Subgroups — C₁₉×#32653 #18038612

Order 4864.

656 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2581	C₂	1	2	yes	maximal
2	C₁₉×#2097	C₂	1	2	yes	maximal
3	C₁₉×#2085	C₂	1	2	yes	maximal
4	C₁₉×#2066	C₂	1	2	yes	maximal
5	C₁₉×#1984	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1969	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1942	C₂	1	1	yes	maximal, characteristic
8	C₁₉×#1878	C₂	1	2	yes	maximal
9	C₁₉×#1863	C₂	1	2	yes	maximal
10	C₂²×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₁₉	C₂²	1	2	yes	—
12	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
13	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
14	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
15	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
16	C₂×C₁₉×#106	C₂²	1	2	yes	—
17	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₁₉×#103	C₂²	1	2	yes	—
19	C₂×C₁₉×#103	C₂²	1	2	yes	—
20	C₂×C₁₉×#103	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
22	C₁₉×#400	—	2	2	no	—
23	C₁₉×#395	—	2	2	no	—
24	C₁₉×#391	C₂²	1	2	yes	—
25	C₁₉×#391	C₂²	1	2	yes	—
26	C₁₉×#389	C₂²	1	2	yes	—
27	C₁₉×#389	C₂²	1	2	yes	—
28	C₁₉×#387	—	2	2	no	—
29	C₁₉×#387	—	2	2	no	—
30	C₁₉×#382	C₂²	1	2	yes	—
31	C₁₉×#340	C₂²	1	2	yes	—
32	C₁₉×#340	C₂²	1	2	yes	—
33	C₁₉×#336	C₂²	1	2	yes	—
34	C₁₉×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
36	C₂²×Q₈×C₁₉	C₂³	1	2	yes	—
37	C₂×Q₁₆×C₁₉	—	4	1	no	—
38	C₂×Q₁₆×C₁₉	—	4	1	no	—
39	C₂×Q₁₆×C₁₉	Q₈	1	4	yes	complemented
40	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
43	C₄×Q₈×C₁₉	—	4	1	no	—
44	C₄×Q₈×C₁₉	—	4	1	no	—
45	C₄×Q₈×C₁₉	—	4	2	no	—
46	C₄×Q₈×C₁₉	—	4	2	no	—
47	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
48	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
49	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
50	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
52	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
55	C₂×C₄²×C₁₉	—	2	2	no	—
56	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
57	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
58	C₁₉×#107	—	2	2	no	—
59	C₁₉×#107	—	4	1	no	—
60	C₁₉×#107	—	4	1	no	—
61	C₁₉×#106	—	4	2	no	—
62	C₁₉×C₄⋊C₈	—	4	1	no	—
63	C₁₉×#103	—	2	4	no	—
64	C₁₉×#103	—	4	2	no	—
65	C₁₉×#103	—	4	2	no	—
66	C₄×C₈×C₁₉	—	4	1	no	—
67	C₁₉×#95	—	2	2	no	—
68	C₁₉×#95	—	2	2	no	—
69	C₁₉×#95	—	2	2	no	—
70	C₁₉×#95	—	2	2	no	—
71	C₁₉×#95	—	2	2	no	—
72	C₁₉×#95	—	2	2	no	—
73	C₁₉×#95	—	2	2	no	—
74	C₁₉×#95	—	2	2	no	—
75	C₂×Q₈×C₁₉	—	2	2	no	—
76	C₂×Q₈×C₁₉	—	2	2	no	—
77	C₂×Q₈×C₁₉	—	2	2	no	—
78	C₂×Q₈×C₁₉	—	2	2	no	—
79	C₂×Q₈×C₁₉	—	2	2	no	—
80	C₂×Q₈×C₁₉	—	2	2	no	—
81	C₂×Q₈×C₁₉	—	4	1	no	—
82	C₂×Q₈×C₁₉	—	4	1	no	—
83	C₂×Q₈×C₁₉	C₂×Q₈	1	4	yes	—
84	C₂²×C₄×C₁₉	—	2	2	no	—
85	C₂²×C₄×C₁₉	—	2	2	no	—
86	C₂²×C₄×C₁₉	—	2	2	no	—
87	C₂²×C₄×C₁₉	—	2	2	no	—
88	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
93	Q₁₆×C₁₉	—	4	4	no	—
94	C₂×C₈×C₁₉	—	2	1	no	—
95	C₂×C₈×C₁₉	—	2	1	no	—
96	C₂×C₈×C₁₉	—	2	1	no	—
97	C₂×C₈×C₁₉	—	2	1	no	—
98	C₂×C₈×C₁₉	—	2	2	no	—
99	C₂×C₈×C₁₉	—	2	2	no	—
100	C₂×C₈×C₁₉	—	2	2	no	—
101	C₂×C₈×C₁₉	C₂×Q₈	1	2	yes	—
102	C₂×C₈×C₁₉	#41	1	2	yes	—
103	C₄⋊C₄×C₁₉	—	2	2	no	—
104	C₄⋊C₄×C₁₉	—	2	2	no	—
105	C₄⋊C₄×C₁₉	—	2	2	no	—
106	C₄⋊C₄×C₁₉	—	2	2	no	—
107	C₄⋊C₄×C₁₉	—	2	2	no	—
108	C₄⋊C₄×C₁₉	—	2	2	no	—
109	C₄⋊C₄×C₁₉	—	4	1	no	—
110	C₄⋊C₄×C₁₉	—	4	2	no	—
111	C₄⋊C₄×C₁₉	—	4	4	no	—
112	C₄⋊C₄×C₁₉	—	4	4	no	—
113	C₄⋊C₄×C₁₉	—	4	4	no	—
114	C₄²×C₁₉	—	2	2	no	—
115	C₄²×C₁₉	—	2	2	no	—
116	C₄²×C₁₉	—	2	1	no	—
117	C₄²×C₁₉	—	2	1	no	—
118	C₄²×C₁₉	—	4	4	no	—
119	#32653	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
121	Q₈×C₁₉	—	2	4	no	—
122	Q₈×C₁₉	—	2	4	no	—
123	Q₈×C₁₉	—	4	4	no	—
124	Q₈×C₁₉	—	4	2	no	—
125	Q₈×C₁₉	—	4	2	no	—
126	C₂×C₄×C₁₉	—	2	1	no	—
127	C₂×C₄×C₁₉	—	2	1	no	—
128	C₂×C₄×C₁₉	—	2	2	no	—
129	C₂×C₄×C₁₉	—	2	2	no	—
130	C₂×C₄×C₁₉	—	2	1	no	—
131	C₂×C₄×C₁₉	—	2	4	no	—
132	C₂×C₄×C₁₉	—	2	2	no	—
133	C₂×C₄×C₁₉	—	2	2	no	—
134	C₂×C₄×C₁₉	—	2	2	no	—
135	C₂×C₄×C₁₉	—	2	1	no	—
136	C₂×C₄×C₁₉	—	2	2	no	—
137	C₂×C₄×C₁₉	—	2	1	no	—
138	C₂×C₄×C₁₉	—	4	2	no	—
139	C₂×C₄×C₁₉	—	4	2	no	—
140	C₂×C₄×C₁₉	—	4	2	no	—
141	C₂×C₄×C₁₉	—	4	2	no	—
142	C₂×C₄×C₁₉	—	4	2	no	—
143	C₂×C₄×C₁₉	—	4	2	no	—
144	C₂×C₄×C₁₉	—	4	2	no	—
145	C₂×C₄×C₁₉	—	4	2	no	—
146	C₂×C₄×C₁₉	—	4	2	no	—
147	C₂×C₄×C₁₉	—	4	2	no	—
148	C₂×C₄×C₁₉	—	4	2	no	—
149	C₂×C₄×C₁₉	#135	1	2	yes	—
150	C₂×C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
151	C₂×C₄×C₁₉	C₂×#41	1	1	yes	characteristic
152	C₂×C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
153	C₂×C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
154	C₂×C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂×C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
156	C₈×C₁₉	—	2	2	no	—
157	C₈×C₁₉	—	2	2	no	—
158	C₈×C₁₉	—	4	2	no	—
159	#2581	C₂×C₁₉	1	2	yes	—
160	#2097	C₂×C₁₉	1	2	yes	—
161	#2085	C₂×C₁₉	1	2	yes	—
162	#2066	C₂×C₁₉	1	2	yes	—
163	#1984	C₂×C₁₉	1	1	yes	characteristic
164	#1969	C₂×C₁₉	1	1	yes	characteristic
165	#1942	C₂×C₁₉	1	1	yes	characteristic
166	#1878	C₂×C₁₉	1	2	yes	—
167	#1863	C₂×C₁₉	1	2	yes	—
168	C₂²×C₁₉	#534	1	1	yes	characteristic
169	C₂²×C₁₉	D₄×Q₈	1	1	yes	characteristic
170	C₂²×C₁₉	#550	1	1	yes	characteristic
171	C₂²×C₁₉	C₂×#135	1	1	yes	characteristic
172	C₂²×C₁₉	#575	1	1	yes	characteristic
173	C₂²×C₁₉	#578	1	1	yes	characteristic
174	C₂²×C₁₉	#578	1	1	yes	characteristic
175	C₄×C₁₉	—	2	2	no	—
176	C₄×C₁₉	—	4	2	no	—
177	C₄×C₁₉	—	4	2	no	—
178	C₄×C₁₉	—	4	1	no	—
179	C₄×C₁₉	—	4	4	no	—
180	C₄×C₁₉	—	8	2	no	—
181	C₄×C₁₉	—	8	2	no	—
182	C₄×C₁₉	—	8	2	no	—
183	C₄×C₁₉	#539	1	1	yes	characteristic
184	C₄×C₁₉	#543	1	1	yes	characteristic
185	C₄×C₁₉	#554	1	1	yes	characteristic
186	C₄×C₁₉	#562	1	1	yes	characteristic
187	C₂²×Q₁₆	C₂²×C₁₉	1	1	yes	characteristic
188	C₂×C₄×Q₈	C₂²×C₁₉	1	2	yes	—
189	C₂×C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
190	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
191	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
192	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
193	C₂×#106	C₂²×C₁₉	1	2	yes	—
194	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
195	C₂×#103	C₂²×C₁₉	1	2	yes	—
196	C₂×#103	C₂²×C₁₉	1	2	yes	—
197	C₂×#103	C₂²×C₁₉	1	2	yes	—
198	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
199	#400	—	2	2	no	—
200	#395	—	2	2	no	—
201	#391	C₂²×C₁₉	1	2	yes	—
202	#391	C₂²×C₁₉	1	2	yes	—
203	#389	C₂²×C₁₉	1	2	yes	—
204	#389	C₂²×C₁₉	1	2	yes	—
205	#387	—	2	2	no	—
206	#387	—	2	2	no	—
207	#382	C₂²×C₁₉	1	2	yes	—
208	#340	C₂²×C₁₉	1	2	yes	—
209	#340	C₂²×C₁₉	1	2	yes	—
210	#336	C₂²×C₁₉	1	2	yes	—
211	#336	C₂²×C₁₉	1	2	yes	—
212	C₂×C₁₉	#2732	1	1	yes	characteristic
213	C₂×C₁₉	#3191	1	1	yes	characteristic
214	C₂×C₁₉	#3202	1	1	yes	characteristic
215	C₂×C₁₉	#3383	1	1	yes	characteristic
216	C₂×C₁₉	#3390	1	1	yes	characteristic
217	C₂×C₁₉	#3393	1	1	yes	characteristic
218	C₂×C₁₉	#3402	1	1	yes	characteristic
219	C₂²×Q₈	D₄×C₁₉	1	1	yes	characteristic
220	C₂²×Q₈	C₂³×C₁₉	1	2	yes	—
221	C₂×Q₁₆	—	4	1	no	—
222	C₂×Q₁₆	—	4	1	no	—
223	C₂×Q₁₆	Q₈×C₁₉	1	4	yes	complemented
224	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
225	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
226	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
227	C₄×Q₈	—	4	1	no	—
228	C₄×Q₈	—	4	1	no	—
229	C₄×Q₈	—	4	2	no	—
230	C₄×Q₈	—	4	2	no	—
231	C₂×C₄⋊C₄	—	2	2	no	—
232	C₂×C₄⋊C₄	—	2	2	no	—
233	C₂×C₄⋊C₄	—	2	2	no	—
234	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
235	C₂×C₄⋊C₄	D₄×C₁₉	1	2	yes	—
236	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
237	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
238	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
239	C₂×C₄²	—	2	2	no	—
240	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
241	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
242	#107	—	2	2	no	—
243	#107	—	4	1	no	—
244	#107	—	4	1	no	—
245	#106	—	4	2	no	—
246	C₄⋊C₈	—	4	1	no	—
247	#103	—	2	4	no	—
248	#103	—	4	2	no	—
249	#103	—	4	2	no	—
250	C₄×C₈	—	4	1	no	—
251	#95	—	2	2	no	—
252	#95	—	2	2	no	—
253	#95	—	2	2	no	—
254	#95	—	2	2	no	—
255	#95	—	2	2	no	—
256	#95	—	2	2	no	—
257	#95	—	2	2	no	—
258	#95	—	2	2	no	—
259	C₁₉	#32653	1	1	yes	characteristic, direct product factor, Sylow
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	—	4	1	no	—
267	C₂×Q₈	—	4	1	no	—
268	C₂×Q₈	C₂×Q₈×C₁₉	1	4	yes	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	—	2	2	no	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
274	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
275	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
276	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
277	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
278	Q₁₆	—	4	4	no	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	2	1	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	C₂×Q₈×C₁₉	1	2	yes	—
287	C₂×C₈	#41×C₁₉	1	2	yes	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	4	1	no	—
295	C₄⋊C₄	—	4	2	no	—
296	C₄⋊C₄	—	4	4	no	—
297	C₄⋊C₄	—	4	4	no	—
298	C₄⋊C₄	—	4	4	no	—
299	C₄²	—	2	1	no	—
300	C₄²	—	2	1	no	—
301	C₄²	—	2	2	no	—
302	C₄²	—	2	2	no	—
303	C₄²	—	4	4	no	—
304	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
305	Q₈	—	2	4	no	—
306	Q₈	—	2	4	no	—
307	Q₈	—	4	4	no	—
308	Q₈	—	4	2	no	—
309	Q₈	—	4	2	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	C₁₉×#135	1	2	yes	—
334	C₂×C₄	C₂²×Q₈×C₁₉	1	1	yes	characteristic
335	C₂×C₄	C₂×#41×C₁₉	1	1	yes	characteristic
336	C₂×C₄	C₁₉×H(5,𝔽₂)	1	1	yes	characteristic
337	C₂×C₄	C₁₉×H(5,𝔽₂)	1	1	yes	characteristic
338	C₂×C₄	C₁₉×H(5,𝔽₂)	1	1	yes	characteristic
339	C₂×C₄	C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
340	C₈	—	2	2	no	—
341	C₈	—	2	2	no	—
342	C₈	—	4	2	no	—
343	C₂²	C₁₉×#534	1	1	yes	characteristic
344	C₂²	D₄×Q₈×C₁₉	1	1	yes	characteristic
345	C₂²	C₁₉×#550	1	1	yes	characteristic
346	C₂²	C₂×C₁₉×#135	1	1	yes	characteristic
347	C₂²	C₁₉×#575	1	1	yes	characteristic
348	C₂²	C₁₉×#578	1	1	yes	characteristic
349	C₂²	C₁₉×#578	1	1	yes	characteristic
350	C₄	—	2	2	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	2	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	4	no	—
355	C₄	—	8	2	no	—
356	C₄	—	8	2	no	—
357	C₄	—	8	2	no	—
358	C₄	C₁₉×#539	1	1	yes	characteristic
359	C₄	C₁₉×#543	1	1	yes	characteristic
360	C₄	C₁₉×#554	1	1	yes	characteristic
361	C₄	C₁₉×#562	1	1	yes	characteristic
362	C₂	C₁₉×#2732	1	1	yes	characteristic
363	C₂	C₁₉×#3191	1	1	yes	characteristic
364	C₂	C₁₉×#3202	1	1	yes	characteristic
365	C₂	C₁₉×#3383	1	1	yes	characteristic
366	C₂	C₁₉×#3390	1	1	yes	characteristic
367	C₂	C₁₉×#3393	1	1	yes	characteristic
368	C₂	C₁₉×#3402	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂×C₄×Q₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
39. C₂×Q₁₆×C₁₉	Q₈	semidirect
119. #32653	C₁₉	direct (N × H)
223. C₂×Q₁₆	Q₈×C₁₉	semidirect
259. C₁₉	#32653	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×#2581	2
C₁₉×#2097	2
C₁₉×#2085	2
C₁₉×#2066	2
C₁₉×#1984	2
C₁₉×#1969	2
C₁₉×#1942	2
C₁₉×#1878	2
C₁₉×#1863	2
#32653	19

Subgroups — C19×#32653 #18038612

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#32653 #18038612