Subgroups — C₁₉×#18374 #18024333

Order 4864.

838 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3502	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#3236	C₂	1	2	yes	maximal
3	C₁₉×#2634	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1617	C₂	1	4	yes	maximal
5	C₁₉×#1611	C₂	1	4	yes	maximal
6	C₁₉×#1515	C₂	1	2	yes	maximal
7	C₁₉×#1467	C₂	1	1	yes	maximal, characteristic
8	Q₈²×C₁₉	—	2	2	no	—
9	C₁₉×#557	—	2	2	no	—
10	C₁₉×#557	—	2	4	no	—
11	C₁₉×#544	—	2	1	no	—
12	C₁₉×#544	C₂²	1	4	yes	—
13	C₁₉×#533	—	2	2	no	—
14	C₁₉×#533	C₂²	1	2	yes	—
15	C₂×C₁₉×#125	C₂²	1	1	yes	characteristic
16	C₁₉×#519	—	2	1	no	—
17	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#501	—	2	2	no	—
19	C₁₉×#491	—	2	2	no	—
20	C₁₉×#484	C₂²	1	4	yes	—
21	C₁₉×#483	C₂²	1	4	yes	—
22	C₁₉×#479	—	2	4	no	—
23	C₁₉×#432	C₂²	1	2	yes	—
24	C₁₉×#400	—	2	1	no	—
25	C₁₉×#400	C₂²	1	2	yes	—
26	C₁₉×#398	C₂²	1	2	yes	—
27	C₁₉×#395	—	2	1	no	—
28	C₁₉×#387	—	2	2	no	—
29	C₁₉×#383	C₂²	1	1	yes	characteristic
30	C₁₉×#328	C₂²	1	8	yes	—
31	C₁₉×#324	C₂²	1	4	yes	—
32	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
33	C₁₉×C₄⋊Q₈	—	2	2	no	—
34	C₁₉×C₄⋊Q₈	—	2	2	no	—
35	C₁₉×C₄⋊Q₈	—	2	4	no	—
36	C₁₉×C₄⋊Q₈	—	2	8	no	—
37	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
38	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
39	C₁₉×#125	—	2	4	no	—
40	C₁₉×#125	—	2	4	no	—
41	C₁₉×#125	—	2	8	no	—
42	C₁₉×#125	—	2	2	no	—
43	C₁₉×#125	—	2	2	no	—
44	C₁₉×#125	—	2	2	no	—
45	C₁₉×C₂²⋊Q₈	—	2	2	no	—
46	C₁₉×C₂²⋊Q₈	—	2	4	no	—
47	C₁₉×C₂²⋊Q₈	C₂³	1	4	yes	—
48	C₄×Q₈×C₁₉	—	2	4	no	—
49	C₄×Q₈×C₁₉	—	2	4	no	—
50	C₄×Q₈×C₁₉	—	2	4	no	—
51	C₄×Q₈×C₁₉	—	2	4	no	—
52	C₄×Q₈×C₁₉	—	2	2	no	—
53	C₄×Q₈×C₁₉	—	2	2	no	—
54	C₄×Q₈×C₁₉	—	2	2	no	—
55	C₄×Q₈×C₁₉	D₄	1	4	yes	—
56	C₁₉×#117	—	2	4	no	—
57	C₁₉×#117	—	2	2	no	—
58	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
59	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
60	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
61	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
65	C₂×C₄²×C₁₉	D₄	1	2	yes	—
66	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
67	C₁₉×#107	—	2	4	no	—
68	C₁₉×#106	—	2	4	no	—
69	C₁₉×C₄⋊C₈	—	2	2	no	—
70	C₁₉×#103	—	2	4	no	—
71	C₁₉×#103	—	2	4	no	—
72	C₁₉×C₂²⋊C₈	C₂³	1	4	yes	—
73	C₁₉×#97	—	2	2	no	—
74	C₁₉×#95	—	2	1	no	—
75	C₁₉×#95	—	2	1	no	—
76	C₁₉×#95	—	2	2	no	—
77	C₁₉×#95	—	2	2	no	—
78	C₁₉×#95	—	2	2	no	—
79	C₁₉×#95	C₂³	1	2	yes	—
80	C₂×Q₈×C₁₉	—	2	4	no	—
81	C₂×Q₈×C₁₉	—	2	2	no	—
82	C₂×Q₈×C₁₉	—	2	2	no	—
83	C₂×Q₈×C₁₉	—	2	4	no	—
84	C₂×Q₈×C₁₉	C₂×D₄	1	4	yes	—
85	C₂²×C₄×C₁₉	—	2	1	no	—
86	C₂²×C₄×C₁₉	—	2	1	no	—
87	C₂²×C₄×C₁₉	—	2	2	no	—
88	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
93	C₂×C₈×C₁₉	—	2	4	no	—
94	C₄⋊C₄×C₁₉	—	2	4	no	—
95	C₄⋊C₄×C₁₉	—	2	4	no	—
96	C₄⋊C₄×C₁₉	—	2	4	no	—
97	C₄⋊C₄×C₁₉	—	2	8	no	—
98	C₄⋊C₄×C₁₉	—	2	2	no	—
99	C₄⋊C₄×C₁₉	—	2	2	no	—
100	C₄⋊C₄×C₁₉	—	2	2	no	—
101	C₄⋊C₄×C₁₉	—	2	2	no	—
102	C₄⋊C₄×C₁₉	—	2	4	no	—
103	C₄⋊C₄×C₁₉	—	2	4	no	—
104	C₄⋊C₄×C₁₉	—	2	4	no	—
105	C₄⋊C₄×C₁₉	—	2	2	no	—
106	C₄⋊C₄×C₁₉	—	2	2	no	—
107	C₄⋊C₄×C₁₉	—	4	1	no	—
108	C₄⋊C₄×C₁₉	—	4	2	no	—
109	C₄⋊C₄×C₁₉	—	4	1	no	—
110	C₄⋊C₄×C₁₉	—	4	4	no	—
111	C₄⋊C₄×C₁₉	C₂×D₄	1	4	yes	—
112	C₂²⋊C₄×C₁₉	—	2	4	no	—
113	C₄²×C₁₉	—	2	8	no	—
114	C₄²×C₁₉	—	2	2	no	—
115	C₄²×C₁₉	—	2	2	no	—
116	C₄²×C₁₉	—	2	4	no	—
117	C₄²×C₁₉	C₂×D₄	1	2	yes	—
118	C₄²×C₁₉	C₂×D₄	1	2	yes	—
119	#18374	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
121	Q₈×C₁₉	—	2	8	no	—
122	Q₈×C₁₉	—	4	4	no	—
123	Q₈×C₁₉	—	4	4	no	—
124	Q₈×C₁₉	—	8	4	no	—
125	C₂×C₄×C₁₉	—	2	2	no	—
126	C₂×C₄×C₁₉	—	2	2	no	—
127	C₂×C₄×C₁₉	—	2	2	no	—
128	C₂×C₄×C₁₉	—	2	4	no	—
129	C₂×C₄×C₁₉	—	2	4	no	—
130	C₂×C₄×C₁₉	—	2	2	no	—
131	C₂×C₄×C₁₉	—	4	1	no	—
132	C₂×C₄×C₁₉	—	4	1	no	—
133	C₂×C₄×C₁₉	—	4	2	no	—
134	C₂×C₄×C₁₉	—	4	2	no	—
135	C₂×C₄×C₁₉	—	4	2	no	—
136	C₂×C₄×C₁₉	—	4	2	no	—
137	C₂×C₄×C₁₉	—	4	1	no	—
138	C₂×C₄×C₁₉	—	4	1	no	—
139	C₂×C₄×C₁₉	—	4	2	no	—
140	C₂×C₄×C₁₉	—	4	2	no	—
141	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
142	C₂×C₄×C₁₉	#120	1	2	yes	—
143	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
144	C₂×C₄×C₁₉	#137	1	4	yes	—
145	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
146	C₈×C₁₉	—	4	4	no	—
147	#3502	C₂×C₁₉	1	1	yes	characteristic
148	#3236	C₂×C₁₉	1	2	yes	—
149	#2634	C₂×C₁₉	1	1	yes	characteristic
150	#1617	C₂×C₁₉	1	4	yes	—
151	#1611	C₂×C₁₉	1	4	yes	—
152	#1515	C₂×C₁₉	1	2	yes	—
153	#1467	C₂×C₁₉	1	1	yes	characteristic
154	C₂²×C₁₉	—	2	1	no	—
155	C₂²×C₁₉	—	2	1	no	—
156	C₂²×C₁₉	C₂×#120	1	1	yes	characteristic
157	C₂²×C₁₉	C₂×#137	1	2	yes	—
158	C₄×C₁₉	—	2	2	no	—
159	C₄×C₁₉	—	2	4	no	—
160	C₄×C₁₉	—	2	1	no	—
161	C₄×C₁₉	—	2	1	no	—
162	C₄×C₁₉	—	2	2	no	—
163	C₄×C₁₉	—	4	4	no	—
164	C₄×C₁₉	—	4	2	no	—
165	C₄×C₁₉	—	4	4	no	—
166	C₄×C₁₉	—	8	2	no	—
167	C₄×C₁₉	—	8	1	no	—
168	C₄×C₁₉	—	8	1	no	—
169	Q₈²	—	2	2	no	—
170	#557	—	2	2	no	—
171	#557	—	2	4	no	—
172	#544	—	2	1	no	—
173	#544	C₂²×C₁₉	1	4	yes	—
174	#533	—	2	2	no	—
175	#533	C₂²×C₁₉	1	2	yes	—
176	C₂×#125	C₂²×C₁₉	1	1	yes	characteristic
177	#519	—	2	1	no	—
178	C₂×C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
179	#501	—	2	2	no	—
180	#491	—	2	2	no	—
181	#484	C₂²×C₁₉	1	4	yes	—
182	#483	C₂²×C₁₉	1	4	yes	—
183	#479	—	2	4	no	—
184	#432	C₂²×C₁₉	1	2	yes	—
185	#400	—	2	1	no	—
186	#400	C₂²×C₁₉	1	2	yes	—
187	#398	C₂²×C₁₉	1	2	yes	—
188	#395	—	2	1	no	—
189	#387	—	2	2	no	—
190	#383	C₂²×C₁₉	1	1	yes	characteristic
191	#328	C₂²×C₁₉	1	8	yes	—
192	#324	C₂²×C₁₉	1	4	yes	—
193	C₂×C₁₉	—	2	2	no	—
194	C₂×C₁₉	#3004	1	1	yes	characteristic
195	C₂×C₁₉	#3020	1	1	yes	characteristic
196	C₂×C₁₉	#3028	1	1	yes	characteristic
197	C₂²×Q₈	D₄×C₁₉	1	1	yes	characteristic
198	C₄⋊Q₈	—	2	2	no	—
199	C₄⋊Q₈	—	2	2	no	—
200	C₄⋊Q₈	—	2	4	no	—
201	C₄⋊Q₈	—	2	8	no	—
202	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
203	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
204	#125	—	2	4	no	—
205	#125	—	2	4	no	—
206	#125	—	2	8	no	—
207	#125	—	2	2	no	—
208	#125	—	2	2	no	—
209	#125	—	2	2	no	—
210	C₂²⋊Q₈	—	2	2	no	—
211	C₂²⋊Q₈	—	2	4	no	—
212	C₂²⋊Q₈	C₂³×C₁₉	1	4	yes	—
213	C₄×Q₈	—	2	4	no	—
214	C₄×Q₈	—	2	4	no	—
215	C₄×Q₈	—	2	4	no	—
216	C₄×Q₈	—	2	4	no	—
217	C₄×Q₈	—	2	2	no	—
218	C₄×Q₈	—	2	2	no	—
219	C₄×Q₈	—	2	2	no	—
220	C₄×Q₈	D₄×C₁₉	1	4	yes	—
221	#117	—	2	4	no	—
222	#117	—	2	2	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	—	2	1	no	—
225	C₂×C₄⋊C₄	—	2	2	no	—
226	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
227	C₂×C₄⋊C₄	C₂³×C₁₉	1	1	yes	characteristic
228	C₂×C₄⋊C₄	C₂³×C₁₉	1	1	yes	characteristic
229	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
230	C₂×C₄²	D₄×C₁₉	1	2	yes	—
231	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
232	#107	—	2	4	no	—
233	#106	—	2	4	no	—
234	C₄⋊C₈	—	2	2	no	—
235	#103	—	2	4	no	—
236	#103	—	2	4	no	—
237	C₂²⋊C₈	C₂³×C₁₉	1	4	yes	—
238	#97	—	2	2	no	—
239	#95	—	2	1	no	—
240	#95	—	2	1	no	—
241	#95	—	2	2	no	—
242	#95	—	2	2	no	—
243	#95	—	2	2	no	—
244	#95	C₂³×C₁₉	1	2	yes	—
245	C₁₉	#18374	1	1	yes	characteristic, direct product factor, Sylow
246	C₂×Q₈	—	2	4	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	4	no	—
250	C₂×Q₈	C₂×D₄×C₁₉	1	4	yes	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
256	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
257	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
258	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
259	C₂×C₈	—	2	4	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	2	8	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	4	no	—
277	C₄⋊C₄	C₂×D₄×C₁₉	1	4	yes	—
278	C₂²⋊C₄	—	2	4	no	—
279	C₄²	—	2	8	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	4	no	—
283	C₄²	C₂×D₄×C₁₉	1	2	yes	—
284	C₄²	C₂×D₄×C₁₉	1	2	yes	—
285	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
286	Q₈	—	2	8	no	—
287	Q₈	—	4	4	no	—
288	Q₈	—	4	4	no	—
289	Q₈	—	8	4	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
307	C₂×C₄	C₁₉×#120	1	2	yes	—
308	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
309	C₂×C₄	C₁₉×#137	1	4	yes	—
310	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
311	C₈	—	4	4	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	C₂×C₁₉×#120	1	1	yes	characteristic
315	C₂²	C₂×C₁₉×#137	1	2	yes	—
316	C₄	—	2	2	no	—
317	C₄	—	2	4	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	2	no	—
321	C₄	—	4	4	no	—
322	C₄	—	4	2	no	—
323	C₄	—	4	4	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	1	no	—
326	C₄	—	8	1	no	—
327	C₂	—	2	2	no	—
328	C₂	C₁₉×#3004	1	1	yes	characteristic
329	C₂	C₁₉×#3020	1	1	yes	characteristic
330	C₂	C₁₉×#3028	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
prime radical	C₂³×C₁₉	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ 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
119. #18374	C₁₉	direct (N × H)
245. C₁₉	#18374	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×#3502	2
C₁₉×#3236	2
C₁₉×#2634	2
C₁₉×#1617	2
C₁₉×#1611	2
C₁₉×#1515	2
C₁₉×#1467	2
#18374	19

Subgroups — C19×#18374 #18024333

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#18374 #18024333