Subgroups — #18082726

Order 4864.

592 conjugacy classes of subgroups.

The subgroup lattice has 358 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	#14561862	C₂	1	1	yes	maximal, characteristic
3	#14561513	C₂	1	4	yes	maximal
4	C₁₉×#2857	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
6	C₂×C₂²⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₁₉⋊#95	—	2	1	no	—
8	C₂×C₁₉⋊#95	—	2	1	no	—
9	C₂×C₁₉⋊#95	C₂²	1	2	yes	—
10	#11926299	—	2	1	no	—
11	C₂×C₁₉×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
12	C₂×C₁₉×C₂²⋊Q₈	—	2	1	no	—
13	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
14	C₁₉×#391	—	2	1	no	—
15	C₁₉×#389	—	2	1	no	—
16	C₁₉×#389	C₄	1	2	yes	complemented
17	C₁₉×#384	C₄	1	1	yes	characteristic, complemented
18	C₁₉×#380	—	2	1	no	—
19	C₁₉×#380	C₂²	1	2	yes	—
20	C₂²×Q₈×C₁₉	—	2	1	no	—
21	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
24	C₂³×Dic₁₉	—	2	2	no	—
25	C₁₉×C₄⋊Q₈	—	4	2	no	—
26	C₁₉×C₂²⋊Q₈	—	4	2	no	—
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	—	2	2	no	—
32	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
34	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
35	C₂×C₄²×C₁₉	—	4	2	no	—
36	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
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	C₂²×C₁₉⋊C₈	—	2	2	no	—
41	C₁₉×#95	—	2	2	no	—
42	C₁₉×#95	—	2	2	no	—
43	C₁₉⋊#95	—	2	2	no	—
44	C₁₉⋊#95	—	4	2	no	—
45	C₁₉⋊#95	—	4	2	no	—
46	C₁₉⋊#95	Q₈	1	4	yes	complemented
47	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
48	C₄⋊(C₁₉⋊C₈)	—	4	2	no	—
49	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
50	C₂×Q₈×C₁₉	—	4	2	no	—
51	C₂×Q₈×C₁₉	—	4	2	no	—
52	C₂×Q₈×C₁₉	—	4	2	no	—
53	C₂²×C₄×C₁₉	—	2	2	no	—
54	C₂²×C₄×C₁₉	—	2	2	no	—
55	C₂²×C₄×C₁₉	—	2	2	no	—
56	C₂²×C₄×C₁₉	—	2	2	no	—
57	C₂²×C₄×C₁₉	—	4	1	no	—
58	C₂²×C₄×C₁₉	—	4	1	no	—
59	C₂²×C₄×C₁₉	—	4	1	no	—
60	C₂²×C₄×C₁₉	—	4	1	no	—
61	C₂²×C₄×C₁₉	—	4	1	no	—
62	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	D₈	1	2	yes	—
67	C₂²×C₄×C₁₉	SD₁₆	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	2	yes	—
71	C₂²×C₄×C₁₉	#41	1	2	yes	—
72	C₂²×Dic₁₉	—	2	4	no	—
73	C₂²×Dic₁₉	—	4	2	no	—
74	C₂²×Dic₁₉	—	4	2	no	—
75	C₂²×Dic₁₉	—	4	2	no	—
76	C₂²×Dic₁₉	—	4	2	no	—
77	C₂²×Dic₁₉	—	4	2	no	—
78	C₂²×Dic₁₉	—	4	2	no	—
79	C₄⋊C₄×C₁₉	—	4	2	no	—
80	C₄⋊C₄×C₁₉	—	4	2	no	—
81	C₄⋊C₄×C₁₉	—	4	2	no	—
82	C₄⋊C₄×C₁₉	—	4	2	no	—
83	C₄⋊C₄×C₁₉	—	4	2	no	—
84	C₄⋊C₄×C₁₉	—	4	4	no	—
85	C₂²⋊C₄×C₁₉	—	4	2	no	—
86	C₂²⋊C₄×C₁₉	—	4	2	no	—
87	C₄²×C₁₉	—	2	2	no	—
88	C₄²×C₁₉	—	2	2	no	—
89	C₄²×C₁₉	—	4	1	no	—
90	C₄²×C₁₉	—	4	1	no	—
91	C₂×C₁₉⋊C₈	—	4	2	no	—
92	C₂×C₁₉⋊C₈	—	4	2	no	—
93	C₂×C₁₉⋊C₈	—	4	2	no	—
94	#9912	—	19	1	no	maximal, Sylow
95	C₂³×C₁₉	—	2	1	no	—
96	C₂³×C₁₉	—	2	1	no	—
97	C₂³×C₁₉	—	2	1	no	—
98	C₂³×C₁₉	—	2	1	no	—
99	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂³×C₁₉	C₂²⋊Q₈	1	2	yes	—
101	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
102	C₂³×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
103	C₂³×C₁₉	#137	1	2	yes	—
104	Q₈×C₁₉	—	8	4	no	—
105	C₂×C₄×C₁₉	—	2	1	no	—
106	C₂×C₄×C₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	—	2	1	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	—	4	2	no	—
119	C₂×C₄×C₁₉	—	4	2	no	—
120	C₂×C₄×C₁₉	—	4	2	no	—
121	C₂×C₄×C₁₉	—	4	2	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	#99	1	2	yes	—
130	C₂×C₄×C₁₉	#102	1	2	yes	—
131	C₂×C₄×C₁₉	#102	1	2	yes	—
132	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
133	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
134	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
135	C₂×Dic₁₉	—	4	4	no	—
136	C₂×Dic₁₉	—	4	4	no	—
137	C₂×Dic₁₉	—	4	4	no	—
138	C₂×Dic₁₉	—	8	2	no	—
139	C₂×Dic₁₉	—	8	2	no	—
140	C₂×Dic₁₉	—	8	2	no	—
141	C₂×Dic₁₉	—	8	2	no	—
142	C₁₉⋊C₈	—	8	2	no	—
143	#2857	D₁₉	1	1	yes	characteristic
144	#1797	—	19	1	no	—
145	#1434	—	19	1	no	—
146	#1294	—	19	4	no	—
147	C₂²×C₁₉	—	2	2	no	—
148	C₂²×C₁₉	—	2	2	no	—
149	C₂²×C₁₉	—	2	2	no	—
150	C₂²×C₁₉	—	2	2	no	—
151	C₂²×C₁₉	—	2	1	no	—
152	C₂²×C₁₉	—	2	1	no	—
153	C₂²×C₁₉	—	2	1	no	—
154	C₂²×C₁₉	—	2	1	no	—
155	C₂²×C₁₉	#391	1	1	yes	characteristic
156	C₂²×C₁₉	C₂×#99	1	1	yes	characteristic
157	C₂²×C₁₉	#410	1	1	yes	characteristic
158	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
159	C₂²×C₁₉	#419	1	1	yes	characteristic
160	C₂²×C₁₉	#421	1	1	yes	characteristic
161	C₂²×C₁₉	#421	1	1	yes	characteristic
162	C₂²×C₁₉	#474	1	2	yes	—
163	C₂²×C₁₉	#475	1	2	yes	—
164	C₄×C₁₉	—	2	1	no	—
165	C₄×C₁₉	—	2	2	no	—
166	C₄×C₁₉	—	2	1	no	—
167	C₄×C₁₉	—	2	2	no	—
168	C₄×C₁₉	—	2	2	no	—
169	C₄×C₁₉	—	2	2	no	—
170	C₄×C₁₉	—	4	1	no	—
171	C₄×C₁₉	—	8	2	no	—
172	C₄×C₁₉	—	8	2	no	—
173	Dic₁₉	—	8	4	no	—
174	C₂×C₄⋊Q₈	Dic₁₉	1	1	yes	characteristic, complemented
175	C₂×C₂²⋊Q₈	—	2	1	no	—
176	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
177	C₂×C₄⋊C₈	—	38	1	no	—
178	C₂×C₂²⋊C₈	—	19	2	no	—
179	#391	—	2	1	no	—
180	#389	—	2	1	no	—
181	#389	Dic₁₉	1	2	yes	complemented
182	#384	Dic₁₉	1	1	yes	characteristic, complemented
183	#380	—	2	1	no	—
184	#380	C₂×D₁₉	1	2	yes	—
185	C₂×#95	—	19	2	no	—
186	C₂×#95	—	38	1	no	—
187	C₂×#95	—	38	1	no	—
188	#336	—	38	1	no	—
189	C₂×C₁₉	—	2	2	no	—
190	C₂×C₁₉	—	2	2	no	—
191	C₂×C₁₉	#1511	1	1	yes	characteristic
192	C₂×C₁₉	#1515	1	1	yes	characteristic
193	C₂×C₁₉	#1517	1	1	yes	characteristic
194	C₂×C₁₉	#1519	1	1	yes	characteristic
195	C₂×C₁₉	#1988	1	1	yes	characteristic
196	C₂×C₁₉	#1995	1	1	yes	characteristic
197	C₂×C₁₉	#1996	1	1	yes	characteristic
198	C₂²×Q₈	—	2	1	no	—
199	C₂³×C₄	—	38	2	no	—
200	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
201	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
202	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
203	C₂²×C₈	—	38	2	no	—
204	C₄⋊Q₈	—	4	2	no	—
205	C₂²⋊Q₈	—	4	2	no	—
206	C₂²⋊Q₈	—	4	2	no	—
207	C₂×C₄⋊C₄	—	2	1	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	C₂×Dic₁₉	1	2	yes	—
212	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
213	C₂×C₂²⋊C₄	—	2	2	no	—
214	C₂×C₄²	—	4	2	no	—
215	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
216	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
217	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
218	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
219	C₄⋊C₈	—	76	2	no	—
220	C₂²⋊C₈	—	38	4	no	—
221	#95	—	2	2	no	—
222	#95	—	2	2	no	—
223	#95	—	19	4	no	—
224	#95	—	38	2	no	—
225	#95	—	76	2	no	—
226	#95	—	76	2	no	—
227	C₁₉	#9912	1	1	yes	characteristic, complemented, Sylow
228	C₂⁴	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
229	C₂×Q₈	—	4	2	no	—
230	C₂×Q₈	—	4	2	no	—
231	C₂×Q₈	—	4	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	—	38	4	no	—
242	C₂²×C₄	—	76	2	no	—
243	C₂²×C₄	—	76	2	no	—
244	C₂²×C₄	—	76	2	no	—
245	C₂²×C₄	—	76	2	no	—
246	C₂²×C₄	—	76	2	no	—
247	C₂²×C₄	—	76	2	no	—
248	C₂²×C₄	#64525	1	2	yes	—
249	C₂²×C₄	#64526	1	2	yes	—
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	C₂²⋊Dic₁₉	1	1	yes	characteristic
254	C₂²×C₄	Q₈×D₁₉	1	2	yes	—
255	C₂²×C₄	#64545	1	2	yes	—
256	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
257	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
258	C₂×C₈	—	76	2	no	—
259	C₂×C₈	—	76	2	no	—
260	C₂×C₈	—	76	2	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	4	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₂²⋊C₄	—	4	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	4	1	no	—
272	C₄²	—	4	1	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	—	2	1	no	—
277	C₂³	C₂×#64525	1	1	yes	characteristic
278	C₂³	C₂×#64526	1	1	yes	characteristic
279	C₂³	#10597388	1	2	yes	—
280	C₂³	#10597392	1	2	yes	—
281	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
282	Q₈	—	8	4	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	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	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	4	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	—	4	2	no	—
307	C₂×C₄	—	76	4	no	—
308	C₂×C₄	—	76	4	no	—
309	C₂×C₄	—	76	4	no	—
310	C₂×C₄	—	152	2	no	—
311	C₂×C₄	—	152	2	no	—
312	C₂×C₄	—	152	2	no	—
313	C₂×C₄	—	152	2	no	—
314	C₂×C₄	#10597289	1	2	yes	—
315	C₂×C₄	#10597289	1	2	yes	—
316	C₂×C₄	#10597291	1	2	yes	—
317	C₂×C₄	#10597390	1	1	yes	characteristic
318	C₂×C₄	Q₈×Dic₁₉	1	2	yes	—
319	C₂×C₄	#10597393	1	1	yes	characteristic
320	C₈	—	152	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	#11926451	1	1	yes	characteristic
330	C₂²	C₂×#10597291	1	1	yes	characteristic
331	C₂²	#11926518	1	1	yes	characteristic
332	C₂²	C₂×#10597289	1	1	yes	characteristic
333	C₂²	#11926553	1	1	yes	characteristic
334	C₂²	#11926561	1	1	yes	characteristic
335	C₂²	#11926561	1	1	yes	characteristic
336	C₂²	#11926821	1	2	yes	—
337	C₂²	#11926826	1	2	yes	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	2	no	—
343	C₄	—	2	2	no	—
344	C₄	—	4	1	no	—
345	C₄	—	8	2	no	—
346	C₄	—	8	2	no	—
347	C₄	—	152	4	no	—
348	C₂	—	2	2	no	—
349	C₂	—	2	2	no	—
350	C₂	#14562168	1	1	yes	characteristic
351	C₂	#14562186	1	1	yes	characteristic
352	C₂	#14562195	1	1	yes	characteristic
353	C₂	#14562203	1	1	yes	characteristic
354	C₂	#14564588	1	1	yes	characteristic
355	C₂	#14564617	1	1	yes	characteristic
356	C₂	#14564621	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₁₉×#2857	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂×D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#2857	D₁₉
hypocenter	C₁₉	#9912
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
11. C₂×C₁₉×C₄⋊Q₈	C₄	semidirect
16. C₁₉×#389	C₄	semidirect
17. C₁₉×#384	C₄	semidirect
46. C₁₉⋊#95	Q₈	semidirect
174. C₂×C₄⋊Q₈	Dic₁₉	semidirect
181. #389	Dic₁₉	semidirect
182. #384	Dic₁₉	semidirect
227. C₁₉	#9912	semidirect

Maximal subgroups

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

subgroup	index
#14563615	2
#14561862	2
#14561513	2
C₁₉×#2857	2
#9912	19