Subgroups — #18081135

Order 4864.

388 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563786	C₂	1	1	yes	maximal, characteristic
2	#14563490	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×#11926294	C₂	1	1	yes	maximal, characteristic
5	#14561470	C₂	1	1	yes	maximal, characteristic
6	#14561469	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₉×#334	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×Dic₁₉⋊C₈	—	2	1	no	—
11	C₂×Dic₁₉⋊C₈	—	2	1	no	—
12	C₂×Dic₁₉⋊C₈	C₄	1	1	yes	characteristic
13	C₂×Dic₁₉⋊C₈	C₄	1	1	yes	characteristic
14	C₂×#10597272	—	2	1	no	—
15	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₄²×Dic₁₉	C₂²	1	1	yes	characteristic
17	#11926298	—	2	1	no	—
18	#11926298	C₄	1	1	yes	characteristic
19	#11926298	C₄	1	1	yes	characteristic
20	#11926294	—	2	2	no	—
21	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
24	C₁₉×#334	—	2	2	no	—
25	C₂²×C₈×C₁₉	—	2	1	no	—
26	C₂²×C₈×C₁₉	—	2	1	no	—
27	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
30	C₂×C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₁₉	D₄	1	1	yes	characteristic
33	C₂×C₄×Dic₁₉	D₄	1	1	yes	characteristic
34	C₂×C₄×Dic₁₉	Q₈	1	1	yes	characteristic
35	C₂×C₄×Dic₁₉	Q₈	1	1	yes	characteristic
36	C₂²×C₁₉⋊C₈	—	2	1	no	—
37	C₂²×C₁₉⋊C₈	—	2	1	no	—
38	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
40	C₁₉×C₄⋊C₈	—	2	1	no	—
41	C₁₉×C₄⋊C₈	—	2	1	no	—
42	C₁₉×C₄⋊C₈	—	2	1	no	—
43	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
44	C₄×C₈×C₁₉	—	2	1	no	—
45	C₄×C₈×C₁₉	—	2	1	no	—
46	#10597272	—	4	2	no	—
47	Dic₁₉⋊C₈	—	2	2	no	—
48	Dic₁₉⋊C₈	—	2	2	no	—
49	Dic₁₉⋊C₈	—	4	2	no	—
50	Dic₁₉⋊C₈	—	4	2	no	—
51	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
52	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
53	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
54	C₄⋊(C₁₉⋊C₈)	Q₈	1	2	yes	—
55	C₄×C₁₉⋊C₈	—	2	1	no	—
56	C₄×C₁₉⋊C₈	—	2	1	no	—
57	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
60	C₂²×Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₉	—	2	1	no	—
65	C₂×C₈×C₁₉	—	2	1	no	—
66	C₂×C₈×C₁₉	—	2	1	no	—
67	C₂×C₈×C₁₉	—	2	2	no	—
68	C₂×C₈×C₁₉	—	2	1	no	—
69	C₂×C₈×C₁₉	—	2	1	no	—
70	C₂×C₈×C₁₉	—	4	1	no	—
71	C₂×C₈×C₁₉	—	4	1	no	—
72	C₂×C₈×C₁₉	—	4	1	no	—
73	C₂×C₈×C₁₉	—	4	1	no	—
74	C₂×C₈×C₁₉	—	4	1	no	—
75	C₂×C₈×C₁₉	M₁₆	1	2	yes	—
76	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
78	C₄²×C₁₉	#41	1	1	yes	characteristic
79	C₄²×C₁₉	#41	1	1	yes	characteristic
80	C₄×Dic₁₉	—	2	1	no	—
81	C₄×Dic₁₉	—	2	1	no	—
82	C₄×Dic₁₉	—	2	1	no	—
83	C₄×Dic₁₉	—	2	1	no	—
84	C₄×Dic₁₉	—	2	1	no	—
85	C₄×Dic₁₉	—	2	1	no	—
86	C₄×Dic₁₉	—	2	2	no	—
87	C₄×Dic₁₉	—	4	1	no	—
88	C₄×Dic₁₉	SD₁₆	1	2	yes	—
89	C₄×Dic₁₉	SD₁₆	1	2	yes	—
90	C₂×C₁₉⋊C₈	—	2	1	no	—
91	C₂×C₁₉⋊C₈	—	2	1	no	—
92	C₂×C₁₉⋊C₈	—	2	1	no	—
93	C₂×C₁₉⋊C₈	—	2	2	no	—
94	C₂×C₁₉⋊C₈	—	2	1	no	—
95	C₂×C₁₉⋊C₈	—	2	1	no	—
96	C₂×C₁₉⋊C₈	—	4	1	no	—
97	C₂×C₁₉⋊C₈	—	4	1	no	—
98	C₂×C₁₉⋊C₈	—	4	1	no	—
99	C₂×C₁₉⋊C₈	—	4	1	no	—
100	C₂×C₁₉⋊C₈	—	4	1	no	—
101	C₂×C₁₉⋊C₈	M₁₆	1	2	yes	—
102	#9605	—	19	1	no	maximal, Sylow
103	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₉	—	2	1	no	—
105	C₂×C₄×C₁₉	—	2	1	no	—
106	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
108	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
109	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂×C₄×C₁₉	#125	1	1	yes	characteristic
111	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
113	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
114	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
115	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
120	C₂×Dic₁₉	—	2	1	no	—
121	C₂×Dic₁₉	—	2	1	no	—
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	—	2	1	no	—
127	C₂×Dic₁₉	—	2	1	no	—
128	C₂×Dic₁₉	—	2	1	no	—
129	C₂×Dic₁₉	#106	1	2	yes	—
130	C₂×Dic₁₉	#106	1	2	yes	—
131	C₂×Dic₁₉	#108	1	2	yes	—
132	C₈×C₁₉	—	2	2	no	—
133	C₈×C₁₉	—	4	1	no	—
134	C₈×C₁₉	—	4	2	no	—
135	C₈×C₁₉	—	8	1	no	—
136	C₁₉⋊C₈	—	2	2	no	—
137	C₁₉⋊C₈	—	4	1	no	—
138	C₁₉⋊C₈	—	4	2	no	—
139	C₁₉⋊C₈	—	8	1	no	—
140	#1842	—	19	1	no	—
141	#1767	—	19	1	no	—
142	C₄×C₄⋊C₈	—	19	1	no	—
143	C₂×#334	—	19	1	no	—
144	C₂×#334	D₁₉	1	1	yes	characteristic
145	#1277	—	19	1	no	—
146	#1277	—	19	1	no	—
147	C₂²×C₁₉	#389	1	1	yes	characteristic
148	C₂²×C₁₉	#423	1	1	yes	characteristic
149	C₂²×C₁₉	#424	1	1	yes	characteristic
150	C₂²×C₁₉	C₂×#106	1	1	yes	characteristic
151	C₂²×C₁₉	#428	1	1	yes	characteristic
152	C₂²×C₁₉	C₂×#108	1	1	yes	characteristic
153	C₂²×C₁₉	#430	1	1	yes	characteristic
154	C₄×C₁₉	—	2	1	no	—
155	C₄×C₁₉	—	2	1	no	—
156	C₄×C₁₉	#435	1	1	yes	characteristic
157	C₄×C₁₉	#435	1	1	yes	characteristic
158	C₄×C₁₉	#446	1	1	yes	characteristic
159	C₄×C₁₉	#446	1	1	yes	characteristic
160	C₄×C₁₉	#460	1	1	yes	characteristic
161	C₄×C₁₉	#461	1	1	yes	characteristic
162	C₄×C₁₉	#475	1	1	yes	characteristic
163	C₄×C₁₉	#476	1	1	yes	characteristic
164	Dic₁₉	—	2	2	no	—
165	Dic₁₉	—	2	2	no	—
166	Dic₁₉	—	2	2	no	—
167	Dic₁₉	—	4	1	no	—
168	C₂×C₄⋊C₈	—	19	1	no	—
169	C₂×C₄⋊C₈	—	19	1	no	—
170	C₂×C₄⋊C₈	—	19	1	no	—
171	C₂×C₄⋊C₈	—	19	1	no	—
172	C₂×C₄⋊C₈	—	38	1	no	—
173	C₂×C₄⋊C₈	—	38	1	no	—
174	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
175	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
176	C₂×#97	—	38	1	no	—
177	C₂×C₄×C₈	—	19	1	no	—
178	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
179	C₄³	—	19	1	no	—
180	#336	—	19	1	no	—
181	#336	—	19	1	no	—
182	#336	—	38	1	no	—
183	#334	—	2	2	no	—
184	#334	—	38	2	no	—
185	C₂×C₁₉	#1566	1	1	yes	characteristic
186	C₂×C₁₉	#1569	1	1	yes	characteristic
187	C₂×C₁₉	#1570	1	1	yes	characteristic
188	C₂×C₁₉	#1573	1	1	yes	characteristic
189	C₂×C₁₉	#1940	1	1	yes	characteristic
190	C₂×C₁₉	#1948	1	1	yes	characteristic
191	C₂×C₁₉	#1950	1	1	yes	characteristic
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	—	2	1	no	—
194	C₂²×C₈	—	19	1	no	—
195	C₂²×C₈	—	19	1	no	—
196	C₂²×C₈	—	38	1	no	—
197	C₂²×C₈	—	38	1	no	—
198	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
199	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
200	C₂×C₄²	—	19	1	no	—
201	C₂×C₄²	—	19	1	no	—
202	C₂×C₄²	—	19	1	no	—
203	C₂×C₄²	—	19	1	no	—
204	C₂×C₄²	—	19	1	no	—
205	C₂×C₄²	—	19	1	no	—
206	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
207	C₄⋊C₈	—	2	1	no	—
208	C₄⋊C₈	—	2	1	no	—
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	—	19	2	no	—
211	C₄⋊C₈	—	38	2	no	—
212	C₄⋊C₈	—	38	2	no	—
213	C₄⋊C₈	—	38	1	no	—
214	C₄⋊C₈	—	38	1	no	—
215	C₄⋊C₈	—	38	1	no	—
216	C₄⋊C₈	—	76	2	no	—
217	C₄⋊C₈	—	76	2	no	—
218	C₄⋊C₈	C₂²⋊D₁₉	1	2	yes	—
219	#97	—	76	2	no	—
220	C₄×C₈	—	2	1	no	—
221	C₄×C₈	—	2	1	no	—
222	C₄×C₈	—	38	1	no	—
223	C₄×C₈	—	38	1	no	—
224	C₁₉	#9605	1	1	yes	characteristic, complemented, Sylow
225	C₂²×C₄	—	19	1	no	—
226	C₂²×C₄	—	19	1	no	—
227	C₂²×C₄	—	19	1	no	—
228	C₂²×C₄	—	19	1	no	—
229	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
230	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
231	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	4	1	no	—
239	C₂×C₈	—	4	1	no	—
240	C₂×C₈	—	4	1	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	19	2	no	—
244	C₂×C₈	—	38	1	no	—
245	C₂×C₈	—	38	1	no	—
246	C₂×C₈	—	38	2	no	—
247	C₂×C₈	—	38	1	no	—
248	C₂×C₈	—	38	1	no	—
249	C₂×C₈	—	38	1	no	—
250	C₂×C₈	—	76	1	no	—
251	C₂×C₈	—	76	1	no	—
252	C₂×C₈	—	76	1	no	—
253	C₂×C₈	—	76	1	no	—
254	C₂×C₈	—	76	1	no	—
255	C₂×C₈	#64515	1	2	yes	—
256	C₄²	—	19	2	no	—
257	C₄²	—	19	2	no	—
258	C₄²	—	38	2	no	—
259	C₄²	—	38	1	no	—
260	C₄²	—	38	1	no	—
261	C₄²	—	38	1	no	—
262	C₄²	—	38	1	no	—
263	C₄²	—	38	1	no	—
264	C₄²	—	38	1	no	—
265	C₄²	—	76	1	no	—
266	C₄²	#64541	1	1	yes	characteristic
267	C₄²	#64543	1	1	yes	characteristic
268	C₄²	Q₈×D₁₉	1	1	yes	characteristic
269	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
270	C₂³	C₄⋊C₄×D₁₉	1	1	yes	characteristic
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	19	2	no	—
274	C₂×C₄	—	19	2	no	—
275	C₂×C₄	—	19	2	no	—
276	C₂×C₄	—	38	1	no	—
277	C₂×C₄	—	38	1	no	—
278	C₂×C₄	—	38	1	no	—
279	C₂×C₄	—	38	1	no	—
280	C₂×C₄	—	38	1	no	—
281	C₂×C₄	—	38	1	no	—
282	C₂×C₄	—	38	1	no	—
283	C₂×C₄	—	38	1	no	—
284	C₂×C₄	—	38	1	no	—
285	C₂×C₄	#10597333	1	1	yes	characteristic
286	C₂×C₄	#10597335	1	1	yes	characteristic
287	C₂×C₄	#10597342	1	1	yes	characteristic
288	C₂×C₄	C₂×#64515	1	1	yes	characteristic
289	C₂×C₄	#10597347	1	1	yes	characteristic
290	C₂×C₄	M₁₆×D₁₉	1	1	yes	characteristic
291	C₂×C₄	#10597353	1	1	yes	characteristic
292	C₂×C₄	SD₁₆×D₁₉	1	1	yes	characteristic
293	C₂×C₄	SD₁₆×D₁₉	1	1	yes	characteristic
294	C₂×C₄	#10597360	1	1	yes	characteristic
295	C₂×C₄	#10597361	1	1	yes	characteristic
296	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
297	C₂×C₄	#10597385	1	1	yes	characteristic
298	C₂×C₄	#10597390	1	1	yes	characteristic
299	C₈	—	2	2	no	—
300	C₈	—	4	1	no	—
301	C₈	—	4	2	no	—
302	C₈	—	8	1	no	—
303	C₈	—	38	2	no	—
304	C₈	—	76	2	no	—
305	C₈	—	76	1	no	—
306	C₈	—	152	1	no	—
307	C₂²	#11926439	1	1	yes	characteristic
308	C₂²	#11926566	1	1	yes	characteristic
309	C₂²	#11926569	1	1	yes	characteristic
310	C₂²	#106×D₁₉	1	1	yes	characteristic
311	C₂²	#11926583	1	1	yes	characteristic
312	C₂²	#108×D₁₉	1	1	yes	characteristic
313	C₂²	#11926591	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	38	2	no	—
317	C₄	—	38	2	no	—
318	C₄	—	38	2	no	—
319	C₄	—	76	1	no	—
320	C₄	#11926613	1	1	yes	characteristic
321	C₄	#11926615	1	1	yes	characteristic
322	C₄	#11926671	1	1	yes	characteristic
323	C₄	#11926673	1	1	yes	characteristic
324	C₄	#11926742	1	1	yes	characteristic
325	C₄	#11926747	1	1	yes	characteristic
326	C₄	#11926828	1	1	yes	characteristic
327	C₄	#11926833	1	1	yes	characteristic
328	C₂	#14562389	1	1	yes	characteristic
329	C₂	#14562399	1	1	yes	characteristic
330	C₂	#14562404	1	1	yes	characteristic
331	C₂	#14562417	1	1	yes	characteristic
332	C₂	#14564355	1	1	yes	characteristic
333	C₂	#14564395	1	1	yes	characteristic
334	C₂	#14564407	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#334	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#334	D₁₉
hypocenter	C₁₉	#9605
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
224. C₁₉	#9605	semidirect

Maximal subgroups

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

subgroup	index
#14563786	2
#14563490	2
C₄×Dic₁₉⋊C₈	2
C₂×#11926294	2
#14561470	2
#14561469	2
C₂×C₁₉×#334	2
#9605	19