Subgroups — C₁₉×#13248 #18019207

Order 4864.

394 conjugacy classes of subgroups.

The subgroup lattice has 326 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×#2155	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#2143	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#2141	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#2110	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#1979	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1976	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₉×#135	C₂²	1	1	yes	characteristic
9	C₂²×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
10	C₁₉×#510	—	2	2	no	—
11	C₁₉×#510	—	2	2	no	—
12	C₂×C₁₉×Q₃₂	C₄	1	2	yes	complemented
13	C₂×C₁₉×SD₃₂	C₄	1	2	yes	complemented
14	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
17	C₁₉×#427	C₂²	1	1	yes	characteristic
18	C₁₉×#421	—	2	1	no	—
19	C₁₉×#419	—	2	1	no	—
20	C₂×C₁₉×#101	—	2	1	no	—
21	C₁₉×#410	—	2	1	no	—
22	C₁₉×#405	C₂²	1	1	yes	characteristic
23	C₁₉×#365	—	2	2	no	—
24	C₁₉×#359	—	2	2	no	—
25	C₁₉×#358	—	2	2	no	—
26	C₁₉×#347	—	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	—	2	1	no	—
32	C₁₉×#135	—	2	1	no	—
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	C₂×M₁₆×C₁₉	—	2	1	no	—
43	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
45	C₁₉×#117	—	2	1	no	—
46	C₁₉×#117	D₄	1	1	yes	characteristic
47	C₁₉×Q₃₂	—	2	4	no	—
48	C₁₉×SD₃₂	—	2	4	no	—
49	C₁₉×M₃₂	—	2	1	no	—
50	C₁₉×M₃₂	—	2	1	no	—
51	C₁₉×M₃₂	D₄	1	2	yes	—
52	C₁₉×M₃₂	D₄	1	2	yes	—
53	C₂×C₁₆×C₁₉	C₂×C₄	1	2	yes	—
54	C₂×C₁₆×C₁₉	D₄	1	2	yes	—
55	C₁₉×#108	—	2	1	no	—
56	C₁₉×#108	—	2	1	no	—
57	C₁₉×#107	—	2	1	no	—
58	C₁₉×#106	—	2	1	no	—
59	C₁₉×#104	—	4	2	no	—
60	C₁₉×#103	—	4	2	no	—
61	C₁₉×#101	—	4	2	no	—
62	C₁₉×#99	—	4	2	no	—
63	C₁₉×#97	—	2	1	no	—
64	C₄×C₈×C₁₉	—	2	1	no	—
65	#41×C₁₉	—	4	1	no	—
66	#41×C₁₉	—	4	1	no	—
67	#41×C₁₉	—	4	1	no	—
68	#41×C₁₉	—	4	1	no	—
69	C₂×Q₈×C₁₉	—	2	1	no	—
70	C₂×Q₈×C₁₉	—	2	2	no	—
71	C₂×Q₈×C₁₉	—	4	1	no	—
72	C₂×Q₈×C₁₉	—	4	1	no	—
73	C₂×Q₈×C₁₉	—	4	1	no	—
74	C₂×Q₈×C₁₉	—	4	1	no	—
75	C₂×Q₈×C₁₉	—	4	1	no	—
76	C₂×D₄×C₁₉	—	2	1	no	—
77	C₂×D₄×C₁₉	—	4	1	no	—
78	C₂²×C₄×C₁₉	—	4	1	no	—
79	C₂²×C₄×C₁₉	—	4	1	no	—
80	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
81	Q₁₆×C₁₉	—	2	2	no	—
82	Q₁₆×C₁₉	—	2	1	no	—
83	Q₁₆×C₁₉	—	2	2	no	—
84	Q₁₆×C₁₉	—	2	1	no	—
85	Q₁₆×C₁₉	—	4	1	no	—
86	Q₁₆×C₁₉	—	4	1	no	—
87	SD₁₆×C₁₉	—	4	1	no	—
88	SD₁₆×C₁₉	—	4	1	no	—
89	D₈×C₁₉	—	2	1	no	—
90	D₈×C₁₉	—	2	1	no	—
91	M₁₆×C₁₉	—	2	1	no	—
92	M₁₆×C₁₉	—	2	1	no	—
93	M₁₆×C₁₉	—	4	1	no	—
94	M₁₆×C₁₉	—	4	1	no	—
95	C₂×C₈×C₁₉	—	2	1	no	—
96	C₂×C₈×C₁₉	—	4	1	no	—
97	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
102	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
103	C₄⋊C₄×C₁₉	—	2	1	no	—
104	C₄⋊C₄×C₁₉	—	4	1	no	—
105	C₂²⋊C₄×C₁₉	—	2	1	no	—
106	C₂²⋊C₄×C₁₉	—	4	1	no	—
107	C₄²×C₁₉	—	2	1	no	—
108	C₄²×C₁₉	—	4	1	no	—
109	C₁₆×C₁₉	—	2	2	no	—
110	C₁₆×C₁₉	C₂²⋊C₄	1	4	yes	—
111	#13248	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂³×C₁₉	—	4	1	no	—
113	C₂³×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
114	Q₈×C₁₉	—	4	1	no	—
115	Q₈×C₁₉	—	4	2	no	—
116	Q₈×C₁₉	—	4	2	no	—
117	Q₈×C₁₉	—	4	1	no	—
118	Q₈×C₁₉	—	8	1	no	—
119	Q₈×C₁₉	—	8	1	no	—
120	D₄×C₁₉	—	4	1	no	—
121	D₄×C₁₉	—	4	1	no	—
122	D₄×C₁₉	—	8	1	no	—
123	D₄×C₁₉	—	8	1	no	—
124	C₂×C₄×C₁₉	—	4	1	no	—
125	C₂×C₄×C₁₉	—	4	2	no	—
126	C₂×C₄×C₁₉	—	4	1	no	—
127	C₂×C₄×C₁₉	—	8	1	no	—
128	C₂×C₄×C₁₉	—	8	1	no	—
129	C₂×C₄×C₁₉	—	8	1	no	—
130	C₂×C₄×C₁₉	—	8	1	no	—
131	C₂×C₄×C₁₉	—	8	1	no	—
132	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
134	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
135	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
137	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₈×C₁₉	—	4	1	no	—
139	C₈×C₁₉	—	8	1	no	—
140	C₈×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₈×C₁₉	C₄×D₄	1	1	yes	characteristic
142	C₈×C₁₉	#121	1	1	yes	characteristic
143	C₈×C₁₉	#124	1	1	yes	characteristic
144	C₂×#510	C₂×C₁₉	1	1	yes	characteristic
145	#2155	C₂×C₁₉	1	1	yes	characteristic, complemented
146	#2143	C₂×C₁₉	1	1	yes	characteristic
147	#2141	C₂×C₁₉	1	1	yes	characteristic, complemented
148	#2110	C₂×C₁₉	1	1	yes	characteristic, complemented
149	#1979	C₂×C₁₉	1	1	yes	characteristic, complemented
150	#1976	C₂×C₁₉	1	1	yes	characteristic
151	C₂²×C₁₉	—	4	1	no	—
152	C₂²×C₁₉	—	8	1	no	—
153	C₂²×C₁₉	—	16	1	no	—
154	C₂²×C₁₉	#442	1	1	yes	characteristic
155	C₂²×C₁₉	#469	1	1	yes	characteristic
156	C₂²×C₁₉	#496	1	1	yes	characteristic
157	C₄×C₁₉	—	8	1	no	—
158	C₄×C₁₉	—	8	1	no	—
159	C₄×C₁₉	—	8	2	no	—
160	C₄×C₁₉	—	16	1	no	—
161	C₄×C₁₉	#390	1	1	yes	characteristic
162	C₄×C₁₉	#442	1	1	yes	characteristic
163	C₄×C₁₉	#469	1	1	yes	characteristic
164	C₄×C₁₉	#496	1	1	yes	characteristic
165	C₂×#135	C₂²×C₁₉	1	1	yes	characteristic
166	C₂²×Q₁₆	C₂²×C₁₉	1	1	yes	characteristic
167	#510	—	2	2	no	—
168	#510	—	2	2	no	—
169	C₂×Q₃₂	C₄×C₁₉	1	2	yes	complemented
170	C₂×SD₃₂	C₄×C₁₉	1	2	yes	complemented
171	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
172	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
173	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
174	#427	C₂²×C₁₉	1	1	yes	characteristic
175	#421	—	2	1	no	—
176	#419	—	2	1	no	—
177	C₂×#101	—	2	1	no	—
178	#410	—	2	1	no	—
179	#405	C₂²×C₁₉	1	1	yes	characteristic
180	#365	—	2	2	no	—
181	#359	—	2	2	no	—
182	#358	—	2	2	no	—
183	#347	—	2	2	no	—
184	C₂×C₁₉	—	2	1	no	—
185	C₂×C₁₉	—	2	1	no	—
186	C₂×C₁₉	—	2	1	no	—
187	C₂×C₁₉	—	16	1	no	—
188	C₂×C₁₉	#1972	1	1	yes	characteristic
189	C₂×#41	—	2	1	no	—
190	C₂²×Q₈	—	2	1	no	—
191	#135	—	2	1	no	—
192	#135	—	2	1	no	—
193	#135	—	2	1	no	—
194	#135	—	2	1	no	—
195	C₂×Q₁₆	—	2	1	no	—
196	C₂×Q₁₆	—	2	1	no	—
197	C₂×Q₁₆	—	2	1	no	—
198	C₂×Q₁₆	—	2	1	no	—
199	C₂×Q₁₆	—	2	1	no	—
200	C₂×Q₁₆	C₂×C₄×C₁₉	1	2	yes	—
201	C₂×Q₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
202	C₂×SD₁₆	—	2	1	no	—
203	C₂×D₈	C₂×C₄×C₁₉	1	1	yes	characteristic
204	C₂×M₁₆	—	2	1	no	—
205	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
206	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
207	#117	—	2	1	no	—
208	#117	D₄×C₁₉	1	1	yes	characteristic
209	Q₃₂	—	2	4	no	—
210	SD₃₂	—	2	4	no	—
211	M₃₂	—	2	1	no	—
212	M₃₂	—	2	1	no	—
213	M₃₂	D₄×C₁₉	1	2	yes	—
214	M₃₂	D₄×C₁₉	1	2	yes	—
215	C₂×C₁₆	C₂×C₄×C₁₉	1	2	yes	—
216	C₂×C₁₆	D₄×C₁₉	1	2	yes	—
217	#108	—	2	1	no	—
218	#108	—	2	1	no	—
219	#107	—	2	1	no	—
220	#106	—	2	1	no	—
221	#104	—	4	2	no	—
222	#103	—	4	2	no	—
223	#101	—	4	2	no	—
224	#99	—	4	2	no	—
225	#97	—	2	1	no	—
226	C₄×C₈	—	2	1	no	—
227	C₁₉	#13248	1	1	yes	characteristic, direct product factor, Sylow
228	#41	—	4	1	no	—
229	#41	—	4	1	no	—
230	#41	—	4	1	no	—
231	#41	—	4	1	no	—
232	C₂×Q₈	—	2	1	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	4	1	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂×Q₈	—	4	1	no	—
237	C₂×Q₈	—	4	1	no	—
238	C₂×Q₈	—	4	1	no	—
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	—	4	1	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
244	Q₁₆	—	2	2	no	—
245	Q₁₆	—	2	1	no	—
246	Q₁₆	—	2	2	no	—
247	Q₁₆	—	2	1	no	—
248	Q₁₆	—	4	1	no	—
249	Q₁₆	—	4	1	no	—
250	SD₁₆	—	4	1	no	—
251	SD₁₆	—	4	1	no	—
252	D₈	—	2	1	no	—
253	D₈	—	2	1	no	—
254	M₁₆	—	2	1	no	—
255	M₁₆	—	2	1	no	—
256	M₁₆	—	4	1	no	—
257	M₁₆	—	4	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
261	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
262	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
263	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
264	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
265	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₂²⋊C₄	—	2	1	no	—
269	C₂²⋊C₄	—	4	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	4	1	no	—
272	C₁₆	—	2	2	no	—
273	C₁₆	C₂²⋊C₄×C₁₉	1	4	yes	—
274	C₂³	—	4	1	no	—
275	C₂³	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
276	Q₈	—	4	1	no	—
277	Q₈	—	4	2	no	—
278	Q₈	—	4	2	no	—
279	Q₈	—	4	1	no	—
280	Q₈	—	8	1	no	—
281	Q₈	—	8	1	no	—
282	D₄	—	4	1	no	—
283	D₄	—	4	1	no	—
284	D₄	—	8	1	no	—
285	D₄	—	8	1	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	8	1	no	—
290	C₂×C₄	—	8	1	no	—
291	C₂×C₄	—	8	1	no	—
292	C₂×C₄	—	8	1	no	—
293	C₂×C₄	—	8	1	no	—
294	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
295	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
296	C₂×C₄	C₁₉×#127	1	1	yes	characteristic
297	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
298	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
299	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
300	C₈	—	4	1	no	—
301	C₈	—	8	1	no	—
302	C₈	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
303	C₈	C₄×D₄×C₁₉	1	1	yes	characteristic
304	C₈	C₁₉×#121	1	1	yes	characteristic
305	C₈	C₁₉×#124	1	1	yes	characteristic
306	C₂²	—	4	1	no	—
307	C₂²	—	8	1	no	—
308	C₂²	—	16	1	no	—
309	C₂²	C₁₉×#442	1	1	yes	characteristic
310	C₂²	C₁₉×#469	1	1	yes	characteristic
311	C₂²	C₁₉×#496	1	1	yes	characteristic
312	C₄	—	8	1	no	—
313	C₄	—	8	1	no	—
314	C₄	—	8	2	no	—
315	C₄	—	16	1	no	—
316	C₄	C₁₉×#390	1	1	yes	characteristic
317	C₄	C₁₉×#442	1	1	yes	characteristic
318	C₄	C₁₉×#469	1	1	yes	characteristic
319	C₄	C₁₉×#496	1	1	yes	characteristic
320	C₂	—	2	1	no	—
321	C₂	—	2	1	no	—
322	C₂	—	2	1	no	—
323	C₂	—	16	1	no	—
324	C₂	C₁₉×#1972	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₉	#1972
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
2. C₁₉×#2155	C₂	semidirect
4. C₁₉×#2141	C₂	semidirect
5. C₁₉×#2110	C₂	semidirect
6. C₁₉×#1979	C₂	semidirect
12. C₂×C₁₉×Q₃₂	C₄	semidirect
13. C₂×C₁₉×SD₃₂	C₄	semidirect
111. #13248	C₁₉	direct (N × H)
145. #2155	C₂×C₁₉	semidirect
147. #2141	C₂×C₁₉	semidirect
148. #2110	C₂×C₁₉	semidirect
149. #1979	C₂×C₁₉	semidirect
169. C₂×Q₃₂	C₄×C₁₉	semidirect
170. C₂×SD₃₂	C₄×C₁₉	semidirect
227. C₁₉	#13248	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×#2155	2
C₁₉×#2143	2
C₁₉×#2141	2
C₁₉×#2110	2
C₁₉×#1979	2
C₁₉×#1976	2
#13248	19

Subgroups — C19×#13248 #18019207

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13248 #18019207