Subgroups — C₁₉×#32887 #18038846

Order 4864.

3592 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₉×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#452	C₂	1	3	yes	maximal, complemented
3	C₂×C₁₉×#451	C₂	1	3	yes	maximal, complemented
4	C₁₉×#2616	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1788	C₂	1	3	yes	maximal, complemented
6	C₁₉×#1787	C₂	1	3	yes	maximal, complemented
7	C₁₉×#1780	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₉×2₋¹⁺⁴	—	2	12	no	—
9	C₂×C₁₉×2₋¹⁺⁴	—	2	12	no	—
10	C₂²×#41×C₁₉	—	2	3	no	—
11	C₂²×#41×C₁₉	C₂²	1	3	yes	—
12	C₂²×#41×C₁₉	C₂²	1	1	yes	characteristic
13	C₂³×Q₈×C₁₉	—	2	1	no	—
14	C₂³×Q₈×C₁₉	C₂²	1	3	yes	—
15	C₂²×Q₁₆×C₁₉	C₂²	1	3	yes	—
16	C₂²×SD₁₆×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×C₁₉×C₂²⋊Q₈	C₂²	1	3	yes	—
18	C₁₉×#452	—	2	12	no	—
19	C₁₉×#451	—	2	12	no	—
20	C₂×C₁₉×#103	C₂²	1	6	yes	—
21	C₂×C₁₉×#103	C₂²	1	3	yes	—
22	C₂×C₁₉×#102	C₂²	1	3	yes	—
23	C₂×C₁₉×C₂²⋊C₈	C₂²	1	3	yes	—
24	C₂×C₁₉×C₂²⋊C₈	C₂²	1	3	yes	—
25	C₁₉×#395	—	2	1	no	—
26	C₁₉×#394	—	2	3	no	—
27	C₁₉×#380	C₂²	1	3	yes	—
28	C₁₉×2₋¹⁺⁴	—	2	32	no	—
29	C₁₉×2₋¹⁺⁴	—	2	96	no	—
30	C₂×#41×C₁₉	—	2	48	no	—
31	C₂×#41×C₁₉	—	2	24	no	—
32	C₂×#41×C₁₉	—	2	12	no	—
33	C₂×#41×C₁₉	—	2	12	no	—
34	C₂×#41×C₁₉	—	2	6	no	—
35	C₂×#41×C₁₉	—	2	6	no	—
36	C₂×#41×C₁₉	—	2	6	no	—
37	C₂×#41×C₁₉	D₄	1	12	yes	—
38	C₂²×Q₈×C₁₉	—	2	24	no	—
39	C₂²×Q₈×C₁₉	—	2	12	no	—
40	C₂²×Q₈×C₁₉	—	2	12	no	—
41	C₂²×Q₈×C₁₉	—	2	3	no	—
42	C₂²×Q₈×C₁₉	—	2	6	no	—
43	C₂²×Q₈×C₁₉	D₄	1	12	yes	—
44	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
45	C₂²×Q₈×C₁₉	C₂³	1	1	yes	characteristic
46	C₂²×Q₈×C₁₉	C₂³	1	3	yes	—
47	C₂²×D₄×C₁₉	—	2	3	no	—
48	C₂²×D₄×C₁₉	D₄	1	3	yes	—
49	C₂²×D₄×C₁₉	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₁₉	—	2	3	no	—
51	C₂³×C₄×C₁₉	—	2	3	no	—
52	C₂³×C₄×C₁₉	C₂³	1	3	yes	—
53	C₂×Q₁₆×C₁₉	—	2	12	no	—
54	C₂×Q₁₆×C₁₉	—	2	6	no	—
55	C₂×SD₁₆×C₁₉	—	2	6	no	—
56	C₂²×C₈×C₁₉	C₂³	1	3	yes	—
57	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
58	C₁₉×C₂²⋊Q₈	—	2	6	no	—
59	C₁₉×C₂²⋊Q₈	—	2	6	no	—
60	C₂×C₄⋊C₄×C₁₉	—	2	3	no	—
61	C₂×C₄⋊C₄×C₁₉	C₂³	1	3	yes	—
62	C₂×C₂²⋊C₄×C₁₉	—	2	3	no	—
63	C₂×C₂²⋊C₄×C₁₉	—	2	3	no	—
64	C₁₉×#103	—	2	12	no	—
65	C₁₉×#103	—	2	6	no	—
66	C₁₉×#102	—	2	6	no	—
67	C₁₉×C₂²⋊C₈	—	2	6	no	—
68	C₁₉×C₂²⋊C₈	—	2	6	no	—
69	C₁₉×#95	—	2	3	no	—
70	C₂⁴×C₁₉	—	2	1	no	—
71	C₂⁴×C₁₉	C₂×D₄	1	3	yes	—
72	#41×C₁₉	—	2	192	no	—
73	#41×C₁₉	—	2	48	no	—
74	#41×C₁₉	—	2	24	no	—
75	#41×C₁₉	—	2	8	no	—
76	#41×C₁₉	—	2	24	no	—
77	#41×C₁₉	—	2	24	no	—
78	C₂×Q₈×C₁₉	—	2	24	no	—
79	C₂×Q₈×C₁₉	—	2	64	no	—
80	C₂×Q₈×C₁₉	—	2	24	no	—
81	C₂×Q₈×C₁₉	—	2	24	no	—
82	C₂×Q₈×C₁₉	—	2	48	no	—
83	C₂×Q₈×C₁₉	—	2	12	no	—
84	C₂×Q₈×C₁₉	—	2	6	no	—
85	C₂×Q₈×C₁₉	—	2	3	no	—
86	C₂×Q₈×C₁₉	—	2	6	no	—
87	C₂×Q₈×C₁₉	—	2	3	no	—
88	C₂×Q₈×C₁₉	—	2	3	no	—
89	C₂×Q₈×C₁₉	C₂×D₄	1	6	yes	—
90	C₂×Q₈×C₁₉	C₂×D₄	1	12	yes	—
91	C₂×D₄×C₁₉	—	2	24	no	—
92	C₂×D₄×C₁₉	—	2	12	no	—
93	C₂×D₄×C₁₉	—	2	12	no	—
94	C₂×D₄×C₁₉	—	2	6	no	—
95	C₂×D₄×C₁₉	—	2	3	no	—
96	C₂×D₄×C₁₉	C₂×D₄	1	6	yes	—
97	C₂²×C₄×C₁₉	—	2	24	no	—
98	C₂²×C₄×C₁₉	—	2	24	no	—
99	C₂²×C₄×C₁₉	—	2	12	no	—
100	C₂²×C₄×C₁₉	—	2	3	no	—
101	C₂²×C₄×C₁₉	—	2	12	no	—
102	C₂²×C₄×C₁₉	—	2	6	no	—
103	C₂²×C₄×C₁₉	—	2	6	no	—
104	C₂²×C₄×C₁₉	—	2	1	no	—
105	C₂²×C₄×C₁₉	—	2	3	no	—
106	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
107	C₂²×C₄×C₁₉	C₂×D₄	1	6	yes	—
108	C₂²×C₄×C₁₉	C₂×D₄	1	6	yes	—
109	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
110	Q₁₆×C₁₉	—	4	12	no	—
111	SD₁₆×C₁₉	—	4	4	no	—
112	C₂×C₈×C₁₉	—	2	3	no	—
113	C₂×C₈×C₁₉	—	2	3	no	—
114	C₂×C₈×C₁₉	—	2	6	no	—
115	C₄⋊C₄×C₁₉	—	2	3	no	—
116	C₄⋊C₄×C₁₉	—	2	3	no	—
117	C₄⋊C₄×C₁₉	—	4	6	no	—
118	C₂²⋊C₄×C₁₉	—	4	6	no	—
119	C₂²⋊C₄×C₁₉	—	8	3	no	—
120	#32887	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂³×C₁₉	—	2	6	no	—
122	C₂³×C₁₉	—	2	6	no	—
123	C₂³×C₁₉	—	2	3	no	—
124	C₂³×C₁₉	—	2	6	no	—
125	C₂³×C₁₉	—	2	3	no	—
126	C₂³×C₁₉	—	4	1	no	—
127	C₂³×C₁₉	—	4	3	no	—
128	C₂³×C₁₉	#120	1	6	yes	—
129	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
130	Q₈×C₁₉	—	2	48	no	—
131	Q₈×C₁₉	—	2	8	no	—
132	Q₈×C₁₉	—	2	12	no	—
133	Q₈×C₁₉	—	2	6	no	—
134	Q₈×C₁₉	—	4	1	no	—
135	Q₈×C₁₉	—	4	3	no	—
136	Q₈×C₁₉	#120	1	12	yes	—
137	D₄×C₁₉	—	2	48	no	—
138	D₄×C₁₉	—	2	24	no	—
139	D₄×C₁₉	—	2	6	no	—
140	D₄×C₁₉	#120	1	4	yes	—
141	C₂×C₄×C₁₉	—	2	48	no	—
142	C₂×C₄×C₁₉	—	2	48	no	—
143	C₂×C₄×C₁₉	—	2	12	no	—
144	C₂×C₄×C₁₉	—	2	6	no	—
145	C₂×C₄×C₁₉	—	2	6	no	—
146	C₂×C₄×C₁₉	—	2	6	no	—
147	C₂×C₄×C₁₉	—	2	12	no	—
148	C₂×C₄×C₁₉	—	2	6	no	—
149	C₂×C₄×C₁₉	—	4	3	no	—
150	C₂×C₄×C₁₉	—	4	3	no	—
151	C₂×C₄×C₁₉	—	4	3	no	—
152	C₂×C₄×C₁₉	—	4	3	no	—
153	C₂×C₄×C₁₉	#120	1	6	yes	—
154	C₂×C₄×C₁₉	C₂²×D₄	1	3	yes	—
155	C₂×C₄×C₁₉	C₂²×D₄	1	3	yes	—
156	C₈×C₁₉	—	4	1	no	—
157	C₈×C₁₉	—	4	3	no	—
158	C₂²×2₋¹⁺⁴	C₂×C₁₉	1	1	yes	characteristic
159	C₂×#452	C₂×C₁₉	1	3	yes	complemented
160	C₂×#451	C₂×C₁₉	1	3	yes	complemented
161	#2616	C₂×C₁₉	1	1	yes	characteristic, complemented
162	#1788	C₂×C₁₉	1	3	yes	complemented
163	#1787	C₂×C₁₉	1	3	yes	complemented
164	#1780	C₂×C₁₉	1	1	yes	characteristic, complemented
165	C₂²×C₁₉	—	2	6	no	—
166	C₂²×C₁₉	—	2	4	no	—
167	C₂²×C₁₉	—	2	6	no	—
168	C₂²×C₁₉	—	2	6	no	—
169	C₂²×C₁₉	—	4	6	no	—
170	C₂²×C₁₉	—	4	6	no	—
171	C₂²×C₁₉	—	4	6	no	—
172	C₂²×C₁₉	—	4	6	no	—
173	C₂²×C₁₉	C₂×#120	1	3	yes	—
174	C₂²×C₁₉	#577	1	1	yes	characteristic
175	C₂²×C₁₉	#578	1	3	yes	—
176	C₄×C₁₉	—	2	12	no	—
177	C₄×C₁₉	—	2	6	no	—
178	C₄×C₁₉	—	8	1	no	—
179	C₄×C₁₉	—	8	3	no	—
180	C₄×C₁₉	C₂×#120	1	3	yes	—
181	C₄×C₁₉	C₂×#120	1	1	yes	characteristic
182	C₂×2₋¹⁺⁴	—	2	12	no	—
183	C₂×2₋¹⁺⁴	—	2	12	no	—
184	C₂²×#41	—	2	3	no	—
185	C₂²×#41	C₂²×C₁₉	1	3	yes	—
186	C₂²×#41	C₂²×C₁₉	1	1	yes	characteristic
187	C₂³×Q₈	—	2	1	no	—
188	C₂³×Q₈	C₂²×C₁₉	1	3	yes	—
189	C₂²×Q₁₆	C₂²×C₁₉	1	3	yes	—
190	C₂²×SD₁₆	C₂²×C₁₉	1	1	yes	characteristic
191	C₂×C₂²⋊Q₈	C₂²×C₁₉	1	3	yes	—
192	#452	—	2	12	no	—
193	#451	—	2	12	no	—
194	C₂×#103	C₂²×C₁₉	1	6	yes	—
195	C₂×#103	C₂²×C₁₉	1	3	yes	—
196	C₂×#102	C₂²×C₁₉	1	3	yes	—
197	C₂×C₂²⋊C₈	C₂²×C₁₉	1	3	yes	—
198	C₂×C₂²⋊C₈	C₂²×C₁₉	1	3	yes	—
199	#395	—	2	1	no	—
200	#394	—	2	3	no	—
201	#380	C₂²×C₁₉	1	3	yes	—
202	C₂×C₁₉	—	4	6	no	—
203	C₂×C₁₉	—	4	4	no	—
204	C₂×C₁₉	#2846	1	1	yes	characteristic
205	C₂×C₁₉	#3011	1	3	yes	—
206	C₂×C₁₉	#3013	1	3	yes	—
207	2₋¹⁺⁴	—	2	32	no	—
208	2₋¹⁺⁴	—	2	96	no	—
209	C₂×#41	—	2	48	no	—
210	C₂×#41	—	2	24	no	—
211	C₂×#41	—	2	12	no	—
212	C₂×#41	—	2	12	no	—
213	C₂×#41	—	2	6	no	—
214	C₂×#41	—	2	6	no	—
215	C₂×#41	—	2	6	no	—
216	C₂×#41	D₄×C₁₉	1	12	yes	—
217	C₂²×Q₈	—	2	24	no	—
218	C₂²×Q₈	—	2	12	no	—
219	C₂²×Q₈	—	2	12	no	—
220	C₂²×Q₈	—	2	3	no	—
221	C₂²×Q₈	—	2	6	no	—
222	C₂²×Q₈	D₄×C₁₉	1	12	yes	—
223	C₂²×Q₈	D₄×C₁₉	1	1	yes	characteristic
224	C₂²×Q₈	C₂³×C₁₉	1	1	yes	characteristic
225	C₂²×Q₈	C₂³×C₁₉	1	3	yes	—
226	C₂²×D₄	—	2	3	no	—
227	C₂²×D₄	D₄×C₁₉	1	3	yes	—
228	C₂²×D₄	C₂³×C₁₉	1	1	yes	characteristic
229	C₂³×C₄	—	2	3	no	—
230	C₂³×C₄	—	2	3	no	—
231	C₂³×C₄	C₂³×C₁₉	1	3	yes	—
232	C₂×Q₁₆	—	2	12	no	—
233	C₂×Q₁₆	—	2	6	no	—
234	C₂×SD₁₆	—	2	6	no	—
235	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
236	C₂²×C₈	C₂³×C₁₉	1	3	yes	—
237	C₂²⋊Q₈	—	2	6	no	—
238	C₂²⋊Q₈	—	2	6	no	—
239	C₂×C₄⋊C₄	—	2	3	no	—
240	C₂×C₄⋊C₄	C₂³×C₁₉	1	3	yes	—
241	C₂×C₂²⋊C₄	—	2	3	no	—
242	C₂×C₂²⋊C₄	—	2	3	no	—
243	#103	—	2	12	no	—
244	#103	—	2	6	no	—
245	#102	—	2	6	no	—
246	C₂²⋊C₈	—	2	6	no	—
247	C₂²⋊C₈	—	2	6	no	—
248	#95	—	2	3	no	—
249	C₁₉	#32887	1	1	yes	characteristic, direct product factor, Sylow
250	C₂⁴	—	2	1	no	—
251	C₂⁴	C₂×D₄×C₁₉	1	3	yes	—
252	#41	—	2	192	no	—
253	#41	—	2	48	no	—
254	#41	—	2	24	no	—
255	#41	—	2	8	no	—
256	#41	—	2	24	no	—
257	#41	—	2	24	no	—
258	C₂×Q₈	—	2	24	no	—
259	C₂×Q₈	—	2	64	no	—
260	C₂×Q₈	—	2	24	no	—
261	C₂×Q₈	—	2	24	no	—
262	C₂×Q₈	—	2	48	no	—
263	C₂×Q₈	—	2	12	no	—
264	C₂×Q₈	—	2	6	no	—
265	C₂×Q₈	—	2	3	no	—
266	C₂×Q₈	—	2	6	no	—
267	C₂×Q₈	—	2	3	no	—
268	C₂×Q₈	—	2	3	no	—
269	C₂×Q₈	C₂×D₄×C₁₉	1	6	yes	—
270	C₂×Q₈	C₂×D₄×C₁₉	1	12	yes	—
271	C₂×D₄	—	2	24	no	—
272	C₂×D₄	—	2	12	no	—
273	C₂×D₄	—	2	12	no	—
274	C₂×D₄	—	2	6	no	—
275	C₂×D₄	—	2	3	no	—
276	C₂×D₄	C₂×D₄×C₁₉	1	6	yes	—
277	C₂²×C₄	—	2	24	no	—
278	C₂²×C₄	—	2	24	no	—
279	C₂²×C₄	—	2	12	no	—
280	C₂²×C₄	—	2	3	no	—
281	C₂²×C₄	—	2	12	no	—
282	C₂²×C₄	—	2	6	no	—
283	C₂²×C₄	—	2	6	no	—
284	C₂²×C₄	—	2	3	no	—
285	C₂²×C₄	—	2	1	no	—
286	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
287	C₂²×C₄	C₂×D₄×C₁₉	1	6	yes	—
288	C₂²×C₄	C₂×D₄×C₁₉	1	6	yes	—
289	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
290	Q₁₆	—	4	12	no	—
291	SD₁₆	—	4	4	no	—
292	C₂×C₈	—	2	3	no	—
293	C₂×C₈	—	2	3	no	—
294	C₂×C₈	—	2	6	no	—
295	C₄⋊C₄	—	2	3	no	—
296	C₄⋊C₄	—	2	3	no	—
297	C₄⋊C₄	—	4	6	no	—
298	C₂²⋊C₄	—	4	6	no	—
299	C₂²⋊C₄	—	8	3	no	—
300	C₂³	—	2	6	no	—
301	C₂³	—	2	6	no	—
302	C₂³	—	2	3	no	—
303	C₂³	—	2	6	no	—
304	C₂³	—	2	3	no	—
305	C₂³	—	4	1	no	—
306	C₂³	—	4	3	no	—
307	C₂³	C₁₉×#120	1	6	yes	—
308	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
309	Q₈	—	2	48	no	—
310	Q₈	—	2	8	no	—
311	Q₈	—	2	12	no	—
312	Q₈	—	2	6	no	—
313	Q₈	—	4	1	no	—
314	Q₈	—	4	3	no	—
315	Q₈	C₁₉×#120	1	12	yes	—
316	D₄	—	2	48	no	—
317	D₄	—	2	24	no	—
318	D₄	—	2	6	no	—
319	D₄	C₁₉×#120	1	4	yes	—
320	C₂×C₄	—	2	48	no	—
321	C₂×C₄	—	2	48	no	—
322	C₂×C₄	—	2	12	no	—
323	C₂×C₄	—	2	6	no	—
324	C₂×C₄	—	2	6	no	—
325	C₂×C₄	—	2	6	no	—
326	C₂×C₄	—	2	12	no	—
327	C₂×C₄	—	2	6	no	—
328	C₂×C₄	—	4	3	no	—
329	C₂×C₄	—	4	3	no	—
330	C₂×C₄	—	4	3	no	—
331	C₂×C₄	—	4	3	no	—
332	C₂×C₄	C₁₉×#120	1	6	yes	—
333	C₂×C₄	C₂²×D₄×C₁₉	1	3	yes	—
334	C₂×C₄	C₂²×D₄×C₁₉	1	3	yes	—
335	C₈	—	4	1	no	—
336	C₈	—	4	3	no	—
337	C₂²	—	2	6	no	—
338	C₂²	—	2	6	no	—
339	C₂²	—	2	6	no	—
340	C₂²	—	2	4	no	—
341	C₂²	—	4	6	no	—
342	C₂²	—	4	6	no	—
343	C₂²	—	4	6	no	—
344	C₂²	—	4	6	no	—
345	C₂²	C₂×C₁₉×#120	1	3	yes	—
346	C₂²	C₁₉×#577	1	1	yes	characteristic
347	C₂²	C₁₉×#578	1	3	yes	—
348	C₄	—	2	12	no	—
349	C₄	—	2	6	no	—
350	C₄	—	8	3	no	—
351	C₄	—	8	1	no	—
352	C₄	C₂×C₁₉×#120	1	3	yes	—
353	C₄	C₂×C₁₉×#120	1	1	yes	characteristic
354	C₂	—	4	6	no	—
355	C₂	—	4	4	no	—
356	C₂	C₁₉×#2846	1	1	yes	characteristic
357	C₂	C₁₉×#3011	1	3	yes	—
358	C₂	C₁₉×#3013	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
prime radical	C₂²×C₁₉×2₋¹⁺⁴	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂²×#41×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₉×#452	C₂	semidirect
3. C₂×C₁₉×#451	C₂	semidirect
4. C₁₉×#2616	C₂	semidirect
5. C₁₉×#1788	C₂	semidirect
6. C₁₉×#1787	C₂	semidirect
7. C₁₉×#1780	C₂	semidirect
120. #32887	C₁₉	direct (N × H)
159. C₂×#452	C₂×C₁₉	semidirect
160. C₂×#451	C₂×C₁₉	semidirect
161. #2616	C₂×C₁₉	semidirect
162. #1788	C₂×C₁₉	semidirect
163. #1787	C₂×C₁₉	semidirect
164. #1780	C₂×C₁₉	semidirect
249. C₁₉	#32887	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₁₉×2₋¹⁺⁴	2
C₂×C₁₉×#452	2
C₂×C₁₉×#451	2
C₁₉×#2616	2
C₁₉×#1788	2
C₁₉×#1787	2
C₁₉×#1780	2
#32887	19

Subgroups — C19×#32887 #18038846

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#32887 #18038846