Subgroups — #18082061

Order 4864.

606 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14564472	C₂	1	2	yes	maximal, complemented
2	#14563664	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×C₂²⋊Dic₁₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#14562203	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₉×#466	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597275	—	2	1	no	—
7	C₂×#10597291	—	2	4	no	—
8	C₂×C₂²⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂×C₈×Dic₁₉	—	2	1	no	—
10	#11926404	C₂²	1	2	yes	—
11	C₄×C₂²⋊Dic₁₉	C₂²	1	1	yes	characteristic
12	#11926307	—	2	1	no	—
13	#11926298	—	2	1	no	—
14	#11926278	—	2	4	no	—
15	C₂²×D₈×C₁₉	C₄	1	1	yes	characteristic, complemented
16	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×C₁₉×#121	C₂²	1	2	yes	—
18	C₁₉×#466	—	2	1	no	—
19	C₁₉×#466	—	2	1	no	—
20	C₁₉×#466	—	2	1	no	—
21	C₁₉×#466	—	2	1	no	—
22	C₂×C₁₉×#107	C₄	1	1	yes	characteristic, complemented
23	C₂×C₁₉×#102	C₄	1	2	yes	complemented
24	C₂²×D₄×C₁₉	—	2	2	no	—
25	C₂²×D₄×C₁₉	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
27	C₂×D₈×C₁₉	—	2	1	no	—
28	C₂×D₈×C₁₉	—	2	1	no	—
29	C₂×D₈×C₁₉	—	4	2	no	—
30	C₂²×C₈×C₁₉	—	4	2	no	—
31	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
37	C₁₉×#121	—	2	2	no	—
38	C₁₉×#121	—	2	2	no	—
39	C₁₉×#121	—	2	2	no	—
40	C₁₉×#121	—	2	2	no	—
41	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
42	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
43	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
44	C₂×C₂²⋊Dic₁₉	D₄	1	2	yes	—
45	C₂×C₄⋊Dic₁₉	—	2	2	no	—
46	C₂×C₄×Dic₁₉	—	2	1	no	—
47	C₂²×C₁₉⋊C₈	—	2	1	no	—
48	C₁₉×#107	—	2	1	no	—
49	C₁₉×#107	—	2	1	no	—
50	C₁₉×#102	—	2	2	no	—
51	C₁₉×#102	—	2	2	no	—
52	#10597291	—	4	8	no	—
53	C₁₉⋊#95	—	2	1	no	—
54	C₁₉⋊#95	—	2	2	no	—
55	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
56	C₂²⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
57	#10597275	—	4	2	no	—
58	C₈×Dic₁₉	—	4	2	no	—
59	C₂⁴×C₁₉	—	2	2	no	—
60	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
61	C₂×D₄×C₁₉	—	2	2	no	—
62	C₂×D₄×C₁₉	—	2	2	no	—
63	C₂×D₄×C₁₉	—	4	2	no	—
64	C₂×D₄×C₁₉	—	4	2	no	—
65	C₂×D₄×C₁₉	—	4	2	no	—
66	C₂×D₄×C₁₉	—	4	2	no	—
67	C₂×D₄×C₁₉	—	4	4	no	—
68	C₂×D₄×C₁₉	—	4	4	no	—
69	C₂²×C₄×C₁₉	—	2	2	no	—
70	C₂²×C₄×C₁₉	—	4	1	no	—
71	C₂²×C₄×C₁₉	—	4	1	no	—
72	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
77	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
78	C₂²×Dic₁₉	—	2	2	no	—
79	C₂²×Dic₁₉	—	4	2	no	—
80	D₈×C₁₉	—	4	2	no	—
81	D₈×C₁₉	—	4	2	no	—
82	C₂×C₈×C₁₉	—	2	2	no	—
83	C₂×C₈×C₁₉	—	2	2	no	—
84	C₂×C₈×C₁₉	—	2	2	no	—
85	C₂×C₈×C₁₉	—	4	1	no	—
86	C₂×C₈×C₁₉	—	4	1	no	—
87	C₂×C₈×C₁₉	—	4	1	no	—
88	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
92	C₄⋊C₄×C₁₉	—	2	2	no	—
93	C₄⋊C₄×C₁₉	—	2	2	no	—
94	C₂²⋊C₄×C₁₉	—	4	2	no	—
95	C₂²⋊C₄×C₁₉	—	4	2	no	—
96	C₂²⋊Dic₁₉	—	2	2	no	—
97	C₂²⋊Dic₁₉	—	4	4	no	—
98	C₂²⋊Dic₁₉	D₈	1	4	yes	complemented
99	C₄⋊Dic₁₉	—	4	2	no	—
100	C₄⋊Dic₁₉	—	4	2	no	—
101	C₄×Dic₁₉	—	4	1	no	—
102	C₄×Dic₁₉	—	4	1	no	—
103	C₂×C₁₉⋊C₈	—	2	2	no	—
104	C₂×C₁₉⋊C₈	—	4	1	no	—
105	C₂×C₁₉⋊C₈	—	4	1	no	—
106	#9783	—	19	1	no	maximal, Sylow
107	C₂³×C₁₉	—	4	1	no	—
108	C₂³×C₁₉	—	4	1	no	—
109	C₂³×C₁₉	—	4	2	no	—
110	C₂³×C₁₉	—	4	4	no	—
111	C₂³×C₁₉	—	8	2	no	—
112	C₂³×C₁₉	—	8	2	no	—
113	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
114	C₂³×C₁₉	#121	1	1	yes	characteristic
115	C₂³×C₁₉	#127	1	1	yes	characteristic
116	C₂³×C₁₉	C₂×D₈	1	2	yes	—
117	C₂³×C₁₉	#135	1	2	yes	—
118	D₄×C₁₉	—	4	2	no	—
119	D₄×C₁₉	—	4	2	no	—
120	D₄×C₁₉	—	8	2	no	—
121	D₄×C₁₉	—	8	4	no	—
122	D₄×C₁₉	—	8	2	no	—
123	D₄×C₁₉	—	8	2	no	—
124	C₂×C₄×C₁₉	—	2	1	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	—	2	1	no	—
129	C₂×C₄×C₁₉	—	2	1	no	—
130	C₂×C₄×C₁₉	—	4	1	no	—
131	C₂×C₄×C₁₉	—	4	1	no	—
132	C₂×C₄×C₁₉	—	4	1	no	—
133	C₂×C₄×C₁₉	—	4	2	no	—
134	C₂×C₄×C₁₉	—	4	4	no	—
135	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
138	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
139	C₂×Dic₁₉	—	2	4	no	—
140	C₂×Dic₁₉	—	4	2	no	—
141	C₂×Dic₁₉	—	4	2	no	—
142	C₂×Dic₁₉	—	8	2	no	—
143	C₂×Dic₁₉	—	8	2	no	—
144	C₂×Dic₁₉	—	8	2	no	—
145	C₈×C₁₉	—	2	1	no	—
146	C₈×C₁₉	—	2	1	no	—
147	C₈×C₁₉	—	2	1	no	—
148	C₈×C₁₉	#99	1	2	yes	complemented
149	C₁₉⋊C₈	—	4	2	no	—
150	C₂×#466	D₁₉	1	1	yes	characteristic
151	#1964	—	19	2	no	—
152	#1808	—	19	1	no	—
153	C₈×C₂²⋊C₄	—	19	1	no	—
154	#1519	—	19	2	no	—
155	C₂²×C₁₉	—	2	1	no	—
156	C₂²×C₁₉	—	2	1	no	—
157	C₂²×C₁₉	—	2	1	no	—
158	C₂²×C₁₉	—	2	1	no	—
159	C₂²×C₁₉	—	2	1	no	—
160	C₂²×C₁₉	—	2	1	no	—
161	C₂²×C₁₉	—	4	2	no	—
162	C₂²×C₁₉	—	4	2	no	—
163	C₂²×C₁₉	—	8	2	no	—
164	C₂²×C₁₉	—	8	2	no	—
165	C₂²×C₁₉	—	8	2	no	—
166	C₂²×C₁₉	—	16	4	no	—
167	C₂²×C₁₉	#390	1	1	yes	characteristic
168	C₂²×C₁₉	C₄×D₈	1	2	yes	—
169	C₂²×C₁₉	#466	1	2	yes	—
170	C₂²×C₁₉	#493	1	1	yes	characteristic
171	C₂²×C₁₉	#495	1	1	yes	characteristic
172	C₄×C₁₉	—	2	1	no	—
173	C₄×C₁₉	—	2	1	no	—
174	C₄×C₁₉	—	2	1	no	—
175	C₄×C₁₉	—	8	2	no	—
176	C₄×C₁₉	C₂×#99	1	1	yes	characteristic
177	C₄×C₁₉	#410	1	1	yes	characteristic
178	Dic₁₉	—	4	4	no	—
179	Dic₁₉	—	16	2	no	—
180	C₂²×D₈	Dic₁₉	1	1	yes	characteristic, complemented
181	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
182	C₂×#121	C₂×D₁₉	1	2	yes	—
183	#466	—	2	1	no	—
184	#466	—	2	1	no	—
185	#466	—	2	1	no	—
186	#466	—	2	1	no	—
187	C₂×#107	—	38	1	no	—
188	C₂×#107	Dic₁₉	1	1	yes	characteristic, complemented
189	C₂×#102	Dic₁₉	1	2	yes	complemented
190	C₂×#99	—	38	4	no	—
191	C₂×C₂²⋊C₈	—	19	1	no	—
192	C₂×C₄×C₈	—	38	1	no	—
193	#380	—	19	2	no	—
194	C₄×C₂²⋊C₄	—	19	1	no	—
195	#340	—	38	1	no	—
196	#336	—	38	1	no	—
197	#327	—	38	4	no	—
198	C₂×C₁₉	—	2	2	no	—
199	C₂×C₁₉	—	2	1	no	—
200	C₂×C₁₉	—	2	2	no	—
201	C₂×C₁₉	—	2	1	no	—
202	C₂×C₁₉	—	16	2	no	—
203	C₂×C₁₉	#1964	1	1	yes	characteristic
204	C₂×C₁₉	#1970	1	1	yes	characteristic
205	C₂×C₁₉	#1976	1	1	yes	characteristic
206	C₂²×D₄	—	2	2	no	—
207	C₂²×D₄	C₂×Dic₁₉	1	2	yes	—
208	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
209	C₂×D₈	—	2	1	no	—
210	C₂×D₈	—	2	1	no	—
211	C₂×D₈	—	4	2	no	—
212	C₂²×C₈	—	4	2	no	—
213	C₂²×C₈	—	38	1	no	—
214	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
215	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
216	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
217	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
218	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
219	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
220	#121	—	2	2	no	—
221	#121	—	2	2	no	—
222	#121	—	2	2	no	—
223	#121	—	2	2	no	—
224	C₂×C₄⋊C₄	—	38	2	no	—
225	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
226	C₂×C₂²⋊C₄	—	2	2	no	—
227	C₂×C₂²⋊C₄	—	19	2	no	—
228	C₂×C₂²⋊C₄	—	38	2	no	—
229	C₂×C₄²	—	38	1	no	—
230	#107	—	2	1	no	—
231	#107	—	2	1	no	—
232	#107	—	76	2	no	—
233	#102	—	2	2	no	—
234	#102	—	2	2	no	—
235	#99	—	76	8	no	—
236	C₂²⋊C₈	—	19	2	no	—
237	C₂²⋊C₈	—	38	1	no	—
238	C₄×C₈	—	76	2	no	—
239	#95	—	38	1	no	—
240	#95	—	38	2	no	—
241	C₁₉	#9783	1	1	yes	characteristic, complemented, Sylow
242	C₂⁴	—	2	2	no	—
243	C₂⁴	D₄×D₁₉	1	1	yes	characteristic
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	2	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	4	no	—
251	C₂×D₄	—	4	4	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	—	38	2	no	—
256	C₂²×C₄	—	76	2	no	—
257	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
258	C₂²×C₄	#64543	1	1	yes	characteristic
259	C₂²×C₄	#64543	1	1	yes	characteristic
260	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
262	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
263	D₈	—	4	2	no	—
264	D₈	—	4	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	4	1	no	—
271	C₂×C₈	—	38	2	no	—
272	C₂×C₈	—	76	1	no	—
273	C₂×C₈	—	76	1	no	—
274	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
275	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
276	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
277	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	76	2	no	—
281	C₄⋊C₄	—	76	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₂²⋊C₄	—	19	4	no	—
285	C₂²⋊C₄	—	38	2	no	—
286	C₂²⋊C₄	—	76	4	no	—
287	C₄²	—	76	1	no	—
288	C₄²	—	76	1	no	—
289	C₂³	—	4	1	no	—
290	C₂³	—	4	1	no	—
291	C₂³	—	4	2	no	—
292	C₂³	—	4	4	no	—
293	C₂³	—	8	2	no	—
294	C₂³	—	8	2	no	—
295	C₂³	D₈×D₁₉	1	2	yes	—
296	C₂³	#10597358	1	2	yes	—
297	C₂³	D₄×Dic₁₉	1	1	yes	characteristic
298	C₂³	#10597384	1	1	yes	characteristic
299	C₂³	#10597386	1	1	yes	characteristic
300	D₄	—	4	2	no	—
301	D₄	—	4	2	no	—
302	D₄	—	8	2	no	—
303	D₄	—	8	2	no	—
304	D₄	—	8	2	no	—
305	D₄	—	8	4	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	4	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	38	4	no	—
318	C₂×C₄	—	76	2	no	—
319	C₂×C₄	—	76	2	no	—
320	C₂×C₄	—	152	2	no	—
321	C₂×C₄	—	152	2	no	—
322	C₂×C₄	—	152	2	no	—
323	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
324	C₂×C₄	#10597382	1	1	yes	characteristic
325	C₂×C₄	#10597384	1	1	yes	characteristic
326	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
327	C₈	—	2	1	no	—
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	—	76	2	no	—
331	C₈	#10597291	1	2	yes	complemented
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	4	2	no	—
339	C₂²	—	4	2	no	—
340	C₂²	—	8	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	16	4	no	—
344	C₂²	#11926447	1	1	yes	characteristic
345	C₂²	D₈×Dic₁₉	1	2	yes	—
346	C₂²	#11926777	1	2	yes	—
347	C₂²	#11926919	1	1	yes	characteristic
348	C₂²	#11926927	1	1	yes	characteristic
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	8	2	no	—
353	C₄	—	304	2	no	—
354	C₄	—	76	4	no	—
355	C₄	C₂×#10597291	1	1	yes	characteristic
356	C₄	#11926518	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	2	no	—
359	C₂	—	2	2	no	—
360	C₂	—	2	1	no	—
361	C₂	—	16	2	no	—
362	C₂	#14564472	1	1	yes	characteristic
363	C₂	#14564502	1	1	yes	characteristic
364	C₂	#14564534	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926447
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting, prime radical	C₂×C₁₉×#466	C₂
socle	C₂²×C₁₉	#390
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×#466	D₁₉
hypocenter	C₁₉	#9783
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#466

Complements

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

N	H ≅ G/N	type
1. #14564472	C₂	semidirect
2. #14563664	C₂	semidirect
3. C₈×C₂²⋊Dic₁₉	C₂	semidirect
4. #14562203	C₂	semidirect
15. C₂²×D₈×C₁₉	C₄	semidirect
22. C₂×C₁₉×#107	C₄	semidirect
23. C₂×C₁₉×#102	C₄	semidirect
98. C₂²⋊Dic₁₉	D₈	semidirect
148. C₈×C₁₉	#99	semidirect
180. C₂²×D₈	Dic₁₉	semidirect
188. C₂×#107	Dic₁₉	semidirect
189. C₂×#102	Dic₁₉	semidirect
241. C₁₉	#9783	semidirect
331. C₈	#10597291	semidirect

Maximal subgroups

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

subgroup	index
#14564472	2
#14563664	2
C₈×C₂²⋊Dic₁₉	2
#14562203	2
C₂×C₁₉×#466	2
#9783	19