Subgroups — C₁₉×#13144 #18019103

Order 4864.

334 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×C₁₉×#361	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#357	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2108	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1978	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1952	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#1387	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1381	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₁₉	C₂²	1	1	yes	characteristic
9	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
12	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
13	C₁₉×#428	—	2	1	no	—
14	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
15	C₁₉×#418	—	2	1	no	—
16	C₂×C₁₉×#100	—	2	1	no	—
17	C₁₉×#405	C₂²	1	1	yes	characteristic
18	C₁₉×#361	C₄	1	4	yes	complemented
19	C₁₉×#357	—	2	1	no	—
20	C₁₉×#357	—	2	1	no	—
21	C₁₉×#346	—	2	1	no	—
22	C₄×C₁₆×C₁₉	—	2	1	no	—
23	C₁₉×#343	—	2	1	no	—
24	C₂²×D₄×C₁₉	—	2	1	no	—
25	C₂×D₈×C₁₉	—	2	1	no	—
26	C₂×D₈×C₁₉	—	2	1	no	—
27	C₂×D₈×C₁₉	—	2	1	no	—
28	C₂×D₈×C₁₉	—	2	1	no	—
29	C₂×D₈×C₁₉	—	2	1	no	—
30	C₂×D₈×C₁₉	C₂×C₄	1	2	yes	—
31	C₂×M₁₆×C₁₉	—	2	1	no	—
32	C₂×M₁₆×C₁₉	—	2	1	no	—
33	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
35	C₁₉×#117	D₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
37	C₁₉×M₃₂	—	2	1	no	—
38	C₁₉×M₃₂	C₂×C₄	1	2	yes	—
39	C₂×C₁₆×C₁₉	—	2	1	no	—
40	C₂×C₁₆×C₁₉	—	2	1	no	—
41	C₂×C₁₆×C₁₉	—	2	1	no	—
42	C₂×C₁₆×C₁₉	D₄	1	1	yes	characteristic
43	C₂×C₁₆×C₁₉	D₄	1	1	yes	characteristic
44	C₁₉×#108	—	2	1	no	—
45	C₁₉×#108	—	2	1	no	—
46	C₁₉×#108	—	2	1	no	—
47	C₁₉×#108	C₂×C₄	1	2	yes	—
48	C₁₉×#107	—	2	1	no	—
49	C₁₉×#107	—	2	1	no	—
50	C₁₉×#107	—	4	1	no	—
51	C₁₉×#106	—	4	1	no	—
52	C₁₉×#102	—	4	1	no	—
53	C₁₉×#102	—	4	1	no	—
54	C₁₉×#100	—	4	2	no	—
55	C₁₉×#97	—	2	1	no	—
56	C₄×C₈×C₁₉	—	2	1	no	—
57	C₂⁴×C₁₉	—	4	1	no	—
58	C₂×D₄×C₁₉	—	2	2	no	—
59	C₂×D₄×C₁₉	—	4	1	no	—
60	C₂×D₄×C₁₉	—	4	1	no	—
61	C₂×D₄×C₁₉	—	4	1	no	—
62	C₂×D₄×C₁₉	—	4	1	no	—
63	C₂×D₄×C₁₉	—	4	1	no	—
64	C₂²×C₄×C₁₉	—	4	1	no	—
65	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
66	D₈×C₁₉	—	2	2	no	—
67	D₈×C₁₉	—	2	2	no	—
68	D₈×C₁₉	—	4	1	no	—
69	D₈×C₁₉	—	4	1	no	—
70	M₁₆×C₁₉	—	2	1	no	—
71	M₁₆×C₁₉	—	2	2	no	—
72	M₁₆×C₁₉	—	4	1	no	—
73	M₁₆×C₁₉	—	4	1	no	—
74	M₁₆×C₁₉	—	4	1	no	—
75	M₁₆×C₁₉	SD₁₆	1	2	yes	—
76	C₂×C₈×C₁₉	—	2	1	no	—
77	C₂×C₈×C₁₉	—	4	1	no	—
78	C₂×C₈×C₁₉	—	4	1	no	—
79	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
82	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
83	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
84	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
85	C₄⋊C₄×C₁₉	—	4	1	no	—
86	C₄⋊C₄×C₁₉	—	4	1	no	—
87	C₄⋊C₄×C₁₉	D₈	1	1	yes	characteristic
88	C₄⋊C₄×C₁₉	D₈	1	1	yes	characteristic
89	C₂²⋊C₄×C₁₉	—	2	1	no	—
90	C₄²×C₁₉	—	2	1	no	—
91	C₁₆×C₁₉	—	2	2	no	—
92	C₁₆×C₁₉	—	2	1	no	—
93	C₁₆×C₁₉	—	2	1	no	—
94	#13144	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂³×C₁₉	—	4	2	no	—
96	C₂³×C₁₉	—	8	1	no	—
97	C₂³×C₁₉	—	8	1	no	—
98	C₂³×C₁₉	—	16	1	no	—
99	C₂³×C₁₉	—	16	1	no	—
100	C₂³×C₁₉	#135	1	1	yes	characteristic
101	D₄×C₁₉	—	4	2	no	—
102	D₄×C₁₉	—	4	2	no	—
103	D₄×C₁₉	—	8	1	no	—
104	D₄×C₁₉	—	8	1	no	—
105	C₂×C₄×C₁₉	—	2	1	no	—
106	C₂×C₄×C₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	—	8	1	no	—
108	C₂×C₄×C₁₉	—	16	1	no	—
109	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
111	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
113	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
114	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
115	C₈×C₁₉	—	2	2	no	—
116	C₈×C₁₉	—	4	2	no	—
117	C₈×C₁₉	—	8	1	no	—
118	C₈×C₁₉	#117	1	1	yes	characteristic
119	C₈×C₁₉	C₄×D₄	1	1	yes	characteristic
120	C₈×C₁₉	#123	1	1	yes	characteristic
121	C₈×C₁₉	#126	1	1	yes	characteristic
122	C₂×#361	C₂×C₁₉	1	1	yes	characteristic
123	C₂×#357	C₂×C₁₉	1	1	yes	characteristic
124	#2108	C₂×C₁₉	1	1	yes	characteristic, complemented
125	#1978	C₂×C₁₉	1	1	yes	characteristic
126	#1952	C₂×C₁₉	1	1	yes	characteristic, complemented
127	#1387	C₂×C₁₉	1	1	yes	characteristic, complemented
128	#1381	C₂×C₁₉	1	1	yes	characteristic, complemented
129	C₂²×C₁₉	—	2	1	no	—
130	C₂²×C₁₉	—	2	1	no	—
131	C₂²×C₁₉	—	8	2	no	—
132	C₂²×C₁₉	—	16	2	no	—
133	C₂²×C₁₉	—	16	2	no	—
134	C₂²×C₁₉	—	16	1	no	—
135	C₂²×C₁₉	—	16	1	no	—
136	C₂²×C₁₉	C₄×SD₁₆	1	1	yes	characteristic
137	C₂²×C₁₉	#466	1	1	yes	characteristic
138	C₂²×C₁₉	#488	1	1	yes	characteristic
139	C₄×C₁₉	—	4	1	no	—
140	C₄×C₁₉	—	4	1	no	—
141	C₄×C₁₉	—	16	1	no	—
142	C₄×C₁₉	#388	1	1	yes	characteristic
143	C₄×C₁₉	C₄×D₈	1	1	yes	characteristic
144	C₄×C₁₉	#465	1	1	yes	characteristic
145	C₄×C₁₉	#486	1	1	yes	characteristic
146	C₂²×D₈	C₂²×C₁₉	1	1	yes	characteristic
147	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
148	C₂²×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
149	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
150	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
151	#428	—	2	1	no	—
152	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
153	#418	—	2	1	no	—
154	C₂×#100	—	2	1	no	—
155	#405	C₂²×C₁₉	1	1	yes	characteristic
156	#361	C₄×C₁₉	1	4	yes	complemented
157	#357	—	2	1	no	—
158	#357	—	2	1	no	—
159	#346	—	2	1	no	—
160	C₄×C₁₆	—	2	1	no	—
161	#343	—	2	1	no	—
162	C₂×C₁₉	—	2	1	no	—
163	C₂×C₁₉	—	2	1	no	—
164	C₂×C₁₉	—	16	2	no	—
165	C₂×C₁₉	#1934	1	1	yes	characteristic
166	C₂×C₁₉	#2221	1	1	yes	characteristic
167	C₂×C₁₉	#2223	1	1	yes	characteristic
168	C₂²×D₄	—	2	1	no	—
169	C₂×D₈	—	2	1	no	—
170	C₂×D₈	—	2	1	no	—
171	C₂×D₈	—	2	1	no	—
172	C₂×D₈	—	2	1	no	—
173	C₂×D₈	—	2	1	no	—
174	C₂×D₈	C₂×C₄×C₁₉	1	2	yes	—
175	C₂×M₁₆	—	2	1	no	—
176	C₂×M₁₆	—	2	1	no	—
177	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
178	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
179	#117	D₄×C₁₉	1	1	yes	characteristic
180	C₂×C₄⋊C₄	—	2	1	no	—
181	M₃₂	—	2	1	no	—
182	M₃₂	C₂×C₄×C₁₉	1	2	yes	—
183	C₂×C₁₆	—	2	1	no	—
184	C₂×C₁₆	—	2	1	no	—
185	C₂×C₁₆	—	2	1	no	—
186	C₂×C₁₆	D₄×C₁₉	1	1	yes	characteristic
187	C₂×C₁₆	D₄×C₁₉	1	1	yes	characteristic
188	#108	—	2	1	no	—
189	#108	—	2	1	no	—
190	#108	—	2	1	no	—
191	#108	C₂×C₄×C₁₉	1	2	yes	—
192	#107	—	2	1	no	—
193	#107	—	2	1	no	—
194	#107	—	4	1	no	—
195	#106	—	4	1	no	—
196	#102	—	4	1	no	—
197	#102	—	4	1	no	—
198	#100	—	4	2	no	—
199	#97	—	2	1	no	—
200	C₄×C₈	—	2	1	no	—
201	C₁₉	#13144	1	1	yes	characteristic, direct product factor, Sylow
202	C₂⁴	—	4	1	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	4	1	no	—
205	C₂×D₄	—	4	1	no	—
206	C₂×D₄	—	4	1	no	—
207	C₂×D₄	—	4	1	no	—
208	C₂×D₄	—	4	1	no	—
209	C₂²×C₄	—	4	1	no	—
210	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
211	D₈	—	2	2	no	—
212	D₈	—	2	2	no	—
213	D₈	—	4	1	no	—
214	D₈	—	4	1	no	—
215	M₁₆	—	2	1	no	—
216	M₁₆	—	2	2	no	—
217	M₁₆	—	4	1	no	—
218	M₁₆	—	4	1	no	—
219	M₁₆	—	4	1	no	—
220	M₁₆	SD₁₆×C₁₉	1	2	yes	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
225	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
226	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
227	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
228	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
229	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
230	C₄⋊C₄	—	4	1	no	—
231	C₄⋊C₄	—	4	1	no	—
232	C₄⋊C₄	D₈×C₁₉	1	1	yes	characteristic
233	C₄⋊C₄	D₈×C₁₉	1	1	yes	characteristic
234	C₂²⋊C₄	—	2	1	no	—
235	C₄²	—	2	1	no	—
236	C₁₆	—	2	2	no	—
237	C₁₆	—	2	1	no	—
238	C₁₆	—	2	1	no	—
239	C₂³	—	4	2	no	—
240	C₂³	—	8	1	no	—
241	C₂³	—	8	1	no	—
242	C₂³	—	16	1	no	—
243	C₂³	—	16	1	no	—
244	C₂³	C₁₉×#135	1	1	yes	characteristic
245	D₄	—	4	2	no	—
246	D₄	—	4	2	no	—
247	D₄	—	8	1	no	—
248	D₄	—	8	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	8	1	no	—
252	C₂×C₄	—	16	1	no	—
253	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
254	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
255	C₂×C₄	C₁₉×#124	1	1	yes	characteristic
256	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
257	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
258	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
259	C₈	—	2	2	no	—
260	C₈	—	4	2	no	—
261	C₈	—	8	1	no	—
262	C₈	C₁₉×#117	1	1	yes	characteristic
263	C₈	C₄×D₄×C₁₉	1	1	yes	characteristic
264	C₈	C₁₉×#123	1	1	yes	characteristic
265	C₈	C₁₉×#126	1	1	yes	characteristic
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	8	2	no	—
269	C₂²	—	16	2	no	—
270	C₂²	—	16	2	no	—
271	C₂²	—	16	1	no	—
272	C₂²	—	16	1	no	—
273	C₂²	C₄×SD₁₆×C₁₉	1	1	yes	characteristic
274	C₂²	C₁₉×#466	1	1	yes	characteristic
275	C₂²	C₁₉×#488	1	1	yes	characteristic
276	C₄	—	4	1	no	—
277	C₄	—	4	1	no	—
278	C₄	—	16	1	no	—
279	C₄	C₁₉×#388	1	1	yes	characteristic
280	C₄	C₄×D₈×C₁₉	1	1	yes	characteristic
281	C₄	C₁₉×#465	1	1	yes	characteristic
282	C₄	C₁₉×#486	1	1	yes	characteristic
283	C₂	—	2	1	no	—
284	C₂	—	2	1	no	—
285	C₂	—	16	2	no	—
286	C₂	C₁₉×#1934	1	1	yes	characteristic
287	C₂	C₁₉×#2221	1	1	yes	characteristic
288	C₂	C₁₉×#2223	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#466
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₂²×D₈×C₁₉	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#405 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₉×#2108	C₂	semidirect
5. C₁₉×#1952	C₂	semidirect
6. C₁₉×#1387	C₂	semidirect
7. C₁₉×#1381	C₂	semidirect
18. C₁₉×#361	C₄	semidirect
94. #13144	C₁₉	direct (N × H)
124. #2108	C₂×C₁₉	semidirect
126. #1952	C₂×C₁₉	semidirect
127. #1387	C₂×C₁₉	semidirect
128. #1381	C₂×C₁₉	semidirect
156. #361	C₄×C₁₉	semidirect
201. C₁₉	#13144	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×#361	2
C₂×C₁₉×#357	2
C₁₉×#2108	2
C₁₉×#1978	2
C₁₉×#1952	2
C₁₉×#1387	2
C₁₉×#1381	2
#13144	19

Subgroups — C19×#13144 #18019103

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13144 #18019103