Subgroups — C₁₉×#13008 #18018967

Order 4864.

516 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3054	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2911	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#2145	C₂	1	2	yes	maximal, complemented
4	C₂×C₁₉×#349	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1384	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₉×#135	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
8	C₁₉×#525	—	2	1	no	—
9	C₂×C₁₉×#117	—	2	1	no	—
10	C₂×C₁₉×M₃₂	C₂²	1	2	yes	—
11	C₁₉×#470	C₄	1	2	yes	complemented
12	C₁₉×#469	C₄	1	2	yes	complemented
13	C₁₉×#468	—	2	2	no	—
14	C₁₉×#428	—	2	4	no	—
15	C₁₉×#427	—	2	1	no	—
16	C₁₉×#427	C₂²	1	1	yes	characteristic
17	C₁₉×#427	C₂²	1	2	yes	—
18	C₁₉×#416	—	2	1	no	—
19	C₁₉×#360	—	2	4	no	—
20	C₁₉×#349	—	2	2	no	—
21	C₂×#41×C₁₉	—	2	1	no	—
22	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
23	C₁₉×#135	—	2	2	no	—
24	C₁₉×#135	—	2	2	no	—
25	C₂×Q₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic, complemented
26	C₂×SD₁₆×C₁₉	—	2	1	no	—
27	C₂×D₈×C₁₉	C₂×C₄	1	1	yes	characteristic, complemented
28	C₂×M₁₆×C₁₉	—	2	2	no	—
29	C₂×M₁₆×C₁₉	—	2	2	no	—
30	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
31	C₂×M₁₆×C₁₉	D₄	1	2	yes	—
32	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
34	C₁₉×#123	—	4	1	no	—
35	C₁₉×C₂²⋊Q₈	—	2	2	no	—
36	C₁₉×#121	—	2	2	no	—
37	C₁₉×#120	—	4	1	no	—
38	C₄×D₄×C₁₉	—	4	2	no	—
39	C₁₉×#117	—	2	2	no	—
40	C₁₉×#117	—	2	1	no	—
41	C₁₉×#117	—	2	2	no	—
42	C₁₉×#117	—	4	1	no	—
43	C₁₉×#117	—	4	1	no	—
44	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
45	C₂×C₂²⋊C₄×C₁₉	—	4	1	no	—
46	C₂×C₄²×C₁₉	—	4	1	no	—
47	C₁₉×M₃₂	—	2	2	no	—
48	C₁₉×M₃₂	—	2	2	no	—
49	C₂×C₁₆×C₁₉	—	2	2	no	—
50	C₁₉×#107	—	2	2	no	—
51	C₁₉×#107	—	2	2	no	—
52	C₁₉×#107	—	2	1	no	—
53	C₁₉×#106	—	2	2	no	—
54	C₁₉×#106	C₂×C₄	1	2	yes	complemented
55	C₁₉×#106	D₄	1	4	yes	complemented
56	C₁₉×#103	—	2	2	no	—
57	C₁₉×#102	—	2	2	no	—
58	C₂⁴×C₁₉	D₈	1	1	yes	characteristic
59	#41×C₁₉	—	4	1	no	—
60	#41×C₁₉	—	4	1	no	—
61	#41×C₁₉	—	4	1	no	—
62	#41×C₁₉	—	4	1	no	—
63	C₂×Q₈×C₁₉	—	2	1	no	—
64	C₂×D₄×C₁₉	—	2	1	no	—
65	C₂×D₄×C₁₉	—	4	2	no	—
66	C₂×D₄×C₁₉	—	4	1	no	—
67	C₂²×C₄×C₁₉	—	2	1	no	—
68	C₂²×C₄×C₁₉	—	2	1	no	—
69	C₂²×C₄×C₁₉	—	2	1	no	—
70	C₂²×C₄×C₁₉	—	2	1	no	—
71	C₂²×C₄×C₁₉	—	4	2	no	—
72	C₂²×C₄×C₁₉	—	4	1	no	—
73	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
76	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
77	Q₁₆×C₁₉	—	2	2	no	—
78	SD₁₆×C₁₉	—	4	2	no	—
79	D₈×C₁₉	—	2	2	no	—
80	M₁₆×C₁₉	—	2	4	no	—
81	M₁₆×C₁₉	—	2	4	no	—
82	C₂×C₈×C₁₉	—	2	1	no	—
83	C₂×C₈×C₁₉	—	2	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	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₁₉	#41	1	2	yes	—
89	C₄⋊C₄×C₁₉	—	2	2	no	—
90	C₄⋊C₄×C₁₉	—	2	4	no	—
91	C₄⋊C₄×C₁₉	—	4	2	no	—
92	C₄⋊C₄×C₁₉	—	4	2	no	—
93	C₂²⋊C₄×C₁₉	—	4	2	no	—
94	C₂²⋊C₄×C₁₉	—	4	2	no	—
95	C₂²⋊C₄×C₁₉	—	4	2	no	—
96	C₂²⋊C₄×C₁₉	—	4	1	no	—
97	C₂²⋊C₄×C₁₉	—	4	2	no	—
98	C₄²×C₁₉	—	4	2	no	—
99	C₄²×C₁₉	—	4	1	no	—
100	C₄²×C₁₉	—	4	2	no	—
101	C₁₆×C₁₉	—	4	2	no	—
102	#13008	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂³×C₁₉	—	2	1	no	—
104	C₂³×C₁₉	—	2	1	no	—
105	C₂³×C₁₉	—	4	1	no	—
106	C₂³×C₁₉	—	8	1	no	—
107	C₂³×C₁₉	#102	1	2	yes	—
108	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
109	Q₈×C₁₉	—	4	1	no	—
110	Q₈×C₁₉	—	4	1	no	—
111	D₄×C₁₉	—	4	1	no	—
112	D₄×C₁₉	—	4	1	no	—
113	D₄×C₁₉	—	8	4	no	—
114	D₄×C₁₉	—	8	1	no	—
115	D₄×C₁₉	—	8	1	no	—
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	2	1	no	—
118	C₂×C₄×C₁₉	—	2	2	no	—
119	C₂×C₄×C₁₉	—	2	2	no	—
120	C₂×C₄×C₁₉	—	2	2	no	—
121	C₂×C₄×C₁₉	—	2	2	no	—
122	C₂×C₄×C₁₉	—	4	2	no	—
123	C₂×C₄×C₁₉	—	4	1	no	—
124	C₂×C₄×C₁₉	—	4	4	no	—
125	C₂×C₄×C₁₉	—	8	4	no	—
126	C₂×C₄×C₁₉	—	8	1	no	—
127	C₂×C₄×C₁₉	—	8	1	no	—
128	C₂×C₄×C₁₉	#102	1	2	yes	—
129	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
131	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
132	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
133	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
135	C₈×C₁₉	—	2	2	no	—
136	C₈×C₁₉	C₄×D₄	1	2	yes	—
137	C₈×C₁₉	#121	1	2	yes	—
138	#3054	C₂×C₁₉	1	1	yes	characteristic
139	#2911	C₂×C₁₉	1	1	yes	characteristic, complemented
140	#2145	C₂×C₁₉	1	2	yes	complemented
141	C₂×#349	C₂×C₁₉	1	1	yes	characteristic, complemented
142	#1384	C₂×C₁₉	1	2	yes	complemented
143	C₂²×C₁₉	—	2	2	no	—
144	C₂²×C₁₉	—	4	1	no	—
145	C₂²×C₁₉	—	4	2	no	—
146	C₂²×C₁₉	—	4	2	no	—
147	C₂²×C₁₉	—	8	1	no	—
148	C₂²×C₁₉	—	8	2	no	—
149	C₂²×C₁₉	—	8	1	no	—
150	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
151	C₂²×C₁₉	#447	1	1	yes	characteristic
152	C₂²×C₁₉	#480	1	1	yes	characteristic
153	C₄×C₁₉	—	2	2	no	—
154	C₄×C₁₉	—	8	4	no	—
155	C₄×C₁₉	—	8	2	no	—
156	C₄×C₁₉	—	8	1	no	—
157	C₄×C₁₉	#386	1	1	yes	characteristic
158	C₄×C₁₉	#417	1	1	yes	characteristic
159	C₄×C₁₉	#448	1	1	yes	characteristic
160	C₄×C₁₉	#481	1	1	yes	characteristic
161	C₂×#135	C₂²×C₁₉	1	1	yes	characteristic
162	C₂²×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
163	#525	—	2	1	no	—
164	C₂×#117	—	2	1	no	—
165	C₂×M₃₂	C₂²×C₁₉	1	2	yes	—
166	#470	C₄×C₁₉	1	2	yes	complemented
167	#469	C₄×C₁₉	1	2	yes	complemented
168	#468	—	2	2	no	—
169	#428	—	2	4	no	—
170	#427	—	2	1	no	—
171	#427	C₂²×C₁₉	1	1	yes	characteristic
172	#427	C₂²×C₁₉	1	2	yes	—
173	#416	—	2	1	no	—
174	#360	—	2	4	no	—
175	#349	—	2	2	no	—
176	C₂×C₁₉	—	2	1	no	—
177	C₂×C₁₉	—	2	1	no	—
178	C₂×C₁₉	—	2	1	no	—
179	C₂×C₁₉	—	4	2	no	—
180	C₂×C₁₉	—	16	1	no	—
181	C₂×C₁₉	#1874	1	1	yes	characteristic
182	C₂×#41	—	2	1	no	—
183	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
184	#135	—	2	2	no	—
185	#135	—	2	2	no	—
186	C₂×Q₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic, complemented
187	C₂×SD₁₆	—	2	1	no	—
188	C₂×D₈	C₂×C₄×C₁₉	1	1	yes	characteristic, complemented
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	C₂×C₄×C₁₉	1	2	yes	—
192	C₂×M₁₆	D₄×C₁₉	1	2	yes	—
193	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
194	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
195	#123	—	4	1	no	—
196	C₂²⋊Q₈	—	2	2	no	—
197	#121	—	2	2	no	—
198	#120	—	4	1	no	—
199	C₄×D₄	—	4	2	no	—
200	#117	—	2	2	no	—
201	#117	—	2	2	no	—
202	#117	—	2	1	no	—
203	#117	—	4	1	no	—
204	#117	—	4	1	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₂²⋊C₄	—	4	1	no	—
207	C₂×C₄²	—	4	1	no	—
208	M₃₂	—	2	2	no	—
209	M₃₂	—	2	2	no	—
210	C₂×C₁₆	—	2	2	no	—
211	#107	—	2	2	no	—
212	#107	—	2	2	no	—
213	#107	—	2	1	no	—
214	#106	—	2	2	no	—
215	#106	C₂×C₄×C₁₉	1	2	yes	complemented
216	#106	D₄×C₁₉	1	4	yes	complemented
217	#103	—	2	2	no	—
218	#102	—	2	2	no	—
219	C₁₉	#13008	1	1	yes	characteristic, direct product factor, Sylow
220	C₂⁴	D₈×C₁₉	1	1	yes	characteristic
221	#41	—	4	1	no	—
222	#41	—	4	1	no	—
223	#41	—	4	1	no	—
224	#41	—	4	1	no	—
225	C₂×Q₈	—	2	1	no	—
226	C₂×D₄	—	2	1	no	—
227	C₂×D₄	—	4	1	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	—	4	1	no	—
235	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
236	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
237	C₂²×C₄	SD₁₆×C₁₉	1	2	yes	—
238	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
239	Q₁₆	—	2	2	no	—
240	SD₁₆	—	4	2	no	—
241	D₈	—	2	2	no	—
242	M₁₆	—	2	4	no	—
243	M₁₆	—	2	4	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	C₂²⋊C₄×C₁₉	1	2	yes	—
247	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
248	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
249	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
250	C₂×C₈	#41×C₁₉	1	2	yes	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₂²⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₂²⋊C₄	—	4	1	no	—
258	C₂²⋊C₄	—	4	2	no	—
259	C₂²⋊C₄	—	4	2	no	—
260	C₄²	—	4	2	no	—
261	C₄²	—	4	1	no	—
262	C₄²	—	4	2	no	—
263	C₁₆	—	4	2	no	—
264	C₂³	—	2	1	no	—
265	C₂³	—	2	1	no	—
266	C₂³	—	4	1	no	—
267	C₂³	—	8	1	no	—
268	C₂³	C₁₉×#102	1	2	yes	—
269	C₂³	C₂×D₈×C₁₉	1	1	yes	characteristic
270	Q₈	—	4	1	no	—
271	Q₈	—	4	1	no	—
272	D₄	—	4	1	no	—
273	D₄	—	4	1	no	—
274	D₄	—	8	4	no	—
275	D₄	—	8	1	no	—
276	D₄	—	8	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	8	4	no	—
287	C₂×C₄	—	8	1	no	—
288	C₂×C₄	—	8	1	no	—
289	C₂×C₄	C₁₉×#102	1	2	yes	—
290	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
291	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
292	C₂×C₄	C₁₉×#123	1	1	yes	characteristic
293	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
294	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
295	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
296	C₈	—	2	2	no	—
297	C₈	C₄×D₄×C₁₉	1	2	yes	—
298	C₈	C₁₉×#121	1	2	yes	—
299	C₂²	—	2	2	no	—
300	C₂²	—	4	1	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	—	8	1	no	—
304	C₂²	—	8	2	no	—
305	C₂²	—	8	1	no	—
306	C₂²	C₂×C₁₉×#102	1	1	yes	characteristic
307	C₂²	C₁₉×#447	1	1	yes	characteristic
308	C₂²	C₁₉×#480	1	1	yes	characteristic
309	C₄	—	2	2	no	—
310	C₄	—	8	4	no	—
311	C₄	—	8	2	no	—
312	C₄	—	8	1	no	—
313	C₄	C₁₉×#386	1	1	yes	characteristic
314	C₄	C₁₉×#417	1	1	yes	characteristic
315	C₄	C₁₉×#448	1	1	yes	characteristic
316	C₄	C₁₉×#481	1	1	yes	characteristic
317	C₂	—	2	1	no	—
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	—	4	2	no	—
321	C₂	—	16	1	no	—
322	C₂	C₁₉×#1874	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₉	#481
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
socle	C₂×C₁₉	#1874
prime radical	C₁₉×#3054	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂²×M₁₆×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#2911	C₂	semidirect
3. C₁₉×#2145	C₂	semidirect
4. C₂×C₁₉×#349	C₂	semidirect
5. C₁₉×#1384	C₂	semidirect
11. C₁₉×#470	C₄	semidirect
12. C₁₉×#469	C₄	semidirect
25. C₂×Q₁₆×C₁₉	C₂×C₄	semidirect
27. C₂×D₈×C₁₉	C₂×C₄	semidirect
54. C₁₉×#106	C₂×C₄	semidirect
55. C₁₉×#106	D₄	semidirect
102. #13008	C₁₉	direct (N × H)
139. #2911	C₂×C₁₉	semidirect
140. #2145	C₂×C₁₉	semidirect
141. C₂×#349	C₂×C₁₉	semidirect
142. #1384	C₂×C₁₉	semidirect
166. #470	C₄×C₁₉	semidirect
167. #469	C₄×C₁₉	semidirect
186. C₂×Q₁₆	C₂×C₄×C₁₉	semidirect
188. C₂×D₈	C₂×C₄×C₁₉	semidirect
215. #106	C₂×C₄×C₁₉	semidirect
216. #106	D₄×C₁₉	semidirect
219. C₁₉	#13008	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×#3054	2
C₁₉×#2911	2
C₁₉×#2145	2
C₂×C₁₉×#349	2
C₁₉×#1384	2
#13008	19

Subgroups — C19×#13008 #18018967

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13008 #18018967