Subgroups — #18075930

Order 4864.

588 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563703	C₂	1	1	yes	maximal, characteristic
2	#14561862	C₂	1	1	yes	maximal, characteristic
3	#14561513	C₂	1	4	yes	maximal
4	C₁₉×#2285	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂²⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
6	C₂×C₁₉⋊#95	—	2	1	no	—
7	C₂×C₁₉⋊#95	—	2	1	no	—
8	C₂×C₁₉⋊#95	C₂²	1	2	yes	—
9	#11926299	—	2	1	no	—
10	#11926299	—	2	1	no	—
11	C₂×C₁₉×#117	—	2	1	no	—
12	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
13	C₁₉×#384	C₄	1	1	yes	characteristic, complemented
14	C₁₉×#384	C₄	1	1	yes	characteristic, complemented
15	C₁₉×#382	—	2	1	no	—
16	C₁₉×#380	—	2	1	no	—
17	C₁₉×#380	C₂²	1	2	yes	—
18	C₄×C₄⋊C₄×C₁₉	—	2	1	no	—
19	C₄×C₄⋊C₄×C₁₉	C₄	1	2	yes	complemented
20	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
23	C₂³×Dic₁₉	—	2	2	no	—
24	C₁₉×#117	—	4	2	no	—
25	C₁₉×#117	—	4	2	no	—
26	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
27	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
28	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
32	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
33	C₂×C₄²×C₁₉	—	2	4	no	—
34	C₂×C₄²×C₁₉	—	2	2	no	—
35	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
39	C₂²×C₁₉⋊C₈	—	2	2	no	—
40	C₁₉×#95	—	2	2	no	—
41	C₁₉×#95	—	2	2	no	—
42	C₁₉⋊#95	—	2	4	no	—
43	C₁₉⋊#95	—	2	4	no	—
44	C₁₉⋊#95	—	4	2	no	—
45	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
46	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₉	—	2	2	no	—
48	C₂²×C₄×C₁₉	—	2	2	no	—
49	C₂²×C₄×C₁₉	—	2	2	no	—
50	C₂²×C₄×C₁₉	—	2	2	no	—
51	C₂²×C₄×C₁₉	—	2	2	no	—
52	C₂²×C₄×C₁₉	—	2	2	no	—
53	C₂²×C₄×C₁₉	—	4	1	no	—
54	C₂²×C₄×C₁₉	—	4	1	no	—
55	C₂²×C₄×C₁₉	—	4	1	no	—
56	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	D₈	1	2	yes	—
61	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
62	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	#41	1	4	yes	—
65	C₂²×Dic₁₉	—	2	4	no	—
66	C₂²×Dic₁₉	—	4	2	no	—
67	C₂²×Dic₁₉	—	4	2	no	—
68	C₂²×Dic₁₉	—	4	2	no	—
69	C₂²×Dic₁₉	—	4	2	no	—
70	C₂²×Dic₁₉	—	4	2	no	—
71	C₂²×Dic₁₉	—	4	2	no	—
72	C₄⋊C₄×C₁₉	—	4	2	no	—
73	C₄⋊C₄×C₁₉	—	4	2	no	—
74	C₄⋊C₄×C₁₉	—	4	2	no	—
75	C₄⋊C₄×C₁₉	—	4	2	no	—
76	C₄⋊C₄×C₁₉	—	4	2	no	—
77	C₄⋊C₄×C₁₉	—	4	2	no	—
78	C₂²⋊C₄×C₁₉	—	4	2	no	—
79	C₂²⋊C₄×C₁₉	—	4	2	no	—
80	C₄²×C₁₉	—	2	2	no	—
81	C₄²×C₁₉	—	2	2	no	—
82	C₄²×C₁₉	—	2	2	no	—
83	C₄²×C₁₉	—	2	2	no	—
84	C₄²×C₁₉	—	4	4	no	—
85	C₂×C₁₉⋊C₈	—	4	2	no	—
86	C₂×C₁₉⋊C₈	—	4	2	no	—
87	C₂×C₁₉⋊C₈	—	4	2	no	—
88	#8716	—	19	1	no	maximal, Sylow
89	C₂³×C₁₉	—	2	1	no	—
90	C₂³×C₁₉	—	2	1	no	—
91	C₂³×C₁₉	—	2	1	no	—
92	C₂³×C₁₉	—	2	1	no	—
93	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂³×C₁₉	#123	1	2	yes	—
95	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
96	C₂³×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
97	C₂³×C₁₉	#137	1	2	yes	—
98	C₂×C₄×C₁₉	—	2	1	no	—
99	C₂×C₄×C₁₉	—	2	1	no	—
100	C₂×C₄×C₁₉	—	2	1	no	—
101	C₂×C₄×C₁₉	—	2	1	no	—
102	C₂×C₄×C₁₉	—	2	1	no	—
103	C₂×C₄×C₁₉	—	2	1	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	4	no	—
109	C₂×C₄×C₁₉	—	2	4	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	—	4	4	no	—
114	C₂×C₄×C₁₉	—	4	4	no	—
115	C₂×C₄×C₁₉	—	4	2	no	—
116	C₂×C₄×C₁₉	—	4	2	no	—
117	C₂×C₄×C₁₉	—	4	2	no	—
118	C₂×C₄×C₁₉	—	4	2	no	—
119	C₂×C₄×C₁₉	—	4	2	no	—
120	C₂×C₄×C₁₉	—	4	2	no	—
121	C₂×C₄×C₁₉	#102	1	2	yes	—
122	C₂×C₄×C₁₉	#102	1	2	yes	—
123	C₂×C₄×C₁₉	#104	1	2	yes	—
124	C₂×C₄×C₁₉	#117	1	2	yes	—
125	C₂×C₄×C₁₉	#123	1	2	yes	—
126	C₂×Dic₁₉	—	4	4	no	—
127	C₂×Dic₁₉	—	4	4	no	—
128	C₂×Dic₁₉	—	4	4	no	—
129	C₂×Dic₁₉	—	8	2	no	—
130	C₂×Dic₁₉	—	8	2	no	—
131	C₂×Dic₁₉	—	8	2	no	—
132	C₂×Dic₁₉	—	8	2	no	—
133	C₁₉⋊C₈	—	8	2	no	—
134	#2285	D₁₉	1	1	yes	characteristic
135	#1817	—	19	1	no	—
136	#1434	—	19	1	no	—
137	#1294	—	19	4	no	—
138	C₂²×C₁₉	—	2	2	no	—
139	C₂²×C₁₉	—	2	2	no	—
140	C₂²×C₁₉	—	2	1	no	—
141	C₂²×C₁₉	—	2	1	no	—
142	C₂²×C₁₉	—	2	2	no	—
143	C₂²×C₁₉	—	2	2	no	—
144	C₂²×C₁₉	—	2	1	no	—
145	C₂²×C₁₉	—	2	1	no	—
146	C₂²×C₁₉	#381	1	1	yes	characteristic
147	C₂²×C₁₉	#410	1	1	yes	characteristic
148	C₂²×C₁₉	#410	1	1	yes	characteristic
149	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
150	C₂²×C₁₉	#419	1	1	yes	characteristic
151	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
152	C₂²×C₁₉	#421	1	1	yes	characteristic
153	C₂²×C₁₉	#480	1	2	yes	—
154	C₂²×C₁₉	#483	1	2	yes	—
155	C₄×C₁₉	—	2	2	no	—
156	C₄×C₁₉	—	2	2	no	—
157	C₄×C₁₉	—	2	2	no	—
158	C₄×C₁₉	—	2	2	no	—
159	C₄×C₁₉	—	2	2	no	—
160	C₄×C₁₉	—	4	1	no	—
161	C₄×C₁₉	—	8	2	no	—
162	C₄×C₁₉	—	8	2	no	—
163	Dic₁₉	—	8	4	no	—
164	C₂×#117	—	2	1	no	—
165	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
166	C₂×C₂²⋊C₈	—	19	2	no	—
167	#384	Dic₁₉	1	1	yes	characteristic, complemented
168	#384	Dic₁₉	1	1	yes	characteristic, complemented
169	#382	—	2	1	no	—
170	#380	—	2	1	no	—
171	#380	C₂×D₁₉	1	2	yes	—
172	C₄×C₄⋊C₄	—	2	1	no	—
173	C₄×C₄⋊C₄	Dic₁₉	1	2	yes	complemented
174	C₂×#95	—	19	2	no	—
175	C₂×#95	—	38	1	no	—
176	C₂×#95	—	38	1	no	—
177	#336	—	38	1	no	—
178	#336	—	38	1	no	—
179	C₂×C₁₉	—	2	2	no	—
180	C₂×C₁₉	—	2	2	no	—
181	C₂×C₁₉	#1502	1	1	yes	characteristic
182	C₂×C₁₉	#1506	1	1	yes	characteristic
183	C₂×C₁₉	#1515	1	1	yes	characteristic
184	C₂×C₁₉	#1517	1	1	yes	characteristic
185	C₂×C₁₉	#1820	1	1	yes	characteristic
186	C₂×C₁₉	#1823	1	1	yes	characteristic
187	C₂×C₁₉	#1826	1	1	yes	characteristic
188	C₂³×C₄	—	38	2	no	—
189	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
190	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
191	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
192	C₂²×C₈	—	38	2	no	—
193	#117	—	4	2	no	—
194	#117	—	4	2	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄⋊C₄	—	2	1	no	—
199	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
200	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
201	C₂×C₂²⋊C₄	—	2	2	no	—
202	C₂×C₄²	—	2	4	no	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
205	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
206	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
207	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
208	C₂²⋊C₈	—	38	4	no	—
209	#95	—	2	2	no	—
210	#95	—	2	2	no	—
211	#95	—	38	4	no	—
212	#95	—	38	4	no	—
213	#95	—	76	2	no	—
214	C₁₉	#8716	1	1	yes	characteristic, complemented, Sylow
215	C₂⁴	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	—	4	1	no	—
224	C₂²×C₄	—	4	1	no	—
225	C₂²×C₄	—	38	4	no	—
226	C₂²×C₄	—	76	2	no	—
227	C₂²×C₄	—	76	2	no	—
228	C₂²×C₄	—	76	2	no	—
229	C₂²×C₄	—	76	2	no	—
230	C₂²×C₄	—	76	2	no	—
231	C₂²×C₄	—	76	2	no	—
232	C₂²×C₄	#64525	1	2	yes	—
233	C₂²×C₄	#64526	1	2	yes	—
234	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
235	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
236	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
237	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
238	C₂²×C₄	#64541	1	4	yes	—
239	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
241	C₂×C₈	—	76	2	no	—
242	C₂×C₈	—	76	2	no	—
243	C₂×C₈	—	76	2	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₂²⋊C₄	—	4	2	no	—
251	C₂²⋊C₄	—	4	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	4	4	no	—
257	C₂³	—	2	1	no	—
258	C₂³	—	2	1	no	—
259	C₂³	—	2	1	no	—
260	C₂³	—	2	1	no	—
261	C₂³	#10597375	1	2	yes	—
262	C₂³	C₂×#64525	1	1	yes	characteristic
263	C₂³	C₂×#64526	1	1	yes	characteristic
264	C₂³	#10597388	1	2	yes	—
265	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	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	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	76	4	no	—
290	C₂×C₄	—	76	4	no	—
291	C₂×C₄	—	76	4	no	—
292	C₂×C₄	—	152	2	no	—
293	C₂×C₄	—	152	2	no	—
294	C₂×C₄	—	152	2	no	—
295	C₂×C₄	—	152	2	no	—
296	C₂×C₄	#10597289	1	2	yes	—
297	C₂×C₄	#10597289	1	2	yes	—
298	C₂×C₄	#10597294	1	2	yes	—
299	C₂×C₄	#10597372	1	2	yes	—
300	C₂×C₄	#10597375	1	2	yes	—
301	C₈	—	152	2	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	2	no	—
310	C₂²	#11926407	1	1	yes	characteristic
311	C₂²	#11926518	1	1	yes	characteristic
312	C₂²	#11926518	1	1	yes	characteristic
313	C₂²	C₂×#10597289	1	1	yes	characteristic
314	C₂²	#11926553	1	1	yes	characteristic
315	C₂²	C₂×#10597294	1	1	yes	characteristic
316	C₂²	#11926561	1	1	yes	characteristic
317	C₂²	#11926857	1	2	yes	—
318	C₂²	#11926874	1	2	yes	—
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	2	2	no	—
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	4	1	no	—
325	C₄	—	8	2	no	—
326	C₄	—	8	2	no	—
327	C₄	—	152	4	no	—
328	C₂	—	2	2	no	—
329	C₂	—	2	2	no	—
330	C₂	#14562130	1	1	yes	characteristic
331	C₂	#14562147	1	1	yes	characteristic
332	C₂	#14562186	1	1	yes	characteristic
333	C₂	#14562195	1	1	yes	characteristic
334	C₂	#14563712	1	1	yes	characteristic
335	C₂	#14563721	1	1	yes	characteristic
336	C₂	#14563734	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₁₉×#2285	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂×D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#2285	D₁₉
hypocenter	C₁₉	#8716
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #2285

Complements

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

N	H ≅ G/N	type
13. C₁₉×#384	C₄	semidirect
14. C₁₉×#384	C₄	semidirect
19. C₄×C₄⋊C₄×C₁₉	C₄	semidirect
167. #384	Dic₁₉	semidirect
168. #384	Dic₁₉	semidirect
173. C₄×C₄⋊C₄	Dic₁₉	semidirect
214. C₁₉	#8716	semidirect

Maximal subgroups

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

subgroup	index
#14563703	2
#14561862	2
#14561513	2
C₁₉×#2285	2
#8716	19