Subgroups — #18075526

Order 4864.

456 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	#14563673	C₂	1	1	yes	maximal, characteristic, complemented
2	#14563660	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×#10597274	—	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	#10597274	—	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	#8629	—	19	1	no	maximal, Sylow
115	C₂³×C₁₉	—	2	2	no	—
116	C₂³×C₁₉	—	2	1	no	—
117	C₂³×C₁₉	—	4	1	no	—
118	C₂³×C₁₉	—	4	1	no	—
119	C₂³×C₁₉	#117	1	1	yes	characteristic
120	C₂³×C₁₉	#121	1	1	yes	characteristic
121	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
122	Q₈×C₁₉	—	4	1	no	—
123	D₄×C₁₉	—	4	1	no	—
124	D₄×C₁₉	—	4	2	no	—
125	D₄×C₁₉	—	4	2	no	—
126	D₄×C₁₉	—	4	2	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	—	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	—	4	1	no	—
137	C₂×C₄×C₁₉	—	4	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	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
144	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
145	C₂×Dic₁₉	—	8	2	no	—
146	C₂×Dic₁₉	—	8	2	no	—
147	C₂×Dic₁₉	—	8	2	no	—
148	C₈×C₁₉	—	2	1	no	—
149	C₈×C₁₉	—	2	1	no	—
150	C₈×C₁₉	—	2	1	no	—
151	C₈×C₁₉	—	4	1	no	—
152	C₈×C₁₉	—	4	1	no	—
153	C₈×C₁₉	#99	1	2	yes	—
154	C₁₉⋊C₈	—	8	2	no	—
155	#2929	D₁₉	1	1	yes	characteristic
156	#1810	—	19	1	no	—
157	#1807	—	19	1	no	—
158	#1513	—	19	2	no	—
159	#1305	—	19	2	no	—
160	C₂²×C₁₉	—	2	1	no	—
161	C₂²×C₁₉	—	2	1	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	—	4	4	no	—
169	C₂²×C₁₉	—	4	1	no	—
170	C₂²×C₁₉	—	4	1	no	—
171	C₂²×C₁₉	—	4	1	no	—
172	C₂²×C₁₉	#380	1	1	yes	characteristic
173	C₂²×C₁₉	#471	1	2	yes	—
174	C₄×C₁₉	—	2	1	no	—
175	C₄×C₁₉	—	2	1	no	—
176	C₄×C₁₉	—	2	1	no	—
177	C₄×C₁₉	—	4	1	no	—
178	C₄×C₁₉	—	4	1	no	—
179	C₄×C₁₉	—	4	1	no	—
180	C₄×C₁₉	C₂×#99	1	1	yes	characteristic
181	C₄×C₁₉	#410	1	1	yes	characteristic
182	Dic₁₉	—	16	2	no	—
183	C₂×#131	Dic₁₉	1	1	yes	characteristic, complemented
184	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
185	#525	C₂×D₁₉	1	1	yes	characteristic
186	#435	—	2	1	no	—
187	#435	—	2	1	no	—
188	C₈×D₄	—	2	1	no	—
189	C₈×D₄	—	2	1	no	—
190	C₂×#108	—	38	1	no	—
191	C₂×#106	—	38	1	no	—
192	#424	Dic₁₉	1	1	yes	characteristic, complemented
193	#408	Dic₁₉	1	1	yes	characteristic, complemented
194	#407	—	19	2	no	—
195	#407	C₂×D₁₉	1	1	yes	characteristic
196	#405	Dic₁₉	1	1	yes	characteristic, complemented
197	#380	—	19	2	no	—
198	#341	—	38	1	no	—
199	#340	—	38	1	no	—
200	#328	—	38	2	no	—
201	#327	—	38	2	no	—
202	C₂×C₁₉	—	2	2	no	—
203	C₂×C₁₉	—	4	1	no	—
204	C₂×C₁₉	—	4	1	no	—
205	C₂×C₁₉	—	8	1	no	—
206	C₂×C₁₉	#1799	1	1	yes	characteristic
207	C₂×C₁₉	#1814	1	1	yes	characteristic
208	C₂×C₁₉	#1815	1	1	yes	characteristic
209	C₂×#41	C₂×Dic₁₉	1	1	yes	characteristic
210	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
211	#131	—	4	2	no	—
212	C₂×M₁₆	—	2	1	no	—
213	C₂×M₁₆	—	38	2	no	—
214	C₂×M₁₆	C₂×Dic₁₉	1	1	yes	characteristic
215	C₂×M₁₆	C₂×Dic₁₉	1	1	yes	characteristic
216	C₂²×C₈	—	2	1	no	—
217	C₂²×C₈	—	4	2	no	—
218	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
219	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
220	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
221	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
222	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
223	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
224	#123	—	2	1	no	—
225	C₂²⋊Q₈	Dic₃₈	1	1	yes	characteristic
226	C₂²⋊Q₈	D₇₆	1	1	yes	characteristic
227	#121	Dic₃₈	1	1	yes	characteristic
228	#121	D₇₆	1	1	yes	characteristic
229	#120	—	2	1	no	—
230	C₄×D₄	—	2	1	no	—
231	C₄×D₄	—	2	1	no	—
232	#117	C₂×Dic₁₉	1	1	yes	characteristic
233	C₂×C₄⋊C₄	—	38	2	no	—
234	C₂×C₂²⋊C₄	—	38	2	no	—
235	#108	—	76	2	no	—
236	#106	—	76	2	no	—
237	C₄⋊C₈	—	2	1	no	—
238	C₄⋊C₈	—	2	1	no	—
239	C₂²⋊C₈	—	2	1	no	—
240	C₂²⋊C₈	—	2	1	no	—
241	C₂²⋊C₈	—	2	1	no	—
242	C₂²⋊C₈	—	2	1	no	—
243	C₂²⋊C₈	—	38	2	no	—
244	C₂²⋊C₈	—	38	2	no	—
245	#97	—	2	1	no	—
246	C₄×C₈	—	2	1	no	—
247	#95	—	38	2	no	—
248	C₁₉	#8629	1	1	yes	characteristic, complemented, Sylow
249	C₂⁴	#64541	1	1	yes	characteristic
250	#41	—	4	1	no	—
251	#41	—	4	1	no	—
252	C₂×Q₈	C₄⋊Dic₁₉	1	1	yes	characteristic
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	—	2	1	no	—
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	C₄⋊Dic₁₉	1	1	yes	characteristic
257	C₂²×C₄	—	2	1	no	—
258	C₂²×C₄	—	4	1	no	—
259	C₂²×C₄	—	4	1	no	—
260	C₂²×C₄	—	76	2	no	—
261	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
263	C₂²×C₄	#64541	1	1	yes	characteristic
264	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
266	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
267	M₁₆	—	4	2	no	—
268	M₁₆	—	4	1	no	—
269	M₁₆	—	4	1	no	—
270	M₁₆	—	76	2	no	—
271	M₁₆	—	76	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	1	no	—
276	C₂×C₈	—	2	1	no	—
277	C₂×C₈	—	4	1	no	—
278	C₂×C₈	—	4	1	no	—
279	C₂×C₈	—	4	1	no	—
280	C₂×C₈	—	4	2	no	—
281	C₂×C₈	—	76	2	no	—
282	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
283	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
284	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
285	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
286	C₄⋊C₄	—	2	1	no	—
287	C₄⋊C₄	—	2	1	no	—
288	C₄⋊C₄	—	76	2	no	—
289	C₄⋊C₄	—	76	2	no	—
290	C₄⋊C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
291	C₄⋊C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
292	C₂²⋊C₄	—	2	1	no	—
293	C₂²⋊C₄	—	2	1	no	—
294	C₂²⋊C₄	—	2	1	no	—
295	C₂²⋊C₄	—	2	1	no	—
296	C₂²⋊C₄	—	2	1	no	—
297	C₂²⋊C₄	—	152	2	no	—
298	C₄²	—	2	1	no	—
299	C₂³	—	2	2	no	—
300	C₂³	—	2	1	no	—
301	C₂³	—	4	1	no	—
302	C₂³	—	4	1	no	—
303	C₂³	#10597370	1	1	yes	characteristic
304	C₂³	#10597372	1	1	yes	characteristic
305	C₂³	#10597376	1	1	yes	characteristic
306	Q₈	—	4	1	no	—
307	D₄	—	4	1	no	—
308	D₄	—	4	2	no	—
309	D₄	—	4	2	no	—
310	D₄	—	4	2	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	152	2	no	—
326	C₂×C₄	—	152	2	no	—
327	C₂×C₄	—	152	2	no	—
328	C₂×C₄	#10597370	1	1	yes	characteristic
329	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
330	C₂×C₄	#10597376	1	1	yes	characteristic
331	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	4	1	no	—
336	C₈	—	4	1	no	—
337	C₈	—	152	2	no	—
338	C₈	#10597291	1	2	yes	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	4	4	no	—
348	C₂²	—	4	1	no	—
349	C₂²	—	4	1	no	—
350	C₂²	—	4	1	no	—
351	C₂²	#11926404	1	1	yes	characteristic
352	C₂²	#11926807	1	2	yes	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	4	1	no	—
358	C₄	—	4	1	no	—
359	C₄	—	304	2	no	—
360	C₄	C₂×#10597291	1	1	yes	characteristic
361	C₄	#11926518	1	1	yes	characteristic
362	C₂	—	2	2	no	—
363	C₂	—	4	1	no	—
364	C₂	—	4	1	no	—
365	C₂	—	8	1	no	—
366	C₂	#14563623	1	1	yes	characteristic
367	C₂	#14563689	1	1	yes	characteristic
368	C₂	#14563694	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	#8629
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. #14563660	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
183. C₂×#131	Dic₁₉	semidirect
192. #424	Dic₁₉	semidirect
193. #408	Dic₁₉	semidirect
196. #405	Dic₁₉	semidirect
248. C₁₉	#8629	semidirect

Maximal subgroups

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

subgroup	index
#14563673	2
#14563660	2
#14562178	2
#14561540	2
C₁₉×#2929	2
#8629	19