Subgroups — C₁₉×#13122 #18019081

Order 4864.

326 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#364	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₉×#350	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2113	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#2108	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1764	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1379	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₉×#131	C₂²	1	1	yes	characteristic
9	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
13	C₂×C₁₉×#104	C₄	1	1	yes	characteristic
14	C₁₉×#416	C₄	1	1	yes	characteristic
15	C₁₉×#405	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#364	—	2	2	no	—
18	C₁₉×#350	—	2	2	no	—
19	C₁₉×C₂²⋊C₁₆	—	2	1	no	—
20	C₁₉×C₂²⋊C₁₆	—	2	1	no	—
21	C₁₉×#346	—	2	1	no	—
22	C₄×C₁₆×C₁₉	—	2	1	no	—
23	C₁₉×#341	C₄	1	1	yes	characteristic
24	C₁₉×#339	C₄	1	1	yes	characteristic
25	C₂×#41×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₁₉×#131	—	2	1	no	—
27	C₁₉×#131	—	2	1	no	—
28	C₁₉×#131	—	2	1	no	—
29	C₁₉×#131	—	2	1	no	—
30	C₂×M₁₆×C₁₉	—	2	1	no	—
31	C₂×M₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
32	C₂×M₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₉	—	2	1	no	—
34	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
36	C₁₉×#117	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
38	C₁₉×M₃₂	—	2	1	no	—
39	C₁₉×M₃₂	—	2	1	no	—
40	C₁₉×M₃₂	—	2	1	no	—
41	C₁₉×M₃₂	D₄	1	2	yes	—
42	C₂×C₁₆×C₁₉	—	2	1	no	—
43	C₂×C₁₆×C₁₉	—	2	1	no	—
44	C₂×C₁₆×C₁₉	—	2	1	no	—
45	C₂×C₁₆×C₁₉	—	2	1	no	—
46	C₂×C₁₆×C₁₉	—	2	1	no	—
47	C₂×C₁₆×C₁₉	—	2	1	no	—
48	C₂×C₁₆×C₁₉	—	2	1	no	—
49	C₂×C₁₆×C₁₉	D₄	1	1	yes	characteristic
50	C₂×C₁₆×C₁₉	D₄	1	1	yes	characteristic
51	C₁₉×#104	—	2	2	no	—
52	C₁₉×#103	C₈	1	1	yes	characteristic
53	C₁₉×#103	C₈	1	1	yes	characteristic
54	C₁₉×#102	C₈	1	1	yes	characteristic
55	C₁₉×#102	C₈	1	1	yes	characteristic
56	C₁₉×#97	—	2	1	no	—
57	C₄×C₈×C₁₉	—	2	1	no	—
58	C₄×C₈×C₁₉	—	2	1	no	—
59	C₄×C₈×C₁₉	—	2	1	no	—
60	#41×C₁₉	—	2	1	no	—
61	#41×C₁₉	—	2	1	no	—
62	#41×C₁₉	—	2	1	no	—
63	#41×C₁₉	—	2	1	no	—
64	C₂×Q₈×C₁₉	C₂×C₈	1	1	yes	characteristic
65	C₂×D₄×C₁₉	—	2	1	no	—
66	C₂×D₄×C₁₉	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	—	2	1	no	—
68	C₂²×C₄×C₁₉	—	2	1	no	—
69	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
70	M₁₆×C₁₉	—	2	1	no	—
71	M₁₆×C₁₉	—	2	1	no	—
72	M₁₆×C₁₉	—	2	1	no	—
73	M₁₆×C₁₉	—	4	1	no	—
74	M₁₆×C₁₉	—	4	1	no	—
75	M₁₆×C₁₉	M₁₆	1	2	yes	—
76	C₂×C₈×C₁₉	—	2	1	no	—
77	C₂×C₈×C₁₉	—	2	1	no	—
78	C₂×C₈×C₁₉	—	2	1	no	—
79	C₂×C₈×C₁₉	—	2	1	no	—
80	C₂×C₈×C₁₉	—	4	1	no	—
81	C₂×C₈×C₁₉	—	4	1	no	—
82	C₂×C₈×C₁₉	C₂×C₈	1	1	yes	characteristic
83	C₂×C₈×C₁₉	C₂×C₈	1	1	yes	characteristic
84	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
87	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
88	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
89	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
90	C₄⋊C₄×C₁₉	C₂×C₈	1	1	yes	characteristic
91	C₄⋊C₄×C₁₉	C₂×C₈	1	1	yes	characteristic
92	C₂²⋊C₄×C₁₉	—	2	1	no	—
93	C₄²×C₁₉	—	2	1	no	—
94	C₄²×C₁₉	—	2	1	no	—
95	C₄²×C₁₉	—	2	1	no	—
96	C₁₆×C₁₉	—	2	2	no	—
97	C₁₆×C₁₉	—	2	1	no	—
98	C₁₆×C₁₉	—	2	1	no	—
99	C₁₆×C₁₉	—	4	1	no	—
100	C₁₆×C₁₉	—	4	1	no	—
101	#13122	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂³×C₁₉	—	2	1	no	—
103	C₂³×C₁₉	#131	1	1	yes	characteristic
104	Q₈×C₁₉	—	2	1	no	—
105	Q₈×C₁₉	—	2	1	no	—
106	D₄×C₁₉	—	2	1	no	—
107	D₄×C₁₉	—	2	1	no	—
108	D₄×C₁₉	—	4	1	no	—
109	D₄×C₁₉	—	4	1	no	—
110	C₂×C₄×C₁₉	—	2	2	no	—
111	C₂×C₄×C₁₉	—	2	1	no	—
112	C₂×C₄×C₁₉	—	2	1	no	—
113	C₂×C₄×C₁₉	—	2	1	no	—
114	C₂×C₄×C₁₉	—	4	1	no	—
115	C₂×C₄×C₁₉	—	4	1	no	—
116	C₂×C₄×C₁₉	—	4	1	no	—
117	C₂×C₄×C₁₉	—	4	1	no	—
118	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
123	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
124	C₈×C₁₉	—	2	1	no	—
125	C₈×C₁₉	—	2	1	no	—
126	C₈×C₁₉	—	2	2	no	—
127	C₈×C₁₉	—	4	1	no	—
128	C₈×C₁₉	#121	1	1	yes	characteristic
129	C₈×C₁₉	#123	1	1	yes	characteristic
130	C₈×C₁₉	#124	1	1	yes	characteristic
131	C₈×C₁₉	#126	1	1	yes	characteristic
132	C₂×#364	C₂×C₁₉	1	1	yes	characteristic, complemented
133	C₂×#350	C₂×C₁₉	1	1	yes	characteristic
134	#2113	C₂×C₁₉	1	1	yes	characteristic
135	#2108	C₂×C₁₉	1	1	yes	characteristic, complemented
136	#1764	C₂×C₁₉	1	1	yes	characteristic
137	#1379	C₂×C₁₉	1	1	yes	characteristic, complemented
138	#1374	C₂×C₁₉	1	1	yes	characteristic, complemented
139	C₂²×C₁₉	—	2	1	no	—
140	C₂²×C₁₉	—	2	1	no	—
141	C₂²×C₁₉	—	4	1	no	—
142	C₂²×C₁₉	—	4	1	no	—
143	C₂²×C₁₉	—	4	1	no	—
144	C₂²×C₁₉	#433	1	1	yes	characteristic
145	C₂²×C₁₉	C₈×D₄	1	1	yes	characteristic
146	C₂²×C₁₉	#436	1	1	yes	characteristic
147	C₄×C₁₉	—	4	2	no	—
148	C₄×C₁₉	—	4	1	no	—
149	C₄×C₁₉	—	4	1	no	—
150	C₄×C₁₉	—	4	1	no	—
151	C₄×C₁₉	#388	1	1	yes	characteristic
152	C₄×C₁₉	#431	1	1	yes	characteristic
153	C₄×C₁₉	C₈×D₄	1	1	yes	characteristic
154	C₄×C₁₉	#435	1	1	yes	characteristic
155	C₂×#131	C₂²×C₁₉	1	1	yes	characteristic
156	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
157	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
158	C₂²×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
159	C₂²×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
160	C₂×#104	C₄×C₁₉	1	1	yes	characteristic
161	#416	C₄×C₁₉	1	1	yes	characteristic
162	#405	C₂²×C₁₉	1	1	yes	characteristic
163	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
164	#364	—	2	2	no	—
165	#350	—	2	2	no	—
166	C₂²⋊C₁₆	—	2	1	no	—
167	C₂²⋊C₁₆	—	2	1	no	—
168	#346	—	2	1	no	—
169	C₄×C₁₆	—	2	1	no	—
170	#341	C₄×C₁₉	1	1	yes	characteristic
171	#339	C₄×C₁₉	1	1	yes	characteristic
172	C₂×C₁₉	—	2	1	no	—
173	C₂×C₁₉	—	2	1	no	—
174	C₂×C₁₉	—	8	1	no	—
175	C₂×C₁₉	#1923	1	1	yes	characteristic
176	C₂×C₁₉	#2170	1	1	yes	characteristic
177	C₂×C₁₉	#2171	1	1	yes	characteristic
178	C₂×#41	C₂×C₄×C₁₉	1	1	yes	characteristic
179	#131	—	2	1	no	—
180	#131	—	2	1	no	—
181	#131	—	2	1	no	—
182	#131	—	2	1	no	—
183	C₂×M₁₆	—	2	1	no	—
184	C₂×M₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
185	C₂×M₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
186	C₂²×C₈	—	2	1	no	—
187	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
188	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
189	#117	C₂×C₄×C₁₉	1	1	yes	characteristic
190	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
191	M₃₂	—	2	1	no	—
192	M₃₂	—	2	1	no	—
193	M₃₂	—	2	1	no	—
194	M₃₂	D₄×C₁₉	1	2	yes	—
195	C₂×C₁₆	—	2	1	no	—
196	C₂×C₁₆	—	2	1	no	—
197	C₂×C₁₆	—	2	1	no	—
198	C₂×C₁₆	—	2	1	no	—
199	C₂×C₁₆	—	2	1	no	—
200	C₂×C₁₆	—	2	1	no	—
201	C₂×C₁₆	—	2	1	no	—
202	C₂×C₁₆	D₄×C₁₉	1	1	yes	characteristic
203	C₂×C₁₆	D₄×C₁₉	1	1	yes	characteristic
204	#104	—	2	2	no	—
205	#103	C₈×C₁₉	1	1	yes	characteristic
206	#103	C₈×C₁₉	1	1	yes	characteristic
207	#102	C₈×C₁₉	1	1	yes	characteristic
208	#102	C₈×C₁₉	1	1	yes	characteristic
209	#97	—	2	1	no	—
210	C₄×C₈	—	2	1	no	—
211	C₄×C₈	—	2	1	no	—
212	C₄×C₈	—	2	1	no	—
213	C₁₉	#13122	1	1	yes	characteristic, direct product factor, Sylow
214	#41	—	2	1	no	—
215	#41	—	2	1	no	—
216	#41	—	2	1	no	—
217	#41	—	2	1	no	—
218	C₂×Q₈	C₂×C₈×C₁₉	1	1	yes	characteristic
219	C₂×D₄	—	2	1	no	—
220	C₂×D₄	C₂×C₈×C₁₉	1	1	yes	characteristic
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
224	M₁₆	—	2	1	no	—
225	M₁₆	—	2	1	no	—
226	M₁₆	—	2	1	no	—
227	M₁₆	—	4	1	no	—
228	M₁₆	—	4	1	no	—
229	M₁₆	M₁₆×C₁₉	1	2	yes	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	4	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	C₂×C₈×C₁₉	1	1	yes	characteristic
237	C₂×C₈	C₂×C₈×C₁₉	1	1	yes	characteristic
238	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
239	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
240	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
241	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
242	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
243	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
244	C₄⋊C₄	C₂×C₈×C₁₉	1	1	yes	characteristic
245	C₄⋊C₄	C₂×C₈×C₁₉	1	1	yes	characteristic
246	C₂²⋊C₄	—	2	1	no	—
247	C₄²	—	2	1	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	2	1	no	—
250	C₁₆	—	2	2	no	—
251	C₁₆	—	2	1	no	—
252	C₁₆	—	2	1	no	—
253	C₁₆	—	4	1	no	—
254	C₁₆	—	4	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	C₁₉×#131	1	1	yes	characteristic
257	Q₈	—	2	1	no	—
258	Q₈	—	2	1	no	—
259	D₄	—	2	1	no	—
260	D₄	—	2	1	no	—
261	D₄	—	4	1	no	—
262	D₄	—	4	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	C₁₉×#117	1	1	yes	characteristic
272	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
273	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₈×C₁₉	1	1	yes	characteristic
275	C₂×C₄	C₂×M₁₆×C₁₉	1	1	yes	characteristic
276	C₂×C₄	C₁₉×#131	1	1	yes	characteristic
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	2	2	no	—
280	C₈	—	4	1	no	—
281	C₈	C₁₉×#121	1	1	yes	characteristic
282	C₈	C₁₉×#123	1	1	yes	characteristic
283	C₈	C₁₉×#124	1	1	yes	characteristic
284	C₈	C₁₉×#126	1	1	yes	characteristic
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	1	no	—
290	C₂²	C₁₉×#433	1	1	yes	characteristic
291	C₂²	C₈×D₄×C₁₉	1	1	yes	characteristic
292	C₂²	C₁₉×#436	1	1	yes	characteristic
293	C₄	—	4	2	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	—	4	1	no	—
297	C₄	C₁₉×#388	1	1	yes	characteristic
298	C₄	C₁₉×#431	1	1	yes	characteristic
299	C₄	C₈×D₄×C₁₉	1	1	yes	characteristic
300	C₄	C₁₉×#435	1	1	yes	characteristic
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	8	1	no	—
304	C₂	C₁₉×#1923	1	1	yes	characteristic
305	C₂	C₁₉×#2170	1	1	yes	characteristic
306	C₂	C₁₉×#2171	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₁₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
socle	C₂²×C₁₉	C₈×D₄
prime radical	C₂×#41×C₁₉	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₉ ◁ C₂²×C₁₆×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₉×#364	C₂	semidirect
4. C₁₉×#2108	C₂	semidirect
6. C₁₉×#1379	C₂	semidirect
7. C₁₉×#1374	C₂	semidirect
101. #13122	C₁₉	direct (N × H)
132. C₂×#364	C₂×C₁₉	semidirect
135. #2108	C₂×C₁₉	semidirect
137. #1379	C₂×C₁₉	semidirect
138. #1374	C₂×C₁₉	semidirect
213. C₁₉	#13122	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×#364	2
C₂×C₁₉×#350	2
C₁₉×#2113	2
C₁₉×#2108	2
C₁₉×#1764	2
C₁₉×#1379	2
C₁₉×#1374	2
#13122	19

Subgroups — C19×#13122 #18019081

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13122 #18019081