Subgroups — C₁₉×#20610 #18026569

Order 4864.

3754 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#585	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#410	C₂	1	3	yes	maximal, complemented
3	C₁₉×#2311	C₂	1	3	yes	maximal, complemented
4	C₁₉×#1790	C₂	1	8	yes	maximal, complemented
5	C₁₉×#585	—	2	12	no	—
6	C₁₉×#585	C₄	1	8	yes	complemented
7	C₂×C₁₉×2₋¹⁺⁴	—	2	3	no	—
8	C₂×C₁₉×H(5,𝔽₂)	—	2	3	no	—
9	C₂×C₁₉×H(5,𝔽₂)	C₂²	1	4	yes	—
10	C₂²×#41×C₁₉	—	2	6	no	—
11	C₂²×#41×C₁₉	C₂²	1	3	yes	—
12	C₂×C₁₉×#117	C₂²	1	3	yes	complemented
13	C₂²×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic, complemented
14	C₁₉×#410	—	2	12	no	—
15	C₂×C₁₉×#99	C₂²	1	12	yes	complemented
16	C₁₉×#391	—	2	3	no	—
17	C₁₉×#390	—	2	3	no	—
18	C₁₉×#386	C₂²	1	12	yes	complemented
19	C₁₉×#380	—	2	6	no	—
20	C₁₉×2₋¹⁺⁴	—	2	24	no	—
21	C₁₉×2₋¹⁺⁴	—	2	24	no	—
22	C₁₉×H(5,𝔽₂)	—	2	24	no	—
23	C₁₉×H(5,𝔽₂)	—	2	24	no	—
24	C₁₉×H(5,𝔽₂)	—	2	24	no	—
25	C₁₉×H(5,𝔽₂)	C₂×C₄	1	16	yes	—
26	C₂×#41×C₁₉	—	2	4	no	—
27	C₂×#41×C₁₉	—	2	12	no	—
28	C₂×#41×C₁₉	—	2	48	no	—
29	C₂×#41×C₁₉	—	2	48	no	—
30	C₂×#41×C₁₉	—	2	6	no	—
31	C₂×#41×C₁₉	—	2	12	no	—
32	C₂×#41×C₁₉	—	2	6	no	—
33	C₂×#41×C₁₉	—	2	12	no	—
34	C₂×#41×C₁₉	C₂×C₄	1	12	yes	—
35	C₂×#41×C₁₉	D₄	1	12	yes	—
36	C₂²×Q₈×C₁₉	—	2	6	no	—
37	C₂²×Q₈×C₁₉	D₄	1	3	yes	—
38	C₂²×D₄×C₁₉	—	2	6	no	—
39	C₂²×D₄×C₁₉	—	2	12	no	—
40	C₂²×D₄×C₁₉	D₄	1	3	yes	—
41	C₂²×D₄×C₁₉	C₂³	1	6	yes	—
42	C₂³×C₄×C₁₉	—	2	4	no	—
43	C₂³×C₄×C₁₉	D₄	1	6	yes	—
44	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
45	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
46	C₁₉×#117	—	2	12	no	—
47	C₂×C₄⋊C₄×C₁₉	—	2	3	no	—
48	C₂×C₄⋊C₄×C₁₉	—	2	3	no	—
49	C₂×C₄⋊C₄×C₁₉	—	2	3	no	—
50	C₂×C₂²⋊C₄×C₁₉	—	2	12	no	—
51	C₂×C₂²⋊C₄×C₁₉	C₂³	1	6	yes	—
52	C₂×C₄²×C₁₉	—	2	3	no	—
53	C₁₉×#99	—	2	24	no	—
54	C₁₉×#95	—	2	12	no	—
55	C₂⁴×C₁₉	—	2	4	no	—
56	C₂⁴×C₁₉	C₂×D₄	1	6	yes	—
57	C₂⁴×C₁₉	C₂⁴	1	1	yes	characteristic
58	#41×C₁₉	—	2	32	no	—
59	#41×C₁₉	—	2	48	no	—
60	#41×C₁₉	—	2	96	no	—
61	#41×C₁₉	—	2	24	no	—
62	#41×C₁₉	—	2	48	no	—
63	#41×C₁₉	—	2	24	no	—
64	#41×C₁₉	—	2	24	no	—
65	#41×C₁₉	—	2	24	no	—
66	C₂×Q₈×C₁₉	—	2	4	no	—
67	C₂×Q₈×C₁₉	—	2	48	no	—
68	C₂×Q₈×C₁₉	—	2	6	no	—
69	C₂×Q₈×C₁₉	—	2	6	no	—
70	C₂×Q₈×C₁₉	C₂²⋊C₄	1	12	yes	complemented
71	C₂×D₄×C₁₉	—	2	12	no	—
72	C₂×D₄×C₁₉	—	2	48	no	—
73	C₂×D₄×C₁₉	—	2	48	no	—
74	C₂×D₄×C₁₉	—	2	48	no	—
75	C₂×D₄×C₁₉	—	2	12	no	—
76	C₂×D₄×C₁₉	—	2	12	no	—
77	C₂×D₄×C₁₉	—	2	6	no	—
78	C₂×D₄×C₁₉	—	2	6	no	—
79	C₂×D₄×C₁₉	C₂²⋊C₄	1	12	yes	complemented
80	C₂×D₄×C₁₉	C₂²×C₄	1	12	yes	—
81	C₂×D₄×C₁₉	C₂×D₄	1	12	yes	—
82	C₂²×C₄×C₁₉	—	2	12	no	—
83	C₂²×C₄×C₁₉	—	2	32	no	—
84	C₂²×C₄×C₁₉	—	2	6	no	—
85	C₂²×C₄×C₁₉	—	2	6	no	—
86	C₂²×C₄×C₁₉	—	2	12	no	—
87	C₂²×C₄×C₁₉	—	2	6	no	—
88	C₂²×C₄×C₁₉	—	2	6	no	—
89	C₂²×C₄×C₁₉	—	8	2	no	—
90	C₂²×C₄×C₁₉	C₂²⋊C₄	1	24	yes	complemented
91	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₁₉	C₂×D₄	1	6	yes	—
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	6	yes	—
96	C₄⋊C₄×C₁₉	—	4	1	no	—
97	C₄⋊C₄×C₁₉	—	4	3	no	—
98	C₄⋊C₄×C₁₉	—	4	6	no	—
99	C₂²⋊C₄×C₁₉	—	2	12	no	—
100	C₂²⋊C₄×C₁₉	—	8	12	no	—
101	C₄²×C₁₉	—	4	6	no	—
102	#20610	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂³×C₁₉	—	2	16	no	—
104	C₂³×C₁₉	—	2	6	no	—
105	C₂³×C₁₉	—	2	6	no	—
106	C₂³×C₁₉	—	4	1	no	—
107	C₂³×C₁₉	—	4	1	no	—
108	C₂³×C₁₉	—	4	6	no	—
109	C₂³×C₁₉	—	4	6	no	—
110	C₂³×C₁₉	—	8	4	no	—
111	C₂³×C₁₉	—	8	4	no	—
112	C₂³×C₁₉	C₂×C₂²⋊C₄	1	12	yes	—
113	C₂³×C₁₉	C₂³×C₄	1	1	yes	characteristic
114	C₂³×C₁₉	C₂²×D₄	1	3	yes	—
115	C₂³×C₁₉	C₂²×D₄	1	3	yes	—
116	Q₈×C₁₉	—	2	16	no	—
117	Q₈×C₁₉	—	2	24	no	—
118	D₄×C₁₉	—	2	48	no	—
119	D₄×C₁₉	—	2	24	no	—
120	D₄×C₁₉	—	2	24	no	—
121	D₄×C₁₉	—	2	24	no	—
122	C₂×C₄×C₁₉	—	2	48	no	—
123	C₂×C₄×C₁₉	—	2	6	no	—
124	C₂×C₄×C₁₉	—	2	6	no	—
125	C₂×C₄×C₁₉	—	2	24	no	—
126	C₂×C₄×C₁₉	—	2	24	no	—
127	C₂×C₄×C₁₉	—	4	6	no	—
128	C₂×C₄×C₁₉	—	4	12	no	—
129	C₂×C₄×C₁₉	—	4	6	no	—
130	C₂×C₄×C₁₉	—	4	6	no	—
131	C₂×C₄×C₁₉	—	4	2	no	—
132	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	12	yes	—
133	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	6	yes	—
134	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	6	yes	—
135	C₂×C₄×C₁₉	#120	1	6	yes	—
136	C₂×C₄×C₁₉	#120	1	6	yes	—
137	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
138	C₂×C₄×C₁₉	#120	1	3	yes	—
139	C₂×#585	C₂×C₁₉	1	1	yes	characteristic
140	C₂×#410	C₂×C₁₉	1	3	yes	complemented
141	#2311	C₂×C₁₉	1	3	yes	complemented
142	#1790	C₂×C₁₉	1	8	yes	complemented
143	C₂²×C₁₉	—	2	3	no	—
144	C₂²×C₁₉	—	2	3	no	—
145	C₂²×C₁₉	—	2	12	no	—
146	C₂²×C₁₉	—	4	1	no	—
147	C₂²×C₁₉	—	4	3	no	—
148	C₂²×C₁₉	—	4	12	no	—
149	C₂²×C₁₉	—	4	6	no	—
150	C₂²×C₁₉	—	4	6	no	—
151	C₂²×C₁₉	—	8	6	no	—
152	C₂²×C₁₉	—	8	16	no	—
153	C₂²×C₁₉	C₂²×C₂²⋊C₄	1	3	yes	—
154	C₂²×C₁₉	C₂×#120	1	3	yes	—
155	C₂²×C₁₉	C₂×#120	1	1	yes	characteristic
156	C₄×C₁₉	—	2	12	no	—
157	C₄×C₁₉	—	8	6	no	—
158	C₄×C₁₉	—	8	2	no	—
159	C₄×C₁₉	#379	1	2	yes	—
160	C₄×C₁₉	#379	1	6	yes	—
161	#585	—	2	12	no	—
162	#585	C₄×C₁₉	1	8	yes	complemented
163	C₂×2₋¹⁺⁴	—	2	3	no	—
164	C₂×H(5,𝔽₂)	—	2	3	no	—
165	C₂×H(5,𝔽₂)	C₂²×C₁₉	1	4	yes	—
166	C₂²×#41	—	2	6	no	—
167	C₂²×#41	C₂²×C₁₉	1	3	yes	—
168	C₂×#117	C₂²×C₁₉	1	3	yes	complemented
169	C₂²×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic, complemented
170	#410	—	2	12	no	—
171	C₂×#99	C₂²×C₁₉	1	12	yes	complemented
172	#391	—	2	3	no	—
173	#390	—	2	3	no	—
174	#386	C₂²×C₁₉	1	12	yes	complemented
175	#380	—	2	6	no	—
176	C₂×C₁₉	—	2	3	no	—
177	C₂×C₁₉	—	2	3	no	—
178	C₂×C₁₉	—	4	12	no	—
179	C₂×C₁₉	C₂×#379	1	1	yes	characteristic
180	C₂×C₁₉	#3027	1	1	yes	characteristic
181	C₂×C₁₉	#3028	1	1	yes	characteristic
182	2₋¹⁺⁴	—	2	24	no	—
183	2₋¹⁺⁴	—	2	24	no	—
184	H(5,𝔽₂)	—	2	24	no	—
185	H(5,𝔽₂)	—	2	24	no	—
186	H(5,𝔽₂)	—	2	24	no	—
187	H(5,𝔽₂)	C₂×C₄×C₁₉	1	16	yes	—
188	C₂×#41	—	2	4	no	—
189	C₂×#41	—	2	12	no	—
190	C₂×#41	—	2	48	no	—
191	C₂×#41	—	2	48	no	—
192	C₂×#41	—	2	6	no	—
193	C₂×#41	—	2	12	no	—
194	C₂×#41	—	2	6	no	—
195	C₂×#41	—	2	12	no	—
196	C₂×#41	C₂×C₄×C₁₉	1	12	yes	—
197	C₂×#41	D₄×C₁₉	1	12	yes	—
198	C₂²×Q₈	—	2	6	no	—
199	C₂²×Q₈	D₄×C₁₉	1	3	yes	—
200	C₂²×D₄	—	2	6	no	—
201	C₂²×D₄	—	2	12	no	—
202	C₂²×D₄	D₄×C₁₉	1	3	yes	—
203	C₂²×D₄	C₂³×C₁₉	1	6	yes	—
204	C₂³×C₄	—	2	4	no	—
205	C₂³×C₄	D₄×C₁₉	1	6	yes	—
206	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
207	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
208	#117	—	2	12	no	—
209	C₂×C₄⋊C₄	—	2	3	no	—
210	C₂×C₄⋊C₄	—	2	3	no	—
211	C₂×C₄⋊C₄	—	2	3	no	—
212	C₂×C₂²⋊C₄	—	2	12	no	—
213	C₂×C₂²⋊C₄	C₂³×C₁₉	1	6	yes	—
214	C₂×C₄²	—	2	3	no	—
215	#99	—	2	24	no	—
216	#95	—	2	12	no	—
217	C₁₉	#20610	1	1	yes	characteristic, direct product factor, Sylow
218	C₂⁴	—	2	4	no	—
219	C₂⁴	C₂×D₄×C₁₉	1	6	yes	—
220	C₂⁴	C₂⁴×C₁₉	1	1	yes	characteristic
221	#41	—	2	32	no	—
222	#41	—	2	48	no	—
223	#41	—	2	96	no	—
224	#41	—	2	24	no	—
225	#41	—	2	48	no	—
226	#41	—	2	24	no	—
227	#41	—	2	24	no	—
228	#41	—	2	24	no	—
229	C₂×Q₈	—	2	4	no	—
230	C₂×Q₈	—	2	48	no	—
231	C₂×Q₈	—	2	6	no	—
232	C₂×Q₈	—	2	6	no	—
233	C₂×Q₈	C₂²⋊C₄×C₁₉	1	12	yes	complemented
234	C₂×D₄	—	2	12	no	—
235	C₂×D₄	—	2	48	no	—
236	C₂×D₄	—	2	48	no	—
237	C₂×D₄	—	2	48	no	—
238	C₂×D₄	—	2	12	no	—
239	C₂×D₄	—	2	12	no	—
240	C₂×D₄	—	2	6	no	—
241	C₂×D₄	—	2	6	no	—
242	C₂×D₄	C₂²⋊C₄×C₁₉	1	12	yes	complemented
243	C₂×D₄	C₂²×C₄×C₁₉	1	12	yes	—
244	C₂×D₄	C₂×D₄×C₁₉	1	12	yes	—
245	C₂²×C₄	—	2	12	no	—
246	C₂²×C₄	—	2	32	no	—
247	C₂²×C₄	—	2	6	no	—
248	C₂²×C₄	—	2	6	no	—
249	C₂²×C₄	—	2	12	no	—
250	C₂²×C₄	—	2	6	no	—
251	C₂²×C₄	—	2	6	no	—
252	C₂²×C₄	—	8	2	no	—
253	C₂²×C₄	C₂²⋊C₄×C₁₉	1	24	yes	complemented
254	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
255	C₂²×C₄	C₂×D₄×C₁₉	1	6	yes	—
256	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
257	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
258	C₂²×C₄	C₂×D₄×C₁₉	1	6	yes	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	4	3	no	—
261	C₄⋊C₄	—	4	6	no	—
262	C₂²⋊C₄	—	2	12	no	—
263	C₂²⋊C₄	—	8	12	no	—
264	C₄²	—	4	6	no	—
265	C₂³	—	2	16	no	—
266	C₂³	—	2	6	no	—
267	C₂³	—	2	6	no	—
268	C₂³	—	4	1	no	—
269	C₂³	—	4	1	no	—
270	C₂³	—	4	6	no	—
271	C₂³	—	4	6	no	—
272	C₂³	—	8	4	no	—
273	C₂³	—	8	4	no	—
274	C₂³	C₂×C₂²⋊C₄×C₁₉	1	12	yes	—
275	C₂³	C₂³×C₄×C₁₉	1	1	yes	characteristic
276	C₂³	C₂²×D₄×C₁₉	1	3	yes	—
277	C₂³	C₂²×D₄×C₁₉	1	3	yes	—
278	Q₈	—	2	16	no	—
279	Q₈	—	2	24	no	—
280	D₄	—	2	48	no	—
281	D₄	—	2	24	no	—
282	D₄	—	2	24	no	—
283	D₄	—	2	24	no	—
284	C₂×C₄	—	2	48	no	—
285	C₂×C₄	—	2	6	no	—
286	C₂×C₄	—	2	6	no	—
287	C₂×C₄	—	2	24	no	—
288	C₂×C₄	—	2	24	no	—
289	C₂×C₄	—	4	6	no	—
290	C₂×C₄	—	4	12	no	—
291	C₂×C₄	—	4	6	no	—
292	C₂×C₄	—	4	6	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	12	yes	—
295	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	6	yes	—
296	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	6	yes	—
297	C₂×C₄	C₁₉×#120	1	6	yes	—
298	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
299	C₂×C₄	C₁₉×#120	1	3	yes	—
300	C₂×C₄	C₁₉×#120	1	6	yes	—
301	C₂²	—	2	3	no	—
302	C₂²	—	2	3	no	—
303	C₂²	—	2	12	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	3	no	—
306	C₂²	—	4	12	no	—
307	C₂²	—	4	6	no	—
308	C₂²	—	4	6	no	—
309	C₂²	—	8	6	no	—
310	C₂²	—	8	16	no	—
311	C₂²	C₂²×C₂²⋊C₄×C₁₉	1	3	yes	—
312	C₂²	C₂×C₁₉×#120	1	3	yes	—
313	C₂²	C₂×C₁₉×#120	1	1	yes	characteristic
314	C₄	—	2	12	no	—
315	C₄	—	8	6	no	—
316	C₄	—	8	2	no	—
317	C₄	C₁₉×#379	1	2	yes	—
318	C₄	C₁₉×#379	1	6	yes	—
319	C₂	—	2	3	no	—
320	C₂	—	2	3	no	—
321	C₂	—	4	12	no	—
322	C₂	C₂×C₁₉×#379	1	1	yes	characteristic
323	C₂	C₁₉×#3027	1	1	yes	characteristic
324	C₂	C₁₉×#3028	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#120
commutator	C₂³	C₂³×C₄×C₁₉
Frattini, cosocle	C₂⁴	C₂⁴×C₁₉
prime radical	C₂×C₁₉×#585	C₂
LCS	G ▷ C₂³ ▷ 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. C₂×C₁₉×#410	C₂	semidirect
3. C₁₉×#2311	C₂	semidirect
4. C₁₉×#1790	C₂	semidirect
6. C₁₉×#585	C₄	semidirect
12. C₂×C₁₉×#117	C₂²	semidirect
13. C₂²×C₄⋊C₄×C₁₉	C₂²	semidirect
15. C₂×C₁₉×#99	C₂²	semidirect
18. C₁₉×#386	C₂²	semidirect
70. C₂×Q₈×C₁₉	C₂²⋊C₄	semidirect
79. C₂×D₄×C₁₉	C₂²⋊C₄	semidirect
90. C₂²×C₄×C₁₉	C₂²⋊C₄	semidirect
102. #20610	C₁₉	direct (N × H)
140. C₂×#410	C₂×C₁₉	semidirect
141. #2311	C₂×C₁₉	semidirect
142. #1790	C₂×C₁₉	semidirect
162. #585	C₄×C₁₉	semidirect
168. C₂×#117	C₂²×C₁₉	semidirect
169. C₂²×C₄⋊C₄	C₂²×C₁₉	semidirect
171. C₂×#99	C₂²×C₁₉	semidirect
174. #386	C₂²×C₁₉	semidirect
217. C₁₉	#20610	direct (N × H)
233. C₂×Q₈	C₂²⋊C₄×C₁₉	semidirect
242. C₂×D₄	C₂²⋊C₄×C₁₉	semidirect
253. C₂²×C₄	C₂²⋊C₄×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₉×#585	2
C₂×C₁₉×#410	2
C₁₉×#2311	2
C₁₉×#1790	2
#20610	19

Subgroups — C19×#20610 #18026569

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#20610 #18026569