Subgroups — C₁₉×#12819 #18018778

Order 4864.

410 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2150	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#362	C₂	1	2	yes	maximal
4	C₁₉×#1848	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1387	C₂	1	2	yes	maximal
6	C₂²×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₁₉	—	2	1	no	—
8	C₂×C₁₉×M₃₂	C₂²	1	2	yes	—
9	C₄×Q₁₆×C₁₉	—	2	1	no	—
10	C₄×Q₁₆×C₁₉	—	2	1	no	—
11	C₄×Q₁₆×C₁₉	C₄	1	2	yes	—
12	C₄×Q₁₆×C₁₉	C₄	1	2	yes	—
13	C₂×C₁₉×#108	—	2	1	no	—
14	C₂×C₁₉×#108	C₂²	1	2	yes	—
15	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
16	C₁₉×#423	—	2	1	no	—
17	C₂×C₁₉×#103	—	2	1	no	—
18	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
19	C₁₉×C₄⋊C₁₆	—	2	1	no	—
20	C₁₉×C₄⋊C₁₆	—	2	1	no	—
21	C₁₉×#362	—	2	4	no	—
22	C₂²×Q₈×C₁₉	—	2	1	no	—
23	C₂×Q₁₆×C₁₉	—	2	1	no	—
24	C₂×Q₁₆×C₁₉	—	2	1	no	—
25	C₂×Q₁₆×C₁₉	—	2	1	no	—
26	C₂×Q₁₆×C₁₉	C₂×C₄	1	2	yes	—
27	C₂×Q₁₆×C₁₉	D₄	1	2	yes	—
28	C₂×Q₁₆×C₁₉	Q₈	1	2	yes	—
29	C₂×M₁₆×C₁₉	—	2	2	no	—
30	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
32	C₄×Q₈×C₁₉	—	2	2	no	—
33	C₄×Q₈×C₁₉	—	2	2	no	—
34	C₄×Q₈×C₁₉	—	4	1	no	—
35	C₄×Q₈×C₁₉	—	4	1	no	—
36	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
37	C₂×C₄⋊C₄×C₁₉	—	4	1	no	—
38	C₂×C₄²×C₁₉	—	4	1	no	—
39	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
40	C₁₉×M₃₂	—	2	2	no	—
41	C₁₉×M₃₂	—	2	2	no	—
42	C₂×C₁₆×C₁₉	—	2	2	no	—
43	C₁₉×#108	—	2	2	no	—
44	C₁₉×#108	—	2	2	no	—
45	C₁₉×#108	—	4	2	no	—
46	C₁₉×#107	—	2	1	no	—
47	C₁₉×#107	C₂×C₄	1	2	yes	—
48	C₁₉×C₄⋊C₈	—	4	1	no	—
49	C₁₉×C₄⋊C₈	—	4	1	no	—
50	C₁₉×#103	—	2	2	no	—
51	C₁₉×#103	—	2	2	no	—
52	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
55	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
56	C₂×Q₈×C₁₉	—	2	2	no	—
57	C₂×Q₈×C₁₉	—	2	2	no	—
58	C₂×Q₈×C₁₉	—	2	2	no	—
59	C₂×Q₈×C₁₉	—	4	1	no	—
60	C₂×Q₈×C₁₉	—	4	1	no	—
61	C₂×Q₈×C₁₉	—	4	1	no	—
62	C₂²×C₄×C₁₉	—	4	1	no	—
63	C₂²×C₄×C₁₉	—	4	1	no	—
64	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
67	Q₁₆×C₁₉	—	2	2	no	—
68	Q₁₆×C₁₉	—	2	2	no	—
69	Q₁₆×C₁₉	—	2	2	no	—
70	Q₁₆×C₁₉	C₄⋊C₄	1	4	yes	—
71	M₁₆×C₁₉	—	4	2	no	—
72	M₁₆×C₁₉	—	4	2	no	—
73	C₂×C₈×C₁₉	—	2	1	no	—
74	C₂×C₈×C₁₉	—	2	1	no	—
75	C₂×C₈×C₁₉	—	4	2	no	—
76	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
82	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
83	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
84	C₄⋊C₄×C₁₉	—	2	2	no	—
85	C₄⋊C₄×C₁₉	—	4	2	no	—
86	C₄⋊C₄×C₁₉	—	4	2	no	—
87	C₄⋊C₄×C₁₉	—	4	1	no	—
88	C₄²×C₁₉	—	4	2	no	—
89	C₄²×C₁₉	—	4	2	no	—
90	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
91	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
92	C₄²×C₁₉	D₈	1	1	yes	characteristic
93	C₄²×C₁₉	D₈	1	1	yes	characteristic
94	C₁₆×C₁₉	—	4	2	no	—
95	#12819	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂³×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
97	Q₈×C₁₉	—	2	4	no	—
98	Q₈×C₁₉	—	4	2	no	—
99	Q₈×C₁₉	—	4	2	no	—
100	Q₈×C₁₉	—	4	2	no	—
101	C₂×C₄×C₁₉	—	2	2	no	—
102	C₂×C₄×C₁₉	—	4	1	no	—
103	C₂×C₄×C₁₉	—	4	2	no	—
104	C₂×C₄×C₁₉	—	4	2	no	—
105	C₂×C₄×C₁₉	—	4	2	no	—
106	C₂×C₄×C₁₉	—	4	2	no	—
107	C₂×C₄×C₁₉	—	8	2	no	—
108	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
109	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
110	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
111	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
118	C₈×C₁₉	—	2	1	no	—
119	C₈×C₁₉	—	2	1	no	—
120	C₈×C₁₉	—	8	2	no	—
121	C₈×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₈×C₁₉	#117	1	1	yes	characteristic
123	C₈×C₁₉	#121	1	1	yes	characteristic
124	C₈×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×Q₁₆	C₂×C₁₉	1	1	yes	characteristic
126	#2150	C₂×C₁₉	1	1	yes	characteristic
127	C₂×#362	C₂×C₁₉	1	2	yes	—
128	#1848	C₂×C₁₉	1	1	yes	characteristic
129	#1387	C₂×C₁₉	1	2	yes	—
130	C₂²×C₁₉	—	2	1	no	—
131	C₂²×C₁₉	—	2	1	no	—
132	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
133	C₂²×C₁₉	#461	1	1	yes	characteristic
134	C₂²×C₁₉	#476	1	1	yes	characteristic
135	C₄×C₁₉	—	2	1	no	—
136	C₄×C₁₉	—	2	1	no	—
137	C₄×C₁₉	—	2	1	no	—
138	C₄×C₁₉	—	4	4	no	—
139	C₄×C₁₉	—	8	2	no	—
140	C₄×C₁₉	#362	1	2	yes	complemented
141	C₄×C₁₉	#380	1	1	yes	characteristic
142	C₄×C₁₉	#417	1	1	yes	characteristic
143	C₄×C₁₉	#459	1	1	yes	characteristic
144	C₄×C₁₉	#474	1	1	yes	characteristic
145	C₂²×Q₁₆	C₂²×C₁₉	1	1	yes	characteristic
146	C₂×C₄×Q₈	—	2	1	no	—
147	C₂×M₃₂	C₂²×C₁₉	1	2	yes	—
148	C₄×Q₁₆	—	2	1	no	—
149	C₄×Q₁₆	—	2	1	no	—
150	C₄×Q₁₆	C₄×C₁₉	1	2	yes	—
151	C₄×Q₁₆	C₄×C₁₉	1	2	yes	—
152	C₂×#108	—	2	1	no	—
153	C₂×#108	C₂²×C₁₉	1	2	yes	—
154	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
155	#423	—	2	1	no	—
156	C₂×#103	—	2	1	no	—
157	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
158	C₄⋊C₁₆	—	2	1	no	—
159	C₄⋊C₁₆	—	2	1	no	—
160	#362	—	2	4	no	—
161	C₂×C₁₉	—	2	1	no	—
162	C₂×C₁₉	—	2	1	no	—
163	C₂×C₁₉	#1802	1	1	yes	characteristic
164	C₂×C₁₉	C₂×#362	1	1	yes	characteristic
165	C₂×C₁₉	#2148	1	1	yes	characteristic
166	C₂²×Q₈	—	2	1	no	—
167	C₂×Q₁₆	—	2	1	no	—
168	C₂×Q₁₆	—	2	1	no	—
169	C₂×Q₁₆	—	2	1	no	—
170	C₂×Q₁₆	C₂×C₄×C₁₉	1	2	yes	—
171	C₂×Q₁₆	D₄×C₁₉	1	2	yes	—
172	C₂×Q₁₆	Q₈×C₁₉	1	2	yes	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
175	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
176	C₄×Q₈	—	2	2	no	—
177	C₄×Q₈	—	2	2	no	—
178	C₄×Q₈	—	4	1	no	—
179	C₄×Q₈	—	4	1	no	—
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₄⋊C₄	—	4	1	no	—
182	C₂×C₄²	—	4	1	no	—
183	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
184	M₃₂	—	2	2	no	—
185	M₃₂	—	2	2	no	—
186	C₂×C₁₆	—	2	2	no	—
187	#108	—	2	2	no	—
188	#108	—	2	2	no	—
189	#108	—	4	2	no	—
190	#107	—	2	1	no	—
191	#107	C₂×C₄×C₁₉	1	2	yes	—
192	C₄⋊C₈	—	4	1	no	—
193	C₄⋊C₈	—	4	1	no	—
194	#103	—	2	2	no	—
195	#103	—	2	2	no	—
196	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
197	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
198	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
199	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
200	C₁₉	#12819	1	1	yes	characteristic, direct product factor, Sylow
201	C₂×Q₈	—	2	2	no	—
202	C₂×Q₈	—	2	2	no	—
203	C₂×Q₈	—	2	2	no	—
204	C₂×Q₈	—	4	1	no	—
205	C₂×Q₈	—	4	1	no	—
206	C₂×Q₈	—	4	1	no	—
207	C₂²×C₄	—	4	1	no	—
208	C₂²×C₄	—	4	1	no	—
209	C₂²×C₄	SD₁₆×C₁₉	1	1	yes	characteristic
210	C₂²×C₄	SD₁₆×C₁₉	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
212	Q₁₆	—	2	2	no	—
213	Q₁₆	—	2	2	no	—
214	Q₁₆	—	2	2	no	—
215	Q₁₆	C₄⋊C₄×C₁₉	1	4	yes	—
216	M₁₆	—	4	2	no	—
217	M₁₆	—	4	2	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
222	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
223	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
224	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
225	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
226	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
227	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
228	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	4	1	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄²	—	4	2	no	—
234	C₄²	—	4	2	no	—
235	C₄²	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
236	C₄²	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
237	C₄²	D₈×C₁₉	1	1	yes	characteristic
238	C₄²	D₈×C₁₉	1	1	yes	characteristic
239	C₁₆	—	4	2	no	—
240	C₂³	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
241	Q₈	—	2	4	no	—
242	Q₈	—	4	2	no	—
243	Q₈	—	4	2	no	—
244	Q₈	—	4	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	8	2	no	—
252	C₂×C₄	C₁₉×#102	1	1	yes	characteristic
253	C₂×C₄	C₁₉×#102	1	1	yes	characteristic
254	C₂×C₄	C₁₉×#102	1	1	yes	characteristic
255	C₂×C₄	C₁₉×#102	1	1	yes	characteristic
256	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
257	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
258	C₂×C₄	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
259	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
260	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
261	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	8	2	no	—
265	C₈	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
266	C₈	C₁₉×#117	1	1	yes	characteristic
267	C₈	C₁₉×#121	1	1	yes	characteristic
268	C₈	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	C₂×C₁₉×#102	1	1	yes	characteristic
272	C₂²	C₁₉×#461	1	1	yes	characteristic
273	C₂²	C₁₉×#476	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	4	4	no	—
278	C₄	—	8	2	no	—
279	C₄	C₁₉×#362	1	2	yes	complemented
280	C₄	C₁₉×#380	1	1	yes	characteristic
281	C₄	C₁₉×#417	1	1	yes	characteristic
282	C₄	C₁₉×#459	1	1	yes	characteristic
283	C₄	C₁₉×#474	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	C₁₉×#1802	1	1	yes	characteristic
287	C₂	C₂×C₁₉×#362	1	1	yes	characteristic
288	C₂	C₁₉×#2148	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₂³×C₁₉	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
95. #12819	C₁₉	direct (N × H)
140. C₄×C₁₉	#362	semidirect
200. C₁₉	#12819	direct (N × H)
279. C₄	C₁₉×#362	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₁₆×C₁₉	2
C₁₉×#2150	2
C₂×C₁₉×#362	2
C₁₉×#1848	2
C₁₉×#1387	2
#12819	19

Subgroups — C19×#12819 #18018778

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12819 #18018778