Subgroups — #18076454

Order 4864.

464 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926668	C₂	1	1	yes	maximal, characteristic
2	#14564351	C₂	1	1	yes	maximal, characteristic
3	C₄×#10597273	C₂	1	1	yes	maximal, characteristic
4	C₂×#11926274	C₂	1	1	yes	maximal, characteristic
5	#14561474	C₂	1	1	yes	maximal, characteristic
6	C₁₉⋊#1273	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₉×#335	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Dic₃₈	C₂²	1	1	yes	characteristic
9	#11926668	—	2	2	no	—
10	#11926668	—	2	2	no	—
11	C₂×Dic₁₉⋊C₈	—	2	1	no	—
12	C₂×Dic₁₉⋊C₈	—	2	1	no	—
13	C₂×#10597273	C₄	1	1	yes	characteristic
14	C₂×#10597273	C₄	1	1	yes	characteristic
15	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₁₉	—	2	1	no	—
18	C₄×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
19	#11926307	C₄	1	1	yes	characteristic
20	#11926307	C₄	1	1	yes	characteristic
21	#11926298	—	2	1	no	—
22	#11926274	—	2	2	no	—
23	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
26	C₁₉×#335	—	2	2	no	—
27	C₂²×Dic₃₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₉	—	2	1	no	—
29	C₂²×C₈×C₁₉	—	2	1	no	—
30	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊Dic₁₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊Dic₁₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₁₉	C₂×C₄	1	1	yes	characteristic
36	C₂×Dic₁₉⋊C₄	—	2	1	no	—
37	C₂×C₄×Dic₁₉	—	2	1	no	—
38	C₂×C₄×Dic₁₉	—	2	1	no	—
39	C₂²×C₁₉⋊C₈	—	2	1	no	—
40	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
41	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
42	C₄×Dic₃₈	—	2	1	no	—
43	C₄×Dic₃₈	—	2	1	no	—
44	C₄×Dic₃₈	D₄	1	2	yes	—
45	C₄×Dic₃₈	Q₈	1	2	yes	—
46	C₁₉×C₄⋊C₈	—	2	1	no	—
47	C₁₉×C₄⋊C₈	—	2	1	no	—
48	C₁₉×C₄⋊C₈	—	2	1	no	—
49	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
50	C₄×C₈×C₁₉	—	2	1	no	—
51	C₄×C₈×C₁₉	—	2	1	no	—
52	C₁₉⋊#95	—	2	1	no	—
53	#10597273	—	2	2	no	—
54	#10597273	C₈	1	4	yes	complemented
55	Dic₁₉⋊C₈	—	2	4	no	—
56	Dic₁₉⋊C₈	—	4	2	no	—
57	C₈×Dic₁₉	—	2	4	no	—
58	C₄×C₁₉⋊C₈	—	2	1	no	—
59	C₄×C₁₉⋊C₈	—	2	1	no	—
60	C₄×C₁₉⋊C₈	—	2	1	no	—
61	C₄×C₁₉⋊C₈	—	2	1	no	—
62	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
65	C₂²×Dic₁₉	—	2	1	no	—
66	C₂²×Dic₁₉	—	2	1	no	—
67	C₂²×Dic₁₉	—	2	1	no	—
68	C₂²×Dic₁₉	—	2	1	no	—
69	C₂×Dic₃₈	—	2	1	no	—
70	C₂×Dic₃₈	—	2	1	no	—
71	C₂×Dic₃₈	—	2	1	no	—
72	C₂×Dic₃₈	C₄⋊C₄	1	2	yes	—
73	C₂×Dic₃₈	C₂×C₈	1	2	yes	—
74	C₂×Dic₃₈	M₁₆	1	2	yes	—
75	C₂×C₈×C₁₉	—	2	1	no	—
76	C₂×C₈×C₁₉	—	2	1	no	—
77	C₂×C₈×C₁₉	—	2	1	no	—
78	C₂×C₈×C₁₉	—	2	2	no	—
79	C₂×C₈×C₁₉	—	2	1	no	—
80	C₂×C₈×C₁₉	—	2	1	no	—
81	C₂×C₈×C₁₉	—	4	1	no	—
82	C₂×C₈×C₁₉	—	4	1	no	—
83	C₂×C₈×C₁₉	—	4	1	no	—
84	C₂×C₈×C₁₉	—	4	1	no	—
85	C₂×C₈×C₁₉	—	4	1	no	—
86	C₂×C₈×C₁₉	C₂×C₈	1	2	yes	—
87	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
90	C₄²×C₁₉	#41	1	1	yes	characteristic
91	C₄⋊Dic₁₉	—	2	1	no	—
92	C₄⋊Dic₁₉	—	2	1	no	—
93	C₄⋊Dic₁₉	—	2	1	no	—
94	C₄⋊Dic₁₉	—	2	1	no	—
95	C₄⋊Dic₁₉	C₄⋊C₄	1	2	yes	—
96	C₄⋊Dic₁₉	C₂×C₈	1	2	yes	—
97	Dic₁₉⋊C₄	—	2	2	no	—
98	Dic₁₉⋊C₄	—	2	2	no	—
99	C₄×Dic₁₉	—	2	2	no	—
100	C₄×Dic₁₉	—	2	2	no	—
101	C₄×Dic₁₉	—	4	1	no	—
102	C₄×Dic₁₉	—	4	1	no	—
103	C₂×C₁₉⋊C₈	—	2	1	no	—
104	C₂×C₁₉⋊C₈	—	2	1	no	—
105	C₂×C₁₉⋊C₈	—	2	2	no	—
106	C₂×C₁₉⋊C₈	—	2	1	no	—
107	C₂×C₁₉⋊C₈	—	2	1	no	—
108	C₂×C₁₉⋊C₈	—	2	1	no	—
109	C₂×C₁₉⋊C₈	—	4	1	no	—
110	C₂×C₁₉⋊C₈	—	4	1	no	—
111	C₂×C₁₉⋊C₈	Q₁₆	1	2	yes	—
112	#8844	—	19	1	no	maximal, Sylow
113	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₉	—	2	1	no	—
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
124	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
125	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
127	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
128	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
130	C₂×Dic₁₉	—	2	2	no	—
131	C₂×Dic₁₉	—	2	2	no	—
132	C₂×Dic₁₉	—	2	2	no	—
133	C₂×Dic₁₉	—	2	2	no	—
134	C₂×Dic₁₉	—	2	2	no	—
135	C₂×Dic₁₉	—	4	1	no	—
136	C₂×Dic₁₉	—	4	1	no	—
137	C₂×Dic₁₉	—	4	1	no	—
138	C₂×Dic₁₉	—	4	1	no	—
139	C₂×Dic₁₉	—	4	1	no	—
140	C₂×Dic₁₉	—	4	1	no	—
141	C₂×Dic₁₉	—	4	1	no	—
142	Dic₃₈	—	2	2	no	—
143	Dic₃₈	—	2	2	no	—
144	Dic₃₈	—	2	2	no	—
145	Dic₃₈	C₄⋊C₈	1	4	yes	complemented
146	C₈×C₁₉	—	2	2	no	—
147	C₈×C₁₉	—	4	1	no	—
148	C₈×C₁₉	—	4	2	no	—
149	C₈×C₁₉	—	8	1	no	—
150	C₁₉⋊C₈	—	2	2	no	—
151	C₁₉⋊C₈	—	4	2	no	—
152	C₁₉⋊C₈	—	4	1	no	—
153	C₂×C₈×Q₈	—	19	1	no	—
154	#1939	—	19	1	no	—
155	C₄×#103	—	19	1	no	—
156	C₂×#335	D₁₉	1	1	yes	characteristic
157	C₂×Q₈⋊C₈	—	19	1	no	—
158	#1279	—	19	1	no	—
159	#1273	—	19	1	no	—
160	C₂²×C₁₉	#385	1	1	yes	characteristic
161	C₂²×C₁₉	C₈×D₄	1	1	yes	characteristic
162	C₂²×C₁₉	#435	1	1	yes	characteristic
163	C₂²×C₁₉	C₄×Q₁₆	1	1	yes	characteristic
164	C₂²×C₁₉	#440	1	1	yes	characteristic
165	C₂²×C₁₉	#443	1	1	yes	characteristic
166	C₂²×C₁₉	#444	1	1	yes	characteristic
167	C₄×C₁₉	—	2	1	no	—
168	C₄×C₁₉	—	2	1	no	—
169	C₄×C₁₉	C₂×C₄⋊C₈	1	1	yes	characteristic
170	C₄×C₁₉	#424	1	1	yes	characteristic
171	C₄×C₁₉	C₈×D₄	1	1	yes	characteristic
172	C₄×C₁₉	#435	1	1	yes	characteristic
173	C₄×C₁₉	#449	1	1	yes	characteristic
174	C₄×C₁₉	#451	1	1	yes	characteristic
175	C₄×C₁₉	#477	1	1	yes	characteristic
176	C₄×C₁₉	#479	1	1	yes	characteristic
177	Dic₁₉	—	2	4	no	—
178	Dic₁₉	—	4	2	no	—
179	Dic₁₉	—	4	2	no	—
180	Dic₁₉	—	8	1	no	—
181	C₂×C₄×Q₈	—	19	1	no	—
182	C₈×Q₈	—	38	2	no	—
183	C₈×Q₈	—	38	2	no	—
184	C₂×C₄⋊C₈	—	38	1	no	—
185	C₂×C₄⋊C₈	—	38	1	no	—
186	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
187	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
188	C₂×#103	—	19	1	no	—
189	C₂×#103	—	19	1	no	—
190	C₂×C₄×C₈	—	19	1	no	—
191	C₂×C₄×C₈	—	19	1	no	—
192	C₂×C₄×C₈	—	38	1	no	—
193	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
194	C₄×C₄⋊C₄	—	19	1	no	—
195	#340	—	19	1	no	—
196	#340	—	19	1	no	—
197	#336	—	38	1	no	—
198	#335	—	2	2	no	—
199	Q₈⋊C₈	—	38	2	no	—
200	C₂×C₁₉	C₈×Q₁₆	1	1	yes	characteristic
201	C₂×C₁₉	SD₁₆⋊C₈	1	1	yes	characteristic
202	C₂×C₁₉	#1584	1	1	yes	characteristic
203	C₂×C₁₉	#1588	1	1	yes	characteristic
204	C₂×C₁₉	#1850	1	1	yes	characteristic
205	C₂×C₁₉	#1863	1	1	yes	characteristic
206	C₂×C₁₉	#1866	1	1	yes	characteristic
207	C₂²×Q₈	—	19	1	no	—
208	C₂²×C₈	—	2	1	no	—
209	C₂²×C₈	—	2	1	no	—
210	C₂²×C₈	—	19	1	no	—
211	C₂²×C₈	—	19	1	no	—
212	C₂²×C₈	—	38	1	no	—
213	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
214	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
215	C₄×Q₈	—	19	2	no	—
216	C₄×Q₈	—	19	2	no	—
217	C₄×Q₈	—	38	1	no	—
218	C₄×Q₈	—	38	1	no	—
219	C₂×C₄⋊C₄	—	19	1	no	—
220	C₂×C₄⋊C₄	—	19	1	no	—
221	C₂×C₄⋊C₄	—	19	1	no	—
222	C₂×C₄⋊C₄	—	38	1	no	—
223	C₂×C₄²	—	38	1	no	—
224	C₂×C₄²	—	38	1	no	—
225	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	—	2	1	no	—
228	C₄⋊C₈	—	2	1	no	—
229	C₄⋊C₈	—	38	4	no	—
230	C₄⋊C₈	—	76	2	no	—
231	C₄⋊C₈	C₂²⋊D₁₉	1	2	yes	—
232	#103	—	19	4	no	—
233	#103	—	38	2	no	—
234	C₄×C₈	—	2	1	no	—
235	C₄×C₈	—	2	1	no	—
236	C₄×C₈	—	38	4	no	—
237	C₄×C₈	—	38	1	no	—
238	C₄×C₈	—	38	1	no	—
239	C₄×C₈	—	38	1	no	—
240	C₄×C₈	—	38	1	no	—
241	#95	—	38	1	no	—
242	C₁₉	#8844	1	1	yes	characteristic, complemented, Sylow
243	C₂×Q₈	—	19	2	no	—
244	C₂×Q₈	—	19	2	no	—
245	C₂×Q₈	—	19	2	no	—
246	C₂×Q₈	—	38	1	no	—
247	C₂×Q₈	—	38	1	no	—
248	C₂×Q₈	—	38	1	no	—
249	C₂²×C₄	—	38	1	no	—
250	C₂²×C₄	—	38	1	no	—
251	C₂²×C₄	—	38	1	no	—
252	C₂²×C₄	—	38	1	no	—
253	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
254	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
255	C₂²×C₄	#64545	1	1	yes	characteristic
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	4	1	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	4	1	no	—
265	C₂×C₈	—	4	1	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	19	2	no	—
268	C₂×C₈	—	38	1	no	—
269	C₂×C₈	—	38	1	no	—
270	C₂×C₈	—	38	2	no	—
271	C₂×C₈	—	38	1	no	—
272	C₂×C₈	—	38	1	no	—
273	C₂×C₈	—	38	1	no	—
274	C₂×C₈	—	76	1	no	—
275	C₂×C₈	—	76	1	no	—
276	C₂×C₈	C₈×D₁₉	1	2	yes	—
277	C₄⋊C₄	—	19	2	no	—
278	C₄⋊C₄	—	19	2	no	—
279	C₄⋊C₄	—	38	2	no	—
280	C₄⋊C₄	—	38	2	no	—
281	C₄⋊C₄	—	38	1	no	—
282	C₄⋊C₄	—	38	1	no	—
283	C₄⋊C₄	—	38	1	no	—
284	C₄⋊C₄	—	38	1	no	—
285	C₄²	—	38	2	no	—
286	C₄²	—	38	2	no	—
287	C₄²	—	76	1	no	—
288	C₄²	—	76	1	no	—
289	C₄²	#64541	1	1	yes	characteristic
290	C₄²	D₄×D₁₉	1	1	yes	characteristic
291	C₄²	Q₈×D₁₉	1	1	yes	characteristic
292	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
293	C₂³	#10597339	1	1	yes	characteristic
294	Q₈	—	19	4	no	—
295	Q₈	—	38	2	no	—
296	Q₈	—	38	2	no	—
297	Q₈	—	38	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	38	2	no	—
301	C₂×C₄	—	38	2	no	—
302	C₂×C₄	—	38	2	no	—
303	C₂×C₄	—	38	2	no	—
304	C₂×C₄	—	38	2	no	—
305	C₂×C₄	—	76	1	no	—
306	C₂×C₄	—	76	1	no	—
307	C₂×C₄	—	76	1	no	—
308	C₂×C₄	—	76	1	no	—
309	C₂×C₄	—	76	1	no	—
310	C₂×C₄	—	76	1	no	—
311	C₂×C₄	—	76	1	no	—
312	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
313	C₂×C₄	#10597340	1	1	yes	characteristic
314	C₂×C₄	#10597342	1	1	yes	characteristic
315	C₂×C₄	C₂×C₈×D₁₉	1	1	yes	characteristic
316	C₂×C₄	#10597347	1	1	yes	characteristic
317	C₂×C₄	M₁₆×D₁₉	1	1	yes	characteristic
318	C₂×C₄	#10597353	1	1	yes	characteristic
319	C₂×C₄	#10597357	1	1	yes	characteristic
320	C₂×C₄	#10597358	1	1	yes	characteristic
321	C₂×C₄	Q₁₆×D₁₉	1	1	yes	characteristic
322	C₂×C₄	#10597364	1	1	yes	characteristic
323	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
324	C₂×C₄	#10597383	1	1	yes	characteristic
325	C₂×C₄	#10597392	1	1	yes	characteristic
326	C₈	—	2	2	no	—
327	C₈	—	4	1	no	—
328	C₈	—	4	2	no	—
329	C₈	—	8	1	no	—
330	C₈	—	38	2	no	—
331	C₈	—	76	2	no	—
332	C₈	—	76	1	no	—
333	C₂²	#11926415	1	1	yes	characteristic
334	C₂²	#11926609	1	1	yes	characteristic
335	C₂²	#11926614	1	1	yes	characteristic
336	C₂²	#11926638	1	1	yes	characteristic
337	C₂²	#11926643	1	1	yes	characteristic
338	C₂²	#11926660	1	1	yes	characteristic
339	C₂²	#11926665	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	38	4	no	—
343	C₄	—	76	2	no	—
344	C₄	—	76	2	no	—
345	C₄	—	152	1	no	—
346	C₄	C₄⋊C₈×D₁₉	1	1	yes	characteristic
347	C₄	#11926569	1	1	yes	characteristic
348	C₄	C₈×C₂²⋊D₁₉	1	1	yes	characteristic
349	C₄	#11926613	1	1	yes	characteristic
350	C₄	#11926686	1	1	yes	characteristic
351	C₄	#11926698	1	1	yes	characteristic
352	C₄	#11926838	1	1	yes	characteristic
353	C₄	#11926850	1	1	yes	characteristic
354	C₂	#14562441	1	1	yes	characteristic
355	C₂	#14562446	1	1	yes	characteristic
356	C₂	#14562484	1	1	yes	characteristic
357	C₂	#14562512	1	1	yes	characteristic
358	C₂	#14563811	1	1	yes	characteristic
359	C₂	#14563886	1	1	yes	characteristic
360	C₂	#14563907	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₉
commutator	C₂×C₄×C₁₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#335	C₂
socle, prime radical	C₂³×C₁₉	C₄×D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#335	D₁₉
hypocenter	C₁₉	#8844
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
54. #10597273	C₈	semidirect
145. Dic₃₈	C₄⋊C₈	semidirect
242. C₁₉	#8844	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926668	2
#14564351	2
C₄×#10597273	2
C₂×#11926274	2
#14561474	2
C₁₉⋊#1273	2
C₂×C₁₉×#335	2
#8844	19