Subgroups — #18076565

Order 4864.

420 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572197	C₂	1	1	yes	maximal, characteristic
2	#14563616	C₂	1	1	yes	maximal, characteristic
3	#14562407	C₂	1	1	yes	maximal, characteristic, complemented
4	#14562406	C₂	1	1	yes	maximal, characteristic, complemented
5	#14561947	C₂	1	1	yes	maximal, characteristic
6	#14561532	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1301	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₁₉	—	2	1	no	—
9	C₂×#10597318	C₂²	1	1	yes	characteristic
10	#11926597	C₂²	1	1	yes	characteristic
11	#11926569	—	2	2	no	—
12	#11926569	—	2	2	no	—
13	#11926568	C₂²	1	1	yes	characteristic
14	#11926567	—	2	1	no	—
15	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×(C₂×D₁₉)⋊C₈	—	2	1	no	—
17	#11926298	—	2	1	no	—
18	#11926295	—	2	2	no	—
19	#11926294	—	2	2	no	—
20	#11926269	C₄	1	2	yes	—
21	#11926267	C₄	1	2	yes	—
22	C₁₉×#431	C₂²	1	1	yes	characteristic
23	C₁₉×#423	C₂²	1	1	yes	characteristic
24	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂²×C₄×D₁₉	D₄	1	1	yes	characteristic
26	C₂×M₁₆×C₁₉	—	2	1	no	—
27	C₂²×C₈×C₁₉	—	2	1	no	—
28	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
29	C₂×(C₂×D₁₉)⋊C₄	C₂×C₄	1	1	yes	characteristic
30	C₂×(C₂×D₁₉)⋊C₄	Q₈	1	1	yes	characteristic
31	C₂×Dic₁₉⋊C₄	C₂×C₄	1	1	yes	characteristic
32	C₂×Dic₁₉⋊C₄	Q₈	1	1	yes	characteristic
33	C₂×C₄×Dic₁₉	D₄	1	1	yes	characteristic
34	C₂×#64520	—	2	1	no	—
35	C₂²×C₁₉⋊C₈	—	2	1	no	—
36	M₁₆×D₁₉	—	4	2	no	—
37	M₁₆×D₁₉	—	4	2	no	—
38	C₂×#64515	—	2	2	no	—
39	C₂×C₈×D₁₉	—	2	2	no	—
40	#10597318	—	2	1	no	—
41	#10597318	—	2	1	no	—
42	#10597318	—	2	1	no	—
43	#10597318	—	2	1	no	—
44	C₁₉×C₄⋊C₈	—	2	1	no	—
45	C₁₉×C₄⋊C₈	—	2	1	no	—
46	C₁₉×C₄⋊C₈	—	2	1	no	—
47	C₁₉×C₄⋊C₈	—	2	1	no	—
48	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
49	C₁₉×C₂²⋊C₈	C₂×C₄	1	2	yes	—
50	C₄×C₈×C₁₉	—	2	1	no	—
51	C₂²⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
52	(C₂×D₁₉)⋊C₈	—	4	2	no	—
53	Dic₁₉⋊C₈	—	2	4	no	—
54	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
55	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
56	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
57	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
58	C₄⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
59	C₄×C₁₉⋊C₈	—	2	1	no	—
60	C₂³×D₁₉	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
64	C₂²×Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×D₁₉	—	2	1	no	—
68	C₂×C₄×D₁₉	—	2	1	no	—
69	C₂×C₄×D₁₉	—	2	1	no	—
70	C₂×C₄×D₁₉	—	2	1	no	—
71	C₂×C₄×D₁₉	SD₁₆	1	2	yes	—
72	C₂×C₄×D₁₉	SD₁₆	1	2	yes	—
73	M₁₆×C₁₉	—	4	1	no	—
74	M₁₆×C₁₉	—	4	1	no	—
75	C₂×C₈×C₁₉	—	2	2	no	—
76	C₂×C₈×C₁₉	—	2	2	no	—
77	C₂×C₈×C₁₉	—	4	1	no	—
78	C₂×C₈×C₁₉	—	4	1	no	—
79	C₂×C₈×C₁₉	—	4	1	no	—
80	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₁₉	#41	1	1	yes	characteristic
83	C₄²×C₁₉	#41	1	1	yes	characteristic
84	(C₂×D₁₉)⋊C₄	—	2	1	no	—
85	(C₂×D₁₉)⋊C₄	—	2	1	no	—
86	(C₂×D₁₉)⋊C₄	—	2	1	no	—
87	(C₂×D₁₉)⋊C₄	—	2	1	no	—
88	Dic₁₉⋊C₄	—	2	1	no	—
89	Dic₁₉⋊C₄	—	2	1	no	—
90	Dic₁₉⋊C₄	—	2	1	no	—
91	Dic₁₉⋊C₄	—	2	1	no	—
92	C₄×Dic₁₉	—	2	1	no	—
93	C₄×Dic₁₉	—	2	1	no	—
94	#64520	—	4	1	no	—
95	#64520	—	4	1	no	—
96	C₂×C₁₉⋊C₈	—	2	2	no	—
97	C₂×C₁₉⋊C₈	—	2	2	no	—
98	C₂×C₁₉⋊C₈	—	4	1	no	—
99	C₂×C₁₉⋊C₈	—	4	1	no	—
100	C₂×C₁₉⋊C₈	—	4	1	no	—
101	#64515	—	4	4	no	—
102	C₈×D₁₉	—	4	4	no	—
103	#8862	—	19	1	no	maximal, Sylow
104	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂²×D₁₉	—	2	1	no	—
106	C₂²×D₁₉	—	2	1	no	—
107	C₂²×D₁₉	—	4	1	no	—
108	C₂²×D₁₉	—	4	1	no	—
109	C₂²×D₁₉	#106	1	2	yes	—
110	C₂×C₄×C₁₉	—	4	1	no	—
111	C₂×C₄×C₁₉	—	4	1	no	—
112	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
115	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
122	C₂×Dic₁₉	—	2	1	no	—
123	C₂×Dic₁₉	—	2	1	no	—
124	C₂×Dic₁₉	—	2	1	no	—
125	C₂×Dic₁₉	—	2	1	no	—
126	C₂×Dic₁₉	—	4	1	no	—
127	C₂×Dic₁₉	—	4	1	no	—
128	C₂×Dic₁₉	#106	1	2	yes	—
129	C₄×D₁₉	—	2	2	no	—
130	C₄×D₁₉	—	2	2	no	—
131	C₄×D₁₉	—	2	2	no	—
132	C₄×D₁₉	—	2	2	no	—
133	C₈×C₁₉	—	4	2	no	—
134	C₈×C₁₉	—	4	2	no	—
135	C₈×C₁₉	—	8	1	no	—
136	C₁₉⋊C₈	—	4	2	no	—
137	C₁₉⋊C₈	—	4	2	no	—
138	C₁₉⋊C₈	—	8	1	no	—
139	#2905	—	19	1	no	—
140	#1798	—	19	1	no	—
141	#1570	—	19	1	no	—
142	#1570	—	19	1	no	—
143	#1458	—	19	1	no	—
144	#1301	—	19	1	no	—
145	#1301	D₁₉	1	1	yes	characteristic, complemented
146	C₂²×C₁₉	—	2	1	no	—
147	C₂²×C₁₉	—	2	1	no	—
148	C₂²×C₁₉	#385	1	1	yes	characteristic
149	C₂²×C₁₉	C₂×#106	1	1	yes	characteristic
150	C₂²×C₁₉	#428	1	1	yes	characteristic
151	C₂×D₁₉	—	2	2	no	—
152	C₂×D₁₉	—	4	1	no	—
153	C₂×D₁₉	—	4	1	no	—
154	C₂×D₁₉	—	4	2	no	—
155	C₂×D₁₉	—	8	1	no	—
156	C₄×C₁₉	—	2	1	no	—
157	C₄×C₁₉	—	2	1	no	—
158	C₄×C₁₉	—	2	1	no	—
159	C₄×C₁₉	—	2	1	no	—
160	C₄×C₁₉	#448	1	1	yes	characteristic
161	C₄×C₁₉	#450	1	1	yes	characteristic
162	C₄×C₁₉	#481	1	1	yes	characteristic
163	C₄×C₁₉	#483	1	1	yes	characteristic
164	Dic₁₉	—	2	2	no	—
165	Dic₁₉	—	4	1	no	—
166	Dic₁₉	—	4	1	no	—
167	C₂²×M₁₆	—	38	1	no	—
168	C₂×#117	—	19	1	no	—
169	#431	—	19	1	no	—
170	#431	C₂×D₁₉	1	1	yes	characteristic
171	#424	—	38	2	no	—
172	#424	—	38	2	no	—
173	#423	—	19	1	no	—
174	#423	—	38	1	no	—
175	#423	C₂×D₁₉	1	1	yes	characteristic
176	C₂×C₄⋊C₈	—	19	1	no	—
177	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
178	C₂×C₂²⋊C₈	—	38	1	no	—
179	#336	—	38	1	no	—
180	#334	—	38	2	no	—
181	#334	—	38	2	no	—
182	#324	—	19	2	no	—
183	#323	—	19	2	no	—
184	C₂×C₁₉	—	2	1	no	—
185	C₂×C₁₉	—	2	1	no	—
186	C₂×C₁₉	#1853	1	1	yes	characteristic
187	C₂×C₁₉	#1858	1	1	yes	characteristic
188	C₂×C₁₉	#1868	1	1	yes	characteristic
189	D₁₉	—	4	2	no	—
190	C₂³×C₄	—	19	1	no	—
191	C₂×M₁₆	—	2	1	no	—
192	C₂×M₁₆	—	38	2	no	—
193	C₂×M₁₆	—	38	1	no	—
194	C₂×M₁₆	—	76	2	no	—
195	C₂×M₁₆	—	76	2	no	—
196	C₂²×C₈	—	2	1	no	—
197	C₂²×C₈	—	38	2	no	—
198	C₂²×C₈	—	38	1	no	—
199	#117	—	38	1	no	—
200	#117	—	38	1	no	—
201	#117	—	38	1	no	—
202	#117	—	38	1	no	—
203	C₂×C₄⋊C₄	—	19	1	no	—
204	C₂×C₄⋊C₄	—	19	1	no	—
205	C₂×C₂²⋊C₄	—	19	1	no	—
206	C₂×C₂²⋊C₄	—	19	1	no	—
207	C₂×C₄²	—	19	1	no	—
208	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	—	2	1	no	—
211	C₄⋊C₈	—	2	1	no	—
212	C₄⋊C₈	—	2	1	no	—
213	C₄⋊C₈	—	19	2	no	—
214	C₄⋊C₈	—	38	4	no	—
215	C₄⋊C₈	—	38	1	no	—
216	C₄⋊C₈	—	38	1	no	—
217	C₄⋊C₈	—	38	1	no	—
218	C₄⋊C₈	—	38	1	no	—
219	C₄⋊C₈	D₇₆	1	2	yes	—
220	C₂²⋊C₈	—	19	2	no	—
221	C₂²⋊C₈	—	76	2	no	—
222	C₂²⋊C₈	C₄×D₁₉	1	2	yes	—
223	C₄×C₈	—	2	1	no	—
224	C₄×C₈	—	38	1	no	—
225	C₁₉	#8862	1	1	yes	characteristic, complemented, Sylow
226	C₂⁴	—	19	1	no	—
227	C₂²×C₄	—	19	1	no	—
228	C₂²×C₄	—	19	2	no	—
229	C₂²×C₄	—	19	2	no	—
230	C₂²×C₄	—	19	1	no	—
231	C₂²×C₄	—	19	1	no	—
232	C₂²×C₄	—	38	1	no	—
233	C₂²×C₄	—	38	1	no	—
234	C₂²×C₄	—	38	1	no	—
235	C₂²×C₄	—	38	1	no	—
236	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
237	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
238	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
239	M₁₆	—	4	1	no	—
240	M₁₆	—	4	1	no	—
241	M₁₆	—	76	4	no	—
242	M₁₆	—	76	1	no	—
243	M₁₆	—	76	1	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	38	2	no	—
250	C₂×C₈	—	38	2	no	—
251	C₂×C₈	—	76	4	no	—
252	C₂×C₈	—	76	1	no	—
253	C₂×C₈	—	76	1	no	—
254	C₂×C₈	—	76	1	no	—
255	C₄⋊C₄	—	38	1	no	—
256	C₄⋊C₄	—	38	1	no	—
257	C₄⋊C₄	—	38	1	no	—
258	C₄⋊C₄	—	38	1	no	—
259	C₂²⋊C₄	—	38	1	no	—
260	C₂²⋊C₄	—	38	1	no	—
261	C₂²⋊C₄	—	38	1	no	—
262	C₂²⋊C₄	—	38	1	no	—
263	C₄²	—	38	1	no	—
264	C₄²	—	38	1	no	—
265	C₄²	C₂×D₇₆	1	1	yes	characteristic
266	C₄²	#64541	1	1	yes	characteristic
267	C₄²	D₄×D₁₉	1	1	yes	characteristic
268	C₄²	#64543	1	1	yes	characteristic
269	C₂³	—	19	2	no	—
270	C₂³	—	38	1	no	—
271	C₂³	—	38	1	no	—
272	C₂³	—	76	1	no	—
273	C₂³	—	76	1	no	—
274	C₂³	C₄⋊C₄×D₁₉	1	1	yes	characteristic
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	19	2	no	—
278	C₂×C₄	—	38	1	no	—
279	C₂×C₄	—	38	1	no	—
280	C₂×C₄	—	38	2	no	—
281	C₂×C₄	—	38	2	no	—
282	C₂×C₄	—	38	2	no	—
283	C₂×C₄	—	38	2	no	—
284	C₂×C₄	—	38	1	no	—
285	C₂×C₄	—	38	1	no	—
286	C₂×C₄	—	76	1	no	—
287	C₂×C₄	—	76	1	no	—
288	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
289	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
290	C₂×C₄	#10597328	1	1	yes	characteristic
291	C₂×C₄	#10597329	1	1	yes	characteristic
292	C₂×C₄	#10597342	1	1	yes	characteristic
293	C₂×C₄	#10597343	1	1	yes	characteristic
294	C₂×C₄	#10597357	1	1	yes	characteristic
295	C₂×C₄	SD₁₆×D₁₉	1	1	yes	characteristic
296	C₂×C₄	SD₁₆×D₁₉	1	1	yes	characteristic
297	C₂×C₄	#10597364	1	1	yes	characteristic
298	C₈	—	4	2	no	—
299	C₈	—	4	2	no	—
300	C₈	—	8	1	no	—
301	C₈	—	76	2	no	—
302	C₈	—	76	2	no	—
303	C₈	—	152	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	38	2	no	—
307	C₂²	—	76	2	no	—
308	C₂²	—	76	1	no	—
309	C₂²	—	76	1	no	—
310	C₂²	—	152	1	no	—
311	C₂²	#11926418	1	1	yes	characteristic
312	C₂²	#106×D₁₉	1	1	yes	characteristic
313	C₂²	#11926585	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	38	2	no	—
319	C₄	—	76	1	no	—
320	C₄	—	76	1	no	—
321	C₄	#11926684	1	1	yes	characteristic
322	C₄	#11926696	1	1	yes	characteristic
323	C₄	#11926861	1	1	yes	characteristic
324	C₄	#11926873	1	1	yes	characteristic
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	—	76	2	no	—
328	C₂	#14563829	1	1	yes	characteristic
329	C₂	#14563856	1	1	yes	characteristic
330	C₂	#14563922	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597328
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1301	C₂
socle	C₂²×C₁₉	#385
prime radical	C₂³×D₁₉	C₄⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1301	D₁₉
hypocenter	C₁₉	#8862
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1301

Complements

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

N	H ≅ G/N	type
3. #14562407	C₂	semidirect
4. #14562406	C₂	semidirect
6. #14561532	C₂	semidirect
7. C₁₉×#1301	C₂	semidirect
145. #1301	D₁₉	semidirect
225. C₁₉	#8862	semidirect

Maximal subgroups

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

subgroup	index
#14572197	2
#14563616	2
#14562407	2
#14562406	2
#14561947	2
#14561532	2
C₁₉×#1301	2
#8862	19