Subgroups — #18082583

Order 4864.

616 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	#14563615	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#14561478	C₂	1	4	yes	maximal
4	C₂×C₁₉×#389	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×C₂²⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
9	#11926299	—	2	1	no	—
10	#11926299	—	2	1	no	—
11	C₂²×C₄⋊C₄×C₁₉	C₄	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₁₉	C₄	1	2	yes	—
13	C₂²×C₄⋊C₄×C₁₉	C₄	1	1	yes	characteristic
14	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
15	C₁₉×#389	—	2	1	no	—
16	C₁₉×#389	—	2	1	no	—
17	C₁₉×#389	—	2	1	no	—
18	C₁₉×#389	—	2	1	no	—
19	C₂×C₁₉×#95	C₂²	1	2	yes	—
20	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
26	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
27	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
28	C₂×C₄⋊C₄×C₁₉	—	2	2	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	—	4	2	no	—
33	C₂×C₄⋊C₄×C₁₉	—	4	4	no	—
34	C₂×C₄⋊C₄×C₁₉	C₈	1	4	yes	complemented
35	C₂×C₄²×C₁₉	—	4	2	no	—
36	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₉	D₄	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	C₂²×C₁₉⋊C₈	—	2	2	no	—
42	C₁₉×#95	—	2	2	no	—
43	C₁₉×#95	—	2	2	no	—
44	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
45	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
46	C₂²⋊(C₁₉⋊C₈)	Q₈	1	4	yes	—
47	C₄⋊(C₁₉⋊C₈)	—	4	2	no	—
48	C₄×C₁₉⋊C₈	—	4	2	no	—
49	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	—	2	2	no	—
51	C₂²×C₄×C₁₉	—	2	2	no	—
52	C₂²×C₄×C₁₉	—	2	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	—	2	4	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	2	no	—
62	C₂²×C₄×C₁₉	—	4	2	no	—
63	C₂²×C₄×C₁₉	—	4	2	no	—
64	C₂²×C₄×C₁₉	—	4	2	no	—
65	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂×C₈	1	2	yes	—
70	C₂²×C₄×C₁₉	C₂×C₈	1	4	yes	—
71	C₂²×C₄×C₁₉	M₁₆	1	2	yes	—
72	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₁₉	#41	1	2	yes	—
76	C₄⋊C₄×C₁₉	—	2	4	no	—
77	C₄⋊C₄×C₁₉	—	2	4	no	—
78	C₄⋊C₄×C₁₉	—	4	2	no	—
79	C₄⋊C₄×C₁₉	—	4	4	no	—
80	C₄⋊C₄×C₁₉	—	4	2	no	—
81	C₄⋊C₄×C₁₉	—	4	4	no	—
82	C₄²×C₁₉	—	2	2	no	—
83	C₄²×C₁₉	—	2	2	no	—
84	C₄²×C₁₉	—	4	1	no	—
85	C₄²×C₁₉	—	4	1	no	—
86	C₂×C₁₉⋊C₈	—	2	4	no	—
87	C₂×C₁₉⋊C₈	—	4	2	no	—
88	C₂×C₁₉⋊C₈	—	4	2	no	—
89	C₂×C₁₉⋊C₈	—	4	2	no	—
90	C₂×C₁₉⋊C₈	—	4	2	no	—
91	C₂×C₁₉⋊C₈	—	4	2	no	—
92	#9879	—	19	1	no	maximal, Sylow
93	C₂³×C₁₉	—	2	1	no	—
94	C₂³×C₁₉	—	2	1	no	—
95	C₂³×C₁₉	—	2	1	no	—
96	C₂³×C₁₉	—	2	1	no	—
97	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂³×C₁₉	C₄×Q₈	1	2	yes	—
99	C₂³×C₁₉	C₂²×C₈	1	1	yes	characteristic
100	C₂³×C₁₉	C₂×M₁₆	1	1	yes	characteristic
101	C₂³×C₁₉	#131	1	2	yes	—
102	C₂×C₄×C₁₉	—	2	1	no	—
103	C₂×C₄×C₁₉	—	2	1	no	—
104	C₂×C₄×C₁₉	—	2	1	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	4	no	—
111	C₂×C₄×C₁₉	—	2	4	no	—
112	C₂×C₄×C₁₉	—	2	4	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	—	4	2	no	—
117	C₂×C₄×C₁₉	—	4	2	no	—
118	C₂×C₄×C₁₉	—	4	4	no	—
119	C₂×C₄×C₁₉	—	4	4	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	—	8	4	no	—
127	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
128	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
129	C₂×C₄×C₁₉	#99	1	2	yes	—
130	C₂×C₄×C₁₉	#100	1	2	yes	—
131	C₂×C₄×C₁₉	#101	1	2	yes	—
132	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
133	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
134	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
135	C₁₉⋊C₈	—	4	4	no	—
136	C₁₉⋊C₈	—	8	2	no	—
137	C₂×#389	D₁₉	1	1	yes	characteristic
138	#1797	—	19	1	no	—
139	C₄×C₂²⋊C₈	—	19	1	no	—
140	#1280	—	19	4	no	—
141	C₂²×C₁₉	—	2	1	no	—
142	C₂²×C₁₉	—	2	1	no	—
143	C₂²×C₁₉	—	2	1	no	—
144	C₂²×C₁₉	—	2	1	no	—
145	C₂²×C₁₉	—	2	2	no	—
146	C₂²×C₁₉	—	2	2	no	—
147	C₂²×C₁₉	—	2	2	no	—
148	C₂²×C₁₉	—	2	2	no	—
149	C₂²×C₁₉	#391	1	1	yes	characteristic
150	C₂²×C₁₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
151	C₂²×C₁₉	#408	1	1	yes	characteristic
152	C₂²×C₁₉	C₂×#99	1	1	yes	characteristic
153	C₂²×C₁₉	#410	1	1	yes	characteristic
154	C₂²×C₁₉	C₂×#100	1	1	yes	characteristic
155	C₂²×C₁₉	C₂×#101	1	1	yes	characteristic
156	C₂²×C₁₉	C₈×Q₈	1	2	yes	—
157	C₂²×C₁₉	#446	1	2	yes	—
158	C₄×C₁₉	—	2	2	no	—
159	C₄×C₁₉	—	2	2	no	—
160	C₄×C₁₉	—	2	2	no	—
161	C₄×C₁₉	—	2	1	no	—
162	C₄×C₁₉	—	2	1	no	—
163	C₄×C₁₉	—	4	4	no	—
164	C₄×C₁₉	—	4	1	no	—
165	C₄×C₁₉	—	8	2	no	—
166	C₄×C₁₉	#324	1	4	yes	—
167	C₂²×C₄⋊C₄	Dic₁₉	1	1	yes	characteristic
168	C₂²×C₄⋊C₄	Dic₁₉	1	2	yes	—
169	C₂²×C₄⋊C₄	Dic₁₉	1	1	yes	characteristic
170	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
171	C₂×C₄⋊C₈	—	38	1	no	—
172	C₂×C₂²⋊C₈	—	19	2	no	—
173	C₂×C₂²⋊C₈	—	19	2	no	—
174	C₂×C₄×C₈	—	38	1	no	—
175	#389	—	2	1	no	—
176	#389	—	2	1	no	—
177	#389	—	2	1	no	—
178	#389	—	2	1	no	—
179	C₂×#95	C₂×D₁₉	1	2	yes	—
180	#336	—	38	1	no	—
181	#336	—	38	1	no	—
182	C₂×C₁₉	—	2	2	no	—
183	C₂×C₁₉	—	2	2	no	—
184	C₂×C₁₉	C₂×#324	1	1	yes	characteristic
185	C₂×C₁₉	#1460	1	1	yes	characteristic
186	C₂×C₁₉	#1470	1	1	yes	characteristic
187	C₂×C₁₉	#1471	1	1	yes	characteristic
188	C₂×C₁₉	#1984	1	1	yes	characteristic
189	C₂×C₁₉	#1996	1	1	yes	characteristic
190	C₂×C₁₉	#1997	1	1	yes	characteristic
191	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
192	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
193	C₂³×C₄	C₂×Dic₁₉	1	2	yes	—
194	C₂³×C₄	C₂×Dic₁₉	1	2	yes	—
195	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
196	C₂²×C₈	—	38	2	no	—
197	C₂²×C₈	—	38	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	1	no	—
200	C₂×C₄⋊C₄	—	2	1	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	1	no	—
205	C₂×C₄⋊C₄	—	4	4	no	—
206	C₂×C₄⋊C₄	—	4	2	no	—
207	C₂×C₄⋊C₄	C₁₉⋊C₈	1	4	yes	complemented
208	C₂×C₄²	—	4	2	no	—
209	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
210	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
211	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
212	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
213	C₄⋊C₈	—	76	2	no	—
214	C₂²⋊C₈	—	19	4	no	—
215	C₂²⋊C₈	—	38	2	no	—
216	C₂²⋊C₈	—	38	4	no	—
217	C₄×C₈	—	76	2	no	—
218	#95	—	2	2	no	—
219	#95	—	2	2	no	—
220	C₁₉	#9879	1	1	yes	characteristic, complemented, Sylow
221	C₂⁴	C₂²×Dic₁₉	1	1	yes	characteristic
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	4	1	no	—
231	C₂²×C₄	—	4	1	no	—
232	C₂²×C₄	—	4	1	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	—	4	2	no	—
236	C₂²×C₄	—	4	2	no	—
237	C₂²×C₄	C₂×C₁₉⋊C₈	1	2	yes	—
238	C₂²×C₄	C₂×C₁₉⋊C₈	1	4	yes	—
239	C₂²×C₄	#64520	1	2	yes	—
240	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
241	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
242	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
243	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
244	C₂²×C₄	Q₈×D₁₉	1	2	yes	—
245	C₂²×C₄	#64545	1	2	yes	—
246	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
248	C₂×C₈	—	38	4	no	—
249	C₂×C₈	—	76	2	no	—
250	C₂×C₈	—	76	2	no	—
251	C₂×C₈	—	76	2	no	—
252	C₂×C₈	—	76	2	no	—
253	C₂×C₈	—	76	2	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	4	no	—
258	C₄⋊C₄	—	4	4	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	4	1	no	—
263	C₄²	—	4	1	no	—
264	C₂³	—	2	1	no	—
265	C₂³	—	2	1	no	—
266	C₂³	—	2	1	no	—
267	C₂³	—	2	1	no	—
268	C₂³	C₂²×C₁₉⋊C₈	1	1	yes	characteristic
269	C₂³	C₂×#64520	1	1	yes	characteristic
270	C₂³	Q₈×Dic₁₉	1	2	yes	—
271	C₂³	#10597394	1	2	yes	—
272	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	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	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	4	no	—
298	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
299	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
300	C₂×C₄	#10597290	1	2	yes	—
301	C₂×C₄	#10597291	1	2	yes	—
302	C₂×C₄	#10597293	1	2	yes	—
303	C₂×C₄	#10597390	1	1	yes	characteristic
304	C₂×C₄	#10597392	1	2	yes	—
305	C₂×C₄	#10597393	1	1	yes	characteristic
306	C₈	—	76	4	no	—
307	C₈	—	152	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	#11926451	1	1	yes	characteristic
317	C₂²	C₂×C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
318	C₂²	#11926511	1	1	yes	characteristic
319	C₂²	C₂×#10597291	1	1	yes	characteristic
320	C₂²	#11926518	1	1	yes	characteristic
321	C₂²	C₂×#10597290	1	1	yes	characteristic
322	C₂²	C₂×#10597293	1	1	yes	characteristic
323	C₂²	Q₈×C₁₉⋊C₈	1	2	yes	—
324	C₂²	#11926675	1	2	yes	—
325	C₄	—	2	2	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	4	4	no	—
331	C₄	—	4	1	no	—
332	C₄	—	8	2	no	—
333	C₄	#11926270	1	4	yes	—
334	C₂	—	2	2	no	—
335	C₂	—	2	2	no	—
336	C₂	C₂×#11926270	1	1	yes	characteristic
337	C₂	#14561956	1	1	yes	characteristic
338	C₂	#14561991	1	1	yes	characteristic
339	C₂	#14561995	1	1	yes	characteristic
340	C₂	#14564572	1	1	yes	characteristic
341	C₂	#14564621	1	1	yes	characteristic
342	C₂	#14564625	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₉
commutator	C₂³×C₁₉	C₂²×C₈
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×#389	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂²×C₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×#389	D₁₉
hypocenter	C₁₉	#9879
LCS	G ▷ C₂³×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#389

Complements

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

N	H ≅ G/N	type
34. C₂×C₄⋊C₄×C₁₉	C₈	semidirect
207. C₂×C₄⋊C₄	C₁₉⋊C₈	semidirect
220. C₁₉	#9879	semidirect

Maximal subgroups

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

subgroup	index
#14563615	2
C₄×C₂²⋊(C₁₉⋊C₈)	2
#14561478	2
C₂×C₁₉×#389	2
#9879	19