Subgroups — #18076880

Order 4864.

454 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₁₆×D₇₆	C₂	1	1	yes	maximal, characteristic
2	#14562390	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562059	C₂	1	1	yes	maximal, characteristic
4	C₂×#11926271	C₂	1	1	yes	maximal, characteristic
5	#14561532	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561473	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1950	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₁₉	—	2	1	no	—
9	C₂×C₄×D₇₆	C₂²	1	1	yes	characteristic
10	#11926623	—	2	1	no	—
11	#11926617	—	2	2	no	—
12	C₈×D₇₆	—	2	1	no	—
13	#11926597	C₂²	1	1	yes	characteristic
14	#11926568	C₂²	1	1	yes	characteristic
15	#11926507	—	2	1	no	—
16	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
17	#11926294	—	2	2	no	—
18	#11926271	—	2	1	no	—
19	#11926271	—	2	1	no	—
20	#11926271	C₄	1	1	yes	characteristic
21	#11926271	C₄	1	1	yes	characteristic
22	#11926271	C₄	1	1	yes	characteristic
23	#11926271	C₄	1	1	yes	characteristic
24	C₂×C₁₉×#108	—	2	1	no	—
25	C₁₉×#423	C₂²	1	1	yes	characteristic
26	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
28	C₁₉×#341	—	2	1	no	—
29	C₂²×D₇₆	Q₈	1	1	yes	characteristic
30	C₂²×C₄×D₁₉	—	2	1	no	—
31	C₂×M₁₆×C₁₉	—	2	1	no	—
32	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
33	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₉	—	2	1	no	—
35	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
36	C₂×(C₂×D₁₉)⋊C₄	—	2	1	no	—
37	C₂×C₄⋊Dic₁₉	Q₈	1	1	yes	characteristic
38	C₂×#64520	—	2	1	no	—
39	C₂²×C₁₉⋊C₈	—	2	1	no	—
40	M₁₆×D₁₉	—	4	2	no	—
41	M₁₆×D₁₉	—	4	2	no	—
42	C₂×#64515	—	2	2	no	—
43	C₂×C₈×D₁₉	—	2	2	no	—
44	C₄×D₇₆	—	2	1	no	—
45	C₄×D₇₆	—	2	1	no	—
46	C₄×D₇₆	C₂×C₄	1	1	yes	characteristic
47	C₄×D₇₆	C₂×C₄	1	1	yes	characteristic
48	C₄×D₇₆	D₄	1	1	yes	characteristic
49	C₄×D₇₆	D₄	1	1	yes	characteristic
50	C₁₉×#108	—	4	2	no	—
51	C₁₉×C₄⋊C₈	—	2	1	no	—
52	C₁₉×C₄⋊C₈	—	2	1	no	—
53	C₁₉×C₄⋊C₈	—	2	1	no	—
54	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
55	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
56	C₁₉×#97	—	2	1	no	—
57	C₄×C₈×C₁₉	—	2	1	no	—
58	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
59	(C₂×D₁₉)⋊C₈	—	2	2	no	—
60	(C₂×D₁₉)⋊C₈	—	2	2	no	—
61	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
62	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
63	C₄⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
64	C₄×C₁₉⋊C₈	—	2	1	no	—
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	C₂³×D₁₉	—	2	1	no	—
69	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
72	C₂²×Dic₁₉	—	2	1	no	—
73	C₂×D₇₆	—	2	2	no	—
74	C₂×D₇₆	—	2	1	no	—
75	C₂×D₇₆	—	4	1	no	—
76	C₂×D₇₆	C₄⋊C₄	1	1	yes	characteristic
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×C₄×D₁₉	—	2	1	no	—
82	C₂×C₄×D₁₉	—	4	1	no	—
83	C₂×C₄×D₁₉	—	4	1	no	—
84	C₂×C₄×D₁₉	—	4	1	no	—
85	C₂×C₄×D₁₉	—	4	1	no	—
86	M₁₆×C₁₉	—	2	1	no	—
87	M₁₆×C₁₉	—	2	1	no	—
88	M₁₆×C₁₉	—	2	1	no	—
89	M₁₆×C₁₉	—	4	1	no	—
90	M₁₆×C₁₉	—	4	1	no	—
91	M₁₆×C₁₉	SD₁₆	1	2	yes	—
92	C₂×C₈×C₁₉	—	2	1	no	—
93	C₂×C₈×C₁₉	—	2	1	no	—
94	C₂×C₈×C₁₉	—	2	1	no	—
95	C₂×C₈×C₁₉	—	2	1	no	—
96	C₂×C₈×C₁₉	—	4	1	no	—
97	C₂×C₈×C₁₉	—	4	1	no	—
98	C₂×C₈×C₁₉	—	4	1	no	—
99	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₁₉	#41	1	1	yes	characteristic
103	(C₂×D₁₉)⋊C₄	—	2	1	no	—
104	(C₂×D₁₉)⋊C₄	—	2	1	no	—
105	(C₂×D₁₉)⋊C₄	—	2	1	no	—
106	(C₂×D₁₉)⋊C₄	—	2	1	no	—
107	C₄⋊Dic₁₉	—	2	1	no	—
108	C₄⋊Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
109	C₄⋊Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
110	#64520	—	4	1	no	—
111	#64520	—	4	1	no	—
112	C₂×C₁₉⋊C₈	—	2	1	no	—
113	C₂×C₁₉⋊C₈	—	2	1	no	—
114	C₂×C₁₉⋊C₈	—	2	2	no	—
115	C₂×C₁₉⋊C₈	—	4	2	no	—
116	C₂×C₁₉⋊C₈	—	4	1	no	—
117	#64515	—	4	4	no	—
118	C₈×D₁₉	—	4	4	no	—
119	#8908	—	19	1	no	maximal, Sylow
120	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂²×D₁₉	—	2	1	no	—
122	C₂²×D₁₉	—	2	1	no	—
123	C₂²×D₁₉	—	4	2	no	—
124	C₂²×D₁₉	—	4	1	no	—
125	C₂²×D₁₉	—	4	1	no	—
126	C₂²×D₁₉	—	8	1	no	—
127	C₂×C₄×C₁₉	—	2	1	no	—
128	C₂×C₄×C₁₉	—	2	1	no	—
129	C₂×C₄×C₁₉	—	4	1	no	—
130	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
134	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
136	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
137	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
138	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₂×Dic₁₉	—	2	1	no	—
141	C₂×Dic₁₉	—	2	1	no	—
142	C₂×Dic₁₉	—	4	1	no	—
143	C₂×Dic₁₉	—	4	1	no	—
144	D₇₆	—	2	1	no	—
145	D₇₆	—	2	1	no	—
146	D₇₆	—	2	1	no	—
147	D₇₆	—	4	2	no	—
148	D₇₆	#108	1	2	yes	complemented
149	C₄×D₁₉	—	4	1	no	—
150	C₄×D₁₉	—	4	1	no	—
151	C₄×D₁₉	—	4	1	no	—
152	C₄×D₁₉	—	4	1	no	—
153	C₄×D₁₉	—	4	1	no	—
154	C₄×D₁₉	—	4	1	no	—
155	C₄×D₁₉	—	4	1	no	—
156	C₄×D₁₉	—	4	1	no	—
157	C₈×C₁₉	—	2	2	no	—
158	C₈×C₁₉	—	4	1	no	—
159	C₈×C₁₉	—	4	1	no	—
160	C₈×C₁₉	—	4	1	no	—
161	C₈×C₁₉	—	8	1	no	—
162	C₁₉⋊C₈	—	2	2	no	—
163	C₁₉⋊C₈	—	4	1	no	—
164	C₁₉⋊C₈	—	4	2	no	—
165	C₁₉⋊C₈	—	8	1	no	—
166	D₄×M₁₆	—	19	1	no	—
167	#1950	D₁₉	1	1	yes	characteristic, complemented
168	#1566	—	19	1	no	—
169	#1486	—	19	1	no	—
170	C₂×D₄⋊C₈	—	19	1	no	—
171	#1301	—	19	1	no	—
172	#1278	—	19	1	no	—
173	C₂²×C₁₉	—	2	1	no	—
174	C₂²×C₁₉	—	2	1	no	—
175	C₂²×C₁₉	#385	1	1	yes	characteristic
176	C₂²×C₁₉	#476	1	1	yes	characteristic
177	C₂²×C₁₉	#478	1	1	yes	characteristic
178	C₂×D₁₉	—	2	2	no	—
179	C₂×D₁₉	—	4	1	no	—
180	C₂×D₁₉	—	4	1	no	—
181	C₂×D₁₉	—	4	1	no	—
182	C₂×D₁₉	—	4	1	no	—
183	C₂×D₁₉	—	4	1	no	—
184	C₂×D₁₉	—	8	2	no	—
185	C₂×D₁₉	—	8	2	no	—
186	C₄×C₁₉	—	2	1	no	—
187	C₄×C₁₉	—	2	1	no	—
188	C₄×C₁₉	—	2	1	no	—
189	C₄×C₁₉	C₂×#108	1	1	yes	characteristic
190	C₄×C₁₉	#430	1	1	yes	characteristic
191	C₄×C₁₉	C₄×SD₁₆	1	1	yes	characteristic
192	C₄×C₁₉	#442	1	1	yes	characteristic
193	C₄×C₁₉	#449	1	1	yes	characteristic
194	C₄×C₁₉	#450	1	1	yes	characteristic
195	Dic₁₉	—	4	1	no	—
196	Dic₁₉	—	4	1	no	—
197	C₂²×M₁₆	—	38	1	no	—
198	C₂×C₄×D₄	—	19	1	no	—
199	#436	—	38	1	no	—
200	#435	—	38	2	no	—
201	C₈×D₄	—	38	1	no	—
202	#431	—	19	1	no	—
203	C₂×#108	—	2	1	no	—
204	#423	—	19	1	no	—
205	#423	C₂×D₁₉	1	1	yes	characteristic
206	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
207	#407	—	38	1	no	—
208	C₄×M₁₆	C₂×D₁₉	1	1	yes	characteristic
209	C₂×C₄×C₈	—	19	1	no	—
210	#341	—	2	1	no	—
211	#334	—	38	2	no	—
212	D₄⋊C₈	—	19	1	no	—
213	D₄⋊C₈	—	19	1	no	—
214	D₄⋊C₈	—	19	1	no	—
215	D₄⋊C₈	—	19	1	no	—
216	D₄⋊C₈	—	38	1	no	—
217	D₄⋊C₈	—	38	1	no	—
218	C₂×C₁₉	—	2	1	no	—
219	C₂×C₁₉	—	2	1	no	—
220	C₂×C₁₉	#1855	1	1	yes	characteristic
221	C₂×C₁₉	#1861	1	1	yes	characteristic
222	C₂×C₁₉	#1864	1	1	yes	characteristic
223	D₁₉	—	4	2	no	—
224	D₁₉	—	8	1	no	—
225	C₂²×D₄	—	19	1	no	—
226	C₂³×C₄	—	38	1	no	—
227	C₂×M₁₆	—	2	1	no	—
228	C₂×M₁₆	—	38	2	no	—
229	C₂×M₁₆	—	38	1	no	—
230	C₂×M₁₆	—	76	2	no	—
231	C₂×M₁₆	—	76	2	no	—
232	C₂×M₁₆	C₂²⋊D₁₉	1	1	yes	characteristic
233	C₂×M₁₆	C₂²⋊D₁₉	1	1	yes	characteristic
234	C₂²×C₈	—	2	1	no	—
235	C₂²×C₈	—	38	2	no	—
236	C₂²×C₈	—	38	1	no	—
237	C₄×D₄	—	19	1	no	—
238	C₄×D₄	—	19	1	no	—
239	C₄×D₄	—	19	1	no	—
240	C₄×D₄	—	19	1	no	—
241	C₄×D₄	—	38	1	no	—
242	C₄×D₄	—	38	1	no	—
243	C₂×C₄⋊C₄	—	19	1	no	—
244	C₂×C₂²⋊C₄	—	38	1	no	—
245	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
246	#108	—	4	2	no	—
247	C₄⋊C₈	—	2	1	no	—
248	C₄⋊C₈	—	2	1	no	—
249	C₄⋊C₈	—	2	1	no	—
250	C₄⋊C₈	—	19	2	no	—
251	C₄⋊C₈	—	38	1	no	—
252	C₄⋊C₈	—	38	1	no	—
253	C₄⋊C₈	C₄×D₁₉	1	1	yes	characteristic
254	C₄⋊C₈	C₄×D₁₉	1	1	yes	characteristic
255	C₂²⋊C₈	—	38	2	no	—
256	C₂²⋊C₈	—	38	2	no	—
257	C₂²⋊C₈	—	38	1	no	—
258	#97	—	2	1	no	—
259	C₄×C₈	—	2	1	no	—
260	C₄×C₈	—	19	1	no	—
261	C₄×C₈	—	19	1	no	—
262	C₄×C₈	—	38	1	no	—
263	C₄×C₈	—	38	1	no	—
264	C₁₉	#8908	1	1	yes	characteristic, complemented, Sylow
265	C₂⁴	—	38	1	no	—
266	C₂×D₄	—	19	1	no	—
267	C₂×D₄	—	19	1	no	—
268	C₂×D₄	—	38	2	no	—
269	C₂×D₄	—	38	1	no	—
270	C₂×D₄	—	76	1	no	—
271	C₂²×C₄	—	38	1	no	—
272	C₂²×C₄	—	38	1	no	—
273	C₂²×C₄	—	38	1	no	—
274	C₂²×C₄	—	38	1	no	—
275	C₂²×C₄	—	38	1	no	—
276	C₂²×C₄	—	76	1	no	—
277	C₂²×C₄	—	76	1	no	—
278	C₂²×C₄	—	76	1	no	—
279	C₂²×C₄	—	76	1	no	—
280	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
281	C₂²×C₄	#64545	1	1	yes	characteristic
282	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
283	M₁₆	—	2	1	no	—
284	M₁₆	—	2	1	no	—
285	M₁₆	—	2	1	no	—
286	M₁₆	—	4	1	no	—
287	M₁₆	—	4	1	no	—
288	M₁₆	—	76	4	no	—
289	M₁₆	—	76	1	no	—
290	M₁₆	—	76	1	no	—
291	M₁₆	#64527	1	2	yes	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	2	1	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	4	1	no	—
297	C₂×C₈	—	4	1	no	—
298	C₂×C₈	—	4	1	no	—
299	C₂×C₈	—	38	1	no	—
300	C₂×C₈	—	38	2	no	—
301	C₂×C₈	—	38	1	no	—
302	C₂×C₈	—	76	4	no	—
303	C₂×C₈	—	76	2	no	—
304	C₂×C₈	—	76	1	no	—
305	C₄⋊C₄	—	19	1	no	—
306	C₄⋊C₄	—	19	1	no	—
307	C₄⋊C₄	—	38	1	no	—
308	C₂²⋊C₄	—	38	1	no	—
309	C₂²⋊C₄	—	38	1	no	—
310	C₂²⋊C₄	—	38	1	no	—
311	C₂²⋊C₄	—	38	1	no	—
312	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
313	C₄²	#64541	1	1	yes	characteristic
314	C₄²	D₄×D₁₉	1	1	yes	characteristic
315	C₄²	D₄×D₁₉	1	1	yes	characteristic
316	C₂³	—	38	1	no	—
317	C₂³	—	38	1	no	—
318	C₂³	—	76	2	no	—
319	C₂³	—	76	1	no	—
320	C₂³	—	76	1	no	—
321	C₂³	—	152	1	no	—
322	C₂³	#10597392	1	1	yes	characteristic
323	D₄	—	19	2	no	—
324	D₄	—	38	1	no	—
325	D₄	—	38	1	no	—
326	D₄	—	38	1	no	—
327	D₄	—	76	2	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	38	1	no	—
332	C₂×C₄	—	38	1	no	—
333	C₂×C₄	—	76	1	no	—
334	C₂×C₄	—	76	1	no	—
335	C₂×C₄	—	76	1	no	—
336	C₂×C₄	—	76	1	no	—
337	C₂×C₄	—	76	1	no	—
338	C₂×C₄	—	76	1	no	—
339	C₂×C₄	—	76	1	no	—
340	C₂×C₄	—	76	1	no	—
341	C₂×C₄	—	76	1	no	—
342	C₂×C₄	—	76	1	no	—
343	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
344	C₂×C₄	#10597339	1	1	yes	characteristic
345	C₂×C₄	#10597340	1	1	yes	characteristic
346	C₂×C₄	#10597342	1	1	yes	characteristic
347	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
348	C₂×C₄	#10597378	1	1	yes	characteristic
349	C₂×C₄	#10597383	1	1	yes	characteristic
350	C₂×C₄	C₂×#64527	1	1	yes	characteristic
351	C₂×C₄	#10597395	1	1	yes	characteristic
352	C₂×C₄	#10597396	1	1	yes	characteristic
353	C₈	—	2	2	no	—
354	C₈	—	4	1	no	—
355	C₈	—	4	1	no	—
356	C₈	—	4	1	no	—
357	C₈	—	8	1	no	—
358	C₈	—	38	2	no	—
359	C₈	—	76	1	no	—
360	C₈	—	76	2	no	—
361	C₈	—	152	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	2	1	no	—
364	C₂²	—	38	2	no	—
365	C₂²	—	76	1	no	—
366	C₂²	—	76	1	no	—
367	C₂²	—	76	1	no	—
368	C₂²	—	76	1	no	—
369	C₂²	—	76	1	no	—
370	C₂²	—	152	2	no	—
371	C₂²	—	152	2	no	—
372	C₂²	#11926415	1	1	yes	characteristic
373	C₂²	#11926831	1	1	yes	characteristic
374	C₂²	#11926841	1	1	yes	characteristic
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	76	1	no	—
379	C₄	—	76	1	no	—
380	C₄	#108×D₁₉	1	1	yes	characteristic
381	C₄	#11926591	1	1	yes	characteristic
382	C₄	C₄×#64527	1	1	yes	characteristic
383	C₄	#11926654	1	1	yes	characteristic
384	C₄	#11926687	1	1	yes	characteristic
385	C₄	#11926694	1	1	yes	characteristic
386	C₂	—	2	1	no	—
387	C₂	—	2	1	no	—
388	C₂	—	76	2	no	—
389	C₂	—	152	1	no	—
390	C₂	#14563838	1	1	yes	characteristic
391	C₂	#14563872	1	1	yes	characteristic
392	C₂	#14563895	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597383
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1950	C₂
socle	C₂²×C₁₉	#385
prime radical	C₂²×D₇₆	Q₈
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1950	D₁₉
hypocenter	C₁₉	#8908
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1950

Complements

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

N	H ≅ G/N	type
2. #14562390	C₂	semidirect
5. #14561532	C₂	semidirect
6. #14561473	C₂	semidirect
7. C₁₉×#1950	C₂	semidirect
148. D₇₆	#108	semidirect
167. #1950	D₁₉	semidirect
264. C₁₉	#8908	semidirect

Maximal subgroups

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

subgroup	index
M₁₆×D₇₆	2
#14562390	2
#14562059	2
C₂×#11926271	2
#14561532	2
#14561473	2
C₁₉×#1950	2
#8908	19