Subgroups — #18075890

Order 4864.

680 conjugacy classes of subgroups.

The subgroup lattice has 336 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	#14561476	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₄⋊C₄×C₁₉	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
6	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
7	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
8	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
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	1	yes	characteristic
13	C₂²×C₄²×C₁₉	—	2	1	no	—
14	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₁₉	—	2	1	no	—
16	C₄×C₄⋊C₄×C₁₉	—	2	1	no	—
17	C₄×C₄⋊C₄×C₁₉	C₄	1	2	yes	—
18	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
19	C₂×C₁₉×#95	—	2	1	no	—
20	C₂³×C₄×C₁₉	—	2	1	no	—
21	C₂³×C₄×C₁₉	—	2	1	no	—
22	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₉	D₄	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	4	no	—
29	C₂×C₄⋊C₄×C₁₉	C₈	1	4	yes	complemented
30	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₉	—	2	4	no	—
33	C₂×C₄²×C₁₉	—	2	2	no	—
34	C₂×C₄²×C₁₉	—	2	2	no	—
35	C₂×C₄²×C₁₉	—	4	4	no	—
36	C₂×C₄²×C₁₉	C₂×C₄	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	C₂³	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	4	no	—
45	C₂²⋊(C₁₉⋊C₈)	—	4	2	no	—
46	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
47	C₄×C₁₉⋊C₈	—	2	4	no	—
48	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	—	2	2	no	—
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	4	no	—
54	C₂²×C₄×C₁₉	—	2	4	no	—
55	C₂²×C₄×C₁₉	—	2	2	no	—
56	C₂²×C₄×C₁₉	—	2	2	no	—
57	C₂²×C₄×C₁₉	—	2	2	no	—
58	C₂²×C₄×C₁₉	—	2	2	no	—
59	C₂²×C₄×C₁₉	—	4	1	no	—
60	C₂²×C₄×C₁₉	—	4	2	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	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂×C₈	1	2	yes	—
68	C₂²×C₄×C₁₉	M₁₆	1	2	yes	—
69	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
70	C₂²×C₄×C₁₉	Q₁₆	1	2	yes	—
71	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
74	C₄⋊C₄×C₁₉	—	2	4	no	—
75	C₄⋊C₄×C₁₉	—	2	4	no	—
76	C₄⋊C₄×C₁₉	—	2	2	no	—
77	C₄⋊C₄×C₁₉	—	2	2	no	—
78	C₄⋊C₄×C₁₉	C₂×C₈	1	4	yes	—
79	C₄⋊C₄×C₁₉	M₁₆	1	4	yes	—
80	C₄²×C₁₉	—	2	2	no	—
81	C₄²×C₁₉	—	2	2	no	—
82	C₄²×C₁₉	—	2	2	no	—
83	C₄²×C₁₉	—	4	4	no	—
84	C₄²×C₁₉	—	4	4	no	—
85	C₄²×C₁₉	#41	1	4	yes	—
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	#8707	—	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	C₂²⋊C₈	1	2	yes	—
98	C₂³×C₁₉	#103	1	2	yes	—
99	C₂³×C₁₉	#104	1	2	yes	—
100	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₉	—	2	1	no	—
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	4	no	—
108	C₂×C₄×C₁₉	—	2	4	no	—
109	C₂×C₄×C₁₉	—	2	4	no	—
110	C₂×C₄×C₁₉	—	2	4	no	—
111	C₂×C₄×C₁₉	—	2	1	no	—
112	C₂×C₄×C₁₉	—	2	4	no	—
113	C₂×C₄×C₁₉	—	2	4	no	—
114	C₂×C₄×C₁₉	—	2	1	no	—
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	4	4	no	—
117	C₂×C₄×C₁₉	—	4	8	no	—
118	C₂×C₄×C₁₉	—	4	8	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	2	no	—
123	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
124	C₂×C₄×C₁₉	#103	1	2	yes	—
125	C₂×C₄×C₁₉	#117	1	2	yes	—
126	C₂×C₄×C₁₉	#123	1	2	yes	—
127	C₂×C₄×C₁₉	#123	1	2	yes	—
128	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
130	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
131	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
132	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
133	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
134	C₁₉⋊C₈	—	4	4	no	—
135	C₁₉⋊C₈	—	8	2	no	—
136	C₂×C₄×C₄⋊C₄	D₁₉	1	1	yes	characteristic
137	#1797	—	19	1	no	—
138	C₄×C₂²⋊C₈	—	19	1	no	—
139	#1279	—	19	4	no	—
140	C₂²×C₁₉	—	2	2	no	—
141	C₂²×C₁₉	—	2	2	no	—
142	C₂²×C₁₉	—	2	2	no	—
143	C₂²×C₁₉	—	2	2	no	—
144	C₂²×C₁₉	—	2	2	no	—
145	C₂²×C₁₉	—	2	2	no	—
146	C₂²×C₁₉	Q₈⋊C₈	1	4	yes	—
147	C₂²×C₁₉	#381	1	1	yes	characteristic
148	C₂²×C₁₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
149	C₂²×C₁₉	#407	1	1	yes	characteristic
150	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
151	C₂²×C₁₉	#418	1	1	yes	characteristic
152	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
153	C₂²×C₁₉	#421	1	1	yes	characteristic
154	C₄×C₁₉	—	2	2	no	—
155	C₄×C₁₉	—	2	2	no	—
156	C₄×C₁₉	—	2	2	no	—
157	C₄×C₁₉	—	4	1	no	—
158	C₄×C₁₉	—	4	4	no	—
159	C₄×C₁₉	—	4	4	no	—
160	C₄×C₁₉	#431	1	2	yes	—
161	C₄×C₁₉	#432	1	2	yes	—
162	C₄×C₁₉	#481	1	2	yes	—
163	C₄×C₁₉	#484	1	2	yes	—
164	C₂²×C₄⋊C₄	Dic₁₉	1	1	yes	characteristic
165	C₂²×C₄⋊C₄	Dic₁₉	1	1	yes	characteristic
166	C₂²×C₄²	—	2	1	no	—
167	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
168	C₂×C₄⋊C₈	—	19	2	no	—
169	C₂×C₂²⋊C₈	—	38	1	no	—
170	C₂×C₂²⋊C₈	—	38	1	no	—
171	C₂×C₄×C₈	—	19	2	no	—
172	C₄×C₄⋊C₄	—	2	1	no	—
173	C₄×C₄⋊C₄	—	2	1	no	—
174	C₄×C₄⋊C₄	Dic₁₉	1	2	yes	—
175	C₄×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
176	C₂×#95	—	2	1	no	—
177	#336	—	38	1	no	—
178	#336	—	38	1	no	—
179	C₂×C₁₉	—	2	4	no	—
180	C₂×C₁₉	C₂×Q₈⋊C₈	1	1	yes	characteristic
181	C₂×C₁₉	#1478	1	1	yes	characteristic
182	C₂×C₁₉	#1487	1	1	yes	characteristic
183	C₂×C₁₉	#1490	1	1	yes	characteristic
184	C₂×C₁₉	#1817	1	1	yes	characteristic
185	C₂×C₁₉	#1824	1	1	yes	characteristic
186	C₂×C₁₉	#1826	1	1	yes	characteristic
187	C₂³×C₄	—	2	1	no	—
188	C₂³×C₄	—	2	1	no	—
189	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
190	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
191	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
192	C₂²×C₈	—	38	2	no	—
193	C₂²×C₈	—	38	2	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	2	4	no	—
198	C₂×C₄⋊C₄	C₁₉⋊C₈	1	4	yes	complemented
199	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
200	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
201	C₂×C₄²	—	2	4	no	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	—	4	4	no	—
205	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
206	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
207	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
208	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
209	C₄⋊C₈	—	38	4	no	—
210	C₂²⋊C₈	—	38	4	no	—
211	C₂²⋊C₈	—	76	2	no	—
212	C₄×C₈	—	38	4	no	—
213	#95	—	2	2	no	—
214	#95	—	2	2	no	—
215	C₁₉	#8707	1	1	yes	characteristic, complemented, Sylow
216	C₂⁴	C₂²⋊Dic₁₉	1	1	yes	characteristic
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	4	1	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	C₂×C₁₉⋊C₈	1	2	yes	—
233	C₂²×C₄	#64520	1	2	yes	—
234	C₂²×C₄	#64527	1	2	yes	—
235	C₂²×C₄	#64528	1	2	yes	—
236	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
237	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
238	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
239	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
242	C₂×C₈	—	38	4	no	—
243	C₂×C₈	—	76	2	no	—
244	C₂×C₈	—	76	2	no	—
245	C₂×C₈	—	76	2	no	—
246	C₂×C₈	—	76	2	no	—
247	C₂×C₈	—	76	2	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	C₂×C₁₉⋊C₈	1	4	yes	—
253	C₄⋊C₄	#64520	1	4	yes	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	4	4	no	—
258	C₄²	—	4	4	no	—
259	C₄²	#64541	1	4	yes	—
260	C₂³	—	2	1	no	—
261	C₂³	—	2	1	no	—
262	C₂³	—	2	1	no	—
263	C₂³	—	2	1	no	—
264	C₂³	C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
265	C₂³	#10597292	1	2	yes	—
266	C₂³	#10597294	1	2	yes	—
267	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	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	—	4	4	no	—
284	C₂×C₄	—	4	8	no	—
285	C₂×C₄	—	4	8	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
291	C₂×C₄	#10597292	1	2	yes	—
292	C₂×C₄	C₂²×C₁₉⋊C₈	1	1	yes	characteristic
293	C₂×C₄	C₂×#64520	1	2	yes	—
294	C₂×C₄	C₂×#64520	1	1	yes	characteristic
295	C₂×C₄	#10597372	1	2	yes	—
296	C₂×C₄	#10597375	1	2	yes	—
297	C₂×C₄	#10597375	1	2	yes	—
298	C₂×C₄	#10597378	1	2	yes	—
299	C₂×C₄	C₂×#64527	1	1	yes	characteristic
300	C₂×C₄	C₂×#64528	1	1	yes	characteristic
301	C₈	—	76	4	no	—
302	C₈	—	152	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	Q₈⋊(C₁₉⋊C₈)	1	4	yes	—
310	C₂²	#11926407	1	1	yes	characteristic
311	C₂²	C₂×C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
312	C₂²	#11926508	1	1	yes	characteristic
313	C₂²	C₂×#10597292	1	1	yes	characteristic
314	C₂²	#11926549	1	1	yes	characteristic
315	C₂²	C₂×#10597294	1	1	yes	characteristic
316	C₂²	#11926561	1	1	yes	characteristic
317	C₄	—	2	2	no	—
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	4	4	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	4	no	—
323	C₄	#11926597	1	2	yes	—
324	C₄	#11926601	1	2	yes	—
325	C₄	#11926862	1	2	yes	—
326	C₄	#11926879	1	2	yes	—
327	C₂	—	2	4	no	—
328	C₂	C₂×Q₈⋊(C₁₉⋊C₈)	1	1	yes	characteristic
329	C₂	#14562026	1	1	yes	characteristic
330	C₂	#14562064	1	1	yes	characteristic
331	C₂	#14562078	1	1	yes	characteristic
332	C₂	#14563703	1	1	yes	characteristic
333	C₂	#14563725	1	1	yes	characteristic
334	C₂	#14563734	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂×C₄×C₁₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₄×C₄⋊C₄×C₁₉	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×C₄×C₄⋊C₄	D₁₉
hypocenter	C₁₉	#8707
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
3. #14561476	C₂	semidirect
29. C₂×C₄⋊C₄×C₁₉	C₈	semidirect
198. C₂×C₄⋊C₄	C₁₉⋊C₈	semidirect
215. C₁₉	#8707	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
#14561476	2
C₂×C₄×C₄⋊C₄×C₁₉	2
#8707	19