Subgroups — #18076505

Order 4864.

558 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	#14563928	C₂	1	1	yes	maximal, characteristic
2	#14563812	C₂	1	2	yes	maximal, complemented
3	C₄×#10597291	C₂	1	1	yes	maximal, characteristic
4	#14561482	C₂	1	2	yes	maximal, complemented
5	C₁₉×#2972	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×Dic₁₉⋊C₈	—	2	2	no	—
8	C₂×#10597291	C₄	1	1	yes	characteristic
9	C₂×#10597291	C₄	1	1	yes	characteristic
10	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
11	C₂×C₂²⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₄×C₂²⋊Dic₁₉	C₂²	1	2	yes	—
13	#11926311	C₄	1	2	yes	—
14	#11926299	—	2	1	no	—
15	#11926298	—	2	2	no	—
16	C₂×C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
17	C₈×D₄×C₁₉	—	2	2	no	—
18	C₈×D₄×C₁₉	—	2	2	no	—
19	C₁₉×#431	—	2	1	no	—
20	C₂×C₁₉×C₂²⋊C₈	—	2	1	no	—
21	C₂×C₁₉×C₂²⋊C₈	C₂²	1	2	yes	—
22	C₂²×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₁₉	—	2	1	no	—
26	C₂²×C₈×C₁₉	—	2	1	no	—
27	C₂²×C₈×C₁₉	D₄	1	2	yes	—
28	C₂²×C₈×C₁₉	D₄	1	2	yes	—
29	C₄×D₄×C₁₉	—	2	2	no	—
30	C₄×D₄×C₁₉	—	2	2	no	—
31	C₂×C₄⋊C₄×C₁₉	Q₈	1	1	yes	characteristic
32	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₁₉	Q₈	1	1	yes	characteristic
34	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
35	C₂×C₂²⋊Dic₁₉	C₂×C₄	1	2	yes	—
36	C₂×C₂²⋊Dic₁₉	C₂×C₄	1	2	yes	—
37	C₂×C₄×Dic₁₉	—	2	2	no	—
38	C₂²×C₁₉⋊C₈	—	2	2	no	—
39	C₂²×C₁₉⋊C₈	—	2	1	no	—
40	C₂²×C₁₉⋊C₈	—	2	1	no	—
41	C₂²×C₁₉⋊C₈	—	4	2	no	—
42	C₁₉×C₄⋊C₈	—	2	1	no	—
43	C₁₉×C₄⋊C₈	—	2	1	no	—
44	C₁₉×C₂²⋊C₈	—	2	2	no	—
45	C₁₉×C₂²⋊C₈	—	2	2	no	—
46	C₁₉×C₂²⋊C₈	—	2	2	no	—
47	C₄×C₈×C₁₉	—	2	2	no	—
48	#10597291	—	2	2	no	—
49	#10597291	C₈	1	4	yes	complemented
50	C₁₉⋊#95	—	2	2	no	—
51	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
52	C₂²⋊(C₁₉⋊C₈)	—	4	2	no	—
53	Dic₁₉⋊C₈	—	4	4	no	—
54	C₂⁴×C₁₉	C₄⋊C₄	1	1	yes	characteristic
55	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×D₄×C₁₉	—	2	2	no	—
57	C₂×D₄×C₁₉	—	2	2	no	—
58	C₂×D₄×C₁₉	C₂×C₈	1	2	yes	—
59	C₂×D₄×C₁₉	M₁₆	1	2	yes	—
60	C₂²×C₄×C₁₉	—	2	1	no	—
61	C₂²×C₄×C₁₉	—	2	1	no	—
62	C₂²×C₄×C₁₉	—	4	1	no	—
63	C₂²×C₄×C₁₉	—	4	1	no	—
64	C₂²×C₄×C₁₉	—	4	1	no	—
65	C₂²×C₄×C₁₉	—	4	1	no	—
66	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	#41	1	2	yes	—
73	C₂²×Dic₁₉	—	2	2	no	—
74	C₂²×Dic₁₉	—	2	2	no	—
75	C₂×C₈×C₁₉	—	2	2	no	—
76	C₂×C₈×C₁₉	—	2	2	no	—
77	C₂×C₈×C₁₉	—	4	2	no	—
78	C₂×C₈×C₁₉	—	4	2	no	—
79	C₂×C₈×C₁₉	—	8	1	no	—
80	C₂×C₈×C₁₉	—	8	1	no	—
81	C₄⋊C₄×C₁₉	—	2	2	no	—
82	C₂²⋊C₄×C₁₉	—	2	2	no	—
83	C₂²⋊C₄×C₁₉	—	2	2	no	—
84	C₄²×C₁₉	—	2	1	no	—
85	C₄²×C₁₉	—	2	1	no	—
86	C₂²⋊Dic₁₉	—	2	2	no	—
87	C₂²⋊Dic₁₉	—	2	2	no	—
88	C₂²⋊Dic₁₉	—	2	2	no	—
89	C₂²⋊Dic₁₉	C₂×C₈	1	4	yes	complemented
90	C₄×Dic₁₉	—	4	2	no	—
91	C₄×Dic₁₉	—	4	2	no	—
92	C₂×C₁₉⋊C₈	—	2	4	no	—
93	C₂×C₁₉⋊C₈	—	2	2	no	—
94	C₂×C₁₉⋊C₈	—	4	1	no	—
95	C₂×C₁₉⋊C₈	—	4	2	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	#8851	—	19	1	no	maximal, Sylow
101	C₂³×C₁₉	—	2	1	no	—
102	C₂³×C₁₉	—	2	1	no	—
103	C₂³×C₁₉	—	4	1	no	—
104	C₂³×C₁₉	—	4	1	no	—
105	C₂³×C₁₉	—	4	1	no	—
106	C₂³×C₁₉	—	4	1	no	—
107	C₂³×C₁₉	C₄⋊C₈	1	2	yes	complemented
108	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂³×C₁₉	C₄×D₄	1	2	yes	—
110	C₂³×C₁₉	C₂²×C₈	1	1	yes	characteristic
111	C₂³×C₁₉	C₂×M₁₆	1	1	yes	characteristic
112	C₂³×C₁₉	#131	1	2	yes	—
113	D₄×C₁₉	—	2	4	no	—
114	D₄×C₁₉	—	2	4	no	—
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	2	1	no	—
118	C₂×C₄×C₁₉	—	2	2	no	—
119	C₂×C₄×C₁₉	—	2	1	no	—
120	C₂×C₄×C₁₉	—	2	2	no	—
121	C₂×C₄×C₁₉	—	2	1	no	—
122	C₂×C₄×C₁₉	—	2	1	no	—
123	C₂×C₄×C₁₉	—	4	1	no	—
124	C₂×C₄×C₁₉	—	4	2	no	—
125	C₂×C₄×C₁₉	—	4	1	no	—
126	C₂×C₄×C₁₉	—	4	1	no	—
127	C₂×C₄×C₁₉	—	4	1	no	—
128	C₂×C₄×C₁₉	—	4	1	no	—
129	C₂×C₄×C₁₉	—	8	1	no	—
130	C₂×C₄×C₁₉	C₄⋊C₈	1	2	yes	complemented
131	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
132	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
133	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
134	C₂×Dic₁₉	—	2	4	no	—
135	C₂×Dic₁₉	—	4	2	no	—
136	C₂×Dic₁₉	—	4	2	no	—
137	C₂×Dic₁₉	—	4	2	no	—
138	C₂×Dic₁₉	—	4	2	no	—
139	C₂×Dic₁₉	—	4	2	no	—
140	C₈×C₁₉	—	4	2	no	—
141	C₈×C₁₉	—	4	2	no	—
142	C₁₉⋊C₈	—	4	2	no	—
143	C₁₉⋊C₈	—	8	1	no	—
144	#2972	D₁₉	1	1	yes	characteristic
145	#1869	—	19	1	no	—
146	#1850	—	19	2	no	—
147	C₄×#99	—	19	1	no	—
148	#1282	—	19	2	no	—
149	C₂²×C₁₉	—	2	2	no	—
150	C₂²×C₁₉	—	2	1	no	—
151	C₂²×C₁₉	—	2	2	no	—
152	C₂²×C₁₉	—	2	1	no	—
153	C₂²×C₁₉	—	2	2	no	—
154	C₂²×C₁₉	—	4	1	no	—
155	C₂²×C₁₉	—	4	1	no	—
156	C₂²×C₁₉	—	4	2	no	—
157	C₂²×C₁₉	—	4	2	no	—
158	C₂²×C₁₉	—	4	1	no	—
159	C₂²×C₁₉	—	4	1	no	—
160	C₂²×C₁₉	—	8	1	no	—
161	C₂²×C₁₉	#385	1	1	yes	characteristic
162	C₂²×C₁₉	C₂×C₄⋊C₈	1	1	yes	characteristic
163	C₂²×C₁₉	#424	1	1	yes	characteristic
164	C₂²×C₁₉	C₈×D₄	1	2	yes	—
165	C₂²×C₁₉	#435	1	2	yes	—
166	C₄×C₁₉	—	2	2	no	—
167	C₄×C₁₉	—	2	1	no	—
168	C₄×C₁₉	—	2	2	no	—
169	C₄×C₁₉	—	4	1	no	—
170	C₄×C₁₉	—	4	1	no	—
171	C₄×C₁₉	B(4,𝔽₂)	1	1	yes	characteristic
172	C₄×C₁₉	#458	1	1	yes	characteristic
173	Dic₁₉	—	4	4	no	—
174	Dic₁₉	—	8	2	no	—
175	C₂³×C₈	—	19	1	no	—
176	C₂×C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
177	C₈×D₄	—	2	2	no	—
178	C₈×D₄	—	2	2	no	—
179	#431	—	2	1	no	—
180	C₂×C₄⋊C₈	—	38	2	no	—
181	C₂×#99	—	19	1	no	—
182	C₂×#99	—	19	1	no	—
183	C₂×C₂²⋊C₈	—	2	1	no	—
184	C₂×C₂²⋊C₈	—	19	1	no	—
185	C₂×C₂²⋊C₈	—	38	1	no	—
186	C₂×C₂²⋊C₈	C₂×D₁₉	1	2	yes	—
187	C₄×C₂²⋊C₄	—	19	2	no	—
188	#342	—	19	2	no	—
189	#336	—	38	2	no	—
190	#336	—	38	1	no	—
191	C₂×C₁₉	—	2	2	no	—
192	C₂×C₁₉	—	2	2	no	—
193	C₂×C₁₉	—	2	1	no	—
194	C₂×C₁₉	—	2	1	no	—
195	C₂×C₁₉	—	4	2	no	—
196	C₂×C₁₉	#1850	1	1	yes	characteristic
197	C₂×C₁₉	#1867	1	1	yes	characteristic
198	C₂×C₁₉	#1868	1	1	yes	characteristic
199	C₂²×D₄	C₄×D₁₉	1	1	yes	characteristic
200	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
201	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
202	C₂²×C₈	—	2	1	no	—
203	C₂²×C₈	—	2	1	no	—
204	C₂²×C₈	—	38	1	no	—
205	C₂²×C₈	—	38	2	no	—
206	C₂²×C₈	—	38	1	no	—
207	C₂²×C₈	—	76	2	no	—
208	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
209	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
210	C₄×D₄	—	2	2	no	—
211	C₄×D₄	—	2	2	no	—
212	C₂×C₄⋊C₄	Dic₃₈	1	1	yes	characteristic
213	C₂×C₂²⋊C₄	—	19	2	no	—
214	C₂×C₂²⋊C₄	—	19	2	no	—
215	C₂×C₂²⋊C₄	Dic₃₈	1	1	yes	characteristic
216	C₂×C₂²⋊C₄	C₄×D₁₉	1	1	yes	characteristic
217	C₂×C₄²	—	38	2	no	—
218	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	2	1	no	—
221	C₄⋊C₈	—	76	4	no	—
222	#99	—	19	4	no	—
223	#99	—	38	2	no	—
224	C₂²⋊C₈	—	2	2	no	—
225	C₂²⋊C₈	—	2	2	no	—
226	C₂²⋊C₈	—	2	2	no	—
227	C₂²⋊C₈	—	38	2	no	—
228	C₂²⋊C₈	—	76	2	no	—
229	C₄×C₈	—	2	2	no	—
230	#95	—	38	2	no	—
231	C₁₉	#8851	1	1	yes	characteristic, complemented, Sylow
232	C₂⁴	Dic₁₉⋊C₄	1	1	yes	characteristic
233	C₂⁴	C₂×C₄×D₁₉	1	1	yes	characteristic
234	C₂×D₄	—	2	2	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	C₈×D₁₉	1	2	yes	—
237	C₂×D₄	#64515	1	2	yes	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	—	38	2	no	—
245	C₂²×C₄	—	38	2	no	—
246	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
247	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
248	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
249	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
250	C₂²×C₄	#64541	1	2	yes	—
251	C₂²×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
252	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	4	2	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	—	8	1	no	—
258	C₂×C₈	—	8	1	no	—
259	C₂×C₈	—	38	2	no	—
260	C₂×C₈	—	38	4	no	—
261	C₂×C₈	—	76	2	no	—
262	C₂×C₈	—	76	1	no	—
263	C₂×C₈	—	76	1	no	—
264	C₂×C₈	—	76	1	no	—
265	C₂×C₈	—	76	1	no	—
266	C₂×C₈	—	76	1	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₂²⋊C₄	—	2	2	no	—
269	C₂²⋊C₄	—	2	2	no	—
270	C₂²⋊C₄	—	19	4	no	—
271	C₂²⋊C₄	—	38	2	no	—
272	C₂²⋊C₄	—	38	2	no	—
273	C₂²⋊C₄	—	38	2	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	2	1	no	—
276	C₄²	—	76	2	no	—
277	C₄²	—	76	2	no	—
278	C₂³	—	2	1	no	—
279	C₂³	—	2	1	no	—
280	C₂³	—	4	1	no	—
281	C₂³	—	4	1	no	—
282	C₂³	—	4	1	no	—
283	C₂³	—	4	1	no	—
284	C₂³	Dic₁₉⋊C₈	1	2	yes	complemented
285	C₂³	C₂×C₈×D₁₉	1	1	yes	characteristic
286	C₂³	C₂×#64515	1	1	yes	characteristic
287	C₂³	#10597347	1	2	yes	—
288	C₂³	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
289	C₂³	C₄×C₂²⋊D₁₉	1	2	yes	—
290	D₄	—	2	4	no	—
291	D₄	—	2	4	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	8	1	no	—
307	C₂×C₄	—	38	4	no	—
308	C₂×C₄	—	76	2	no	—
309	C₂×C₄	—	76	2	no	—
310	C₂×C₄	—	76	2	no	—
311	C₂×C₄	—	76	2	no	—
312	C₂×C₄	—	76	2	no	—
313	C₂×C₄	Dic₁₉⋊C₈	1	2	yes	complemented
314	C₂×C₄	#10597370	1	2	yes	—
315	C₂×C₄	#10597375	1	1	yes	characteristic
316	C₂×C₄	#10597399	1	1	yes	characteristic
317	C₈	—	4	2	no	—
318	C₈	—	4	2	no	—
319	C₈	—	76	2	no	—
320	C₈	—	152	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	2	no	—
329	C₂²	—	4	2	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	8	1	no	—
333	C₂²	#11926417	1	1	yes	characteristic
334	C₂²	C₂×Dic₁₉⋊C₈	1	1	yes	characteristic
335	C₂²	#11926570	1	1	yes	characteristic
336	C₂²	C₈×C₂²⋊D₁₉	1	2	yes	—
337	C₂²	#11926615	1	2	yes	—
338	C₄	—	2	1	no	—
339	C₄	—	2	2	no	—
340	C₄	—	2	2	no	—
341	C₄	—	4	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	76	4	no	—
344	C₄	—	152	2	no	—
345	C₄	#11926731	1	1	yes	characteristic
346	C₄	#11926734	1	1	yes	characteristic
347	C₂	—	2	2	no	—
348	C₂	—	2	2	no	—
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	—	4	2	no	—
352	C₂	#14563813	1	1	yes	characteristic
353	C₂	#14563916	1	1	yes	characteristic
354	C₂	#14563921	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597399
commutator	C₂³×C₁₉	C₂²×C₈
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₁₉×#2972	C₂
socle	C₂²×C₁₉	#385
prime radical	C₂²×D₄×C₁₉	C₂×C₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#2972	D₁₉
hypocenter	C₁₉	#8851
LCS	G ▷ C₂³×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #2972

Complements

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

N	H ≅ G/N	type
2. #14563812	C₂	semidirect
4. #14561482	C₂	semidirect
49. #10597291	C₈	semidirect
89. C₂²⋊Dic₁₉	C₂×C₈	semidirect
107. C₂³×C₁₉	C₄⋊C₈	semidirect
130. C₂×C₄×C₁₉	C₄⋊C₈	semidirect
231. C₁₉	#8851	semidirect
284. C₂³	Dic₁₉⋊C₈	semidirect
313. C₂×C₄	Dic₁₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#14563928	2
#14563812	2
C₄×#10597291	2
#14561482	2
C₁₉×#2972	2
#8851	19