Subgroups — C₁₉×#3178 #14561035

Order 2432.

350 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#542	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#541	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#501	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#497	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#488	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#487	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#470	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₉×#468	C₂	1	1	yes	maximal, characteristic
9	C₁₉×#467	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₉×#465	C₂	1	1	yes	maximal, characteristic
11	C₁₉×#441	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₉×#440	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×Q₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×SD₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#405	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
18	C₂×SD₁₆×C₁₉	C₂²	1	1	yes	characteristic
19	C₂×SD₁₆×C₁₉	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₁₉	C₂²	1	1	yes	characteristic
22	C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
23	C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
24	C₁₉×#126	—	2	1	no	—
25	C₁₉×#126	—	2	1	no	—
26	C₁₉×#126	—	2	1	no	—
27	C₁₉×#125	—	2	1	no	—
28	C₁₉×#125	—	2	1	no	—
29	C₁₉×#125	C₂²	1	1	yes	characteristic
30	C₁₉×#124	—	2	1	no	—
31	C₁₉×#124	C₂²	1	1	yes	characteristic
32	C₁₉×#123	—	2	1	no	—
33	C₁₉×C₂²⋊Q₈	—	2	1	no	—
34	C₁₉×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
35	C₁₉×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
36	C₁₉×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
37	C₁₉×#121	C₂²	1	1	yes	characteristic
38	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
39	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
40	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
41	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
42	C₁₉×#117	C₂²	1	1	yes	characteristic
43	C₁₉×#107	C₂²	1	1	yes	characteristic
44	C₁₉×#107	C₂²	1	1	yes	characteristic
45	C₁₉×#106	C₂²	1	1	yes	characteristic
46	C₁₉×#106	C₂²	1	1	yes	characteristic
47	C₁₉×#103	C₂²	1	1	yes	characteristic
48	C₁₉×#103	C₂²	1	1	yes	characteristic
49	C₁₉×#103	C₂²	1	1	yes	characteristic
50	C₁₉×#103	C₂²	1	1	yes	characteristic
51	C₁₉×#103	C₂²	1	1	yes	characteristic
52	C₁₉×#103	C₂²	1	1	yes	characteristic
53	C₁₉×#102	C₂²	1	1	yes	characteristic
54	C₁₉×#102	C₂²	1	1	yes	characteristic
55	C₁₉×#97	C₂²	1	1	yes	characteristic
56	C₁₉×#97	C₂²	1	1	yes	characteristic
57	C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₉	—	2	1	no	—
60	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
61	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
62	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
63	C₂×D₄×C₁₉	—	2	1	no	—
64	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	—	2	1	no	—
66	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
67	Q₁₆×C₁₉	—	2	2	no	—
68	Q₁₆×C₁₉	—	2	2	no	—
69	SD₁₆×C₁₉	—	2	2	no	—
70	SD₁₆×C₁₉	—	2	2	no	—
71	M₁₆×C₁₉	—	2	2	no	—
72	C₂×C₈×C₁₉	—	2	2	no	—
73	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₁₉	—	2	1	no	—
78	C₄⋊C₄×C₁₉	—	2	1	no	—
79	C₄⋊C₄×C₁₉	—	2	1	no	—
80	C₄⋊C₄×C₁₉	—	2	1	no	—
81	C₄⋊C₄×C₁₉	—	2	1	no	—
82	C₄⋊C₄×C₁₉	—	2	1	no	—
83	C₄⋊C₄×C₁₉	—	2	1	no	—
84	C₄⋊C₄×C₁₉	—	2	1	no	—
85	C₄⋊C₄×C₁₉	—	2	1	no	—
86	C₄⋊C₄×C₁₉	—	2	1	no	—
87	C₄⋊C₄×C₁₉	—	2	1	no	—
88	C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
89	C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
90	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
91	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
92	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
93	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
94	C₂²⋊C₄×C₁₉	—	2	1	no	—
95	C₂²⋊C₄×C₁₉	—	2	1	no	—
96	C₂²⋊C₄×C₁₉	—	2	1	no	—
97	C₂²⋊C₄×C₁₉	—	2	1	no	—
98	C₂²⋊C₄×C₁₉	—	2	1	no	—
99	C₂²⋊C₄×C₁₉	—	2	1	no	—
100	C₂²⋊C₄×C₁₉	—	2	1	no	—
101	C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
102	C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
103	C₄²×C₁₉	—	2	1	no	—
104	C₄²×C₁₉	—	2	1	no	—
105	C₄²×C₁₉	—	2	1	no	—
106	C₄²×C₁₉	C₂³	1	1	yes	characteristic
107	C₄²×C₁₉	C₂³	1	1	yes	characteristic
108	C₂³×C₁₉	—	2	1	no	—
109	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₁₉	—	2	1	no	—
111	Q₈×C₁₉	—	2	1	no	—
112	Q₈×C₁₉	—	2	1	no	—
113	Q₈×C₁₉	—	2	1	no	—
114	Q₈×C₁₉	—	2	1	no	—
115	Q₈×C₁₉	—	2	1	no	—
116	Q₈×C₁₉	—	8	1	no	—
117	D₄×C₁₉	—	2	1	no	—
118	D₄×C₁₉	—	2	1	no	—
119	D₄×C₁₉	—	8	1	no	—
120	C₂×C₄×C₁₉	—	2	1	no	—
121	C₂×C₄×C₁₉	—	2	1	no	—
122	C₂×C₄×C₁₉	—	2	1	no	—
123	C₂×C₄×C₁₉	—	2	1	no	—
124	C₂×C₄×C₁₉	—	2	1	no	—
125	C₂×C₄×C₁₉	—	2	1	no	—
126	C₂×C₄×C₁₉	—	2	1	no	—
127	C₂×C₄×C₁₉	—	2	1	no	—
128	C₂×C₄×C₁₉	—	2	1	no	—
129	C₂×C₄×C₁₉	—	8	1	no	—
130	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
136	C₈×C₁₉	—	2	1	no	—
137	C₈×C₁₉	—	2	1	no	—
138	C₈×C₁₉	#41	1	2	yes	—
139	C₈×C₁₉	#41	1	2	yes	—
140	#3178	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂²×C₁₉	—	2	1	no	—
142	C₂²×C₁₉	—	2	1	no	—
143	C₂²×C₁₉	—	2	1	no	—
144	C₂²×C₁₉	—	4	1	no	—
145	C₂²×C₁₉	—	4	1	no	—
146	C₂²×C₁₉	—	4	1	no	—
147	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
148	C₄×C₁₉	—	2	1	no	—
149	C₄×C₁₉	—	2	1	no	—
150	C₄×C₁₉	—	2	1	no	—
151	C₄×C₁₉	—	2	1	no	—
152	C₄×C₁₉	—	2	1	no	—
153	C₄×C₁₉	—	4	1	no	—
154	C₄×C₁₉	—	4	1	no	—
155	C₄×C₁₉	—	4	1	no	—
156	C₄×C₁₉	—	4	1	no	—
157	C₄×C₁₉	—	4	1	no	—
158	C₄×C₁₉	—	4	1	no	—
159	C₄×C₁₉	—	4	1	no	—
160	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
161	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
162	#542	C₂×C₁₉	1	1	yes	characteristic
163	#541	C₂×C₁₉	1	1	yes	characteristic, complemented
164	#501	C₂×C₁₉	1	1	yes	characteristic, complemented
165	#497	C₂×C₁₉	1	1	yes	characteristic, complemented
166	#488	C₂×C₁₉	1	1	yes	characteristic, complemented
167	#487	C₂×C₁₉	1	1	yes	characteristic, complemented
168	#470	C₂×C₁₉	1	1	yes	characteristic, complemented
169	#468	C₂×C₁₉	1	1	yes	characteristic
170	#467	C₂×C₁₉	1	1	yes	characteristic, complemented
171	#465	C₂×C₁₉	1	1	yes	characteristic
172	#441	C₂×C₁₉	1	1	yes	characteristic, complemented
173	#440	C₂×C₁₉	1	1	yes	characteristic, complemented
174	C₄×Q₁₆	C₂×C₁₉	1	1	yes	characteristic, complemented
175	C₄×SD₁₆	C₂×C₁₉	1	1	yes	characteristic, complemented
176	#405	C₂×C₁₉	1	1	yes	characteristic, complemented
177	C₂×C₁₉	—	4	1	no	—
178	C₂×C₁₉	—	8	1	no	—
179	C₂×C₁₉	#532	1	1	yes	characteristic
180	C₂×C₁₉	#577	1	1	yes	characteristic
181	C₂×C₁₉	#578	1	1	yes	characteristic
182	C₂×Q₁₆	C₂²×C₁₉	1	1	yes	characteristic
183	C₂×Q₁₆	C₂²×C₁₉	1	1	yes	characteristic
184	C₂×SD₁₆	C₂²×C₁₉	1	1	yes	characteristic
185	C₂×SD₁₆	C₂²×C₁₉	1	1	yes	characteristic
186	C₂×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
187	C₂²×C₈	C₂²×C₁₉	1	1	yes	characteristic
188	C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
189	C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
190	#126	—	2	1	no	—
191	#126	—	2	1	no	—
192	#126	—	2	1	no	—
193	#125	—	2	1	no	—
194	#125	—	2	1	no	—
195	#125	C₂²×C₁₉	1	1	yes	characteristic
196	#124	—	2	1	no	—
197	#124	C₂²×C₁₉	1	1	yes	characteristic
198	#123	—	2	1	no	—
199	C₂²⋊Q₈	—	2	1	no	—
200	C₂²⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
201	C₂²⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
202	C₂²⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
203	#121	C₂²×C₁₉	1	1	yes	characteristic
204	C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
205	C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
206	C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
207	C₄×D₄	C₂²×C₁₉	1	1	yes	characteristic
208	#117	C₂²×C₁₉	1	1	yes	characteristic
209	#107	C₂²×C₁₉	1	1	yes	characteristic
210	#107	C₂²×C₁₉	1	1	yes	characteristic
211	#106	C₂²×C₁₉	1	1	yes	characteristic
212	#106	C₂²×C₁₉	1	1	yes	characteristic
213	#103	C₂²×C₁₉	1	1	yes	characteristic
214	#103	C₂²×C₁₉	1	1	yes	characteristic
215	#103	C₂²×C₁₉	1	1	yes	characteristic
216	#103	C₂²×C₁₉	1	1	yes	characteristic
217	#103	C₂²×C₁₉	1	1	yes	characteristic
218	#103	C₂²×C₁₉	1	1	yes	characteristic
219	#102	C₂²×C₁₉	1	1	yes	characteristic
220	#102	C₂²×C₁₉	1	1	yes	characteristic
221	#97	C₂²×C₁₉	1	1	yes	characteristic
222	#97	C₂²×C₁₉	1	1	yes	characteristic
223	C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
224	C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
225	C₁₉	#3178	1	1	yes	characteristic, direct product factor, Sylow
226	C₂×Q₈	—	2	1	no	—
227	C₂×Q₈	C₂³×C₁₉	1	1	yes	characteristic
228	C₂×Q₈	C₂³×C₁₉	1	1	yes	characteristic
229	C₂×Q₈	C₂³×C₁₉	1	1	yes	characteristic
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	C₂³×C₁₉	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
234	Q₁₆	—	2	2	no	—
235	Q₁₆	—	2	2	no	—
236	SD₁₆	—	2	2	no	—
237	SD₁₆	—	2	2	no	—
238	M₁₆	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	C₂³×C₁₉	1	1	yes	characteristic
241	C₂×C₈	C₂³×C₁₉	1	1	yes	characteristic
242	C₂×C₈	C₂³×C₁₉	1	1	yes	characteristic
243	C₂×C₈	C₂³×C₁₉	1	1	yes	characteristic
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
256	C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
257	C₄⋊C₄	C₂³×C₁₉	1	1	yes	characteristic
258	C₄⋊C₄	C₂³×C₁₉	1	1	yes	characteristic
259	C₄⋊C₄	C₂³×C₁₉	1	1	yes	characteristic
260	C₄⋊C₄	C₂³×C₁₉	1	1	yes	characteristic
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	D₄×C₁₉	1	1	yes	characteristic
269	C₂²⋊C₄	D₄×C₁₉	1	1	yes	characteristic
270	C₄²	—	2	1	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	2	1	no	—
273	C₄²	C₂³×C₁₉	1	1	yes	characteristic
274	C₄²	C₂³×C₁₉	1	1	yes	characteristic
275	C₂³	—	2	1	no	—
276	C₂³	C₂×D₄×C₁₉	1	1	yes	characteristic
277	Q₈	—	2	1	no	—
278	Q₈	—	2	1	no	—
279	Q₈	—	2	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	8	1	no	—
284	D₄	—	2	1	no	—
285	D₄	—	2	1	no	—
286	D₄	—	8	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	8	1	no	—
297	C₂×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
298	C₂×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
299	C₂×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
300	C₂×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
301	C₂×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
302	C₂×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	#41×C₁₉	1	2	yes	—
306	C₈	#41×C₁₉	1	2	yes	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	C₂²×D₄×C₁₉	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	C₂×#41×C₁₉	1	1	yes	characteristic
327	C₄	C₂×#41×C₁₉	1	1	yes	characteristic
328	C₂	—	4	1	no	—
329	C₂	—	8	1	no	—
330	C₂	C₁₉×#532	1	1	yes	characteristic
331	C₂	C₁₉×#577	1	1	yes	characteristic
332	C₂	C₁₉×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₉
prime radical	C₁₉×#121	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₁₉×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#541	C₂	semidirect
3. C₁₉×#501	C₂	semidirect
4. C₁₉×#497	C₂	semidirect
5. C₁₉×#488	C₂	semidirect
6. C₁₉×#487	C₂	semidirect
7. C₁₉×#470	C₂	semidirect
9. C₁₉×#467	C₂	semidirect
11. C₁₉×#441	C₂	semidirect
12. C₁₉×#440	C₂	semidirect
13. C₄×Q₁₆×C₁₉	C₂	semidirect
14. C₄×SD₁₆×C₁₉	C₂	semidirect
15. C₁₉×#405	C₂	semidirect
140. #3178	C₁₉	direct (N × H)
163. #541	C₂×C₁₉	semidirect
164. #501	C₂×C₁₉	semidirect
165. #497	C₂×C₁₉	semidirect
166. #488	C₂×C₁₉	semidirect
167. #487	C₂×C₁₉	semidirect
168. #470	C₂×C₁₉	semidirect
170. #467	C₂×C₁₉	semidirect
172. #441	C₂×C₁₉	semidirect
173. #440	C₂×C₁₉	semidirect
174. C₄×Q₁₆	C₂×C₁₉	semidirect
175. C₄×SD₁₆	C₂×C₁₉	semidirect
176. #405	C₂×C₁₉	semidirect
225. C₁₉	#3178	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×#542	2
C₁₉×#541	2
C₁₉×#501	2
C₁₉×#497	2
C₁₉×#488	2
C₁₉×#487	2
C₁₉×#470	2
C₁₉×#468	2
C₁₉×#467	2
C₁₉×#465	2
C₁₉×#441	2
C₁₉×#440	2
C₄×Q₁₆×C₁₉	2
C₄×SD₁₆×C₁₉	2
C₁₉×#405	2
#3178	19

Subgroups — C19×#3178 #14561035

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#3178 #14561035