Subgroups — C₁₉×#23158 #18029117

Order 4864.

750 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#500	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2369	C₂	1	2	yes	maximal
3	C₁₉×#1987	C₂	1	2	yes	maximal
4	C₁₉×#1942	C₂	1	2	yes	maximal
5	C₁₉×#1921	C₂	1	4	yes	maximal
6	C₄×C₁₉×#107	C₂	1	1	yes	maximal, characteristic
7	C₄×C₁₉×#107	C₂	1	2	yes	maximal
8	C₈×C₄⋊C₄×C₁₉	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₉×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₁₉×#500	C₄	1	8	yes	complemented
11	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
12	C₂×C₁₉×#107	C₂²	1	2	yes	—
13	C₂×C₁₉×#107	C₂²	1	2	yes	—
14	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
15	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#391	—	2	4	no	—
19	C₁₉×#389	—	2	2	no	—
20	C₁₉×#389	C₂²	1	4	yes	—
21	C₄×C₄⋊C₄×C₁₉	—	2	2	no	—
22	C₄×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
23	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
25	C₁₉×#340	C₂²	1	2	yes	—
26	C₁₉×#340	C₂²	1	4	yes	—
27	C₁₉×#340	C₂²	1	4	yes	—
28	C₁₉×#336	C₂²	1	2	yes	—
29	C₂²×Q₈×C₁₉	—	2	2	no	—
30	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
31	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
33	C₁₉×C₄⋊Q₈	—	2	4	no	—
34	C₁₉×C₄⋊Q₈	C₂×C₄	1	8	yes	—
35	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
36	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
37	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
41	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
42	C₂×C₄²×C₁₉	—	2	2	no	—
43	C₂×C₄²×C₁₉	—	2	4	no	—
44	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
45	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
46	C₁₉×#107	—	2	2	no	—
47	C₁₉×#107	C₂×C₄	1	8	yes	—
48	C₁₉×#107	C₂×C₄	1	8	yes	—
49	C₁₉×#107	Q₈	1	4	yes	complemented
50	C₁₉×C₄⋊C₈	—	2	2	no	—
51	C₄×C₈×C₁₉	—	2	4	no	—
52	C₄×C₈×C₁₉	C₂×C₄	1	4	yes	—
53	C₁₉×#95	—	2	4	no	—
54	C₁₉×#95	—	2	4	no	—
55	C₁₉×#95	—	2	2	no	—
56	C₁₉×#95	D₄	1	2	yes	—
57	C₂×Q₈×C₁₉	—	2	8	no	—
58	C₂×Q₈×C₁₉	—	8	2	no	—
59	C₂×Q₈×C₁₉	—	8	2	no	—
60	C₂²×C₄×C₁₉	—	2	2	no	—
61	C₂²×C₄×C₁₉	—	2	4	no	—
62	C₂²×C₄×C₁₉	—	2	2	no	—
63	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
68	C₂×C₈×C₁₉	—	2	1	no	—
69	C₂×C₈×C₁₉	—	2	1	no	—
70	C₂×C₈×C₁₉	—	2	2	no	—
71	C₂×C₈×C₁₉	—	2	2	no	—
72	C₂×C₈×C₁₉	—	2	2	no	—
73	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₁₉	C₂×Q₈	1	2	yes	—
76	C₂×C₈×C₁₉	#41	1	2	yes	—
77	C₄⋊C₄×C₁₉	—	2	8	no	—
78	C₄⋊C₄×C₁₉	—	2	2	no	—
79	C₄⋊C₄×C₁₉	—	2	4	no	—
80	C₄⋊C₄×C₁₉	—	2	2	no	—
81	C₄⋊C₄×C₁₉	—	2	2	no	—
82	C₄⋊C₄×C₁₉	—	2	2	no	—
83	C₄⋊C₄×C₁₉	—	8	4	no	—
84	C₄⋊C₄×C₁₉	Q₁₆	1	4	yes	—
85	C₄⋊C₄×C₁₉	C₂²×C₄	1	8	yes	—
86	C₄⋊C₄×C₁₉	C₂×Q₈	1	4	yes	—
87	C₄²×C₁₉	—	2	2	no	—
88	C₄²×C₁₉	—	2	2	no	—
89	C₄²×C₁₉	—	4	4	no	—
90	C₄²×C₁₉	—	8	4	no	—
91	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
92	C₄²×C₁₉	#41	1	2	yes	—
93	#23158	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
95	Q₈×C₁₉	—	8	8	no	—
96	C₂×C₄×C₁₉	—	2	2	no	—
97	C₂×C₄×C₁₉	—	2	2	no	—
98	C₂×C₄×C₁₉	—	2	1	no	—
99	C₂×C₄×C₁₉	—	2	1	no	—
100	C₂×C₄×C₁₉	—	2	8	no	—
101	C₂×C₄×C₁₉	—	2	2	no	—
102	C₂×C₄×C₁₉	—	2	4	no	—
103	C₂×C₄×C₁₉	—	2	2	no	—
104	C₂×C₄×C₁₉	—	2	2	no	—
105	C₂×C₄×C₁₉	—	4	2	no	—
106	C₂×C₄×C₁₉	—	4	2	no	—
107	C₂×C₄×C₁₉	—	4	4	no	—
108	C₂×C₄×C₁₉	—	4	4	no	—
109	C₂×C₄×C₁₉	—	4	2	no	—
110	C₂×C₄×C₁₉	—	4	2	no	—
111	C₂×C₄×C₁₉	—	4	2	no	—
112	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₁₉	C₂×Q₁₆	1	4	yes	—
115	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
116	C₂×C₄×C₁₉	#135	1	2	yes	—
117	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₂×#41	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂×#41	1	1	yes	characteristic
121	C₂×C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
122	C₂×C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
123	C₈×C₁₉	—	2	2	no	—
124	C₈×C₁₉	—	4	2	no	—
125	C₈×C₁₉	C₄×Q₈	1	4	yes	—
126	C₂×#500	C₂×C₁₉	1	1	yes	characteristic
127	#2369	C₂×C₁₉	1	2	yes	—
128	#1987	C₂×C₁₉	1	2	yes	—
129	#1942	C₂×C₁₉	1	2	yes	—
130	#1921	C₂×C₁₉	1	4	yes	—
131	C₄×#107	C₂×C₁₉	1	1	yes	characteristic
132	C₄×#107	C₂×C₁₉	1	2	yes	—
133	C₈×C₄⋊C₄	C₂×C₁₉	1	1	yes	characteristic
134	C₂²×C₁₉	C₂×C₄×D₄	1	1	yes	characteristic
135	C₂²×C₁₉	#547	1	1	yes	characteristic
136	C₂²×C₁₉	D₄×Q₈	1	1	yes	characteristic
137	C₂²×C₁₉	C₂²×Q₁₆	1	1	yes	characteristic
138	C₂²×C₁₉	C₂×#135	1	1	yes	characteristic
139	C₂²×C₁₉	#576	1	1	yes	characteristic
140	C₂²×C₁₉	#576	1	1	yes	characteristic
141	C₄×C₁₉	—	2	4	no	—
142	C₄×C₁₉	—	2	2	no	—
143	C₄×C₁₉	—	4	2	no	—
144	C₄×C₁₉	—	4	4	no	—
145	C₄×C₁₉	—	4	8	no	—
146	C₄×C₁₉	—	8	2	no	—
147	C₄×C₁₉	C₄×Q₁₆	1	4	yes	—
148	C₄×C₁₉	C₂×C₄×Q₈	1	1	yes	characteristic
149	C₄×C₁₉	#519	1	1	yes	characteristic
150	C₄×C₁₉	#556	1	1	yes	characteristic
151	C₄×C₁₉	#557	1	1	yes	characteristic
152	C₂×C₄⋊Q₈	C₂²×C₁₉	1	2	yes	—
153	#500	C₄×C₁₉	1	8	yes	complemented
154	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
155	C₂×#107	C₂²×C₁₉	1	2	yes	—
156	C₂×#107	C₂²×C₁₉	1	2	yes	—
157	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
158	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂²×C₁₉	1	2	yes	—
160	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
161	#391	—	2	4	no	—
162	#389	—	2	2	no	—
163	#389	C₂²×C₁₉	1	4	yes	—
164	C₄×C₄⋊C₄	—	2	2	no	—
165	C₄×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
166	C₄×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
167	C₄×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
168	#340	C₂²×C₁₉	1	2	yes	—
169	#340	C₂²×C₁₉	1	4	yes	—
170	#340	C₂²×C₁₉	1	4	yes	—
171	#336	C₂²×C₁₉	1	2	yes	—
172	C₂×C₁₉	#2365	1	1	yes	characteristic
173	C₂×C₁₉	C₂×C₄×Q₁₆	1	1	yes	characteristic
174	C₂×C₁₉	#2948	1	1	yes	characteristic
175	C₂×C₁₉	#3338	1	1	yes	characteristic
176	C₂×C₁₉	#3341	1	1	yes	characteristic
177	C₂×C₁₉	#3380	1	1	yes	characteristic
178	C₂×C₁₉	Q₈×Q₁₆	1	1	yes	characteristic
179	C₂²×Q₈	—	2	2	no	—
180	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
181	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
182	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
183	C₄⋊Q₈	—	2	4	no	—
184	C₄⋊Q₈	C₂×C₄×C₁₉	1	8	yes	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	4	no	—
187	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
188	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
189	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
190	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
191	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	—	2	4	no	—
194	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
195	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
196	#107	—	2	2	no	—
197	#107	C₂×C₄×C₁₉	1	8	yes	—
198	#107	C₂×C₄×C₁₉	1	8	yes	—
199	#107	Q₈×C₁₉	1	4	yes	complemented
200	C₄⋊C₈	—	2	2	no	—
201	C₄×C₈	—	2	4	no	—
202	C₄×C₈	C₂×C₄×C₁₉	1	4	yes	—
203	#95	—	2	4	no	—
204	#95	—	2	4	no	—
205	#95	—	2	2	no	—
206	#95	D₄×C₁₉	1	2	yes	—
207	C₁₉	#23158	1	1	yes	characteristic, direct product factor, Sylow
208	C₂×Q₈	—	2	8	no	—
209	C₂×Q₈	—	8	2	no	—
210	C₂×Q₈	—	8	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
215	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
217	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
218	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
225	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
226	C₂×C₈	C₂×Q₈×C₁₉	1	2	yes	—
227	C₂×C₈	#41×C₁₉	1	2	yes	—
228	C₄⋊C₄	—	2	8	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	8	4	no	—
235	C₄⋊C₄	Q₁₆×C₁₉	1	4	yes	—
236	C₄⋊C₄	C₂²×C₄×C₁₉	1	8	yes	—
237	C₄⋊C₄	C₂×Q₈×C₁₉	1	4	yes	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	4	4	no	—
241	C₄²	—	8	4	no	—
242	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
243	C₄²	#41×C₁₉	1	2	yes	—
244	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
245	Q₈	—	8	8	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	C₄×D₄×C₁₉	1	2	yes	—
263	C₂×C₄	C₄×D₄×C₁₉	1	2	yes	—
264	C₂×C₄	C₂×Q₁₆×C₁₉	1	4	yes	—
265	C₂×C₄	C₂×Q₁₆×C₁₉	1	2	yes	—
266	C₂×C₄	C₁₉×#135	1	2	yes	—
267	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
268	C₂×C₄	C₂²×Q₈×C₁₉	1	1	yes	characteristic
269	C₂×C₄	C₂×#41×C₁₉	1	1	yes	characteristic
270	C₂×C₄	C₂×#41×C₁₉	1	1	yes	characteristic
271	C₂×C₄	C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
272	C₂×C₄	C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
273	C₈	—	2	2	no	—
274	C₈	—	4	2	no	—
275	C₈	C₄×Q₈×C₁₉	1	4	yes	—
276	C₂²	C₂×C₄×D₄×C₁₉	1	1	yes	characteristic
277	C₂²	C₁₉×#547	1	1	yes	characteristic
278	C₂²	D₄×Q₈×C₁₉	1	1	yes	characteristic
279	C₂²	C₂²×Q₁₆×C₁₉	1	1	yes	characteristic
280	C₂²	C₂×C₁₉×#135	1	1	yes	characteristic
281	C₂²	C₁₉×#576	1	1	yes	characteristic
282	C₂²	C₁₉×#576	1	1	yes	characteristic
283	C₄	—	2	2	no	—
284	C₄	—	2	4	no	—
285	C₄	—	4	8	no	—
286	C₄	—	4	2	no	—
287	C₄	—	4	4	no	—
288	C₄	—	8	2	no	—
289	C₄	C₄×Q₁₆×C₁₉	1	4	yes	—
290	C₄	C₂×C₄×Q₈×C₁₉	1	1	yes	characteristic
291	C₄	C₁₉×#519	1	1	yes	characteristic
292	C₄	C₁₉×#556	1	1	yes	characteristic
293	C₄	C₁₉×#557	1	1	yes	characteristic
294	C₂	C₁₉×#2365	1	1	yes	characteristic
295	C₂	C₂×C₄×Q₁₆×C₁₉	1	1	yes	characteristic
296	C₂	C₁₉×#2948	1	1	yes	characteristic
297	C₂	C₁₉×#3338	1	1	yes	characteristic
298	C₂	C₁₉×#3341	1	1	yes	characteristic
299	C₂	C₁₉×#3380	1	1	yes	characteristic
300	C₂	Q₈×Q₁₆×C₁₉	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
10. C₁₉×#500	C₄	semidirect
49. C₁₉×#107	Q₈	semidirect
93. #23158	C₁₉	direct (N × H)
153. #500	C₄×C₁₉	semidirect
199. #107	Q₈×C₁₉	semidirect
207. C₁₉	#23158	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₉×#500	2
C₁₉×#2369	2
C₁₉×#1987	2
C₁₉×#1942	2
C₁₉×#1921	2
C₄×C₁₉×#107	2
C₈×C₄⋊C₄×C₁₉	2
#23158	19

Subgroups — C19×#23158 #18029117

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#23158 #18029117