Subgroups — C₁₉×#9580 #18015539

Order 4864.

338 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#1994	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1571	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1542	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1481	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1301	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1276	C₂	1	1	yes	maximal, characteristic
8	C₁₉×#533	C₂²	1	1	yes	characteristic
9	C₁₉×#431	C₂²	1	1	yes	characteristic
10	C₁₉×#431	C₂²	1	1	yes	characteristic
11	C₁₉×#423	C₂²	1	1	yes	characteristic
12	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₉	—	2	1	no	—
15	C₄³×C₁₉	C₂²	1	1	yes	characteristic
16	C₁₉×#339	—	2	1	no	—
17	C₁₉×#339	—	2	1	no	—
18	C₁₉×#336	—	2	1	no	—
19	C₁₉×#335	—	2	2	no	—
20	C₁₉×#333	C₄	1	2	yes	complemented
21	C₁₉×#332	C₄	1	2	yes	complemented
22	C₁₉×Q₈⋊C₈	—	2	2	no	—
23	C₂×M₁₆×C₁₉	—	2	1	no	—
24	C₂²×C₈×C₁₉	—	2	1	no	—
25	C₂²×C₈×C₁₉	—	2	1	no	—
26	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
27	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
28	C₁₉×#125	—	2	1	no	—
29	C₁₉×C₂²⋊Q₈	—	2	1	no	—
30	C₁₉×C₂²⋊Q₈	—	2	1	no	—
31	C₄×Q₈×C₁₉	—	2	1	no	—
32	C₄×Q₈×C₁₉	—	2	1	no	—
33	C₁₉×#117	—	2	1	no	—
34	C₂×C₄²×C₁₉	—	2	1	no	—
35	C₂×C₄²×C₁₉	—	2	1	no	—
36	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
39	C₁₉×C₄⋊C₈	—	2	1	no	—
40	C₁₉×C₄⋊C₈	—	2	1	no	—
41	C₁₉×C₄⋊C₈	—	2	1	no	—
42	C₁₉×C₄⋊C₈	—	2	1	no	—
43	C₁₉×C₄⋊C₈	—	2	1	no	—
44	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
45	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
46	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
47	C₁₉×C₂²⋊C₈	—	2	1	no	—
48	C₁₉×C₂²⋊C₈	—	2	1	no	—
49	C₄×C₈×C₁₉	—	2	1	no	—
50	C₄×C₈×C₁₉	—	2	1	no	—
51	C₄×C₈×C₁₉	—	4	2	no	—
52	C₂×Q₈×C₁₉	—	2	1	no	—
53	C₂×Q₈×C₁₉	—	2	1	no	—
54	C₂²×C₄×C₁₉	—	2	1	no	—
55	C₂²×C₄×C₁₉	—	2	1	no	—
56	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
59	M₁₆×C₁₉	—	4	1	no	—
60	M₁₆×C₁₉	—	4	1	no	—
61	C₂×C₈×C₁₉	—	2	2	no	—
62	C₂×C₈×C₁₉	—	2	2	no	—
63	C₂×C₈×C₁₉	—	2	2	no	—
64	C₂×C₈×C₁₉	—	4	1	no	—
65	C₂×C₈×C₁₉	—	4	1	no	—
66	C₂×C₈×C₁₉	—	4	1	no	—
67	C₂×C₈×C₁₉	—	4	1	no	—
68	C₂×C₈×C₁₉	—	4	1	no	—
69	C₂×C₈×C₁₉	—	4	1	no	—
70	C₄⋊C₄×C₁₉	—	2	1	no	—
71	C₄⋊C₄×C₁₉	—	2	1	no	—
72	C₄⋊C₄×C₁₉	—	2	1	no	—
73	C₄⋊C₄×C₁₉	—	2	1	no	—
74	C₄⋊C₄×C₁₉	—	4	1	no	—
75	C₄⋊C₄×C₁₉	—	4	1	no	—
76	C₂²⋊C₄×C₁₉	—	4	1	no	—
77	C₄²×C₁₉	—	2	4	no	—
78	C₄²×C₁₉	—	2	1	no	—
79	C₄²×C₁₉	—	2	1	no	—
80	C₄²×C₁₉	—	2	1	no	—
81	C₄²×C₁₉	—	4	2	no	—
82	C₄²×C₁₉	—	4	1	no	—
83	C₄²×C₁₉	Q₁₆	1	2	yes	—
84	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₁₉	#41	1	1	yes	characteristic
87	C₄²×C₁₉	#41	1	1	yes	characteristic
88	#9580	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂³×C₁₉	#123	1	1	yes	characteristic
90	Q₈×C₁₉	—	4	2	no	—
91	Q₈×C₁₉	—	8	1	no	—
92	C₂×C₄×C₁₉	—	2	2	no	—
93	C₂×C₄×C₁₉	—	2	2	no	—
94	C₂×C₄×C₁₉	—	2	2	no	—
95	C₂×C₄×C₁₉	—	2	2	no	—
96	C₂×C₄×C₁₉	—	2	2	no	—
97	C₂×C₄×C₁₉	—	4	1	no	—
98	C₂×C₄×C₁₉	—	4	1	no	—
99	C₂×C₄×C₁₉	—	4	1	no	—
100	C₂×C₄×C₁₉	—	4	1	no	—
101	C₂×C₄×C₁₉	—	4	1	no	—
102	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
103	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
106	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
107	C₂×C₄×C₁₉	#126	1	1	yes	characteristic
108	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
109	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
110	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
112	C₈×C₁₉	—	4	2	no	—
113	C₈×C₁₉	—	4	2	no	—
114	C₈×C₁₉	—	4	2	no	—
115	C₈×C₁₉	—	8	1	no	—
116	C₈×C₁₉	—	8	1	no	—
117	#1994	C₂×C₁₉	1	1	yes	characteristic
118	C₄×C₄⋊C₈	C₂×C₁₉	1	1	yes	characteristic
119	#1571	C₂×C₁₉	1	1	yes	characteristic
120	#1542	C₂×C₁₉	1	1	yes	characteristic
121	#1481	C₂×C₁₉	1	1	yes	characteristic
122	#1301	C₂×C₁₉	1	1	yes	characteristic
123	#1276	C₂×C₁₉	1	1	yes	characteristic
124	C₂²×C₁₉	—	2	1	no	—
125	C₂²×C₁₉	—	2	1	no	—
126	C₂²×C₁₉	#388	1	1	yes	characteristic
127	C₂²×C₁₉	#484	1	1	yes	characteristic
128	C₂²×C₁₉	#485	1	1	yes	characteristic
129	C₄×C₁₉	—	2	4	no	—
130	C₄×C₁₉	—	2	1	no	—
131	C₄×C₁₉	—	2	1	no	—
132	C₄×C₁₉	—	2	1	no	—
133	C₄×C₁₉	—	4	2	no	—
134	C₄×C₁₉	—	8	1	no	—
135	C₄×C₁₉	—	8	1	no	—
136	C₄×C₁₉	C₄×Q₁₆	1	1	yes	characteristic
137	C₄×C₁₉	#440	1	1	yes	characteristic
138	C₄×C₁₉	#443	1	2	yes	—
139	C₄×C₁₉	#462	1	1	yes	characteristic
140	C₄×C₁₉	#463	1	1	yes	characteristic
141	#533	C₂²×C₁₉	1	1	yes	characteristic
142	#431	C₂²×C₁₉	1	1	yes	characteristic
143	#431	C₂²×C₁₉	1	1	yes	characteristic
144	#423	C₂²×C₁₉	1	1	yes	characteristic
145	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
146	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
147	C₂×C₄×C₈	—	2	1	no	—
148	C₄³	C₂²×C₁₉	1	1	yes	characteristic
149	#339	—	2	1	no	—
150	#339	—	2	1	no	—
151	#336	—	2	1	no	—
152	#335	—	2	2	no	—
153	#333	C₄×C₁₉	1	2	yes	complemented
154	#332	C₄×C₁₉	1	2	yes	complemented
155	Q₈⋊C₈	—	2	2	no	—
156	C₂×C₁₉	—	2	1	no	—
157	C₂×C₁₉	—	2	1	no	—
158	C₂×C₁₉	#1928	1	1	yes	characteristic
159	C₂×C₁₉	#1932	1	1	yes	characteristic
160	C₂×C₁₉	#1937	1	1	yes	characteristic
161	C₂×M₁₆	—	2	1	no	—
162	C₂²×C₈	—	2	1	no	—
163	C₂²×C₈	—	2	1	no	—
164	C₄⋊Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
165	C₄⋊Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
166	#125	—	2	1	no	—
167	C₂²⋊Q₈	—	2	1	no	—
168	C₂²⋊Q₈	—	2	1	no	—
169	C₄×Q₈	—	2	1	no	—
170	C₄×Q₈	—	2	1	no	—
171	#117	—	2	1	no	—
172	C₂×C₄²	—	2	1	no	—
173	C₂×C₄²	—	2	1	no	—
174	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
175	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
176	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
177	C₄⋊C₈	—	2	1	no	—
178	C₄⋊C₈	—	2	1	no	—
179	C₄⋊C₈	—	2	1	no	—
180	C₄⋊C₈	—	2	1	no	—
181	C₄⋊C₈	—	2	1	no	—
182	C₄⋊C₈	C₂×C₄×C₁₉	1	2	yes	—
183	C₄⋊C₈	C₂×C₄×C₁₉	1	2	yes	—
184	C₄⋊C₈	D₄×C₁₉	1	2	yes	—
185	C₂²⋊C₈	—	2	1	no	—
186	C₂²⋊C₈	—	2	1	no	—
187	C₄×C₈	—	2	1	no	—
188	C₄×C₈	—	2	1	no	—
189	C₄×C₈	—	4	2	no	—
190	C₁₉	#9580	1	1	yes	characteristic, direct product factor, Sylow
191	C₂×Q₈	—	2	1	no	—
192	C₂×Q₈	—	2	1	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
196	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
197	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
198	M₁₆	—	4	1	no	—
199	M₁₆	—	4	1	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	—	4	1	no	—
206	C₂×C₈	—	4	1	no	—
207	C₂×C₈	—	4	1	no	—
208	C₂×C₈	—	4	1	no	—
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	2	1	no	—
213	C₄⋊C₄	—	4	1	no	—
214	C₄⋊C₄	—	4	1	no	—
215	C₂²⋊C₄	—	4	1	no	—
216	C₄²	—	2	4	no	—
217	C₄²	—	2	1	no	—
218	C₄²	—	2	1	no	—
219	C₄²	—	2	1	no	—
220	C₄²	—	4	2	no	—
221	C₄²	—	4	1	no	—
222	C₄²	Q₁₆×C₁₉	1	2	yes	—
223	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
224	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
225	C₄²	#41×C₁₉	1	1	yes	characteristic
226	C₄²	#41×C₁₉	1	1	yes	characteristic
227	C₂³	C₁₉×#123	1	1	yes	characteristic
228	Q₈	—	4	2	no	—
229	Q₈	—	8	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	—	4	1	no	—
240	C₂×C₄	C₁₉×#117	1	1	yes	characteristic
241	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
242	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
243	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
244	C₂×C₄	C₁₉×#124	1	1	yes	characteristic
245	C₂×C₄	C₁₉×#126	1	1	yes	characteristic
246	C₂×C₄	C₂×Q₁₆×C₁₉	1	1	yes	characteristic
247	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
248	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
249	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
250	C₈	—	4	2	no	—
251	C₈	—	4	2	no	—
252	C₈	—	4	2	no	—
253	C₈	—	8	1	no	—
254	C₈	—	8	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	C₁₉×#388	1	1	yes	characteristic
258	C₂²	C₁₉×#484	1	1	yes	characteristic
259	C₂²	C₁₉×#485	1	1	yes	characteristic
260	C₄	—	2	4	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	4	2	no	—
265	C₄	—	8	1	no	—
266	C₄	—	8	1	no	—
267	C₄	C₄×Q₁₆×C₁₉	1	1	yes	characteristic
268	C₄	C₁₉×#440	1	1	yes	characteristic
269	C₄	C₁₉×#443	1	2	yes	—
270	C₄	C₁₉×#462	1	1	yes	characteristic
271	C₄	C₁₉×#463	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	1	no	—
274	C₂	C₁₉×#1928	1	1	yes	characteristic
275	C₂	C₁₉×#1932	1	1	yes	characteristic
276	C₂	C₁₉×#1937	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₉	#121
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
socle	C₂²×C₁₉	#388
prime radical	C₂³×C₁₉	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
20. C₁₉×#333	C₄	semidirect
21. C₁₉×#332	C₄	semidirect
88. #9580	C₁₉	direct (N × H)
153. #333	C₄×C₁₉	semidirect
154. #332	C₄×C₁₉	semidirect
190. C₁₉	#9580	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×#1994	2
C₄×C₁₉×C₄⋊C₈	2
C₁₉×#1571	2
C₁₉×#1542	2
C₁₉×#1481	2
C₁₉×#1301	2
C₁₉×#1276	2
#9580	19

Subgroups — C19×#9580 #18015539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#9580 #18015539