Subgroups — C₁₉×#13035 #18018994

Order 4864.

456 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	C₂×C₁₉×#510	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2957	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#362	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#2143	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1979	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#1385	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1378	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	C₁₉×#568	—	2	1	no	—
11	C₁₉×#510	—	2	2	no	—
12	C₁₉×#510	—	2	2	no	—
13	C₂×C₁₉×Q₃₂	C₄	1	2	yes	complemented
14	C₂×C₁₉×SD₃₂	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×#444	—	2	2	no	—
18	C₁₉×#443	—	2	2	no	—
19	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
20	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
21	C₁₉×#421	—	2	1	no	—
22	C₁₉×#419	—	2	1	no	—
23	C₂×C₁₉×#101	—	2	1	no	—
24	C₁₉×#405	C₂²	1	1	yes	characteristic
25	C₁₉×#362	—	2	2	no	—
26	C₁₉×#359	—	2	2	no	—
27	C₂×#41×C₁₉	—	2	1	no	—
28	C₂²×Q₈×C₁₉	—	2	1	no	—
29	C₁₉×#135	—	2	1	no	—
30	C₁₉×#135	—	2	1	no	—
31	C₁₉×#135	D₄	1	2	yes	—
32	C₁₉×#135	D₄	1	2	yes	—
33	C₂×Q₁₆×C₁₉	—	2	1	no	—
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	C₂×C₄	1	2	yes	—
39	C₂×Q₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
40	C₂×SD₁₆×C₁₉	—	2	1	no	—
41	C₂×D₈×C₁₉	C₂×C₄	1	1	yes	characteristic
42	C₁₉×#131	—	2	4	no	—
43	C₁₉×#131	—	4	2	no	—
44	C₂×M₁₆×C₁₉	—	2	1	no	—
45	C₂×M₁₆×C₁₉	—	2	1	no	—
46	C₂×M₁₆×C₁₉	—	4	2	no	—
47	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
49	C₁₉×#117	D₄	1	1	yes	characteristic
50	C₁₉×Q₃₂	—	2	4	no	—
51	C₁₉×SD₃₂	—	2	4	no	—
52	C₁₉×M₃₂	—	2	1	no	—
53	C₁₉×M₃₂	—	2	1	no	—
54	C₁₉×M₃₂	—	2	1	no	—
55	C₁₉×M₃₂	—	2	1	no	—
56	C₂×C₁₆×C₁₉	—	2	1	no	—
57	C₂×C₁₆×C₁₉	C₂×C₄	1	2	yes	—
58	C₁₉×#108	—	2	1	no	—
59	C₁₉×#108	—	2	1	no	—
60	C₁₉×#108	—	2	1	no	—
61	C₁₉×#108	D₄	1	2	yes	—
62	C₁₉×#104	—	2	4	no	—
63	C₁₉×#103	—	4	2	no	—
64	C₁₉×#101	—	4	2	no	—
65	C₁₉×#97	—	2	1	no	—
66	C₄×C₈×C₁₉	—	2	1	no	—
67	#41×C₁₉	—	2	2	no	—
68	#41×C₁₉	—	2	2	no	—
69	#41×C₁₉	—	4	1	no	—
70	#41×C₁₉	—	4	1	no	—
71	C₂×Q₈×C₁₉	—	2	1	no	—
72	C₂×Q₈×C₁₉	—	2	2	no	—
73	C₂×Q₈×C₁₉	—	4	1	no	—
74	C₂×Q₈×C₁₉	—	4	1	no	—
75	C₂×Q₈×C₁₉	—	4	1	no	—
76	C₂×Q₈×C₁₉	—	4	1	no	—
77	C₂×Q₈×C₁₉	—	4	1	no	—
78	C₂×D₄×C₁₉	—	2	1	no	—
79	C₂×D₄×C₁₉	—	4	1	no	—
80	C₂²×C₄×C₁₉	—	4	1	no	—
81	C₂²×C₄×C₁₉	—	4	1	no	—
82	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
83	Q₁₆×C₁₉	—	2	2	no	—
84	Q₁₆×C₁₉	—	2	1	no	—
85	Q₁₆×C₁₉	—	2	2	no	—
86	Q₁₆×C₁₉	—	4	1	no	—
87	Q₁₆×C₁₉	—	4	1	no	—
88	Q₁₆×C₁₉	C₂²⋊C₄	1	2	yes	—
89	SD₁₆×C₁₉	—	2	2	no	—
90	SD₁₆×C₁₉	—	2	2	no	—
91	D₈×C₁₉	—	2	1	no	—
92	D₈×C₁₉	C₂²⋊C₄	1	2	yes	—
93	M₁₆×C₁₉	—	2	1	no	—
94	M₁₆×C₁₉	—	2	2	no	—
95	M₁₆×C₁₉	—	4	4	no	—
96	M₁₆×C₁₉	—	4	1	no	—
97	M₁₆×C₁₉	—	4	1	no	—
98	M₁₆×C₁₉	—	4	1	no	—
99	M₁₆×C₁₉	D₈	1	2	yes	—
100	C₂×C₈×C₁₉	—	2	1	no	—
101	C₂×C₈×C₁₉	—	4	4	no	—
102	C₂×C₈×C₁₉	—	4	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	1	no	—
111	C₂²⋊C₄×C₁₉	—	2	1	no	—
112	C₄²×C₁₉	—	2	1	no	—
113	C₁₆×C₁₉	—	2	2	no	—
114	C₁₆×C₁₉	—	4	1	no	—
115	#13035	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂³×C₁₉	—	4	1	no	—
117	C₂³×C₁₉	#135	1	1	yes	characteristic
118	Q₈×C₁₉	—	2	2	no	—
119	Q₈×C₁₉	—	4	1	no	—
120	Q₈×C₁₉	—	4	2	no	—
121	Q₈×C₁₉	—	4	2	no	—
122	Q₈×C₁₉	—	8	1	no	—
123	Q₈×C₁₉	—	8	1	no	—
124	D₄×C₁₉	—	2	2	no	—
125	D₄×C₁₉	—	4	2	no	—
126	D₄×C₁₉	—	4	2	no	—
127	D₄×C₁₉	—	4	1	no	—
128	C₂×C₄×C₁₉	—	4	1	no	—
129	C₂×C₄×C₁₉	—	4	2	no	—
130	C₂×C₄×C₁₉	—	4	2	no	—
131	C₂×C₄×C₁₉	—	4	2	no	—
132	C₂×C₄×C₁₉	—	4	1	no	—
133	C₂×C₄×C₁₉	—	8	1	no	—
134	C₂×C₄×C₁₉	—	8	1	no	—
135	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
137	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
138	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₈×C₁₉	—	2	2	no	—
142	C₈×C₁₉	—	4	2	no	—
143	C₈×C₁₉	—	8	1	no	—
144	C₈×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₈×C₁₉	C₄×D₄	1	1	yes	characteristic
146	C₈×C₁₉	#121	1	1	yes	characteristic
147	C₈×C₁₉	#123	1	1	yes	characteristic
148	C₂×#510	C₂×C₁₉	1	1	yes	characteristic
149	#2957	C₂×C₁₉	1	1	yes	characteristic
150	C₂×#362	C₂×C₁₉	1	1	yes	characteristic, complemented
151	#2143	C₂×C₁₉	1	1	yes	characteristic
152	#1979	C₂×C₁₉	1	1	yes	characteristic, complemented
153	#1385	C₂×C₁₉	1	1	yes	characteristic, complemented
154	#1378	C₂×C₁₉	1	1	yes	characteristic, complemented
155	C₂²×C₁₉	—	4	1	no	—
156	C₂²×C₁₉	—	4	2	no	—
157	C₂²×C₁₉	—	16	1	no	—
158	C₂²×C₁₉	C₄×D₈	1	1	yes	characteristic
159	C₂²×C₁₉	#449	1	1	yes	characteristic
160	C₂²×C₁₉	#463	1	1	yes	characteristic
161	C₄×C₁₉	—	4	2	no	—
162	C₄×C₁₉	—	8	1	no	—
163	C₄×C₁₉	—	8	2	no	—
164	C₄×C₁₉	#386	1	1	yes	characteristic
165	C₄×C₁₉	#442	1	1	yes	characteristic
166	C₄×C₁₉	#447	1	1	yes	characteristic
167	C₄×C₁₉	#459	1	1	yes	characteristic
168	C₂×#135	C₂²×C₁₉	1	1	yes	characteristic
169	C₂²×Q₁₆	C₂²×C₁₉	1	1	yes	characteristic
170	#568	—	2	1	no	—
171	#510	—	2	2	no	—
172	#510	—	2	2	no	—
173	C₂×Q₃₂	C₄×C₁₉	1	2	yes	complemented
174	C₂×SD₃₂	C₄×C₁₉	1	2	yes	complemented
175	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
176	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
177	#444	—	2	2	no	—
178	#443	—	2	2	no	—
179	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
180	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
181	#421	—	2	1	no	—
182	#419	—	2	1	no	—
183	C₂×#101	—	2	1	no	—
184	#405	C₂²×C₁₉	1	1	yes	characteristic
185	#362	—	2	2	no	—
186	#359	—	2	2	no	—
187	C₂×C₁₉	—	2	1	no	—
188	C₂×C₁₉	—	2	1	no	—
189	C₂×C₁₉	—	2	1	no	—
190	C₂×C₁₉	—	8	2	no	—
191	C₂×C₁₉	#1879	1	1	yes	characteristic
192	C₂×#41	—	2	1	no	—
193	C₂²×Q₈	—	2	1	no	—
194	#135	—	2	1	no	—
195	#135	—	2	1	no	—
196	#135	D₄×C₁₉	1	2	yes	—
197	#135	D₄×C₁₉	1	2	yes	—
198	C₂×Q₁₆	—	2	1	no	—
199	C₂×Q₁₆	—	2	1	no	—
200	C₂×Q₁₆	—	2	1	no	—
201	C₂×Q₁₆	—	2	1	no	—
202	C₂×Q₁₆	—	2	1	no	—
203	C₂×Q₁₆	C₂×C₄×C₁₉	1	2	yes	—
204	C₂×Q₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
205	C₂×SD₁₆	—	2	1	no	—
206	C₂×D₈	C₂×C₄×C₁₉	1	1	yes	characteristic
207	#131	—	2	4	no	—
208	#131	—	4	2	no	—
209	C₂×M₁₆	—	2	1	no	—
210	C₂×M₁₆	—	2	1	no	—
211	C₂×M₁₆	—	4	2	no	—
212	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
213	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
214	#117	D₄×C₁₉	1	1	yes	characteristic
215	Q₃₂	—	2	4	no	—
216	SD₃₂	—	2	4	no	—
217	M₃₂	—	2	1	no	—
218	M₃₂	—	2	1	no	—
219	M₃₂	—	2	1	no	—
220	M₃₂	—	2	1	no	—
221	C₂×C₁₆	—	2	1	no	—
222	C₂×C₁₆	C₂×C₄×C₁₉	1	2	yes	—
223	#108	—	2	1	no	—
224	#108	—	2	1	no	—
225	#108	—	2	1	no	—
226	#108	D₄×C₁₉	1	2	yes	—
227	#104	—	2	4	no	—
228	#103	—	4	2	no	—
229	#101	—	4	2	no	—
230	#97	—	2	1	no	—
231	C₄×C₈	—	2	1	no	—
232	C₁₉	#13035	1	1	yes	characteristic, direct product factor, Sylow
233	#41	—	2	2	no	—
234	#41	—	2	2	no	—
235	#41	—	4	1	no	—
236	#41	—	4	1	no	—
237	C₂×Q₈	—	2	1	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×D₄	—	2	1	no	—
245	C₂×D₄	—	4	1	no	—
246	C₂²×C₄	—	4	1	no	—
247	C₂²×C₄	—	4	1	no	—
248	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
249	Q₁₆	—	2	2	no	—
250	Q₁₆	—	2	1	no	—
251	Q₁₆	—	2	2	no	—
252	Q₁₆	—	4	1	no	—
253	Q₁₆	—	4	1	no	—
254	Q₁₆	C₂²⋊C₄×C₁₉	1	2	yes	—
255	SD₁₆	—	2	2	no	—
256	SD₁₆	—	2	2	no	—
257	D₈	—	2	1	no	—
258	D₈	C₂²⋊C₄×C₁₉	1	2	yes	—
259	M₁₆	—	2	1	no	—
260	M₁₆	—	2	2	no	—
261	M₁₆	—	4	4	no	—
262	M₁₆	—	4	1	no	—
263	M₁₆	—	4	1	no	—
264	M₁₆	—	4	1	no	—
265	M₁₆	D₈×C₁₉	1	2	yes	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	4	4	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
271	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
272	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
273	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
274	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
275	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
276	C₄⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	2	1	no	—
278	C₄²	—	2	1	no	—
279	C₁₆	—	2	2	no	—
280	C₁₆	—	4	1	no	—
281	C₂³	—	4	1	no	—
282	C₂³	C₁₉×#135	1	1	yes	characteristic
283	Q₈	—	2	2	no	—
284	Q₈	—	4	1	no	—
285	Q₈	—	4	2	no	—
286	Q₈	—	4	2	no	—
287	Q₈	—	8	1	no	—
288	Q₈	—	8	1	no	—
289	D₄	—	2	2	no	—
290	D₄	—	4	2	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	8	1	no	—
299	C₂×C₄	—	8	1	no	—
300	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
301	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
302	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
303	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
304	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
305	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	—	4	2	no	—
308	C₈	—	8	1	no	—
309	C₈	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
310	C₈	C₄×D₄×C₁₉	1	1	yes	characteristic
311	C₈	C₁₉×#121	1	1	yes	characteristic
312	C₈	C₁₉×#123	1	1	yes	characteristic
313	C₂²	—	4	1	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	16	1	no	—
316	C₂²	C₄×D₈×C₁₉	1	1	yes	characteristic
317	C₂²	C₁₉×#449	1	1	yes	characteristic
318	C₂²	C₁₉×#463	1	1	yes	characteristic
319	C₄	—	4	2	no	—
320	C₄	—	8	1	no	—
321	C₄	—	8	2	no	—
322	C₄	C₁₉×#386	1	1	yes	characteristic
323	C₄	C₁₉×#442	1	1	yes	characteristic
324	C₄	C₁₉×#447	1	1	yes	characteristic
325	C₄	C₁₉×#459	1	1	yes	characteristic
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	2	1	no	—
329	C₂	—	8	2	no	—
330	C₂	C₁₉×#1879	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₉	#1879
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 ◁ C₂×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#405 ◁ 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×#1979	C₂	semidirect
6. C₁₉×#1385	C₂	semidirect
7. C₁₉×#1378	C₂	semidirect
13. C₂×C₁₉×Q₃₂	C₄	semidirect
14. C₂×C₁₉×SD₃₂	C₄	semidirect
115. #13035	C₁₉	direct (N × H)
150. C₂×#362	C₂×C₁₉	semidirect
152. #1979	C₂×C₁₉	semidirect
153. #1385	C₂×C₁₉	semidirect
154. #1378	C₂×C₁₉	semidirect
173. C₂×Q₃₂	C₄×C₁₉	semidirect
174. C₂×SD₃₂	C₄×C₁₉	semidirect
232. C₁₉	#13035	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
C₁₉×#2957	2
C₂×C₁₉×#362	2
C₁₉×#2143	2
C₁₉×#1979	2
C₁₉×#1385	2
C₁₉×#1378	2
#13035	19

Subgroups — C19×#13035 #18018994

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13035 #18018994