Subgroups — C₁₉×#23598 #18029557

Order 4864.

1200 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3459	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2322	C₂	1	3	yes	maximal, complemented
3	C₁₉×#2062	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#2044	C₂	1	3	yes	maximal, complemented
5	C₁₉×#2030	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#2022	C₂	1	3	yes	maximal, complemented
7	C₄×C₁₉×#99	C₂	1	3	yes	maximal, complemented
8	C₁₉×#543	C₂²	1	1	yes	characteristic
9	C₁₉×#539	—	2	3	no	—
10	C₁₉×#538	—	2	3	no	—
11	C₁₉×#534	C₂²	1	3	yes	—
12	C₁₉×#529	—	2	3	no	—
13	C₁₉×#525	—	2	3	no	—
14	C₂×C₁₉×#123	C₂²	1	3	yes	—
15	C₂×C₄×D₄×C₁₉	C₂²	1	3	yes	—
16	C₂²×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×C₁₉×#99	C₂²	1	6	yes	—
18	C₁₉×#389	—	2	3	no	—
19	C₁₉×#388	—	2	3	no	—
20	C₁₉×#387	—	2	3	no	—
21	C₁₉×#386	C₂²	1	3	yes	—
22	C₁₉×#385	—	2	3	no	—
23	C₁₉×#385	C₂²	1	3	yes	—
24	C₁₉×#381	C₂²	1	3	yes	—
25	C₄×C₂²⋊C₄×C₁₉	C₂²	1	3	yes	—
26	C₁₉×#342	C₂²	1	3	yes	—
27	C₁₉×#342	C₂²	1	3	yes	—
28	C₂×#41×C₁₉	—	2	1	no	—
29	C₂²×D₄×C₁₉	C₂³	1	3	yes	—
30	C₂³×C₄×C₁₉	D₄	1	3	yes	—
31	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
32	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
34	C₁₉×#126	—	2	6	no	—
35	C₁₉×#125	—	2	3	no	—
36	C₁₉×#124	—	2	3	no	—
37	C₁₉×#123	—	2	6	no	—
38	C₁₉×#123	—	2	6	no	—
39	C₁₉×#123	—	2	6	no	—
40	C₁₉×#123	—	2	6	no	—
41	C₁₉×C₂²⋊Q₈	—	2	3	no	—
42	C₁₉×C₂²⋊Q₈	—	2	3	no	—
43	C₁₉×C₂²⋊Q₈	—	2	3	no	—
44	C₁₉×#121	—	2	3	no	—
45	C₁₉×#121	—	2	3	no	—
46	C₁₉×#121	—	2	3	no	—
47	C₁₉×#120	—	2	6	no	—
48	C₄×Q₈×C₁₉	—	2	1	no	—
49	C₄×D₄×C₁₉	—	2	3	no	—
50	C₄×D₄×C₁₉	—	2	6	no	—
51	C₄×D₄×C₁₉	—	2	6	no	—
52	C₁₉×#117	—	2	3	no	—
53	C₂×C₄⋊C₄×C₁₉	—	2	3	no	—
54	C₂×C₄⋊C₄×C₁₉	—	2	3	no	—
55	C₂×C₄⋊C₄×C₁₉	—	2	3	no	—
56	C₂×C₄⋊C₄×C₁₉	—	2	3	no	—
57	C₂×C₄⋊C₄×C₁₉	D₄	1	3	yes	—
58	C₂×C₂²⋊C₄×C₁₉	—	2	3	no	—
59	C₂×C₂²⋊C₄×C₁₉	—	2	3	no	—
60	C₂×C₂²⋊C₄×C₁₉	D₄	1	3	yes	—
61	C₂×C₂²⋊C₄×C₁₉	C₂³	1	3	yes	—
62	C₂×C₂²⋊C₄×C₁₉	C₂³	1	3	yes	—
63	C₂×C₂²⋊C₄×C₁₉	C₂³	1	3	yes	—
64	C₂×C₄²×C₁₉	—	2	3	no	—
65	C₂×C₄²×C₁₉	D₄	1	3	yes	—
66	C₁₉×#99	—	2	12	no	—
67	C₁₉×#95	—	2	3	no	—
68	C₁₉×#95	—	2	3	no	—
69	C₁₉×#95	—	2	3	no	—
70	C₁₉×#95	—	2	3	no	—
71	C₁₉×#95	—	2	3	no	—
72	C₂⁴×C₁₉	C₂×D₄	1	3	yes	—
73	C₂⁴×C₁₉	C₂⁴	1	1	yes	characteristic
74	#41×C₁₉	—	8	2	no	—
75	C₂×Q₈×C₁₉	—	2	1	no	—
76	C₂×Q₈×C₁₉	—	2	1	no	—
77	C₂×D₄×C₁₉	—	2	6	no	—
78	C₂×D₄×C₁₉	—	2	3	no	—
79	C₂×D₄×C₁₉	—	2	3	no	—
80	C₂×D₄×C₁₉	—	2	6	no	—
81	C₂×D₄×C₁₉	—	2	3	no	—
82	C₂×D₄×C₁₉	—	2	3	no	—
83	C₂²×C₄×C₁₉	—	2	3	no	—
84	C₂²×C₄×C₁₉	—	2	3	no	—
85	C₂²×C₄×C₁₉	—	2	3	no	—
86	C₂²×C₄×C₁₉	—	2	3	no	—
87	C₂²×C₄×C₁₉	—	2	3	no	—
88	C₂²×C₄×C₁₉	—	2	3	no	—
89	C₂²×C₄×C₁₉	—	2	3	no	—
90	C₂²×C₄×C₁₉	—	8	1	no	—
91	C₂²×C₄×C₁₉	—	8	1	no	—
92	C₂²×C₄×C₁₉	—	8	3	no	—
93	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
94	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
95	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
96	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
97	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
98	C₂²×C₄×C₁₉	#41	1	2	yes	complemented
99	C₄⋊C₄×C₁₉	—	2	12	no	—
100	C₄⋊C₄×C₁₉	—	2	3	no	—
101	C₄⋊C₄×C₁₉	—	2	3	no	—
102	C₄⋊C₄×C₁₉	—	2	3	no	—
103	C₄⋊C₄×C₁₉	—	2	3	no	—
104	C₄⋊C₄×C₁₉	—	2	6	no	—
105	C₄⋊C₄×C₁₉	—	4	1	no	—
106	C₄⋊C₄×C₁₉	—	4	3	no	—
107	C₄⋊C₄×C₁₉	—	8	3	no	—
108	C₂²⋊C₄×C₁₉	—	2	12	no	—
109	C₂²⋊C₄×C₁₉	—	2	6	no	—
110	C₂²⋊C₄×C₁₉	—	2	6	no	—
111	C₂²⋊C₄×C₁₉	—	2	6	no	—
112	C₂²⋊C₄×C₁₉	—	2	3	no	—
113	C₂²⋊C₄×C₁₉	—	2	6	no	—
114	C₂²⋊C₄×C₁₉	—	2	6	no	—
115	C₂²⋊C₄×C₁₉	—	8	3	no	—
116	C₂²⋊C₄×C₁₉	#41	1	6	yes	complemented
117	C₄²×C₁₉	—	2	3	no	—
118	C₄²×C₁₉	—	2	3	no	—
119	C₄²×C₁₉	—	8	3	no	—
120	#23598	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂³×C₁₉	—	2	3	no	—
122	C₂³×C₁₉	—	2	3	no	—
123	C₂³×C₁₉	—	2	3	no	—
124	C₂³×C₁₉	—	4	1	no	—
125	C₂³×C₁₉	—	4	1	no	—
126	C₂³×C₁₉	—	4	3	no	—
127	C₂³×C₁₉	—	4	3	no	—
128	C₂³×C₁₉	C₂²×D₄	1	3	yes	—
129	C₂³×C₁₉	C₂×#41	1	1	yes	characteristic
130	C₂³×C₁₉	C₂×#41	1	3	yes	—
131	Q₈×C₁₉	—	8	1	no	—
132	Q₈×C₁₉	—	8	1	no	—
133	D₄×C₁₉	—	4	6	no	—
134	D₄×C₁₉	—	4	6	no	—
135	D₄×C₁₉	—	8	3	no	—
136	D₄×C₁₉	—	8	3	no	—
137	C₂×C₄×C₁₉	—	2	3	no	—
138	C₂×C₄×C₁₉	—	2	3	no	—
139	C₂×C₄×C₁₉	—	2	3	no	—
140	C₂×C₄×C₁₉	—	2	6	no	—
141	C₂×C₄×C₁₉	—	2	6	no	—
142	C₂×C₄×C₁₉	—	2	3	no	—
143	C₂×C₄×C₁₉	—	2	3	no	—
144	C₂×C₄×C₁₉	—	4	3	no	—
145	C₂×C₄×C₁₉	—	4	3	no	—
146	C₂×C₄×C₁₉	—	4	6	no	—
147	C₂×C₄×C₁₉	—	4	3	no	—
148	C₂×C₄×C₁₉	—	4	3	no	—
149	C₂×C₄×C₁₉	—	4	1	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	—	4	3	no	—
154	C₂×C₄×C₁₉	—	4	3	no	—
155	C₂×C₄×C₁₉	—	4	3	no	—
156	C₂×C₄×C₁₉	—	4	3	no	—
157	C₂×C₄×C₁₉	—	4	1	no	—
158	C₂×C₄×C₁₉	—	4	3	no	—
159	C₂×C₄×C₁₉	—	8	3	no	—
160	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
161	C₂×C₄×C₁₉	#120	1	3	yes	—
162	#3459	C₂×C₁₉	1	1	yes	characteristic
163	#2322	C₂×C₁₉	1	3	yes	complemented
164	#2062	C₂×C₁₉	1	1	yes	characteristic, complemented
165	#2044	C₂×C₁₉	1	3	yes	complemented
166	#2030	C₂×C₁₉	1	1	yes	characteristic, complemented
167	#2022	C₂×C₁₉	1	3	yes	complemented
168	C₄×#99	C₂×C₁₉	1	3	yes	complemented
169	C₂²×C₁₉	—	2	3	no	—
170	C₂²×C₁₉	—	2	3	no	—
171	C₂²×C₁₉	—	4	1	no	—
172	C₂²×C₁₉	—	4	3	no	—
173	C₂²×C₁₉	—	4	6	no	—
174	C₂²×C₁₉	—	4	3	no	—
175	C₂²×C₁₉	—	4	3	no	—
176	C₂²×C₁₉	—	4	3	no	—
177	C₂²×C₁₉	—	8	3	no	—
178	C₂²×C₁₉	C₂×#120	1	1	yes	characteristic
179	C₂²×C₁₉	#525	1	3	yes	—
180	C₂²×C₁₉	#532	1	3	yes	—
181	C₄×C₁₉	—	2	1	no	—
182	C₄×C₁₉	—	2	3	no	—
183	C₄×C₁₉	—	4	6	no	—
184	C₄×C₁₉	—	4	2	no	—
185	C₄×C₁₉	—	4	3	no	—
186	C₄×C₁₉	—	4	3	no	—
187	C₄×C₁₉	—	4	3	no	—
188	C₄×C₁₉	—	8	3	no	—
189	C₄×C₁₉	—	8	1	no	—
190	#543	C₂²×C₁₉	1	1	yes	characteristic
191	#539	—	2	3	no	—
192	#538	—	2	3	no	—
193	#534	C₂²×C₁₉	1	3	yes	—
194	#529	—	2	3	no	—
195	#525	—	2	3	no	—
196	C₂×#123	C₂²×C₁₉	1	3	yes	—
197	C₂×C₄×D₄	C₂²×C₁₉	1	3	yes	—
198	C₂²×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
199	C₂×#99	C₂²×C₁₉	1	6	yes	—
200	#389	—	2	3	no	—
201	#388	—	2	3	no	—
202	#387	—	2	3	no	—
203	#386	C₂²×C₁₉	1	3	yes	—
204	#385	—	2	3	no	—
205	#385	C₂²×C₁₉	1	3	yes	—
206	#381	C₂²×C₁₉	1	3	yes	—
207	C₄×C₂²⋊C₄	C₂²×C₁₉	1	3	yes	—
208	#342	C₂²×C₁₉	1	3	yes	—
209	#342	C₂²×C₁₉	1	3	yes	—
210	C₂×C₁₉	—	2	3	no	—
211	C₂×C₁₉	—	2	3	no	—
212	C₂×C₁₉	—	8	3	no	—
213	C₂×C₁₉	#2388	1	1	yes	characteristic
214	C₂×C₁₉	#3026	1	1	yes	characteristic
215	C₂×C₁₉	#3028	1	1	yes	characteristic
216	C₂×#41	—	2	1	no	—
217	C₂²×D₄	C₂³×C₁₉	1	3	yes	—
218	C₂³×C₄	D₄×C₁₉	1	3	yes	—
219	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
220	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
221	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
222	#126	—	2	6	no	—
223	#125	—	2	3	no	—
224	#124	—	2	3	no	—
225	#123	—	2	6	no	—
226	#123	—	2	6	no	—
227	#123	—	2	6	no	—
228	#123	—	2	6	no	—
229	C₂²⋊Q₈	—	2	3	no	—
230	C₂²⋊Q₈	—	2	3	no	—
231	C₂²⋊Q₈	—	2	3	no	—
232	#121	—	2	3	no	—
233	#121	—	2	3	no	—
234	#121	—	2	3	no	—
235	#120	—	2	6	no	—
236	C₄×Q₈	—	2	1	no	—
237	C₄×D₄	—	2	3	no	—
238	C₄×D₄	—	2	6	no	—
239	C₄×D₄	—	2	6	no	—
240	#117	—	2	3	no	—
241	C₂×C₄⋊C₄	—	2	3	no	—
242	C₂×C₄⋊C₄	—	2	3	no	—
243	C₂×C₄⋊C₄	—	2	3	no	—
244	C₂×C₄⋊C₄	—	2	3	no	—
245	C₂×C₄⋊C₄	D₄×C₁₉	1	3	yes	—
246	C₂×C₂²⋊C₄	—	2	3	no	—
247	C₂×C₂²⋊C₄	—	2	3	no	—
248	C₂×C₂²⋊C₄	D₄×C₁₉	1	3	yes	—
249	C₂×C₂²⋊C₄	C₂³×C₁₉	1	3	yes	—
250	C₂×C₂²⋊C₄	C₂³×C₁₉	1	3	yes	—
251	C₂×C₂²⋊C₄	C₂³×C₁₉	1	3	yes	—
252	C₂×C₄²	—	2	3	no	—
253	C₂×C₄²	D₄×C₁₉	1	3	yes	—
254	#99	—	2	12	no	—
255	#95	—	2	3	no	—
256	#95	—	2	3	no	—
257	#95	—	2	3	no	—
258	#95	—	2	3	no	—
259	#95	—	2	3	no	—
260	C₁₉	#23598	1	1	yes	characteristic, direct product factor, Sylow
261	C₂⁴	C₂×D₄×C₁₉	1	3	yes	—
262	C₂⁴	C₂⁴×C₁₉	1	1	yes	characteristic
263	#41	—	8	2	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	2	1	no	—
266	C₂×D₄	—	2	6	no	—
267	C₂×D₄	—	2	6	no	—
268	C₂×D₄	—	2	3	no	—
269	C₂×D₄	—	2	3	no	—
270	C₂×D₄	—	2	3	no	—
271	C₂×D₄	—	2	3	no	—
272	C₂²×C₄	—	2	3	no	—
273	C₂²×C₄	—	2	3	no	—
274	C₂²×C₄	—	2	3	no	—
275	C₂²×C₄	—	2	3	no	—
276	C₂²×C₄	—	2	3	no	—
277	C₂²×C₄	—	2	3	no	—
278	C₂²×C₄	—	2	3	no	—
279	C₂²×C₄	—	8	1	no	—
280	C₂²×C₄	—	8	1	no	—
281	C₂²×C₄	—	8	3	no	—
282	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
283	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
284	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
285	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
286	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
287	C₂²×C₄	#41×C₁₉	1	2	yes	complemented
288	C₄⋊C₄	—	2	12	no	—
289	C₄⋊C₄	—	2	3	no	—
290	C₄⋊C₄	—	2	3	no	—
291	C₄⋊C₄	—	2	3	no	—
292	C₄⋊C₄	—	2	3	no	—
293	C₄⋊C₄	—	2	6	no	—
294	C₄⋊C₄	—	4	1	no	—
295	C₄⋊C₄	—	4	3	no	—
296	C₄⋊C₄	—	8	3	no	—
297	C₂²⋊C₄	—	2	12	no	—
298	C₂²⋊C₄	—	2	6	no	—
299	C₂²⋊C₄	—	2	6	no	—
300	C₂²⋊C₄	—	2	6	no	—
301	C₂²⋊C₄	—	2	3	no	—
302	C₂²⋊C₄	—	2	6	no	—
303	C₂²⋊C₄	—	2	6	no	—
304	C₂²⋊C₄	—	8	3	no	—
305	C₂²⋊C₄	#41×C₁₉	1	6	yes	complemented
306	C₄²	—	2	3	no	—
307	C₄²	—	2	3	no	—
308	C₄²	—	8	3	no	—
309	C₂³	—	2	3	no	—
310	C₂³	—	2	3	no	—
311	C₂³	—	2	3	no	—
312	C₂³	—	4	1	no	—
313	C₂³	—	4	1	no	—
314	C₂³	—	4	3	no	—
315	C₂³	—	4	3	no	—
316	C₂³	C₂²×D₄×C₁₉	1	3	yes	—
317	C₂³	C₂×#41×C₁₉	1	1	yes	characteristic
318	C₂³	C₂×#41×C₁₉	1	3	yes	—
319	Q₈	—	8	1	no	—
320	Q₈	—	8	1	no	—
321	D₄	—	4	6	no	—
322	D₄	—	4	6	no	—
323	D₄	—	8	3	no	—
324	D₄	—	8	3	no	—
325	C₂×C₄	—	2	3	no	—
326	C₂×C₄	—	2	3	no	—
327	C₂×C₄	—	2	3	no	—
328	C₂×C₄	—	2	6	no	—
329	C₂×C₄	—	2	6	no	—
330	C₂×C₄	—	2	3	no	—
331	C₂×C₄	—	2	3	no	—
332	C₂×C₄	—	4	3	no	—
333	C₂×C₄	—	4	3	no	—
334	C₂×C₄	—	4	6	no	—
335	C₂×C₄	—	4	3	no	—
336	C₂×C₄	—	4	3	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	4	3	no	—
339	C₂×C₄	—	4	3	no	—
340	C₂×C₄	—	4	3	no	—
341	C₂×C₄	—	4	3	no	—
342	C₂×C₄	—	4	3	no	—
343	C₂×C₄	—	4	3	no	—
344	C₂×C₄	—	4	3	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	4	3	no	—
347	C₂×C₄	—	8	3	no	—
348	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
349	C₂×C₄	C₁₉×#120	1	3	yes	—
350	C₂²	—	2	3	no	—
351	C₂²	—	2	3	no	—
352	C₂²	—	4	1	no	—
353	C₂²	—	4	3	no	—
354	C₂²	—	4	6	no	—
355	C₂²	—	4	3	no	—
356	C₂²	—	4	3	no	—
357	C₂²	—	4	3	no	—
358	C₂²	—	8	3	no	—
359	C₂²	C₂×C₁₉×#120	1	1	yes	characteristic
360	C₂²	C₁₉×#525	1	3	yes	—
361	C₂²	C₁₉×#532	1	3	yes	—
362	C₄	—	2	1	no	—
363	C₄	—	2	3	no	—
364	C₄	—	4	6	no	—
365	C₄	—	4	3	no	—
366	C₄	—	4	3	no	—
367	C₄	—	4	2	no	—
368	C₄	—	4	3	no	—
369	C₄	—	8	3	no	—
370	C₄	—	8	1	no	—
371	C₂	—	2	3	no	—
372	C₂	—	2	3	no	—
373	C₂	—	8	3	no	—
374	C₂	C₁₉×#2388	1	1	yes	characteristic
375	C₂	C₁₉×#3026	1	1	yes	characteristic
376	C₂	C₁₉×#3028	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₁₉
prime radical	C₁₉×#3459	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂³×C₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#2322	C₂	semidirect
3. C₁₉×#2062	C₂	semidirect
4. C₁₉×#2044	C₂	semidirect
5. C₁₉×#2030	C₂	semidirect
6. C₁₉×#2022	C₂	semidirect
7. C₄×C₁₉×#99	C₂	semidirect
98. C₂²×C₄×C₁₉	#41	semidirect
116. C₂²⋊C₄×C₁₉	#41	semidirect
120. #23598	C₁₉	direct (N × H)
163. #2322	C₂×C₁₉	semidirect
164. #2062	C₂×C₁₉	semidirect
165. #2044	C₂×C₁₉	semidirect
166. #2030	C₂×C₁₉	semidirect
167. #2022	C₂×C₁₉	semidirect
168. C₄×#99	C₂×C₁₉	semidirect
260. C₁₉	#23598	direct (N × H)
287. C₂²×C₄	#41×C₁₉	semidirect
305. C₂²⋊C₄	#41×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₉×#3459	2
C₁₉×#2322	2
C₁₉×#2062	2
C₁₉×#2044	2
C₁₉×#2030	2
C₁₉×#2022	2
C₄×C₁₉×#99	2
#23598	19

Subgroups — C19×#23598 #18029557

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#23598 #18029557