Subgroups — C₁₉×#9372 #18015331

Order 4864.

390 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3235	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2985	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#1552	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1536	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1480	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1290	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1287	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₉×#533	C₂²	1	1	yes	characteristic
9	C₁₉×#532	C₂²	1	1	yes	characteristic
10	C₁₉×#501	C₄	1	2	yes	—
11	C₁₉×#490	C₄	1	2	yes	—
12	C₁₉×#482	—	2	2	no	—
13	C₁₉×#476	—	2	1	no	—
14	C₁₉×#474	—	2	1	no	—
15	C₁₉×#446	—	2	2	no	—
16	C₁₉×#446	—	2	1	no	—
17	C₈×Q₈×C₁₉	—	2	1	no	—
18	C₁₉×#433	—	2	2	no	—
19	C₁₉×#432	C₂²	1	1	yes	characteristic
20	C₁₉×#432	C₂²	1	1	yes	characteristic
21	C₁₉×#432	C₂²	1	1	yes	characteristic
22	C₁₉×#431	C₂²	1	1	yes	characteristic
23	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₁₉×#332	—	2	1	no	—
25	C₁₉×#332	—	2	1	no	—
26	C₁₉×#329	—	2	2	no	—
27	C₁₉×D₄⋊C₈	—	2	1	no	—
28	C₁₉×D₄⋊C₈	—	2	1	no	—
29	C₂×#41×C₁₉	—	2	1	no	—
30	C₂²×C₈×C₁₉	—	2	1	no	—
31	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
32	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
33	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
34	C₁₉×#127	C₂×C₄	1	1	yes	characteristic
35	C₁₉×#125	C₂×C₄	1	2	yes	—
36	C₁₉×#124	—	2	1	no	—
37	C₁₉×C₂²⋊Q₈	—	2	1	no	—
38	C₁₉×C₂²⋊Q₈	—	2	1	no	—
39	C₁₉×#121	—	2	1	no	—
40	C₁₉×#121	—	2	1	no	—
41	C₄×Q₈×C₁₉	—	2	1	no	—
42	C₄×Q₈×C₁₉	—	2	1	no	—
43	C₄×D₄×C₁₉	—	2	1	no	—
44	C₄×D₄×C₁₉	—	2	1	no	—
45	C₁₉×#117	D₄	1	2	yes	—
46	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
47	C₁₉×#107	—	2	2	no	—
48	C₁₉×#106	—	2	2	no	—
49	C₁₉×C₄⋊C₈	—	2	2	no	—
50	C₁₉×C₄⋊C₈	—	2	2	no	—
51	C₁₉×C₄⋊C₈	—	2	1	no	—
52	C₁₉×C₄⋊C₈	—	2	1	no	—
53	C₁₉×C₄⋊C₈	—	2	1	no	—
54	C₁₉×C₄⋊C₈	—	2	1	no	—
55	C₁₉×C₄⋊C₈	—	2	1	no	—
56	C₁₉×C₄⋊C₈	—	2	1	no	—
57	C₁₉×#102	—	2	2	no	—
58	C₁₉×#102	—	2	2	no	—
59	C₁₉×C₂²⋊C₈	—	2	1	no	—
60	C₁₉×C₂²⋊C₈	—	2	1	no	—
61	C₁₉×C₂²⋊C₈	—	2	1	no	—
62	C₁₉×C₂²⋊C₈	—	2	1	no	—
63	C₁₉×#97	—	2	2	no	—
64	C₁₉×#97	—	2	1	no	—
65	C₁₉×#97	C₂×C₄	1	2	yes	—
66	C₁₉×#97	D₄	1	2	yes	—
67	C₄×C₈×C₁₉	—	2	2	no	—
68	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
69	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
70	#41×C₁₉	—	8	1	no	—
71	#41×C₁₉	—	8	1	no	—
72	C₂×Q₈×C₁₉	—	2	1	no	—
73	C₂×Q₈×C₁₉	—	2	1	no	—
74	C₂×Q₈×C₁₉	—	2	1	no	—
75	C₂×D₄×C₁₉	—	2	1	no	—
76	C₂×D₄×C₁₉	—	2	1	no	—
77	C₂×D₄×C₁₉	—	2	1	no	—
78	C₂×D₄×C₁₉	—	4	1	no	—
79	C₂²×C₄×C₁₉	—	4	1	no	—
80	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
82	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
83	C₂×C₈×C₁₉	—	2	2	no	—
84	C₂×C₈×C₁₉	—	2	1	no	—
85	C₂×C₈×C₁₉	—	2	2	no	—
86	C₂×C₈×C₁₉	—	2	1	no	—
87	C₂×C₈×C₁₉	—	4	1	no	—
88	C₂×C₈×C₁₉	—	4	1	no	—
89	C₂×C₈×C₁₉	—	8	1	no	—
90	C₄⋊C₄×C₁₉	—	2	2	no	—
91	C₄⋊C₄×C₁₉	—	2	2	no	—
92	C₄⋊C₄×C₁₉	—	2	1	no	—
93	C₄⋊C₄×C₁₉	—	2	1	no	—
94	C₄⋊C₄×C₁₉	—	2	1	no	—
95	C₄⋊C₄×C₁₉	—	2	1	no	—
96	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
97	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
98	C₂²⋊C₄×C₁₉	—	2	2	no	—
99	C₂²⋊C₄×C₁₉	—	4	1	no	—
100	C₂²⋊C₄×C₁₉	—	4	1	no	—
101	C₄²×C₁₉	—	2	2	no	—
102	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
106	#9372	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂³×C₁₉	—	4	1	no	—
108	C₂³×C₁₉	#123	1	1	yes	characteristic
109	Q₈×C₁₉	—	4	1	no	—
110	Q₈×C₁₉	—	4	1	no	—
111	Q₈×C₁₉	—	8	1	no	—
112	Q₈×C₁₉	—	8	1	no	—
113	D₄×C₁₉	—	4	2	no	—
114	D₄×C₁₉	—	4	2	no	—
115	D₄×C₁₉	—	8	1	no	—
116	D₄×C₁₉	—	8	2	no	—
117	C₂×C₄×C₁₉	—	2	1	no	—
118	C₂×C₄×C₁₉	—	2	1	no	—
119	C₂×C₄×C₁₉	—	2	2	no	—
120	C₂×C₄×C₁₉	—	2	2	no	—
121	C₂×C₄×C₁₉	—	4	1	no	—
122	C₂×C₄×C₁₉	—	4	1	no	—
123	C₂×C₄×C₁₉	—	4	1	no	—
124	C₂×C₄×C₁₉	—	8	1	no	—
125	C₂×C₄×C₁₉	—	8	1	no	—
126	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
130	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
131	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
132	C₈×C₁₉	—	4	1	no	—
133	C₈×C₁₉	—	4	2	no	—
134	C₈×C₁₉	—	4	1	no	—
135	C₈×C₁₉	—	4	2	no	—
136	C₈×C₁₉	—	8	1	no	—
137	C₈×C₁₉	—	8	1	no	—
138	#3235	C₂×C₁₉	1	1	yes	characteristic
139	#2985	C₂×C₁₉	1	1	yes	characteristic, complemented
140	#1552	C₂×C₁₉	1	1	yes	characteristic, complemented
141	#1536	C₂×C₁₉	1	1	yes	characteristic
142	#1480	C₂×C₁₉	1	1	yes	characteristic
143	#1290	C₂×C₁₉	1	1	yes	characteristic, complemented
144	#1287	C₂×C₁₉	1	1	yes	characteristic, complemented
145	C₂²×C₁₉	—	2	1	no	—
146	C₂²×C₁₉	—	2	1	no	—
147	C₂²×C₁₉	—	2	1	no	—
148	C₂²×C₁₉	—	8	1	no	—
149	C₂²×C₁₉	—	8	1	no	—
150	C₂²×C₁₉	—	8	1	no	—
151	C₂²×C₁₉	#386	1	1	yes	characteristic
152	C₄×C₁₉	—	2	1	no	—
153	C₄×C₁₉	—	2	1	no	—
154	C₄×C₁₉	—	2	1	no	—
155	C₄×C₁₉	—	2	1	no	—
156	C₄×C₁₉	—	2	1	no	—
157	C₄×C₁₉	—	4	2	no	—
158	C₄×C₁₉	—	4	2	no	—
159	C₄×C₁₉	—	8	1	no	—
160	C₄×C₁₉	#471	1	1	yes	characteristic
161	C₄×C₁₉	#472	1	1	yes	characteristic
162	#533	C₂²×C₁₉	1	1	yes	characteristic
163	#532	C₂²×C₁₉	1	1	yes	characteristic
164	#501	C₄×C₁₉	1	2	yes	—
165	#490	C₄×C₁₉	1	2	yes	—
166	#482	—	2	2	no	—
167	#476	—	2	1	no	—
168	#474	—	2	1	no	—
169	#446	—	2	2	no	—
170	#446	—	2	1	no	—
171	C₈×Q₈	—	2	1	no	—
172	#433	—	2	2	no	—
173	#432	C₂²×C₁₉	1	1	yes	characteristic
174	#432	C₂²×C₁₉	1	1	yes	characteristic
175	#432	C₂²×C₁₉	1	1	yes	characteristic
176	#431	C₂²×C₁₉	1	1	yes	characteristic
177	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
178	#332	—	2	1	no	—
179	#332	—	2	1	no	—
180	#329	—	2	2	no	—
181	D₄⋊C₈	—	2	1	no	—
182	D₄⋊C₈	—	2	1	no	—
183	C₂×C₁₉	—	4	1	no	—
184	C₂×C₁₉	—	16	1	no	—
185	C₂×C₁₉	#1881	1	1	yes	characteristic
186	C₂×C₁₉	#1907	1	1	yes	characteristic
187	C₂×C₁₉	#1912	1	1	yes	characteristic
188	C₂×#41	—	2	1	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₄⋊Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
191	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
192	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
193	#127	C₂×C₄×C₁₉	1	1	yes	characteristic
194	#125	C₂×C₄×C₁₉	1	2	yes	—
195	#124	—	2	1	no	—
196	C₂²⋊Q₈	—	2	1	no	—
197	C₂²⋊Q₈	—	2	1	no	—
198	#121	—	2	1	no	—
199	#121	—	2	1	no	—
200	C₄×Q₈	—	2	1	no	—
201	C₄×Q₈	—	2	1	no	—
202	C₄×D₄	—	2	1	no	—
203	C₄×D₄	—	2	1	no	—
204	#117	D₄×C₁₉	1	2	yes	—
205	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
206	#107	—	2	2	no	—
207	#106	—	2	2	no	—
208	C₄⋊C₈	—	2	2	no	—
209	C₄⋊C₈	—	2	2	no	—
210	C₄⋊C₈	—	2	1	no	—
211	C₄⋊C₈	—	2	1	no	—
212	C₄⋊C₈	—	2	1	no	—
213	C₄⋊C₈	—	2	1	no	—
214	C₄⋊C₈	—	2	1	no	—
215	C₄⋊C₈	—	2	1	no	—
216	#102	—	2	2	no	—
217	#102	—	2	2	no	—
218	C₂²⋊C₈	—	2	1	no	—
219	C₂²⋊C₈	—	2	1	no	—
220	C₂²⋊C₈	—	2	1	no	—
221	C₂²⋊C₈	—	2	1	no	—
222	#97	—	2	2	no	—
223	#97	—	2	1	no	—
224	#97	C₂×C₄×C₁₉	1	2	yes	—
225	#97	D₄×C₁₉	1	2	yes	—
226	C₄×C₈	—	2	2	no	—
227	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
228	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
229	C₁₉	#9372	1	1	yes	characteristic, direct product factor, Sylow
230	#41	—	8	1	no	—
231	#41	—	8	1	no	—
232	C₂×Q₈	—	2	1	no	—
233	C₂×Q₈	—	2	1	no	—
234	C₂×Q₈	—	2	1	no	—
235	C₂×D₄	—	2	1	no	—
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	—	2	1	no	—
238	C₂×D₄	—	4	1	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
241	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
242	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	8	1	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
257	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
258	C₂²⋊C₄	—	2	2	no	—
259	C₂²⋊C₄	—	4	1	no	—
260	C₂²⋊C₄	—	4	1	no	—
261	C₄²	—	2	2	no	—
262	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
263	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
264	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
265	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
266	C₂³	—	4	1	no	—
267	C₂³	C₁₉×#123	1	1	yes	characteristic
268	Q₈	—	4	1	no	—
269	Q₈	—	4	1	no	—
270	Q₈	—	8	1	no	—
271	Q₈	—	8	1	no	—
272	D₄	—	4	2	no	—
273	D₄	—	4	2	no	—
274	D₄	—	8	1	no	—
275	D₄	—	8	2	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	8	1	no	—
284	C₂×C₄	—	8	1	no	—
285	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
286	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
287	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
288	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
289	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
290	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
291	C₈	—	4	1	no	—
292	C₈	—	4	2	no	—
293	C₈	—	4	1	no	—
294	C₈	—	4	2	no	—
295	C₈	—	8	1	no	—
296	C₈	—	8	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	8	1	no	—
301	C₂²	—	8	1	no	—
302	C₂²	—	8	1	no	—
303	C₂²	C₁₉×#386	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	4	2	no	—
310	C₄	—	4	2	no	—
311	C₄	—	8	1	no	—
312	C₄	C₁₉×#471	1	1	yes	characteristic
313	C₄	C₁₉×#472	1	1	yes	characteristic
314	C₂	—	4	1	no	—
315	C₂	—	16	1	no	—
316	C₂	C₁₉×#1881	1	1	yes	characteristic
317	C₂	C₁₉×#1907	1	1	yes	characteristic
318	C₂	C₁₉×#1912	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#386
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
prime radical	C₁₉×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#2985	C₂	semidirect
3. C₁₉×#1552	C₂	semidirect
6. C₁₉×#1290	C₂	semidirect
7. C₁₉×#1287	C₂	semidirect
106. #9372	C₁₉	direct (N × H)
139. #2985	C₂×C₁₉	semidirect
140. #1552	C₂×C₁₉	semidirect
143. #1290	C₂×C₁₉	semidirect
144. #1287	C₂×C₁₉	semidirect
229. C₁₉	#9372	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×#3235	2
C₁₉×#2985	2
C₁₉×#1552	2
C₁₉×#1536	2
C₁₉×#1480	2
C₁₉×#1290	2
C₁₉×#1287	2
#9372	19

Subgroups — C19×#9372 #18015331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#9372 #18015331