Subgroups — C₁₉×#20788 #18026747

Order 4864.

608 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2875	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1841	C₂	1	2	yes	maximal, complemented
3	C₁₉×#1833	C₂	1	2	yes	maximal, complemented
4	C₁₉×#1827	C₂	1	2	yes	maximal
5	C₁₉×#1813	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1812	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1773	C₂	1	2	yes	maximal, complemented
8	C₁₉×#1757	C₂	1	2	yes	maximal
9	C₁₉×#1747	C₂	1	2	yes	maximal, complemented
10	C₂×C₁₉×#131	C₂²	1	1	yes	characteristic
11	C₂×C₁₉×#131	C₂²	1	1	yes	characteristic
12	C₄×#41×C₁₉	C₂²	1	1	yes	characteristic
13	C₂×C₁₉×#108	C₂²	1	2	yes	—
14	C₁₉×#427	C₂²	1	2	yes	—
15	C₁₉×#413	C₂²	1	2	yes	—
16	C₁₉×#410	C₂²	1	2	yes	—
17	C₁₉×#405	C₄	1	2	yes	complemented
18	C₁₉×#405	C₄	1	2	yes	complemented
19	C₁₉×#405	C₂²	1	4	yes	—
20	C₁₉×#405	C₂²	1	1	yes	characteristic
21	C₁₉×#405	C₂²	1	1	yes	characteristic
22	C₄×M₁₆×C₁₉	C₄	1	2	yes	complemented
23	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
24	C₂×C₁₉×#97	C₄	1	2	yes	complemented
25	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
26	C₁₉×#344	C₂²	1	4	yes	—
27	C₁₉×#339	C₂²	1	4	yes	—
28	C₁₉×#338	C₂²	1	2	yes	—
29	C₁₉×H(3,ℤ/4)	C₂²	1	2	yes	—
30	C₁₉×H(3,ℤ/4)	C₂²	1	4	yes	—
31	C₂×#41×C₁₉	C₂³	1	1	yes	characteristic
32	C₁₉×#131	—	2	4	no	—
33	C₁₉×#131	—	2	2	no	—
34	C₁₉×#131	—	2	2	no	—
35	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
36	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
37	C₂×M₁₆×C₁₉	C₂³	1	4	yes	—
38	C₂×M₁₆×C₁₉	C₂³	1	1	yes	characteristic
39	C₂×M₁₆×C₁₉	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
44	C₄×Q₈×C₁₉	—	2	1	no	—
45	C₄×D₄×C₁₉	—	2	2	no	—
46	C₄×D₄×C₁₉	—	2	1	no	—
47	C₁₉×#117	C₂×C₄	1	2	yes	—
48	C₁₉×#117	C₂³	1	4	yes	—
49	C₁₉×#117	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
52	C₁₉×#108	—	2	2	no	—
53	C₁₉×#108	—	2	2	no	—
54	C₁₉×#107	—	2	2	no	—
55	C₁₉×#106	—	2	2	no	—
56	C₁₉×#101	—	2	2	no	—
57	C₁₉×#100	—	2	2	no	—
58	C₁₉×#99	—	2	4	no	—
59	C₁₉×#97	—	2	4	no	—
60	C₁₉×#97	C₂×C₄	1	1	yes	characteristic
61	C₁₉×#97	C₂×C₄	1	2	yes	—
62	C₁₉×#97	C₂×C₄	1	2	yes	—
63	C₁₉×#97	C₂×C₄	1	2	yes	—
64	C₁₉×#97	C₂×C₄	1	1	yes	characteristic
65	C₁₉×#97	C₂×C₄	1	1	yes	characteristic
66	C₁₉×#97	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈×C₁₉	—	2	4	no	—
68	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
70	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
71	#41×C₁₉	—	2	2	no	—
72	#41×C₁₉	—	2	2	no	—
73	C₂×Q₈×C₁₉	#41	1	1	yes	characteristic
74	C₂×D₄×C₁₉	#41	1	2	yes	—
75	C₂×D₄×C₁₉	#41	1	1	yes	characteristic
76	C₂²×C₄×C₁₉	—	2	2	no	—
77	C₂²×C₄×C₁₉	—	2	1	no	—
78	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
80	M₁₆×C₁₉	—	2	2	no	—
81	M₁₆×C₁₉	—	2	2	no	—
82	M₁₆×C₁₉	—	2	4	no	—
83	M₁₆×C₁₉	—	2	4	no	—
84	M₁₆×C₁₉	—	2	4	no	—
85	M₁₆×C₁₉	—	2	4	no	—
86	C₂×C₈×C₁₉	—	2	2	no	—
87	C₂×C₈×C₁₉	—	2	2	no	—
88	C₂×C₈×C₁₉	—	2	4	no	—
89	C₂×C₈×C₁₉	—	2	4	no	—
90	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₁₉	#41	1	4	yes	—
97	C₂×C₈×C₁₉	#41	1	2	yes	—
98	C₂×C₈×C₁₉	#41	1	2	yes	—
99	C₄⋊C₄×C₁₉	—	2	2	no	—
100	C₄⋊C₄×C₁₉	—	2	1	no	—
101	C₄⋊C₄×C₁₉	—	2	4	no	—
102	C₂²⋊C₄×C₁₉	—	2	2	no	—
103	C₂²⋊C₄×C₁₉	—	2	1	no	—
104	C₂²⋊C₄×C₁₉	—	2	4	no	—
105	C₄²×C₁₉	—	2	2	no	—
106	C₄²×C₁₉	—	2	1	no	—
107	C₄²×C₁₉	—	2	4	no	—
108	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
110	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
111	#20788	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂³×C₁₉	#117	1	2	yes	—
113	C₂³×C₁₉	C₂×#41	1	1	yes	characteristic
114	Q₈×C₁₉	—	2	2	no	—
115	D₄×C₁₉	—	2	4	no	—
116	D₄×C₁₉	—	2	2	no	—
117	C₂×C₄×C₁₉	—	2	2	no	—
118	C₂×C₄×C₁₉	—	2	2	no	—
119	C₂×C₄×C₁₉	—	2	1	no	—
120	C₂×C₄×C₁₉	—	2	2	no	—
121	C₂×C₄×C₁₉	—	2	1	no	—
122	C₂×C₄×C₁₉	—	4	4	no	—
123	C₂×C₄×C₁₉	—	8	1	no	—
124	C₂×C₄×C₁₉	—	8	2	no	—
125	C₂×C₄×C₁₉	#117	1	2	yes	—
126	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₉	C₂×#41	1	2	yes	—
128	C₂×C₄×C₁₉	C₂×#41	1	2	yes	—
129	C₂×C₄×C₁₉	C₂×#41	1	1	yes	characteristic
130	C₈×C₁₉	—	2	2	no	—
131	C₈×C₁₉	—	2	2	no	—
132	C₈×C₁₉	—	4	4	no	—
133	C₈×C₁₉	#117	1	1	yes	characteristic
134	C₈×C₁₉	#117	1	1	yes	characteristic
135	C₈×C₁₉	#117	1	2	yes	—
136	C₈×C₁₉	#117	1	2	yes	—
137	C₈×C₁₉	#117	1	2	yes	—
138	#2875	C₂×C₁₉	1	1	yes	characteristic
139	#1841	C₂×C₁₉	1	2	yes	complemented
140	#1833	C₂×C₁₉	1	2	yes	complemented
141	#1827	C₂×C₁₉	1	2	yes	—
142	#1813	C₂×C₁₉	1	1	yes	characteristic
143	#1812	C₂×C₁₉	1	1	yes	characteristic
144	#1773	C₂×C₁₉	1	2	yes	complemented
145	#1757	C₂×C₁₉	1	2	yes	—
146	#1747	C₂×C₁₉	1	2	yes	complemented
147	C₂²×C₁₉	—	2	2	no	—
148	C₂²×C₁₉	—	4	1	no	—
149	C₂²×C₁₉	—	4	2	no	—
150	C₂²×C₁₉	C₂×#117	1	1	yes	characteristic
151	C₂²×C₁₉	#529	1	2	yes	—
152	C₄×C₁₉	—	2	2	no	—
153	C₄×C₁₉	—	4	2	no	—
154	C₄×C₁₉	—	4	1	no	—
155	C₄×C₁₉	—	4	1	no	—
156	C₄×C₁₉	—	8	4	no	—
157	C₄×C₁₉	C₂×#117	1	1	yes	characteristic
158	C₄×C₁₉	C₂×#117	1	1	yes	characteristic
159	C₄×C₁₉	#529	1	2	yes	—
160	C₂×#131	C₂²×C₁₉	1	1	yes	characteristic
161	C₂×#131	C₂²×C₁₉	1	1	yes	characteristic
162	C₄×#41	C₂²×C₁₉	1	1	yes	characteristic
163	C₂×#108	C₂²×C₁₉	1	2	yes	—
164	#427	C₂²×C₁₉	1	2	yes	—
165	#413	C₂²×C₁₉	1	2	yes	—
166	#410	C₂²×C₁₉	1	2	yes	—
167	#405	C₄×C₁₉	1	2	yes	complemented
168	#405	C₄×C₁₉	1	2	yes	complemented
169	#405	C₂²×C₁₉	1	4	yes	—
170	#405	C₂²×C₁₉	1	1	yes	characteristic
171	#405	C₂²×C₁₉	1	1	yes	characteristic
172	C₄×M₁₆	C₄×C₁₉	1	2	yes	complemented
173	C₄×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
174	C₂×#97	C₄×C₁₉	1	2	yes	complemented
175	C₂×#97	C₂²×C₁₉	1	1	yes	characteristic
176	#344	C₂²×C₁₉	1	4	yes	—
177	#339	C₂²×C₁₉	1	4	yes	—
178	#338	C₂²×C₁₉	1	2	yes	—
179	H(3,ℤ/4)	C₂²×C₁₉	1	2	yes	—
180	H(3,ℤ/4)	C₂²×C₁₉	1	4	yes	—
181	C₂×C₁₉	—	2	1	no	—
182	C₂×C₁₉	—	2	2	no	—
183	C₂×C₁₉	—	4	2	no	—
184	C₂×C₁₉	#2283	1	1	yes	characteristic
185	C₂×#41	C₂³×C₁₉	1	1	yes	characteristic
186	#131	—	2	4	no	—
187	#131	—	2	2	no	—
188	#131	—	2	2	no	—
189	C₂×M₁₆	C₂×C₄×C₁₉	1	2	yes	—
190	C₂×M₁₆	C₂×C₄×C₁₉	1	2	yes	—
191	C₂×M₁₆	C₂³×C₁₉	1	4	yes	—
192	C₂×M₁₆	C₂³×C₁₉	1	1	yes	characteristic
193	C₂×M₁₆	C₂³×C₁₉	1	1	yes	characteristic
194	C₂²×C₈	C₂×C₄×C₁₉	1	2	yes	—
195	C₂²×C₈	C₂×C₄×C₁₉	1	2	yes	—
196	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
197	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
198	C₄×Q₈	—	2	1	no	—
199	C₄×D₄	—	2	2	no	—
200	C₄×D₄	—	2	1	no	—
201	#117	C₂×C₄×C₁₉	1	2	yes	—
202	#117	C₂³×C₁₉	1	4	yes	—
203	#117	C₂³×C₁₉	1	1	yes	characteristic
204	C₂×C₄²	C₂×C₄×C₁₉	1	2	yes	—
205	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
206	#108	—	2	2	no	—
207	#108	—	2	2	no	—
208	#107	—	2	2	no	—
209	#106	—	2	2	no	—
210	#101	—	2	2	no	—
211	#100	—	2	2	no	—
212	#99	—	2	4	no	—
213	#97	—	2	4	no	—
214	#97	C₂×C₄×C₁₉	1	1	yes	characteristic
215	#97	C₂×C₄×C₁₉	1	2	yes	—
216	#97	C₂×C₄×C₁₉	1	2	yes	—
217	#97	C₂×C₄×C₁₉	1	2	yes	—
218	#97	C₂×C₄×C₁₉	1	1	yes	characteristic
219	#97	C₂×C₄×C₁₉	1	1	yes	characteristic
220	#97	C₂×C₄×C₁₉	1	1	yes	characteristic
221	C₄×C₈	—	2	4	no	—
222	C₄×C₈	C₂×C₄×C₁₉	1	2	yes	—
223	C₄×C₈	C₂×C₄×C₁₉	1	2	yes	—
224	C₄×C₈	C₂×C₄×C₁₉	1	2	yes	—
225	C₁₉	#20788	1	1	yes	characteristic, direct product factor, Sylow
226	#41	—	2	2	no	—
227	#41	—	2	2	no	—
228	C₂×Q₈	#41×C₁₉	1	1	yes	characteristic
229	C₂×D₄	#41×C₁₉	1	2	yes	—
230	C₂×D₄	#41×C₁₉	1	1	yes	characteristic
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
234	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
235	M₁₆	—	2	2	no	—
236	M₁₆	—	2	2	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	2	4	no	—
240	M₁₆	—	2	4	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
246	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
247	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
248	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
249	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
250	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
251	C₂×C₈	#41×C₁₉	1	4	yes	—
252	C₂×C₈	#41×C₁₉	1	2	yes	—
253	C₂×C₈	#41×C₁₉	1	2	yes	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₂²⋊C₄	—	2	2	no	—
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	2	4	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	1	no	—
262	C₄²	—	2	4	no	—
263	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
264	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
265	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
266	C₂³	C₁₉×#117	1	2	yes	—
267	C₂³	C₂×#41×C₁₉	1	1	yes	characteristic
268	Q₈	—	2	2	no	—
269	D₄	—	2	4	no	—
270	D₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	8	1	no	—
278	C₂×C₄	—	8	2	no	—
279	C₂×C₄	C₁₉×#117	1	2	yes	—
280	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
281	C₂×C₄	C₂×#41×C₁₉	1	2	yes	—
282	C₂×C₄	C₂×#41×C₁₉	1	2	yes	—
283	C₂×C₄	C₂×#41×C₁₉	1	1	yes	characteristic
284	C₈	—	2	2	no	—
285	C₈	—	2	2	no	—
286	C₈	—	4	4	no	—
287	C₈	C₁₉×#117	1	1	yes	characteristic
288	C₈	C₁₉×#117	1	1	yes	characteristic
289	C₈	C₁₉×#117	1	2	yes	—
290	C₈	C₁₉×#117	1	2	yes	—
291	C₈	C₁₉×#117	1	2	yes	—
292	C₂²	—	2	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	2	no	—
295	C₂²	C₂×C₁₉×#117	1	1	yes	characteristic
296	C₂²	C₁₉×#529	1	2	yes	—
297	C₄	—	2	2	no	—
298	C₄	—	4	1	no	—
299	C₄	—	4	2	no	—
300	C₄	—	4	1	no	—
301	C₄	—	8	4	no	—
302	C₄	C₂×C₁₉×#117	1	1	yes	characteristic
303	C₄	C₂×C₁₉×#117	1	1	yes	characteristic
304	C₄	C₁₉×#529	1	2	yes	—
305	C₂	—	2	1	no	—
306	C₂	—	2	2	no	—
307	C₂	—	4	2	no	—
308	C₂	C₁₉×#2283	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₉	C₂×#117
commutator	C₂×C₄	C₂³×C₄×C₁₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
socle	C₂×C₁₉	#2283
prime radical	C₂×#41×C₁₉	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₁₉×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#1841	C₂	semidirect
3. C₁₉×#1833	C₂	semidirect
7. C₁₉×#1773	C₂	semidirect
9. C₁₉×#1747	C₂	semidirect
17. C₁₉×#405	C₄	semidirect
18. C₁₉×#405	C₄	semidirect
22. C₄×M₁₆×C₁₉	C₄	semidirect
24. C₂×C₁₉×#97	C₄	semidirect
111. #20788	C₁₉	direct (N × H)
139. #1841	C₂×C₁₉	semidirect
140. #1833	C₂×C₁₉	semidirect
144. #1773	C₂×C₁₉	semidirect
146. #1747	C₂×C₁₉	semidirect
167. #405	C₄×C₁₉	semidirect
168. #405	C₄×C₁₉	semidirect
172. C₄×M₁₆	C₄×C₁₉	semidirect
174. C₂×#97	C₄×C₁₉	semidirect
225. C₁₉	#20788	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×#2875	2
C₁₉×#1841	2
C₁₉×#1833	2
C₁₉×#1827	2
C₁₉×#1813	2
C₁₉×#1812	2
C₁₉×#1773	2
C₁₉×#1757	2
C₁₉×#1747	2
#20788	19

Subgroups — C19×#20788 #18026747

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#20788 #18026747