Subgroups — #18081891

Order 4864.

428 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14573957	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562381	C₂	1	1	yes	maximal, characteristic, complemented
4	#14562133	C₂	1	1	yes	maximal, characteristic, complemented
5	#14562039	C₂	1	1	yes	maximal, characteristic, complemented
6	#14562034	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1502	C₂	1	1	yes	maximal, characteristic
8	#11927101	C₂²	1	1	yes	characteristic
9	#11926898	—	2	1	no	—
10	#11926898	—	2	1	no	—
11	#11926894	C₄	1	1	yes	characteristic, complemented
12	#11926894	C₄	1	1	yes	characteristic, complemented
13	#11926890	C₄	1	1	yes	characteristic, complemented
14	#11926890	C₄	1	1	yes	characteristic, complemented
15	#11926597	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
17	#11926540	—	2	1	no	—
18	#11926540	—	2	1	no	—
19	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
20	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
21	#11926299	—	2	1	no	—
22	#11926296	—	2	1	no	—
23	#11926294	—	2	1	no	—
24	#11926280	—	2	2	no	—
25	#11926276	—	2	2	no	—
26	D₄⋊(C₁₉⋊C₈)	—	2	2	no	—
27	C₁₉×#532	C₂²	1	1	yes	characteristic
28	C₁₉×#431	C₂²	1	1	yes	characteristic
29	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
30	C₁₉×#327	—	2	2	no	—
31	C₂×#41×C₁₉	—	2	1	no	—
32	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
33	C₁₉×#127	C₂×C₄	1	1	yes	characteristic
34	C₁₉×#124	—	2	1	no	—
35	C₁₉×#121	—	2	1	no	—
36	C₁₉×#121	—	2	1	no	—
37	C₄×D₄×C₁₉	—	2	1	no	—
38	C₄×D₄×C₁₉	—	2	1	no	—
39	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
40	C₂×C₄²×C₁₉	—	2	1	no	—
41	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
42	#10597372	—	2	1	no	—
43	#10597370	—	2	1	no	—
44	#10597370	—	2	1	no	—
45	C₂²×C₁₉⋊C₈	—	2	1	no	—
46	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
47	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
48	#10597316	—	2	1	no	—
49	#10597315	C₂×C₄	1	1	yes	characteristic
50	#10597315	C₂×C₄	1	1	yes	characteristic
51	C₄×Dic₃₈	—	2	1	no	—
52	C₄×Dic₃₈	—	2	1	no	—
53	C₁₉×C₄⋊C₈	—	2	1	no	—
54	C₁₉×C₂²⋊C₈	—	2	1	no	—
55	C₄×C₈×C₁₉	—	2	1	no	—
56	C₁₉×#95	D₄	1	2	yes	—
57	#10597292	—	2	1	no	—
58	#10597292	—	2	1	no	—
59	#10597292	—	2	1	no	—
60	#10597292	—	2	1	no	—
61	#10597289	—	2	1	no	—
62	#10597289	—	2	1	no	—
63	#10597289	—	2	1	no	—
64	#10597289	—	2	1	no	—
65	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
66	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
67	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
68	C₄⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
69	C₄×C₁₉⋊C₈	—	2	1	no	—
70	C₄×C₁₉⋊C₈	—	4	2	no	—
71	C₄×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
72	C₄×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
73	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
74	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
75	#41×C₁₉	—	4	1	no	—
76	#41×C₁₉	—	4	1	no	—
77	#41×C₁₉	—	4	1	no	—
78	#41×C₁₉	—	4	1	no	—
79	C₂×Q₈×C₁₉	—	2	1	no	—
80	C₂×D₄×C₁₉	—	2	1	no	—
81	C₂×D₄×C₁₉	—	2	1	no	—
82	C₂×D₄×C₁₉	—	2	1	no	—
83	C₂×D₄×C₁₉	—	4	1	no	—
84	C₂²×C₄×C₁₉	—	2	2	no	—
85	C₂²×C₄×C₁₉	—	4	1	no	—
86	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
89	C₂×Dic₃₈	—	2	1	no	—
90	C₂×Dic₃₈	—	2	1	no	—
91	C₂×C₈×C₁₉	—	4	1	no	—
92	C₄⋊C₄×C₁₉	—	2	4	no	—
93	C₄⋊C₄×C₁₉	—	2	1	no	—
94	C₄⋊C₄×C₁₉	—	2	1	no	—
95	C₂²⋊C₄×C₁₉	—	4	1	no	—
96	C₂²⋊C₄×C₁₉	—	4	1	no	—
97	C₄²×C₁₉	—	4	1	no	—
98	C₄²×C₁₉	—	4	1	no	—
99	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₁₉	#41	1	1	yes	characteristic
103	C₂²⋊Dic₁₉	—	4	1	no	—
104	C₄⋊Dic₁₉	—	2	1	no	—
105	C₄⋊Dic₁₉	—	2	1	no	—
106	C₄⋊Dic₁₉	—	2	1	no	—
107	C₄⋊Dic₁₉	—	2	1	no	—
108	Dic₁₉⋊C₄	—	4	1	no	—
109	Dic₁₉⋊C₄	—	4	1	no	—
110	C₄×Dic₁₉	—	4	1	no	—
111	C₂×C₁₉⋊C₈	—	2	1	no	—
112	C₂×C₁₉⋊C₈	—	2	1	no	—
113	C₂×C₁₉⋊C₈	—	2	1	no	—
114	C₂×C₁₉⋊C₈	—	2	1	no	—
115	C₂×C₁₉⋊C₈	—	2	2	no	—
116	C₂×C₁₉⋊C₈	—	4	1	no	—
117	C₂×C₁₉⋊C₈	—	4	1	no	—
118	C₂×C₁₉⋊C₈	—	4	1	no	—
119	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
123	#9751	—	19	1	no	maximal, Sylow
124	C₂³×C₁₉	—	4	1	no	—
125	C₂³×C₁₉	#121	1	1	yes	characteristic
126	Q₈×C₁₉	—	4	1	no	—
127	Q₈×C₁₉	—	4	1	no	—
128	D₄×C₁₉	—	4	1	no	—
129	D₄×C₁₉	—	4	1	no	—
130	D₄×C₁₉	—	4	2	no	—
131	D₄×C₁₉	—	8	1	no	—
132	D₄×C₁₉	—	8	1	no	—
133	D₄×C₁₉	—	8	1	no	—
134	C₂×C₄×C₁₉	—	2	2	no	—
135	C₂×C₄×C₁₉	—	2	4	no	—
136	C₂×C₄×C₁₉	—	4	1	no	—
137	C₂×C₄×C₁₉	—	4	2	no	—
138	C₂×C₄×C₁₉	—	4	2	no	—
139	C₂×C₄×C₁₉	—	4	1	no	—
140	C₂×C₄×C₁₉	—	8	1	no	—
141	C₂×C₄×C₁₉	—	8	1	no	—
142	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
146	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
147	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
148	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
149	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
150	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
152	C₂×Dic₁₉	—	4	1	no	—
153	C₂×Dic₁₉	—	4	1	no	—
154	Dic₃₈	—	4	2	no	—
155	Dic₃₈	—	8	1	no	—
156	C₈×C₁₉	—	8	1	no	—
157	C₁₉⋊C₈	—	2	1	no	—
158	C₁₉⋊C₈	—	2	1	no	—
159	C₁₉⋊C₈	—	2	1	no	—
160	C₁₉⋊C₈	—	2	1	no	—
161	C₁₉⋊C₈	—	4	2	no	—
162	C₁₉⋊C₈	—	8	1	no	—
163	#3131	—	19	1	no	—
164	C₈×C₄⋊C₄	—	19	1	no	—
165	#1564	—	19	1	no	—
166	#1503	—	19	1	no	—
167	#1502	D₁₉	1	1	yes	characteristic
168	#1481	—	19	1	no	—
169	#1480	—	19	1	no	—
170	C₂²×C₁₉	—	2	1	no	—
171	C₂²×C₁₉	—	2	1	no	—
172	C₂²×C₁₉	—	8	1	no	—
173	C₂²×C₁₉	—	8	1	no	—
174	C₂²×C₁₉	—	8	1	no	—
175	C₂²×C₁₉	#390	1	1	yes	characteristic
176	C₂²×C₁₉	#463	1	1	yes	characteristic
177	C₂²×C₁₉	#464	1	1	yes	characteristic
178	C₄×C₁₉	—	2	1	no	—
179	C₄×C₁₉	—	2	1	no	—
180	C₄×C₁₉	—	2	1	no	—
181	C₄×C₁₉	—	4	4	no	—
182	C₄×C₁₉	—	8	1	no	—
183	C₄×C₁₉	#416	1	1	yes	characteristic
184	C₄×C₁₉	#417	1	1	yes	characteristic
185	C₄×C₁₉	#443	1	2	yes	—
186	C₄×C₁₉	#463	1	1	yes	characteristic
187	C₄×C₁₉	#464	1	1	yes	characteristic
188	Dic₁₉	—	8	1	no	—
189	Dic₁₉	—	8	1	no	—
190	#533	—	19	1	no	—
191	#532	C₂×D₁₉	1	1	yes	characteristic
192	#488	—	38	1	no	—
193	#488	—	38	1	no	—
194	#487	—	19	1	no	—
195	#487	—	19	1	no	—
196	#486	—	19	1	no	—
197	#486	—	19	1	no	—
198	#431	—	19	1	no	—
199	#431	C₂×D₁₉	1	1	yes	characteristic
200	C₂×C₄⋊C₈	—	19	1	no	—
201	#416	—	38	1	no	—
202	#416	—	38	1	no	—
203	C₂×C₄×C₈	—	19	1	no	—
204	C₂×C₄×C₈	—	38	1	no	—
205	C₄×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
206	#336	—	38	1	no	—
207	#335	—	38	1	no	—
208	#334	—	38	1	no	—
209	#328	—	38	2	no	—
210	#327	—	2	2	no	—
211	Q₈⋊C₈	—	38	2	no	—
212	D₄⋊C₈	—	38	2	no	—
213	C₂×C₁₉	—	2	1	no	—
214	C₂×C₁₉	—	2	1	no	—
215	C₂×C₁₉	—	16	1	no	—
216	C₂×C₁₉	#1974	1	1	yes	characteristic
217	C₂×C₁₉	#1976	1	1	yes	characteristic
218	C₂×C₁₉	#1983	1	1	yes	characteristic
219	C₂×#41	—	2	1	no	—
220	C₂²×C₈	—	19	1	no	—
221	C₂²×C₈	—	19	1	no	—
222	C₂²×C₈	—	38	1	no	—
223	C₄⋊Q₈	—	19	1	no	—
224	C₄⋊Q₈	—	19	1	no	—
225	C₄⋊Q₈	C₄×D₁₉	1	1	yes	characteristic
226	#127	C₄×D₁₉	1	1	yes	characteristic
227	#125	—	38	1	no	—
228	#124	—	2	1	no	—
229	C₂²⋊Q₈	—	38	1	no	—
230	C₂²⋊Q₈	—	38	1	no	—
231	#121	—	2	1	no	—
232	#121	—	2	1	no	—
233	C₄×Q₈	—	38	1	no	—
234	C₄×Q₈	—	38	1	no	—
235	C₄×D₄	—	2	1	no	—
236	C₄×D₄	—	2	1	no	—
237	#117	—	38	1	no	—
238	C₂×C₄⋊C₄	—	2	1	no	—
239	C₂×C₄²	—	2	1	no	—
240	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
241	C₄⋊C₈	—	2	1	no	—
242	C₄⋊C₈	—	19	2	no	—
243	C₄⋊C₈	—	38	1	no	—
244	C₄⋊C₈	—	38	1	no	—
245	#103	—	38	1	no	—
246	#103	—	38	1	no	—
247	#103	—	38	1	no	—
248	#103	—	38	1	no	—
249	#102	—	38	1	no	—
250	#102	—	38	1	no	—
251	#102	—	38	1	no	—
252	#102	—	38	1	no	—
253	C₂²⋊C₈	—	2	1	no	—
254	C₂²⋊C₈	—	38	1	no	—
255	C₄×C₈	—	2	1	no	—
256	C₄×C₈	—	19	1	no	—
257	C₄×C₈	—	19	1	no	—
258	C₄×C₈	—	19	1	no	—
259	C₄×C₈	—	19	1	no	—
260	C₄×C₈	—	38	1	no	—
261	C₄×C₈	—	76	2	no	—
262	#95	C₂²⋊D₁₉	1	2	yes	—
263	C₁₉	#9751	1	1	yes	characteristic, complemented, Sylow
264	#41	—	4	1	no	—
265	#41	—	4	1	no	—
266	#41	—	4	1	no	—
267	#41	—	4	1	no	—
268	C₂×Q₈	—	2	1	no	—
269	C₂×Q₈	—	38	1	no	—
270	C₂×Q₈	—	38	1	no	—
271	C₂×D₄	—	2	1	no	—
272	C₂×D₄	—	2	1	no	—
273	C₂×D₄	—	2	1	no	—
274	C₂×D₄	—	4	1	no	—
275	C₂²×C₄	—	2	2	no	—
276	C₂²×C₄	—	4	1	no	—
277	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
278	C₂²×C₄	#64543	1	1	yes	characteristic
279	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
280	C₂×C₈	—	4	1	no	—
281	C₂×C₈	—	19	1	no	—
282	C₂×C₈	—	19	1	no	—
283	C₂×C₈	—	19	1	no	—
284	C₂×C₈	—	19	1	no	—
285	C₂×C₈	—	38	1	no	—
286	C₂×C₈	—	38	1	no	—
287	C₂×C₈	—	38	1	no	—
288	C₂×C₈	—	38	1	no	—
289	C₂×C₈	—	38	2	no	—
290	C₂×C₈	—	76	1	no	—
291	C₂×C₈	—	76	1	no	—
292	C₂×C₈	—	76	1	no	—
293	C₄⋊C₄	—	2	4	no	—
294	C₄⋊C₄	—	2	1	no	—
295	C₄⋊C₄	—	2	1	no	—
296	C₄⋊C₄	—	38	1	no	—
297	C₄⋊C₄	—	38	1	no	—
298	C₄⋊C₄	—	38	1	no	—
299	C₄⋊C₄	—	38	1	no	—
300	C₄⋊C₄	—	76	1	no	—
301	C₄⋊C₄	—	76	1	no	—
302	C₂²⋊C₄	—	4	1	no	—
303	C₂²⋊C₄	—	4	1	no	—
304	C₂²⋊C₄	—	76	1	no	—
305	C₄²	—	4	1	no	—
306	C₄²	—	4	1	no	—
307	C₄²	—	76	1	no	—
308	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
309	C₄²	#64541	1	1	yes	characteristic
310	C₄²	D₄×D₁₉	1	1	yes	characteristic
311	C₄²	D₄×D₁₉	1	1	yes	characteristic
312	C₂³	—	4	1	no	—
313	C₂³	#10597385	1	1	yes	characteristic
314	Q₈	—	4	1	no	—
315	Q₈	—	4	1	no	—
316	Q₈	—	76	2	no	—
317	Q₈	—	152	1	no	—
318	D₄	—	4	1	no	—
319	D₄	—	4	1	no	—
320	D₄	—	4	2	no	—
321	D₄	—	8	1	no	—
322	D₄	—	8	1	no	—
323	D₄	—	8	1	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	4	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	8	1	no	—
331	C₂×C₄	—	8	1	no	—
332	C₂×C₄	—	76	1	no	—
333	C₂×C₄	—	76	1	no	—
334	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
335	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
336	C₂×C₄	#10597330	1	1	yes	characteristic
337	C₂×C₄	#10597331	1	1	yes	characteristic
338	C₂×C₄	#10597357	1	1	yes	characteristic
339	C₂×C₄	#10597358	1	1	yes	characteristic
340	C₂×C₄	#10597361	1	1	yes	characteristic
341	C₂×C₄	#10597362	1	1	yes	characteristic
342	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
343	C₂×C₄	#10597386	1	1	yes	characteristic
344	C₈	—	8	1	no	—
345	C₈	—	38	1	no	—
346	C₈	—	38	1	no	—
347	C₈	—	38	1	no	—
348	C₈	—	38	1	no	—
349	C₈	—	76	2	no	—
350	C₈	—	152	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	8	1	no	—
354	C₂²	—	8	1	no	—
355	C₂²	—	8	1	no	—
356	C₂²	#11926444	1	1	yes	characteristic
357	C₂²	#11926757	1	1	yes	characteristic
358	C₂²	#11926764	1	1	yes	characteristic
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	4	4	no	—
363	C₄	—	8	1	no	—
364	C₄	—	152	1	no	—
365	C₄	—	152	1	no	—
366	C₄	#11926537	1	1	yes	characteristic
367	C₄	#11926542	1	1	yes	characteristic
368	C₄	#11926658	1	2	yes	—
369	C₄	#11926758	1	1	yes	characteristic
370	C₄	#11926765	1	1	yes	characteristic
371	C₂	—	2	1	no	—
372	C₂	—	2	1	no	—
373	C₂	—	16	1	no	—
374	C₂	#14564520	1	1	yes	characteristic
375	C₂	#14564530	1	1	yes	characteristic
376	C₂	#14564566	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597330
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1502	C₂
socle	C₂²×C₁₉	#390
prime radical	C₁₉×#532	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1502	D₁₉
hypocenter	C₁₉	#9751
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1502

Complements

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

N	H ≅ G/N	type
2. C₄⋊C₄×C₁₉⋊C₈	C₂	semidirect
3. #14562381	C₂	semidirect
4. #14562133	C₂	semidirect
5. #14562039	C₂	semidirect
11. #11926894	C₄	semidirect
12. #11926894	C₄	semidirect
13. #11926890	C₄	semidirect
14. #11926890	C₄	semidirect
263. C₁₉	#9751	semidirect

Maximal subgroups

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

subgroup	index
#14573957	2
C₄⋊C₄×C₁₉⋊C₈	2
#14562381	2
#14562133	2
#14562039	2
#14562034	2
C₁₉×#1502	2
#9751	19