Subgroups — C₁₉×#13028 #18018987

Order 4864.

476 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	C₂×C₁₉×#510	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#443	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#362	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#2139	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1969	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#1387	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1376	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₉×#135	C₂²	1	1	yes	characteristic
9	C₂²×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
10	C₂×C₁₉×#131	—	2	1	no	—
11	C₁₉×#510	—	2	2	no	—
12	C₁₉×#510	C₄	1	4	yes	complemented
13	C₂×C₁₉×Q₃₂	—	2	1	no	—
14	C₂×C₁₉×SD₃₂	—	2	1	no	—
15	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
17	C₁₉×#443	—	2	2	no	—
18	C₁₉×#443	—	2	2	no	—
19	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
20	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
21	C₂×C₁₉×#104	—	2	1	no	—
22	C₂×C₁₉×#103	—	2	1	no	—
23	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
24	C₁₉×#391	—	2	1	no	—
25	C₁₉×#362	—	2	2	no	—
26	C₁₉×#358	—	2	1	no	—
27	C₁₉×#357	—	2	1	no	—
28	C₂×#41×C₁₉	—	2	1	no	—
29	C₂²×Q₈×C₁₉	—	2	1	no	—
30	C₁₉×#135	—	2	1	no	—
31	C₁₉×#135	—	2	1	no	—
32	C₁₉×#135	C₂×C₄	1	2	yes	—
33	C₁₉×#135	D₄	1	2	yes	—
34	C₂×Q₁₆×C₁₉	—	2	1	no	—
35	C₂×Q₁₆×C₁₉	—	2	1	no	—
36	C₂×Q₁₆×C₁₉	—	2	1	no	—
37	C₂×Q₁₆×C₁₉	—	2	1	no	—
38	C₂×Q₁₆×C₁₉	—	2	1	no	—
39	C₂×Q₁₆×C₁₉	C₂×C₄	1	2	yes	—
40	C₂×Q₁₆×C₁₉	D₄	1	1	yes	characteristic
41	C₂×SD₁₆×C₁₉	—	2	1	no	—
42	C₂×D₈×C₁₉	D₄	1	1	yes	characteristic
43	C₁₉×#131	—	2	4	no	—
44	C₁₉×#131	—	4	2	no	—
45	C₂×M₁₆×C₁₉	—	2	1	no	—
46	C₂×M₁₆×C₁₉	—	4	1	no	—
47	C₂²×C₈×C₁₉	—	4	1	no	—
48	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
51	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
52	C₁₉×Q₃₂	—	2	4	no	—
53	C₁₉×SD₃₂	—	2	4	no	—
54	C₁₉×M₃₂	—	2	1	no	—
55	C₁₉×M₃₂	—	2	1	no	—
56	C₁₉×M₃₂	—	2	1	no	—
57	C₁₉×M₃₂	C₂×C₄	1	2	yes	—
58	C₂×C₁₆×C₁₉	—	2	1	no	—
59	C₂×C₁₆×C₁₉	—	2	1	no	—
60	C₁₉×#108	—	2	1	no	—
61	C₁₉×#108	D₄	1	2	yes	—
62	C₁₉×#107	—	2	1	no	—
63	C₁₉×#107	—	2	1	no	—
64	C₁₉×#104	—	2	4	no	—
65	C₁₉×#103	—	4	1	no	—
66	C₁₉×#103	—	4	1	no	—
67	C₄×C₈×C₁₉	—	2	1	no	—
68	C₄×C₈×C₁₉	—	2	1	no	—
69	C₁₉×#95	—	4	2	no	—
70	#41×C₁₉	—	2	2	no	—
71	#41×C₁₉	—	2	2	no	—
72	#41×C₁₉	—	4	1	no	—
73	#41×C₁₉	—	4	1	no	—
74	C₂×Q₈×C₁₉	—	2	2	no	—
75	C₂×Q₈×C₁₉	—	2	1	no	—
76	C₂×Q₈×C₁₉	—	4	1	no	—
77	C₂×Q₈×C₁₉	—	4	1	no	—
78	C₂×Q₈×C₁₉	—	4	1	no	—
79	C₂×Q₈×C₁₉	—	4	1	no	—
80	C₂×Q₈×C₁₉	—	4	1	no	—
81	C₂×D₄×C₁₉	—	2	1	no	—
82	C₂×D₄×C₁₉	—	4	1	no	—
83	C₂²×C₄×C₁₉	—	2	1	no	—
84	C₂²×C₄×C₁₉	—	4	1	no	—
85	C₂²×C₄×C₁₉	—	4	1	no	—
86	C₂²×C₄×C₁₉	—	4	1	no	—
87	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
88	Q₁₆×C₁₉	—	2	2	no	—
89	Q₁₆×C₁₉	—	2	2	no	—
90	Q₁₆×C₁₉	—	2	1	no	—
91	Q₁₆×C₁₉	—	4	1	no	—
92	Q₁₆×C₁₉	—	4	1	no	—
93	Q₁₆×C₁₉	C₂²⋊C₄	1	2	yes	—
94	SD₁₆×C₁₉	—	2	2	no	—
95	SD₁₆×C₁₉	—	2	2	no	—
96	D₈×C₁₉	—	2	1	no	—
97	D₈×C₁₉	C₂²⋊C₄	1	2	yes	—
98	M₁₆×C₁₉	—	2	2	no	—
99	M₁₆×C₁₉	—	4	4	no	—
100	M₁₆×C₁₉	—	4	1	no	—
101	C₂×C₈×C₁₉	—	2	1	no	—
102	C₂×C₈×C₁₉	—	2	1	no	—
103	C₂×C₈×C₁₉	—	4	4	no	—
104	C₂×C₈×C₁₉	—	4	1	no	—
105	C₂×C₈×C₁₉	D₈	1	1	yes	characteristic
106	C₂×C₈×C₁₉	D₈	1	1	yes	characteristic
107	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
112	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
113	C₄⋊C₄×C₁₉	—	4	1	no	—
114	C₄⋊C₄×C₁₉	—	4	1	no	—
115	C₄²×C₁₉	—	2	1	no	—
116	C₄²×C₁₉	—	2	1	no	—
117	C₁₆×C₁₉	—	2	2	no	—
118	C₁₆×C₁₉	—	4	1	no	—
119	#13028	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂³×C₁₉	—	4	1	no	—
121	C₂³×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
122	Q₈×C₁₉	—	2	2	no	—
123	Q₈×C₁₉	—	4	1	no	—
124	Q₈×C₁₉	—	4	2	no	—
125	Q₈×C₁₉	—	4	2	no	—
126	Q₈×C₁₉	—	8	1	no	—
127	Q₈×C₁₉	—	8	1	no	—
128	D₄×C₁₉	—	2	2	no	—
129	D₄×C₁₉	—	4	2	no	—
130	D₄×C₁₉	—	4	2	no	—
131	D₄×C₁₉	—	4	1	no	—
132	C₂×C₄×C₁₉	—	2	2	no	—
133	C₂×C₄×C₁₉	—	4	1	no	—
134	C₂×C₄×C₁₉	—	4	1	no	—
135	C₂×C₄×C₁₉	—	4	1	no	—
136	C₂×C₄×C₁₉	—	4	2	no	—
137	C₂×C₄×C₁₉	—	4	2	no	—
138	C₂×C₄×C₁₉	—	4	2	no	—
139	C₂×C₄×C₁₉	—	8	1	no	—
140	C₂×C₄×C₁₉	—	8	1	no	—
141	C₂×C₄×C₁₉	—	8	2	no	—
142	C₂×C₄×C₁₉	—	8	1	no	—
143	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
145	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
146	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
148	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₈×C₁₉	—	2	1	no	—
150	C₈×C₁₉	—	2	1	no	—
151	C₈×C₁₉	—	4	2	no	—
152	C₈×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
153	C₈×C₁₉	C₄×D₄	1	1	yes	characteristic
154	C₈×C₁₉	#121	1	1	yes	characteristic
155	C₈×C₁₉	#123	1	1	yes	characteristic
156	C₂×#510	C₂×C₁₉	1	1	yes	characteristic
157	C₂×#443	C₂×C₁₉	1	1	yes	characteristic
158	C₂×#362	C₂×C₁₉	1	1	yes	characteristic, complemented
159	#2139	C₂×C₁₉	1	1	yes	characteristic
160	#1969	C₂×C₁₉	1	1	yes	characteristic, complemented
161	#1387	C₂×C₁₉	1	1	yes	characteristic, complemented
162	#1376	C₂×C₁₉	1	1	yes	characteristic, complemented
163	C₂²×C₁₉	—	2	1	no	—
164	C₂²×C₁₉	—	2	1	no	—
165	C₂²×C₁₉	—	4	2	no	—
166	C₂²×C₁₉	—	8	1	no	—
167	C₂²×C₁₉	—	8	1	no	—
168	C₂²×C₁₉	#442	1	1	yes	characteristic
169	C₂²×C₁₉	#449	1	1	yes	characteristic
170	C₂²×C₁₉	#459	1	1	yes	characteristic
171	C₄×C₁₉	—	4	2	no	—
172	C₄×C₁₉	—	4	2	no	—
173	C₄×C₁₉	—	8	2	no	—
174	C₄×C₁₉	—	16	1	no	—
175	C₄×C₁₉	#386	1	1	yes	characteristic
176	C₄×C₁₉	C₄×D₈	1	1	yes	characteristic
177	C₄×C₁₉	#447	1	1	yes	characteristic
178	C₄×C₁₉	#463	1	1	yes	characteristic
179	C₂×#135	C₂²×C₁₉	1	1	yes	characteristic
180	C₂²×Q₁₆	C₂²×C₁₉	1	1	yes	characteristic
181	C₂×#131	—	2	1	no	—
182	#510	—	2	2	no	—
183	#510	C₄×C₁₉	1	4	yes	complemented
184	C₂×Q₃₂	—	2	1	no	—
185	C₂×SD₃₂	—	2	1	no	—
186	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
187	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
188	#443	—	2	2	no	—
189	#443	—	2	2	no	—
190	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
191	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
192	C₂×#104	—	2	1	no	—
193	C₂×#103	—	2	1	no	—
194	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
195	#391	—	2	1	no	—
196	#362	—	2	2	no	—
197	#358	—	2	1	no	—
198	#357	—	2	1	no	—
199	C₂×C₁₉	—	2	1	no	—
200	C₂×C₁₉	—	2	1	no	—
201	C₂×C₁₉	—	8	2	no	—
202	C₂×C₁₉	#1879	1	1	yes	characteristic
203	C₂×C₁₉	#2212	1	1	yes	characteristic
204	C₂×C₁₉	#2214	1	1	yes	characteristic
205	C₂×#41	—	2	1	no	—
206	C₂²×Q₈	—	2	1	no	—
207	#135	—	2	1	no	—
208	#135	—	2	1	no	—
209	#135	C₂×C₄×C₁₉	1	2	yes	—
210	#135	D₄×C₁₉	1	2	yes	—
211	C₂×Q₁₆	—	2	1	no	—
212	C₂×Q₁₆	—	2	1	no	—
213	C₂×Q₁₆	—	2	1	no	—
214	C₂×Q₁₆	—	2	1	no	—
215	C₂×Q₁₆	—	2	1	no	—
216	C₂×Q₁₆	C₂×C₄×C₁₉	1	2	yes	—
217	C₂×Q₁₆	D₄×C₁₉	1	1	yes	characteristic
218	C₂×SD₁₆	—	2	1	no	—
219	C₂×D₈	D₄×C₁₉	1	1	yes	characteristic
220	#131	—	2	4	no	—
221	#131	—	4	2	no	—
222	C₂×M₁₆	—	2	1	no	—
223	C₂×M₁₆	—	4	1	no	—
224	C₂²×C₈	—	4	1	no	—
225	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
226	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
227	C₂×C₄⋊C₄	—	2	1	no	—
228	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
229	Q₃₂	—	2	4	no	—
230	SD₃₂	—	2	4	no	—
231	M₃₂	—	2	1	no	—
232	M₃₂	—	2	1	no	—
233	M₃₂	—	2	1	no	—
234	M₃₂	C₂×C₄×C₁₉	1	2	yes	—
235	C₂×C₁₆	—	2	1	no	—
236	C₂×C₁₆	—	2	1	no	—
237	#108	—	2	1	no	—
238	#108	D₄×C₁₉	1	2	yes	—
239	#107	—	2	1	no	—
240	#107	—	2	1	no	—
241	#104	—	2	4	no	—
242	#103	—	4	1	no	—
243	#103	—	4	1	no	—
244	C₄×C₈	—	2	1	no	—
245	C₄×C₈	—	2	1	no	—
246	#95	—	4	2	no	—
247	C₁₉	#13028	1	1	yes	characteristic, direct product factor, Sylow
248	#41	—	2	2	no	—
249	#41	—	2	2	no	—
250	#41	—	4	1	no	—
251	#41	—	4	1	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	—	2	1	no	—
254	C₂×Q₈	—	4	1	no	—
255	C₂×Q₈	—	4	1	no	—
256	C₂×Q₈	—	4	1	no	—
257	C₂×Q₈	—	4	1	no	—
258	C₂×Q₈	—	4	1	no	—
259	C₂×D₄	—	2	1	no	—
260	C₂×D₄	—	4	1	no	—
261	C₂²×C₄	—	2	1	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	4	1	no	—
264	C₂²×C₄	—	4	1	no	—
265	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
266	Q₁₆	—	2	2	no	—
267	Q₁₆	—	2	2	no	—
268	Q₁₆	—	2	1	no	—
269	Q₁₆	—	4	1	no	—
270	Q₁₆	—	4	1	no	—
271	Q₁₆	C₂²⋊C₄×C₁₉	1	2	yes	—
272	SD₁₆	—	2	2	no	—
273	SD₁₆	—	2	2	no	—
274	D₈	—	2	1	no	—
275	D₈	C₂²⋊C₄×C₁₉	1	2	yes	—
276	M₁₆	—	2	2	no	—
277	M₁₆	—	4	4	no	—
278	M₁₆	—	4	1	no	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	2	1	no	—
281	C₂×C₈	—	4	4	no	—
282	C₂×C₈	—	4	1	no	—
283	C₂×C₈	D₈×C₁₉	1	1	yes	characteristic
284	C₂×C₈	D₈×C₁₉	1	1	yes	characteristic
285	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
286	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
287	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
288	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
289	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
290	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄²	—	2	1	no	—
294	C₄²	—	2	1	no	—
295	C₁₆	—	2	2	no	—
296	C₁₆	—	4	1	no	—
297	C₂³	—	4	1	no	—
298	C₂³	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
299	Q₈	—	2	2	no	—
300	Q₈	—	4	1	no	—
301	Q₈	—	4	2	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	8	1	no	—
304	Q₈	—	8	1	no	—
305	D₄	—	2	2	no	—
306	D₄	—	4	2	no	—
307	D₄	—	4	2	no	—
308	D₄	—	4	1	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	8	1	no	—
317	C₂×C₄	—	8	1	no	—
318	C₂×C₄	—	8	2	no	—
319	C₂×C₄	—	8	1	no	—
320	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
321	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
322	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
323	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
324	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
325	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
326	C₈	—	2	1	no	—
327	C₈	—	2	1	no	—
328	C₈	—	4	2	no	—
329	C₈	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
330	C₈	C₄×D₄×C₁₉	1	1	yes	characteristic
331	C₈	C₁₉×#121	1	1	yes	characteristic
332	C₈	C₁₉×#123	1	1	yes	characteristic
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	4	2	no	—
336	C₂²	—	8	1	no	—
337	C₂²	—	8	1	no	—
338	C₂²	C₁₉×#442	1	1	yes	characteristic
339	C₂²	C₁₉×#449	1	1	yes	characteristic
340	C₂²	C₁₉×#459	1	1	yes	characteristic
341	C₄	—	4	2	no	—
342	C₄	—	4	2	no	—
343	C₄	—	8	2	no	—
344	C₄	—	16	1	no	—
345	C₄	C₁₉×#386	1	1	yes	characteristic
346	C₄	C₄×D₈×C₁₉	1	1	yes	characteristic
347	C₄	C₁₉×#447	1	1	yes	characteristic
348	C₄	C₁₉×#463	1	1	yes	characteristic
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	—	8	2	no	—
352	C₂	C₁₉×#1879	1	1	yes	characteristic
353	C₂	C₁₉×#2212	1	1	yes	characteristic
354	C₂	C₁₉×#2214	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#459
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₂×C₁₉×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₁₉×#362	C₂	semidirect
5. C₁₉×#1969	C₂	semidirect
6. C₁₉×#1387	C₂	semidirect
7. C₁₉×#1376	C₂	semidirect
12. C₁₉×#510	C₄	semidirect
119. #13028	C₁₉	direct (N × H)
158. C₂×#362	C₂×C₁₉	semidirect
160. #1969	C₂×C₁₉	semidirect
161. #1387	C₂×C₁₉	semidirect
162. #1376	C₂×C₁₉	semidirect
183. #510	C₄×C₁₉	semidirect
247. C₁₉	#13028	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₉×#510	2
C₂×C₁₉×#443	2
C₂×C₁₉×#362	2
C₁₉×#2139	2
C₁₉×#1969	2
C₁₉×#1387	2
C₁₉×#1376	2
#13028	19

Subgroups — C19×#13028 #18018987

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13028 #18018987