Subgroups — C₁₉×#32863 #18038822

Order 4864.

600 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2863	C₂	1	2	yes	maximal
2	C₁₉×#2088	C₂	1	2	yes	maximal
3	C₁₉×#2086	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#2068	C₂	1	2	yes	maximal
5	C₁₉×#2066	C₂	1	2	yes	maximal
6	C₁₉×#2066	C₂	1	2	yes	maximal
7	C₁₉×#2056	C₂	1	2	yes	maximal
8	C₁₉×#2056	C₂	1	1	yes	maximal, characteristic
9	C₁₉×#1842	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
12	C₂×C₁₉×#106	C₂²	1	2	yes	—
13	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₁₉×#103	C₂²	1	2	yes	—
18	C₂×C₁₉×#103	C₂²	1	2	yes	—
19	C₂×C₁₉×#103	C₂²	1	2	yes	—
20	C₁₉×#391	C₂²	1	2	yes	—
21	C₁₉×#391	C₂²	1	2	yes	—
22	C₁₉×#391	C₂²	1	2	yes	—
23	C₁₉×#389	C₂²	1	2	yes	—
24	C₁₉×#389	C₂²	1	2	yes	—
25	C₁₉×#389	C₂²	1	2	yes	—
26	C₁₉×#387	—	2	2	no	—
27	C₁₉×#387	—	2	2	no	—
28	C₁₉×#387	—	2	2	no	—
29	C₁₉×#383	—	2	2	no	—
30	C₄³×C₁₉	C₂²	1	1	yes	characteristic
31	C₁₉×#340	C₂²	1	2	yes	—
32	C₁₉×#340	C₂²	1	1	yes	characteristic
33	C₁₉×#340	C₂²	1	1	yes	characteristic
34	C₁₉×#340	C₂²	1	2	yes	—
35	C₂²×Q₈×C₁₉	C₂³	1	2	yes	—
36	C₂×Q₁₆×C₁₉	—	4	2	no	—
37	C₂×Q₁₆×C₁₉	—	4	1	no	—
38	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
40	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
47	C₂×C₄²×C₁₉	D₄	1	2	yes	—
48	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
51	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
52	C₁₉×#107	—	4	1	no	—
53	C₁₉×#106	—	4	2	no	—
54	C₁₉×C₄⋊C₈	—	4	2	no	—
55	C₁₉×C₄⋊C₈	—	4	1	no	—
56	C₁₉×C₄⋊C₈	—	4	1	no	—
57	C₁₉×C₄⋊C₈	—	4	2	no	—
58	C₁₉×#103	—	4	2	no	—
59	C₁₉×#103	—	4	2	no	—
60	C₁₉×#103	—	4	2	no	—
61	C₁₉×#95	—	2	2	no	—
62	C₁₉×#95	—	2	2	no	—
63	C₁₉×#95	—	2	2	no	—
64	C₁₉×#95	—	2	2	no	—
65	C₁₉×#95	—	2	2	no	—
66	C₁₉×#95	—	2	2	no	—
67	C₁₉×#95	—	2	2	no	—
68	C₁₉×#95	—	2	2	no	—
69	C₁₉×#95	—	2	2	no	—
70	C₂×Q₈×C₁₉	—	2	2	no	—
71	C₂×Q₈×C₁₉	—	2	2	no	—
72	C₂×Q₈×C₁₉	—	2	2	no	—
73	C₂×Q₈×C₁₉	—	2	2	no	—
74	C₂×Q₈×C₁₉	—	2	2	no	—
75	C₂×Q₈×C₁₉	—	2	2	no	—
76	C₂²×C₄×C₁₉	—	2	2	no	—
77	C₂²×C₄×C₁₉	—	2	2	no	—
78	C₂²×C₄×C₁₉	—	2	2	no	—
79	C₂²×C₄×C₁₉	—	2	2	no	—
80	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
85	Q₁₆×C₁₉	—	4	4	no	—
86	C₂×C₈×C₁₉	—	2	2	no	—
87	C₂×C₈×C₁₉	—	2	1	no	—
88	C₂×C₈×C₁₉	—	2	2	no	—
89	C₂×C₈×C₁₉	—	2	2	no	—
90	C₂×C₈×C₁₉	—	2	2	no	—
91	C₂×C₈×C₁₉	—	2	2	no	—
92	C₂×C₈×C₁₉	—	2	1	no	—
93	C₄⋊C₄×C₁₉	—	2	2	no	—
94	C₄⋊C₄×C₁₉	—	2	2	no	—
95	C₄⋊C₄×C₁₉	—	2	2	no	—
96	C₄⋊C₄×C₁₉	—	2	2	no	—
97	C₄⋊C₄×C₁₉	—	2	2	no	—
98	C₄⋊C₄×C₁₉	—	2	2	no	—
99	C₄⋊C₄×C₁₉	—	4	4	no	—
100	C₄⋊C₄×C₁₉	—	4	4	no	—
101	C₄⋊C₄×C₁₉	—	4	4	no	—
102	C₄²×C₁₉	—	2	2	no	—
103	C₄²×C₁₉	—	2	4	no	—
104	C₄²×C₁₉	—	2	2	no	—
105	C₄²×C₁₉	—	2	1	no	—
106	C₄²×C₁₉	—	2	2	no	—
107	C₄²×C₁₉	—	2	1	no	—
108	C₄²×C₁₉	—	4	1	no	—
109	#32863	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
111	Q₈×C₁₉	—	4	2	no	—
112	Q₈×C₁₉	—	4	2	no	—
113	Q₈×C₁₉	—	4	2	no	—
114	Q₈×C₁₉	—	4	2	no	—
115	C₂×C₄×C₁₉	—	2	2	no	—
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	2	2	no	—
118	C₂×C₄×C₁₉	—	2	1	no	—
119	C₂×C₄×C₁₉	—	2	2	no	—
120	C₂×C₄×C₁₉	—	2	1	no	—
121	C₂×C₄×C₁₉	—	2	2	no	—
122	C₂×C₄×C₁₉	—	2	2	no	—
123	C₂×C₄×C₁₉	—	2	2	no	—
124	C₂×C₄×C₁₉	—	4	2	no	—
125	C₂×C₄×C₁₉	—	4	2	no	—
126	C₂×C₄×C₁₉	—	4	2	no	—
127	C₂×C₄×C₁₉	—	4	2	no	—
128	C₂×C₄×C₁₉	—	4	2	no	—
129	C₂×C₄×C₁₉	—	4	2	no	—
130	C₂×C₄×C₁₉	—	4	2	no	—
131	C₂×C₄×C₁₉	—	4	2	no	—
132	C₂×C₄×C₁₉	—	4	2	no	—
133	C₂×C₄×C₁₉	—	4	2	no	—
134	C₂×C₄×C₁₉	—	4	2	no	—
135	C₂×C₄×C₁₉	—	4	2	no	—
136	C₂×C₄×C₁₉	#135	1	2	yes	—
137	C₂×C₄×C₁₉	#137	1	4	yes	—
138	C₂×C₄×C₁₉	H(5,𝔽₂)	1	2	yes	—
139	C₂×C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
140	C₂×C₄×C₁₉	2₋¹⁺⁴	1	2	yes	—
141	C₂×C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
142	C₈×C₁₉	—	4	2	no	—
143	C₈×C₁₉	—	4	1	no	—
144	C₈×C₁₉	—	4	1	no	—
145	#2863	C₂×C₁₉	1	2	yes	—
146	#2088	C₂×C₁₉	1	2	yes	—
147	#2086	C₂×C₁₉	1	1	yes	characteristic
148	#2068	C₂×C₁₉	1	2	yes	—
149	#2066	C₂×C₁₉	1	2	yes	—
150	#2066	C₂×C₁₉	1	2	yes	—
151	#2056	C₂×C₁₉	1	2	yes	—
152	#2056	C₂×C₁₉	1	1	yes	characteristic
153	#1842	C₂×C₁₉	1	1	yes	characteristic
154	C₂²×C₁₉	#537	1	2	yes	—
155	C₂²×C₁₉	#537	1	1	yes	characteristic
156	C₂²×C₁₉	C₂×#135	1	1	yes	characteristic
157	C₂²×C₁₉	C₂×#137	1	2	yes	—
158	C₂²×C₁₉	#575	1	1	yes	characteristic
159	C₄×C₁₉	—	2	4	no	—
160	C₄×C₁₉	—	2	2	no	—
161	C₄×C₁₉	—	4	2	no	—
162	C₄×C₁₉	—	4	1	no	—
163	C₄×C₁₉	—	8	2	no	—
164	C₄×C₁₉	—	8	2	no	—
165	C₄×C₁₉	—	8	2	no	—
166	C₄×C₁₉	—	8	2	no	—
167	C₄×C₁₉	#560	1	1	yes	characteristic
168	C₄×C₁₉	#563	1	2	yes	—
169	C₄×C₁₉	#563	1	1	yes	characteristic
170	C₂²×Q₁₆	C₂²×C₁₉	1	1	yes	characteristic
171	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
172	C₂×#106	C₂²×C₁₉	1	2	yes	—
173	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
174	C₂×C₄⋊C₈	C₂²×C₁₉	1	2	yes	—
175	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
176	C₂×C₄⋊C₈	C₂²×C₁₉	1	2	yes	—
177	C₂×#103	C₂²×C₁₉	1	2	yes	—
178	C₂×#103	C₂²×C₁₉	1	2	yes	—
179	C₂×#103	C₂²×C₁₉	1	2	yes	—
180	#391	C₂²×C₁₉	1	2	yes	—
181	#391	C₂²×C₁₉	1	2	yes	—
182	#391	C₂²×C₁₉	1	2	yes	—
183	#389	C₂²×C₁₉	1	2	yes	—
184	#389	C₂²×C₁₉	1	2	yes	—
185	#389	C₂²×C₁₉	1	2	yes	—
186	#387	—	2	2	no	—
187	#387	—	2	2	no	—
188	#387	—	2	2	no	—
189	#383	—	2	2	no	—
190	C₄³	C₂²×C₁₉	1	1	yes	characteristic
191	#340	C₂²×C₁₉	1	2	yes	—
192	#340	C₂²×C₁₉	1	1	yes	characteristic
193	#340	C₂²×C₁₉	1	1	yes	characteristic
194	#340	C₂²×C₁₉	1	2	yes	—
195	C₂×C₁₉	#2823	1	1	yes	characteristic
196	C₂×C₁₉	#3248	1	2	yes	—
197	C₂×C₁₉	#3250	1	1	yes	characteristic
198	C₂×C₁₉	#3252	1	2	yes	—
199	C₂×C₁₉	#3255	1	1	yes	characteristic
200	C₂²×Q₈	C₂³×C₁₉	1	2	yes	—
201	C₂×Q₁₆	—	4	2	no	—
202	C₂×Q₁₆	—	4	1	no	—
203	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
204	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
205	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
210	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
211	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
212	C₂×C₄²	D₄×C₁₉	1	2	yes	—
213	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
214	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
215	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
216	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
217	#107	—	4	1	no	—
218	#106	—	4	2	no	—
219	C₄⋊C₈	—	4	1	no	—
220	C₄⋊C₈	—	4	2	no	—
221	C₄⋊C₈	—	4	2	no	—
222	C₄⋊C₈	—	4	1	no	—
223	#103	—	4	2	no	—
224	#103	—	4	2	no	—
225	#103	—	4	2	no	—
226	#95	—	2	2	no	—
227	#95	—	2	2	no	—
228	#95	—	2	2	no	—
229	#95	—	2	2	no	—
230	#95	—	2	2	no	—
231	#95	—	2	2	no	—
232	#95	—	2	2	no	—
233	#95	—	2	2	no	—
234	#95	—	2	2	no	—
235	C₁₉	#32863	1	1	yes	characteristic, direct product factor, Sylow
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
248	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
250	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
251	Q₁₆	—	4	4	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	1	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	4	no	—
266	C₄⋊C₄	—	4	4	no	—
267	C₄⋊C₄	—	4	4	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	4	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	1	no	—
274	C₄²	—	4	1	no	—
275	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
276	Q₈	—	4	2	no	—
277	Q₈	—	4	2	no	—
278	Q₈	—	4	2	no	—
279	Q₈	—	4	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	C₁₉×#135	1	2	yes	—
302	C₂×C₄	C₁₉×#137	1	4	yes	—
303	C₂×C₄	C₁₉×H(5,𝔽₂)	1	2	yes	—
304	C₂×C₄	C₁₉×H(5,𝔽₂)	1	1	yes	characteristic
305	C₂×C₄	C₁₉×2₋¹⁺⁴	1	2	yes	—
306	C₂×C₄	C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
307	C₈	—	4	1	no	—
308	C₈	—	4	2	no	—
309	C₈	—	4	1	no	—
310	C₂²	C₁₉×#537	1	2	yes	—
311	C₂²	C₁₉×#537	1	1	yes	characteristic
312	C₂²	C₂×C₁₉×#135	1	1	yes	characteristic
313	C₂²	C₂×C₁₉×#137	1	2	yes	—
314	C₂²	C₁₉×#575	1	1	yes	characteristic
315	C₄	—	2	4	no	—
316	C₄	—	2	2	no	—
317	C₄	—	4	2	no	—
318	C₄	—	4	1	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	2	no	—
322	C₄	—	8	2	no	—
323	C₄	C₁₉×#560	1	1	yes	characteristic
324	C₄	C₁₉×#563	1	2	yes	—
325	C₄	C₁₉×#563	1	1	yes	characteristic
326	C₂	C₁₉×#2823	1	1	yes	characteristic
327	C₂	C₁₉×#3248	1	2	yes	—
328	C₂	C₁₉×#3250	1	1	yes	characteristic
329	C₂	C₁₉×#3252	1	2	yes	—
330	C₂	C₁₉×#3255	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 ◁ C₄³×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
109. #32863	C₁₉	direct (N × H)
235. C₁₉	#32863	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×#2863	2
C₁₉×#2088	2
C₁₉×#2086	2
C₁₉×#2068	2
C₁₉×#2066	2
C₁₉×#2056	2
C₁₉×#1842	2
#32863	19

Subgroups — C19×#32863 #18038822

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#32863 #18038822