Subgroups — #18081141

Order 4864.

368 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	#14563488	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926296	C₂	1	1	yes	maximal, characteristic
4	C₂×#11926294	C₂	1	1	yes	maximal, characteristic
5	#14561469	C₂	1	1	yes	maximal, characteristic
6	#14561469	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1831	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊(C₁₉⋊C₈)	C₄	1	1	yes	characteristic
9	C₂×C₄⋊(C₁₉⋊C₈)	C₄	1	1	yes	characteristic
10	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×#10597260	—	2	1	no	—
13	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
15	#11926299	—	2	1	no	—
16	#11926299	C₄	1	1	yes	characteristic
17	#11926299	C₄	1	1	yes	characteristic
18	#11926296	—	2	2	no	—
19	#11926294	—	2	2	no	—
20	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₄³×C₁₉	C₂²	1	1	yes	characteristic
23	C₁₉×#336	—	2	1	no	—
24	C₁₉×#336	—	2	1	no	—
25	C₂²×C₈×C₁₉	—	2	1	no	—
26	C₂²×C₈×C₁₉	—	2	1	no	—
27	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₉	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₁₉	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
34	C₂²×C₁₉⋊C₈	—	2	1	no	—
35	C₂²×C₁₉⋊C₈	—	2	1	no	—
36	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
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	—	2	1	no	—
44	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
45	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
46	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
47	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
48	C₄⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
49	C₄⋊(C₁₉⋊C₈)	Q₈	1	2	yes	—
50	#10597260	—	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	—	2	1	no	—
55	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₁₉	—	4	1	no	—
63	C₂×C₈×C₁₉	—	4	1	no	—
64	C₂×C₈×C₁₉	—	4	1	no	—
65	C₂×C₈×C₁₉	—	4	1	no	—
66	C₂×C₈×C₁₉	—	4	1	no	—
67	C₂×C₈×C₁₉	—	4	1	no	—
68	C₄²×C₁₉	—	2	2	no	—
69	C₄²×C₁₉	—	2	1	no	—
70	C₄²×C₁₉	—	2	1	no	—
71	C₄²×C₁₉	—	2	1	no	—
72	C₄²×C₁₉	—	2	1	no	—
73	C₄²×C₁₉	—	2	1	no	—
74	C₄²×C₁₉	—	2	1	no	—
75	C₄²×C₁₉	—	2	1	no	—
76	C₄²×C₁₉	—	2	1	no	—
77	C₄²×C₁₉	—	4	1	no	—
78	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
80	C₄²×C₁₉	#41	1	1	yes	characteristic
81	C₄²×C₁₉	#41	1	1	yes	characteristic
82	C₂×C₁₉⋊C₈	—	2	1	no	—
83	C₂×C₁₉⋊C₈	—	2	1	no	—
84	C₂×C₁₉⋊C₈	—	2	1	no	—
85	C₂×C₁₉⋊C₈	—	2	1	no	—
86	C₂×C₁₉⋊C₈	—	2	1	no	—
87	C₂×C₁₉⋊C₈	—	2	1	no	—
88	C₂×C₁₉⋊C₈	—	2	2	no	—
89	C₂×C₁₉⋊C₈	—	2	2	no	—
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	1	no	—
94	C₂×C₁₉⋊C₈	—	4	1	no	—
95	C₂×C₁₉⋊C₈	—	4	1	no	—
96	C₂×C₁₉⋊C₈	—	4	1	no	—
97	C₂×C₁₉⋊C₈	—	4	1	no	—
98	C₂×C₁₉⋊C₈	M₁₆	1	2	yes	—
99	C₂×C₁₉⋊C₈	M₁₆	1	2	yes	—
100	#9606	—	19	1	no	maximal, Sylow
101	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₉	—	2	1	no	—
103	C₂×C₄×C₁₉	—	2	1	no	—
104	C₂×C₄×C₁₉	—	2	1	no	—
105	C₂×C₄×C₁₉	—	2	1	no	—
106	C₂×C₄×C₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	—	2	1	no	—
108	C₂×C₄×C₁₉	—	2	1	no	—
109	C₂×C₄×C₁₉	—	2	1	no	—
110	C₂×C₄×C₁₉	—	2	1	no	—
111	C₂×C₄×C₁₉	—	2	1	no	—
112	C₂×C₄×C₁₉	—	2	1	no	—
113	C₂×C₄×C₁₉	—	2	1	no	—
114	C₂×C₄×C₁₉	—	2	1	no	—
115	C₂×C₄×C₁₉	#108	1	2	yes	—
116	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₁₉	#125	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂×M₁₆	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	#135	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	C₈×C₁₉	—	8	1	no	—
131	C₈×C₁₉	—	8	1	no	—
132	C₁₉⋊C₈	—	2	2	no	—
133	C₁₉⋊C₈	—	2	2	no	—
134	C₁₉⋊C₈	—	4	1	no	—
135	C₁₉⋊C₈	—	4	1	no	—
136	C₁₉⋊C₈	—	4	2	no	—
137	C₁₉⋊C₈	—	4	2	no	—
138	#1831	D₁₉	1	1	yes	characteristic
139	#1767	—	19	1	no	—
140	C₄×C₄⋊C₈	—	19	1	no	—
141	C₂×#335	—	19	1	no	—
142	C₂×#334	—	19	1	no	—
143	#1277	—	19	1	no	—
144	#1277	—	19	1	no	—
145	C₂²×C₁₉	#389	1	1	yes	characteristic
146	C₂²×C₁₉	#423	1	1	yes	characteristic
147	C₂²×C₁₉	#424	1	1	yes	characteristic
148	C₂²×C₁₉	#427	1	1	yes	characteristic
149	C₂²×C₁₉	#428	1	1	yes	characteristic
150	C₂²×C₁₉	C₂×#108	1	1	yes	characteristic
151	C₂²×C₁₉	#430	1	1	yes	characteristic
152	C₄×C₁₉	—	2	2	no	—
153	C₄×C₁₉	—	2	1	no	—
154	C₄×C₁₉	—	2	1	no	—
155	C₄×C₁₉	—	4	1	no	—
156	C₄×C₁₉	—	4	1	no	—
157	C₄×C₁₉	—	4	1	no	—
158	C₄×C₁₉	#435	1	1	yes	characteristic
159	C₄×C₁₉	#435	1	1	yes	characteristic
160	C₄×C₁₉	#446	1	1	yes	characteristic
161	C₄×C₁₉	#446	1	1	yes	characteristic
162	C₄×C₁₉	#463	1	1	yes	characteristic
163	C₄×C₁₉	#464	1	1	yes	characteristic
164	C₄×C₁₉	#478	1	1	yes	characteristic
165	C₄×C₁₉	#479	1	1	yes	characteristic
166	C₂×C₄⋊C₈	—	19	1	no	—
167	C₂×C₄⋊C₈	—	19	1	no	—
168	C₂×C₄⋊C₈	—	19	1	no	—
169	C₂×C₄⋊C₈	—	19	1	no	—
170	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
171	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
172	C₂×#97	—	38	1	no	—
173	C₂×C₄×C₈	—	19	1	no	—
174	C₂×C₄×C₈	—	19	1	no	—
175	C₄³	C₂×D₁₉	1	1	yes	characteristic
176	#336	—	2	1	no	—
177	#336	—	2	1	no	—
178	#336	—	19	1	no	—
179	#336	—	19	1	no	—
180	#336	—	38	1	no	—
181	#335	—	38	2	no	—
182	#334	—	38	2	no	—
183	C₂×C₁₉	#1568	1	1	yes	characteristic
184	C₂×C₁₉	#1569	1	1	yes	characteristic
185	C₂×C₁₉	#1572	1	1	yes	characteristic
186	C₂×C₁₉	#1574	1	1	yes	characteristic
187	C₂×C₁₉	#1940	1	1	yes	characteristic
188	C₂×C₁₉	#1949	1	1	yes	characteristic
189	C₂×C₁₉	#1950	1	1	yes	characteristic
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	—	2	1	no	—
192	C₂²×C₈	—	19	1	no	—
193	C₂²×C₈	—	19	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	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
199	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
200	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
201	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
202	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
203	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
204	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
205	C₄⋊C₈	—	2	1	no	—
206	C₄⋊C₈	—	2	1	no	—
207	C₄⋊C₈	—	2	1	no	—
208	C₄⋊C₈	—	2	1	no	—
209	C₄⋊C₈	—	19	2	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	#97	—	76	2	no	—
216	C₄×C₈	—	38	1	no	—
217	C₄×C₈	—	38	1	no	—
218	C₄×C₈	—	38	1	no	—
219	C₄×C₈	—	38	1	no	—
220	C₁₉	#9606	1	1	yes	characteristic, complemented, Sylow
221	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
222	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
223	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
224	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
225	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	4	1	no	—
234	C₂×C₈	—	19	2	no	—
235	C₂×C₈	—	19	2	no	—
236	C₂×C₈	—	38	1	no	—
237	C₂×C₈	—	38	1	no	—
238	C₂×C₈	—	38	1	no	—
239	C₂×C₈	—	38	1	no	—
240	C₂×C₈	—	38	2	no	—
241	C₂×C₈	—	38	2	no	—
242	C₂×C₈	—	38	1	no	—
243	C₂×C₈	—	38	1	no	—
244	C₂×C₈	—	38	1	no	—
245	C₂×C₈	—	38	1	no	—
246	C₂×C₈	—	38	1	no	—
247	C₂×C₈	—	38	1	no	—
248	C₂×C₈	—	76	1	no	—
249	C₂×C₈	—	76	1	no	—
250	C₂×C₈	—	76	1	no	—
251	C₂×C₈	—	76	1	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	2	1	no	—
261	C₄²	—	4	1	no	—
262	C₄²	D₄×D₁₉	1	1	yes	characteristic
263	C₄²	#64543	1	1	yes	characteristic
264	C₄²	Q₈×D₁₉	1	1	yes	characteristic
265	C₄²	#64545	1	1	yes	characteristic
266	C₂³	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	#10597279	1	2	yes	—
281	C₂×C₄	#10597333	1	1	yes	characteristic
282	C₂×C₄	#10597334	1	1	yes	characteristic
283	C₂×C₄	#10597336	1	1	yes	characteristic
284	C₂×C₄	#10597339	1	1	yes	characteristic
285	C₂×C₄	#10597347	1	1	yes	characteristic
286	C₂×C₄	#10597347	1	1	yes	characteristic
287	C₂×C₄	M₁₆×D₁₉	1	1	yes	characteristic
288	C₂×C₄	M₁₆×D₁₉	1	1	yes	characteristic
289	C₂×C₄	#10597384	1	1	yes	characteristic
290	C₂×C₄	#10597392	1	1	yes	characteristic
291	C₂×C₄	#10597395	1	1	yes	characteristic
292	C₂×C₄	#10597396	1	1	yes	characteristic
293	C₂×C₄	#10597396	1	1	yes	characteristic
294	C₂×C₄	#10597397	1	1	yes	characteristic
295	C₈	—	8	1	no	—
296	C₈	—	8	1	no	—
297	C₈	—	38	2	no	—
298	C₈	—	38	2	no	—
299	C₈	—	76	2	no	—
300	C₈	—	76	1	no	—
301	C₈	—	76	1	no	—
302	C₈	—	76	2	no	—
303	C₂²	#11926440	1	1	yes	characteristic
304	C₂²	#11926567	1	1	yes	characteristic
305	C₂²	#11926570	1	1	yes	characteristic
306	C₂²	#11926580	1	1	yes	characteristic
307	C₂²	#11926587	1	1	yes	characteristic
308	C₂²	C₂×#10597279	1	1	yes	characteristic
309	C₂²	#11926592	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	2	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	1	no	—
316	C₄	#11926614	1	1	yes	characteristic
317	C₄	#11926614	1	1	yes	characteristic
318	C₄	#11926673	1	1	yes	characteristic
319	C₄	#11926673	1	1	yes	characteristic
320	C₄	#11926756	1	1	yes	characteristic
321	C₄	#11926763	1	1	yes	characteristic
322	C₄	#11926841	1	1	yes	characteristic
323	C₄	#11926848	1	1	yes	characteristic
324	C₂	#14562396	1	1	yes	characteristic
325	C₂	#14562403	1	1	yes	characteristic
326	C₂	#14562413	1	1	yes	characteristic
327	C₂	#14562423	1	1	yes	characteristic
328	C₂	#14564356	1	1	yes	characteristic
329	C₂	#14564401	1	1	yes	characteristic
330	C₂	#14564408	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1831	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1831	D₁₉
hypocenter	C₁₉	#9606
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
220. C₁₉	#9606	semidirect

Maximal subgroups

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

subgroup	index
#14563488	2
C₄×C₄⋊(C₁₉⋊C₈)	2
C₂×#11926296	2
C₂×#11926294	2
#14561469	2
C₁₉×#1831	2
#9606	19