Subgroups — #18075247

Order 4864.

492 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563787	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)	—	2	1	no	—
8	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
9	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
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)	—	4	2	no	—
57	C₄⋊(C₁₉⋊C₈)	—	4	2	no	—
58	C₄×C₁₉⋊C₈	—	2	2	no	—
59	C₄×C₁₉⋊C₈	—	2	2	no	—
60	C₂×Q₈×C₁₉	—	2	1	no	—
61	C₂×Q₈×C₁₉	—	2	2	no	—
62	C₂×Q₈×C₁₉	—	2	2	no	—
63	C₂×Q₈×C₁₉	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₉	—	2	1	no	—
65	C₂²×C₄×C₁₉	—	2	1	no	—
66	C₂²×C₄×C₁₉	—	2	1	no	—
67	C₂²×C₄×C₁₉	—	2	1	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
75	C₄⋊C₄×C₁₉	—	2	1	no	—
76	C₄⋊C₄×C₁₉	—	2	2	no	—
77	C₄⋊C₄×C₁₉	—	2	2	no	—
78	C₄⋊C₄×C₁₉	—	2	2	no	—
79	C₄⋊C₄×C₁₉	—	2	2	no	—
80	C₄⋊C₄×C₁₉	—	4	2	no	—
81	C₄⋊C₄×C₁₉	—	4	2	no	—
82	C₄⋊C₄×C₁₉	C₄⋊C₄	1	2	yes	—
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	—	2	1	no	—
89	C₄²×C₁₉	—	2	1	no	—
90	C₄²×C₁₉	—	4	1	no	—
91	C₄²×C₁₉	—	4	1	no	—
92	C₄²×C₁₉	—	4	1	no	—
93	C₄²×C₁₉	SD₁₆	1	2	yes	—
94	C₄²×C₁₉	Q₁₆	1	2	yes	—
95	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
97	C₄²×C₁₉	#41	1	1	yes	characteristic
98	C₄²×C₁₉	#41	1	1	yes	characteristic
99	C₂×C₁₉⋊C₈	—	2	4	no	—
100	C₂×C₁₉⋊C₈	—	4	2	no	—
101	C₂×C₁₉⋊C₈	—	4	2	no	—
102	C₂×C₁₉⋊C₈	—	4	2	no	—
103	C₂×C₁₉⋊C₈	—	4	2	no	—
104	C₂×C₁₉⋊C₈	—	4	2	no	—
105	#8568	—	19	1	no	maximal, Sylow
106	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	Q₈×C₁₉	—	2	4	no	—
108	Q₈×C₁₉	—	4	2	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	1	no	—
117	C₂×C₄×C₁₉	—	2	2	no	—
118	C₂×C₄×C₁₉	—	2	2	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	—	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	—	4	1	no	—
130	C₂×C₄×C₁₉	—	4	1	no	—
131	C₂×C₄×C₁₉	—	4	1	no	—
132	C₂×C₄×C₁₉	#103	1	2	yes	—
133	C₂×C₄×C₁₉	#103	1	2	yes	—
134	C₂×C₄×C₁₉	#104	1	2	yes	—
135	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
137	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
138	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
139	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
140	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
141	C₂×C₄×C₁₉	#131	1	2	yes	—
142	C₂×C₄×C₁₉	#131	1	2	yes	—
143	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
144	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
145	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
147	C₁₉⋊C₈	—	4	4	no	—
148	C₁₉⋊C₈	—	8	2	no	—
149	#2295	D₁₉	1	1	yes	characteristic
150	#1842	—	19	1	no	—
151	C₄×C₄⋊C₈	—	19	1	no	—
152	C₂×Q₈⋊C₈	—	19	2	no	—
153	#1279	—	19	2	no	—
154	C₂²×C₁₉	#380	1	1	yes	characteristic
155	C₂²×C₁₉	#408	1	1	yes	characteristic
156	C₂²×C₁₉	#408	1	1	yes	characteristic
157	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
158	C₂²×C₁₉	#417	1	1	yes	characteristic
159	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
160	C₂²×C₁₉	#421	1	1	yes	characteristic
161	C₄×C₁₉	—	2	1	no	—
162	C₄×C₁₉	—	2	1	no	—
163	C₄×C₁₉	—	2	2	no	—
164	C₄×C₁₉	—	2	2	no	—
165	C₄×C₁₉	—	2	2	no	—
166	C₄×C₁₉	—	4	2	no	—
167	C₄×C₁₉	—	4	2	no	—
168	C₄×C₁₉	—	4	1	no	—
169	C₄×C₁₉	—	8	1	no	—
170	C₄×C₁₉	—	8	1	no	—
171	C₄×C₁₉	#424	1	2	yes	—
172	C₄×C₁₉	#433	1	2	yes	—
173	C₄×C₁₉	#460	1	1	yes	characteristic
174	C₄×C₁₉	#462	1	1	yes	characteristic
175	C₄×C₁₉	#475	1	1	yes	characteristic
176	C₄×C₁₉	#477	1	1	yes	characteristic
177	C₂×C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
178	C₂×C₄⋊C₈	—	19	2	no	—
179	C₂×C₄⋊C₈	—	38	1	no	—
180	C₂×C₄⋊C₈	—	38	1	no	—
181	C₂×C₄⋊C₈	—	38	1	no	—
182	C₂×C₄×C₈	—	19	2	no	—
183	#391	Dic₁₉	1	1	yes	characteristic
184	#391	Dic₁₉	1	1	yes	characteristic
185	#389	Dic₁₉	1	1	yes	characteristic
186	#389	Dic₁₉	1	1	yes	characteristic
187	#387	—	2	1	no	—
188	#387	—	2	1	no	—
189	C₄×C₄⋊C₄	—	2	1	no	—
190	C₄×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
191	#376	—	2	1	no	—
192	C₄³	C₂×D₁₉	1	1	yes	characteristic
193	#336	—	38	1	no	—
194	Q₈⋊C₈	—	38	4	no	—
195	C₂×C₁₉	#1481	1	1	yes	characteristic
196	C₂×C₁₉	#1483	1	1	yes	characteristic
197	C₂×C₁₉	#1493	1	1	yes	characteristic
198	C₂×C₁₉	#1495	1	1	yes	characteristic
199	C₂×C₁₉	#1798	1	1	yes	characteristic
200	C₂×C₁₉	#1803	1	1	yes	characteristic
201	C₂×C₁₉	#1806	1	1	yes	characteristic
202	C₂²×Q₈	C₂×Dic₁₉	1	1	yes	characteristic
203	C₂²×C₈	—	38	2	no	—
204	C₂²×C₈	—	38	2	no	—
205	C₄×Q₈	—	2	1	no	—
206	C₄×Q₈	—	2	1	no	—
207	C₄×Q₈	Dic₃₈	1	2	yes	—
208	C₄×Q₈	D₇₆	1	2	yes	—
209	C₂×C₄⋊C₄	—	2	1	no	—
210	C₂×C₄⋊C₄	—	2	1	no	—
211	C₂×C₄⋊C₄	—	2	1	no	—
212	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
213	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
214	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
215	C₂×C₄²	—	2	1	no	—
216	C₂×C₄²	—	2	1	no	—
217	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
218	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
219	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
220	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
221	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
222	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
223	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
224	C₄⋊C₈	—	38	4	no	—
225	C₄⋊C₈	—	38	2	no	—
226	C₄⋊C₈	—	38	2	no	—
227	C₄⋊C₈	—	76	2	no	—
228	C₄⋊C₈	—	76	2	no	—
229	C₄×C₈	—	38	2	no	—
230	C₄×C₈	—	38	2	no	—
231	#95	—	2	1	no	—
232	#95	—	2	1	no	—
233	#95	—	2	1	no	—
234	#95	—	2	1	no	—
235	#95	—	2	1	no	—
236	#95	—	2	1	no	—
237	#95	—	2	1	no	—
238	C₁₉	#8568	1	1	yes	characteristic, complemented, Sylow
239	C₂×Q₈	—	2	1	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	C₄⋊Dic₁₉	1	2	yes	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
248	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
249	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
250	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
251	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
254	C₂×C₈	—	38	4	no	—
255	C₂×C₈	—	76	2	no	—
256	C₂×C₈	—	76	2	no	—
257	C₂×C₈	—	76	2	no	—
258	C₂×C₈	—	76	2	no	—
259	C₂×C₈	—	76	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	C₄⋊Dic₁₉	1	2	yes	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	2	1	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	4	1	no	—
276	C₄²	—	4	1	no	—
277	C₄²	—	4	1	no	—
278	C₄²	#64527	1	2	yes	—
279	C₄²	#64528	1	2	yes	—
280	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
281	C₄²	C₂×D₇₆	1	1	yes	characteristic
282	C₄²	#64541	1	1	yes	characteristic
283	C₄²	#64541	1	1	yes	characteristic
284	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
285	Q₈	—	2	4	no	—
286	Q₈	—	4	2	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	#10597292	1	2	yes	—
311	C₂×C₄	#10597292	1	2	yes	—
312	C₂×C₄	#10597294	1	2	yes	—
313	C₂×C₄	#10597370	1	1	yes	characteristic
314	C₂×C₄	#10597370	1	1	yes	characteristic
315	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
316	C₂×C₄	#10597372	1	1	yes	characteristic
317	C₂×C₄	#10597376	1	1	yes	characteristic
318	C₂×C₄	#10597376	1	1	yes	characteristic
319	C₂×C₄	C₂×#64527	1	1	yes	characteristic
320	C₂×C₄	C₂×#64528	1	1	yes	characteristic
321	C₂×C₄	#10597394	1	2	yes	—
322	C₂×C₄	#10597394	1	2	yes	—
323	C₂×C₄	#10597395	1	1	yes	characteristic
324	C₂×C₄	#10597397	1	1	yes	characteristic
325	C₈	—	76	4	no	—
326	C₈	—	152	2	no	—
327	C₂²	#11926404	1	1	yes	characteristic
328	C₂²	#11926511	1	1	yes	characteristic
329	C₂²	#11926511	1	1	yes	characteristic
330	C₂²	C₂×#10597292	1	1	yes	characteristic
331	C₂²	#11926545	1	1	yes	characteristic
332	C₂²	C₂×#10597294	1	1	yes	characteristic
333	C₂²	#11926561	1	1	yes	characteristic
334	C₄	—	2	2	no	—
335	C₄	—	2	2	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	2	no	—
339	C₄	—	4	2	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	2	no	—
342	C₄	—	8	1	no	—
343	C₄	—	8	1	no	—
344	C₄	#11926571	1	2	yes	—
345	C₄	#11926605	1	2	yes	—
346	C₄	#11926744	1	1	yes	characteristic
347	C₄	#11926754	1	1	yes	characteristic
348	C₄	#11926829	1	1	yes	characteristic
349	C₄	#11926839	1	1	yes	characteristic
350	C₂	#14562038	1	1	yes	characteristic
351	C₂	#14562048	1	1	yes	characteristic
352	C₂	#14562090	1	1	yes	characteristic
353	C₂	#14562100	1	1	yes	characteristic
354	C₂	#14563619	1	1	yes	characteristic
355	C₂	#14563640	1	1	yes	characteristic
356	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	#8568
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
238. C₁₉	#8568	semidirect

Maximal subgroups

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

subgroup	index
#14563787	2
C₄×C₄⋊(C₁₉⋊C₈)	2
C₂×Q₈⋊(C₁₉⋊C₈)	2
#14561476	2
C₁₉×#2295	2
#8568	19