Subgroups — #18080299

Order 4864.

738 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572659	C₂	1	1	yes	maximal, characteristic
2	#14563623	C₂	1	1	yes	maximal, characteristic
3	#14563570	C₂	1	2	yes	maximal, complemented
4	#14561502	C₂	1	2	yes	maximal, complemented
5	C₁₉×#2409	C₂	1	1	yes	maximal, characteristic
6	D₄×C₁₉⋊C₈	—	2	4	no	—
7	#11926597	—	2	1	no	—
8	C₂×#10597292	—	2	4	no	—
9	C₂×#10597291	—	2	1	no	—
10	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
11	C₂×C₂²⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
12	#11926404	C₂²	1	2	yes	—
13	#11926311	—	2	1	no	—
14	C₂³×Q₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₂×C₁₉×C₂²⋊Q₈	—	2	1	no	—
16	C₂×C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#395	C₄	1	2	yes	complemented
18	C₁₉×#393	C₄	1	2	yes	complemented
19	C₁₉×#391	—	2	1	no	—
20	C₁₉×#386	—	2	1	no	—
21	C₁₉×#385	—	2	1	no	—
22	C₁₉×#380	C₂²	1	1	yes	characteristic
23	C₂²×Q₈×C₁₉	—	2	1	no	—
24	C₂²×Q₈×C₁₉	—	2	2	no	—
25	C₂²×Q₈×C₁₉	—	2	2	no	—
26	C₂²×Q₈×C₁₉	—	4	4	no	—
27	C₂²×Q₈×C₁₉	C₂×C₄	1	2	yes	—
28	C₂²×D₄×C₁₉	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₁₉	—	2	1	no	—
30	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
32	C₁₉×C₂²⋊Q₈	—	4	4	no	—
33	C₄×D₄×C₁₉	—	2	2	no	—
34	C₄×D₄×C₁₉	—	2	2	no	—
35	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
36	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
37	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
39	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
40	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
43	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
44	C₂×C₄⋊Dic₁₉	—	2	2	no	—
45	C₂²×C₁₉⋊C₈	—	2	2	no	—
46	C₂²×C₁₉⋊C₈	D₄	1	4	yes	—
47	C₁₉×#95	—	2	2	no	—
48	C₁₉×#95	—	2	2	no	—
49	C₁₉×#95	C₂×C₄	1	2	yes	—
50	#10597292	—	4	8	no	—
51	#10597291	—	4	2	no	—
52	C₁₉⋊#95	—	2	2	no	—
53	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
54	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
55	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
56	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
57	C₄×C₁₉⋊C₈	—	2	2	no	—
58	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
60	C₂×Q₈×C₁₉	—	2	4	no	—
61	C₂×Q₈×C₁₉	—	2	8	no	—
62	C₂×Q₈×C₁₉	—	2	4	no	—
63	C₂×Q₈×C₁₉	—	4	4	no	—
64	C₂×Q₈×C₁₉	—	4	4	no	—
65	C₂×Q₈×C₁₉	—	4	2	no	—
66	C₂×Q₈×C₁₉	—	4	8	no	—
67	C₂×Q₈×C₁₉	—	4	1	no	—
68	C₂×Q₈×C₁₉	—	4	1	no	—
69	C₂×D₄×C₁₉	—	2	2	no	—
70	C₂×D₄×C₁₉	—	2	2	no	—
71	C₂×D₄×C₁₉	SD₁₆	1	2	yes	—
72	C₂×D₄×C₁₉	Q₁₆	1	2	yes	—
73	C₂²×C₄×C₁₉	—	2	2	no	—
74	C₂²×C₄×C₁₉	—	2	1	no	—
75	C₂²×C₄×C₁₉	—	2	1	no	—
76	C₂²×C₄×C₁₉	—	2	2	no	—
77	C₂²×C₄×C₁₉	—	4	2	no	—
78	C₂²×C₄×C₁₉	—	4	4	no	—
79	C₂²×C₄×C₁₉	—	4	2	no	—
80	C₂²×C₄×C₁₉	—	4	2	no	—
81	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₉	#41	1	2	yes	—
88	C₂²×Dic₁₉	—	4	2	no	—
89	C₄⋊C₄×C₁₉	—	2	2	no	—
90	C₄⋊C₄×C₁₉	—	4	2	no	—
91	C₄⋊C₄×C₁₉	—	4	4	no	—
92	C₂²⋊C₄×C₁₉	—	2	2	no	—
93	C₂²⋊C₄×C₁₉	—	2	2	no	—
94	C₂²⋊C₄×C₁₉	—	4	4	no	—
95	C₂²⋊C₄×C₁₉	—	4	2	no	—
96	C₄²×C₁₉	—	2	1	no	—
97	C₄²×C₁₉	—	2	1	no	—
98	C₂²⋊Dic₁₉	—	4	2	no	—
99	C₂²⋊Dic₁₉	—	4	2	no	—
100	C₄⋊Dic₁₉	—	4	4	no	—
101	C₂×C₁₉⋊C₈	—	2	4	no	—
102	C₂×C₁₉⋊C₈	—	4	4	no	—
103	C₂×C₁₉⋊C₈	—	8	2	no	—
104	#9467	—	19	1	no	maximal, Sylow
105	C₂³×C₁₉	—	2	1	no	—
106	C₂³×C₁₉	—	2	1	no	—
107	C₂³×C₁₉	—	4	1	no	—
108	C₂³×C₁₉	—	4	1	no	—
109	C₂³×C₁₉	—	4	1	no	—
110	C₂³×C₁₉	—	4	1	no	—
111	C₂³×C₁₉	#103	1	2	yes	complemented
112	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂³×C₁₉	#121	1	2	yes	—
114	C₂³×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
115	C₂³×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
116	C₂³×C₁₉	#135	1	2	yes	—
117	Q₈×C₁₉	—	2	8	no	—
118	Q₈×C₁₉	—	4	4	no	—
119	Q₈×C₁₉	—	4	4	no	—
120	Q₈×C₁₉	—	4	4	no	—
121	D₄×C₁₉	—	2	4	no	—
122	D₄×C₁₉	—	2	4	no	—
123	C₂×C₄×C₁₉	—	2	1	no	—
124	C₂×C₄×C₁₉	—	2	1	no	—
125	C₂×C₄×C₁₉	—	2	4	no	—
126	C₂×C₄×C₁₉	—	2	2	no	—
127	C₂×C₄×C₁₉	—	2	4	no	—
128	C₂×C₄×C₁₉	—	2	1	no	—
129	C₂×C₄×C₁₉	—	2	1	no	—
130	C₂×C₄×C₁₉	—	4	1	no	—
131	C₂×C₄×C₁₉	—	4	2	no	—
132	C₂×C₄×C₁₉	—	4	2	no	—
133	C₂×C₄×C₁₉	—	4	8	no	—
134	C₂×C₄×C₁₉	—	4	2	no	—
135	C₂×C₄×C₁₉	—	4	2	no	—
136	C₂×C₄×C₁₉	—	4	1	no	—
137	C₂×C₄×C₁₉	—	4	1	no	—
138	C₂×C₄×C₁₉	—	4	2	no	—
139	C₂×C₄×C₁₉	—	4	2	no	—
140	C₂×C₄×C₁₉	—	4	2	no	—
141	C₂×C₄×C₁₉	—	8	1	no	—
142	C₂×C₄×C₁₉	#103	1	2	yes	complemented
143	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
144	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
145	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
146	C₂×Dic₁₉	—	8	2	no	—
147	C₂×Dic₁₉	—	8	2	no	—
148	C₂×Dic₁₉	—	8	2	no	—
149	C₁₉⋊C₈	—	4	4	no	—
150	#2972	—	19	1	no	—
151	#2409	D₁₉	1	1	yes	characteristic
152	#1799	—	19	1	no	—
153	#1787	—	19	2	no	—
154	#1289	—	19	2	no	—
155	C₂²×C₁₉	—	2	2	no	—
156	C₂²×C₁₉	—	2	1	no	—
157	C₂²×C₁₉	—	2	2	no	—
158	C₂²×C₁₉	—	2	1	no	—
159	C₂²×C₁₉	—	2	2	no	—
160	C₂²×C₁₉	—	4	1	no	—
161	C₂²×C₁₉	—	4	1	no	—
162	C₂²×C₁₉	—	4	2	no	—
163	C₂²×C₁₉	—	4	2	no	—
164	C₂²×C₁₉	—	4	1	no	—
165	C₂²×C₁₉	—	4	1	no	—
166	C₂²×C₁₉	—	8	1	no	—
167	C₂²×C₁₉	#386	1	1	yes	characteristic
168	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
169	C₂²×C₁₉	#416	1	1	yes	characteristic
170	C₂²×C₁₉	#465	1	2	yes	—
171	C₂²×C₁₉	#467	1	2	yes	—
172	C₄×C₁₉	—	2	1	no	—
173	C₄×C₁₉	—	2	2	no	—
174	C₄×C₁₉	—	2	2	no	—
175	C₄×C₁₉	—	4	4	no	—
176	C₄×C₁₉	—	4	1	no	—
177	C₄×C₁₉	—	4	1	no	—
178	C₄×C₁₉	—	8	2	no	—
179	C₄×C₁₉	B(4,𝔽₂)	1	2	yes	—
180	Dic₁₉	—	16	2	no	—
181	C₂³×Q₈	C₂×D₁₉	1	1	yes	characteristic
182	C₂×C₂²⋊Q₈	—	2	1	no	—
183	C₂×C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
184	C₈×D₄	—	38	4	no	—
185	#431	—	38	1	no	—
186	C₂×#103	—	38	4	no	—
187	C₂×#99	—	38	1	no	—
188	C₂×C₂²⋊C₈	—	19	2	no	—
189	C₂×C₂²⋊C₈	—	38	1	no	—
190	#395	Dic₁₉	1	2	yes	complemented
191	#393	Dic₁₉	1	2	yes	complemented
192	#391	—	2	1	no	—
193	#386	—	2	1	no	—
194	#385	—	2	1	no	—
195	#380	—	19	2	no	—
196	#380	C₂×D₁₉	1	1	yes	characteristic
197	#342	—	38	1	no	—
198	C₂×C₁₉	—	2	2	no	—
199	C₂×C₁₉	—	2	1	no	—
200	C₂×C₁₉	—	2	1	no	—
201	C₂×C₁₉	—	2	2	no	—
202	C₂×C₁₉	—	4	2	no	—
203	C₂×C₁₉	#1889	1	1	yes	characteristic
204	C₂×C₁₉	#1892	1	1	yes	characteristic
205	C₂×C₁₉	#1909	1	1	yes	characteristic
206	C₂²×Q₈	—	2	1	no	—
207	C₂²×Q₈	—	2	2	no	—
208	C₂²×Q₈	—	2	2	no	—
209	C₂²×Q₈	—	4	4	no	—
210	C₂²×Q₈	C₂×Dic₁₉	1	2	yes	—
211	C₂²×D₄	C₂²⋊D₁₉	1	1	yes	characteristic
212	C₂³×C₄	—	2	1	no	—
213	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
214	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
215	C₂²×C₈	—	19	4	no	—
216	C₂²×C₈	—	38	2	no	—
217	C₂²⋊Q₈	—	4	4	no	—
218	C₄×D₄	—	2	2	no	—
219	C₄×D₄	—	2	2	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	—	38	2	no	—
223	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
224	C₂×C₂²⋊C₄	—	2	2	no	—
225	C₂×C₂²⋊C₄	—	2	1	no	—
226	C₂×C₂²⋊C₄	—	38	2	no	—
227	C₂×C₂²⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
228	C₂×C₂²⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
229	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
230	C₄⋊C₈	—	38	2	no	—
231	#103	—	76	8	no	—
232	#99	—	76	2	no	—
233	C₂²⋊C₈	—	38	2	no	—
234	C₂²⋊C₈	—	38	2	no	—
235	C₂²⋊C₈	—	38	2	no	—
236	C₄×C₈	—	38	2	no	—
237	#95	—	2	2	no	—
238	#95	—	2	2	no	—
239	#95	—	38	2	no	—
240	#95	C₂×Dic₁₉	1	2	yes	—
241	C₁₉	#9467	1	1	yes	characteristic, complemented, Sylow
242	C₂⁴	C₂²⋊Dic₁₉	1	1	yes	characteristic
243	C₂⁴	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
244	C₂×Q₈	—	2	4	no	—
245	C₂×Q₈	—	2	8	no	—
246	C₂×Q₈	—	2	4	no	—
247	C₂×Q₈	—	4	4	no	—
248	C₂×Q₈	—	4	2	no	—
249	C₂×Q₈	—	4	4	no	—
250	C₂×Q₈	—	4	8	no	—
251	C₂×Q₈	—	4	1	no	—
252	C₂×Q₈	—	4	1	no	—
253	C₂×D₄	—	2	2	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	#64527	1	2	yes	—
256	C₂×D₄	#64528	1	2	yes	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	2	1	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	4	2	no	—
262	C₂²×C₄	—	4	4	no	—
263	C₂²×C₄	—	4	2	no	—
264	C₂²×C₄	—	4	2	no	—
265	C₂²×C₄	—	76	2	no	—
266	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
267	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
268	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
269	C₂²×C₄	D₄×D₁₉	1	2	yes	—
270	C₂²×C₄	#64543	1	2	yes	—
271	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
272	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
273	C₂×C₈	—	38	4	no	—
274	C₂×C₈	—	76	4	no	—
275	C₂×C₈	—	152	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	4	no	—
279	C₄⋊C₄	—	76	4	no	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₂²⋊C₄	—	2	2	no	—
282	C₂²⋊C₄	—	4	4	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₂²⋊C₄	—	76	2	no	—
285	C₂²⋊C₄	—	76	2	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	—	2	1	no	—
290	C₂³	—	4	1	no	—
291	C₂³	—	4	1	no	—
292	C₂³	—	4	1	no	—
293	C₂³	—	4	1	no	—
294	C₂³	#10597292	1	2	yes	complemented
295	C₂³	#10597385	1	2	yes	—
296	C₂³	C₂×#64527	1	1	yes	characteristic
297	C₂³	C₂×#64528	1	1	yes	characteristic
298	C₂³	#10597396	1	2	yes	—
299	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
300	Q₈	—	2	8	no	—
301	Q₈	—	4	4	no	—
302	Q₈	—	4	4	no	—
303	Q₈	—	4	4	no	—
304	D₄	—	2	4	no	—
305	D₄	—	2	4	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	4	no	—
310	C₂×C₄	—	2	4	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	8	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	8	1	no	—
325	C₂×C₄	—	152	2	no	—
326	C₂×C₄	—	152	2	no	—
327	C₂×C₄	—	152	2	no	—
328	C₂×C₄	#10597292	1	2	yes	complemented
329	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
330	C₂×C₄	#10597381	1	1	yes	characteristic
331	C₂×C₄	#10597383	1	1	yes	characteristic
332	C₈	—	76	4	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	2	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	4	1	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	4	2	no	—
341	C₂²	—	4	2	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	8	1	no	—
345	C₂²	#11926424	1	1	yes	characteristic
346	C₂²	C₂×#10597292	1	1	yes	characteristic
347	C₂²	#11926540	1	1	yes	characteristic
348	C₂²	#11926775	1	2	yes	—
349	C₂²	#11926783	1	2	yes	—
350	C₄	—	2	1	no	—
351	C₄	—	2	2	no	—
352	C₄	—	2	2	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	4	no	—
355	C₄	—	4	1	no	—
356	C₄	—	8	2	no	—
357	C₄	—	304	2	no	—
358	C₄	#11926730	1	2	yes	—
359	C₂	—	2	1	no	—
360	C₂	—	2	2	no	—
361	C₂	—	2	1	no	—
362	C₂	—	2	2	no	—
363	C₂	—	4	2	no	—
364	C₂	#14564044	1	1	yes	characteristic
365	C₂	#14564059	1	1	yes	characteristic
366	C₂	#14564158	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926424
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₁₉×#2409	C₂
socle	C₂²×C₁₉	#386
prime radical	C₂²×D₄×C₁₉	D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#2409	D₁₉
hypocenter	C₁₉	#9467
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2409

Complements

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

N	H ≅ G/N	type
3. #14563570	C₂	semidirect
4. #14561502	C₂	semidirect
17. C₁₉×#395	C₄	semidirect
18. C₁₉×#393	C₄	semidirect
111. C₂³×C₁₉	#103	semidirect
142. C₂×C₄×C₁₉	#103	semidirect
190. #395	Dic₁₉	semidirect
191. #393	Dic₁₉	semidirect
241. C₁₉	#9467	semidirect
294. C₂³	#10597292	semidirect
328. C₂×C₄	#10597292	semidirect

Maximal subgroups

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

subgroup	index
#14572659	2
#14563623	2
#14563570	2
#14561502	2
C₁₉×#2409	2
#9467	19