Subgroups — C₁₉×#19052 #18025011

Order 4864.

1078 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3549	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₉×#3214	C₂	1	2	yes	maximal, complemented
3	C₁₉×#3150	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1659	C₂	1	4	yes	maximal, complemented
5	C₁₉×#1655	C₂	1	4	yes	maximal, complemented
6	C₁₉×#1536	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1530	C₂	1	2	yes	maximal, complemented
8	C₂×C₁₉×#135	—	2	2	no	—
9	C₁₉×H(3,𝔽₄)	C₂²	1	4	yes	—
10	C₁₉×#555	—	2	2	no	—
11	C₁₉×#555	—	2	1	no	—
12	D₄²×C₁₉	—	2	2	no	—
13	D₄²×C₁₉	—	2	1	no	—
14	C₁₉×#535	—	2	2	no	—
15	C₁₉×#535	—	2	4	no	—
16	C₁₉×#532	C₂²	1	2	yes	—
17	C₁₉×#532	C₂²	1	1	yes	characteristic
18	C₁₉×#532	C₂²	1	1	yes	characteristic, complemented
19	C₁₉×#497	—	2	1	no	—
20	C₁₉×#497	C₂²	1	2	yes	complemented
21	C₁₉×#496	—	2	1	no	—
22	C₁₉×#496	C₂²	1	4	yes	complemented
23	C₁₉×#496	C₂²	1	2	yes	complemented
24	C₁₉×#489	C₂²	1	4	yes	complemented
25	C₁₉×#461	—	2	2	no	—
26	C₁₉×#459	—	2	2	no	—
27	C₁₉×#449	—	2	4	no	—
28	C₁₉×#432	C₂²	1	2	yes	complemented
29	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic, complemented
30	C₁₉×#329	C₂²	1	8	yes	complemented
31	C₁₉×#329	C₂²	1	4	yes	complemented
32	C₂×#41×C₁₉	—	2	1	no	—
33	C₂×#41×C₁₉	D₄	1	2	yes	—
34	C₂×#41×C₁₉	D₄	1	4	yes	—
35	C₂²×D₄×C₁₉	—	2	4	no	—
36	C₂²×D₄×C₁₉	—	2	2	no	—
37	C₁₉×#135	—	4	8	no	—
38	C₂×Q₁₆×C₁₉	—	2	2	no	—
39	C₂×SD₁₆×C₁₉	—	2	2	no	—
40	C₂×SD₁₆×C₁₉	—	2	2	no	—
41	C₂×SD₁₆×C₁₉	—	2	4	no	—
42	C₂×D₈×C₁₉	—	2	2	no	—
43	C₂×D₈×C₁₉	—	2	4	no	—
44	C₂²×C₈×C₁₉	—	2	1	no	—
45	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
46	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
47	C₁₉×C₄⋊Q₈	C₂³	1	1	yes	characteristic
48	C₁₉×#127	—	2	4	no	—
49	C₁₉×#127	D₄	1	2	yes	—
50	C₁₉×#127	D₄	1	1	yes	characteristic
51	C₁₉×#127	C₂³	1	1	yes	characteristic
52	C₁₉×#124	—	2	4	no	—
53	C₁₉×#124	—	2	8	no	—
54	C₁₉×#124	—	2	1	no	—
55	C₁₉×#124	C₂³	1	4	yes	—
56	C₁₉×#124	C₂³	1	2	yes	—
57	C₁₉×#121	—	2	4	no	—
58	C₁₉×#121	—	2	4	no	—
59	C₁₉×#121	—	2	4	no	—
60	C₁₉×#121	—	2	2	no	—
61	C₁₉×#121	—	2	2	no	—
62	C₁₉×#121	—	2	2	no	—
63	C₁₉×#121	—	2	2	no	—
64	C₁₉×#120	—	2	4	no	—
65	C₁₉×#120	—	2	4	no	—
66	C₁₉×#120	—	2	8	no	—
67	C₄×D₄×C₁₉	—	2	2	no	—
68	C₄×D₄×C₁₉	—	2	2	no	—
69	C₄×D₄×C₁₉	—	2	2	no	—
70	C₄×D₄×C₁₉	—	2	2	no	—
71	C₁₉×#117	—	2	4	no	—
72	C₁₉×#117	—	2	2	no	—
73	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
74	C₁₉×C₄⋊C₈	—	2	2	no	—
75	C₁₉×#103	—	2	4	no	—
76	C₁₉×#102	—	2	4	no	—
77	C₁₉×C₂²⋊C₈	—	2	2	no	—
78	C₁₉×#97	—	2	1	no	—
79	C₁₉×#97	C₂³	1	2	yes	—
80	C₁₉×#97	C₂³	1	4	yes	—
81	C₂⁴×C₁₉	—	2	4	no	—
82	#41×C₁₉	—	4	4	no	—
83	#41×C₁₉	—	4	4	no	—
84	#41×C₁₉	—	8	4	no	—
85	C₂×Q₈×C₁₉	—	2	1	no	—
86	C₂×Q₈×C₁₉	C₂×D₄	1	2	yes	—
87	C₂×Q₈×C₁₉	C₂×D₄	1	4	yes	—
88	C₂×D₄×C₁₉	—	2	4	no	—
89	C₂×D₄×C₁₉	—	2	8	no	—
90	C₂×D₄×C₁₉	—	2	2	no	—
91	C₂×D₄×C₁₉	—	2	4	no	—
92	C₂×D₄×C₁₉	—	2	2	no	—
93	C₂×D₄×C₁₉	—	2	2	no	—
94	C₂×D₄×C₁₉	—	2	2	no	—
95	C₂×D₄×C₁₉	—	2	2	no	—
96	C₂×D₄×C₁₉	—	2	1	no	—
97	C₂×D₄×C₁₉	—	4	1	no	—
98	C₂×D₄×C₁₉	—	8	8	no	—
99	C₂×D₄×C₁₉	—	8	4	no	—
100	C₂×D₄×C₁₉	C₂×D₄	1	2	yes	—
101	C₂×D₄×C₁₉	C₂×D₄	1	4	yes	—
102	C₂²×C₄×C₁₉	—	2	4	no	—
103	C₂²×C₄×C₁₉	—	2	2	no	—
104	C₂²×C₄×C₁₉	—	4	1	no	—
105	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
107	Q₁₆×C₁₉	—	4	4	no	—
108	SD₁₆×C₁₉	—	4	4	no	—
109	SD₁₆×C₁₉	—	4	4	no	—
110	SD₁₆×C₁₉	—	8	4	no	—
111	D₈×C₁₉	—	4	4	no	—
112	D₈×C₁₉	—	8	4	no	—
113	C₂×C₈×C₁₉	—	2	4	no	—
114	C₂×C₈×C₁₉	—	2	2	no	—
115	C₂×C₈×C₁₉	—	2	4	no	—
116	C₄⋊C₄×C₁₉	—	2	2	no	—
117	C₄⋊C₄×C₁₉	—	2	2	no	—
118	C₄⋊C₄×C₁₉	—	2	2	no	—
119	C₄⋊C₄×C₁₉	—	2	2	no	—
120	C₂²⋊C₄×C₁₉	—	2	8	no	—
121	C₂²⋊C₄×C₁₉	—	2	4	no	—
122	C₂²⋊C₄×C₁₉	—	2	4	no	—
123	C₂²⋊C₄×C₁₉	—	2	4	no	—
124	C₂²⋊C₄×C₁₉	—	2	4	no	—
125	C₂²⋊C₄×C₁₉	—	4	2	no	—
126	C₄²×C₁₉	—	2	4	no	—
127	C₄²×C₁₉	C₂×D₄	1	2	yes	—
128	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
130	#19052	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂³×C₁₉	—	2	2	no	—
132	C₂³×C₁₉	—	2	4	no	—
133	C₂³×C₁₉	—	4	1	no	—
134	C₂³×C₁₉	—	8	4	no	—
135	C₂³×C₁₉	—	8	8	no	—
136	C₂³×C₁₉	#120	1	1	yes	characteristic
137	Q₈×C₁₉	—	4	2	no	—
138	Q₈×C₁₉	—	4	4	no	—
139	Q₈×C₁₉	—	4	2	no	—
140	D₄×C₁₉	—	4	2	no	—
141	D₄×C₁₉	—	4	4	no	—
142	D₄×C₁₉	—	4	4	no	—
143	D₄×C₁₉	—	4	4	no	—
144	D₄×C₁₉	—	4	2	no	—
145	D₄×C₁₉	—	4	2	no	—
146	D₄×C₁₉	—	4	2	no	—
147	D₄×C₁₉	—	4	4	no	—
148	D₄×C₁₉	—	8	2	no	—
149	D₄×C₁₉	—	8	4	no	—
150	D₄×C₁₉	—	8	4	no	—
151	D₄×C₁₉	—	8	8	no	—
152	D₄×C₁₉	—	8	2	no	—
153	C₂×C₄×C₁₉	—	2	2	no	—
154	C₂×C₄×C₁₉	—	2	2	no	—
155	C₂×C₄×C₁₉	—	2	4	no	—
156	C₂×C₄×C₁₉	—	4	2	no	—
157	C₂×C₄×C₁₉	—	4	8	no	—
158	C₂×C₄×C₁₉	—	4	4	no	—
159	C₂×C₄×C₁₉	—	4	1	no	—
160	C₂×C₄×C₁₉	—	8	2	no	—
161	C₂×C₄×C₁₉	#120	1	2	yes	—
162	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
163	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
164	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
165	C₈×C₁₉	—	2	4	no	—
166	C₈×C₁₉	—	4	4	no	—
167	#3549	C₂×C₁₉	1	1	yes	characteristic, complemented
168	#3214	C₂×C₁₉	1	2	yes	complemented
169	#3150	C₂×C₁₉	1	1	yes	characteristic, complemented
170	#1659	C₂×C₁₉	1	4	yes	complemented
171	#1655	C₂×C₁₉	1	4	yes	complemented
172	#1536	C₂×C₁₉	1	1	yes	characteristic, complemented
173	#1530	C₂×C₁₉	1	2	yes	complemented
174	C₂²×C₁₉	—	2	1	no	—
175	C₂²×C₁₉	—	2	2	no	—
176	C₂²×C₁₉	—	4	2	no	—
177	C₂²×C₁₉	—	4	4	no	—
178	C₂²×C₁₉	—	4	4	no	—
179	C₂²×C₁₉	—	4	4	no	—
180	C₂²×C₁₉	—	4	4	no	—
181	C₂²×C₁₉	—	8	1	no	—
182	C₂²×C₁₉	—	8	2	no	—
183	C₂²×C₁₉	—	16	8	no	—
184	C₂²×C₁₉	C₂×#120	1	1	yes	characteristic
185	C₄×C₁₉	—	2	1	no	—
186	C₄×C₁₉	—	2	2	no	—
187	C₄×C₁₉	—	2	2	no	—
188	C₄×C₁₉	—	4	2	no	—
189	C₄×C₁₉	—	4	4	no	—
190	C₄×C₁₉	—	8	1	no	—
191	C₄×C₁₉	#453	1	2	yes	—
192	C₂×#135	—	2	2	no	—
193	H(3,𝔽₄)	C₂²×C₁₉	1	4	yes	—
194	#555	—	2	2	no	—
195	#555	—	2	1	no	—
196	D₄²	—	2	2	no	—
197	D₄²	—	2	1	no	—
198	#535	—	2	2	no	—
199	#535	—	2	4	no	—
200	#532	C₂²×C₁₉	1	1	yes	characteristic
201	#532	C₂²×C₁₉	1	2	yes	—
202	#532	C₂²×C₁₉	1	1	yes	characteristic, complemented
203	#497	—	2	1	no	—
204	#497	C₂²×C₁₉	1	2	yes	complemented
205	#496	—	2	1	no	—
206	#496	C₂²×C₁₉	1	2	yes	complemented
207	#496	C₂²×C₁₉	1	4	yes	complemented
208	#489	C₂²×C₁₉	1	4	yes	complemented
209	#461	—	2	2	no	—
210	#459	—	2	2	no	—
211	#449	—	2	4	no	—
212	#432	C₂²×C₁₉	1	2	yes	complemented
213	C₂×#97	C₂²×C₁₉	1	1	yes	characteristic, complemented
214	#329	C₂²×C₁₉	1	4	yes	complemented
215	#329	C₂²×C₁₉	1	8	yes	complemented
216	C₂×C₁₉	—	4	1	no	—
217	C₂×C₁₉	—	8	2	no	—
218	C₂×C₁₉	—	8	4	no	—
219	C₂×C₁₉	—	16	1	no	—
220	C₂×C₁₉	C₂×#453	1	1	yes	characteristic
221	C₂×C₁₉	#3019	1	2	yes	—
222	C₂×#41	—	2	1	no	—
223	C₂×#41	D₄×C₁₉	1	2	yes	—
224	C₂×#41	D₄×C₁₉	1	4	yes	—
225	C₂²×D₄	—	2	4	no	—
226	C₂²×D₄	—	2	2	no	—
227	#135	—	4	8	no	—
228	C₂×Q₁₆	—	2	2	no	—
229	C₂×SD₁₆	—	2	2	no	—
230	C₂×SD₁₆	—	2	2	no	—
231	C₂×SD₁₆	—	2	4	no	—
232	C₂×D₈	—	2	2	no	—
233	C₂×D₈	—	2	4	no	—
234	C₂²×C₈	—	2	1	no	—
235	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
236	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
237	C₄⋊Q₈	C₂³×C₁₉	1	1	yes	characteristic
238	#127	—	2	4	no	—
239	#127	D₄×C₁₉	1	2	yes	—
240	#127	D₄×C₁₉	1	1	yes	characteristic
241	#127	C₂³×C₁₉	1	1	yes	characteristic
242	#124	—	2	8	no	—
243	#124	—	2	4	no	—
244	#124	—	2	1	no	—
245	#124	C₂³×C₁₉	1	4	yes	—
246	#124	C₂³×C₁₉	1	2	yes	—
247	#121	—	2	4	no	—
248	#121	—	2	4	no	—
249	#121	—	2	4	no	—
250	#121	—	2	2	no	—
251	#121	—	2	2	no	—
252	#121	—	2	2	no	—
253	#121	—	2	2	no	—
254	#120	—	2	4	no	—
255	#120	—	2	8	no	—
256	#120	—	2	4	no	—
257	C₄×D₄	—	2	2	no	—
258	C₄×D₄	—	2	2	no	—
259	C₄×D₄	—	2	2	no	—
260	C₄×D₄	—	2	2	no	—
261	#117	—	2	4	no	—
262	#117	—	2	2	no	—
263	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
264	C₄⋊C₈	—	2	2	no	—
265	#103	—	2	4	no	—
266	#102	—	2	4	no	—
267	C₂²⋊C₈	—	2	2	no	—
268	#97	—	2	1	no	—
269	#97	C₂³×C₁₉	1	2	yes	—
270	#97	C₂³×C₁₉	1	4	yes	—
271	C₁₉	#19052	1	1	yes	characteristic, direct product factor, Sylow
272	C₂⁴	—	2	4	no	—
273	#41	—	4	4	no	—
274	#41	—	4	4	no	—
275	#41	—	8	4	no	—
276	C₂×Q₈	—	2	1	no	—
277	C₂×Q₈	C₂×D₄×C₁₉	1	4	yes	—
278	C₂×Q₈	C₂×D₄×C₁₉	1	2	yes	—
279	C₂×D₄	—	2	4	no	—
280	C₂×D₄	—	2	8	no	—
281	C₂×D₄	—	2	2	no	—
282	C₂×D₄	—	2	4	no	—
283	C₂×D₄	—	2	2	no	—
284	C₂×D₄	—	2	2	no	—
285	C₂×D₄	—	2	2	no	—
286	C₂×D₄	—	2	2	no	—
287	C₂×D₄	—	2	1	no	—
288	C₂×D₄	—	4	1	no	—
289	C₂×D₄	—	8	8	no	—
290	C₂×D₄	—	8	4	no	—
291	C₂×D₄	C₂×D₄×C₁₉	1	4	yes	—
292	C₂×D₄	C₂×D₄×C₁₉	1	2	yes	—
293	C₂²×C₄	—	2	4	no	—
294	C₂²×C₄	—	2	2	no	—
295	C₂²×C₄	—	4	1	no	—
296	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
297	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
298	Q₁₆	—	4	4	no	—
299	SD₁₆	—	4	4	no	—
300	SD₁₆	—	4	4	no	—
301	SD₁₆	—	8	4	no	—
302	D₈	—	4	4	no	—
303	D₈	—	8	4	no	—
304	C₂×C₈	—	2	4	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	2	4	no	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₂²⋊C₄	—	2	8	no	—
312	C₂²⋊C₄	—	2	4	no	—
313	C₂²⋊C₄	—	2	4	no	—
314	C₂²⋊C₄	—	2	4	no	—
315	C₂²⋊C₄	—	2	4	no	—
316	C₂²⋊C₄	—	4	2	no	—
317	C₄²	—	2	4	no	—
318	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
319	C₄²	C₂×D₄×C₁₉	1	2	yes	—
320	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
321	C₂³	—	2	2	no	—
322	C₂³	—	2	4	no	—
323	C₂³	—	4	1	no	—
324	C₂³	—	8	4	no	—
325	C₂³	—	8	8	no	—
326	C₂³	C₁₉×#120	1	1	yes	characteristic
327	Q₈	—	4	2	no	—
328	Q₈	—	4	4	no	—
329	Q₈	—	4	2	no	—
330	D₄	—	4	2	no	—
331	D₄	—	4	4	no	—
332	D₄	—	4	4	no	—
333	D₄	—	4	4	no	—
334	D₄	—	4	2	no	—
335	D₄	—	4	2	no	—
336	D₄	—	4	2	no	—
337	D₄	—	4	4	no	—
338	D₄	—	8	2	no	—
339	D₄	—	8	4	no	—
340	D₄	—	8	4	no	—
341	D₄	—	8	8	no	—
342	D₄	—	8	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	4	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	8	no	—
348	C₂×C₄	—	4	4	no	—
349	C₂×C₄	—	4	1	no	—
350	C₂×C₄	—	8	2	no	—
351	C₂×C₄	C₁₉×#120	1	2	yes	—
352	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
353	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
354	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
355	C₈	—	2	4	no	—
356	C₈	—	4	4	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	2	no	—
359	C₂²	—	4	2	no	—
360	C₂²	—	4	4	no	—
361	C₂²	—	4	4	no	—
362	C₂²	—	4	4	no	—
363	C₂²	—	4	4	no	—
364	C₂²	—	8	1	no	—
365	C₂²	—	8	2	no	—
366	C₂²	—	16	8	no	—
367	C₂²	C₂×C₁₉×#120	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	2	2	no	—
370	C₄	—	2	2	no	—
371	C₄	—	4	2	no	—
372	C₄	—	4	4	no	—
373	C₄	—	8	1	no	—
374	C₄	C₁₉×#453	1	2	yes	—
375	C₂	—	4	1	no	—
376	C₂	—	8	2	no	—
377	C₂	—	8	4	no	—
378	C₂	—	16	1	no	—
379	C₂	C₂×C₁₉×#453	1	1	yes	characteristic
380	C₂	C₁₉×#3019	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₉
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₉×#3549	C₂	semidirect
2. C₁₉×#3214	C₂	semidirect
3. C₁₉×#3150	C₂	semidirect
4. C₁₉×#1659	C₂	semidirect
5. C₁₉×#1655	C₂	semidirect
6. C₁₉×#1536	C₂	semidirect
7. C₁₉×#1530	C₂	semidirect
18. C₁₉×#532	C₂²	semidirect
20. C₁₉×#497	C₂²	semidirect
22. C₁₉×#496	C₂²	semidirect
23. C₁₉×#496	C₂²	semidirect
24. C₁₉×#489	C₂²	semidirect
28. C₁₉×#432	C₂²	semidirect
29. C₂×C₁₉×#97	C₂²	semidirect
30. C₁₉×#329	C₂²	semidirect
31. C₁₉×#329	C₂²	semidirect
130. #19052	C₁₉	direct (N × H)
167. #3549	C₂×C₁₉	semidirect
168. #3214	C₂×C₁₉	semidirect
169. #3150	C₂×C₁₉	semidirect
170. #1659	C₂×C₁₉	semidirect
171. #1655	C₂×C₁₉	semidirect
172. #1536	C₂×C₁₉	semidirect
173. #1530	C₂×C₁₉	semidirect
202. #532	C₂²×C₁₉	semidirect
204. #497	C₂²×C₁₉	semidirect
206. #496	C₂²×C₁₉	semidirect
207. #496	C₂²×C₁₉	semidirect
208. #489	C₂²×C₁₉	semidirect
212. #432	C₂²×C₁₉	semidirect
213. C₂×#97	C₂²×C₁₉	semidirect
214. #329	C₂²×C₁₉	semidirect
215. #329	C₂²×C₁₉	semidirect
271. C₁₉	#19052	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×#3549	2
C₁₉×#3214	2
C₁₉×#3150	2
C₁₉×#1659	2
C₁₉×#1655	2
C₁₉×#1536	2
C₁₉×#1530	2
#19052	19

Subgroups — C19×#19052 #18025011

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#19052 #18025011