Subgroups — C₁₉×#12649 #18018608

Order 4864.

428 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2943	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2145	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2143	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₉×#347	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1773	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#1389	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1384	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₉×#135	C₂²	1	1	yes	characteristic
9	C₄×#41×C₁₉	—	2	1	no	—
10	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
12	C₁₉×#442	C₄	1	2	yes	complemented
13	C₁₉×#441	C₄	1	2	yes	complemented
14	C₁₉×#440	—	2	2	no	—
15	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
16	C₁₉×#427	C₂²	1	1	yes	characteristic
17	C₁₉×#427	C₂²	1	1	yes	characteristic
18	C₁₉×#416	—	2	1	no	—
19	C₁₉×#405	—	2	1	no	—
20	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
21	C₁₉×#360	C₄	1	4	yes	complemented
22	C₁₉×#359	C₄	1	4	yes	complemented
23	C₁₉×#347	—	2	2	no	—
24	C₁₉×#339	—	2	1	no	—
25	C₂×#41×C₁₉	—	2	1	no	—
26	C₁₉×#135	—	2	1	no	—
27	C₁₉×#135	—	2	1	no	—
28	C₁₉×#135	C₂×C₄	1	2	yes	—
29	C₁₉×#135	C₂×C₄	1	2	yes	—
30	C₂×Q₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂×SD₁₆×C₁₉	—	2	1	no	—
32	C₂×D₈×C₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂×M₁₆×C₁₉	—	2	1	no	—
34	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
36	C₄×Q₈×C₁₉	—	2	2	no	—
37	C₄×D₄×C₁₉	—	2	2	no	—
38	C₄×D₄×C₁₉	—	4	2	no	—
39	C₁₉×#117	—	2	1	no	—
40	C₁₉×#117	—	2	1	no	—
41	C₁₉×#117	—	4	1	no	—
42	C₂×C₄²×C₁₉	—	4	1	no	—
43	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
44	C₁₉×M₃₂	—	2	1	no	—
45	C₁₉×M₃₂	—	2	1	no	—
46	C₁₉×M₃₂	—	2	1	no	—
47	C₁₉×M₃₂	C₂×C₄	1	2	yes	complemented
48	C₂×C₁₆×C₁₉	—	2	1	no	—
49	C₂×C₁₆×C₁₉	C₂×C₄	1	2	yes	complemented
50	C₁₉×#108	—	2	1	no	—
51	C₁₉×#108	C₂×C₄	1	2	yes	complemented
52	C₁₉×#107	—	2	1	no	—
53	C₁₉×#107	—	2	1	no	—
54	C₁₉×#106	C₂×C₄	1	2	yes	—
55	C₁₉×#106	C₂×C₄	1	2	yes	complemented
56	C₁₉×#103	—	2	2	no	—
57	C₁₉×#102	—	2	2	no	—
58	C₁₉×#97	—	4	1	no	—
59	C₁₉×#97	C₂×C₄	1	2	yes	—
60	C₁₉×#97	D₄	1	2	yes	—
61	C₄×C₈×C₁₉	—	4	1	no	—
62	#41×C₁₉	—	2	2	no	—
63	#41×C₁₉	—	2	2	no	—
64	#41×C₁₉	—	4	1	no	—
65	#41×C₁₉	—	4	1	no	—
66	C₂×Q₈×C₁₉	—	2	1	no	—
67	C₂×D₄×C₁₉	—	2	1	no	—
68	C₂×D₄×C₁₉	—	4	1	no	—
69	C₂²×C₄×C₁₉	—	2	1	no	—
70	C₂²×C₄×C₁₉	—	4	2	no	—
71	C₂²×C₄×C₁₉	—	4	1	no	—
72	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
73	Q₁₆×C₁₉	—	2	1	no	—
74	Q₁₆×C₁₉	C₄²	1	2	yes	—
75	SD₁₆×C₁₉	—	2	2	no	—
76	SD₁₆×C₁₉	—	2	2	no	—
77	D₈×C₁₉	—	2	1	no	—
78	D₈×C₁₉	C₄²	1	2	yes	—
79	M₁₆×C₁₉	—	2	2	no	—
80	M₁₆×C₁₉	—	4	1	no	—
81	C₂×C₈×C₁₉	—	2	1	no	—
82	C₂×C₈×C₁₉	—	2	1	no	—
83	C₂×C₈×C₁₉	—	4	1	no	—
84	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
85	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
90	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
91	C₄⋊C₄×C₁₉	—	2	2	no	—
92	C₄⋊C₄×C₁₉	—	2	2	no	—
93	C₄⋊C₄×C₁₉	—	4	2	no	—
94	C₂²⋊C₄×C₁₉	—	4	2	no	—
95	C₂²⋊C₄×C₁₉	—	4	1	no	—
96	C₂²⋊C₄×C₁₉	—	4	1	no	—
97	C₄²×C₁₉	—	2	1	no	—
98	C₄²×C₁₉	—	4	2	no	—
99	C₄²×C₁₉	—	4	1	no	—
100	C₄²×C₁₉	—	4	1	no	—
101	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
102	C₁₆×C₁₉	—	2	2	no	—
103	C₁₆×C₁₉	—	2	2	no	—
104	#12649	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂³×C₁₉	—	4	1	no	—
106	C₂³×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
107	Q₈×C₁₉	—	2	2	no	—
108	Q₈×C₁₉	—	4	1	no	—
109	D₄×C₁₉	—	2	2	no	—
110	D₄×C₁₉	—	4	2	no	—
111	D₄×C₁₉	—	4	2	no	—
112	D₄×C₁₉	—	4	1	no	—
113	C₂×C₄×C₁₉	—	2	2	no	—
114	C₂×C₄×C₁₉	—	4	1	no	—
115	C₂×C₄×C₁₉	—	4	2	no	—
116	C₂×C₄×C₁₉	—	4	2	no	—
117	C₂×C₄×C₁₉	—	4	2	no	—
118	C₂×C₄×C₁₉	—	4	2	no	—
119	C₂×C₄×C₁₉	—	8	1	no	—
120	C₂×C₄×C₁₉	—	8	4	no	—
121	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
127	C₈×C₁₉	—	2	2	no	—
128	C₈×C₁₉	—	4	2	no	—
129	C₈×C₁₉	C₂×C₄²	1	1	yes	characteristic
130	C₈×C₁₉	#117	1	1	yes	characteristic
131	C₈×C₁₉	C₄×D₄	1	1	yes	characteristic
132	C₈×C₁₉	C₄×D₄	1	1	yes	characteristic
133	#2943	C₂×C₁₉	1	1	yes	characteristic
134	#2145	C₂×C₁₉	1	1	yes	characteristic
135	#2143	C₂×C₁₉	1	1	yes	characteristic
136	C₂×#347	C₂×C₁₉	1	1	yes	characteristic, complemented
137	#1773	C₂×C₁₉	1	1	yes	characteristic, complemented
138	#1389	C₂×C₁₉	1	1	yes	characteristic, complemented
139	#1384	C₂×C₁₉	1	1	yes	characteristic, complemented
140	C₂²×C₁₉	—	4	1	no	—
141	C₂²×C₁₉	—	4	2	no	—
142	C₂²×C₁₉	—	8	2	no	—
143	C₂²×C₁₉	#418	1	1	yes	characteristic
144	C₂²×C₁₉	#440	1	1	yes	characteristic
145	C₂²×C₁₉	#442	1	1	yes	characteristic
146	C₄×C₁₉	—	4	2	no	—
147	C₄×C₁₉	—	4	2	no	—
148	C₄×C₁₉	—	8	2	no	—
149	C₄×C₁₉	—	8	2	no	—
150	C₄×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
151	C₄×C₁₉	#417	1	1	yes	characteristic
152	C₄×C₁₉	#440	1	1	yes	characteristic
153	C₄×C₁₉	#442	1	1	yes	characteristic
154	C₂×#135	C₂²×C₁₉	1	1	yes	characteristic
155	C₄×#41	—	2	1	no	—
156	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
157	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
158	#442	C₄×C₁₉	1	2	yes	complemented
159	#441	C₄×C₁₉	1	2	yes	complemented
160	#440	—	2	2	no	—
161	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
162	#427	C₂²×C₁₉	1	1	yes	characteristic
163	#427	C₂²×C₁₉	1	1	yes	characteristic
164	#416	—	2	1	no	—
165	#405	—	2	1	no	—
166	C₂×#97	C₂²×C₁₉	1	1	yes	characteristic
167	#360	C₄×C₁₉	1	4	yes	complemented
168	#359	C₄×C₁₉	1	4	yes	complemented
169	#347	—	2	2	no	—
170	#339	—	2	1	no	—
171	C₂×C₁₉	—	2	1	no	—
172	C₂×C₁₉	—	2	1	no	—
173	C₂×C₁₉	—	2	1	no	—
174	C₂×C₁₉	—	8	2	no	—
175	C₂×C₁₉	#1762	1	1	yes	characteristic
176	C₂×#41	—	2	1	no	—
177	#135	—	2	1	no	—
178	#135	—	2	1	no	—
179	#135	C₂×C₄×C₁₉	1	2	yes	—
180	#135	C₂×C₄×C₁₉	1	2	yes	—
181	C₂×Q₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
182	C₂×SD₁₆	—	2	1	no	—
183	C₂×D₈	C₂×C₄×C₁₉	1	1	yes	characteristic
184	C₂×M₁₆	—	2	1	no	—
185	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
186	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
187	C₄×Q₈	—	2	2	no	—
188	C₄×D₄	—	2	2	no	—
189	C₄×D₄	—	4	2	no	—
190	#117	—	2	1	no	—
191	#117	—	2	1	no	—
192	#117	—	4	1	no	—
193	C₂×C₄²	—	4	1	no	—
194	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
195	M₃₂	—	2	1	no	—
196	M₃₂	—	2	1	no	—
197	M₃₂	—	2	1	no	—
198	M₃₂	C₂×C₄×C₁₉	1	2	yes	complemented
199	C₂×C₁₆	—	2	1	no	—
200	C₂×C₁₆	C₂×C₄×C₁₉	1	2	yes	complemented
201	#108	—	2	1	no	—
202	#108	C₂×C₄×C₁₉	1	2	yes	complemented
203	#107	—	2	1	no	—
204	#107	—	2	1	no	—
205	#106	C₂×C₄×C₁₉	1	2	yes	—
206	#106	C₂×C₄×C₁₉	1	2	yes	complemented
207	#103	—	2	2	no	—
208	#102	—	2	2	no	—
209	#97	—	4	1	no	—
210	#97	C₂×C₄×C₁₉	1	2	yes	—
211	#97	D₄×C₁₉	1	2	yes	—
212	C₄×C₈	—	4	1	no	—
213	C₁₉	#12649	1	1	yes	characteristic, direct product factor, Sylow
214	#41	—	2	2	no	—
215	#41	—	2	2	no	—
216	#41	—	4	1	no	—
217	#41	—	4	1	no	—
218	C₂×Q₈	—	2	1	no	—
219	C₂×D₄	—	2	1	no	—
220	C₂×D₄	—	4	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	—	4	1	no	—
224	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
225	Q₁₆	—	2	1	no	—
226	Q₁₆	C₄²×C₁₉	1	2	yes	—
227	SD₁₆	—	2	2	no	—
228	SD₁₆	—	2	2	no	—
229	D₈	—	2	1	no	—
230	D₈	C₄²×C₁₉	1	2	yes	—
231	M₁₆	—	2	2	no	—
232	M₁₆	—	4	1	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	C₂²⋊C₄×C₁₉	1	2	yes	—
237	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
238	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
239	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
240	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
241	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
242	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₂²⋊C₄	—	4	2	no	—
247	C₂²⋊C₄	—	4	1	no	—
248	C₂²⋊C₄	—	4	1	no	—
249	C₄²	—	2	1	no	—
250	C₄²	—	4	2	no	—
251	C₄²	—	4	1	no	—
252	C₄²	—	4	1	no	—
253	C₄²	C₂²⋊C₄×C₁₉	1	2	yes	—
254	C₁₆	—	2	2	no	—
255	C₁₆	—	2	2	no	—
256	C₂³	—	4	1	no	—
257	C₂³	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
258	Q₈	—	2	2	no	—
259	Q₈	—	4	1	no	—
260	D₄	—	2	2	no	—
261	D₄	—	4	2	no	—
262	D₄	—	4	2	no	—
263	D₄	—	4	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	8	1	no	—
271	C₂×C₄	—	8	4	no	—
272	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
273	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
274	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
275	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
276	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
277	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
278	C₈	—	2	2	no	—
279	C₈	—	4	2	no	—
280	C₈	C₂×C₄²×C₁₉	1	1	yes	characteristic
281	C₈	C₁₉×#117	1	1	yes	characteristic
282	C₈	C₄×D₄×C₁₉	1	1	yes	characteristic
283	C₈	C₄×D₄×C₁₉	1	1	yes	characteristic
284	C₂²	—	4	1	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	8	2	no	—
287	C₂²	C₁₉×#418	1	1	yes	characteristic
288	C₂²	C₁₉×#440	1	1	yes	characteristic
289	C₂²	C₁₉×#442	1	1	yes	characteristic
290	C₄	—	4	2	no	—
291	C₄	—	4	2	no	—
292	C₄	—	8	2	no	—
293	C₄	—	8	2	no	—
294	C₄	C₄×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
295	C₄	C₁₉×#417	1	1	yes	characteristic
296	C₄	C₁₉×#440	1	1	yes	characteristic
297	C₄	C₁₉×#442	1	1	yes	characteristic
298	C₂	—	2	1	no	—
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	—	8	2	no	—
302	C₂	C₁₉×#1762	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₉	#442
commutator	C₈	C₂×C₄²×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
socle	C₂×C₁₉	#1762
prime radical	C₂×C₁₉×#135	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₁₉×#97 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₁₉×#347	C₂	semidirect
5. C₁₉×#1773	C₂	semidirect
6. C₁₉×#1389	C₂	semidirect
7. C₁₉×#1384	C₂	semidirect
12. C₁₉×#442	C₄	semidirect
13. C₁₉×#441	C₄	semidirect
21. C₁₉×#360	C₄	semidirect
22. C₁₉×#359	C₄	semidirect
47. C₁₉×M₃₂	C₂×C₄	semidirect
49. C₂×C₁₆×C₁₉	C₂×C₄	semidirect
51. C₁₉×#108	C₂×C₄	semidirect
55. C₁₉×#106	C₂×C₄	semidirect
104. #12649	C₁₉	direct (N × H)
136. C₂×#347	C₂×C₁₉	semidirect
137. #1773	C₂×C₁₉	semidirect
138. #1389	C₂×C₁₉	semidirect
139. #1384	C₂×C₁₉	semidirect
158. #442	C₄×C₁₉	semidirect
159. #441	C₄×C₁₉	semidirect
167. #360	C₄×C₁₉	semidirect
168. #359	C₄×C₁₉	semidirect
198. M₃₂	C₂×C₄×C₁₉	semidirect
200. C₂×C₁₆	C₂×C₄×C₁₉	semidirect
202. #108	C₂×C₄×C₁₉	semidirect
206. #106	C₂×C₄×C₁₉	semidirect
213. C₁₉	#12649	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₉×#2943	2
C₁₉×#2145	2
C₁₉×#2143	2
C₂×C₁₉×#347	2
C₁₉×#1773	2
C₁₉×#1389	2
C₁₉×#1384	2
#12649	19

Subgroups — C19×#12649 #18018608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12649 #18018608