Subgroups — #18077628

Order 4864.

528 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14566903	C₂	1	1	yes	maximal, characteristic
2	#14562063	C₂	1	2	yes	maximal, complemented
3	#14561949	C₂	1	1	yes	maximal, characteristic
4	#14561536	C₂	1	2	yes	maximal, complemented
5	C₁₉×#2905	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₃₈	C₂²	1	1	yes	characteristic
7	#11926597	C₂²	1	2	yes	—
8	#11926483	C₄	1	1	yes	characteristic
9	#11926480	C₄	1	1	yes	characteristic
10	#11926477	—	2	1	no	—
11	#11926467	—	2	1	no	—
12	#11926464	C₄	1	1	yes	characteristic
13	#11926456	C₄	1	1	yes	characteristic
14	C₄×Dic₁₉⋊C₄	—	2	1	no	—
15	C₄×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
16	C₄×C₂²⋊Dic₁₉	—	2	1	no	—
17	#11926274	—	2	4	no	—
18	#11926270	—	2	1	no	—
19	#11926268	—	2	1	no	—
20	C₂²×M₁₆×C₁₉	—	2	1	no	—
21	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
22	C₁₉×#424	—	2	4	no	—
23	C₁₉×#423	—	2	1	no	—
24	C₁₉×#423	C₂²	1	2	yes	—
25	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
26	C₂²×Dic₃₈	C₂×C₄	1	1	yes	characteristic
27	C₂×M₁₆×C₁₉	—	2	2	no	—
28	C₂×M₁₆×C₁₉	—	2	2	no	—
29	C₂×M₁₆×C₁₉	—	4	2	no	—
30	C₂×M₁₆×C₁₉	—	4	2	no	—
31	C₂²×C₈×C₁₉	—	2	2	no	—
32	C₁₉×#117	—	2	2	no	—
33	C₁₉×#117	—	2	2	no	—
34	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×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	C₂×C₂²⋊Dic₁₉	—	2	1	no	—
41	C₂×C₂²⋊Dic₁₉	—	2	1	no	—
42	C₂×C₄⋊Dic₁₉	—	2	1	no	—
43	C₂×C₄⋊Dic₁₉	C₂×C₄	1	1	yes	characteristic
44	C₂×Dic₁₉⋊C₄	—	2	1	no	—
45	C₂×Dic₁₉⋊C₄	—	2	1	no	—
46	C₂×Dic₁₉⋊C₄	—	2	1	no	—
47	C₂×C₄×Dic₁₉	—	2	1	no	—
48	C₂×C₄×Dic₁₉	—	2	1	no	—
49	C₄×Dic₃₈	—	2	2	no	—
50	C₄×Dic₃₈	—	2	2	no	—
51	C₁₉×C₄⋊C₈	—	2	4	no	—
52	C₁₉×C₄⋊C₈	—	2	2	no	—
53	C₁₉×C₄⋊C₈	D₄	1	4	yes	—
54	C₁₉⋊#95	—	2	1	no	—
55	C₁₉⋊#95	—	2	1	no	—
56	C₁₉⋊#95	—	2	1	no	—
57	C₁₉⋊#95	—	2	1	no	—
58	C₁₉⋊#95	C₂×C₄	1	1	yes	characteristic
59	C₁₉⋊#95	C₂×C₄	1	1	yes	characteristic
60	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
61	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
62	C₄×C₁₉⋊C₈	—	2	2	no	—
63	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
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	SD₁₆	1	2	yes	—
72	C₂²×C₄×C₁₉	Q₁₆	1	2	yes	—
73	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×Dic₁₉	—	2	1	no	—
77	C₂²×Dic₁₉	—	2	1	no	—
78	C₂²×Dic₁₉	—	2	1	no	—
79	C₂²×Dic₁₉	—	2	1	no	—
80	C₂×Dic₃₈	—	2	2	no	—
81	C₂×Dic₃₈	—	4	2	no	—
82	M₁₆×C₁₉	—	4	4	no	—
83	M₁₆×C₁₉	—	4	2	no	—
84	M₁₆×C₁₉	—	4	2	no	—
85	C₂×C₈×C₁₉	—	2	4	no	—
86	C₂×C₈×C₁₉	—	4	4	no	—
87	C₄⋊C₄×C₁₉	—	2	2	no	—
88	C₄⋊C₄×C₁₉	—	2	2	no	—
89	C₂²⋊C₄×C₁₉	—	2	2	no	—
90	C₂²⋊C₄×C₁₉	—	2	2	no	—
91	C₄²×C₁₉	—	2	1	no	—
92	C₄²×C₁₉	—	2	1	no	—
93	C₄²×C₁₉	C₂×D₄	1	2	yes	—
94	C₄²×C₁₉	#41	1	2	yes	—
95	C₂²⋊Dic₁₉	—	4	2	no	—
96	C₂²⋊Dic₁₉	—	4	2	no	—
97	C₄⋊Dic₁₉	—	2	2	no	—
98	C₄⋊Dic₁₉	—	4	2	no	—
99	Dic₁₉⋊C₄	—	4	2	no	—
100	Dic₁₉⋊C₄	—	4	2	no	—
101	Dic₁₉⋊C₄	—	4	2	no	—
102	C₄×Dic₁₉	—	4	1	no	—
103	C₄×Dic₁₉	—	4	1	no	—
104	C₄×Dic₁₉	—	4	1	no	—
105	C₄×Dic₁₉	—	4	1	no	—
106	C₂×C₁₉⋊C₈	—	4	2	no	—
107	#9033	—	19	1	no	maximal, Sylow
108	C₂³×C₁₉	—	2	1	no	—
109	C₂³×C₁₉	—	2	1	no	—
110	C₂³×C₁₉	—	4	1	no	—
111	C₂³×C₁₉	—	4	1	no	—
112	C₂³×C₁₉	#103	1	2	yes	—
113	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₉	—	2	1	no	—
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	2	2	no	—
117	C₂×C₄×C₁₉	—	2	2	no	—
118	C₂×C₄×C₁₉	—	2	2	no	—
119	C₂×C₄×C₁₉	—	2	2	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	#103	1	2	yes	—
127	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
129	C₂×C₄×C₁₉	#121	1	2	yes	—
130	C₂×C₄×C₁₉	#123	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	1	yes	characteristic
134	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
135	C₂×Dic₁₉	—	2	2	no	—
136	C₂×Dic₁₉	—	2	2	no	—
137	C₂×Dic₁₉	—	4	1	no	—
138	C₂×Dic₁₉	—	4	1	no	—
139	C₂×Dic₁₉	—	4	1	no	—
140	C₂×Dic₁₉	—	4	1	no	—
141	C₂×Dic₁₉	—	4	1	no	—
142	C₂×Dic₁₉	—	4	1	no	—
143	C₂×Dic₁₉	—	4	1	no	—
144	C₂×Dic₁₉	—	4	1	no	—
145	C₂×Dic₁₉	—	4	1	no	—
146	C₂×Dic₁₉	—	4	1	no	—
147	Dic₃₈	—	2	4	no	—
148	Dic₃₈	—	4	2	no	—
149	C₈×C₁₉	—	4	4	no	—
150	C₁₉⋊C₈	—	8	2	no	—
151	#2905	D₁₉	1	1	yes	characteristic
152	#2396	—	19	1	no	—
153	#1487	—	19	2	no	—
154	#1458	—	19	1	no	—
155	#1303	—	19	2	no	—
156	C₂²×C₁₉	—	2	1	no	—
157	C₂²×C₁₉	—	2	1	no	—
158	C₂²×C₁₉	—	2	2	no	—
159	C₂²×C₁₉	—	4	2	no	—
160	C₂²×C₁₉	—	4	1	no	—
161	C₂²×C₁₉	—	4	1	no	—
162	C₂²×C₁₉	—	8	1	no	—
163	C₂²×C₁₉	#386	1	1	yes	characteristic
164	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
165	C₂²×C₁₉	#417	1	1	yes	characteristic
166	C₄×C₁₉	—	2	2	no	—
167	C₄×C₁₉	—	2	2	no	—
168	C₄×C₁₉	—	2	2	no	—
169	C₄×C₁₉	—	4	1	no	—
170	C₄×C₁₉	—	4	1	no	—
171	C₄×C₁₉	#450	1	1	yes	characteristic
172	C₄×C₁₉	#451	1	1	yes	characteristic
173	C₄×C₁₉	#483	1	1	yes	characteristic
174	C₄×C₁₉	#484	1	1	yes	characteristic
175	Dic₁₉	—	4	2	no	—
176	Dic₁₉	—	4	2	no	—
177	Dic₁₉	—	8	1	no	—
178	Dic₁₉	—	8	1	no	—
179	C₂²×M₁₆	—	2	1	no	—
180	C₂×C₄×Q₈	—	19	1	no	—
181	C₂×#117	C₂×D₁₉	1	1	yes	characteristic
182	#431	—	19	2	no	—
183	#424	—	2	4	no	—
184	#423	—	2	1	no	—
185	#423	C₂×D₁₉	1	2	yes	—
186	#400	—	19	1	no	—
187	#399	—	19	1	no	—
188	#398	—	38	1	no	—
189	#396	—	38	1	no	—
190	#395	—	19	1	no	—
191	#393	—	19	1	no	—
192	C₄×C₄⋊C₄	—	19	1	no	—
193	C₄×C₄⋊C₄	—	38	1	no	—
194	C₄×C₂²⋊C₄	—	38	1	no	—
195	Q₈⋊C₈	—	38	4	no	—
196	#324	—	38	1	no	—
197	#323	—	38	1	no	—
198	C₂×C₁₉	—	2	1	no	—
199	C₂×C₁₉	—	2	1	no	—
200	C₂×C₁₉	—	4	2	no	—
201	C₂×C₁₉	#1872	1	1	yes	characteristic
202	C₂×C₁₉	#1884	1	1	yes	characteristic
203	C₂×C₁₉	#1891	1	1	yes	characteristic
204	C₂²×Q₈	—	19	1	no	—
205	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
206	C₂×M₁₆	—	2	2	no	—
207	C₂×M₁₆	—	2	2	no	—
208	C₂×M₁₆	—	4	2	no	—
209	C₂×M₁₆	—	4	2	no	—
210	C₂²×C₈	—	2	2	no	—
211	C₄×Q₈	—	38	2	no	—
212	C₄×Q₈	—	38	2	no	—
213	#117	—	2	2	no	—
214	#117	—	2	2	no	—
215	C₂×C₄⋊C₄	—	19	1	no	—
216	C₂×C₄⋊C₄	—	38	1	no	—
217	C₂×C₄⋊C₄	—	38	1	no	—
218	C₂×C₄⋊C₄	—	38	1	no	—
219	C₂×C₄⋊C₄	—	38	1	no	—
220	C₂×C₄⋊C₄	C₄×D₁₉	1	1	yes	characteristic
221	C₂×C₄⋊C₄	D₇₆	1	1	yes	characteristic
222	C₂×C₂²⋊C₄	—	38	1	no	—
223	C₂×C₂²⋊C₄	—	38	1	no	—
224	C₂×C₂²⋊C₄	C₄×D₁₉	1	1	yes	characteristic
225	C₂×C₂²⋊C₄	D₇₆	1	1	yes	characteristic
226	C₂×C₄²	—	38	1	no	—
227	C₂×C₄²	—	38	1	no	—
228	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
229	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
230	C₄⋊C₈	—	2	4	no	—
231	C₄⋊C₈	—	2	2	no	—
232	C₄⋊C₈	—	38	2	no	—
233	C₄⋊C₈	C₂²⋊D₁₉	1	4	yes	—
234	C₂²⋊C₈	—	38	2	no	—
235	C₄×C₈	—	38	2	no	—
236	#95	—	19	1	no	—
237	#95	—	19	1	no	—
238	#95	—	38	1	no	—
239	#95	—	38	1	no	—
240	#95	—	38	1	no	—
241	#95	—	38	1	no	—
242	C₁₉	#9033	1	1	yes	characteristic, complemented, Sylow
243	C₂⁴	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
244	C₂×Q₈	—	38	2	no	—
245	C₂×Q₈	—	76	2	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	38	1	no	—
251	C₂²×C₄	—	38	1	no	—
252	C₂²×C₄	—	38	1	no	—
253	C₂²×C₄	—	38	1	no	—
254	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
255	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
256	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
257	C₂²×C₄	#64526	1	2	yes	—
258	C₂²×C₄	#64528	1	2	yes	—
259	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
260	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
262	M₁₆	—	4	4	no	—
263	M₁₆	—	4	2	no	—
264	M₁₆	—	4	2	no	—
265	C₂×C₈	—	2	4	no	—
266	C₂×C₈	—	4	4	no	—
267	C₂×C₈	—	76	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	38	2	no	—
271	C₄⋊C₄	—	76	2	no	—
272	C₄⋊C₄	—	76	2	no	—
273	C₄⋊C₄	—	76	2	no	—
274	C₄⋊C₄	—	76	2	no	—
275	C₂²⋊C₄	—	2	2	no	—
276	C₂²⋊C₄	—	2	2	no	—
277	C₂²⋊C₄	—	76	2	no	—
278	C₂²⋊C₄	—	76	2	no	—
279	C₄²	—	2	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	76	1	no	—
282	C₄²	—	76	1	no	—
283	C₄²	—	76	1	no	—
284	C₄²	—	76	1	no	—
285	C₄²	#64541	1	2	yes	—
286	C₄²	C₂×C₂²⋊D₁₉	1	2	yes	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	—	4	1	no	—
290	C₂³	—	4	1	no	—
291	C₂³	#10597267	1	2	yes	—
292	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
293	Q₈	—	38	4	no	—
294	Q₈	—	76	2	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	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	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	—	38	2	no	—
308	C₂×C₄	—	38	2	no	—
309	C₂×C₄	—	76	1	no	—
310	C₂×C₄	—	76	1	no	—
311	C₂×C₄	—	76	1	no	—
312	C₂×C₄	—	76	1	no	—
313	C₂×C₄	—	76	1	no	—
314	C₂×C₄	—	76	1	no	—
315	C₂×C₄	—	76	1	no	—
316	C₂×C₄	—	76	1	no	—
317	C₂×C₄	—	76	1	no	—
318	C₂×C₄	—	76	1	no	—
319	C₂×C₄	#10597267	1	2	yes	—
320	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
321	C₂×C₄	#10597375	1	1	yes	characteristic
322	C₂×C₄	#10597376	1	2	yes	—
323	C₂×C₄	#10597378	1	1	yes	characteristic
324	C₂×C₄	C₂×#64526	1	1	yes	characteristic
325	C₂×C₄	#10597388	1	1	yes	characteristic
326	C₂×C₄	C₂×#64528	1	1	yes	characteristic
327	C₂×C₄	#10597399	1	1	yes	characteristic
328	C₈	—	4	4	no	—
329	C₈	—	152	2	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	2	no	—
333	C₂²	—	4	2	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	8	1	no	—
337	C₂²	#11926423	1	1	yes	characteristic
338	C₂²	C₂×#10597267	1	1	yes	characteristic
339	C₂²	#11926543	1	1	yes	characteristic
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	2	no	—
343	C₄	—	4	1	no	—
344	C₄	—	4	1	no	—
345	C₄	—	76	2	no	—
346	C₄	—	76	2	no	—
347	C₄	—	152	1	no	—
348	C₄	—	152	1	no	—
349	C₄	#11926695	1	1	yes	characteristic
350	C₄	#11926700	1	1	yes	characteristic
351	C₄	#11926874	1	1	yes	characteristic
352	C₄	#11926879	1	1	yes	characteristic
353	C₂	—	2	1	no	—
354	C₂	—	2	1	no	—
355	C₂	—	4	2	no	—
356	C₂	#14563942	1	1	yes	characteristic
357	C₂	#14564018	1	1	yes	characteristic
358	C₂	#14564053	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597399
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2905	C₂
socle	C₂²×C₁₉	#386
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2905	D₁₉
hypocenter	C₁₉	#9033
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2905

Complements

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

N	H ≅ G/N	type
2. #14562063	C₂	semidirect
4. #14561536	C₂	semidirect
242. C₁₉	#9033	semidirect

Maximal subgroups

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

subgroup	index
#14566903	2
#14562063	2
#14561949	2
#14561536	2
C₁₉×#2905	2
#9033	19