Subgroups — #18070968

Order 4864.

488 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563397	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#14561461	C₂	1	2	yes	maximal, complemented
4	#14561458	C₂	1	2	yes	maximal, complemented
5	C₁₉×#2855	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
7	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
8	C₂×#10597260	—	2	1	no	—
9	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
10	#11926299	C₂²	1	2	yes	—
11	#11926299	C₂²	1	2	yes	—
12	C₂×C₁₉×C₂²⋊Q₈	C₄	1	1	yes	characteristic
13	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
14	C₁₉×#391	C₄	1	1	yes	characteristic
15	C₁₉×#391	C₂²	1	1	yes	characteristic
16	C₁₉×#389	C₄	1	1	yes	characteristic
17	C₁₉×#389	C₂²	1	1	yes	characteristic
18	C₁₉×#387	—	2	1	no	—
19	C₁₉×#387	—	2	1	no	—
20	C₁₉×#385	—	2	1	no	—
21	C₁₉×#381	—	2	1	no	—
22	C₁₉×#380	C₄	1	1	yes	characteristic
23	C₂²×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
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×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	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
35	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
36	C₂×C₄²×C₁₉	—	4	2	no	—
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	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
41	C₂²×C₁₉⋊C₈	—	2	2	no	—
42	C₂²×C₁₉⋊C₈	—	2	2	no	—
43	C₁₉×#95	—	2	1	no	—
44	C₁₉×#95	—	2	1	no	—
45	C₁₉×#95	—	2	1	no	—
46	C₁₉×#95	—	2	1	no	—
47	C₁₉×#95	—	2	1	no	—
48	C₁₉×#95	—	2	1	no	—
49	C₁₉×#95	—	2	1	no	—
50	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
51	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
52	#10597260	—	4	2	no	—
53	C₄×C₁₉⋊C₈	—	4	2	no	—
54	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Q₈×C₁₉	—	2	2	no	—
56	C₂×Q₈×C₁₉	—	4	1	no	—
57	C₂×Q₈×C₁₉	—	4	1	no	—
58	C₂²×C₄×C₁₉	—	2	2	no	—
59	C₂²×C₄×C₁₉	—	2	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	1	no	—
67	C₂²×C₄×C₁₉	—	2	1	no	—
68	C₂²×C₄×C₁₉	—	4	1	no	—
69	C₂²×C₄×C₁₉	—	4	1	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	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
77	C₄⋊C₄×C₁₉	—	2	2	no	—
78	C₄⋊C₄×C₁₉	—	4	1	no	—
79	C₄⋊C₄×C₁₉	—	4	1	no	—
80	C₄⋊C₄×C₁₉	—	4	2	no	—
81	C₄⋊C₄×C₁₉	—	4	2	no	—
82	C₄⋊C₄×C₁₉	—	4	2	no	—
83	C₂²⋊C₄×C₁₉	—	4	2	no	—
84	C₂²⋊C₄×C₁₉	—	4	2	no	—
85	C₄²×C₁₉	—	2	2	no	—
86	C₄²×C₁₉	—	2	2	no	—
87	C₄²×C₁₉	—	4	1	no	—
88	C₄²×C₁₉	—	4	1	no	—
89	C₂×C₁₉⋊C₈	—	2	4	no	—
90	C₂×C₁₉⋊C₈	—	2	4	no	—
91	C₂×C₁₉⋊C₈	—	4	2	no	—
92	C₂×C₁₉⋊C₈	—	4	2	no	—
93	C₂×C₁₉⋊C₈	—	4	2	no	—
94	C₂×C₁₉⋊C₈	—	4	2	no	—
95	#7675	—	19	1	no	maximal, Sylow
96	C₂³×C₁₉	—	2	1	no	—
97	C₂³×C₁₉	—	2	1	no	—
98	C₂³×C₁₉	—	2	1	no	—
99	C₂³×C₁₉	—	2	1	no	—
100	C₂³×C₁₉	—	2	1	no	—
101	C₂³×C₁₉	—	2	1	no	—
102	C₂³×C₁₉	#104	1	2	yes	—
103	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	Q₈×C₁₉	—	8	2	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	4	no	—
119	C₂×C₄×C₁₉	—	4	4	no	—
120	C₂×C₄×C₁₉	—	4	4	no	—
121	C₂×C₄×C₁₉	—	4	1	no	—
122	C₂×C₄×C₁₉	—	4	1	no	—
123	C₂×C₄×C₁₉	—	4	1	no	—
124	C₂×C₄×C₁₉	—	4	1	no	—
125	C₂×C₄×C₁₉	—	4	1	no	—
126	C₂×C₄×C₁₉	—	4	1	no	—
127	C₂×C₄×C₁₉	—	4	1	no	—
128	C₂×C₄×C₁₉	—	4	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	—	4	1	no	—
133	C₂×C₄×C₁₉	#99	1	2	yes	—
134	C₂×C₄×C₁₉	#104	1	2	yes	—
135	C₂×C₄×C₁₉	#131	1	2	yes	—
136	C₂×C₄×C₁₉	#135	1	2	yes	—
137	C₂×C₄×C₁₉	#137	1	2	yes	—
138	C₁₉⋊C₈	—	4	4	no	—
139	C₁₉⋊C₈	—	4	4	no	—
140	#2855	D₁₉	1	1	yes	characteristic
141	#1749	—	19	1	no	—
142	C₄×C₂²⋊C₈	—	19	1	no	—
143	#1272	—	19	2	no	—
144	#1271	—	19	2	no	—
145	C₂²×C₁₉	—	2	2	no	—
146	C₂²×C₁₉	—	2	2	no	—
147	C₂²×C₁₉	—	2	2	no	—
148	C₂²×C₁₉	—	4	1	no	—
149	C₂²×C₁₉	—	4	1	no	—
150	C₂²×C₁₉	—	4	1	no	—
151	C₂²×C₁₉	—	4	1	no	—
152	C₂²×C₁₉	#408	1	1	yes	characteristic
153	C₂²×C₁₉	C₂×#99	1	1	yes	characteristic
154	C₂²×C₁₉	#413	1	1	yes	characteristic
155	C₂²×C₁₉	#416	1	1	yes	characteristic
156	C₂²×C₁₉	#419	1	1	yes	characteristic
157	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
158	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
159	C₄×C₁₉	—	2	2	no	—
160	C₄×C₁₉	—	2	2	no	—
161	C₄×C₁₉	—	2	2	no	—
162	C₄×C₁₉	—	2	2	no	—
163	C₄×C₁₉	—	2	2	no	—
164	C₄×C₁₉	—	4	1	no	—
165	C₄×C₁₉	—	8	1	no	—
166	C₄×C₁₉	—	8	1	no	—
167	C₄×C₁₉	—	8	1	no	—
168	C₄×C₁₉	—	8	1	no	—
169	C₂×C₂²⋊Q₈	Dic₁₉	1	1	yes	characteristic
170	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
171	C₂×C₂²⋊C₈	—	38	1	no	—
172	C₂×C₂²⋊C₈	—	38	1	no	—
173	C₂×#97	—	38	1	no	—
174	C₂×C₄×C₈	—	38	1	no	—
175	#391	Dic₁₉	1	1	yes	characteristic
176	#391	C₂×D₁₉	1	1	yes	characteristic
177	#389	Dic₁₉	1	1	yes	characteristic
178	#389	C₂×D₁₉	1	1	yes	characteristic
179	#387	—	2	1	no	—
180	#387	—	2	1	no	—
181	#385	—	2	1	no	—
182	#381	—	2	1	no	—
183	#380	Dic₁₉	1	1	yes	characteristic
184	#336	—	19	2	no	—
185	#336	—	19	2	no	—
186	C₂×C₁₉	—	4	2	no	—
187	C₂×C₁₉	#1468	1	1	yes	characteristic
188	C₂×C₁₉	#1485	1	1	yes	characteristic
189	C₂×C₁₉	#1494	1	1	yes	characteristic
190	C₂×C₁₉	#1501	1	1	yes	characteristic
191	C₂×C₁₉	#1508	1	1	yes	characteristic
192	C₂×C₁₉	#1524	1	1	yes	characteristic
193	C₂×C₁₉	#1554	1	1	yes	characteristic
194	C₂²×Q₈	C₂×Dic₁₉	1	1	yes	characteristic
195	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
196	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
197	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
198	C₂²×C₈	—	38	2	no	—
199	C₂²×C₈	—	38	2	no	—
200	C₂²⋊Q₈	—	4	2	no	—
201	C₂×C₄⋊C₄	—	2	1	no	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
205	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
206	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
207	C₂×C₂²⋊C₄	—	2	1	no	—
208	C₂×C₂²⋊C₄	—	2	1	no	—
209	C₂×C₄²	—	4	2	no	—
210	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
211	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
212	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
213	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
214	C₂²⋊C₈	—	38	4	no	—
215	C₂²⋊C₈	—	38	4	no	—
216	#97	—	76	2	no	—
217	C₄×C₈	—	76	2	no	—
218	#95	—	2	1	no	—
219	#95	—	2	1	no	—
220	#95	—	2	1	no	—
221	#95	—	2	1	no	—
222	#95	—	2	1	no	—
223	#95	—	2	1	no	—
224	#95	—	2	1	no	—
225	C₁₉	#7675	1	1	yes	characteristic, complemented, Sylow
226	C₂⁴	C₂²⋊Dic₁₉	1	1	yes	characteristic
227	C₂×Q₈	—	2	2	no	—
228	C₂×Q₈	—	4	1	no	—
229	C₂×Q₈	—	4	1	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	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	—	2	1	no	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
244	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
245	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
246	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
249	C₂×C₈	—	38	4	no	—
250	C₂×C₈	—	38	4	no	—
251	C₂×C₈	—	76	2	no	—
252	C₂×C₈	—	76	2	no	—
253	C₂×C₈	—	76	2	no	—
254	C₂×C₈	—	76	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₄⋊C₄	—	4	1	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₂²⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	4	1	no	—
266	C₄²	—	4	1	no	—
267	C₂³	—	2	1	no	—
268	C₂³	—	2	1	no	—
269	C₂³	—	2	1	no	—
270	C₂³	—	2	1	no	—
271	C₂³	—	2	1	no	—
272	C₂³	—	2	1	no	—
273	C₂³	#10597294	1	2	yes	—
274	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
275	Q₈	—	8	2	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	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	—	4	4	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	#10597291	1	2	yes	—
305	C₂×C₄	#10597294	1	2	yes	—
306	C₂×C₄	#10597388	1	2	yes	—
307	C₂×C₄	#10597394	1	2	yes	—
308	C₂×C₄	#10597396	1	2	yes	—
309	C₈	—	76	4	no	—
310	C₈	—	76	4	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	#11926511	1	1	yes	characteristic
319	C₂²	C₂×#10597291	1	1	yes	characteristic
320	C₂²	#11926528	1	1	yes	characteristic
321	C₂²	#11926540	1	1	yes	characteristic
322	C₂²	#11926553	1	1	yes	characteristic
323	C₂²	C₂×#10597294	1	1	yes	characteristic
324	C₂²	C₂×#10597294	1	1	yes	characteristic
325	C₄	—	2	2	no	—
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	4	1	no	—
331	C₄	—	8	1	no	—
332	C₄	—	8	1	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	1	no	—
335	C₂	—	4	2	no	—
336	C₂	#14561983	1	1	yes	characteristic
337	C₂	#14562056	1	1	yes	characteristic
338	C₂	#14562095	1	1	yes	characteristic
339	C₂	#14562126	1	1	yes	characteristic
340	C₂	#14562155	1	1	yes	characteristic
341	C₂	#14562223	1	1	yes	characteristic
342	C₂	#14562345	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×#2855	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2855	D₁₉
hypocenter	C₁₉	#7675
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #2855

Complements

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

N	H ≅ G/N	type
3. #14561461	C₂	semidirect
4. #14561458	C₂	semidirect
225. C₁₉	#7675	semidirect

Maximal subgroups

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

subgroup	index
#14563397	2
C₄×C₂²⋊(C₁₉⋊C₈)	2
#14561461	2
#14561458	2
C₁₉×#2855	2
#7675	19