Subgroups — C₁₉×#12928 #18018887

Order 4864.

420 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₉×#108	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#365	C₂	1	2	yes	maximal, complemented
3	C₁₉×#2112	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1807	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1383	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₁₉	—	2	1	no	—
7	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
8	C₂×C₁₉×M₃₂	C₂²	1	2	yes	—
9	C₂×C₁₉×#108	—	2	4	no	—
10	C₂×C₁₉×#108	—	2	1	no	—
11	C₂×C₁₉×#108	C₂²	1	2	yes	—
12	C₂×C₁₉×#106	C₂²	1	1	yes	characteristic
13	C₂×C₁₉×#106	C₂²	1	1	yes	characteristic
14	C₁₉×#380	—	2	1	no	—
15	C₁₉×#365	—	2	4	no	—
16	C₁₉×C₂²⋊C₁₆	C₄	1	2	yes	complemented
17	C₁₉×C₂²⋊C₁₆	C₄	1	2	yes	complemented
18	C₁₉×#340	—	2	1	no	—
19	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
20	C₂×M₁₆×C₁₉	—	2	4	no	—
21	C₂×M₁₆×C₁₉	—	2	2	no	—
22	C₂×M₁₆×C₁₉	—	4	2	no	—
23	C₂×M₁₆×C₁₉	—	4	1	no	—
24	C₂²×C₈×C₁₉	—	2	2	no	—
25	C₂²×C₈×C₁₉	—	2	2	no	—
26	C₂²×C₈×C₁₉	—	4	1	no	—
27	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₉	Q₈	1	1	yes	characteristic
31	C₂²×C₈×C₁₉	Q₈	1	1	yes	characteristic
32	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
34	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
35	C₂×C₂²⋊C₄×C₁₉	—	4	1	no	—
36	C₁₉×M₃₂	—	2	2	no	—
37	C₁₉×M₃₂	—	2	2	no	—
38	C₂×C₁₆×C₁₉	C₂×C₄	1	4	yes	—
39	C₁₉×#108	—	2	4	no	—
40	C₁₉×#108	—	2	2	no	—
41	C₁₉×#108	D₄	1	4	yes	—
42	C₁₉×#106	—	2	2	no	—
43	C₁₉×#106	—	2	2	no	—
44	C₁₉×#95	—	4	1	no	—
45	C₂⁴×C₁₉	SD₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₁₉	—	2	1	no	—
47	C₂²×C₄×C₁₉	—	2	1	no	—
48	C₂²×C₄×C₁₉	—	2	1	no	—
49	C₂²×C₄×C₁₉	—	2	1	no	—
50	C₂²×C₄×C₁₉	—	4	1	no	—
51	C₂²×C₄×C₁₉	—	4	1	no	—
52	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
58	M₁₆×C₁₉	—	2	4	no	—
59	M₁₆×C₁₉	—	4	4	no	—
60	M₁₆×C₁₉	—	4	2	no	—
61	C₂×C₈×C₁₉	—	2	1	no	—
62	C₂×C₈×C₁₉	—	2	1	no	—
63	C₂×C₈×C₁₉	—	2	2	no	—
64	C₂×C₈×C₁₉	—	2	2	no	—
65	C₂×C₈×C₁₉	—	2	2	no	—
66	C₂×C₈×C₁₉	—	2	2	no	—
67	C₂×C₈×C₁₉	—	4	4	no	—
68	C₂×C₈×C₁₉	—	4	2	no	—
69	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₈×C₁₉	#41	1	2	yes	—
75	C₄⋊C₄×C₁₉	—	4	1	no	—
76	C₄⋊C₄×C₁₉	—	4	1	no	—
77	C₄⋊C₄×C₁₉	—	4	1	no	—
78	C₄⋊C₄×C₁₉	—	4	1	no	—
79	C₂²⋊C₄×C₁₉	—	8	2	no	—
80	C₁₆×C₁₉	—	2	4	no	—
81	#12928	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂³×C₁₉	—	2	2	no	—
83	C₂³×C₁₉	—	2	1	no	—
84	C₂³×C₁₉	—	2	1	no	—
85	C₂³×C₁₉	—	4	1	no	—
86	C₂³×C₁₉	#106	1	1	yes	characteristic
87	C₂³×C₁₉	#106	1	1	yes	characteristic
88	C₂³×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
89	C₂×C₄×C₁₉	—	2	1	no	—
90	C₂×C₄×C₁₉	—	2	1	no	—
91	C₂×C₄×C₁₉	—	2	1	no	—
92	C₂×C₄×C₁₉	—	2	1	no	—
93	C₂×C₄×C₁₉	—	2	1	no	—
94	C₂×C₄×C₁₉	—	2	1	no	—
95	C₂×C₄×C₁₉	—	2	1	no	—
96	C₂×C₄×C₁₉	—	2	1	no	—
97	C₂×C₄×C₁₉	—	2	1	no	—
98	C₂×C₄×C₁₉	—	2	1	no	—
99	C₂×C₄×C₁₉	—	8	1	no	—
100	C₂×C₄×C₁₉	—	8	1	no	—
101	C₂×C₄×C₁₉	—	8	1	no	—
102	C₂×C₄×C₁₉	—	8	1	no	—
103	C₂×C₄×C₁₉	—	8	1	no	—
104	C₂×C₄×C₁₉	—	8	1	no	—
105	C₂×C₄×C₁₉	#106	1	1	yes	characteristic
106	C₂×C₄×C₁₉	#106	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
109	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
110	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
111	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
113	C₈×C₁₉	—	2	1	no	—
114	C₈×C₁₉	—	2	1	no	—
115	C₈×C₁₉	—	4	4	no	—
116	C₈×C₁₉	C₄×D₄	1	2	yes	—
117	C₈×C₁₉	C₂²⋊Q₈	1	2	yes	—
118	C₂²×#108	C₂×C₁₉	1	1	yes	characteristic
119	C₂×#365	C₂×C₁₉	1	2	yes	complemented
120	#2112	C₂×C₁₉	1	1	yes	characteristic
121	#1807	C₂×C₁₉	1	1	yes	characteristic
122	#1383	C₂×C₁₉	1	2	yes	complemented
123	C₂²×C₁₉	—	2	1	no	—
124	C₂²×C₁₉	—	2	1	no	—
125	C₂²×C₁₉	—	2	1	no	—
126	C₂²×C₁₉	—	2	1	no	—
127	C₂²×C₁₉	—	2	1	no	—
128	C₂²×C₁₉	—	2	1	no	—
129	C₂²×C₁₉	—	2	1	no	—
130	C₂²×C₁₉	—	4	2	no	—
131	C₂²×C₁₉	—	4	2	no	—
132	C₂²×C₁₉	#365	1	2	yes	complemented
133	C₂²×C₁₉	C₂×#106	1	1	yes	characteristic
134	C₂²×C₁₉	#448	1	1	yes	characteristic
135	C₂²×C₁₉	#481	1	1	yes	characteristic
136	C₄×C₁₉	—	2	1	no	—
137	C₄×C₁₉	—	2	1	no	—
138	C₄×C₁₉	—	16	1	no	—
139	C₄×C₁₉	—	16	1	no	—
140	C₄×C₁₉	#385	1	1	yes	characteristic
141	C₄×C₁₉	#428	1	1	yes	characteristic
142	C₄×C₁₉	#450	1	1	yes	characteristic
143	C₄×C₁₉	#483	1	1	yes	characteristic
144	C₂²×M₁₆	—	2	1	no	—
145	C₂³×C₈	C₂²×C₁₉	1	1	yes	characteristic
146	C₂×M₃₂	C₂²×C₁₉	1	2	yes	—
147	C₂×#108	—	2	1	no	—
148	C₂×#108	—	2	4	no	—
149	C₂×#108	C₂²×C₁₉	1	2	yes	—
150	C₂×#106	C₂²×C₁₉	1	1	yes	characteristic
151	C₂×#106	C₂²×C₁₉	1	1	yes	characteristic
152	#380	—	2	1	no	—
153	#365	—	2	4	no	—
154	C₂²⋊C₁₆	C₄×C₁₉	1	2	yes	complemented
155	C₂²⋊C₁₆	C₄×C₁₉	1	2	yes	complemented
156	#340	—	2	1	no	—
157	C₂×C₁₉	—	2	1	no	—
158	C₂×C₁₉	—	2	1	no	—
159	C₂×C₁₉	—	2	2	no	—
160	C₂×C₁₉	—	4	1	no	—
161	C₂×C₁₉	#1853	1	1	yes	characteristic
162	C₂×C₁₉	C₂×#365	1	1	yes	characteristic
163	C₂×C₁₉	#2155	1	1	yes	characteristic
164	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
165	C₂×M₁₆	—	2	4	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	4	2	no	—
168	C₂×M₁₆	—	4	1	no	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	4	1	no	—
172	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
173	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
174	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
175	C₂²×C₈	Q₈×C₁₉	1	1	yes	characteristic
176	C₂²×C₈	Q₈×C₁₉	1	1	yes	characteristic
177	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
178	C₂×C₄⋊C₄	—	2	1	no	—
179	C₂×C₄⋊C₄	—	2	1	no	—
180	C₂×C₂²⋊C₄	—	4	1	no	—
181	M₃₂	—	2	2	no	—
182	M₃₂	—	2	2	no	—
183	C₂×C₁₆	C₂×C₄×C₁₉	1	4	yes	—
184	#108	—	2	4	no	—
185	#108	—	2	2	no	—
186	#108	D₄×C₁₉	1	4	yes	—
187	#106	—	2	2	no	—
188	#106	—	2	2	no	—
189	#95	—	4	1	no	—
190	C₁₉	#12928	1	1	yes	characteristic, direct product factor, Sylow
191	C₂⁴	SD₁₆×C₁₉	1	1	yes	characteristic
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	4	1	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	C₄⋊C₄×C₁₉	1	1	yes	characteristic
199	C₂²×C₄	C₄⋊C₄×C₁₉	1	1	yes	characteristic
200	C₂²×C₄	SD₁₆×C₁₉	1	1	yes	characteristic
201	C₂²×C₄	SD₁₆×C₁₉	1	1	yes	characteristic
202	C₂²×C₄	SD₁₆×C₁₉	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
204	M₁₆	—	2	4	no	—
205	M₁₆	—	4	4	no	—
206	M₁₆	—	4	2	no	—
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	4	4	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	C₄⋊C₄×C₁₉	1	1	yes	characteristic
216	C₂×C₈	C₄⋊C₄×C₁₉	1	1	yes	characteristic
217	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
218	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
219	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
220	C₂×C₈	#41×C₁₉	1	2	yes	—
221	C₄⋊C₄	—	4	1	no	—
222	C₄⋊C₄	—	4	1	no	—
223	C₄⋊C₄	—	4	1	no	—
224	C₄⋊C₄	—	4	1	no	—
225	C₂²⋊C₄	—	8	2	no	—
226	C₁₆	—	2	4	no	—
227	C₂³	—	2	2	no	—
228	C₂³	—	2	1	no	—
229	C₂³	—	2	1	no	—
230	C₂³	—	4	1	no	—
231	C₂³	C₁₉×#106	1	1	yes	characteristic
232	C₂³	C₁₉×#106	1	1	yes	characteristic
233	C₂³	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	8	1	no	—
245	C₂×C₄	—	8	1	no	—
246	C₂×C₄	—	8	1	no	—
247	C₂×C₄	—	8	1	no	—
248	C₂×C₄	—	8	1	no	—
249	C₂×C₄	—	8	1	no	—
250	C₂×C₄	C₁₉×#106	1	1	yes	characteristic
251	C₂×C₄	C₁₉×#106	1	1	yes	characteristic
252	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
253	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
254	C₂×C₄	C₁₉×#123	1	1	yes	characteristic
255	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
256	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
257	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	4	4	no	—
261	C₈	C₄×D₄×C₁₉	1	2	yes	—
262	C₈	C₁₉×C₂²⋊Q₈	1	2	yes	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	4	2	no	—
271	C₂²	—	4	2	no	—
272	C₂²	C₁₉×#365	1	2	yes	complemented
273	C₂²	C₂×C₁₉×#106	1	1	yes	characteristic
274	C₂²	C₁₉×#448	1	1	yes	characteristic
275	C₂²	C₁₉×#481	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	16	1	no	—
279	C₄	—	16	1	no	—
280	C₄	C₁₉×#385	1	1	yes	characteristic
281	C₄	C₁₉×#428	1	1	yes	characteristic
282	C₄	C₁₉×#450	1	1	yes	characteristic
283	C₄	C₁₉×#483	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	2	2	no	—
287	C₂	—	4	1	no	—
288	C₂	C₁₉×#1853	1	1	yes	characteristic
289	C₂	C₂×C₁₉×#365	1	1	yes	characteristic
290	C₂	C₁₉×#2155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#106
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₂⁴×C₁₉	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂³×C₄×C₁₉ ◁ C₂³×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₉×#365	C₂	semidirect
5. C₁₉×#1383	C₂	semidirect
16. C₁₉×C₂²⋊C₁₆	C₄	semidirect
17. C₁₉×C₂²⋊C₁₆	C₄	semidirect
81. #12928	C₁₉	direct (N × H)
119. C₂×#365	C₂×C₁₉	semidirect
122. #1383	C₂×C₁₉	semidirect
132. C₂²×C₁₉	#365	semidirect
154. C₂²⋊C₁₆	C₄×C₁₉	semidirect
155. C₂²⋊C₁₆	C₄×C₁₉	semidirect
190. C₁₉	#12928	direct (N × H)
272. C₂²	C₁₉×#365	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₉×#108	2
C₂×C₁₉×#365	2
C₁₉×#2112	2
C₁₉×#1807	2
C₁₉×#1383	2
#12928	19

Subgroups — C19×#12928 #18018887

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12928 #18018887