Subgroups — #18082384

Order 4864.

562 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14564597	C₂	1	1	yes	maximal, characteristic, complemented
2	#14564442	C₂	1	1	yes	maximal, characteristic
3	#14564442	C₂	1	1	yes	maximal, characteristic
4	#14564438	C₂	1	1	yes	maximal, characteristic, complemented
5	#14563763	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11926265	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₉×#486	C₂	1	1	yes	maximal, characteristic
8	C₂×#10597292	—	2	2	no	—
9	C₂×#10597289	—	2	2	no	—
10	C₂×#10597289	—	2	2	no	—
11	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
13	#11926414	C₂²	1	1	yes	characteristic
14	#11926414	C₂²	1	1	yes	characteristic
15	#11926307	—	2	2	no	—
16	#11926306	—	2	2	no	—
17	#11926265	—	2	2	no	—
18	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₂×C₁₉×#127	C₂²	1	1	yes	characteristic
20	C₁₉×#486	—	2	2	no	—
21	C₁₉×#486	C₄	1	4	yes	complemented
22	C₂×C₁₉×#102	—	2	2	no	—
23	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
24	C₂²×Q₈×C₁₉	—	2	1	no	—
25	C₂²×D₄×C₁₉	—	2	1	no	—
26	C₂²×D₄×C₁₉	—	2	1	no	—
27	C₂²×D₄×C₁₉	—	2	1	no	—
28	C₂²×C₈×C₁₉	D₄	1	2	yes	—
29	C₁₉×C₄⋊Q₈	—	2	1	no	—
30	C₁₉×C₄⋊Q₈	—	2	1	no	—
31	C₁₉×C₄⋊Q₈	—	2	1	no	—
32	C₁₉×C₄⋊Q₈	C₂×C₄	1	2	yes	—
33	C₁₉×#127	—	2	1	no	—
34	C₁₉×#127	—	2	1	no	—
35	C₁₉×#127	—	2	1	no	—
36	C₁₉×#127	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
38	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
39	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
40	C₂×C₄⋊Dic₁₉	—	2	1	no	—
41	C₂×C₄⋊Dic₁₉	—	2	1	no	—
42	C₂×C₄⋊Dic₁₉	—	2	1	no	—
43	C₂×C₄⋊Dic₁₉	—	2	1	no	—
44	C₂×C₄⋊Dic₁₉	—	2	1	no	—
45	C₂×C₄⋊Dic₁₉	—	2	1	no	—
46	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
47	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
48	C₁₉×#102	—	2	8	no	—
49	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
50	C₄×C₈×C₁₉	D₄	1	2	yes	—
51	#10597292	—	4	4	no	—
52	#10597289	—	4	4	no	—
53	#10597289	—	4	4	no	—
54	C₄×C₁₉⋊C₈	—	2	1	no	—
55	C₄×C₁₉⋊C₈	—	2	1	no	—
56	C₄×C₁₉⋊C₈	—	2	1	no	—
57	C₄×C₁₉⋊C₈	—	2	1	no	—
58	C₂⁴×C₁₉	—	4	1	no	—
59	C₂×Q₈×C₁₉	—	2	2	no	—
60	C₂×Q₈×C₁₉	—	4	1	no	—
61	C₂×Q₈×C₁₉	—	4	2	no	—
62	C₂×Q₈×C₁₉	—	4	2	no	—
63	C₂×D₄×C₁₉	—	2	2	no	—
64	C₂×D₄×C₁₉	—	2	2	no	—
65	C₂×D₄×C₁₉	—	2	2	no	—
66	C₂×D₄×C₁₉	—	4	1	no	—
67	C₂×D₄×C₁₉	—	4	1	no	—
68	C₂×D₄×C₁₉	—	4	1	no	—
69	C₂×D₄×C₁₉	—	4	2	no	—
70	C₂×D₄×C₁₉	—	4	2	no	—
71	C₂×D₄×C₁₉	—	4	2	no	—
72	C₂×D₄×C₁₉	—	4	2	no	—
73	C₂×D₄×C₁₉	—	4	2	no	—
74	C₂×D₄×C₁₉	—	4	2	no	—
75	C₂²×C₄×C₁₉	—	4	1	no	—
76	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×Dic₁₉	—	4	1	no	—
80	C₂²×Dic₁₉	—	4	1	no	—
81	C₂×C₈×C₁₉	—	2	2	no	—
82	C₂×C₈×C₁₉	—	2	2	no	—
83	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
84	C₄⋊C₄×C₁₉	—	2	2	no	—
85	C₄⋊C₄×C₁₉	—	2	2	no	—
86	C₄⋊C₄×C₁₉	—	4	1	no	—
87	C₄⋊C₄×C₁₉	—	4	1	no	—
88	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₉	#41	1	1	yes	characteristic
91	C₄²×C₁₉	#41	1	1	yes	characteristic
92	C₄⋊Dic₁₉	—	4	2	no	—
93	C₄⋊Dic₁₉	—	4	2	no	—
94	C₄⋊Dic₁₉	—	4	2	no	—
95	C₄⋊Dic₁₉	—	4	2	no	—
96	C₄⋊Dic₁₉	—	4	2	no	—
97	C₄⋊Dic₁₉	—	4	2	no	—
98	C₂×C₁₉⋊C₈	—	2	2	no	—
99	C₂×C₁₉⋊C₈	—	2	2	no	—
100	C₂×C₁₉⋊C₈	—	2	2	no	—
101	C₂×C₁₉⋊C₈	—	2	2	no	—
102	C₂×C₁₉⋊C₈	—	2	2	no	—
103	C₂×C₁₉⋊C₈	—	2	2	no	—
104	#9838	—	19	1	no	maximal, Sylow
105	C₂³×C₁₉	—	4	2	no	—
106	C₂³×C₁₉	—	8	1	no	—
107	C₂³×C₁₉	—	8	1	no	—
108	C₂³×C₁₉	—	8	1	no	—
109	C₂³×C₁₉	—	8	1	no	—
110	C₂³×C₁₉	—	8	1	no	—
111	C₂³×C₁₉	—	8	1	no	—
112	C₂³×C₁₉	#127	1	1	yes	characteristic
113	Q₈×C₁₉	—	4	4	no	—
114	Q₈×C₁₉	—	8	2	no	—
115	D₄×C₁₉	—	4	4	no	—
116	D₄×C₁₉	—	4	4	no	—
117	D₄×C₁₉	—	4	4	no	—
118	D₄×C₁₉	—	8	2	no	—
119	D₄×C₁₉	—	8	2	no	—
120	D₄×C₁₉	—	8	2	no	—
121	C₂×C₄×C₁₉	—	4	2	no	—
122	C₂×C₄×C₁₉	—	8	1	no	—
123	C₂×C₄×C₁₉	—	8	1	no	—
124	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
128	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
129	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
130	C₂×C₄×C₁₉	#135	1	2	yes	—
131	C₂×C₄×C₁₉	#135	1	2	yes	—
132	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
133	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
134	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
135	C₂×C₄×C₁₉	#137	1	2	yes	—
136	C₂×Dic₁₉	—	8	1	no	—
137	C₂×Dic₁₉	—	8	1	no	—
138	C₂×Dic₁₉	—	8	1	no	—
139	C₂×Dic₁₉	—	8	1	no	—
140	C₂×Dic₁₉	—	8	1	no	—
141	C₂×Dic₁₉	—	8	1	no	—
142	C₈×C₁₉	—	2	4	no	—
143	C₁₉⋊C₈	—	4	2	no	—
144	C₁₉⋊C₈	—	4	2	no	—
145	C₂×#486	D₁₉	1	1	yes	characteristic
146	#1990	—	19	1	no	—
147	#1957	—	19	1	no	—
148	#1957	—	19	1	no	—
149	#1956	—	19	1	no	—
150	#1834	—	19	1	no	—
151	C₂×#322	—	19	1	no	—
152	C₂²×C₁₉	—	8	2	no	—
153	C₂²×C₁₉	—	8	2	no	—
154	C₂²×C₁₉	—	8	2	no	—
155	C₂²×C₁₉	—	16	1	no	—
156	C₂²×C₁₉	—	16	1	no	—
157	C₂²×C₁₉	—	16	1	no	—
158	C₂²×C₁₉	—	16	1	no	—
159	C₂²×C₁₉	#390	1	1	yes	characteristic
160	C₂²×C₁₉	#495	1	1	yes	characteristic
161	C₂²×C₁₉	#495	1	1	yes	characteristic
162	C₂²×C₁₉	#496	1	1	yes	characteristic
163	C₂²×C₁₉	#496	1	1	yes	characteristic
164	C₂²×C₁₉	#496	1	1	yes	characteristic
165	C₂²×C₁₉	#497	1	1	yes	characteristic
166	C₄×C₁₉	—	8	2	no	—
167	C₄×C₁₉	#416	1	2	yes	—
168	C₄×C₁₉	#440	1	2	yes	—
169	C₄×C₁₉	#442	1	2	yes	—
170	C₄×C₁₉	#468	1	2	yes	—
171	C₄×C₁₉	#469	1	2	yes	—
172	C₄×C₁₉	#488	1	2	yes	—
173	Dic₁₉	—	16	1	no	—
174	Dic₁₉	—	16	1	no	—
175	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
176	C₂×#127	C₂×D₁₉	1	1	yes	characteristic
177	#486	—	2	2	no	—
178	#486	Dic₁₉	1	4	yes	complemented
179	C₂×#103	—	38	2	no	—
180	C₂×#102	—	2	2	no	—
181	C₂×#102	—	38	2	no	—
182	C₂×#102	—	38	2	no	—
183	C₂×C₄×C₈	—	19	1	no	—
184	C₂×C₄×C₈	—	19	1	no	—
185	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
186	#384	—	19	1	no	—
187	#384	—	19	1	no	—
188	#340	—	38	2	no	—
189	#340	—	38	2	no	—
190	#322	—	38	2	no	—
191	C₂×C₁₉	—	16	2	no	—
192	C₂×C₁₉	#1717	1	1	yes	characteristic
193	C₂×C₁₉	#1717	1	1	yes	characteristic
194	C₂×C₁₉	#1719	1	1	yes	characteristic
195	C₂×C₁₉	#1720	1	1	yes	characteristic
196	C₂×C₁₉	#1958	1	1	yes	characteristic
197	C₂×C₁₉	#1972	1	1	yes	characteristic
198	C₂×C₁₉	#1973	1	1	yes	characteristic
199	C₂²×Q₈	—	2	1	no	—
200	C₂²×D₄	—	2	1	no	—
201	C₂²×D₄	—	2	1	no	—
202	C₂²×D₄	—	2	1	no	—
203	C₂²×C₈	—	19	2	no	—
204	C₂²×C₈	—	19	2	no	—
205	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
206	C₄⋊Q₈	—	2	1	no	—
207	C₄⋊Q₈	—	2	1	no	—
208	C₄⋊Q₈	—	2	1	no	—
209	C₄⋊Q₈	C₂×Dic₁₉	1	2	yes	—
210	#127	—	2	1	no	—
211	#127	—	2	1	no	—
212	#127	—	2	1	no	—
213	#127	C₂×Dic₁₉	1	2	yes	—
214	C₂×C₄⋊C₄	—	2	1	no	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄⋊C₄	—	38	1	no	—
217	C₂×C₄⋊C₄	—	38	1	no	—
218	C₂×C₄⋊C₄	—	38	1	no	—
219	C₂×C₄⋊C₄	—	38	1	no	—
220	C₂×C₄⋊C₄	—	38	1	no	—
221	C₂×C₄⋊C₄	—	38	1	no	—
222	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
223	#103	—	76	4	no	—
224	#102	—	2	8	no	—
225	#102	—	76	4	no	—
226	#102	—	76	4	no	—
227	C₄×C₈	—	38	1	no	—
228	C₄×C₈	—	38	1	no	—
229	C₄×C₈	—	38	1	no	—
230	C₄×C₈	—	38	1	no	—
231	C₄×C₈	C₂×Dic₁₉	1	2	yes	—
232	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
233	C₁₉	#9838	1	1	yes	characteristic, complemented, Sylow
234	C₂⁴	—	4	1	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	4	1	no	—
237	C₂×Q₈	—	4	2	no	—
238	C₂×Q₈	—	4	2	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂×D₄	—	4	1	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂×D₄	—	4	2	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	4	2	no	—
249	C₂×D₄	—	4	2	no	—
250	C₂×D₄	—	4	2	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	—	76	1	no	—
253	C₂²×C₄	—	76	1	no	—
254	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
255	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	38	2	no	—
260	C₂×C₈	—	38	2	no	—
261	C₂×C₈	—	38	2	no	—
262	C₂×C₈	—	38	2	no	—
263	C₂×C₈	—	38	2	no	—
264	C₂×C₈	—	38	2	no	—
265	C₂×C₈	C₂²⋊Dic₁₉	1	4	yes	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	76	2	no	—
271	C₄⋊C₄	—	76	2	no	—
272	C₄⋊C₄	—	76	2	no	—
273	C₄⋊C₄	—	76	2	no	—
274	C₄⋊C₄	—	76	2	no	—
275	C₄⋊C₄	—	76	2	no	—
276	C₄²	#64543	1	1	yes	characteristic
277	C₄²	#64543	1	1	yes	characteristic
278	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
279	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
280	C₂³	—	4	2	no	—
281	C₂³	—	8	1	no	—
282	C₂³	—	8	1	no	—
283	C₂³	—	8	1	no	—
284	C₂³	—	8	1	no	—
285	C₂³	—	8	1	no	—
286	C₂³	—	8	1	no	—
287	C₂³	#10597386	1	1	yes	characteristic
288	Q₈	—	4	4	no	—
289	Q₈	—	8	2	no	—
290	D₄	—	4	4	no	—
291	D₄	—	4	4	no	—
292	D₄	—	4	4	no	—
293	D₄	—	8	2	no	—
294	D₄	—	8	2	no	—
295	D₄	—	8	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	1	no	—
298	C₂×C₄	—	8	1	no	—
299	C₂×C₄	—	152	1	no	—
300	C₂×C₄	—	152	1	no	—
301	C₂×C₄	—	152	1	no	—
302	C₂×C₄	—	152	1	no	—
303	C₂×C₄	—	152	1	no	—
304	C₂×C₄	—	152	1	no	—
305	C₂×C₄	#10597357	1	2	yes	—
306	C₂×C₄	#10597357	1	2	yes	—
307	C₂×C₄	#10597360	1	2	yes	—
308	C₂×C₄	#10597361	1	2	yes	—
309	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
310	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
311	C₂×C₄	#10597382	1	1	yes	characteristic
312	C₂×C₄	#10597384	1	1	yes	characteristic
313	C₂×C₄	#10597384	1	1	yes	characteristic
314	C₂×C₄	#10597396	1	2	yes	—
315	C₂×C₄	#10597396	1	2	yes	—
316	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
317	C₈	—	2	4	no	—
318	C₈	—	76	2	no	—
319	C₈	—	76	2	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	16	1	no	—
324	C₂²	—	16	1	no	—
325	C₂²	—	16	1	no	—
326	C₂²	—	16	1	no	—
327	C₂²	#11926447	1	1	yes	characteristic
328	C₂²	#11926926	1	1	yes	characteristic
329	C₂²	#11926926	1	1	yes	characteristic
330	C₂²	#11926930	1	1	yes	characteristic
331	C₂²	#11926932	1	1	yes	characteristic
332	C₂²	#11926932	1	1	yes	characteristic
333	C₂²	#11926935	1	1	yes	characteristic
334	C₄	—	8	2	no	—
335	C₄	—	304	1	no	—
336	C₄	—	304	1	no	—
337	C₄	#11926540	1	2	yes	—
338	C₄	#11926645	1	2	yes	—
339	C₄	#11926656	1	2	yes	—
340	C₄	#11926787	1	2	yes	—
341	C₄	#11926794	1	2	yes	—
342	C₄	#11926898	1	2	yes	—
343	C₂	—	16	2	no	—
344	C₂	#14563271	1	1	yes	characteristic
345	C₂	#14563275	1	1	yes	characteristic
346	C₂	#14563285	1	1	yes	characteristic
347	C₂	#14563296	1	1	yes	characteristic
348	C₂	#14564447	1	1	yes	characteristic
349	C₂	#14564512	1	1	yes	characteristic
350	C₂	#14564518	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597386
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#486	C₂
socle	C₂³×C₁₉	#127
prime radical	C₂×C₁₉×#127	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#486	D₁₉
hypocenter	C₁₉	#9838
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#486

Complements

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

N	H ≅ G/N	type
1. #14564597	C₂	semidirect
4. #14564438	C₂	semidirect
5. #14563763	C₂	semidirect
6. C₂×#11926265	C₂	semidirect
21. C₁₉×#486	C₄	semidirect
178. #486	Dic₁₉	semidirect
233. C₁₉	#9838	semidirect

Maximal subgroups

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

subgroup	index
#14564597	2
#14564442	2
#14564438	2
#14563763	2
C₂×#11926265	2
C₂×C₁₉×#486	2
#9838	19