Subgroups — C₁₉×#13034 #18018993

Order 4864.

462 conjugacy classes of subgroups.

The subgroup lattice has 336 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×#2950	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#2145	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#2141	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1976	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1384	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1378	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₉×#135	C₂²	1	1	yes	characteristic
9	C₂²×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
10	C₁₉×#519	—	2	1	no	—
11	C₁₉×#510	—	2	2	no	—
12	C₁₉×#510	—	2	2	no	—
13	C₂×C₁₉×Q₃₂	C₄	1	2	yes	complemented
14	C₂×C₁₉×SD₃₂	C₄	1	2	yes	complemented
15	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
17	C₁₉×#441	—	2	2	no	—
18	C₄×Q₁₆×C₁₉	—	2	2	no	—
19	C₁₉×#427	C₂²	1	1	yes	characteristic
20	C₁₉×#427	C₂²	1	1	yes	characteristic
21	C₁₉×#421	—	2	1	no	—
22	C₁₉×#419	—	2	1	no	—
23	C₁₉×#410	—	2	1	no	—
24	C₁₉×#405	C₂²	1	1	yes	characteristic
25	C₁₉×#360	—	2	2	no	—
26	C₁₉×#358	—	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	C₂×C₄	1	1	yes	characteristic
37	C₂×Q₁₆×C₁₉	C₂×C₄	1	2	yes	—
38	C₂×Q₁₆×C₁₉	D₄	1	2	yes	—
39	C₂×Q₁₆×C₁₉	D₄	1	2	yes	—
40	C₂×SD₁₆×C₁₉	—	2	1	no	—
41	C₂×D₈×C₁₉	C₂×C₄	1	1	yes	characteristic
42	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
44	C₄×Q₈×C₁₉	—	2	4	no	—
45	C₄×Q₈×C₁₉	—	4	2	no	—
46	C₁₉×#117	—	2	1	no	—
47	C₁₉×#117	—	2	1	no	—
48	C₁₉×#117	—	4	2	no	—
49	C₁₉×#117	D₄	1	1	yes	characteristic
50	C₁₉×Q₃₂	—	2	4	no	—
51	C₁₉×SD₃₂	—	2	4	no	—
52	C₁₉×M₃₂	—	2	1	no	—
53	C₁₉×M₃₂	—	2	1	no	—
54	C₁₉×M₃₂	—	2	1	no	—
55	C₁₉×M₃₂	—	2	1	no	—
56	C₂×C₁₆×C₁₉	—	2	1	no	—
57	C₂×C₁₆×C₁₉	C₂×C₄	1	2	yes	—
58	C₁₉×#107	—	2	1	no	—
59	C₁₉×#107	—	2	1	no	—
60	C₁₉×#106	—	2	1	no	—
61	C₁₉×#106	D₄	1	2	yes	—
62	C₁₉×#104	—	4	2	no	—
63	C₁₉×#103	—	2	4	no	—
64	C₁₉×#99	—	4	2	no	—
65	C₁₉×#97	—	2	1	no	—
66	C₄×C₈×C₁₉	—	2	1	no	—
67	#41×C₁₉	—	4	1	no	—
68	#41×C₁₉	—	4	1	no	—
69	#41×C₁₉	—	4	1	no	—
70	#41×C₁₉	—	4	1	no	—
71	C₂×Q₈×C₁₉	—	2	2	no	—
72	C₂×Q₈×C₁₉	—	2	1	no	—
73	C₂×Q₈×C₁₉	—	2	2	no	—
74	C₂×Q₈×C₁₉	—	2	2	no	—
75	C₂×Q₈×C₁₉	—	4	1	no	—
76	C₂×Q₈×C₁₉	—	4	1	no	—
77	C₂×Q₈×C₁₉	—	4	1	no	—
78	C₂×D₄×C₁₉	—	2	1	no	—
79	C₂×D₄×C₁₉	—	4	1	no	—
80	C₂²×C₄×C₁₉	—	4	1	no	—
81	C₂²×C₄×C₁₉	—	4	1	no	—
82	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
83	Q₁₆×C₁₉	—	2	1	no	—
84	Q₁₆×C₁₉	—	2	2	no	—
85	Q₁₆×C₁₉	—	2	1	no	—
86	Q₁₆×C₁₉	—	2	2	no	—
87	Q₁₆×C₁₉	—	2	2	no	—
88	Q₁₆×C₁₉	C₂²⋊C₄	1	4	yes	—
89	SD₁₆×C₁₉	—	4	1	no	—
90	SD₁₆×C₁₉	—	4	1	no	—
91	D₈×C₁₉	—	2	1	no	—
92	D₈×C₁₉	—	2	1	no	—
93	M₁₆×C₁₉	—	2	1	no	—
94	M₁₆×C₁₉	—	2	1	no	—
95	C₂×C₈×C₁₉	—	2	1	no	—
96	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₁₉	C₂×D₄	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	#41	1	1	yes	characteristic
101	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
102	C₄⋊C₄×C₁₉	—	2	1	no	—
103	C₄⋊C₄×C₁₉	—	2	2	no	—
104	C₄⋊C₄×C₁₉	—	4	4	no	—
105	C₄⋊C₄×C₁₉	—	4	1	no	—
106	C₂²⋊C₄×C₁₉	—	4	1	no	—
107	C₂²⋊C₄×C₁₉	—	4	1	no	—
108	C₂²⋊C₄×C₁₉	D₈	1	2	yes	—
109	C₄²×C₁₉	—	2	1	no	—
110	C₄²×C₁₉	—	4	4	no	—
111	C₄²×C₁₉	—	4	1	no	—
112	C₄²×C₁₉	—	4	1	no	—
113	C₁₆×C₁₉	—	2	2	no	—
114	C₁₆×C₁₉	—	4	1	no	—
115	#13034	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂³×C₁₉	—	4	1	no	—
117	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
118	Q₈×C₁₉	—	2	4	no	—
119	Q₈×C₁₉	—	4	2	no	—
120	Q₈×C₁₉	—	4	2	no	—
121	Q₈×C₁₉	—	4	2	no	—
122	Q₈×C₁₉	—	4	1	no	—
123	Q₈×C₁₉	—	4	1	no	—
124	D₄×C₁₉	—	4	1	no	—
125	D₄×C₁₉	—	4	1	no	—
126	D₄×C₁₉	—	8	1	no	—
127	D₄×C₁₉	—	8	1	no	—
128	C₂×C₄×C₁₉	—	2	2	no	—
129	C₂×C₄×C₁₉	—	4	2	no	—
130	C₂×C₄×C₁₉	—	4	2	no	—
131	C₂×C₄×C₁₉	—	4	2	no	—
132	C₂×C₄×C₁₉	—	4	1	no	—
133	C₂×C₄×C₁₉	—	4	2	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	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
139	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
140	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
141	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₈×C₁₉	—	4	1	no	—
144	C₈×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₈×C₁₉	C₄×D₄	1	1	yes	characteristic
146	C₈×C₁₉	#121	1	1	yes	characteristic
147	C₈×C₁₉	#123	1	1	yes	characteristic
148	C₂×#510	C₂×C₁₉	1	1	yes	characteristic
149	#2950	C₂×C₁₉	1	1	yes	characteristic, complemented
150	#2145	C₂×C₁₉	1	1	yes	characteristic
151	#2141	C₂×C₁₉	1	1	yes	characteristic, complemented
152	#1976	C₂×C₁₉	1	1	yes	characteristic
153	#1384	C₂×C₁₉	1	1	yes	characteristic, complemented
154	#1378	C₂×C₁₉	1	1	yes	characteristic, complemented
155	C₂²×C₁₉	—	4	1	no	—
156	C₂²×C₁₉	—	8	1	no	—
157	C₂²×C₁₉	—	16	1	no	—
158	C₂²×C₁₉	C₄×D₈	1	1	yes	characteristic
159	C₂²×C₁₉	#447	1	1	yes	characteristic
160	C₂²×C₁₉	#459	1	1	yes	characteristic
161	C₄×C₁₉	—	4	2	no	—
162	C₄×C₁₉	—	4	4	no	—
163	C₄×C₁₉	—	8	2	no	—
164	C₄×C₁₉	—	8	1	no	—
165	C₄×C₁₉	—	16	1	no	—
166	C₄×C₁₉	#386	1	1	yes	characteristic
167	C₄×C₁₉	#442	1	1	yes	characteristic
168	C₄×C₁₉	#449	1	1	yes	characteristic
169	C₄×C₁₉	#463	1	1	yes	characteristic
170	C₂×#135	C₂²×C₁₉	1	1	yes	characteristic
171	C₂²×Q₁₆	C₂²×C₁₉	1	1	yes	characteristic
172	#519	—	2	1	no	—
173	#510	—	2	2	no	—
174	#510	—	2	2	no	—
175	C₂×Q₃₂	C₄×C₁₉	1	2	yes	complemented
176	C₂×SD₃₂	C₄×C₁₉	1	2	yes	complemented
177	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
178	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
179	#441	—	2	2	no	—
180	C₄×Q₁₆	—	2	2	no	—
181	#427	C₂²×C₁₉	1	1	yes	characteristic
182	#427	C₂²×C₁₉	1	1	yes	characteristic
183	#421	—	2	1	no	—
184	#419	—	2	1	no	—
185	#410	—	2	1	no	—
186	#405	C₂²×C₁₉	1	1	yes	characteristic
187	#360	—	2	2	no	—
188	#358	—	2	2	no	—
189	C₂×C₁₉	—	2	1	no	—
190	C₂×C₁₉	—	2	1	no	—
191	C₂×C₁₉	—	2	1	no	—
192	C₂×C₁₉	—	16	1	no	—
193	C₂×C₁₉	#1879	1	1	yes	characteristic
194	C₂×#41	—	2	1	no	—
195	C₂²×Q₈	—	2	1	no	—
196	#135	—	2	1	no	—
197	#135	—	2	1	no	—
198	#135	—	2	1	no	—
199	#135	—	2	1	no	—
200	C₂×Q₁₆	—	2	1	no	—
201	C₂×Q₁₆	—	2	1	no	—
202	C₂×Q₁₆	—	2	1	no	—
203	C₂×Q₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
204	C₂×Q₁₆	C₂×C₄×C₁₉	1	2	yes	—
205	C₂×Q₁₆	D₄×C₁₉	1	2	yes	—
206	C₂×Q₁₆	D₄×C₁₉	1	2	yes	—
207	C₂×SD₁₆	—	2	1	no	—
208	C₂×D₈	C₂×C₄×C₁₉	1	1	yes	characteristic
209	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
210	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
211	C₄×Q₈	—	2	4	no	—
212	C₄×Q₈	—	4	2	no	—
213	#117	—	2	1	no	—
214	#117	—	2	1	no	—
215	#117	—	4	2	no	—
216	#117	D₄×C₁₉	1	1	yes	characteristic
217	Q₃₂	—	2	4	no	—
218	SD₃₂	—	2	4	no	—
219	M₃₂	—	2	1	no	—
220	M₃₂	—	2	1	no	—
221	M₃₂	—	2	1	no	—
222	M₃₂	—	2	1	no	—
223	C₂×C₁₆	—	2	1	no	—
224	C₂×C₁₆	C₂×C₄×C₁₉	1	2	yes	—
225	#107	—	2	1	no	—
226	#107	—	2	1	no	—
227	#106	—	2	1	no	—
228	#106	D₄×C₁₉	1	2	yes	—
229	#104	—	4	2	no	—
230	#103	—	2	4	no	—
231	#99	—	4	2	no	—
232	#97	—	2	1	no	—
233	C₄×C₈	—	2	1	no	—
234	C₁₉	#13034	1	1	yes	characteristic, direct product factor, Sylow
235	#41	—	4	1	no	—
236	#41	—	4	1	no	—
237	#41	—	4	1	no	—
238	#41	—	4	1	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	2	1	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×Q₈	—	4	1	no	—
245	C₂×Q₈	—	4	1	no	—
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	—	4	1	no	—
248	C₂²×C₄	—	4	1	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
251	Q₁₆	—	2	1	no	—
252	Q₁₆	—	2	2	no	—
253	Q₁₆	—	2	1	no	—
254	Q₁₆	—	2	2	no	—
255	Q₁₆	—	2	2	no	—
256	Q₁₆	C₂²⋊C₄×C₁₉	1	4	yes	—
257	SD₁₆	—	4	1	no	—
258	SD₁₆	—	4	1	no	—
259	D₈	—	2	1	no	—
260	D₈	—	2	1	no	—
261	M₁₆	—	2	1	no	—
262	M₁₆	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
265	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
266	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
267	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
268	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
269	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
270	C₄⋊C₄	—	2	1	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	4	4	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₂²⋊C₄	—	4	1	no	—
275	C₂²⋊C₄	—	4	1	no	—
276	C₂²⋊C₄	D₈×C₁₉	1	2	yes	—
277	C₄²	—	2	1	no	—
278	C₄²	—	4	4	no	—
279	C₄²	—	4	1	no	—
280	C₄²	—	4	1	no	—
281	C₁₆	—	2	2	no	—
282	C₁₆	—	4	1	no	—
283	C₂³	—	4	1	no	—
284	C₂³	C₂×D₈×C₁₉	1	1	yes	characteristic
285	Q₈	—	2	4	no	—
286	Q₈	—	4	2	no	—
287	Q₈	—	4	2	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	4	1	no	—
291	D₄	—	4	1	no	—
292	D₄	—	4	1	no	—
293	D₄	—	8	1	no	—
294	D₄	—	8	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	8	1	no	—
302	C₂×C₄	—	8	1	no	—
303	C₂×C₄	—	8	1	no	—
304	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
305	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
306	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
307	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
308	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
309	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
310	C₈	—	4	1	no	—
311	C₈	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
312	C₈	C₄×D₄×C₁₉	1	1	yes	characteristic
313	C₈	C₁₉×#121	1	1	yes	characteristic
314	C₈	C₁₉×#123	1	1	yes	characteristic
315	C₂²	—	4	1	no	—
316	C₂²	—	8	1	no	—
317	C₂²	—	16	1	no	—
318	C₂²	C₄×D₈×C₁₉	1	1	yes	characteristic
319	C₂²	C₁₉×#447	1	1	yes	characteristic
320	C₂²	C₁₉×#459	1	1	yes	characteristic
321	C₄	—	4	2	no	—
322	C₄	—	4	4	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	1	no	—
325	C₄	—	16	1	no	—
326	C₄	C₁₉×#386	1	1	yes	characteristic
327	C₄	C₁₉×#442	1	1	yes	characteristic
328	C₄	C₁₉×#449	1	1	yes	characteristic
329	C₄	C₁₉×#463	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	16	1	no	—
334	C₂	C₁₉×#1879	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₉	#1879
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×#2950	C₂	semidirect
4. C₁₉×#2141	C₂	semidirect
6. C₁₉×#1384	C₂	semidirect
7. C₁₉×#1378	C₂	semidirect
13. C₂×C₁₉×Q₃₂	C₄	semidirect
14. C₂×C₁₉×SD₃₂	C₄	semidirect
115. #13034	C₁₉	direct (N × H)
149. #2950	C₂×C₁₉	semidirect
151. #2141	C₂×C₁₉	semidirect
153. #1384	C₂×C₁₉	semidirect
154. #1378	C₂×C₁₉	semidirect
175. C₂×Q₃₂	C₄×C₁₉	semidirect
176. C₂×SD₃₂	C₄×C₁₉	semidirect
234. C₁₉	#13034	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×#2950	2
C₁₉×#2145	2
C₁₉×#2141	2
C₁₉×#1976	2
C₁₉×#1384	2
C₁₉×#1378	2
#13034	19

Subgroups — C19×#13034 #18018993

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13034 #18018993