Subgroups — #18090459

Order 4864.

842 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	C₂×#11926568	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926303	C₂	1	2	yes	maximal
3	C₂×#11926303	C₂	1	2	yes	maximal
4	C₄×C₁₉⋊#95	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#2296	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	2	yes	—
7	C₂²×C₄×Dic₁₉	C₂²	1	2	yes	—
8	#11926568	—	2	4	no	—
9	C₂×C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
10	C₂×C₁₉⋊#95	—	2	2	no	—
11	#11926303	—	2	4	no	—
12	#11926303	—	2	4	no	—
13	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
14	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
16	C₁₉×#389	C₄	1	2	yes	complemented
17	C₁₉×#389	C₄	1	2	yes	complemented
18	C₁₉×#385	—	2	2	no	—
19	C₄×C₄⋊C₄×C₁₉	—	2	2	no	—
20	C₂³×C₄×C₁₉	D₄	1	2	yes	—
21	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
22	C₂³×Dic₁₉	—	2	2	no	—
23	C₁₉×#117	—	2	1	no	—
24	C₁₉×#117	—	2	1	no	—
25	C₁₉×#117	—	2	2	no	—
26	C₁₉×#117	—	2	2	no	—
27	C₁₉×#117	D₄	1	2	yes	—
28	C₁₉×#117	Q₈	1	2	yes	—
29	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
30	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
31	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
33	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
34	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
35	C₂×C₄²×C₁₉	—	2	1	no	—
36	C₂×C₄²×C₁₉	—	2	2	no	—
37	C₂×C₄²×C₁₉	—	2	2	no	—
38	C₂×C₄²×C₁₉	—	2	1	no	—
39	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₉	D₄	1	2	yes	—
42	C₂×C₄×Dic₁₉	—	2	2	no	—
43	C₂×C₄×Dic₁₉	—	2	2	no	—
44	C₂×C₄×Dic₁₉	—	2	2	no	—
45	C₂×C₄×Dic₁₉	—	2	2	no	—
46	C₂×C₄×Dic₁₉	—	2	2	no	—
47	C₂×C₄×Dic₁₉	Q₈	1	4	yes	—
48	C₂×#64520	—	2	2	no	—
49	C₂×#64520	—	2	2	no	—
50	C₂×#64520	—	2	2	no	—
51	C₂×#64520	—	2	2	no	—
52	C₂×#64520	—	2	2	no	—
53	C₂×#64520	D₄	1	4	yes	—
54	C₂²×C₁₉⋊C₈	—	2	2	no	—
55	C₁₉×#95	—	2	2	no	—
56	C₁₉×#95	—	2	2	no	—
57	C₁₉⋊#95	—	2	8	no	—
58	C₄⋊(C₁₉⋊C₈)	—	2	8	no	—
59	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	—	2	2	no	—
61	C₂²×C₄×C₁₉	—	2	4	no	—
62	C₂²×C₄×C₁₉	—	2	4	no	—
63	C₂²×C₄×C₁₉	—	2	2	no	—
64	C₂²×C₄×C₁₉	—	4	2	no	—
65	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
74	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	#41	1	2	yes	—
76	C₂²×C₄×C₁₉	#41	1	2	yes	—
77	C₂²×Dic₁₉	—	2	8	no	—
78	C₂²×Dic₁₉	—	2	4	no	—
79	C₂²×Dic₁₉	—	4	4	no	—
80	C₂²×Dic₁₉	—	4	2	no	—
81	C₂²×Dic₁₉	—	4	2	no	—
82	C₄⋊C₄×C₁₉	—	2	2	no	—
83	C₄⋊C₄×C₁₉	—	2	4	no	—
84	C₄⋊C₄×C₁₉	—	4	4	no	—
85	C₄⋊C₄×C₁₉	—	4	4	no	—
86	C₄⋊C₄×C₁₉	C₄⋊C₄	1	4	yes	—
87	C₂²⋊C₄×C₁₉	—	2	2	no	—
88	C₂²⋊C₄×C₁₉	—	2	2	no	—
89	C₂²⋊C₄×C₁₉	—	4	2	no	—
90	C₄²×C₁₉	—	2	2	no	—
91	C₄²×C₁₉	—	2	2	no	—
92	C₄²×C₁₉	—	2	2	no	—
93	C₄²×C₁₉	—	2	2	no	—
94	C₄²×C₁₉	—	2	2	no	—
95	C₄²×C₁₉	—	2	2	no	—
96	C₄²×C₁₉	—	4	1	no	—
97	C₄²×C₁₉	—	4	1	no	—
98	C₄×Dic₁₉	—	2	4	no	—
99	C₄×Dic₁₉	—	2	4	no	—
100	C₄×Dic₁₉	—	4	2	no	—
101	C₄×Dic₁₉	—	4	2	no	—
102	#64520	—	2	4	no	—
103	#64520	—	2	4	no	—
104	#64520	—	4	2	no	—
105	#64520	—	4	2	no	—
106	C₂×C₁₉⋊C₈	—	2	4	no	—
107	C₂×C₁₉⋊C₈	—	4	2	no	—
108	C₂×C₁₉⋊C₈	—	4	2	no	—
109	#11209	—	19	1	no	maximal, Sylow
110	C₂³×C₁₉	—	2	2	no	—
111	C₂³×C₁₉	—	2	2	no	—
112	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂³×C₁₉	#121	1	1	yes	characteristic
114	C₂³×C₁₉	#121	1	2	yes	—
115	C₂³×C₁₉	C₂²⋊Q₈	1	2	yes	—
116	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₁₉	—	2	2	no	—
118	C₂×C₄×C₁₉	—	2	2	no	—
119	C₂×C₄×C₁₉	—	2	4	no	—
120	C₂×C₄×C₁₉	—	2	2	no	—
121	C₂×C₄×C₁₉	—	2	2	no	—
122	C₂×C₄×C₁₉	—	2	2	no	—
123	C₂×C₄×C₁₉	—	4	8	no	—
124	C₂×C₄×C₁₉	—	4	2	no	—
125	C₂×C₄×C₁₉	—	4	2	no	—
126	C₂×C₄×C₁₉	—	4	2	no	—
127	C₂×C₄×C₁₉	—	4	2	no	—
128	C₂×C₄×C₁₉	—	4	2	no	—
129	C₂×C₄×C₁₉	#104	1	4	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	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
134	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
135	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
136	C₂×C₄×C₁₉	#121	1	2	yes	—
137	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
138	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
139	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
140	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
141	C₂×C₄×C₁₉	#123	1	2	yes	—
142	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
143	C₂×C₄×C₁₉	#125	1	2	yes	—
144	C₂×C₄×C₁₉	#125	1	2	yes	—
145	C₂×C₄×C₁₉	#126	1	2	yes	—
146	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
147	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
148	C₂×Dic₁₉	—	2	8	no	—
149	C₂×Dic₁₉	—	4	8	no	—
150	C₂×Dic₁₉	—	4	8	no	—
151	C₂×Dic₁₉	—	4	4	no	—
152	C₂×Dic₁₉	—	4	4	no	—
153	C₁₉⋊C₈	—	4	4	no	—
154	C₂×#423	—	19	1	no	—
155	#2296	D₁₉	1	1	yes	characteristic
156	C₂×#339	—	19	2	no	—
157	C₂×#339	—	19	2	no	—
158	C₄×#95	—	19	1	no	—
159	C₂²×C₁₉	—	2	2	no	—
160	C₂²×C₁₉	—	2	2	no	—
161	C₂²×C₁₉	—	2	4	no	—
162	C₂²×C₁₉	—	2	2	no	—
163	C₂²×C₁₉	—	2	2	no	—
164	C₂²×C₁₉	#380	1	1	yes	characteristic
165	C₂²×C₁₉	#386	1	1	yes	characteristic
166	C₂²×C₁₉	#391	1	1	yes	characteristic
167	C₂²×C₁₉	#396	1	2	yes	—
168	C₂²×C₁₉	#398	1	2	yes	—
169	C₂²×C₁₉	C₂×#104	1	2	yes	—
170	C₂²×C₁₉	#421	1	2	yes	—
171	C₄×C₁₉	—	2	4	no	—
172	C₄×C₁₉	—	4	4	no	—
173	C₄×C₁₉	—	4	2	no	—
174	C₄×C₁₉	—	8	2	no	—
175	C₄×C₁₉	#387	1	2	yes	—
176	C₄×C₁₉	#389	1	2	yes	—
177	C₄×C₁₉	#394	1	1	yes	characteristic
178	C₄×C₁₉	#395	1	1	yes	characteristic
179	C₄×C₁₉	#398	1	1	yes	characteristic
180	C₄×C₁₉	#400	1	1	yes	characteristic
181	Dic₁₉	—	4	8	no	—
182	C₂²×M₁₆	—	19	2	no	—
183	C₂×#117	C₂×D₁₉	1	1	yes	characteristic
184	C₂×#117	C₂×D₁₉	1	1	yes	characteristic
185	C₂²×C₄²	—	19	2	no	—
186	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
187	#423	—	38	4	no	—
188	C₂×C₄⋊C₈	—	38	2	no	—
189	#389	Dic₁₉	1	2	yes	complemented
190	#389	Dic₁₉	1	2	yes	complemented
191	#385	—	2	2	no	—
192	C₄×C₄⋊C₄	—	2	2	no	—
193	C₂×#95	—	38	2	no	—
194	#339	—	38	4	no	—
195	#339	—	38	4	no	—
196	C₂×C₁₉	—	2	2	no	—
197	C₂×C₁₉	—	2	2	no	—
198	C₂×C₁₉	#1441	1	1	yes	characteristic
199	C₂×C₁₉	#1806	1	2	yes	—
200	C₂×C₁₉	#1871	1	2	yes	—
201	C₂×C₁₉	#1994	1	1	yes	characteristic
202	C₂×C₁₉	#1995	1	1	yes	characteristic
203	C₂³×C₄	—	38	2	no	—
204	C₂³×C₄	C₂²⋊D₁₉	1	2	yes	—
205	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
206	C₂×M₁₆	—	19	4	no	—
207	C₂×M₁₆	—	38	2	no	—
208	C₂×M₁₆	—	38	2	no	—
209	C₂×M₁₆	—	38	2	no	—
210	C₂×M₁₆	—	38	2	no	—
211	C₂×M₁₆	—	38	2	no	—
212	C₂²×C₈	—	38	2	no	—
213	#117	—	2	1	no	—
214	#117	—	2	1	no	—
215	#117	—	2	2	no	—
216	#117	—	2	2	no	—
217	#117	Dic₃₈	1	2	yes	—
218	#117	D₇₆	1	2	yes	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
222	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
223	C₂×C₂²⋊C₄	—	2	1	no	—
224	C₂×C₂²⋊C₄	—	2	1	no	—
225	C₂×C₄²	—	2	2	no	—
226	C₂×C₄²	—	2	2	no	—
227	C₂×C₄²	—	2	1	no	—
228	C₂×C₄²	—	2	1	no	—
229	C₂×C₄²	—	19	4	no	—
230	C₂×C₄²	—	38	2	no	—
231	C₂×C₄²	—	38	2	no	—
232	C₂×C₄²	—	38	2	no	—
233	C₂×C₄²	—	38	2	no	—
234	C₂×C₄²	—	38	2	no	—
235	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
236	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
237	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
238	C₄⋊C₈	—	38	8	no	—
239	#95	—	2	2	no	—
240	#95	—	2	2	no	—
241	#95	—	38	8	no	—
242	C₁₉	#11209	1	1	yes	characteristic, complemented, Sylow
243	C₂⁴	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	4	2	no	—
249	C₂²×C₄	—	38	8	no	—
250	C₂²×C₄	—	38	4	no	—
251	C₂²×C₄	—	76	4	no	—
252	C₂²×C₄	—	76	2	no	—
253	C₂²×C₄	—	76	2	no	—
254	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
255	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
256	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
258	C₂²×C₄	#64541	1	1	yes	characteristic
259	C₂²×C₄	#64541	1	1	yes	characteristic
260	C₂²×C₄	D₄×D₁₉	1	2	yes	—
261	C₂²×C₄	#64543	1	2	yes	—
262	C₂²×C₄	Q₈×D₁₉	1	2	yes	—
263	C₂²×C₄	#64545	1	2	yes	—
264	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
266	M₁₆	—	38	4	no	—
267	M₁₆	—	38	4	no	—
268	M₁₆	—	76	2	no	—
269	M₁₆	—	76	2	no	—
270	C₂×C₈	—	38	4	no	—
271	C₂×C₈	—	76	2	no	—
272	C₂×C₈	—	76	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	4	no	—
275	C₄⋊C₄	—	4	4	no	—
276	C₄⋊C₄	—	4	4	no	—
277	C₄⋊C₄	C₄⋊Dic₁₉	1	4	yes	—
278	C₂²⋊C₄	—	2	2	no	—
279	C₂²⋊C₄	—	2	2	no	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	2	2	no	—
287	C₄²	—	4	1	no	—
288	C₄²	—	4	1	no	—
289	C₄²	—	38	4	no	—
290	C₄²	—	38	4	no	—
291	C₄²	—	76	2	no	—
292	C₄²	—	76	2	no	—
293	C₂³	—	2	2	no	—
294	C₂³	—	2	2	no	—
295	C₂³	#10597370	1	1	yes	characteristic
296	C₂³	#10597376	1	1	yes	characteristic
297	C₂³	#10597385	1	2	yes	—
298	C₂³	#10597392	1	2	yes	—
299	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	4	8	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	38	8	no	—
313	C₂×C₄	—	76	4	no	—
314	C₂×C₄	—	76	8	no	—
315	C₂×C₄	—	76	8	no	—
316	C₂×C₄	—	76	4	no	—
317	C₂×C₄	#10597294	1	4	yes	—
318	C₂×C₄	#10597334	1	2	yes	—
319	C₂×C₄	#10597335	1	2	yes	—
320	C₂×C₄	#10597336	1	2	yes	—
321	C₂×C₄	#10597340	1	2	yes	—
322	C₂×C₄	#10597341	1	2	yes	—
323	C₂×C₄	#10597342	1	2	yes	—
324	C₂×C₄	#10597343	1	2	yes	—
325	C₂×C₄	#10597344	1	2	yes	—
326	C₂×C₄	#10597370	1	1	yes	characteristic
327	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
328	C₂×C₄	#10597372	1	1	yes	characteristic
329	C₂×C₄	#10597376	1	1	yes	characteristic
330	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
331	C₂×C₄	#10597381	1	1	yes	characteristic
332	C₂×C₄	#10597383	1	1	yes	characteristic
333	C₂×C₄	#10597390	1	1	yes	characteristic
334	C₂×C₄	Q₈×Dic₁₉	1	2	yes	—
335	C₂×C₄	#10597393	1	1	yes	characteristic
336	C₈	—	76	4	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	2	2	no	—
341	C₂²	—	2	4	no	—
342	C₂²	#11926404	1	1	yes	characteristic
343	C₂²	#11926424	1	1	yes	characteristic
344	C₂²	#11926451	1	1	yes	characteristic
345	C₂²	#11926466	1	2	yes	—
346	C₂²	#11926477	1	2	yes	—
347	C₂²	C₂×#10597294	1	2	yes	—
348	C₂²	#11926561	1	2	yes	—
349	C₄	—	2	4	no	—
350	C₄	—	4	4	no	—
351	C₄	—	4	2	no	—
352	C₄	—	8	2	no	—
353	C₄	—	76	8	no	—
354	C₄	#11926430	1	2	yes	—
355	C₄	#11926443	1	2	yes	—
356	C₄	#11926461	1	1	yes	characteristic
357	C₄	#11926464	1	1	yes	characteristic
358	C₄	#11926474	1	1	yes	characteristic
359	C₄	#11926483	1	1	yes	characteristic
360	C₂	—	2	2	no	—
361	C₂	—	2	2	no	—
362	C₂	#14561888	1	1	yes	characteristic
363	C₂	#14563655	1	2	yes	—
364	C₂	#14563938	1	2	yes	—
365	C₂	#14564613	1	1	yes	characteristic
366	C₂	#14564617	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₁₉×#2296	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂×D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#2296	D₁₉
hypocenter	C₁₉	#11209
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #2296

Complements

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

N	H ≅ G/N	type
16. C₁₉×#389	C₄	semidirect
17. C₁₉×#389	C₄	semidirect
189. #389	Dic₁₉	semidirect
190. #389	Dic₁₉	semidirect
242. C₁₉	#11209	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926568	2
C₂×#11926303	2
C₄×C₁₉⋊#95	2
C₁₉×#2296	2
#11209	19