Subgroups — C₁₉×#13037 #18018996

Order 4864.

538 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#509	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#442	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₉×#361	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₉×#360	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1975	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#1383	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1375	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₉×#135	C₂²	1	1	yes	characteristic
9	C₂²×D₈×C₁₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₁₉	—	2	1	no	—
11	C₁₉×#509	—	2	2	no	—
12	C₁₉×#509	—	2	2	no	—
13	C₂×C₁₉×SD₃₂	C₄	1	2	yes	complemented
14	C₂×C₁₉×D₁₆	C₄	1	2	yes	complemented
15	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
17	C₁₉×#442	—	2	2	no	—
18	C₁₉×#442	—	2	2	no	—
19	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
20	C₂×C₁₉×#106	C₂²	1	1	yes	characteristic
21	C₂×C₁₉×#104	—	2	1	no	—
22	C₂×C₁₉×#102	—	2	1	no	—
23	C₁₉×#408	—	2	1	no	—
24	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
25	C₁₉×#361	—	2	2	no	—
26	C₁₉×#360	—	2	2	no	—
27	C₂×#41×C₁₉	—	2	1	no	—
28	C₂²×D₄×C₁₉	—	2	1	no	—
29	C₂³×C₄×C₁₉	—	4	1	no	—
30	C₁₉×#135	—	2	1	no	—
31	C₁₉×#135	—	2	1	no	—
32	C₁₉×#135	—	2	1	no	—
33	C₁₉×#135	—	2	1	no	—
34	C₂×Q₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic, complemented
35	C₂×SD₁₆×C₁₉	—	2	1	no	—
36	C₂×D₈×C₁₉	—	2	1	no	—
37	C₂×D₈×C₁₉	—	2	1	no	—
38	C₂×D₈×C₁₉	—	2	1	no	—
39	C₂×D₈×C₁₉	C₂×C₄	1	2	yes	—
40	C₂×D₈×C₁₉	C₂×C₄	1	1	yes	characteristic, complemented
41	C₂×D₈×C₁₉	D₄	1	2	yes	—
42	C₂×D₈×C₁₉	D₄	1	2	yes	—
43	C₂×M₁₆×C₁₉	—	2	1	no	—
44	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
46	C₄×D₄×C₁₉	—	2	4	no	—
47	C₄×D₄×C₁₉	—	4	2	no	—
48	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
49	C₂×C₂²⋊C₄×C₁₉	—	4	1	no	—
50	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
51	C₁₉×SD₃₂	—	2	4	no	—
52	C₁₉×D₁₆	—	2	4	no	—
53	C₁₉×M₃₂	—	2	1	no	—
54	C₁₉×M₃₂	—	2	1	no	—
55	C₁₉×M₃₂	—	2	1	no	—
56	C₁₉×M₃₂	—	2	1	no	—
57	C₂×C₁₆×C₁₉	—	2	1	no	—
58	C₂×C₁₆×C₁₉	C₂×C₄	1	2	yes	complemented
59	C₁₉×#108	—	2	1	no	—
60	C₁₉×#108	—	2	1	no	—
61	C₁₉×#106	—	2	1	no	—
62	C₁₉×#106	D₄	1	2	yes	—
63	C₁₉×#104	—	4	2	no	—
64	C₁₉×#102	—	2	4	no	—
65	C₁₉×C₂²⋊C₈	—	4	2	no	—
66	C₁₉×#97	—	2	1	no	—
67	C₁₉×#97	—	2	1	no	—
68	C₂⁴×C₁₉	—	4	1	no	—
69	#41×C₁₉	—	4	1	no	—
70	#41×C₁₉	—	4	1	no	—
71	#41×C₁₉	—	4	1	no	—
72	#41×C₁₉	—	4	1	no	—
73	C₂×Q₈×C₁₉	—	2	1	no	—
74	C₂×D₄×C₁₉	—	2	2	no	—
75	C₂×D₄×C₁₉	—	2	1	no	—
76	C₂×D₄×C₁₉	—	2	2	no	—
77	C₂×D₄×C₁₉	—	2	2	no	—
78	C₂×D₄×C₁₉	—	4	1	no	—
79	C₂×D₄×C₁₉	—	4	1	no	—
80	C₂×D₄×C₁₉	—	4	1	no	—
81	C₂×D₄×C₁₉	—	4	1	no	—
82	C₂²×C₄×C₁₉	—	2	1	no	—
83	C₂²×C₄×C₁₉	—	4	4	no	—
84	C₂²×C₄×C₁₉	—	4	1	no	—
85	C₂²×C₄×C₁₉	—	4	1	no	—
86	C₂²×C₄×C₁₉	—	8	4	no	—
87	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
88	Q₁₆×C₁₉	—	2	1	no	—
89	Q₁₆×C₁₉	—	2	1	no	—
90	SD₁₆×C₁₉	—	4	1	no	—
91	SD₁₆×C₁₉	—	4	1	no	—
92	D₈×C₁₉	—	2	1	no	—
93	D₈×C₁₉	—	2	2	no	—
94	D₈×C₁₉	—	2	1	no	—
95	D₈×C₁₉	—	2	2	no	—
96	D₈×C₁₉	—	2	2	no	—
97	D₈×C₁₉	C₂²⋊C₄	1	4	yes	—
98	M₁₆×C₁₉	—	4	1	no	—
99	M₁₆×C₁₉	—	4	1	no	—
100	C₂×C₈×C₁₉	—	2	1	no	—
101	C₂×C₈×C₁₉	—	2	1	no	—
102	C₂×C₈×C₁₉	—	2	1	no	—
103	C₂×C₈×C₁₉	—	4	1	no	—
104	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
109	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
110	C₄⋊C₄×C₁₉	—	2	2	no	—
111	C₄⋊C₄×C₁₉	—	4	1	no	—
112	C₂²⋊C₄×C₁₉	—	4	4	no	—
113	C₄²×C₁₉	—	2	1	no	—
114	C₄²×C₁₉	D₈	1	2	yes	—
115	C₁₆×C₁₉	—	2	2	no	—
116	C₁₆×C₁₉	—	4	1	no	—
117	#13037	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂³×C₁₉	—	4	2	no	—
119	C₂³×C₁₉	—	4	2	no	—
120	C₂³×C₁₉	—	4	2	no	—
121	C₂³×C₁₉	—	4	1	no	—
122	C₂³×C₁₉	—	8	2	no	—
123	C₂³×C₁₉	—	8	2	no	—
124	C₂³×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
125	Q₈×C₁₉	—	4	1	no	—
126	Q₈×C₁₉	—	4	1	no	—
127	D₄×C₁₉	—	2	4	no	—
128	D₄×C₁₉	—	4	1	no	—
129	D₄×C₁₉	—	4	2	no	—
130	D₄×C₁₉	—	4	1	no	—
131	D₄×C₁₉	—	4	2	no	—
132	D₄×C₁₉	—	4	2	no	—
133	D₄×C₁₉	—	8	1	no	—
134	D₄×C₁₉	—	8	1	no	—
135	C₂×C₄×C₁₉	—	2	2	no	—
136	C₂×C₄×C₁₉	—	4	2	no	—
137	C₂×C₄×C₁₉	—	4	1	no	—
138	C₂×C₄×C₁₉	—	4	2	no	—
139	C₂×C₄×C₁₉	—	8	8	no	—
140	C₂×C₄×C₁₉	—	8	2	no	—
141	C₂×C₄×C₁₉	—	8	1	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	—	4	1	no	—
150	C₈×C₁₉	—	8	1	no	—
151	C₈×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
152	C₈×C₁₉	C₄×D₄	1	1	yes	characteristic
153	C₈×C₁₉	#121	1	1	yes	characteristic
154	C₈×C₁₉	#123	1	1	yes	characteristic
155	C₂×#509	C₂×C₁₉	1	1	yes	characteristic
156	C₂×#442	C₂×C₁₉	1	1	yes	characteristic, complemented
157	C₂×#361	C₂×C₁₉	1	1	yes	characteristic, complemented
158	C₂×#360	C₂×C₁₉	1	1	yes	characteristic, complemented
159	#1975	C₂×C₁₉	1	1	yes	characteristic, complemented
160	#1383	C₂×C₁₉	1	1	yes	characteristic, complemented
161	#1375	C₂×C₁₉	1	1	yes	characteristic, complemented
162	C₂²×C₁₉	—	2	2	no	—
163	C₂²×C₁₉	—	4	4	no	—
164	C₂²×C₁₉	—	8	2	no	—
165	C₂²×C₁₉	—	8	2	no	—
166	C₂²×C₁₉	—	8	4	no	—
167	C₂²×C₁₉	—	8	4	no	—
168	C₂²×C₁₉	—	8	1	no	—
169	C₂²×C₁₉	—	8	2	no	—
170	C₂²×C₁₉	#442	1	1	yes	characteristic
171	C₂²×C₁₉	#447	1	1	yes	characteristic
172	C₂²×C₁₉	#463	1	1	yes	characteristic
173	C₄×C₁₉	—	4	2	no	—
174	C₄×C₁₉	—	8	2	no	—
175	C₄×C₁₉	—	8	1	no	—
176	C₄×C₁₉	#386	1	1	yes	characteristic
177	C₄×C₁₉	C₄×D₈	1	1	yes	characteristic
178	C₄×C₁₉	#449	1	1	yes	characteristic
179	C₄×C₁₉	#459	1	1	yes	characteristic
180	C₂×#135	C₂²×C₁₉	1	1	yes	characteristic
181	C₂²×D₈	C₂²×C₁₉	1	1	yes	characteristic
182	C₂×C₄×D₄	—	2	1	no	—
183	#509	—	2	2	no	—
184	#509	—	2	2	no	—
185	C₂×SD₃₂	C₄×C₁₉	1	2	yes	complemented
186	C₂×D₁₆	C₄×C₁₉	1	2	yes	complemented
187	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
188	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
189	#442	—	2	2	no	—
190	#442	—	2	2	no	—
191	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
192	C₂×#106	C₂²×C₁₉	1	1	yes	characteristic
193	C₂×#104	—	2	1	no	—
194	C₂×#102	—	2	1	no	—
195	#408	—	2	1	no	—
196	C₂×#97	C₂²×C₁₉	1	1	yes	characteristic
197	#361	—	2	2	no	—
198	#360	—	2	2	no	—
199	C₂×C₁₉	—	2	1	no	—
200	C₂×C₁₉	—	2	1	no	—
201	C₂×C₁₉	—	8	4	no	—
202	C₂×C₁₉	—	16	1	no	—
203	C₂×C₁₉	#1879	1	1	yes	characteristic
204	C₂×C₁₉	AGL(1,ℤ/16)	1	2	yes	—
205	C₂×#41	—	2	1	no	—
206	C₂²×D₄	—	2	1	no	—
207	C₂³×C₄	—	4	1	no	—
208	#135	—	2	1	no	—
209	#135	—	2	1	no	—
210	#135	—	2	1	no	—
211	#135	—	2	1	no	—
212	C₂×Q₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic, complemented
213	C₂×SD₁₆	—	2	1	no	—
214	C₂×D₈	—	2	1	no	—
215	C₂×D₈	—	2	1	no	—
216	C₂×D₈	—	2	1	no	—
217	C₂×D₈	C₂×C₄×C₁₉	1	2	yes	—
218	C₂×D₈	C₂×C₄×C₁₉	1	1	yes	characteristic, complemented
219	C₂×D₈	D₄×C₁₉	1	2	yes	—
220	C₂×D₈	D₄×C₁₉	1	2	yes	—
221	C₂×M₁₆	—	2	1	no	—
222	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
223	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
224	C₄×D₄	—	2	4	no	—
225	C₄×D₄	—	4	2	no	—
226	C₂×C₄⋊C₄	—	2	1	no	—
227	C₂×C₂²⋊C₄	—	4	1	no	—
228	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
229	SD₃₂	—	2	4	no	—
230	D₁₆	—	2	4	no	—
231	M₃₂	—	2	1	no	—
232	M₃₂	—	2	1	no	—
233	M₃₂	—	2	1	no	—
234	M₃₂	—	2	1	no	—
235	C₂×C₁₆	—	2	1	no	—
236	C₂×C₁₆	C₂×C₄×C₁₉	1	2	yes	complemented
237	#108	—	2	1	no	—
238	#108	—	2	1	no	—
239	#106	—	2	1	no	—
240	#106	D₄×C₁₉	1	2	yes	—
241	#104	—	4	2	no	—
242	#102	—	2	4	no	—
243	C₂²⋊C₈	—	4	2	no	—
244	#97	—	2	1	no	—
245	#97	—	2	1	no	—
246	C₁₉	#13037	1	1	yes	characteristic, direct product factor, Sylow
247	C₂⁴	—	4	1	no	—
248	#41	—	4	1	no	—
249	#41	—	4	1	no	—
250	#41	—	4	1	no	—
251	#41	—	4	1	no	—
252	C₂×Q₈	—	2	1	no	—
253	C₂×D₄	—	2	2	no	—
254	C₂×D₄	—	2	1	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂×D₄	—	4	1	no	—
260	C₂×D₄	—	4	1	no	—
261	C₂²×C₄	—	2	1	no	—
262	C₂²×C₄	—	4	4	no	—
263	C₂²×C₄	—	4	1	no	—
264	C₂²×C₄	—	4	1	no	—
265	C₂²×C₄	—	8	4	no	—
266	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
267	Q₁₆	—	2	1	no	—
268	Q₁₆	—	2	1	no	—
269	SD₁₆	—	4	1	no	—
270	SD₁₆	—	4	1	no	—
271	D₈	—	2	1	no	—
272	D₈	—	2	2	no	—
273	D₈	—	2	1	no	—
274	D₈	—	2	2	no	—
275	D₈	—	2	2	no	—
276	D₈	C₂²⋊C₄×C₁₉	1	4	yes	—
277	M₁₆	—	4	1	no	—
278	M₁₆	—	4	1	no	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	2	1	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	4	1	no	—
283	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
284	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
285	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
286	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
287	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
288	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₂²⋊C₄	—	4	4	no	—
292	C₄²	—	2	1	no	—
293	C₄²	D₈×C₁₉	1	2	yes	—
294	C₁₆	—	2	2	no	—
295	C₁₆	—	4	1	no	—
296	C₂³	—	4	2	no	—
297	C₂³	—	4	2	no	—
298	C₂³	—	4	2	no	—
299	C₂³	—	4	1	no	—
300	C₂³	—	8	2	no	—
301	C₂³	—	8	2	no	—
302	C₂³	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
303	Q₈	—	4	1	no	—
304	Q₈	—	4	1	no	—
305	D₄	—	2	4	no	—
306	D₄	—	4	1	no	—
307	D₄	—	4	2	no	—
308	D₄	—	4	1	no	—
309	D₄	—	4	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	8	1	no	—
312	D₄	—	8	1	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	8	8	no	—
318	C₂×C₄	—	8	2	no	—
319	C₂×C₄	—	8	1	no	—
320	C₂×C₄	—	8	1	no	—
321	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
322	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
323	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
324	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
325	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
326	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
327	C₈	—	4	1	no	—
328	C₈	—	8	1	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	2	no	—
334	C₂²	—	4	4	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	8	2	no	—
337	C₂²	—	8	4	no	—
338	C₂²	—	8	4	no	—
339	C₂²	—	8	2	no	—
340	C₂²	—	8	1	no	—
341	C₂²	C₁₉×#442	1	1	yes	characteristic
342	C₂²	C₁₉×#447	1	1	yes	characteristic
343	C₂²	C₁₉×#463	1	1	yes	characteristic
344	C₄	—	4	2	no	—
345	C₄	—	8	2	no	—
346	C₄	—	8	1	no	—
347	C₄	C₁₉×#386	1	1	yes	characteristic
348	C₄	C₄×D₈×C₁₉	1	1	yes	characteristic
349	C₄	C₁₉×#449	1	1	yes	characteristic
350	C₄	C₁₉×#459	1	1	yes	characteristic
351	C₂	—	2	1	no	—
352	C₂	—	2	1	no	—
353	C₂	—	8	4	no	—
354	C₂	—	16	1	no	—
355	C₂	C₁₉×#1879	1	1	yes	characteristic
356	C₂	C₁₉×AGL(1,ℤ/16)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉	#442
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₂×C₁₉×#509	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₁₉×#97 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₉×#442	C₂	semidirect
3. C₂×C₁₉×#361	C₂	semidirect
4. C₂×C₁₉×#360	C₂	semidirect
5. C₁₉×#1975	C₂	semidirect
6. C₁₉×#1383	C₂	semidirect
7. C₁₉×#1375	C₂	semidirect
13. C₂×C₁₉×SD₃₂	C₄	semidirect
14. C₂×C₁₉×D₁₆	C₄	semidirect
34. C₂×Q₁₆×C₁₉	C₂×C₄	semidirect
40. C₂×D₈×C₁₉	C₂×C₄	semidirect
58. C₂×C₁₆×C₁₉	C₂×C₄	semidirect
117. #13037	C₁₉	direct (N × H)
156. C₂×#442	C₂×C₁₉	semidirect
157. C₂×#361	C₂×C₁₉	semidirect
158. C₂×#360	C₂×C₁₉	semidirect
159. #1975	C₂×C₁₉	semidirect
160. #1383	C₂×C₁₉	semidirect
161. #1375	C₂×C₁₉	semidirect
185. C₂×SD₃₂	C₄×C₁₉	semidirect
186. C₂×D₁₆	C₄×C₁₉	semidirect
212. C₂×Q₁₆	C₂×C₄×C₁₉	semidirect
218. C₂×D₈	C₂×C₄×C₁₉	semidirect
236. C₂×C₁₆	C₂×C₄×C₁₉	semidirect
246. C₁₉	#13037	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×#509	2
C₂×C₁₉×#442	2
C₂×C₁₉×#361	2
C₂×C₁₉×#360	2
C₁₉×#1975	2
C₁₉×#1383	2
C₁₉×#1375	2
#13037	19

Subgroups — C19×#13037 #18018996

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13037 #18018996