Subgroups — #18075257

Order 4864.

472 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	#14563752	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊(C₁₉⋊C₈)	C₂	1	2	yes	maximal
4	#14561476	C₂	1	2	yes	maximal
5	C₁₉×#2295	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
7	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
9	#11926299	—	2	1	no	—
10	#11926299	—	2	1	no	—
11	#11926299	—	2	1	no	—
12	Q₈⋊(C₁₉⋊C₈)	—	2	4	no	—
13	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
14	C₁₉×#391	C₄	1	1	yes	characteristic
15	C₁₉×#391	C₄	1	1	yes	characteristic
16	C₁₉×#389	C₄	1	1	yes	characteristic
17	C₁₉×#389	C₄	1	1	yes	characteristic
18	C₁₉×#387	—	2	1	no	—
19	C₁₉×#387	—	2	1	no	—
20	C₄×C₄⋊C₄×C₁₉	—	2	1	no	—
21	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
22	C₁₉×#376	—	2	1	no	—
23	C₄³×C₁₉	C₂²	1	1	yes	characteristic
24	C₂²×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₄×Q₈×C₁₉	—	2	1	no	—
26	C₄×Q₈×C₁₉	—	2	1	no	—
27	C₄×Q₈×C₁₉	D₄	1	2	yes	—
28	C₄×Q₈×C₁₉	Q₈	1	2	yes	—
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×C₄²×C₁₉	—	2	1	no	—
36	C₂×C₄²×C₁₉	—	2	1	no	—
37	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
44	C₂²×C₁₉⋊C₈	—	2	2	no	—
45	C₂²×C₁₉⋊C₈	—	2	2	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	1	no	—
51	C₁₉×#95	—	2	1	no	—
52	C₁₉×#95	—	2	1	no	—
53	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
54	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
55	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
56	C₄×C₁₉⋊C₈	—	2	2	no	—
57	C₄×C₁₉⋊C₈	—	2	2	no	—
58	C₂×Q₈×C₁₉	—	2	1	no	—
59	C₂×Q₈×C₁₉	—	2	2	no	—
60	C₂×Q₈×C₁₉	—	2	2	no	—
61	C₂×Q₈×C₁₉	C₄⋊C₄	1	2	yes	—
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	—	2	1	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
73	C₄⋊C₄×C₁₉	—	2	1	no	—
74	C₄⋊C₄×C₁₉	—	2	2	no	—
75	C₄⋊C₄×C₁₉	—	2	2	no	—
76	C₄⋊C₄×C₁₉	—	2	2	no	—
77	C₄⋊C₄×C₁₉	—	2	2	no	—
78	C₄⋊C₄×C₁₉	—	4	2	no	—
79	C₄⋊C₄×C₁₉	—	4	2	no	—
80	C₄⋊C₄×C₁₉	C₄⋊C₄	1	2	yes	—
81	C₄²×C₁₉	—	2	2	no	—
82	C₄²×C₁₉	—	2	2	no	—
83	C₄²×C₁₉	—	2	2	no	—
84	C₄²×C₁₉	—	2	2	no	—
85	C₄²×C₁₉	—	2	2	no	—
86	C₄²×C₁₉	—	2	2	no	—
87	C₄²×C₁₉	—	2	2	no	—
88	C₄²×C₁₉	—	4	1	no	—
89	C₄²×C₁₉	—	4	2	no	—
90	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
92	C₄²×C₁₉	#41	1	1	yes	characteristic
93	C₄²×C₁₉	#41	1	1	yes	characteristic
94	C₂×C₁₉⋊C₈	—	2	4	no	—
95	C₂×C₁₉⋊C₈	—	4	2	no	—
96	C₂×C₁₉⋊C₈	—	4	2	no	—
97	C₂×C₁₉⋊C₈	—	4	2	no	—
98	C₂×C₁₉⋊C₈	—	4	2	no	—
99	C₂×C₁₉⋊C₈	—	4	2	no	—
100	#8570	—	19	1	no	maximal, Sylow
101	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	Q₈×C₁₉	—	2	4	no	—
103	Q₈×C₁₉	—	4	2	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	2	no	—
109	C₂×C₄×C₁₉	—	2	2	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	—	2	1	no	—
119	C₂×C₄×C₁₉	—	4	1	no	—
120	C₂×C₄×C₁₉	—	4	1	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	#104	1	2	yes	—
130	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
132	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
133	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
134	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
136	C₂×C₄×C₁₉	#131	1	2	yes	—
137	C₂×C₄×C₁₉	#131	1	2	yes	—
138	C₂×C₄×C₁₉	#135	1	2	yes	—
139	C₂×C₄×C₁₉	#137	1	2	yes	—
140	C₁₉⋊C₈	—	4	4	no	—
141	C₁₉⋊C₈	—	8	2	no	—
142	#2295	D₁₉	1	1	yes	characteristic
143	#1831	—	19	1	no	—
144	C₄×C₄⋊C₈	—	19	1	no	—
145	C₂×Q₈⋊C₈	—	19	2	no	—
146	#1279	—	19	2	no	—
147	C₂²×C₁₉	#380	1	1	yes	characteristic
148	C₂²×C₁₉	#408	1	1	yes	characteristic
149	C₂²×C₁₉	#408	1	1	yes	characteristic
150	C₂²×C₁₉	#416	1	1	yes	characteristic
151	C₂²×C₁₉	#419	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	1	no	—
155	C₄×C₁₉	—	2	1	no	—
156	C₄×C₁₉	—	2	2	no	—
157	C₄×C₁₉	—	4	2	no	—
158	C₄×C₁₉	—	4	2	no	—
159	C₄×C₁₉	—	4	1	no	—
160	C₄×C₁₉	—	4	1	no	—
161	C₄×C₁₉	—	4	1	no	—
162	C₄×C₁₉	—	8	1	no	—
163	C₄×C₁₉	—	8	1	no	—
164	C₄×C₁₉	#424	1	2	yes	—
165	C₄×C₁₉	#433	1	2	yes	—
166	C₄×C₁₉	#464	1	2	yes	—
167	C₄×C₁₉	#479	1	2	yes	—
168	C₂×C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
169	C₂×C₄⋊C₈	—	19	2	no	—
170	C₂×C₄⋊C₈	—	38	1	no	—
171	C₂×C₄×C₈	—	19	2	no	—
172	#391	Dic₁₉	1	1	yes	characteristic
173	#391	Dic₁₉	1	1	yes	characteristic
174	#389	Dic₁₉	1	1	yes	characteristic
175	#389	Dic₁₉	1	1	yes	characteristic
176	#387	—	2	1	no	—
177	#387	—	2	1	no	—
178	C₄×C₄⋊C₄	—	2	1	no	—
179	C₄×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
180	#376	—	2	1	no	—
181	C₄³	C₂×D₁₉	1	1	yes	characteristic
182	#336	—	38	1	no	—
183	#336	—	38	1	no	—
184	#336	—	38	1	no	—
185	Q₈⋊C₈	—	38	4	no	—
186	C₂×C₁₉	#1482	1	1	yes	characteristic
187	C₂×C₁₉	#1485	1	1	yes	characteristic
188	C₂×C₁₉	#1494	1	1	yes	characteristic
189	C₂×C₁₉	#1497	1	1	yes	characteristic
190	C₂×C₁₉	#1798	1	1	yes	characteristic
191	C₂×C₁₉	#1805	1	1	yes	characteristic
192	C₂×C₁₉	#1806	1	1	yes	characteristic
193	C₂²×Q₈	C₂×Dic₁₉	1	1	yes	characteristic
194	C₂²×C₈	—	38	2	no	—
195	C₂²×C₈	—	38	2	no	—
196	C₄×Q₈	—	2	1	no	—
197	C₄×Q₈	—	2	1	no	—
198	C₄×Q₈	Dic₃₈	1	2	yes	—
199	C₄×Q₈	D₇₆	1	2	yes	—
200	C₂×C₄⋊C₄	—	2	1	no	—
201	C₂×C₄⋊C₄	—	2	1	no	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
204	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
205	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
206	C₂×C₄²	—	2	1	no	—
207	C₂×C₄²	—	2	1	no	—
208	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
209	C₂×C₄²	C₂×Dic₁₉	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	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
214	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
215	C₄⋊C₈	—	38	4	no	—
216	C₄⋊C₈	—	38	2	no	—
217	C₄⋊C₈	—	38	2	no	—
218	C₄×C₈	—	38	2	no	—
219	C₄×C₈	—	38	2	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	#95	—	2	1	no	—
226	#95	—	2	1	no	—
227	C₁₉	#8570	1	1	yes	characteristic, complemented, Sylow
228	C₂×Q₈	—	2	1	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	C₄⋊Dic₁₉	1	2	yes	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	1	no	—
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	C₂²×Dic₁₉	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
243	C₂×C₈	—	38	4	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	C₂×C₈	—	76	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	C₄⋊Dic₁₉	1	2	yes	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	4	1	no	—
265	C₄²	—	4	2	no	—
266	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
267	C₄²	C₂×D₇₆	1	1	yes	characteristic
268	C₄²	#64541	1	1	yes	characteristic
269	C₄²	#64541	1	1	yes	characteristic
270	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
271	Q₈	—	2	4	no	—
272	Q₈	—	4	2	no	—
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	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	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	#10597294	1	2	yes	—
299	C₂×C₄	#10597370	1	1	yes	characteristic
300	C₂×C₄	#10597370	1	1	yes	characteristic
301	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
302	C₂×C₄	#10597372	1	1	yes	characteristic
303	C₂×C₄	#10597376	1	1	yes	characteristic
304	C₂×C₄	#10597376	1	1	yes	characteristic
305	C₂×C₄	#10597388	1	2	yes	—
306	C₂×C₄	#10597394	1	2	yes	—
307	C₂×C₄	#10597394	1	2	yes	—
308	C₂×C₄	#10597396	1	2	yes	—
309	C₈	—	76	4	no	—
310	C₈	—	152	2	no	—
311	C₂²	#11926404	1	1	yes	characteristic
312	C₂²	#11926511	1	1	yes	characteristic
313	C₂²	#11926511	1	1	yes	characteristic
314	C₂²	#11926540	1	1	yes	characteristic
315	C₂²	#11926553	1	1	yes	characteristic
316	C₂²	C₂×#10597294	1	1	yes	characteristic
317	C₂²	#11926561	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	2	no	—
323	C₄	—	4	1	no	—
324	C₄	—	4	2	no	—
325	C₄	—	4	1	no	—
326	C₄	—	8	1	no	—
327	C₄	—	8	1	no	—
328	C₄	#11926571	1	2	yes	—
329	C₄	#11926605	1	2	yes	—
330	C₄	#11926766	1	2	yes	—
331	C₄	#11926851	1	2	yes	—
332	C₂	#14562043	1	1	yes	characteristic
333	C₂	#14562056	1	1	yes	characteristic
334	C₂	#14562095	1	1	yes	characteristic
335	C₂	#14562108	1	1	yes	characteristic
336	C₂	#14563619	1	1	yes	characteristic
337	C₂	#14563650	1	1	yes	characteristic
338	C₂	#14563655	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2295	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2295	D₁₉
hypocenter	C₁₉	#8570
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #2295

Complements

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

N	H ≅ G/N	type
227. C₁₉	#8570	semidirect

Maximal subgroups

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

subgroup	index
#14563752	2
C₄×C₄⋊(C₁₉⋊C₈)	2
C₂×Q₈⋊(C₁₉⋊C₈)	2
#14561476	2
C₁₉×#2295	2
#8570	19