Subgroups — #18076840

Order 4864.

438 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572204	C₂	1	1	yes	maximal, characteristic
2	#14563656	C₂	1	1	yes	maximal, characteristic
3	#14562410	C₂	1	1	yes	maximal, characteristic, complemented
4	#14562393	C₂	1	1	yes	maximal, characteristic, complemented
5	#14562015	C₂	1	1	yes	maximal, characteristic
6	#14561476	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1303	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10597353	—	2	1	no	—
9	C₄×#64541	C₂²	1	1	yes	characteristic
10	#11926572	—	2	1	no	—
11	#11926569	—	2	2	no	—
12	#11926568	C₂²	1	1	yes	characteristic
13	#11926566	—	2	1	no	—
14	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×D₁₉	—	2	1	no	—
16	C₂×#10597262	—	2	1	no	—
17	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
18	#11926304	—	2	1	no	—
19	#11926297	—	2	1	no	—
20	#11926297	—	2	1	no	—
21	#11926296	—	2	2	no	—
22	#11926274	C₄	1	1	yes	characteristic, complemented
23	#11926274	C₄	1	1	yes	characteristic, complemented
24	#11926271	C₄	1	1	yes	characteristic, complemented
25	#11926271	C₄	1	1	yes	characteristic, complemented
26	C₁₉×#431	C₂²	1	1	yes	characteristic
27	C₁₉×#423	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
29	C₂×#64541	D₄	1	1	yes	characteristic
30	C₂×M₁₆×C₁₉	—	2	1	no	—
31	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
32	C₂×C₄²×C₁₉	—	2	1	no	—
33	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
34	C₄×C₂²⋊D₁₉	—	2	1	no	—
35	C₄×C₂²⋊D₁₉	—	2	1	no	—
36	#10597372	D₄	1	1	yes	characteristic
37	C₂×#64520	—	2	1	no	—
38	C₂²×C₁₉⋊C₈	—	2	1	no	—
39	C₂²×C₁₉⋊C₈	—	2	1	no	—
40	#10597353	—	4	2	no	—
41	#10597353	—	4	2	no	—
42	C₂×#64515	—	2	2	no	—
43	C₂×C₈×D₁₉	—	2	2	no	—
44	C₄×D₇₆	C₂×C₄	1	1	yes	characteristic
45	C₄×D₇₆	Q₈	1	1	yes	characteristic
46	#10597318	—	2	1	no	—
47	C₄²×D₁₉	—	2	1	no	—
48	C₄×Dic₃₈	C₂×C₄	1	1	yes	characteristic
49	C₄×Dic₃₈	Q₈	1	1	yes	characteristic
50	C₁₉×C₄⋊C₈	—	2	1	no	—
51	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
52	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
54	C₁₉×C₂²⋊C₈	—	2	1	no	—
55	C₄×C₈×C₁₉	—	2	1	no	—
56	C₁₉×#95	—	2	1	no	—
57	Dic₁₉⋊C₈	—	2	2	no	—
58	Dic₁₉⋊C₈	—	2	2	no	—
59	#10597262	—	4	2	no	—
60	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
61	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
62	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
63	C₄⋊(C₁₉⋊C₈)	D₄	1	1	yes	characteristic
64	C₄⋊(C₁₉⋊C₈)	D₄	1	1	yes	characteristic
65	C₄×C₁₉⋊C₈	—	2	1	no	—
66	C₄×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
67	C₄×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	—	4	1	no	—
69	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
72	C₂×C₂²⋊D₁₉	—	2	1	no	—
73	#64541	—	2	1	no	—
74	#64541	—	2	1	no	—
75	#64541	D₈	1	2	yes	—
76	#64541	Q₁₆	1	2	yes	—
77	C₂×D₇₆	C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×D₁₉	—	2	1	no	—
79	C₂×C₄×D₁₉	—	2	1	no	—
80	C₂×C₄×D₁₉	—	2	1	no	—
81	C₂×Dic₃₈	C₄⋊C₄	1	1	yes	characteristic
82	M₁₆×C₁₉	—	4	1	no	—
83	M₁₆×C₁₉	—	4	1	no	—
84	C₂×C₈×C₁₉	—	2	1	no	—
85	C₂×C₈×C₁₉	—	2	1	no	—
86	C₂×C₈×C₁₉	—	2	2	no	—
87	C₄⋊C₄×C₁₉	—	2	2	no	—
88	C₄⋊C₄×C₁₉	—	4	1	no	—
89	C₄²×C₁₉	—	4	1	no	—
90	C₄²×C₁₉	—	4	1	no	—
91	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
95	C₂²⋊Dic₁₉	—	2	1	no	—
96	(C₂×D₁₉)⋊C₄	—	2	1	no	—
97	(C₂×D₁₉)⋊C₄	—	2	1	no	—
98	C₄⋊Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
99	C₄⋊Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
100	Dic₁₉⋊C₄	—	2	1	no	—
101	Dic₁₉⋊C₄	—	2	1	no	—
102	C₄×Dic₁₉	—	2	1	no	—
103	C₄×Dic₁₉	—	2	1	no	—
104	C₄×Dic₁₉	—	2	1	no	—
105	#64520	—	4	1	no	—
106	#64520	—	4	1	no	—
107	C₂×C₁₉⋊C₈	—	2	1	no	—
108	C₂×C₁₉⋊C₈	—	2	1	no	—
109	C₂×C₁₉⋊C₈	—	2	1	no	—
110	C₂×C₁₉⋊C₈	—	2	1	no	—
111	C₂×C₁₉⋊C₈	—	4	2	no	—
112	C₂×C₁₉⋊C₈	—	4	1	no	—
113	C₂×C₁₉⋊C₈	—	4	1	no	—
114	C₂×C₁₉⋊C₈	—	4	1	no	—
115	#64515	—	4	4	no	—
116	C₈×D₁₉	—	4	4	no	—
117	#8902	—	19	1	no	maximal, Sylow
118	C₂³×C₁₉	#123	1	1	yes	characteristic
119	C₂²×D₁₉	—	2	1	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	2	no	—
124	C₂×C₄×C₁₉	—	8	1	no	—
125	C₂×C₄×C₁₉	—	8	1	no	—
126	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
130	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
134	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
136	C₂²⋊D₁₉	—	2	2	no	—
137	C₂²⋊D₁₉	—	2	2	no	—
138	C₂×Dic₁₉	—	2	1	no	—
139	C₂×Dic₁₉	—	2	1	no	—
140	C₂×Dic₁₉	—	2	1	no	—
141	D₇₆	—	2	1	no	—
142	D₇₆	#107	1	2	yes	—
143	C₄×D₁₉	—	2	2	no	—
144	C₄×D₁₉	—	2	2	no	—
145	C₄×D₁₉	—	4	1	no	—
146	C₄×D₁₉	—	4	1	no	—
147	C₄×D₁₉	—	8	1	no	—
148	Dic₃₈	—	2	1	no	—
149	Dic₃₈	#107	1	2	yes	—
150	C₈×C₁₉	—	4	1	no	—
151	C₈×C₁₉	—	4	1	no	—
152	C₈×C₁₉	—	4	2	no	—
153	C₁₉⋊C₈	—	2	2	no	—
154	C₁₉⋊C₈	—	4	1	no	—
155	C₁₉⋊C₈	—	4	1	no	—
156	C₁₉⋊C₈	—	4	1	no	—
157	C₁₉⋊C₈	—	8	1	no	—
158	#2906	—	19	1	no	—
159	#1806	—	19	1	no	—
160	#1571	—	19	1	no	—
161	#1567	—	19	1	no	—
162	#1476	—	19	1	no	—
163	#1303	D₁₉	1	1	yes	characteristic, complemented
164	#1279	—	19	1	no	—
165	C₂²×C₁₉	—	2	1	no	—
166	C₂²×C₁₉	—	2	1	no	—
167	C₂²×C₁₉	#385	1	1	yes	characteristic
168	C₂²×C₁₉	#480	1	1	yes	characteristic
169	C₂²×C₁₉	#484	1	1	yes	characteristic
170	C₂×D₁₉	—	2	2	no	—
171	C₂×D₁₉	—	4	1	no	—
172	C₂×D₁₉	—	4	1	no	—
173	C₄×C₁₉	—	2	1	no	—
174	C₄×C₁₉	—	2	1	no	—
175	C₄×C₁₉	—	2	1	no	—
176	C₄×C₁₉	—	8	2	no	—
177	C₄×C₁₉	C₂×#107	1	1	yes	characteristic
178	C₄×C₁₉	#428	1	1	yes	characteristic
179	C₄×C₁₉	#443	1	1	yes	characteristic
180	C₄×C₁₉	#444	1	1	yes	characteristic
181	C₄×C₁₉	#447	1	1	yes	characteristic
182	C₄×C₁₉	#451	1	1	yes	characteristic
183	Dic₁₉	—	2	2	no	—
184	Dic₁₉	—	4	1	no	—
185	Dic₁₉	—	4	1	no	—
186	C₂×#131	—	38	1	no	—
187	C₄×#41	—	19	1	no	—
188	#431	C₂×D₁₉	1	1	yes	characteristic
189	#424	—	38	1	no	—
190	#424	—	38	2	no	—
191	#423	—	19	1	no	—
192	#423	—	38	1	no	—
193	#423	C₂×D₁₉	1	1	yes	characteristic
194	C₂×C₄⋊C₈	—	19	1	no	—
195	C₂×C₄⋊C₈	—	38	1	no	—
196	C₂×#104	—	38	1	no	—
197	C₂×C₄×C₈	—	19	1	no	—
198	C₄×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
199	#339	—	38	1	no	—
200	#335	—	38	1	no	—
201	#335	—	38	1	no	—
202	#335	—	38	2	no	—
203	Q₈⋊C₈	—	19	1	no	—
204	Q₈⋊C₈	—	19	1	no	—
205	D₄⋊C₈	—	19	1	no	—
206	D₄⋊C₈	—	19	1	no	—
207	C₂×C₁₉	—	2	1	no	—
208	C₂×C₁₉	—	2	1	no	—
209	C₂×C₁₉	#1852	1	1	yes	characteristic
210	C₂×C₁₉	#1861	1	1	yes	characteristic
211	C₂×C₁₉	#1866	1	1	yes	characteristic
212	D₁₉	—	4	2	no	—
213	C₂×#41	—	19	1	no	—
214	#131	—	76	2	no	—
215	#131	—	76	2	no	—
216	C₂×M₁₆	—	2	1	no	—
217	C₂×M₁₆	—	38	2	no	—
218	C₂×M₁₆	—	38	1	no	—
219	C₂²×C₈	—	38	2	no	—
220	C₂²×C₈	—	38	1	no	—
221	C₂²×C₈	—	38	1	no	—
222	C₄×Q₈	—	19	1	no	—
223	C₄×Q₈	—	19	1	no	—
224	C₄×D₄	—	19	1	no	—
225	C₄×D₄	—	19	1	no	—
226	C₄×D₄	—	38	1	no	—
227	C₄×D₄	—	38	1	no	—
228	#117	—	19	1	no	—
229	#117	—	38	1	no	—
230	C₂×C₄⋊C₄	—	2	1	no	—
231	C₂×C₄²	—	2	1	no	—
232	C₂×C₄²	—	38	1	no	—
233	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
234	C₄⋊C₈	—	2	1	no	—
235	C₄⋊C₈	—	19	1	no	—
236	C₄⋊C₈	—	19	1	no	—
237	C₄⋊C₈	—	38	2	no	—
238	C₄⋊C₈	—	38	2	no	—
239	C₄⋊C₈	—	38	1	no	—
240	C₄⋊C₈	—	38	1	no	—
241	C₄⋊C₈	—	38	1	no	—
242	C₄⋊C₈	C₄×D₁₉	1	1	yes	characteristic
243	C₄⋊C₈	C₄×D₁₉	1	1	yes	characteristic
244	C₄⋊C₈	D₇₆	1	2	yes	—
245	#104	—	76	2	no	—
246	C₂²⋊C₈	—	2	1	no	—
247	C₄×C₈	—	2	1	no	—
248	C₄×C₈	—	19	1	no	—
249	C₄×C₈	—	19	1	no	—
250	C₄×C₈	—	38	1	no	—
251	#95	—	2	1	no	—
252	C₁₉	#8902	1	1	yes	characteristic, complemented, Sylow
253	#41	—	19	2	no	—
254	#41	—	19	2	no	—
255	#41	—	38	1	no	—
256	#41	—	38	1	no	—
257	C₂×Q₈	—	19	1	no	—
258	C₂×D₄	—	19	1	no	—
259	C₂×D₄	—	38	1	no	—
260	C₂²×C₄	—	4	1	no	—
261	C₂²×C₄	—	38	1	no	—
262	C₂²×C₄	—	38	1	no	—
263	C₂²×C₄	—	38	1	no	—
264	C₂²×C₄	#64541	1	1	yes	characteristic
265	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
266	C₂²×C₄	#64543	1	1	yes	characteristic
267	M₁₆	—	4	1	no	—
268	M₁₆	—	4	1	no	—
269	M₁₆	—	76	4	no	—
270	M₁₆	—	76	1	no	—
271	M₁₆	—	76	1	no	—
272	C₂×C₈	—	2	1	no	—
273	C₂×C₈	—	2	1	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	38	1	no	—
276	C₂×C₈	—	38	1	no	—
277	C₂×C₈	—	38	1	no	—
278	C₂×C₈	—	38	1	no	—
279	C₂×C₈	—	76	4	no	—
280	C₂×C₈	—	76	2	no	—
281	C₂×C₈	—	76	1	no	—
282	C₂×C₈	—	76	1	no	—
283	C₂×C₈	—	76	1	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₄⋊C₄	—	19	1	no	—
287	C₄⋊C₄	—	19	1	no	—
288	C₄⋊C₄	—	38	1	no	—
289	C₄⋊C₄	—	38	1	no	—
290	C₂²⋊C₄	—	38	1	no	—
291	C₂²⋊C₄	—	38	1	no	—
292	C₂²⋊C₄	—	38	1	no	—
293	C₄²	—	4	1	no	—
294	C₄²	—	4	1	no	—
295	C₄²	—	38	1	no	—
296	C₄²	—	38	1	no	—
297	C₄²	—	38	1	no	—
298	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
299	C₄²	C₂×D₇₆	1	1	yes	characteristic
300	C₄²	D₄×D₁₉	1	1	yes	characteristic
301	C₄²	Q₈×D₁₉	1	1	yes	characteristic
302	C₂³	—	38	1	no	—
303	C₂³	#10597329	1	1	yes	characteristic
304	Q₈	—	19	2	no	—
305	Q₈	—	38	1	no	—
306	D₄	—	19	2	no	—
307	D₄	—	38	2	no	—
308	D₄	—	38	2	no	—
309	D₄	—	38	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	8	1	no	—
315	C₂×C₄	—	8	1	no	—
316	C₂×C₄	—	38	1	no	—
317	C₂×C₄	—	38	2	no	—
318	C₂×C₄	—	38	2	no	—
319	C₂×C₄	—	38	1	no	—
320	C₂×C₄	—	38	1	no	—
321	C₂×C₄	—	76	1	no	—
322	C₂×C₄	—	76	1	no	—
323	C₂×C₄	—	152	1	no	—
324	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
325	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
326	C₂×C₄	#10597328	1	1	yes	characteristic
327	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
328	C₂×C₄	#10597342	1	1	yes	characteristic
329	C₂×C₄	#10597343	1	1	yes	characteristic
330	C₂×C₄	D₈×D₁₉	1	1	yes	characteristic
331	C₂×C₄	#10597360	1	1	yes	characteristic
332	C₂×C₄	#10597361	1	1	yes	characteristic
333	C₂×C₄	Q₁₆×D₁₉	1	1	yes	characteristic
334	C₈	—	4	1	no	—
335	C₈	—	4	1	no	—
336	C₈	—	4	2	no	—
337	C₈	—	38	2	no	—
338	C₈	—	76	1	no	—
339	C₈	—	76	1	no	—
340	C₈	—	76	1	no	—
341	C₈	—	152	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	38	2	no	—
345	C₂²	—	76	1	no	—
346	C₂²	—	76	1	no	—
347	C₂²	#11926418	1	1	yes	characteristic
348	C₂²	#11926856	1	1	yes	characteristic
349	C₂²	#11926878	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	8	2	no	—
354	C₄	—	38	2	no	—
355	C₄	—	76	1	no	—
356	C₄	—	76	1	no	—
357	C₄	#107×D₁₉	1	1	yes	characteristic
358	C₄	#11926583	1	1	yes	characteristic
359	C₄	#11926658	1	1	yes	characteristic
360	C₄	#11926663	1	1	yes	characteristic
361	C₄	#11926679	1	1	yes	characteristic
362	C₄	#11926701	1	1	yes	characteristic
363	C₂	—	2	1	no	—
364	C₂	—	2	1	no	—
365	C₂	—	76	2	no	—
366	C₂	#14563824	1	1	yes	characteristic
367	C₂	#14563878	1	1	yes	characteristic
368	C₂	#14563911	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597328
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1303	C₂
socle	C₂²×C₁₉	#385
prime radical	C₂×#64541	D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1303	D₁₉
hypocenter	C₁₉	#8902
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1303

Complements

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

N	H ≅ G/N	type
3. #14562410	C₂	semidirect
4. #14562393	C₂	semidirect
6. #14561476	C₂	semidirect
7. C₁₉×#1303	C₂	semidirect
22. #11926274	C₄	semidirect
23. #11926274	C₄	semidirect
24. #11926271	C₄	semidirect
25. #11926271	C₄	semidirect
163. #1303	D₁₉	semidirect
252. C₁₉	#8902	semidirect

Maximal subgroups

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

subgroup	index
#14572204	2
#14563656	2
#14562410	2
#14562393	2
#14562015	2
#14561476	2
C₁₉×#1303	2
#8902	19