Subgroups — #18082614

Order 4864.

566 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	C₄×C₂²⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	#14561862	C₂	1	1	yes	maximal, characteristic
3	#14561513	C₂	1	2	yes	maximal
4	#14561513	C₂	1	2	yes	maximal
5	C₁₉×#2851	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂²⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
8	C₂×C₁₉⋊#95	—	2	1	no	—
9	C₂×C₁₉⋊#95	—	2	1	no	—
10	C₂×C₁₉⋊#95	C₂²	1	2	yes	—
11	#11926299	—	2	1	no	—
12	C₂×C₁₉×#125	C₄	1	1	yes	characteristic, complemented
13	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
14	C₁₉×#389	C₄	1	1	yes	characteristic, complemented
15	C₁₉×#389	C₄	1	1	yes	characteristic, complemented
16	C₁₉×#387	—	2	1	no	—
17	C₁₉×#387	—	2	1	no	—
18	C₁₉×#385	—	2	1	no	—
19	C₁₉×#385	—	2	1	no	—
20	C₁₉×#382	C₄	1	1	yes	characteristic, complemented
21	C₁₉×#380	C₂²	1	1	yes	characteristic
22	C₁₉×#380	C₂²	1	1	yes	characteristic
23	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
26	C₂³×Dic₁₉	—	2	2	no	—
27	C₁₉×#125	—	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	1	no	—
32	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
37	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
38	C₂×C₄²×C₁₉	—	4	2	no	—
39	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
43	C₂²×C₁₉⋊C₈	—	2	2	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	C₁₉⋊#95	—	2	4	no	—
51	C₁₉⋊#95	—	2	4	no	—
52	C₁₉⋊#95	—	4	2	no	—
53	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
54	C₂²⋊(C₁₉⋊C₈)	Q₈	1	4	yes	—
55	C₄×C₁₉⋊C₈	—	4	2	no	—
56	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	—	2	2	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	1	no	—
62	C₂²×C₄×C₁₉	—	2	1	no	—
63	C₂²×C₄×C₁₉	—	2	1	no	—
64	C₂²×C₄×C₁₉	—	2	1	no	—
65	C₂²×C₄×C₁₉	—	4	1	no	—
66	C₂²×C₄×C₁₉	—	4	1	no	—
67	C₂²×C₄×C₁₉	—	4	1	no	—
68	C₂²×C₄×C₁₉	—	4	1	no	—
69	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₁₉	#41	1	2	yes	—
77	C₂²×Dic₁₉	—	2	4	no	—
78	C₂²×Dic₁₉	—	4	2	no	—
79	C₂²×Dic₁₉	—	4	2	no	—
80	C₂²×Dic₁₉	—	4	2	no	—
81	C₂²×Dic₁₉	—	4	2	no	—
82	C₂²×Dic₁₉	—	4	2	no	—
83	C₂²×Dic₁₉	—	4	2	no	—
84	C₄⋊C₄×C₁₉	—	4	1	no	—
85	C₄⋊C₄×C₁₉	—	4	1	no	—
86	C₄⋊C₄×C₁₉	—	4	1	no	—
87	C₄⋊C₄×C₁₉	—	4	1	no	—
88	C₄⋊C₄×C₁₉	—	4	2	no	—
89	C₄⋊C₄×C₁₉	—	4	2	no	—
90	C₄⋊C₄×C₁₉	—	4	2	no	—
91	C₄⋊C₄×C₁₉	—	4	2	no	—
92	C₂²⋊C₄×C₁₉	—	4	2	no	—
93	C₂²⋊C₄×C₁₉	—	4	2	no	—
94	C₄²×C₁₉	—	2	2	no	—
95	C₄²×C₁₉	—	2	2	no	—
96	C₄²×C₁₉	—	4	1	no	—
97	C₄²×C₁₉	—	4	1	no	—
98	C₂×C₁₉⋊C₈	—	2	4	no	—
99	C₂×C₁₉⋊C₈	—	4	2	no	—
100	C₂×C₁₉⋊C₈	—	4	2	no	—
101	#9886	—	19	1	no	maximal, Sylow
102	C₂³×C₁₉	—	2	1	no	—
103	C₂³×C₁₉	—	2	1	no	—
104	C₂³×C₁₉	—	2	1	no	—
105	C₂³×C₁₉	—	2	1	no	—
106	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂³×C₁₉	#124	1	1	yes	characteristic
108	C₂³×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
109	C₂³×C₁₉	#135	1	2	yes	—
110	C₂³×C₁₉	#135	1	2	yes	—
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	—	2	1	no	—
119	C₂×C₄×C₁₉	—	2	1	no	—
120	C₂×C₄×C₁₉	—	2	1	no	—
121	C₂×C₄×C₁₉	—	2	1	no	—
122	C₂×C₄×C₁₉	—	2	1	no	—
123	C₂×C₄×C₁₉	—	2	1	no	—
124	C₂×C₄×C₁₉	—	2	1	no	—
125	C₂×C₄×C₁₉	—	4	2	no	—
126	C₂×C₄×C₁₉	—	4	2	no	—
127	C₂×C₄×C₁₉	—	4	4	no	—
128	C₂×C₄×C₁₉	—	4	4	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	—	4	1	no	—
134	C₂×C₄×C₁₉	—	4	1	no	—
135	C₂×C₄×C₁₉	—	4	1	no	—
136	C₂×C₄×C₁₉	—	4	1	no	—
137	C₂×C₄×C₁₉	—	4	1	no	—
138	C₂×C₄×C₁₉	—	4	1	no	—
139	C₂×C₄×C₁₉	—	4	1	no	—
140	C₂×C₄×C₁₉	—	4	1	no	—
141	C₂×C₄×C₁₉	#99	1	2	yes	—
142	C₂×C₄×C₁₉	#104	1	2	yes	—
143	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
144	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
145	C₂×Dic₁₉	—	4	4	no	—
146	C₂×Dic₁₉	—	4	4	no	—
147	C₂×Dic₁₉	—	4	4	no	—
148	C₂×Dic₁₉	—	8	2	no	—
149	C₂×Dic₁₉	—	8	2	no	—
150	C₂×Dic₁₉	—	8	2	no	—
151	C₂×Dic₁₉	—	8	2	no	—
152	C₁₉⋊C₈	—	4	4	no	—
153	#2851	D₁₉	1	1	yes	characteristic
154	C₄×C₂²⋊C₈	—	19	1	no	—
155	#1434	—	19	1	no	—
156	#1294	—	19	2	no	—
157	#1294	—	19	2	no	—
158	C₂²×C₁₉	—	2	1	no	—
159	C₂²×C₁₉	—	2	1	no	—
160	C₂²×C₁₉	—	2	1	no	—
161	C₂²×C₁₉	—	2	1	no	—
162	C₂²×C₁₉	—	2	2	no	—
163	C₂²×C₁₉	—	2	2	no	—
164	C₂²×C₁₉	—	2	2	no	—
165	C₂²×C₁₉	—	2	2	no	—
166	C₂²×C₁₉	#391	1	1	yes	characteristic
167	C₂²×C₁₉	C₂×#99	1	1	yes	characteristic
168	C₂²×C₁₉	#410	1	1	yes	characteristic
169	C₂²×C₁₉	#416	1	1	yes	characteristic
170	C₂²×C₁₉	#416	1	1	yes	characteristic
171	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
172	C₂²×C₁₉	#421	1	1	yes	characteristic
173	C₂²×C₁₉	#488	1	2	yes	—
174	C₂²×C₁₉	#499	1	2	yes	—
175	C₄×C₁₉	—	2	2	no	—
176	C₄×C₁₉	—	2	2	no	—
177	C₄×C₁₉	—	2	2	no	—
178	C₄×C₁₉	—	2	2	no	—
179	C₄×C₁₉	—	4	1	no	—
180	C₄×C₁₉	—	4	1	no	—
181	C₄×C₁₉	—	8	1	no	—
182	C₄×C₁₉	—	8	1	no	—
183	C₄×C₁₉	—	8	1	no	—
184	C₄×C₁₉	—	8	1	no	—
185	Dic₁₉	—	8	4	no	—
186	C₂×#125	Dic₁₉	1	1	yes	characteristic, complemented
187	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
188	C₂×C₂²⋊C₈	—	19	2	no	—
189	C₂×C₄×C₈	—	38	1	no	—
190	#389	Dic₁₉	1	1	yes	characteristic, complemented
191	#389	Dic₁₉	1	1	yes	characteristic, complemented
192	#387	—	2	1	no	—
193	#387	—	2	1	no	—
194	#385	—	2	1	no	—
195	#385	—	2	1	no	—
196	#382	Dic₁₉	1	1	yes	characteristic, complemented
197	#380	C₂×D₁₉	1	1	yes	characteristic
198	#380	C₂×D₁₉	1	1	yes	characteristic
199	C₂×#95	—	19	2	no	—
200	C₂×#95	—	38	1	no	—
201	C₂×#95	—	38	1	no	—
202	#336	—	38	1	no	—
203	C₂×C₁₉	—	2	2	no	—
204	C₂×C₁₉	—	2	2	no	—
205	C₂×C₁₉	#1500	1	1	yes	characteristic
206	C₂×C₁₉	#1501	1	1	yes	characteristic
207	C₂×C₁₉	#1516	1	1	yes	characteristic
208	C₂×C₁₉	#1521	1	1	yes	characteristic
209	C₂×C₁₉	#1986	1	1	yes	characteristic
210	C₂×C₁₉	#1991	1	1	yes	characteristic
211	C₂×C₁₉	#1996	1	1	yes	characteristic
212	C₂³×C₄	—	38	2	no	—
213	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
214	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
215	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
216	C₂²×C₈	—	38	2	no	—
217	#125	—	4	2	no	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
223	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
224	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
225	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
226	C₂×C₂²⋊C₄	—	2	1	no	—
227	C₂×C₂²⋊C₄	—	2	1	no	—
228	C₂×C₄²	—	4	2	no	—
229	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
230	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
231	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
232	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
233	C₂²⋊C₈	—	19	4	no	—
234	C₂²⋊C₈	—	38	2	no	—
235	C₄×C₈	—	76	2	no	—
236	#95	—	2	1	no	—
237	#95	—	2	1	no	—
238	#95	—	2	1	no	—
239	#95	—	2	1	no	—
240	#95	—	2	1	no	—
241	#95	—	2	1	no	—
242	#95	—	38	4	no	—
243	#95	—	38	4	no	—
244	#95	—	76	2	no	—
245	C₁₉	#9886	1	1	yes	characteristic, complemented, Sylow
246	C₂⁴	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	1	no	—
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	—	4	1	no	—
257	C₂²×C₄	—	4	1	no	—
258	C₂²×C₄	—	4	1	no	—
259	C₂²×C₄	—	38	4	no	—
260	C₂²×C₄	—	76	2	no	—
261	C₂²×C₄	—	76	2	no	—
262	C₂²×C₄	—	76	2	no	—
263	C₂²×C₄	—	76	2	no	—
264	C₂²×C₄	—	76	2	no	—
265	C₂²×C₄	—	76	2	no	—
266	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
267	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
268	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
269	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
270	C₂²×C₄	Q₈×D₁₉	1	2	yes	—
271	C₂²×C₄	#64545	1	2	yes	—
272	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
273	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
274	C₂×C₈	—	38	4	no	—
275	C₂×C₈	—	76	2	no	—
276	C₂×C₈	—	76	2	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	2	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	—	4	2	no	—
284	C₄⋊C₄	—	4	2	no	—
285	C₂²⋊C₄	—	4	2	no	—
286	C₂²⋊C₄	—	4	2	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	4	1	no	—
290	C₄²	—	4	1	no	—
291	C₂³	—	2	1	no	—
292	C₂³	—	2	1	no	—
293	C₂³	—	2	1	no	—
294	C₂³	—	2	1	no	—
295	C₂³	#10597390	1	1	yes	characteristic
296	C₂³	#10597393	1	1	yes	characteristic
297	C₂³	#10597396	1	2	yes	—
298	C₂³	#10597396	1	2	yes	—
299	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	4	no	—
317	C₂×C₄	—	4	4	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	76	4	no	—
331	C₂×C₄	—	76	4	no	—
332	C₂×C₄	—	76	4	no	—
333	C₂×C₄	—	152	2	no	—
334	C₂×C₄	—	152	2	no	—
335	C₂×C₄	—	152	2	no	—
336	C₂×C₄	—	152	2	no	—
337	C₂×C₄	#10597291	1	2	yes	—
338	C₂×C₄	#10597294	1	2	yes	—
339	C₂×C₄	Q₈×Dic₁₉	1	2	yes	—
340	C₂×C₄	#10597392	1	2	yes	—
341	C₈	—	76	4	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	2	no	—
345	C₂²	—	2	2	no	—
346	C₂²	—	2	2	no	—
347	C₂²	—	2	2	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	#11926451	1	1	yes	characteristic
351	C₂²	C₂×#10597291	1	1	yes	characteristic
352	C₂²	#11926518	1	1	yes	characteristic
353	C₂²	#11926540	1	1	yes	characteristic
354	C₂²	#11926540	1	1	yes	characteristic
355	C₂²	C₂×#10597294	1	1	yes	characteristic
356	C₂²	#11926561	1	1	yes	characteristic
357	C₂²	#11926901	1	2	yes	—
358	C₂²	#11926943	1	2	yes	—
359	C₄	—	2	2	no	—
360	C₄	—	2	2	no	—
361	C₄	—	2	2	no	—
362	C₄	—	2	2	no	—
363	C₄	—	4	1	no	—
364	C₄	—	4	1	no	—
365	C₄	—	8	1	no	—
366	C₄	—	8	1	no	—
367	C₄	—	8	1	no	—
368	C₄	—	8	1	no	—
369	C₄	—	152	4	no	—
370	C₂	—	2	2	no	—
371	C₂	—	2	2	no	—
372	C₂	#14562121	1	1	yes	characteristic
373	C₂	#14562126	1	1	yes	characteristic
374	C₂	#14562191	1	1	yes	characteristic
375	C₂	#14562212	1	1	yes	characteristic
376	C₂	#14564580	1	1	yes	characteristic
377	C₂	#14564601	1	1	yes	characteristic
378	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×#2851	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂×D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#2851	D₁₉
hypocenter	C₁₉	#9886
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #2851

Complements

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

N	H ≅ G/N	type
12. C₂×C₁₉×#125	C₄	semidirect
14. C₁₉×#389	C₄	semidirect
15. C₁₉×#389	C₄	semidirect
20. C₁₉×#382	C₄	semidirect
186. C₂×#125	Dic₁₉	semidirect
190. #389	Dic₁₉	semidirect
191. #389	Dic₁₉	semidirect
196. #382	Dic₁₉	semidirect
245. C₁₉	#9886	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₂²⋊(C₁₉⋊C₈)	2
#14561862	2
#14561513	2
C₁₉×#2851	2
#9886	19