Subgroups — C₁₉×#13369 #18019328

Order 4864.

378 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3164	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₉×#2140	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#357	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1978	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1387	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#1382	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1379	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₁₉	C₂²	1	1	yes	characteristic
9	C₁₉×#544	—	2	1	no	—
10	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
12	C₁₉×#501	—	2	2	no	—
13	C₁₉×#500	—	2	1	no	—
14	C₁₉×#499	—	2	1	no	—
15	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
16	C₁₉×#428	—	2	1	no	—
17	C₁₉×#427	C₂²	1	1	yes	characteristic
18	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
19	C₁₉×#418	—	2	1	no	—
20	C₂×C₁₉×#100	—	2	1	no	—
21	C₁₉×#405	C₂²	1	1	yes	characteristic
22	C₁₉×#357	—	2	2	no	—
23	C₁₉×#357	—	2	1	no	—
24	C₁₉×#357	—	2	1	no	—
25	C₂²×D₄×C₁₉	—	2	1	no	—
26	C₂×D₈×C₁₉	—	2	1	no	—
27	C₂×D₈×C₁₉	—	2	1	no	—
28	C₂×D₈×C₁₉	—	2	1	no	—
29	C₂×D₈×C₁₉	—	2	1	no	—
30	C₂×D₈×C₁₉	—	2	1	no	—
31	C₂×D₈×C₁₉	Q₈	1	2	yes	complemented
32	C₂×M₁₆×C₁₉	—	2	1	no	—
33	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
35	C₁₉×C₄⋊Q₈	—	2	2	no	—
36	C₁₉×C₄⋊Q₈	—	4	1	no	—
37	C₁₉×#125	—	2	2	no	—
38	C₁₉×#125	—	4	1	no	—
39	C₁₉×C₂²⋊Q₈	—	4	2	no	—
40	C₁₉×#117	—	2	1	no	—
41	C₁₉×#117	D₄	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
43	C₁₉×M₃₂	—	2	1	no	—
44	C₁₉×M₃₂	—	2	1	no	—
45	C₂×C₁₆×C₁₉	—	2	1	no	—
46	C₂×C₁₆×C₁₉	—	2	1	no	—
47	C₂×C₁₆×C₁₉	—	2	1	no	—
48	C₂×C₁₆×C₁₉	—	2	1	no	—
49	C₁₉×#108	—	2	1	no	—
50	C₁₉×#108	—	2	1	no	—
51	C₁₉×#107	—	2	2	no	—
52	C₁₉×#107	—	2	1	no	—
53	C₁₉×#107	D₄	1	2	yes	—
54	C₁₉×#107	D₄	1	1	yes	characteristic
55	C₁₉×#107	D₄	1	1	yes	characteristic
56	C₁₉×#106	—	2	2	no	—
57	C₁₉×#106	—	2	1	no	—
58	C₁₉×#102	—	4	1	no	—
59	C₁₉×#102	—	4	1	no	—
60	C₁₉×#100	—	4	2	no	—
61	C₁₉×#97	—	2	1	no	—
62	C₄×C₈×C₁₉	—	2	1	no	—
63	C₂⁴×C₁₉	—	4	1	no	—
64	C₂×Q₈×C₁₉	—	4	2	no	—
65	C₂×D₄×C₁₉	—	2	2	no	—
66	C₂×D₄×C₁₉	—	4	1	no	—
67	C₂×D₄×C₁₉	—	4	1	no	—
68	C₂×D₄×C₁₉	—	4	1	no	—
69	C₂×D₄×C₁₉	—	4	1	no	—
70	C₂×D₄×C₁₉	—	4	1	no	—
71	C₂²×C₄×C₁₉	—	4	1	no	—
72	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
73	D₈×C₁₉	—	2	2	no	—
74	D₈×C₁₉	—	2	2	no	—
75	D₈×C₁₉	—	4	1	no	—
76	D₈×C₁₉	—	4	1	no	—
77	M₁₆×C₁₉	—	2	1	no	—
78	M₁₆×C₁₉	—	4	1	no	—
79	M₁₆×C₁₉	—	4	1	no	—
80	M₁₆×C₁₉	SD₁₆	1	2	yes	—
81	C₂×C₈×C₁₉	—	2	1	no	—
82	C₂×C₈×C₁₉	—	4	1	no	—
83	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
87	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
88	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
89	C₄⋊C₄×C₁₉	—	2	2	no	—
90	C₄⋊C₄×C₁₉	—	2	1	no	—
91	C₄⋊C₄×C₁₉	—	2	1	no	—
92	C₄⋊C₄×C₁₉	—	4	2	no	—
93	C₄⋊C₄×C₁₉	—	4	2	no	—
94	C₄⋊C₄×C₁₉	—	4	2	no	—
95	C₄⋊C₄×C₁₉	—	4	1	no	—
96	C₄⋊C₄×C₁₉	D₈	1	1	yes	characteristic
97	C₄⋊C₄×C₁₉	D₈	1	1	yes	characteristic
98	C₂²⋊C₄×C₁₉	—	2	1	no	—
99	C₂²⋊C₄×C₁₉	—	4	1	no	—
100	C₄²×C₁₉	—	2	1	no	—
101	C₄²×C₁₉	—	4	1	no	—
102	C₁₆×C₁₉	—	4	1	no	—
103	C₁₆×C₁₉	—	4	1	no	—
104	#13369	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂³×C₁₉	—	4	2	no	—
106	C₂³×C₁₉	—	8	1	no	—
107	C₂³×C₁₉	—	8	1	no	—
108	C₂³×C₁₉	—	16	1	no	—
109	C₂³×C₁₉	—	16	1	no	—
110	C₂³×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
111	Q₈×C₁₉	—	16	2	no	—
112	D₄×C₁₉	—	4	2	no	—
113	D₄×C₁₉	—	4	2	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	—	4	1	no	—
119	C₂×C₄×C₁₉	—	4	1	no	—
120	C₂×C₄×C₁₉	—	4	2	no	—
121	C₂×C₄×C₁₉	—	16	1	no	—
122	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
128	C₈×C₁₉	—	2	2	no	—
129	C₈×C₁₉	—	8	1	no	—
130	C₈×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
131	C₈×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
132	C₈×C₁₉	#124	1	1	yes	characteristic
133	C₈×C₁₉	#124	1	1	yes	characteristic
134	#3164	C₂×C₁₉	1	1	yes	characteristic, complemented
135	#2140	C₂×C₁₉	1	1	yes	characteristic
136	C₂×#357	C₂×C₁₉	1	1	yes	characteristic
137	#1978	C₂×C₁₉	1	1	yes	characteristic
138	#1387	C₂×C₁₉	1	1	yes	characteristic, complemented
139	#1382	C₂×C₁₉	1	1	yes	characteristic, complemented
140	#1379	C₂×C₁₉	1	1	yes	characteristic, complemented
141	C₂²×C₁₉	—	2	1	no	—
142	C₂²×C₁₉	—	2	1	no	—
143	C₂²×C₁₉	—	8	2	no	—
144	C₂²×C₁₉	—	16	2	no	—
145	C₂²×C₁₉	—	16	2	no	—
146	C₂²×C₁₉	—	16	1	no	—
147	C₂²×C₁₉	—	16	1	no	—
148	C₂²×C₁₉	#447	1	1	yes	characteristic
149	C₂²×C₁₉	#476	1	1	yes	characteristic
150	C₂²×C₁₉	#489	1	1	yes	characteristic
151	C₄×C₁₉	—	4	1	no	—
152	C₄×C₁₉	—	4	1	no	—
153	C₄×C₁₉	—	8	2	no	—
154	C₄×C₁₉	—	8	1	no	—
155	C₄×C₁₉	—	8	1	no	—
156	C₄×C₁₉	#393	1	1	yes	characteristic
157	C₄×C₁₉	#448	1	1	yes	characteristic
158	C₄×C₁₉	#474	1	1	yes	characteristic
159	C₄×C₁₉	#486	1	1	yes	characteristic
160	C₂²×D₈	C₂²×C₁₉	1	1	yes	characteristic
161	#544	—	2	1	no	—
162	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
163	C₂²×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
164	#501	—	2	2	no	—
165	#500	—	2	1	no	—
166	#499	—	2	1	no	—
167	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
168	#428	—	2	1	no	—
169	#427	C₂²×C₁₉	1	1	yes	characteristic
170	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
171	#418	—	2	1	no	—
172	C₂×#100	—	2	1	no	—
173	#405	C₂²×C₁₉	1	1	yes	characteristic
174	#357	—	2	2	no	—
175	#357	—	2	1	no	—
176	#357	—	2	1	no	—
177	C₂×C₁₉	—	2	1	no	—
178	C₂×C₁₉	—	2	1	no	—
179	C₂×C₁₉	—	16	2	no	—
180	C₂×C₁₉	#2023	1	1	yes	characteristic
181	C₂×C₁₉	#2192	1	1	yes	characteristic
182	C₂×C₁₉	#2193	1	1	yes	characteristic
183	C₂²×D₄	—	2	1	no	—
184	C₂×D₈	—	2	1	no	—
185	C₂×D₈	—	2	1	no	—
186	C₂×D₈	—	2	1	no	—
187	C₂×D₈	—	2	1	no	—
188	C₂×D₈	—	2	1	no	—
189	C₂×D₈	Q₈×C₁₉	1	2	yes	complemented
190	C₂×M₁₆	—	2	1	no	—
191	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
192	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
193	C₄⋊Q₈	—	2	2	no	—
194	C₄⋊Q₈	—	4	1	no	—
195	#125	—	2	2	no	—
196	#125	—	4	1	no	—
197	C₂²⋊Q₈	—	4	2	no	—
198	#117	—	2	1	no	—
199	#117	D₄×C₁₉	1	1	yes	characteristic
200	C₂×C₄⋊C₄	—	2	1	no	—
201	M₃₂	—	2	1	no	—
202	M₃₂	—	2	1	no	—
203	C₂×C₁₆	—	2	1	no	—
204	C₂×C₁₆	—	2	1	no	—
205	C₂×C₁₆	—	2	1	no	—
206	C₂×C₁₆	—	2	1	no	—
207	#108	—	2	1	no	—
208	#108	—	2	1	no	—
209	#107	—	2	2	no	—
210	#107	—	2	1	no	—
211	#107	D₄×C₁₉	1	2	yes	—
212	#107	D₄×C₁₉	1	1	yes	characteristic
213	#107	D₄×C₁₉	1	1	yes	characteristic
214	#106	—	2	2	no	—
215	#106	—	2	1	no	—
216	#102	—	4	1	no	—
217	#102	—	4	1	no	—
218	#100	—	4	2	no	—
219	#97	—	2	1	no	—
220	C₄×C₈	—	2	1	no	—
221	C₁₉	#13369	1	1	yes	characteristic, direct product factor, Sylow
222	C₂⁴	—	4	1	no	—
223	C₂×Q₈	—	4	2	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	—	4	1	no	—
226	C₂×D₄	—	4	1	no	—
227	C₂×D₄	—	4	1	no	—
228	C₂×D₄	—	4	1	no	—
229	C₂×D₄	—	4	1	no	—
230	C₂²×C₄	—	4	1	no	—
231	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
232	D₈	—	2	2	no	—
233	D₈	—	2	2	no	—
234	D₈	—	4	1	no	—
235	D₈	—	4	1	no	—
236	M₁₆	—	2	1	no	—
237	M₁₆	—	4	1	no	—
238	M₁₆	—	4	1	no	—
239	M₁₆	SD₁₆×C₁₉	1	2	yes	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
243	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
244	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
245	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
246	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
247	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	4	1	no	—
255	C₄⋊C₄	D₈×C₁₉	1	1	yes	characteristic
256	C₄⋊C₄	D₈×C₁₉	1	1	yes	characteristic
257	C₂²⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	4	1	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	4	1	no	—
261	C₁₆	—	4	1	no	—
262	C₁₆	—	4	1	no	—
263	C₂³	—	4	2	no	—
264	C₂³	—	8	1	no	—
265	C₂³	—	8	1	no	—
266	C₂³	—	16	1	no	—
267	C₂³	—	16	1	no	—
268	C₂³	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
269	Q₈	—	16	2	no	—
270	D₄	—	4	2	no	—
271	D₄	—	4	2	no	—
272	D₄	—	8	1	no	—
273	D₄	—	8	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	16	1	no	—
280	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
281	C₂×C₄	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
282	C₂×C₄	C₁₉×#124	1	1	yes	characteristic
283	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
284	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
285	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
286	C₈	—	2	2	no	—
287	C₈	—	8	1	no	—
288	C₈	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
289	C₈	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
290	C₈	C₁₉×#124	1	1	yes	characteristic
291	C₈	C₁₉×#124	1	1	yes	characteristic
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	8	2	no	—
295	C₂²	—	16	2	no	—
296	C₂²	—	16	2	no	—
297	C₂²	—	16	1	no	—
298	C₂²	—	16	1	no	—
299	C₂²	C₁₉×#447	1	1	yes	characteristic
300	C₂²	C₁₉×#476	1	1	yes	characteristic
301	C₂²	C₁₉×#489	1	1	yes	characteristic
302	C₄	—	4	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	8	2	no	—
305	C₄	—	8	1	no	—
306	C₄	—	8	1	no	—
307	C₄	C₁₉×#393	1	1	yes	characteristic
308	C₄	C₁₉×#448	1	1	yes	characteristic
309	C₄	C₁₉×#474	1	1	yes	characteristic
310	C₄	C₁₉×#486	1	1	yes	characteristic
311	C₂	—	2	1	no	—
312	C₂	—	2	1	no	—
313	C₂	—	16	2	no	—
314	C₂	C₁₉×#2023	1	1	yes	characteristic
315	C₂	C₁₉×#2192	1	1	yes	characteristic
316	C₂	C₁₉×#2193	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₂²×D₈×C₁₉	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₉×#3164	C₂	semidirect
5. C₁₉×#1387	C₂	semidirect
6. C₁₉×#1382	C₂	semidirect
7. C₁₉×#1379	C₂	semidirect
31. C₂×D₈×C₁₉	Q₈	semidirect
104. #13369	C₁₉	direct (N × H)
134. #3164	C₂×C₁₉	semidirect
138. #1387	C₂×C₁₉	semidirect
139. #1382	C₂×C₁₉	semidirect
140. #1379	C₂×C₁₉	semidirect
189. C₂×D₈	Q₈×C₁₉	semidirect
221. C₁₉	#13369	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×#3164	2
C₁₉×#2140	2
C₂×C₁₉×#357	2
C₁₉×#1978	2
C₁₉×#1387	2
C₁₉×#1382	2
C₁₉×#1379	2
#13369	19

Subgroups — C19×#13369 #18019328

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13369 #18019328