Subgroups — #18075521

Order 4864.

456 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563673	C₂	1	1	yes	maximal, characteristic, complemented
2	#14563664	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562178	C₂	1	2	yes	maximal
4	#14561540	C₂	1	2	yes	maximal, complemented
5	C₁₉×#2929	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597276	—	2	1	no	—
7	C₂×#10597275	—	2	1	no	—
8	#11926508	C₂²	1	2	yes	—
9	#11926404	C₂²	1	2	yes	—
10	#11926309	—	2	1	no	—
11	#11926307	—	2	1	no	—
12	#11926281	—	2	2	no	—
13	#11926278	—	2	2	no	—
14	C₂×C₁₉×#131	C₄	1	1	yes	characteristic, complemented
15	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₁₉×#525	C₂²	1	1	yes	characteristic
17	C₁₉×#435	—	2	1	no	—
18	C₁₉×#435	—	2	1	no	—
19	C₈×D₄×C₁₉	—	2	1	no	—
20	C₈×D₄×C₁₉	—	2	1	no	—
21	C₁₉×#424	C₄	1	1	yes	characteristic, complemented
22	C₁₉×#408	C₄	1	1	yes	characteristic, complemented
23	C₁₉×#407	C₂²	1	1	yes	characteristic
24	C₁₉×#405	C₄	1	1	yes	characteristic, complemented
25	C₂×#41×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
27	C₁₉×#131	—	4	2	no	—
28	C₂×M₁₆×C₁₉	—	2	1	no	—
29	C₂×M₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₂×M₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₉	—	2	1	no	—
32	C₂²×C₈×C₁₉	—	4	2	no	—
33	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
39	C₁₉×#123	—	2	1	no	—
40	C₁₉×C₂²⋊Q₈	D₄	1	1	yes	characteristic
41	C₁₉×C₂²⋊Q₈	Q₈	1	1	yes	characteristic
42	C₁₉×#121	D₄	1	1	yes	characteristic
43	C₁₉×#121	Q₈	1	1	yes	characteristic
44	C₁₉×#120	—	2	1	no	—
45	C₄×D₄×C₁₉	—	2	1	no	—
46	C₄×D₄×C₁₉	—	2	1	no	—
47	C₁₉×#117	C₂×C₄	1	1	yes	characteristic
48	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
49	C₂×C₄⋊Dic₁₉	—	2	2	no	—
50	C₂×#64520	—	2	2	no	—
51	C₁₉×C₄⋊C₈	—	2	1	no	—
52	C₁₉×C₄⋊C₈	—	2	1	no	—
53	C₁₉×C₂²⋊C₈	—	2	1	no	—
54	C₁₉×C₂²⋊C₈	—	2	1	no	—
55	C₁₉×C₂²⋊C₈	—	2	1	no	—
56	C₁₉×C₂²⋊C₈	—	2	1	no	—
57	C₁₉×#97	—	2	1	no	—
58	C₄×C₈×C₁₉	—	2	1	no	—
59	C₁₉⋊#95	—	2	2	no	—
60	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
61	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
62	#10597276	—	4	2	no	—
63	#10597275	—	4	2	no	—
64	C₂⁴×C₁₉	#41	1	1	yes	characteristic
65	#41×C₁₉	—	4	1	no	—
66	#41×C₁₉	—	4	1	no	—
67	C₂×Q₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
68	C₂×D₄×C₁₉	—	2	1	no	—
69	C₂×D₄×C₁₉	—	2	1	no	—
70	C₂×D₄×C₁₉	—	2	1	no	—
71	C₂×D₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	—	2	1	no	—
73	C₂²×C₄×C₁₉	—	4	1	no	—
74	C₂²×C₄×C₁₉	—	4	1	no	—
75	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
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×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
80	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
81	C₂²×Dic₁₉	—	4	2	no	—
82	M₁₆×C₁₉	—	4	2	no	—
83	M₁₆×C₁₉	—	4	1	no	—
84	M₁₆×C₁₉	—	4	1	no	—
85	C₂×C₈×C₁₉	—	2	2	no	—
86	C₂×C₈×C₁₉	—	2	2	no	—
87	C₂×C₈×C₁₉	—	2	2	no	—
88	C₂×C₈×C₁₉	—	2	1	no	—
89	C₂×C₈×C₁₉	—	2	1	no	—
90	C₂×C₈×C₁₉	—	4	1	no	—
91	C₂×C₈×C₁₉	—	4	1	no	—
92	C₂×C₈×C₁₉	—	4	1	no	—
93	C₂×C₈×C₁₉	—	4	2	no	—
94	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
98	C₄⋊C₄×C₁₉	—	2	1	no	—
99	C₄⋊C₄×C₁₉	—	2	1	no	—
100	C₄⋊C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
101	C₄⋊C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
102	C₂²⋊C₄×C₁₉	—	2	1	no	—
103	C₂²⋊C₄×C₁₉	—	2	1	no	—
104	C₂²⋊C₄×C₁₉	—	2	1	no	—
105	C₂²⋊C₄×C₁₉	—	2	1	no	—
106	C₂²⋊C₄×C₁₉	—	2	1	no	—
107	C₄²×C₁₉	—	2	1	no	—
108	C₂²⋊Dic₁₉	—	8	2	no	—
109	C₄⋊Dic₁₉	—	4	2	no	—
110	C₄⋊Dic₁₉	—	4	2	no	—
111	#64520	—	4	2	no	—
112	#64520	—	4	2	no	—
113	C₂×C₁₉⋊C₈	—	4	2	no	—
114	#8628	—	19	1	no	maximal, Sylow
115	C₂³×C₁₉	—	2	1	no	—
116	C₂³×C₁₉	—	2	1	no	—
117	C₂³×C₁₉	—	2	1	no	—
118	C₂³×C₁₉	—	4	1	no	—
119	C₂³×C₁₉	—	4	1	no	—
120	C₂³×C₁₉	#117	1	1	yes	characteristic
121	C₂³×C₁₉	#121	1	1	yes	characteristic
122	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
123	Q₈×C₁₉	—	4	1	no	—
124	D₄×C₁₉	—	4	1	no	—
125	D₄×C₁₉	—	4	2	no	—
126	D₄×C₁₉	—	4	2	no	—
127	D₄×C₁₉	—	4	1	no	—
128	D₄×C₁₉	—	4	1	no	—
129	C₂×C₄×C₁₉	—	2	1	no	—
130	C₂×C₄×C₁₉	—	2	1	no	—
131	C₂×C₄×C₁₉	—	2	1	no	—
132	C₂×C₄×C₁₉	—	2	1	no	—
133	C₂×C₄×C₁₉	—	2	1	no	—
134	C₂×C₄×C₁₉	—	2	1	no	—
135	C₂×C₄×C₁₉	—	2	1	no	—
136	C₂×C₄×C₁₉	—	2	1	no	—
137	C₂×C₄×C₁₉	—	2	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	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
146	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
147	C₂×Dic₁₉	—	8	2	no	—
148	C₂×Dic₁₉	—	8	2	no	—
149	C₂×Dic₁₉	—	8	2	no	—
150	C₈×C₁₉	—	2	1	no	—
151	C₈×C₁₉	—	2	1	no	—
152	C₈×C₁₉	—	2	1	no	—
153	C₈×C₁₉	—	4	1	no	—
154	C₈×C₁₉	—	4	1	no	—
155	C₈×C₁₉	#99	1	2	yes	—
156	C₁₉⋊C₈	—	8	2	no	—
157	#2929	D₁₉	1	1	yes	characteristic
158	#1810	—	19	1	no	—
159	#1808	—	19	1	no	—
160	#1513	—	19	2	no	—
161	#1305	—	19	2	no	—
162	C₂²×C₁₉	—	2	1	no	—
163	C₂²×C₁₉	—	2	1	no	—
164	C₂²×C₁₉	—	2	1	no	—
165	C₂²×C₁₉	—	2	1	no	—
166	C₂²×C₁₉	—	2	1	no	—
167	C₂²×C₁₉	—	2	1	no	—
168	C₂²×C₁₉	—	2	1	no	—
169	C₂²×C₁₉	—	2	1	no	—
170	C₂²×C₁₉	—	4	2	no	—
171	C₂²×C₁₉	—	4	2	no	—
172	C₂²×C₁₉	—	4	1	no	—
173	C₂²×C₁₉	—	4	1	no	—
174	C₂²×C₁₉	—	4	1	no	—
175	C₂²×C₁₉	#380	1	1	yes	characteristic
176	C₂²×C₁₉	#472	1	1	yes	characteristic
177	C₂²×C₁₉	#473	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	1	no	—
182	C₄×C₁₉	—	4	1	no	—
183	C₄×C₁₉	—	4	1	no	—
184	C₄×C₁₉	C₂×#99	1	1	yes	characteristic
185	C₄×C₁₉	#410	1	1	yes	characteristic
186	Dic₁₉	—	16	2	no	—
187	C₂×#131	Dic₁₉	1	1	yes	characteristic, complemented
188	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
189	#525	C₂×D₁₉	1	1	yes	characteristic
190	#435	—	2	1	no	—
191	#435	—	2	1	no	—
192	C₈×D₄	—	2	1	no	—
193	C₈×D₄	—	2	1	no	—
194	C₂×#108	—	38	1	no	—
195	C₂×#107	—	38	1	no	—
196	#424	Dic₁₉	1	1	yes	characteristic, complemented
197	#408	Dic₁₉	1	1	yes	characteristic, complemented
198	#407	—	19	2	no	—
199	#407	C₂×D₁₉	1	1	yes	characteristic
200	#405	Dic₁₉	1	1	yes	characteristic, complemented
201	#380	—	19	2	no	—
202	#341	—	38	1	no	—
203	#340	—	38	1	no	—
204	#328	—	38	2	no	—
205	#327	—	38	2	no	—
206	C₂×C₁₉	—	2	1	no	—
207	C₂×C₁₉	—	2	1	no	—
208	C₂×C₁₉	—	4	1	no	—
209	C₂×C₁₉	—	4	1	no	—
210	C₂×C₁₉	—	8	1	no	—
211	C₂×C₁₉	#1799	1	1	yes	characteristic
212	C₂×C₁₉	#1814	1	1	yes	characteristic
213	C₂×C₁₉	#1816	1	1	yes	characteristic
214	C₂×#41	C₂×Dic₁₉	1	1	yes	characteristic
215	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
216	#131	—	4	2	no	—
217	C₂×M₁₆	—	2	1	no	—
218	C₂×M₁₆	—	38	2	no	—
219	C₂×M₁₆	C₂×Dic₁₉	1	1	yes	characteristic
220	C₂×M₁₆	C₂×Dic₁₉	1	1	yes	characteristic
221	C₂²×C₈	—	2	1	no	—
222	C₂²×C₈	—	4	2	no	—
223	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
224	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
225	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
226	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
227	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
228	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
229	#123	—	2	1	no	—
230	C₂²⋊Q₈	Dic₃₈	1	1	yes	characteristic
231	C₂²⋊Q₈	D₇₆	1	1	yes	characteristic
232	#121	Dic₃₈	1	1	yes	characteristic
233	#121	D₇₆	1	1	yes	characteristic
234	#120	—	2	1	no	—
235	C₄×D₄	—	2	1	no	—
236	C₄×D₄	—	2	1	no	—
237	#117	C₂×Dic₁₉	1	1	yes	characteristic
238	C₂×C₄⋊C₄	—	38	2	no	—
239	C₂×C₂²⋊C₄	—	38	2	no	—
240	#108	—	76	2	no	—
241	#107	—	76	2	no	—
242	C₄⋊C₈	—	2	1	no	—
243	C₄⋊C₈	—	2	1	no	—
244	C₂²⋊C₈	—	2	1	no	—
245	C₂²⋊C₈	—	2	1	no	—
246	C₂²⋊C₈	—	2	1	no	—
247	C₂²⋊C₈	—	2	1	no	—
248	C₂²⋊C₈	—	38	2	no	—
249	C₂²⋊C₈	—	38	2	no	—
250	#97	—	2	1	no	—
251	C₄×C₈	—	2	1	no	—
252	#95	—	38	2	no	—
253	C₁₉	#8628	1	1	yes	characteristic, complemented, Sylow
254	C₂⁴	#64541	1	1	yes	characteristic
255	#41	—	4	1	no	—
256	#41	—	4	1	no	—
257	C₂×Q₈	C₄⋊Dic₁₉	1	1	yes	characteristic
258	C₂×D₄	—	2	1	no	—
259	C₂×D₄	—	2	1	no	—
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	C₄⋊Dic₁₉	1	1	yes	characteristic
262	C₂²×C₄	—	2	1	no	—
263	C₂²×C₄	—	4	1	no	—
264	C₂²×C₄	—	4	1	no	—
265	C₂²×C₄	—	76	2	no	—
266	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
268	C₂²×C₄	#64541	1	1	yes	characteristic
269	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
270	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
271	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
272	M₁₆	—	4	2	no	—
273	M₁₆	—	4	1	no	—
274	M₁₆	—	4	1	no	—
275	M₁₆	—	76	2	no	—
276	M₁₆	—	76	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	1	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	4	1	no	—
283	C₂×C₈	—	4	1	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	4	2	no	—
286	C₂×C₈	—	76	2	no	—
287	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
288	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
289	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
290	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	2	1	no	—
293	C₄⋊C₄	—	76	2	no	—
294	C₄⋊C₄	—	76	2	no	—
295	C₄⋊C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
296	C₄⋊C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
297	C₂²⋊C₄	—	2	1	no	—
298	C₂²⋊C₄	—	2	1	no	—
299	C₂²⋊C₄	—	2	1	no	—
300	C₂²⋊C₄	—	2	1	no	—
301	C₂²⋊C₄	—	2	1	no	—
302	C₂²⋊C₄	—	152	2	no	—
303	C₄²	—	2	1	no	—
304	C₂³	—	2	1	no	—
305	C₂³	—	2	1	no	—
306	C₂³	—	2	1	no	—
307	C₂³	—	4	1	no	—
308	C₂³	—	4	1	no	—
309	C₂³	#10597370	1	1	yes	characteristic
310	C₂³	#10597372	1	1	yes	characteristic
311	C₂³	#10597376	1	1	yes	characteristic
312	Q₈	—	4	1	no	—
313	D₄	—	4	1	no	—
314	D₄	—	4	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	1	no	—
317	D₄	—	4	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	152	2	no	—
333	C₂×C₄	—	152	2	no	—
334	C₂×C₄	—	152	2	no	—
335	C₂×C₄	#10597370	1	1	yes	characteristic
336	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
337	C₂×C₄	#10597376	1	1	yes	characteristic
338	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
339	C₈	—	2	1	no	—
340	C₈	—	2	1	no	—
341	C₈	—	2	1	no	—
342	C₈	—	4	1	no	—
343	C₈	—	4	1	no	—
344	C₈	—	152	2	no	—
345	C₈	#10597291	1	2	yes	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	4	2	no	—
355	C₂²	—	4	2	no	—
356	C₂²	—	4	1	no	—
357	C₂²	—	4	1	no	—
358	C₂²	—	4	1	no	—
359	C₂²	#11926404	1	1	yes	characteristic
360	C₂²	#11926814	1	1	yes	characteristic
361	C₂²	#11926819	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	4	1	no	—
366	C₄	—	4	1	no	—
367	C₄	—	4	1	no	—
368	C₄	—	304	2	no	—
369	C₄	C₂×#10597291	1	1	yes	characteristic
370	C₄	#11926518	1	1	yes	characteristic
371	C₂	—	2	1	no	—
372	C₂	—	2	1	no	—
373	C₂	—	4	1	no	—
374	C₂	—	4	1	no	—
375	C₂	—	8	1	no	—
376	C₂	#14563623	1	1	yes	characteristic
377	C₂	#14563689	1	1	yes	characteristic
378	C₂	#14563699	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926404
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₁₉×#2929	C₂
socle	C₂²×C₁₉	#380
prime radical	C₁₉×#525	C₂²
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#2929	D₁₉
hypocenter	C₁₉	#8628
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929

Complements

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

N	H ≅ G/N	type
1. #14563673	C₂	semidirect
2. #14563664	C₂	semidirect
4. #14561540	C₂	semidirect
14. C₂×C₁₉×#131	C₄	semidirect
21. C₁₉×#424	C₄	semidirect
22. C₁₉×#408	C₄	semidirect
24. C₁₉×#405	C₄	semidirect
187. C₂×#131	Dic₁₉	semidirect
196. #424	Dic₁₉	semidirect
197. #408	Dic₁₉	semidirect
200. #405	Dic₁₉	semidirect
253. C₁₉	#8628	semidirect

Maximal subgroups

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

subgroup	index
#14563673	2
#14563664	2
#14562178	2
#14561540	2
C₁₉×#2929	2
#8628	19