no	—
17	C₂×(C₂×D₁₉)⋊C₈	—	2	2	no	—
18	C₄×#64520	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
20	#11926312	C₄	1	1	yes	characteristic
21	#11926312	C₄	1	1	yes	characteristic
22	#11926296	—	2	1	no	—
23	#11926294	—	2	1	no	—
24	C₁₉×#431	C₂²	1	1	yes	characteristic
25	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
27	C₁₉×#335	—	2	1	no	—
28	C₁₉×#334	—	2	1	no	—
29	C₂²×D₇₆	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×D₁₉	D₄	1	2	yes	—
31	C₂×M₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
32	C₂×M₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₉	—	2	1	no	—
34	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
35	C₂×(C₂×D₁₉)⋊C₄	Q₈	1	2	yes	—
36	C₂×C₄⋊Dic₁₉	C₂×C₄	1	1	yes	characteristic
37	C₂×#64520	C₂×C₄	1	1	yes	characteristic
38	C₂×#64520	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₁₉⋊C₈	—	2	1	no	—
40	C₂×#64515	—	2	2	no	—
41	C₂×#64515	—	2	2	no	—
42	C₄×D₇₆	—	2	1	no	—
43	C₄×D₇₆	—	2	1	no	—
44	C₄×D₇₆	—	2	1	no	—
45	C₄×D₇₆	—	2	1	no	—
46	C₁₉×C₄⋊C₈	—	2	1	no	—
47	C₁₉×C₄⋊C₈	—	2	1	no	—
48	C₁₉×C₄⋊C₈	D₄	1	1	yes	characteristic
49	C₁₉×C₄⋊C₈	D₄	1	1	yes	characteristic
50	C₁₉×C₂²⋊C₈	—	2	1	no	—
51	C₁₉×#97	—	2	1	no	—
52	C₄×C₈×C₁₉	—	2	1	no	—
53	C₄×C₈×C₁₉	—	2	1	no	—
54	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
55	#10597280	—	2	2	no	—
56	#10597280	—	2	2	no	—
57	(C₂×D₁₉)⋊C₈	—	2	2	no	—
58	(C₂×D₁₉)⋊C₈	—	2	2	no	—
59	(C₂×D₁₉)⋊C₈	—	4	4	no	—
60	C₄⋊(C₁₉⋊C₈)	D₄	1	1	yes	characteristic
61	C₄⋊(C₁₉⋊C₈)	D₄	1	1	yes	characteristic
62	#10597260	—	2	1	no	—
63	C₄×C₁₉⋊C₈	—	2	1	no	—
64	C₄×C₁₉⋊C₈	—	2	1	no	—
65	C₄×C₁₉⋊C₈	—	2	1	no	—
66	C₄×C₁₉⋊C₈	—	2	1	no	—
67	C₂³×D₁₉	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
71	C₂²×Dic₁₉	C₄⋊C₄	1	2	yes	—
72	C₂×D₇₆	—	2	1	no	—
73	C₂×D₇₆	—	2	1	no	—
74	C₂×D₇₆	—	2	1	no	—
75	C₂×D₇₆	—	2	1	no	—
76	C₂×D₇₆	—	2	1	no	—
77	C₂×D₇₆	—	2	1	no	—
78	C₂×C₄×D₁₉	—	2	2	no	—
79	C₂×C₄×D₁₉	—	2	2	no	—
80	C₂×C₄×D₁₉	—	4	2	no	—
81	C₂×C₄×D₁₉	—	4	2	no	—
82	M₁₆×C₁₉	—	2	1	no	—
83	M₁₆×C₁₉	—	2	1	no	—
84	M₁₆×C₁₉	—	2	1	no	—
85	M₁₆×C₁₉	M₁₆	1	2	yes	complemented
86	C₂×C₈×C₁₉	—	2	1	no	—
87	C₂×C₈×C₁₉	—	2	1	no	—
88	C₂×C₈×C₁₉	—	2	1	no	—
89	C₂×C₈×C₁₉	—	2	1	no	—
90	C₂×C₈×C₁₉	—	4	2	no	—
91	C₂×C₈×C₁₉	—	4	1	no	—
92	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₁₉	#41	1	1	yes	characteristic
95	C₄²×C₁₉	#41	1	1	yes	characteristic
96	(C₂×D₁₉)⋊C₄	—	2	2	no	—
97	(C₂×D₁₉)⋊C₄	—	2	2	no	—
98	C₄⋊Dic₁₉	—	2	1	no	—
99	C₄⋊Dic₁₉	—	2	1	no	—
100	#64520	—	2	1	no	—
101	#64520	—	2	1	no	—
102	#64520	—	2	1	no	—
103	#64520	M₁₆	1	2	yes	complemented
104	C₂×C₁₉⋊C₈	—	2	1	no	—
105	C₂×C₁₉⋊C₈	—	2	2	no	—
106	C₂×C₁₉⋊C₈	—	2	1	no	—
107	C₂×C₁₉⋊C₈	—	2	1	no	—
108	C₂×C₁₉⋊C₈	—	2	1	no	—
109	C₂×C₁₉⋊C₈	—	4	1	no	—
110	C₂×C₁₉⋊C₈	—	4	1	no	—
111	#64515	—	8	2	no	—
112	#64515	—	8	2	no	—
113	#8837	—	19	1	no	maximal, Sylow
114	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂²×D₁₉	—	2	2	no	—
116	C₂²×D₁₉	—	2	2	no	—
117	C₂²×D₁₉	—	2	2	no	—
118	C₂²×D₁₉	—	4	2	no	—
119	C₂²×D₁₉	—	4	2	no	—
120	C₂×C₄×C₁₉	—	2	1	no	—
121	C₂×C₄×C₁₉	—	2	1	no	—
122	C₂×C₄×C₁₉	—	4	1	no	—
123	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
129	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
130	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
131	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
132	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
133	C₂×Dic₁₉	—	2	2	no	—
134	C₂×Dic₁₉	—	4	2	no	—
135	D₇₆	—	4	1	no	—
136	D₇₆	—	4	1	no	—
137	D₇₆	—	4	1	no	—
138	D₇₆	—	4	1	no	—
139	C₄×D₁₉	—	4	2	no	—
140	C₄×D₁₉	—	4	2	no	—
141	C₄×D₁₉	—	8	2	no	—
142	C₈×C₁₉	—	2	2	no	—
143	C₈×C₁₉	—	4	1	no	—
144	C₈×C₁₉	—	4	1	no	—
145	C₈×C₁₉	—	4	1	no	—
146	C₈×C₁₉	—	8	1	no	—
147	C₁₉⋊C₈	—	2	2	no	—
148	C₁₉⋊C₈	—	4	2	no	—
149	C₁₉⋊C₈	—	4	1	no	—
150	C₁₉⋊C₈	—	4	1	no	—
151	C₁₉⋊C₈	—	4	1	no	—
152	#2975	—	19	1	no	—
153	#1954	—	19	1	no	—
154	C₄×#100	—	19	1	no	—
155	#1565	—	19	1	no	—
156	#1565	D₁₉	1	1	yes	characteristic, complemented
157	#1278	—	19	1	no	—
158	#1278	—	19	1	no	—
159	C₂²×C₁₉	—	2	1	no	—
160	C₂²×C₁₉	—	2	1	no	—
161	C₂²×C₁₉	#385	1	1	yes	characteristic
162	C₂²×C₁₉	#423	1	1	yes	characteristic
163	C₂²×C₁₉	#424	1	1	yes	characteristic
164	C₂×D₁₉	—	4	4	no	—
165	C₂×D₁₉	—	4	2	no	—
166	C₂×D₁₉	—	4	2	no	—
167	C₂×D₁₉	—	4	2	no	—
168	C₂×D₁₉	—	8	2	no	—
169	C₄×C₁₉	—	2	1	no	—
170	C₄×C₁₉	—	2	1	no	—
171	C₄×C₁₉	—	2	1	no	—
172	C₄×C₁₉	#435	1	1	yes	characteristic
173	C₄×C₁₉	#435	1	1	yes	characteristic
174	C₄×C₁₉	#436	1	1	yes	characteristic
175	C₄×C₁₉	#436	1	1	yes	characteristic
176	C₄×C₁₉	#453	1	1	yes	characteristic
177	C₄×C₁₉	#455	1	1	yes	characteristic
178	Dic₁₉	—	4	2	no	—
179	C₂×C₄×D₄	—	19	1	no	—
180	#436	—	38	1	no	—
181	#436	—	38	1	no	—
182	#436	—	38	1	no	—
183	#436	—	38	1	no	—
184	#431	—	19	1	no	—
185	#431	C₂×D₁₉	1	1	yes	characteristic
186	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
187	C₂×#100	—	19	1	no	—
188	C₂×#100	—	19	1	no	—
189	#407	—	38	2	no	—
190	C₂×C₂²⋊C₈	—	38	2	no	—
191	C₄×M₁₆	—	19	1	no	—
192	C₄×M₁₆	C₂×D₁₉	1	1	yes	characteristic
193	C₂×C₄×C₈	—	19	1	no	—
194	#343	—	19	1	no	—
195	#343	—	19	1	no	—
196	#335	—	2	1	no	—
197	#335	—	38	1	no	—
198	#334	—	2	1	no	—
199	#334	—	38	1	no	—
200	C₂×C₁₉	—	2	1	no	—
201	C₂×C₁₉	—	2	1	no	—
202	C₂×C₁₉	#1851	1	1	yes	characteristic
203	C₂×C₁₉	#1856	1	1	yes	characteristic
204	C₂×C₁₉	#1858	1	1	yes	characteristic
205	D₁₉	—	8	2	no	—
206	C₂²×D₄	—	19	1	no	—
207	C₂³×C₄	—	19	2	no	—
208	C₂×M₁₆	—	19	1	no	—
209	C₂×M₁₆	—	19	1	no	—
210	C₂×M₁₆	—	38	2	no	—
211	C₂×M₁₆	—	38	2	no	—
212	C₂×M₁₆	C₄×D₁₉	1	1	yes	characteristic
213	C₂×M₁₆	C₄×D₁₉	1	1	yes	characteristic
214	C₂²×C₈	—	2	1	no	—
215	C₂²×C₈	—	38	1	no	—
216	C₄×D₄	—	38	1	no	—
217	C₄×D₄	—	38	1	no	—
218	C₄×D₄	—	38	1	no	—
219	C₄×D₄	—	38	1	no	—
220	C₂×C₄⋊C₄	—	19	1	no	—
221	C₂×C₂²⋊C₄	—	19	2	no	—
222	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
223	C₄⋊C₈	—	2	1	no	—
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	19	1	no	—
226	C₄⋊C₈	—	19	1	no	—
227	C₄⋊C₈	C₂²⋊D₁₉	1	1	yes	characteristic
228	C₄⋊C₈	C₂²⋊D₁₉	1	1	yes	characteristic
229	#100	—	38	2	no	—
230	#100	—	38	2	no	—
231	C₂²⋊C₈	—	2	1	no	—
232	C₂²⋊C₈	—	38	2	no	—
233	C₂²⋊C₈	—	38	2	no	—
234	C₂²⋊C₈	—	38	1	no	—
235	C₂²⋊C₈	—	76	4	no	—
236	#97	—	2	1	no	—
237	#97	—	38	1	no	—
238	C₄×C₈	—	2	1	no	—
239	C₄×C₈	—	2	1	no	—
240	C₄×C₈	—	38	1	no	—
241	C₄×C₈	—	38	1	no	—
242	C₄×C₈	—	38	1	no	—
243	C₄×C₈	—	38	1	no	—
244	C₁₉	#8837	1	1	yes	characteristic, complemented, Sylow
245	C₂⁴	—	19	2	no	—
246	C₂×D₄	—	38	1	no	—
247	C₂×D₄	—	38	1	no	—
248	C₂×D₄	—	38	1	no	—
249	C₂×D₄	—	38	1	no	—
250	C₂×D₄	—	38	1	no	—
251	C₂×D₄	—	38	1	no	—
252	C₂²×C₄	—	19	2	no	—
253	C₂²×C₄	—	38	2	no	—
254	C₂²×C₄	—	38	2	no	—
255	C₂²×C₄	—	76	2	no	—
256	C₂²×C₄	—	76	2	no	—
257	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
258	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
259	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
260	M₁₆	—	2	1	no	—
261	M₁₆	—	2	1	no	—
262	M₁₆	—	2	1	no	—
263	M₁₆	—	19	2	no	—
264	M₁₆	—	38	1	no	—
265	M₁₆	—	38	1	no	—
266	M₁₆	—	38	1	no	—
267	M₁₆	—	152	2	no	—
268	M₁₆	—	152	2	no	—
269	M₁₆	#64515	1	2	yes	complemented
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	2	1	no	—
273	C₂×C₈	—	2	1	no	—
274	C₂×C₈	—	4	2	no	—
275	C₂×C₈	—	4	1	no	—
276	C₂×C₈	—	38	1	no	—
277	C₂×C₈	—	38	2	no	—
278	C₂×C₈	—	38	1	no	—
279	C₂×C₈	—	38	1	no	—
280	C₂×C₈	—	38	1	no	—
281	C₂×C₈	—	76	1	no	—
282	C₂×C₈	—	76	1	no	—
283	C₄⋊C₄	—	38	1	no	—
284	C₄⋊C₄	—	38	1	no	—
285	C₂²⋊C₄	—	38	2	no	—
286	C₂²⋊C₄	—	38	2	no	—
287	C₄²	#64541	1	1	yes	characteristic
288	C₄²	D₄×D₁₉	1	1	yes	characteristic
289	C₄²	#64545	1	1	yes	characteristic
290	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
291	C₂³	—	38	2	no	—
292	C₂³	—	38	2	no	—
293	C₂³	—	38	2	no	—
294	C₂³	—	76	2	no	—
295	C₂³	—	76	2	no	—
296	C₂³	C₄⋊C₄×D₁₉	1	1	yes	characteristic
297	D₄	—	76	1	no	—
298	D₄	—	76	1	no	—
299	D₄	—	76	1	no	—
300	D₄	—	76	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	38	2	no	—
305	C₂×C₄	—	76	2	no	—
306	C₂×C₄	—	76	2	no	—
307	C₂×C₄	—	76	2	no	—
308	C₂×C₄	—	152	2	no	—
309	C₂×C₄	#10597339	1	1	yes	characteristic
310	C₂×C₄	#10597340	1	1	yes	characteristic
311	C₂×C₄	#10597342	1	1	yes	characteristic
312	C₂×C₄	C₂×#64515	1	1	yes	characteristic
313	C₂×C₄	#10597347	1	1	yes	characteristic
314	C₂×C₄	M₁₆×D₁₉	1	1	yes	characteristic
315	C₂×C₄	#10597353	1	1	yes	characteristic
316	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
317	C₂×C₄	#10597383	1	1	yes	characteristic
318	C₂×C₄	#10597392	1	1	yes	characteristic
319	C₈	—	2	2	no	—
320	C₈	—	4	1	no	—
321	C₈	—	4	1	no	—
322	C₈	—	4	1	no	—
323	C₈	—	8	1	no	—
324	C₈	—	38	2	no	—
325	C₈	—	76	2	no	—
326	C₈	—	76	1	no	—
327	C₈	—	76	1	no	—
328	C₈	—	76	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	76	4	no	—
332	C₂²	—	76	2	no	—
333	C₂²	—	76	2	no	—
334	C₂²	—	76	2	no	—
335	C₂²	—	152	2	no	—
336	C₂²	#11926415	1	1	yes	characteristic
337	C₂²	#11926566	1	1	yes	characteristic
338	C₂²	#11926569	1	1	yes	characteristic
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	76	2	no	—
343	C₄	#11926614	1	1	yes	characteristic
344	C₄	#11926615	1	1	yes	characteristic
345	C₄	#11926620	1	1	yes	characteristic
346	C₄	#11926621	1	1	yes	characteristic
347	C₄	#11926710	1	1	yes	characteristic
348	C₄	#11926720	1	1	yes	characteristic
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	—	152	2	no	—
352	C₂	#14563816	1	1	yes	characteristic
353	C₂	#14563843	1	1	yes	characteristic
354	C₂	#14563853	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597383
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1565	C₂
socle	C₂²×C₁₉	#385
prime radical	C₂²×D₇₆	C₂×C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1565	D₁₉
hypocenter	C₁₉	#8837
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1565

Complements

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

N	H ≅ G/N	type
4. #14562385	C₂	semidirect
5. #14561473	C₂	semidirect
6. #14561471	C₂	semidirect
7. C₁₉×#1565	C₂	semidirect
85. M₁₆×C₁₉	M₁₆	semidirect
103. #64520	M₁₆	semidirect
156. #1565	D₁₉	semidirect
244. C₁₉	#8837	semidirect
269. M₁₆	#64515	semidirect

Maximal subgroups

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

subgroup	index
#14572675	2
#14564428	2
C₄×#10597280	2
#14562385	2
#14561473	2
#14561471	2
C₁₉×#1565	2
#8837	19