Subgroups — #18070849

Order 4864.

500 conjugacy classes of subgroups.

The subgroup lattice has 372 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	#14563615	C₂	1	1	yes	maximal, characteristic
3	#14561461	C₂	1	2	yes	maximal, complemented
4	#14561458	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₉×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
7	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
8	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
9	#11926299	—	2	1	no	—
10	#11926299	C₂²	1	2	yes	—
11	#11926299	C₂²	1	2	yes	—
12	C₂×C₁₉×C₂²⋊Q₈	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₁₉	C₄	1	1	yes	characteristic
14	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
15	C₁₉×#391	C₂²	1	1	yes	characteristic
16	C₁₉×#389	C₂²	1	1	yes	characteristic
17	C₁₉×#387	—	2	1	no	—
18	C₁₉×#385	—	2	1	no	—
19	C₁₉×#380	C₄	1	1	yes	characteristic
20	C₄×C₄⋊C₄×C₁₉	—	2	1	no	—
21	C₄×C₄⋊C₄×C₁₉	C₄	1	1	yes	characteristic
22	C₄×C₂²⋊C₄×C₁₉	—	2	1	no	—
23	C₂²×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
27	C₁₉×C₂²⋊Q₈	—	2	2	no	—
28	C₁₉×C₂²⋊Q₈	—	2	2	no	—
29	C₄×Q₈×C₁₉	—	4	2	no	—
30	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
31	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
32	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
33	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
37	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
38	C₂×C₄²×C₁₉	—	2	1	no	—
39	C₂×C₄²×C₁₉	—	2	1	no	—
40	C₂×C₄²×C₁₉	—	4	2	no	—
41	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
45	C₂²×C₁₉⋊C₈	—	2	2	no	—
46	C₂²×C₁₉⋊C₈	—	2	2	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	C₂²⋊(C₁₉⋊C₈)	—	4	2	no	—
53	C₂²⋊(C₁₉⋊C₈)	—	4	2	no	—
54	C₄⋊(C₁₉⋊C₈)	—	4	2	no	—
55	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Q₈×C₁₉	—	2	1	no	—
57	C₂×Q₈×C₁₉	—	4	1	no	—
58	C₂×Q₈×C₁₉	—	4	1	no	—
59	C₂×Q₈×C₁₉	M₁₆	1	2	yes	—
60	C₂²×C₄×C₁₉	—	2	1	no	—
61	C₂²×C₄×C₁₉	—	2	1	no	—
62	C₂²×C₄×C₁₉	—	2	2	no	—
63	C₂²×C₄×C₁₉	—	2	1	no	—
64	C₂²×C₄×C₁₉	—	2	1	no	—
65	C₂²×C₄×C₁₉	—	2	2	no	—
66	C₂²×C₄×C₁₉	—	2	1	no	—
67	C₂²×C₄×C₁₉	—	2	1	no	—
68	C₂²×C₄×C₁₉	—	4	1	no	—
69	C₂²×C₄×C₁₉	—	4	1	no	—
70	C₂²×C₄×C₁₉	—	4	1	no	—
71	C₂²×C₄×C₁₉	—	4	1	no	—
72	C₂²×C₄×C₁₉	—	4	1	no	—
73	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
77	C₂²×C₄×C₁₉	Q₁₆	1	2	yes	—
78	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
81	C₄⋊C₄×C₁₉	—	2	4	no	—
82	C₄⋊C₄×C₁₉	—	2	1	no	—
83	C₄⋊C₄×C₁₉	—	4	1	no	—
84	C₄⋊C₄×C₁₉	—	4	1	no	—
85	C₄⋊C₄×C₁₉	—	4	2	no	—
86	C₄⋊C₄×C₁₉	—	4	2	no	—
87	C₄⋊C₄×C₁₉	M₁₆	1	2	yes	—
88	C₂²⋊C₄×C₁₉	—	2	4	no	—
89	C₂²⋊C₄×C₁₉	—	4	2	no	—
90	C₄²×C₁₉	—	2	2	no	—
91	C₄²×C₁₉	—	2	2	no	—
92	C₄²×C₁₉	—	4	1	no	—
93	C₄²×C₁₉	—	4	1	no	—
94	C₄²×C₁₉	—	4	2	no	—
95	C₄²×C₁₉	—	4	2	no	—
96	C₂×C₁₉⋊C₈	—	4	2	no	—
97	C₂×C₁₉⋊C₈	—	4	2	no	—
98	C₂×C₁₉⋊C₈	—	4	2	no	—
99	C₂×C₁₉⋊C₈	—	4	2	no	—
100	C₂×C₁₉⋊C₈	—	4	2	no	—
101	C₂×C₁₉⋊C₈	—	4	2	no	—
102	#7648	—	19	1	no	maximal, Sylow
103	C₂³×C₁₉	—	2	1	no	—
104	C₂³×C₁₉	—	2	1	no	—
105	C₂³×C₁₉	—	2	1	no	—
106	C₂³×C₁₉	—	2	1	no	—
107	C₂³×C₁₉	—	2	1	no	—
108	C₂³×C₁₉	—	2	1	no	—
109	C₂³×C₁₉	#103	1	2	yes	—
110	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	Q₈×C₁₉	—	4	2	no	—
112	Q₈×C₁₉	—	4	2	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	1	no	—
118	C₂×C₄×C₁₉	—	2	1	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	—	2	1	no	—
123	C₂×C₄×C₁₉	—	2	2	no	—
124	C₂×C₄×C₁₉	—	2	2	no	—
125	C₂×C₄×C₁₉	—	2	1	no	—
126	C₂×C₄×C₁₉	—	2	1	no	—
127	C₂×C₄×C₁₉	—	2	1	no	—
128	C₂×C₄×C₁₉	—	4	2	no	—
129	C₂×C₄×C₁₉	—	4	2	no	—
130	C₂×C₄×C₁₉	—	4	4	no	—
131	C₂×C₄×C₁₉	—	4	2	no	—
132	C₂×C₄×C₁₉	—	4	2	no	—
133	C₂×C₄×C₁₉	—	4	1	no	—
134	C₂×C₄×C₁₉	—	4	1	no	—
135	C₂×C₄×C₁₉	—	4	1	no	—
136	C₂×C₄×C₁₉	—	4	1	no	—
137	C₂×C₄×C₁₉	—	4	1	no	—
138	C₂×C₄×C₁₉	—	4	1	no	—
139	C₂×C₄×C₁₉	—	4	1	no	—
140	C₂×C₄×C₁₉	—	4	1	no	—
141	C₂×C₄×C₁₉	—	4	1	no	—
142	C₂×C₄×C₁₉	—	4	1	no	—
143	C₂×C₄×C₁₉	#99	1	2	yes	—
144	C₂×C₄×C₁₉	#103	1	2	yes	—
145	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
146	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
147	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
148	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
149	C₂×C₄×C₁₉	#135	1	2	yes	—
150	C₁₉⋊C₈	—	8	2	no	—
151	C₁₉⋊C₈	—	8	2	no	—
152	C₄×C₂²⋊Q₈	D₁₉	1	1	yes	characteristic
153	#1817	—	19	1	no	—
154	#1797	—	19	1	no	—
155	#1272	—	19	2	no	—
156	#1271	—	19	2	no	—
157	C₂²×C₁₉	—	2	2	no	—
158	C₂²×C₁₉	—	2	2	no	—
159	C₂²×C₁₉	—	2	2	no	—
160	C₂²×C₁₉	—	4	1	no	—
161	C₂²×C₁₉	—	4	1	no	—
162	C₂²×C₁₉	—	4	1	no	—
163	C₂²×C₁₉	—	4	1	no	—
164	C₂²×C₁₉	#407	1	1	yes	characteristic
165	C₂²×C₁₉	C₂×#99	1	1	yes	characteristic
166	C₂²×C₁₉	#413	1	1	yes	characteristic
167	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
168	C₂²×C₁₉	#416	1	1	yes	characteristic
169	C₂²×C₁₉	#421	1	1	yes	characteristic
170	C₂²×C₁₉	#421	1	1	yes	characteristic
171	C₄×C₁₉	—	2	1	no	—
172	C₄×C₁₉	—	2	1	no	—
173	C₄×C₁₉	—	2	2	no	—
174	C₄×C₁₉	—	2	2	no	—
175	C₄×C₁₉	—	2	1	no	—
176	C₄×C₁₉	—	2	2	no	—
177	C₄×C₁₉	—	2	1	no	—
178	C₄×C₁₉	—	4	1	no	—
179	C₄×C₁₉	—	4	2	no	—
180	C₄×C₁₉	—	4	2	no	—
181	C₄×C₁₉	—	8	1	no	—
182	C₄×C₁₉	—	8	1	no	—
183	C₂×C₂²⋊Q₈	Dic₁₉	1	1	yes	characteristic
184	C₂×C₄×Q₈	Dic₁₉	1	1	yes	characteristic
185	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
186	C₂×C₄⋊C₈	—	38	1	no	—
187	C₂×C₂²⋊C₈	—	38	1	no	—
188	C₂×C₂²⋊C₈	—	38	1	no	—
189	#391	C₂×D₁₉	1	1	yes	characteristic
190	#389	C₂×D₁₉	1	1	yes	characteristic
191	#387	—	2	1	no	—
192	#385	—	2	1	no	—
193	#380	Dic₁₉	1	1	yes	characteristic
194	C₄×C₄⋊C₄	—	2	1	no	—
195	C₄×C₄⋊C₄	Dic₁₉	1	1	yes	characteristic
196	C₄×C₂²⋊C₄	—	2	1	no	—
197	#336	—	19	2	no	—
198	#336	—	19	2	no	—
199	#336	—	38	1	no	—
200	C₂×C₁₉	—	4	2	no	—
201	C₂×C₁₉	#1465	1	1	yes	characteristic
202	C₂×C₁₉	#1487	1	1	yes	characteristic
203	C₂×C₁₉	#1488	1	1	yes	characteristic
204	C₂×C₁₉	#1516	1	1	yes	characteristic
205	C₂×C₁₉	#1520	1	1	yes	characteristic
206	C₂×C₁₉	#1535	1	1	yes	characteristic
207	C₂×C₁₉	#1542	1	1	yes	characteristic
208	C₂²×Q₈	C₂×Dic₁₉	1	1	yes	characteristic
209	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
210	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
211	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
212	C₂²×C₈	—	38	2	no	—
213	C₂²×C₈	—	38	2	no	—
214	C₂²⋊Q₈	—	2	2	no	—
215	C₂²⋊Q₈	—	2	2	no	—
216	C₄×Q₈	—	4	2	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
221	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
222	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
223	C₂×C₂²⋊C₄	—	2	1	no	—
224	C₂×C₂²⋊C₄	—	2	1	no	—
225	C₂×C₄²	—	2	1	no	—
226	C₂×C₄²	—	2	1	no	—
227	C₂×C₄²	—	4	2	no	—
228	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
229	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
230	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
231	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
232	C₄⋊C₈	—	76	2	no	—
233	C₂²⋊C₈	—	76	2	no	—
234	C₂²⋊C₈	—	76	2	no	—
235	#95	—	2	1	no	—
236	#95	—	2	1	no	—
237	#95	—	2	1	no	—
238	#95	—	2	1	no	—
239	#95	—	2	1	no	—
240	C₁₉	#7648	1	1	yes	characteristic, complemented, Sylow
241	C₂⁴	C₂²⋊Dic₁₉	1	1	yes	characteristic
242	C₂×Q₈	—	2	1	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×Q₈	—	4	1	no	—
245	C₂×Q₈	#64520	1	2	yes	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	2	1	no	—
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	—	4	1	no	—
257	C₂²×C₄	—	4	1	no	—
258	C₂²×C₄	—	4	1	no	—
259	C₂²×C₄	#64527	1	2	yes	—
260	C₂²×C₄	#64528	1	2	yes	—
261	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
262	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
263	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
264	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
266	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
267	C₂×C₈	—	76	2	no	—
268	C₂×C₈	—	76	2	no	—
269	C₂×C₈	—	76	2	no	—
270	C₂×C₈	—	76	2	no	—
271	C₂×C₈	—	76	2	no	—
272	C₂×C₈	—	76	2	no	—
273	C₄⋊C₄	—	2	4	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	#64520	1	2	yes	—
280	C₂²⋊C₄	—	2	4	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	4	1	no	—
285	C₄²	—	4	1	no	—
286	C₄²	—	4	2	no	—
287	C₄²	—	4	2	no	—
288	C₂³	—	2	1	no	—
289	C₂³	—	2	1	no	—
290	C₂³	—	2	1	no	—
291	C₂³	—	2	1	no	—
292	C₂³	—	2	1	no	—
293	C₂³	—	2	1	no	—
294	C₂³	#10597292	1	2	yes	—
295	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
296	Q₈	—	4	2	no	—
297	Q₈	—	4	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	4	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	#10597291	1	2	yes	—
329	C₂×C₄	#10597292	1	2	yes	—
330	C₂×C₄	C₂×#64520	1	1	yes	characteristic
331	C₂×C₄	C₂×#64520	1	1	yes	characteristic
332	C₂×C₄	C₂×#64527	1	1	yes	characteristic
333	C₂×C₄	C₂×#64528	1	1	yes	characteristic
334	C₂×C₄	#10597396	1	2	yes	—
335	C₈	—	152	2	no	—
336	C₈	—	152	2	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	#11926508	1	1	yes	characteristic
345	C₂²	C₂×#10597291	1	1	yes	characteristic
346	C₂²	#11926528	1	1	yes	characteristic
347	C₂²	C₂×#10597292	1	1	yes	characteristic
348	C₂²	#11926540	1	1	yes	characteristic
349	C₂²	#11926561	1	1	yes	characteristic
350	C₂²	#11926561	1	1	yes	characteristic
351	C₄	—	2	2	no	—
352	C₄	—	2	2	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	2	no	—
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	4	2	no	—
359	C₄	—	4	1	no	—
360	C₄	—	4	2	no	—
361	C₄	—	8	1	no	—
362	C₄	—	8	1	no	—
363	C₂	—	4	2	no	—
364	C₂	#14561973	1	1	yes	characteristic
365	C₂	#14562064	1	1	yes	characteristic
366	C₂	#14562069	1	1	yes	characteristic
367	C₂	#14562191	1	1	yes	characteristic
368	C₂	#14562207	1	1	yes	characteristic
369	C₂	#14562266	1	1	yes	characteristic
370	C₂	#14562292	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₄×C₁₉×C₂²⋊Q₈	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄×C₂²⋊Q₈	D₁₉
hypocenter	C₁₉	#7648
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₄×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
3. #14561461	C₂	semidirect
4. #14561458	C₂	semidirect
240. C₁₉	#7648	semidirect

Maximal subgroups

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

subgroup	index
#14563703	2
#14563615	2
#14561461	2
#14561458	2
C₄×C₁₉×C₂²⋊Q₈	2
#7648	19