Subgroups — C₁₉×#18693 #18024652

Order 4864.

868 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3550	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#3255	C₂	1	2	yes	maximal, complemented
3	C₁₉×#3131	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1645	C₂	1	4	yes	maximal, complemented
5	C₁₉×#1640	C₂	1	4	yes	maximal, complemented
6	C₁₉×#1560	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1493	C₂	1	2	yes	maximal, complemented
8	C₁₉×#563	—	2	2	no	—
9	C₁₉×#559	C₂²	1	2	yes	—
10	Q₈²×C₁₉	C₂²	1	2	yes	—
11	C₁₉×#556	—	2	2	no	—
12	D₄×Q₈×C₁₉	—	2	2	no	—
13	C₁₉×#542	—	2	4	no	—
14	C₁₉×#536	—	2	2	no	—
15	C₁₉×#533	C₂²	1	1	yes	characteristic
16	C₁₉×#533	C₂²	1	2	yes	—
17	C₁₉×#532	C₂²	1	1	yes	characteristic
18	C₁₉×#488	—	2	2	no	—
19	C₁₉×#487	C₂²	1	2	yes	—
20	C₁₉×#486	C₂²	1	2	yes	—
21	C₁₉×#475	—	2	2	no	—
22	C₁₉×#475	C₂²	1	4	yes	—
23	C₁₉×#470	—	2	4	no	—
24	C₁₉×#462	—	2	2	no	—
25	C₁₉×#462	C₂²	1	4	yes	—
26	C₁₉×#423	C₂²	1	2	yes	—
27	C₁₉×#416	—	2	2	no	—
28	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
29	C₁₉×#333	C₂²	1	2	yes	—
30	C₁₉×#331	C₂²	1	2	yes	—
31	C₁₉×Q₈⋊C₈	C₂²	1	8	yes	—
32	C₂×#41×C₁₉	D₄	1	2	yes	—
33	C₂²×Q₈×C₁₉	—	2	2	no	—
34	C₂×Q₁₆×C₁₉	—	2	4	no	—
35	C₂×M₁₆×C₁₉	—	2	2	no	—
36	C₂²×C₈×C₁₉	—	2	1	no	—
37	C₁₉×C₄⋊Q₈	—	2	4	no	—
38	C₁₉×C₄⋊Q₈	—	2	4	no	—
39	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
40	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
41	C₁₉×C₄⋊Q₈	C₂³	1	2	yes	—
42	C₁₉×C₄⋊Q₈	C₂³	1	1	yes	characteristic
43	C₁₉×#127	C₂³	1	1	yes	characteristic
44	C₁₉×#126	—	2	8	no	—
45	C₁₉×#125	—	2	2	no	—
46	C₁₉×#125	—	2	1	no	—
47	C₁₉×#124	—	2	4	no	—
48	C₁₉×#124	—	2	4	no	—
49	C₁₉×#124	D₄	1	2	yes	—
50	C₁₉×#123	—	2	4	no	—
51	C₁₉×#123	—	2	4	no	—
52	C₁₉×C₂²⋊Q₈	—	2	4	no	—
53	C₁₉×C₂²⋊Q₈	—	2	8	no	—
54	C₁₉×C₂²⋊Q₈	—	2	2	no	—
55	C₁₉×C₂²⋊Q₈	—	2	2	no	—
56	C₁₉×C₂²⋊Q₈	D₄	1	4	yes	—
57	C₁₉×#121	—	2	2	no	—
58	C₄×Q₈×C₁₉	—	2	4	no	—
59	C₄×Q₈×C₁₉	—	2	2	no	—
60	C₄×Q₈×C₁₉	—	2	2	no	—
61	C₄×Q₈×C₁₉	C₂³	1	4	yes	—
62	C₄×D₄×C₁₉	—	2	4	no	—
63	C₄×D₄×C₁₉	—	2	2	no	—
64	C₁₉×#117	—	2	2	no	—
65	C₁₉×#117	—	2	2	no	—
66	C₁₉×#117	—	2	1	no	—
67	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
68	C₁₉×#106	—	2	4	no	—
69	C₁₉×C₄⋊C₈	—	2	2	no	—
70	C₁₉×C₄⋊C₈	C₂³	1	4	yes	—
71	C₁₉×#103	—	2	4	no	—
72	C₁₉×#103	—	2	4	no	—
73	C₁₉×#103	—	2	4	no	—
74	C₁₉×#102	—	2	4	no	—
75	C₄×C₈×C₁₉	—	2	1	no	—
76	C₄×C₈×C₁₉	C₂³	1	2	yes	—
77	#41×C₁₉	—	4	4	no	—
78	C₂×Q₈×C₁₉	—	2	4	no	—
79	C₂×Q₈×C₁₉	—	2	2	no	—
80	C₂×Q₈×C₁₉	—	2	2	no	—
81	C₂×Q₈×C₁₉	—	8	4	no	—
82	C₂×Q₈×C₁₉	C₂×D₄	1	4	yes	—
83	C₂×Q₈×C₁₉	C₂×D₄	1	2	yes	—
84	C₂×D₄×C₁₉	—	2	2	no	—
85	C₂×D₄×C₁₉	—	2	2	no	—
86	C₂×D₄×C₁₉	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₁₉	—	2	4	no	—
88	C₂²×C₄×C₁₉	—	2	2	no	—
89	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
91	Q₁₆×C₁₉	—	8	4	no	—
92	M₁₆×C₁₉	—	8	2	no	—
93	C₂×C₈×C₁₉	—	2	2	no	—
94	C₂×C₈×C₁₉	—	2	4	no	—
95	C₂×C₈×C₁₉	—	4	2	no	—
96	C₄⋊C₄×C₁₉	—	2	8	no	—
97	C₄⋊C₄×C₁₉	—	2	2	no	—
98	C₄⋊C₄×C₁₉	—	2	2	no	—
99	C₄⋊C₄×C₁₉	—	2	2	no	—
100	C₄⋊C₄×C₁₉	—	2	4	no	—
101	C₄⋊C₄×C₁₉	—	2	2	no	—
102	C₄⋊C₄×C₁₉	—	2	4	no	—
103	C₄⋊C₄×C₁₉	—	2	2	no	—
104	C₄⋊C₄×C₁₉	—	2	2	no	—
105	C₄⋊C₄×C₁₉	—	4	2	no	—
106	C₄⋊C₄×C₁₉	C₂×D₄	1	4	yes	—
107	C₂²⋊C₄×C₁₉	—	2	4	no	—
108	C₂²⋊C₄×C₁₉	—	2	8	no	—
109	C₂²⋊C₄×C₁₉	—	2	4	no	—
110	C₂²⋊C₄×C₁₉	—	2	4	no	—
111	C₂²⋊C₄×C₁₉	—	4	1	no	—
112	C₄²×C₁₉	—	2	4	no	—
113	C₄²×C₁₉	—	2	4	no	—
114	C₄²×C₁₉	—	4	1	no	—
115	C₄²×C₁₉	C₂×D₄	1	2	yes	—
116	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
118	#18693	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂³×C₁₉	—	2	2	no	—
120	C₂³×C₁₉	#120	1	1	yes	characteristic
121	Q₈×C₁₉	—	2	4	no	—
122	Q₈×C₁₉	—	2	4	no	—
123	Q₈×C₁₉	—	4	2	no	—
124	Q₈×C₁₉	—	4	4	no	—
125	Q₈×C₁₉	—	4	2	no	—
126	Q₈×C₁₉	—	4	2	no	—
127	Q₈×C₁₉	—	8	4	no	—
128	D₄×C₁₉	—	4	2	no	—
129	D₄×C₁₉	—	4	2	no	—
130	D₄×C₁₉	—	4	2	no	—
131	D₄×C₁₉	—	8	2	no	—
132	C₂×C₄×C₁₉	—	2	2	no	—
133	C₂×C₄×C₁₉	—	2	2	no	—
134	C₂×C₄×C₁₉	—	2	2	no	—
135	C₂×C₄×C₁₉	—	2	4	no	—
136	C₂×C₄×C₁₉	—	2	2	no	—
137	C₂×C₄×C₁₉	—	2	4	no	—
138	C₂×C₄×C₁₉	—	4	4	no	—
139	C₂×C₄×C₁₉	—	4	2	no	—
140	C₂×C₄×C₁₉	—	8	4	no	—
141	C₂×C₄×C₁₉	#120	1	2	yes	—
142	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
143	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
145	C₈×C₁₉	—	4	2	no	—
146	C₈×C₁₉	—	4	4	no	—
147	#3550	C₂×C₁₉	1	1	yes	characteristic
148	#3255	C₂×C₁₉	1	2	yes	complemented
149	#3131	C₂×C₁₉	1	1	yes	characteristic
150	#1645	C₂×C₁₉	1	4	yes	complemented
151	#1640	C₂×C₁₉	1	4	yes	complemented
152	#1560	C₂×C₁₉	1	1	yes	characteristic
153	#1493	C₂×C₁₉	1	2	yes	complemented
154	C₂²×C₁₉	—	2	2	no	—
155	C₂²×C₁₉	—	2	1	no	—
156	C₂²×C₁₉	—	4	4	no	—
157	C₂²×C₁₉	—	4	2	no	—
158	C₂²×C₁₉	C₂×#120	1	1	yes	characteristic
159	C₄×C₁₉	—	2	2	no	—
160	C₄×C₁₉	—	2	1	no	—
161	C₄×C₁₉	—	4	4	no	—
162	C₄×C₁₉	—	4	2	no	—
163	C₄×C₁₉	—	4	4	no	—
164	C₄×C₁₉	—	8	2	no	—
165	C₄×C₁₉	#455	1	2	yes	—
166	C₄×C₁₉	#577	1	2	yes	—
167	C₄×C₁₉	#578	1	2	yes	—
168	#563	—	2	2	no	—
169	#559	C₂²×C₁₉	1	2	yes	—
170	Q₈²	C₂²×C₁₉	1	2	yes	—
171	#556	—	2	2	no	—
172	D₄×Q₈	—	2	2	no	—
173	#542	—	2	4	no	—
174	#536	—	2	2	no	—
175	#533	C₂²×C₁₉	1	1	yes	characteristic
176	#533	C₂²×C₁₉	1	2	yes	—
177	#532	C₂²×C₁₉	1	1	yes	characteristic
178	#488	—	2	2	no	—
179	#487	C₂²×C₁₉	1	2	yes	—
180	#486	C₂²×C₁₉	1	2	yes	—
181	#475	—	2	2	no	—
182	#475	C₂²×C₁₉	1	4	yes	—
183	#470	—	2	4	no	—
184	#462	—	2	2	no	—
185	#462	C₂²×C₁₉	1	4	yes	—
186	#423	C₂²×C₁₉	1	2	yes	—
187	#416	—	2	2	no	—
188	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
189	#333	C₂²×C₁₉	1	2	yes	—
190	#331	C₂²×C₁₉	1	2	yes	—
191	Q₈⋊C₈	C₂²×C₁₉	1	8	yes	—
192	C₂×C₁₉	—	4	1	no	—
193	C₂×C₁₉	—	8	2	no	—
194	C₂×C₁₉	#3013	1	2	yes	—
195	C₂×C₁₉	C₂×#455	1	1	yes	characteristic
196	C₂×#41	D₄×C₁₉	1	2	yes	—
197	C₂²×Q₈	—	2	2	no	—
198	C₂×Q₁₆	—	2	4	no	—
199	C₂×M₁₆	—	2	2	no	—
200	C₂²×C₈	—	2	1	no	—
201	C₄⋊Q₈	—	2	4	no	—
202	C₄⋊Q₈	—	2	4	no	—
203	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
204	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
205	C₄⋊Q₈	C₂³×C₁₉	1	2	yes	—
206	C₄⋊Q₈	C₂³×C₁₉	1	1	yes	characteristic
207	#127	C₂³×C₁₉	1	1	yes	characteristic
208	#126	—	2	8	no	—
209	#125	—	2	2	no	—
210	#125	—	2	1	no	—
211	#124	—	2	4	no	—
212	#124	—	2	4	no	—
213	#124	D₄×C₁₉	1	2	yes	—
214	#123	—	2	4	no	—
215	#123	—	2	4	no	—
216	C₂²⋊Q₈	—	2	4	no	—
217	C₂²⋊Q₈	—	2	8	no	—
218	C₂²⋊Q₈	—	2	2	no	—
219	C₂²⋊Q₈	—	2	2	no	—
220	C₂²⋊Q₈	D₄×C₁₉	1	4	yes	—
221	#121	—	2	2	no	—
222	C₄×Q₈	—	2	4	no	—
223	C₄×Q₈	—	2	2	no	—
224	C₄×Q₈	—	2	2	no	—
225	C₄×Q₈	C₂³×C₁₉	1	4	yes	—
226	C₄×D₄	—	2	4	no	—
227	C₄×D₄	—	2	2	no	—
228	#117	—	2	2	no	—
229	#117	—	2	2	no	—
230	#117	—	2	1	no	—
231	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
232	#106	—	2	4	no	—
233	C₄⋊C₈	—	2	2	no	—
234	C₄⋊C₈	C₂³×C₁₉	1	4	yes	—
235	#103	—	2	4	no	—
236	#103	—	2	4	no	—
237	#103	—	2	4	no	—
238	#102	—	2	4	no	—
239	C₄×C₈	—	2	1	no	—
240	C₄×C₈	C₂³×C₁₉	1	2	yes	—
241	C₁₉	#18693	1	1	yes	characteristic, direct product factor, Sylow
242	#41	—	4	4	no	—
243	C₂×Q₈	—	2	4	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	8	4	no	—
247	C₂×Q₈	C₂×D₄×C₁₉	1	4	yes	—
248	C₂×Q₈	C₂×D₄×C₁₉	1	2	yes	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	C₂×D₄×C₁₉	1	2	yes	—
252	C₂²×C₄	—	2	4	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
255	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
256	Q₁₆	—	8	4	no	—
257	M₁₆	—	8	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	4	2	no	—
261	C₄⋊C₄	—	2	8	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	C₂×D₄×C₁₉	1	4	yes	—
272	C₂²⋊C₄	—	2	4	no	—
273	C₂²⋊C₄	—	2	8	no	—
274	C₂²⋊C₄	—	2	4	no	—
275	C₂²⋊C₄	—	2	4	no	—
276	C₂²⋊C₄	—	4	1	no	—
277	C₄²	—	2	4	no	—
278	C₄²	—	2	4	no	—
279	C₄²	—	4	1	no	—
280	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
281	C₄²	C₂×D₄×C₁₉	1	2	yes	—
282	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
283	C₂³	—	2	2	no	—
284	C₂³	C₁₉×#120	1	1	yes	characteristic
285	Q₈	—	2	4	no	—
286	Q₈	—	2	4	no	—
287	Q₈	—	4	2	no	—
288	Q₈	—	4	4	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	8	4	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	2	no	—
294	D₄	—	4	2	no	—
295	D₄	—	8	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	8	4	no	—
305	C₂×C₄	C₁₉×#120	1	2	yes	—
306	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
307	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
308	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
309	C₈	—	4	2	no	—
310	C₈	—	4	4	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	4	4	no	—
314	C₂²	—	4	2	no	—
315	C₂²	C₂×C₁₉×#120	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	4	no	—
319	C₄	—	4	2	no	—
320	C₄	—	4	4	no	—
321	C₄	—	8	2	no	—
322	C₄	C₁₉×#455	1	2	yes	—
323	C₄	C₁₉×#577	1	2	yes	—
324	C₄	C₁₉×#578	1	2	yes	—
325	C₂	—	4	1	no	—
326	C₂	—	8	2	no	—
327	C₂	C₁₉×#3013	1	2	yes	—
328	C₂	C₂×C₁₉×#455	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×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×#3255	C₂	semidirect
4. C₁₉×#1645	C₂	semidirect
5. C₁₉×#1640	C₂	semidirect
7. C₁₉×#1493	C₂	semidirect
118. #18693	C₁₉	direct (N × H)
148. #3255	C₂×C₁₉	semidirect
150. #1645	C₂×C₁₉	semidirect
151. #1640	C₂×C₁₉	semidirect
153. #1493	C₂×C₁₉	semidirect
241. C₁₉	#18693	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×#3550	2
C₁₉×#3255	2
C₁₉×#3131	2
C₁₉×#1645	2
C₁₉×#1640	2
C₁₉×#1560	2
C₁₉×#1493	2
#18693	19

Subgroups — C19×#18693 #18024652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#18693 #18024652