Subgroups — C₁₉×#17723 #18023682

Order 4864.

964 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3503	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2988	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2906	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1550	C₂	1	3	yes	maximal, complemented
5	C₁₉×#1549	C₂	1	3	yes	maximal, complemented
6	C₁₉×#1542	C₂	1	3	yes	maximal, complemented
7	C₂×C₁₉×#332	C₂	1	3	yes	maximal, complemented
8	C₂×C₁₉×#131	—	2	1	no	—
9	Q₈²×C₁₉	C₄	1	2	yes	—
10	C₁₉×#557	—	2	1	no	—
11	C₁₉×#554	C₄	1	6	yes	—
12	D₄×Q₈×C₁₉	—	2	3	no	—
13	C₁₉×#544	—	2	3	no	—
14	C₁₉×#533	C₂²	1	3	yes	—
15	C₂×C₁₉×C₄⋊Q₈	C₂²	1	3	yes	—
16	C₁₉×#520	—	2	1	no	—
17	C₄×#41×C₁₉	C₂²	1	1	yes	characteristic
18	C₈×Q₈×C₁₉	—	2	1	no	—
19	C₈×D₄×C₁₉	—	2	3	no	—
20	C₁₉×#431	C₂²	1	3	yes	—
21	C₁₉×#424	—	2	3	no	—
22	C₁₉×#423	C₂²	1	3	yes	—
23	C₂×C₁₉×C₄⋊C₈	C₂²	1	3	yes	—
24	C₂×C₁₉×C₄⋊C₈	C₂²	1	3	yes	—
25	C₁₉×#333	C₂²	1	1	yes	characteristic
26	C₁₉×#333	C₂²	1	3	yes	—
27	C₁₉×#332	C₂²	1	3	yes	—
28	C₁₉×#332	C₂²	1	6	yes	—
29	C₁₉×#332	C₂²	1	3	yes	—
30	C₂×#41×C₁₉	D₄	1	1	yes	characteristic
31	C₂²×Q₈×C₁₉	—	2	3	no	—
32	C₁₉×#131	—	4	4	no	—
33	C₂×M₁₆×C₁₉	—	2	3	no	—
34	C₂²×C₈×C₁₉	—	2	3	no	—
35	C₂²×C₈×C₁₉	—	2	3	no	—
36	C₁₉×C₄⋊Q₈	—	2	6	no	—
37	C₁₉×C₄⋊Q₈	—	2	3	no	—
38	C₁₉×C₄⋊Q₈	—	2	3	no	—
39	C₁₉×C₄⋊Q₈	C₂×C₄	1	6	yes	—
40	C₁₉×C₄⋊Q₈	C₂³	1	1	yes	characteristic
41	C₁₉×C₄⋊Q₈	C₂³	1	3	yes	—
42	C₁₉×#125	—	2	6	no	—
43	C₁₉×#125	—	2	3	no	—
44	C₁₉×C₂²⋊Q₈	—	2	3	no	—
45	C₁₉×C₂²⋊Q₈	—	2	12	no	—
46	C₁₉×C₂²⋊Q₈	C₂×C₄	1	6	yes	—
47	C₄×Q₈×C₁₉	—	2	2	no	—
48	C₄×Q₈×C₁₉	—	2	3	no	—
49	C₄×Q₈×C₁₉	C₂×C₄	1	6	yes	—
50	C₄×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
51	C₄×Q₈×C₁₉	D₄	1	1	yes	characteristic
52	C₄×D₄×C₁₉	—	2	6	no	—
53	C₄×D₄×C₁₉	C₂×C₄	1	3	yes	—
54	C₄×D₄×C₁₉	D₄	1	3	yes	—
55	C₁₉×#117	—	2	3	no	—
56	C₁₉×#117	D₄	1	3	yes	—
57	C₂×C₄⋊C₄×C₁₉	—	2	3	no	—
58	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	6	yes	—
59	C₂×C₄²×C₁₉	C₂³	1	3	yes	—
60	C₁₉×C₄⋊C₈	—	2	3	no	—
61	C₁₉×C₄⋊C₈	—	2	3	no	—
62	C₁₉×C₄⋊C₈	—	2	3	no	—
63	C₁₉×C₄⋊C₈	C₂³	1	1	yes	characteristic
64	C₁₉×C₄⋊C₈	C₂³	1	3	yes	—
65	C₁₉×C₄⋊C₈	C₂³	1	3	yes	—
66	C₁₉×C₄⋊C₈	C₂³	1	1	yes	characteristic
67	C₁₉×C₂²⋊C₈	—	2	3	no	—
68	C₄×C₈×C₁₉	—	2	3	no	—
69	#41×C₁₉	SD₁₆	1	4	yes	—
70	#41×C₁₉	Q₁₆	1	4	yes	—
71	C₂×Q₈×C₁₉	—	2	6	no	—
72	C₂×Q₈×C₁₉	—	2	3	no	—
73	C₂×Q₈×C₁₉	—	8	3	no	—
74	C₂×Q₈×C₁₉	—	8	3	no	—
75	C₂×Q₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×Q₈×C₁₉	C₂²×C₄	1	2	yes	—
77	C₂×D₄×C₁₉	C₂²⋊C₄	1	3	yes	—
78	C₂²×C₄×C₁₉	—	2	6	no	—
79	C₂²×C₄×C₁₉	C₂²×C₄	1	3	yes	—
80	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
81	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
82	M₁₆×C₁₉	—	4	6	no	—
83	C₂×C₈×C₁₉	—	2	1	no	—
84	C₂×C₈×C₁₉	—	2	3	no	—
85	C₂×C₈×C₁₉	—	2	3	no	—
86	C₂×C₈×C₁₉	—	2	1	no	—
87	C₂×C₈×C₁₉	—	4	6	no	—
88	C₂×C₈×C₁₉	—	8	3	no	—
89	C₄⋊C₄×C₁₉	—	2	6	no	—
90	C₄⋊C₄×C₁₉	—	2	6	no	—
91	C₄⋊C₄×C₁₉	—	2	6	no	—
92	C₄⋊C₄×C₁₉	—	2	6	no	—
93	C₄⋊C₄×C₁₉	—	2	3	no	—
94	C₄⋊C₄×C₁₉	—	2	1	no	—
95	C₄⋊C₄×C₁₉	—	2	1	no	—
96	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	3	yes	—
97	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	3	yes	—
98	C₄⋊C₄×C₁₉	C₂²×C₄	1	6	yes	—
99	C₂²⋊C₄×C₁₉	—	2	6	no	—
100	C₂²⋊C₄×C₁₉	C₂²⋊C₄	1	6	yes	—
101	C₄²×C₁₉	—	2	6	no	—
102	C₄²×C₁₉	C₂²×C₄	1	3	yes	—
103	C₄²×C₁₉	C₂×D₄	1	3	yes	—
104	C₄²×C₁₉	C₂×D₄	1	3	yes	—
105	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
106	#17723	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂³×C₁₉	C₂×C₂²⋊C₄	1	3	yes	—
108	Q₈×C₁₉	—	4	2	no	—
109	Q₈×C₁₉	—	4	3	no	—
110	Q₈×C₁₉	—	4	3	no	—
111	Q₈×C₁₉	—	8	6	no	—
112	Q₈×C₁₉	#103	1	4	yes	—
113	D₄×C₁₉	#103	1	12	yes	—
114	C₂×C₄×C₁₉	—	2	3	no	—
115	C₂×C₄×C₁₉	—	2	3	no	—
116	C₂×C₄×C₁₉	—	2	2	no	—
117	C₂×C₄×C₁₉	—	2	6	no	—
118	C₂×C₄×C₁₉	—	4	3	no	—
119	C₂×C₄×C₁₉	—	8	6	no	—
120	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	3	yes	—
121	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	3	yes	—
122	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	3	yes	—
123	C₂×C₄×C₁₉	C₂×SD₁₆	1	6	yes	—
124	C₂×C₄×C₁₉	C₂×Q₁₆	1	6	yes	—
125	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
128	C₈×C₁₉	—	4	1	no	—
129	C₈×C₁₉	—	4	3	no	—
130	C₈×C₁₉	—	4	3	no	—
131	C₈×C₁₉	—	4	1	no	—
132	#3503	C₂×C₁₉	1	1	yes	characteristic
133	#2988	C₂×C₁₉	1	1	yes	characteristic
134	#2906	C₂×C₁₉	1	1	yes	characteristic
135	#1550	C₂×C₁₉	1	3	yes	complemented
136	#1549	C₂×C₁₉	1	3	yes	complemented
137	#1542	C₂×C₁₉	1	3	yes	complemented
138	C₂×#332	C₂×C₁₉	1	3	yes	complemented
139	C₂²×C₁₉	—	2	3	no	—
140	C₂²×C₁₉	—	2	3	no	—
141	C₂²×C₁₉	C₂×#103	1	6	yes	—
142	C₂²×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
143	C₄×C₁₉	—	2	1	no	—
144	C₄×C₁₉	—	2	3	no	—
145	C₄×C₁₉	—	2	3	no	—
146	C₄×C₁₉	—	4	2	no	—
147	C₄×C₁₉	—	4	6	no	—
148	C₄×C₁₉	C₂×#103	1	6	yes	—
149	C₄×C₁₉	C₂²×SD₁₆	1	1	yes	characteristic
150	C₄×C₁₉	C₂²×Q₁₆	1	1	yes	characteristic
151	C₄×C₁₉	#576	1	1	yes	characteristic
152	C₄×C₁₉	#577	1	1	yes	characteristic
153	C₂×#131	—	2	1	no	—
154	Q₈²	C₄×C₁₉	1	2	yes	—
155	#557	—	2	1	no	—
156	#554	C₄×C₁₉	1	6	yes	—
157	D₄×Q₈	—	2	3	no	—
158	#544	—	2	3	no	—
159	#533	C₂²×C₁₉	1	3	yes	—
160	C₂×C₄⋊Q₈	C₂²×C₁₉	1	3	yes	—
161	#520	—	2	1	no	—
162	C₄×#41	C₂²×C₁₉	1	1	yes	characteristic
163	C₈×Q₈	—	2	1	no	—
164	C₈×D₄	—	2	3	no	—
165	#431	C₂²×C₁₉	1	3	yes	—
166	#424	—	2	3	no	—
167	#423	C₂²×C₁₉	1	3	yes	—
168	C₂×C₄⋊C₈	C₂²×C₁₉	1	3	yes	—
169	C₂×C₄⋊C₈	C₂²×C₁₉	1	3	yes	—
170	#333	C₂²×C₁₉	1	1	yes	characteristic
171	#333	C₂²×C₁₉	1	3	yes	—
172	#332	C₂²×C₁₉	1	3	yes	—
173	#332	C₂²×C₁₉	1	6	yes	—
174	#332	C₂²×C₁₉	1	3	yes	—
175	C₂×C₁₉	—	2	6	no	—
176	C₂×C₁₉	#2889	1	1	yes	characteristic
177	C₂×C₁₉	C₂²×#103	1	1	yes	characteristic
178	C₂×C₁₉	#2897	1	1	yes	characteristic
179	C₂×#41	D₄×C₁₉	1	1	yes	characteristic
180	C₂²×Q₈	—	2	3	no	—
181	#131	—	4	4	no	—
182	C₂×M₁₆	—	2	3	no	—
183	C₂²×C₈	—	2	3	no	—
184	C₂²×C₈	—	2	3	no	—
185	C₄⋊Q₈	—	2	6	no	—
186	C₄⋊Q₈	—	2	3	no	—
187	C₄⋊Q₈	—	2	3	no	—
188	C₄⋊Q₈	C₂×C₄×C₁₉	1	6	yes	—
189	C₄⋊Q₈	C₂³×C₁₉	1	1	yes	characteristic
190	C₄⋊Q₈	C₂³×C₁₉	1	3	yes	—
191	#125	—	2	6	no	—
192	#125	—	2	3	no	—
193	C₂²⋊Q₈	—	2	3	no	—
194	C₂²⋊Q₈	—	2	12	no	—
195	C₂²⋊Q₈	C₂×C₄×C₁₉	1	6	yes	—
196	C₄×Q₈	—	2	2	no	—
197	C₄×Q₈	—	2	3	no	—
198	C₄×Q₈	C₂×C₄×C₁₉	1	6	yes	—
199	C₄×Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
200	C₄×Q₈	D₄×C₁₉	1	1	yes	characteristic
201	C₄×D₄	—	2	6	no	—
202	C₄×D₄	C₂×C₄×C₁₉	1	3	yes	—
203	C₄×D₄	D₄×C₁₉	1	3	yes	—
204	#117	—	2	3	no	—
205	#117	D₄×C₁₉	1	3	yes	—
206	C₂×C₄⋊C₄	—	2	3	no	—
207	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	6	yes	—
208	C₂×C₄²	C₂³×C₁₉	1	3	yes	—
209	C₄⋊C₈	—	2	3	no	—
210	C₄⋊C₈	—	2	3	no	—
211	C₄⋊C₈	—	2	3	no	—
212	C₄⋊C₈	C₂³×C₁₉	1	1	yes	characteristic
213	C₄⋊C₈	C₂³×C₁₉	1	3	yes	—
214	C₄⋊C₈	C₂³×C₁₉	1	3	yes	—
215	C₄⋊C₈	C₂³×C₁₉	1	1	yes	characteristic
216	C₂²⋊C₈	—	2	3	no	—
217	C₄×C₈	—	2	3	no	—
218	C₁₉	#17723	1	1	yes	characteristic, direct product factor, Sylow
219	#41	SD₁₆×C₁₉	1	4	yes	—
220	#41	Q₁₆×C₁₉	1	4	yes	—
221	C₂×Q₈	—	2	6	no	—
222	C₂×Q₈	—	2	3	no	—
223	C₂×Q₈	—	8	3	no	—
224	C₂×Q₈	—	8	3	no	—
225	C₂×Q₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
226	C₂×Q₈	C₂²×C₄×C₁₉	1	2	yes	—
227	C₂×D₄	C₂²⋊C₄×C₁₉	1	3	yes	—
228	C₂²×C₄	—	2	6	no	—
229	C₂²×C₄	C₂²×C₄×C₁₉	1	3	yes	—
230	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
231	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
232	M₁₆	—	4	6	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	3	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	3	no	—
237	C₂×C₈	—	4	6	no	—
238	C₂×C₈	—	8	3	no	—
239	C₄⋊C₄	—	2	6	no	—
240	C₄⋊C₄	—	2	6	no	—
241	C₄⋊C₄	—	2	6	no	—
242	C₄⋊C₄	—	2	6	no	—
243	C₄⋊C₄	—	2	3	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	3	yes	—
247	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	3	yes	—
248	C₄⋊C₄	C₂²×C₄×C₁₉	1	6	yes	—
249	C₂²⋊C₄	—	2	6	no	—
250	C₂²⋊C₄	C₂²⋊C₄×C₁₉	1	6	yes	—
251	C₄²	—	2	6	no	—
252	C₄²	C₂²×C₄×C₁₉	1	3	yes	—
253	C₄²	C₂×D₄×C₁₉	1	3	yes	—
254	C₄²	C₂×D₄×C₁₉	1	3	yes	—
255	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
256	C₂³	C₂×C₂²⋊C₄×C₁₉	1	3	yes	—
257	Q₈	—	4	2	no	—
258	Q₈	—	4	3	no	—
259	Q₈	—	4	3	no	—
260	Q₈	—	8	6	no	—
261	Q₈	C₁₉×#103	1	4	yes	—
262	D₄	C₁₉×#103	1	12	yes	—
263	C₂×C₄	—	2	3	no	—
264	C₂×C₄	—	2	3	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	6	no	—
267	C₂×C₄	—	4	3	no	—
268	C₂×C₄	—	8	6	no	—
269	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	3	yes	—
270	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	3	yes	—
271	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	3	yes	—
272	C₂×C₄	C₂×SD₁₆×C₁₉	1	6	yes	—
273	C₂×C₄	C₂×Q₁₆×C₁₉	1	6	yes	—
274	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
275	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
276	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
277	C₈	—	4	1	no	—
278	C₈	—	4	3	no	—
279	C₈	—	4	3	no	—
280	C₈	—	4	1	no	—
281	C₂²	—	2	3	no	—
282	C₂²	—	2	3	no	—
283	C₂²	C₂×C₁₉×#103	1	6	yes	—
284	C₂²	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
285	C₄	—	2	1	no	—
286	C₄	—	2	3	no	—
287	C₄	—	2	3	no	—
288	C₄	—	4	2	no	—
289	C₄	—	4	6	no	—
290	C₄	C₂×C₁₉×#103	1	6	yes	—
291	C₄	C₂²×SD₁₆×C₁₉	1	1	yes	characteristic
292	C₄	C₂²×Q₁₆×C₁₉	1	1	yes	characteristic
293	C₄	C₁₉×#576	1	1	yes	characteristic
294	C₄	C₁₉×#577	1	1	yes	characteristic
295	C₂	—	2	6	no	—
296	C₂	C₁₉×#2889	1	1	yes	characteristic
297	C₂	C₂²×C₁₉×#103	1	1	yes	characteristic
298	C₂	C₁₉×#2897	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₁₉
Frattini, cosocle	C₄²	C₂⁴×C₁₉
prime radical	C₂×#41×C₁₉	D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₄×#41×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₉×#1550	C₂	semidirect
5. C₁₉×#1549	C₂	semidirect
6. C₁₉×#1542	C₂	semidirect
7. C₂×C₁₉×#332	C₂	semidirect
106. #17723	C₁₉	direct (N × H)
135. #1550	C₂×C₁₉	semidirect
136. #1549	C₂×C₁₉	semidirect
137. #1542	C₂×C₁₉	semidirect
138. C₂×#332	C₂×C₁₉	semidirect
218. C₁₉	#17723	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×#3503	2
C₁₉×#2988	2
C₁₉×#2906	2
C₁₉×#1550	2
C₁₉×#1549	2
C₁₉×#1542	2
C₂×C₁₉×#332	2
#17723	19

Subgroups — C19×#17723 #18023682

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#17723 #18023682