Subgroups — #18082597

Order 4864.

664 conjugacy classes of subgroups.

The subgroup lattice has 340 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	#14563397	C₂	1	1	yes	maximal, characteristic
3	#14561478	C₂	1	4	yes	maximal
4	C₂×C₁₉×#391	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×#10597260	—	2	1	no	—
9	#11926299	—	2	1	no	—
10	#11926299	—	2	1	no	—
11	C₂³×Q₈×C₁₉	C₄	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₁₉	C₄	1	1	yes	characteristic
13	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
14	C₁₉×#391	—	2	1	no	—
15	C₁₉×#391	—	2	1	no	—
16	C₁₉×#391	—	2	1	no	—
17	C₁₉×#391	—	2	1	no	—
18	C₂×C₁₉×#95	C₄	1	2	yes	—
19	C₂×C₁₉×#95	C₂²	1	2	yes	—
20	C₂²×Q₈×C₁₉	—	2	1	no	—
21	C₂²×Q₈×C₁₉	—	2	1	no	—
22	C₂²×Q₈×C₁₉	—	2	4	no	—
23	C₂²×Q₈×C₁₉	—	4	2	no	—
24	C₂²×Q₈×C₁₉	—	4	2	no	—
25	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
31	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
32	C₂×C₄⋊C₄×C₁₉	—	4	2	no	—
33	C₂×C₄²×C₁₉	—	4	2	no	—
34	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
38	C₂²×C₁₉⋊C₈	—	2	2	no	—
39	C₂²×C₁₉⋊C₈	—	2	2	no	—
40	C₁₉×#95	—	2	2	no	—
41	C₁₉×#95	—	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	complemented
47	C₄⋊(C₁₉⋊C₈)	—	4	2	no	—
48	#10597260	—	4	2	no	—
49	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×Q₈×C₁₉	—	2	4	no	—
51	C₂×Q₈×C₁₉	—	2	4	no	—
52	C₂×Q₈×C₁₉	—	4	16	no	—
53	C₂×Q₈×C₁₉	—	4	2	no	—
54	C₂×Q₈×C₁₉	—	4	2	no	—
55	C₂×Q₈×C₁₉	—	4	2	no	—
56	C₂×Q₈×C₁₉	—	4	2	no	—
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	2	no	—
62	C₂²×C₄×C₁₉	—	2	2	no	—
63	C₂²×C₄×C₁₉	—	2	2	no	—
64	C₂²×C₄×C₁₉	—	2	2	no	—
65	C₂²×C₄×C₁₉	—	2	4	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	—	4	2	no	—
70	C₂²×C₄×C₁₉	—	4	2	no	—
71	C₂²×C₄×C₁₉	—	4	2	no	—
72	C₂²×C₄×C₁₉	—	4	2	no	—
73	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₉	M₁₆	1	4	yes	—
78	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₁₉	#41	1	2	yes	—
82	C₄⋊C₄×C₁₉	—	4	2	no	—
83	C₄⋊C₄×C₁₉	—	4	2	no	—
84	C₄²×C₁₉	—	2	2	no	—
85	C₄²×C₁₉	—	2	2	no	—
86	C₄²×C₁₉	—	4	1	no	—
87	C₄²×C₁₉	—	4	1	no	—
88	C₂×C₁₉⋊C₈	—	2	4	no	—
89	C₂×C₁₉⋊C₈	—	4	2	no	—
90	C₂×C₁₉⋊C₈	—	4	2	no	—
91	C₂×C₁₉⋊C₈	—	4	2	no	—
92	C₂×C₁₉⋊C₈	—	4	2	no	—
93	C₂×C₁₉⋊C₈	—	4	2	no	—
94	#9882	—	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	C₄×Q₈	1	2	yes	—
101	C₂³×C₁₉	C₂×M₁₆	1	2	yes	—
102	C₂³×C₁₉	#131	1	2	yes	—
103	Q₈×C₁₉	—	4	8	no	—
104	Q₈×C₁₉	—	4	8	no	—
105	C₂×C₄×C₁₉	—	2	8	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	4	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	—	4	2	no	—
121	C₂×C₄×C₁₉	—	4	2	no	—
122	C₂×C₄×C₁₉	—	4	4	no	—
123	C₂×C₄×C₁₉	—	4	4	no	—
124	C₂×C₄×C₁₉	—	4	2	no	—
125	C₂×C₄×C₁₉	—	4	2	no	—
126	C₂×C₄×C₁₉	—	4	4	no	—
127	C₂×C₄×C₁₉	—	4	2	no	—
128	C₂×C₄×C₁₉	—	4	2	no	—
129	C₂×C₄×C₁₉	—	4	2	no	—
130	C₂×C₄×C₁₉	—	8	4	no	—
131	C₂×C₄×C₁₉	#99	1	2	yes	—
132	C₂×C₄×C₁₉	#100	1	2	yes	—
133	C₂×C₄×C₁₉	#101	1	2	yes	—
134	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
135	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
136	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
137	C₁₉⋊C₈	—	4	4	no	—
138	C₁₉⋊C₈	—	8	2	no	—
139	C₂×#391	D₁₉	1	1	yes	characteristic
140	#1797	—	19	1	no	—
141	#1749	—	19	1	no	—
142	#1280	—	19	4	no	—
143	C₂²×C₁₉	—	2	2	no	—
144	C₂²×C₁₉	—	2	2	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	#407	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	#446	1	4	yes	—
157	C₄×C₁₉	—	2	2	no	—
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	C₂³×Q₈	Dic₁₉	1	1	yes	characteristic
167	C₂²×C₄⋊C₄	Dic₁₉	1	1	yes	characteristic
168	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
169	C₂×C₄⋊C₈	—	38	1	no	—
170	C₂×C₂²⋊C₈	—	19	2	no	—
171	C₂×C₂²⋊C₈	—	19	2	no	—
172	C₂×#97	—	38	1	no	—
173	#391	—	2	1	no	—
174	#391	—	2	1	no	—
175	#391	—	2	1	no	—
176	#391	—	2	1	no	—
177	C₂×#95	Dic₁₉	1	2	yes	—
178	C₂×#95	C₂×D₁₉	1	2	yes	—
179	#336	—	38	1	no	—
180	#336	—	38	1	no	—
181	C₂×C₁₉	—	2	4	no	—
182	C₂×C₁₉	#1462	1	1	yes	characteristic
183	C₂×C₁₉	#1467	1	1	yes	characteristic
184	C₂×C₁₉	#1469	1	1	yes	characteristic
185	C₂×C₁₉	#1472	1	1	yes	characteristic
186	C₂×C₁₉	#1985	1	1	yes	characteristic
187	C₂×C₁₉	#1996	1	1	yes	characteristic
188	C₂×C₁₉	#1997	1	1	yes	characteristic
189	C₂²×Q₈	—	2	4	no	—
190	C₂²×Q₈	—	2	1	no	—
191	C₂²×Q₈	—	2	1	no	—
192	C₂²×Q₈	—	4	2	no	—
193	C₂²×Q₈	—	4	2	no	—
194	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
195	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
196	C₂³×C₄	C₂×Dic₁₉	1	2	yes	—
197	C₂³×C₄	C₂×Dic₁₉	1	2	yes	—
198	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
199	C₂²×C₈	—	38	2	no	—
200	C₂²×C₈	—	38	2	no	—
201	C₂×C₄⋊C₄	—	2	1	no	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄⋊C₄	—	4	2	no	—
204	C₂×C₄²	—	4	2	no	—
205	C₂×C₄²	C₂²⋊D₁₉	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	—	76	2	no	—
210	C₂²⋊C₈	—	19	4	no	—
211	C₂²⋊C₈	—	38	2	no	—
212	C₂²⋊C₈	—	38	4	no	—
213	#97	—	76	2	no	—
214	#95	—	2	2	no	—
215	#95	—	2	2	no	—
216	#95	—	2	2	no	—
217	#95	—	2	2	no	—
218	C₁₉	#9882	1	1	yes	characteristic, complemented, Sylow
219	C₂⁴	C₂²×Dic₁₉	1	1	yes	characteristic
220	C₂×Q₈	—	2	4	no	—
221	C₂×Q₈	—	2	4	no	—
222	C₂×Q₈	—	4	16	no	—
223	C₂×Q₈	—	4	2	no	—
224	C₂×Q₈	—	4	2	no	—
225	C₂×Q₈	—	4	2	no	—
226	C₂×Q₈	—	4	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	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	4	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	—	4	2	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	—	4	2	no	—
243	C₂²×C₄	#64520	1	4	yes	—
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	C₂²⋊Dic₁₉	1	1	yes	characteristic
248	C₂²×C₄	Q₈×D₁₉	1	2	yes	—
249	C₂²×C₄	#64545	1	2	yes	—
250	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
252	C₂×C₈	—	38	4	no	—
253	C₂×C₈	—	76	2	no	—
254	C₂×C₈	—	76	2	no	—
255	C₂×C₈	—	76	2	no	—
256	C₂×C₈	—	76	2	no	—
257	C₂×C₈	—	76	2	no	—
258	C₄⋊C₄	—	4	2	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	C₂×#64520	1	2	yes	—
269	C₂³	Q₈×Dic₁₉	1	2	yes	—
270	C₂³	#10597394	1	2	yes	—
271	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
272	Q₈	—	4	8	no	—
273	Q₈	—	4	8	no	—
274	C₂×C₄	—	2	8	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	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	8	4	no	—
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	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	#11926451	1	1	yes	characteristic
315	C₂²	#11926508	1	1	yes	characteristic
316	C₂²	#11926511	1	1	yes	characteristic
317	C₂²	C₂×#10597291	1	1	yes	characteristic
318	C₂²	#11926518	1	1	yes	characteristic
319	C₂²	C₂×#10597290	1	1	yes	characteristic
320	C₂²	C₂×#10597293	1	1	yes	characteristic
321	C₂²	#11926675	1	4	yes	—
322	C₄	—	2	2	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	2	no	—
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	4	4	no	—
329	C₄	—	4	1	no	—
330	C₄	—	8	2	no	—
331	C₂	—	2	4	no	—
332	C₂	#14561962	1	1	yes	characteristic
333	C₂	#14561979	1	1	yes	characteristic
334	C₂	#14561987	1	1	yes	characteristic
335	C₂	#14561999	1	1	yes	characteristic
336	C₂	#14564576	1	1	yes	characteristic
337	C₂	#14564621	1	1	yes	characteristic
338	C₂	#14564625	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₉
commutator, prime radical	C₂⁴×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×#391	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×#391	D₁₉
hypocenter	C₁₉	#9882
LCS	G ▷ C₂⁴×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#391

Complements

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

N	H ≅ G/N	type
46. C₂²⋊(C₁₉⋊C₈)	Q₈	semidirect
218. C₁₉	#9882	semidirect

Maximal subgroups

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

subgroup	index
#14563615	2
#14563397	2
#14561478	2
C₂×C₁₉×#391	2
#9882	19