Subgroups — C₁₉×#13418 #18019377

Order 4864.

422 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3156	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2148	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#362	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₉×#362	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#2143	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1953	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1378	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₉×#137	—	2	1	no	—
9	C₂×C₁₉×#135	C₂²	1	1	yes	characteristic
10	C₂²×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
11	C₁₉×#536	—	2	1	no	—
12	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
14	C₁₉×#497	—	2	2	no	—
15	C₁₉×#495	—	2	1	no	—
16	C₁₉×#494	—	2	1	no	—
17	C₁₉×#430	—	2	1	no	—
18	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
19	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
20	C₁₉×#405	C₂²	1	1	yes	characteristic
21	C₁₉×#362	—	2	2	no	—
22	C₁₉×#362	—	2	2	no	—
23	C₁₉×#362	—	2	1	no	—
24	C₁₉×#361	—	2	1	no	—
25	C₁₉×#359	—	2	2	no	—
26	C₁₉×#344	—	2	1	no	—
27	C₂×#41×C₁₉	—	2	1	no	—
28	C₂²×Q₈×C₁₉	—	2	1	no	—
29	C₁₉×#137	—	4	2	no	—
30	C₁₉×#137	—	4	2	no	—
31	C₁₉×#135	—	2	1	no	—
32	C₁₉×#135	—	2	1	no	—
33	C₁₉×#135	—	2	1	no	—
34	C₁₉×#135	—	2	1	no	—
35	C₂×Q₁₆×C₁₉	—	2	2	no	—
36	C₂×Q₁₆×C₁₉	—	2	1	no	—
37	C₂×Q₁₆×C₁₉	—	2	1	no	—
38	C₂×Q₁₆×C₁₉	—	2	1	no	—
39	C₂×Q₁₆×C₁₉	—	2	1	no	—
40	C₂×Q₁₆×C₁₉	—	2	1	no	—
41	C₂×Q₁₆×C₁₉	D₄	1	1	yes	characteristic
42	C₂×Q₁₆×C₁₉	D₄	1	2	yes	—
43	C₂×SD₁₆×C₁₉	—	2	2	no	—
44	C₂×SD₁₆×C₁₉	—	2	1	no	—
45	C₂×D₈×C₁₉	D₄	1	1	yes	characteristic
46	C₂×M₁₆×C₁₉	—	2	1	no	—
47	C₂×M₁₆×C₁₉	—	2	1	no	—
48	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
50	C₁₉×C₄⋊Q₈	—	2	2	no	—
51	C₁₉×#124	—	2	2	no	—
52	C₁₉×#123	—	4	2	no	—
53	C₁₉×C₂²⋊Q₈	—	4	2	no	—
54	C₁₉×#117	D₄	1	1	yes	characteristic
55	C₁₉×M₃₂	—	2	1	no	—
56	C₁₉×M₃₂	—	2	1	no	—
57	C₁₉×M₃₂	—	2	1	no	—
58	C₁₉×M₃₂	—	2	1	no	—
59	C₂×C₁₆×C₁₉	—	2	1	no	—
60	C₂×C₁₆×C₁₉	—	2	1	no	—
61	C₁₉×#108	—	2	1	no	—
62	C₁₉×#108	—	2	1	no	—
63	C₁₉×#108	—	2	1	no	—
64	C₁₉×#108	—	4	2	no	—
65	C₁₉×#108	D₄	1	2	yes	—
66	C₁₉×#97	—	2	1	no	—
67	C₄×C₈×C₁₉	—	2	1	no	—
68	#41×C₁₉	—	4	1	no	—
69	#41×C₁₉	—	4	1	no	—
70	#41×C₁₉	—	4	1	no	—
71	#41×C₁₉	—	4	1	no	—
72	C₂×Q₈×C₁₉	—	2	2	no	—
73	C₂×Q₈×C₁₉	—	2	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×Q₈×C₁₉	—	4	1	no	—
79	C₂×D₄×C₁₉	—	2	1	no	—
80	C₂×D₄×C₁₉	—	4	1	no	—
81	C₂²×C₄×C₁₉	—	4	1	no	—
82	C₂²×C₄×C₁₉	—	4	1	no	—
83	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
84	Q₁₆×C₁₉	—	2	2	no	—
85	Q₁₆×C₁₉	—	2	1	no	—
86	Q₁₆×C₁₉	—	2	2	no	—
87	Q₁₆×C₁₉	—	2	1	no	—
88	Q₁₆×C₁₉	—	4	4	no	—
89	Q₁₆×C₁₉	—	4	1	no	—
90	Q₁₆×C₁₉	—	4	1	no	—
91	SD₁₆×C₁₉	—	4	4	no	—
92	SD₁₆×C₁₉	—	4	1	no	—
93	SD₁₆×C₁₉	—	4	1	no	—
94	D₈×C₁₉	—	2	1	no	—
95	D₈×C₁₉	—	2	1	no	—
96	M₁₆×C₁₉	—	2	1	no	—
97	M₁₆×C₁₉	—	2	2	no	—
98	M₁₆×C₁₉	—	4	1	no	—
99	M₁₆×C₁₉	—	4	1	no	—
100	M₁₆×C₁₉	—	4	1	no	—
101	M₁₆×C₁₉	D₈	1	2	yes	—
102	C₂×C₈×C₁₉	—	2	1	no	—
103	C₂×C₈×C₁₉	—	4	1	no	—
104	C₂×C₈×C₁₉	—	4	1	no	—
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	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
111	C₄⋊C₄×C₁₉	—	2	1	no	—
112	C₄⋊C₄×C₁₉	—	4	4	no	—
113	C₂²⋊C₄×C₁₉	—	4	4	no	—
114	C₂²⋊C₄×C₁₉	SD₁₆	1	2	yes	—
115	C₄²×C₁₉	—	2	1	no	—
116	C₁₆×C₁₉	—	4	1	no	—
117	C₁₆×C₁₉	—	4	1	no	—
118	#13418	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂³×C₁₉	—	4	1	no	—
120	C₂³×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
121	Q₈×C₁₉	—	4	1	no	—
122	Q₈×C₁₉	—	4	2	no	—
123	Q₈×C₁₉	—	4	2	no	—
124	Q₈×C₁₉	—	4	1	no	—
125	Q₈×C₁₉	—	8	1	no	—
126	Q₈×C₁₉	—	8	1	no	—
127	D₄×C₁₉	—	4	1	no	—
128	D₄×C₁₉	—	4	1	no	—
129	D₄×C₁₉	—	8	1	no	—
130	D₄×C₁₉	—	8	1	no	—
131	C₂×C₄×C₁₉	—	2	2	no	—
132	C₂×C₄×C₁₉	—	4	2	no	—
133	C₂×C₄×C₁₉	—	4	1	no	—
134	C₂×C₄×C₁₉	—	8	1	no	—
135	C₂×C₄×C₁₉	—	8	1	no	—
136	C₂×C₄×C₁₉	—	8	1	no	—
137	C₂×C₄×C₁₉	—	8	1	no	—
138	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
139	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
140	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
141	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₈×C₁₉	—	2	2	no	—
145	C₈×C₁₉	—	4	2	no	—
146	C₈×C₁₉	—	8	1	no	—
147	C₈×C₁₉	#121	1	1	yes	characteristic
148	C₈×C₁₉	#121	1	1	yes	characteristic
149	C₈×C₁₉	#123	1	1	yes	characteristic
150	C₈×C₁₉	#123	1	1	yes	characteristic
151	#3156	C₂×C₁₉	1	1	yes	characteristic
152	#2148	C₂×C₁₉	1	1	yes	characteristic
153	C₂×#362	C₂×C₁₉	1	1	yes	characteristic, complemented
154	C₂×#362	C₂×C₁₉	1	1	yes	characteristic, complemented
155	#2143	C₂×C₁₉	1	1	yes	characteristic
156	#1953	C₂×C₁₉	1	1	yes	characteristic, complemented
157	#1378	C₂×C₁₉	1	1	yes	characteristic, complemented
158	C₂²×C₁₉	—	4	1	no	—
159	C₂²×C₁₉	—	8	2	no	—
160	C₂²×C₁₉	—	8	1	no	—
161	C₂²×C₁₉	#450	1	1	yes	characteristic
162	C₂²×C₁₉	#460	1	1	yes	characteristic
163	C₂²×C₁₉	#486	1	1	yes	characteristic
164	C₄×C₁₉	—	4	2	no	—
165	C₄×C₁₉	—	8	1	no	—
166	C₄×C₁₉	—	8	2	no	—
167	C₄×C₁₉	#394	1	1	yes	characteristic
168	C₄×C₁₉	#447	1	1	yes	characteristic
169	C₄×C₁₉	#459	1	1	yes	characteristic
170	C₄×C₁₉	#490	1	1	yes	characteristic
171	C₂×#137	—	2	1	no	—
172	C₂×#135	C₂²×C₁₉	1	1	yes	characteristic
173	C₂²×Q₁₆	C₂²×C₁₉	1	1	yes	characteristic
174	#536	—	2	1	no	—
175	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
176	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
177	#497	—	2	2	no	—
178	#495	—	2	1	no	—
179	#494	—	2	1	no	—
180	#430	—	2	1	no	—
181	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
182	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
183	#405	C₂²×C₁₉	1	1	yes	characteristic
184	#362	—	2	2	no	—
185	#362	—	2	1	no	—
186	#362	—	2	2	no	—
187	#361	—	2	1	no	—
188	#359	—	2	2	no	—
189	#344	—	2	1	no	—
190	C₂×C₁₉	—	2	1	no	—
191	C₂×C₁₉	—	2	1	no	—
192	C₂×C₁₉	—	2	1	no	—
193	C₂×C₁₉	—	16	1	no	—
194	C₂×C₁₉	#2041	1	1	yes	characteristic
195	C₂×#41	—	2	1	no	—
196	C₂²×Q₈	—	2	1	no	—
197	#137	—	4	2	no	—
198	#137	—	4	2	no	—
199	#135	—	2	1	no	—
200	#135	—	2	1	no	—
201	#135	—	2	1	no	—
202	#135	—	2	1	no	—
203	C₂×Q₁₆	—	2	2	no	—
204	C₂×Q₁₆	—	2	1	no	—
205	C₂×Q₁₆	—	2	1	no	—
206	C₂×Q₁₆	—	2	1	no	—
207	C₂×Q₁₆	—	2	1	no	—
208	C₂×Q₁₆	—	2	1	no	—
209	C₂×Q₁₆	D₄×C₁₉	1	2	yes	—
210	C₂×Q₁₆	D₄×C₁₉	1	1	yes	characteristic
211	C₂×SD₁₆	—	2	2	no	—
212	C₂×SD₁₆	—	2	1	no	—
213	C₂×D₈	D₄×C₁₉	1	1	yes	characteristic
214	C₂×M₁₆	—	2	1	no	—
215	C₂×M₁₆	—	2	1	no	—
216	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
217	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
218	C₄⋊Q₈	—	2	2	no	—
219	#124	—	2	2	no	—
220	#123	—	4	2	no	—
221	C₂²⋊Q₈	—	4	2	no	—
222	#117	D₄×C₁₉	1	1	yes	characteristic
223	M₃₂	—	2	1	no	—
224	M₃₂	—	2	1	no	—
225	M₃₂	—	2	1	no	—
226	M₃₂	—	2	1	no	—
227	C₂×C₁₆	—	2	1	no	—
228	C₂×C₁₆	—	2	1	no	—
229	#108	—	2	1	no	—
230	#108	—	2	1	no	—
231	#108	—	2	1	no	—
232	#108	—	4	2	no	—
233	#108	D₄×C₁₉	1	2	yes	—
234	#97	—	2	1	no	—
235	C₄×C₈	—	2	1	no	—
236	C₁₉	#13418	1	1	yes	characteristic, direct product factor, Sylow
237	#41	—	4	1	no	—
238	#41	—	4	1	no	—
239	#41	—	4	1	no	—
240	#41	—	4	1	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	1	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×Q₈	—	4	1	no	—
245	C₂×Q₈	—	4	1	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×D₄	—	2	1	no	—
249	C₂×D₄	—	4	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
253	Q₁₆	—	2	2	no	—
254	Q₁₆	—	2	1	no	—
255	Q₁₆	—	2	2	no	—
256	Q₁₆	—	2	1	no	—
257	Q₁₆	—	4	4	no	—
258	Q₁₆	—	4	1	no	—
259	Q₁₆	—	4	1	no	—
260	SD₁₆	—	4	4	no	—
261	SD₁₆	—	4	1	no	—
262	SD₁₆	—	4	1	no	—
263	D₈	—	2	1	no	—
264	D₈	—	2	1	no	—
265	M₁₆	—	2	1	no	—
266	M₁₆	—	2	2	no	—
267	M₁₆	—	4	1	no	—
268	M₁₆	—	4	1	no	—
269	M₁₆	—	4	1	no	—
270	M₁₆	D₈×C₁₉	1	2	yes	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
275	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
276	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
277	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
278	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
279	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
280	C₄⋊C₄	—	2	1	no	—
281	C₄⋊C₄	—	4	4	no	—
282	C₂²⋊C₄	—	4	4	no	—
283	C₂²⋊C₄	SD₁₆×C₁₉	1	2	yes	—
284	C₄²	—	2	1	no	—
285	C₁₆	—	4	1	no	—
286	C₁₆	—	4	1	no	—
287	C₂³	—	4	1	no	—
288	C₂³	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
289	Q₈	—	4	1	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	4	2	no	—
292	Q₈	—	4	1	no	—
293	Q₈	—	8	1	no	—
294	Q₈	—	8	1	no	—
295	D₄	—	4	1	no	—
296	D₄	—	4	1	no	—
297	D₄	—	8	1	no	—
298	D₄	—	8	1	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	8	1	no	—
303	C₂×C₄	—	8	1	no	—
304	C₂×C₄	—	8	1	no	—
305	C₂×C₄	—	8	1	no	—
306	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
307	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
308	C₂×C₄	C₁₉×#124	1	1	yes	characteristic
309	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
310	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
311	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
312	C₈	—	2	2	no	—
313	C₈	—	4	2	no	—
314	C₈	—	8	1	no	—
315	C₈	C₁₉×#121	1	1	yes	characteristic
316	C₈	C₁₉×#121	1	1	yes	characteristic
317	C₈	C₁₉×#123	1	1	yes	characteristic
318	C₈	C₁₉×#123	1	1	yes	characteristic
319	C₂²	—	4	1	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	8	1	no	—
322	C₂²	C₁₉×#450	1	1	yes	characteristic
323	C₂²	C₁₉×#460	1	1	yes	characteristic
324	C₂²	C₁₉×#486	1	1	yes	characteristic
325	C₄	—	4	2	no	—
326	C₄	—	8	1	no	—
327	C₄	—	8	2	no	—
328	C₄	C₁₉×#394	1	1	yes	characteristic
329	C₄	C₁₉×#447	1	1	yes	characteristic
330	C₄	C₁₉×#459	1	1	yes	characteristic
331	C₄	C₁₉×#490	1	1	yes	characteristic
332	C₂	—	2	1	no	—
333	C₂	—	2	1	no	—
334	C₂	—	2	1	no	—
335	C₂	—	16	1	no	—
336	C₂	C₁₉×#2041	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₉	#2041
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₂×C₁₉×#135	C₂²
LCS	G ▷ C₂²×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
4. C₂×C₁₉×#362	C₂	semidirect
6. C₁₉×#1953	C₂	semidirect
7. C₁₉×#1378	C₂	semidirect
118. #13418	C₁₉	direct (N × H)
153. C₂×#362	C₂×C₁₉	semidirect
154. C₂×#362	C₂×C₁₉	semidirect
156. #1953	C₂×C₁₉	semidirect
157. #1378	C₂×C₁₉	semidirect
236. C₁₉	#13418	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×#3156	2
C₁₉×#2148	2
C₂×C₁₉×#362	2
C₁₉×#2143	2
C₁₉×#1953	2
C₁₉×#1378	2
#13418	19

Subgroups — C19×#13418 #18019377

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13418 #18019377