Subgroups — #18073034

Order 4864.

554 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂²⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926266	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11926265	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉⋊#1273	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉⋊#1273	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₉×#431	C₂	1	1	yes	maximal, characteristic
8	C₂×C₂²⋊(C₁₉⋊C₈)	C₄	1	2	yes	—
9	C₂×C₂²⋊(C₁₉⋊C₈)	C₄	1	2	yes	—
10	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
14	#11926299	C₄	1	2	yes	—
15	#11926299	C₄	1	2	yes	—
16	#11926266	—	2	2	no	—
17	#11926265	—	2	2	no	—
18	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
19	C₁₉×#431	—	2	2	no	—
20	C₁₉×#431	—	2	2	no	—
21	C₂×C₁₉×C₄⋊C₈	C₄	1	2	yes	—
22	C₂×C₁₉×C₂²⋊C₈	C₄	1	2	yes	—
23	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
25	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₉	—	2	1	no	—
31	C₂×C₄²×C₁₉	—	2	1	no	—
32	C₂×C₄²×C₁₉	—	2	1	no	—
33	C₂×C₄²×C₁₉	—	2	1	no	—
34	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
38	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
39	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
40	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
41	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
42	C₁₉×C₄⋊C₈	—	2	4	no	—
43	C₁₉×C₂²⋊C₈	—	2	4	no	—
44	C₄×C₈×C₁₉	—	2	1	no	—
45	C₄×C₈×C₁₉	—	2	1	no	—
46	C₄×C₈×C₁₉	D₄	1	2	yes	—
47	C₄×C₈×C₁₉	D₄	1	2	yes	—
48	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
49	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
50	C₄×C₁₉⋊C₈	—	2	1	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	C₄×C₁₉⋊C₈	—	2	1	no	—
56	C₄×C₁₉⋊C₈	—	2	1	no	—
57	C₄×C₁₉⋊C₈	—	2	1	no	—
58	C₂⁴×C₁₉	C₄²	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	—	2	1	no	—
60	C₂²×C₄×C₁₉	—	2	1	no	—
61	C₂²×C₄×C₁₉	—	2	2	no	—
62	C₂²×C₄×C₁₉	—	2	1	no	—
63	C₂²×C₄×C₁₉	—	2	1	no	—
64	C₂²×C₄×C₁₉	—	2	1	no	—
65	C₂²×C₄×C₁₉	—	2	1	no	—
66	C₂²×C₄×C₁₉	—	2	2	no	—
67	C₂²×C₄×C₁₉	—	2	2	no	—
68	C₂²×C₄×C₁₉	—	2	2	no	—
69	C₂²×C₄×C₁₉	—	2	2	no	—
70	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	M₁₆	1	4	yes	—
74	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₉	—	2	2	no	—
78	C₂×C₈×C₁₉	—	2	2	no	—
79	C₂×C₈×C₁₉	—	2	2	no	—
80	C₂×C₈×C₁₉	—	2	2	no	—
81	C₂×C₈×C₁₉	—	2	2	no	—
82	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
83	C₄²×C₁₉	—	2	1	no	—
84	C₄²×C₁₉	—	2	1	no	—
85	C₄²×C₁₉	—	2	1	no	—
86	C₄²×C₁₉	—	2	1	no	—
87	C₄²×C₁₉	—	2	1	no	—
88	C₄²×C₁₉	—	2	1	no	—
89	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₁₉	#41	1	1	yes	characteristic
92	C₄²×C₁₉	#41	1	1	yes	characteristic
93	C₂×C₁₉⋊C₈	—	2	2	no	—
94	C₂×C₁₉⋊C₈	—	2	2	no	—
95	C₂×C₁₉⋊C₈	—	2	2	no	—
96	C₂×C₁₉⋊C₈	—	2	2	no	—
97	C₂×C₁₉⋊C₈	—	2	2	no	—
98	C₂×C₁₉⋊C₈	—	2	2	no	—
99	C₂×C₁₉⋊C₈	—	2	2	no	—
100	C₂×C₁₉⋊C₈	—	2	2	no	—
101	C₂×C₁₉⋊C₈	—	2	2	no	—
102	C₂×C₁₉⋊C₈	—	2	2	no	—
103	C₂×C₁₉⋊C₈	—	2	2	no	—
104	C₂×C₁₉⋊C₈	M₁₆	1	4	yes	—
105	#8149	—	19	1	no	maximal, Sylow
106	C₂³×C₁₉	—	2	1	no	—
107	C₂³×C₁₉	—	2	1	no	—
108	C₂³×C₁₉	—	2	1	no	—
109	C₂³×C₁₉	—	2	1	no	—
110	C₂³×C₁₉	—	2	1	no	—
111	C₂³×C₁₉	—	2	1	no	—
112	C₂³×C₁₉	#97	1	2	yes	—
113	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₁₉	—	2	1	no	—
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	2	4	no	—
117	C₂×C₄×C₁₉	—	2	4	no	—
118	C₂×C₄×C₁₉	—	2	1	no	—
119	C₂×C₄×C₁₉	—	2	1	no	—
120	C₂×C₄×C₁₉	—	2	1	no	—
121	C₂×C₄×C₁₉	—	2	1	no	—
122	C₂×C₄×C₁₉	—	2	1	no	—
123	C₂×C₄×C₁₉	—	2	1	no	—
124	C₂×C₄×C₁₉	—	4	2	no	—
125	C₂×C₄×C₁₉	—	4	2	no	—
126	C₂×C₄×C₁₉	—	4	2	no	—
127	C₂×C₄×C₁₉	—	4	2	no	—
128	C₂×C₄×C₁₉	#97	1	2	yes	—
129	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₉	#117	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	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
136	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
137	C₂×C₄×C₁₉	#131	1	2	yes	—
138	C₂×C₄×C₁₉	#131	1	2	yes	—
139	C₂×C₄×C₁₉	#131	1	2	yes	—
140	C₂×C₄×C₁₉	#131	1	2	yes	—
141	C₈×C₁₉	—	2	4	no	—
142	C₈×C₁₉	—	4	2	no	—
143	C₁₉⋊C₈	—	2	4	no	—
144	C₁₉⋊C₈	—	4	2	no	—
145	C₁₉⋊C₈	—	4	2	no	—
146	C₁₉⋊C₈	—	4	2	no	—
147	C₂×#431	D₁₉	1	1	yes	characteristic
148	C₄×C₂²⋊C₈	—	19	1	no	—
149	C₄×C₂²⋊C₈	—	19	1	no	—
150	C₂×#322	—	19	1	no	—
151	C₂×#322	—	19	1	no	—
152	#1273	—	19	1	no	—
153	#1273	—	19	1	no	—
154	C₂²×C₁₉	—	2	2	no	—
155	C₂²×C₁₉	—	2	2	no	—
156	C₂²×C₁₉	—	2	2	no	—
157	C₂²×C₁₉	—	4	1	no	—
158	C₂²×C₁₉	—	4	1	no	—
159	C₂²×C₁₉	—	4	1	no	—
160	C₂²×C₁₉	—	4	1	no	—
161	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
162	C₂²×C₁₉	C₂×#97	1	1	yes	characteristic
163	C₂²×C₁₉	C₄×M₁₆	1	1	yes	characteristic
164	C₂²×C₁₉	#405	1	1	yes	characteristic
165	C₂²×C₁₉	#405	1	1	yes	characteristic
166	C₂²×C₁₉	#405	1	1	yes	characteristic
167	C₂²×C₁₉	#405	1	1	yes	characteristic
168	C₄×C₁₉	—	2	2	no	—
169	C₄×C₁₉	—	4	1	no	—
170	C₄×C₁₉	—	4	1	no	—
171	C₄×C₁₉	#408	1	2	yes	—
172	C₄×C₁₉	#433	1	2	yes	—
173	C₄×C₁₉	#435	1	2	yes	—
174	C₄×C₁₉	#435	1	2	yes	—
175	C₄×C₁₉	#435	1	2	yes	—
176	C₄×C₁₉	#436	1	2	yes	—
177	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
178	#431	—	2	2	no	—
179	#431	—	2	2	no	—
180	C₂×C₄⋊C₈	Dic₁₉	1	2	yes	—
181	C₂×C₂²⋊C₈	—	19	2	no	—
182	C₂×C₂²⋊C₈	—	19	2	no	—
183	C₂×C₂²⋊C₈	Dic₁₉	1	2	yes	—
184	C₂×C₄×C₈	—	19	1	no	—
185	C₂×C₄×C₈	—	19	1	no	—
186	C₂×C₄×C₈	—	19	1	no	—
187	C₂×C₄×C₈	—	19	1	no	—
188	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
189	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
190	#336	—	19	2	no	—
191	#336	—	19	2	no	—
192	#322	—	38	2	no	—
193	#322	—	38	2	no	—
194	C₂×C₁₉	—	4	2	no	—
195	C₂×C₁₉	#1451	1	1	yes	characteristic
196	C₂×C₁₉	#1452	1	1	yes	characteristic
197	C₂×C₁₉	#1453	1	1	yes	characteristic
198	C₂×C₁₉	#1454	1	1	yes	characteristic
199	C₂×C₁₉	#1750	1	1	yes	characteristic
200	C₂×C₁₉	#1752	1	1	yes	characteristic
201	C₂×C₁₉	#1753	1	1	yes	characteristic
202	C₂³×C₄	C₄×D₁₉	1	1	yes	characteristic
203	C₂³×C₄	C₄×D₁₉	1	1	yes	characteristic
204	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
205	C₂²×C₈	—	19	2	no	—
206	C₂²×C₈	—	19	2	no	—
207	C₂²×C₈	—	19	2	no	—
208	C₂²×C₈	—	19	2	no	—
209	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
210	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
211	C₂×C₄²	—	2	1	no	—
212	C₂×C₄²	—	2	1	no	—
213	C₂×C₄²	—	2	1	no	—
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
216	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
217	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
218	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
219	C₄⋊C₈	—	2	4	no	—
220	C₂²⋊C₈	—	2	4	no	—
221	C₂²⋊C₈	—	38	4	no	—
222	C₂²⋊C₈	—	38	4	no	—
223	C₄×C₈	—	2	1	no	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	—	38	1	no	—
226	C₄×C₈	—	38	1	no	—
227	C₄×C₈	—	38	1	no	—
228	C₄×C₈	—	38	1	no	—
229	C₄×C₈	—	38	1	no	—
230	C₄×C₈	—	38	1	no	—
231	C₄×C₈	—	38	1	no	—
232	C₄×C₈	—	38	1	no	—
233	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
234	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
235	C₁₉	#8149	1	1	yes	characteristic, complemented, Sylow
236	C₂⁴	C₄×Dic₁₉	1	1	yes	characteristic
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	#64515	1	4	yes	—
249	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
250	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
251	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
254	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	19	4	no	—
261	C₂×C₈	—	38	2	no	—
262	C₂×C₈	—	38	2	no	—
263	C₂×C₈	—	38	2	no	—
264	C₂×C₈	—	38	2	no	—
265	C₂×C₈	—	38	2	no	—
266	C₂×C₈	—	38	2	no	—
267	C₂×C₈	—	38	2	no	—
268	C₂×C₈	—	38	2	no	—
269	C₂×C₈	—	38	2	no	—
270	C₂×C₈	—	38	2	no	—
271	C₂×C₈	—	38	2	no	—
272	C₂×C₈	C₂²⋊Dic₁₉	1	4	yes	—
273	C₄²	—	2	1	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	2	1	no	—
276	C₄²	—	2	1	no	—
277	C₄²	—	2	1	no	—
278	C₄²	—	2	1	no	—
279	C₄²	#64541	1	1	yes	characteristic
280	C₄²	#64541	1	1	yes	characteristic
281	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
282	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
283	C₂³	—	2	1	no	—
284	C₂³	—	2	1	no	—
285	C₂³	—	2	1	no	—
286	C₂³	—	2	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	#10597272	1	2	yes	—
290	C₂³	C₂×C₄×Dic₁₉	1	1	yes	characteristic
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	#10597272	1	2	yes	—
306	C₂×C₄	C₂×#64515	1	2	yes	—
307	C₂×C₄	#10597347	1	2	yes	—
308	C₂×C₄	M₁₆×D₁₉	1	2	yes	—
309	C₂×C₄	#10597353	1	2	yes	—
310	C₂×C₄	#10597372	1	1	yes	characteristic
311	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
312	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
313	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
314	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
315	C₂×C₄	#10597394	1	2	yes	—
316	C₂×C₄	#10597394	1	2	yes	—
317	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
318	C₈	—	2	4	no	—
319	C₈	—	4	2	no	—
320	C₈	—	38	4	no	—
321	C₈	—	76	2	no	—
322	C₈	—	76	2	no	—
323	C₈	—	76	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	C₄×C₂²⋊Dic₁₉	1	1	yes	characteristic
332	C₂²	C₂×#10597272	1	1	yes	characteristic
333	C₂²	M₁₆×Dic₁₉	1	1	yes	characteristic
334	C₂²	#11926499	1	1	yes	characteristic
335	C₂²	#11926500	1	1	yes	characteristic
336	C₂²	#11926500	1	1	yes	characteristic
337	C₂²	#11926502	1	1	yes	characteristic
338	C₄	—	2	2	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	#11926511	1	2	yes	—
342	C₄	#11926605	1	2	yes	—
343	C₄	#11926613	1	2	yes	—
344	C₄	#11926615	1	2	yes	—
345	C₄	#11926615	1	2	yes	—
346	C₄	#11926620	1	2	yes	—
347	C₂	—	4	2	no	—
348	C₂	#14561925	1	1	yes	characteristic
349	C₂	#14561928	1	1	yes	characteristic
350	C₂	#14561930	1	1	yes	characteristic
351	C₂	#14561935	1	1	yes	characteristic
352	C₂	#14563401	1	1	yes	characteristic
353	C₂	#14563411	1	1	yes	characteristic
354	C₂	#14563416	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₁₉
commutator, socle	C₂³×C₁₉	C₂×C₄²
Fitting	C₂×C₁₉×#431	C₂
prime radical	C₂⁴×C₁₉	C₄²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#431	D₁₉
hypocenter	C₁₉	#8149
LCS	G ▷ C₂³×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. C₂×#11926266	C₂	semidirect
4. C₂×#11926265	C₂	semidirect
5. C₁₉⋊#1273	C₂	semidirect
6. C₁₉⋊#1273	C₂	semidirect
235. C₁₉	#8149	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₂²⋊(C₁₉⋊C₈)	2
C₂×#11926266	2
C₂×#11926265	2
C₁₉⋊#1273	2
C₂×C₁₉×#431	2
#8149	19