Subgroups — C₁₉×#10157 #18016116

Order 4864.

476 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3230	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#3165	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#487	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1997	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1549	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1493	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1278	C₂	1	1	yes	maximal, characteristic
8	C₁₉×#533	C₂²	1	1	yes	characteristic
9	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₁₉×#501	—	2	2	no	—
12	C₁₉×#500	—	2	1	no	—
13	C₁₉×#500	—	2	1	no	—
14	C₁₉×#498	—	2	1	no	—
15	C₁₉×#487	—	2	2	no	—
16	C₁₉×#487	—	2	1	no	—
17	C₁₉×#487	—	2	2	no	—
18	C₁₉×#484	—	2	2	no	—
19	C₁₉×#477	—	2	2	no	—
20	C₁₉×#431	C₂²	1	1	yes	characteristic
21	C₁₉×#428	—	2	2	no	—
22	C₁₉×#423	C₂²	1	1	yes	characteristic
23	C₂×C₁₉×#103	—	2	2	no	—
24	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
26	C₁₉×#343	—	2	2	no	—
27	C₁₉×#332	—	2	2	no	—
28	C₁₉×Q₈⋊C₈	—	2	2	no	—
29	C₂²×Q₈×C₁₉	—	2	1	no	—
30	C₂²×Q₈×C₁₉	—	2	1	no	—
31	C₂×M₁₆×C₁₉	—	2	1	no	—
32	C₂×M₁₆×C₁₉	D₄	1	2	yes	—
33	C₂²×C₈×C₁₉	—	2	1	no	—
34	C₁₉×C₄⋊Q₈	—	2	1	no	—
35	C₁₉×C₄⋊Q₈	—	2	1	no	—
36	C₁₉×C₄⋊Q₈	—	2	1	no	—
37	C₁₉×C₄⋊Q₈	—	2	1	no	—
38	C₁₉×C₄⋊Q₈	—	2	1	no	—
39	C₁₉×C₄⋊Q₈	—	2	1	no	—
40	C₁₉×C₄⋊Q₈	—	2	1	no	—
41	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
42	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
43	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
44	C₁₉×#125	—	2	1	no	—
45	C₁₉×C₂²⋊Q₈	—	2	1	no	—
46	C₁₉×C₂²⋊Q₈	—	2	1	no	—
47	C₄×Q₈×C₁₉	—	2	1	no	—
48	C₄×Q₈×C₁₉	—	2	1	no	—
49	C₁₉×#117	—	2	1	no	—
50	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
51	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
52	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
53	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
54	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
55	C₁₉×#107	—	2	4	no	—
56	C₁₉×#107	—	2	4	no	—
57	C₁₉×#106	—	2	4	no	—
58	C₁₉×C₄⋊C₈	—	2	1	no	—
59	C₁₉×C₄⋊C₈	—	2	1	no	—
60	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
61	C₁₉×#103	—	2	8	no	—
62	C₁₉×#103	—	2	4	no	—
63	C₁₉×C₂²⋊C₈	—	2	1	no	—
64	C₁₉×#97	—	2	1	no	—
65	C₄×C₈×C₁₉	—	2	1	no	—
66	C₄×C₈×C₁₉	—	2	1	no	—
67	C₄×C₈×C₁₉	—	2	1	no	—
68	C₄×C₈×C₁₉	D₄	1	2	yes	—
69	C₂×Q₈×C₁₉	—	2	1	no	—
70	C₂×Q₈×C₁₉	—	2	2	no	—
71	C₂×Q₈×C₁₉	—	2	1	no	—
72	C₂×Q₈×C₁₉	—	4	4	no	—
73	C₂×Q₈×C₁₉	—	4	1	no	—
74	C₂×Q₈×C₁₉	—	4	1	no	—
75	C₂×Q₈×C₁₉	—	4	1	no	—
76	C₂×Q₈×C₁₉	—	8	2	no	—
77	C₂²×C₄×C₁₉	—	2	2	no	—
78	C₂²×C₄×C₁₉	—	4	1	no	—
79	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
82	M₁₆×C₁₉	—	2	2	no	—
83	M₁₆×C₁₉	—	4	2	no	—
84	M₁₆×C₁₉	Q₁₆	1	4	yes	—
85	C₂×C₈×C₁₉	—	2	2	no	—
86	C₂×C₈×C₁₉	—	2	2	no	—
87	C₂×C₈×C₁₉	—	2	2	no	—
88	C₂×C₈×C₁₉	—	4	2	no	—
89	C₂×C₈×C₁₉	—	4	1	no	—
90	C₄⋊C₄×C₁₉	—	2	2	no	—
91	C₄⋊C₄×C₁₉	—	2	2	no	—
92	C₄⋊C₄×C₁₉	—	2	2	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	—	2	2	no	—
97	C₄⋊C₄×C₁₉	—	2	1	no	—
98	C₄⋊C₄×C₁₉	—	4	1	no	—
99	C₄⋊C₄×C₁₉	—	4	1	no	—
100	C₄⋊C₄×C₁₉	—	4	1	no	—
101	C₄⋊C₄×C₁₉	—	4	1	no	—
102	C₄⋊C₄×C₁₉	—	4	1	no	—
103	C₄⋊C₄×C₁₉	—	4	1	no	—
104	C₂²⋊C₄×C₁₉	—	4	1	no	—
105	C₄²×C₁₉	—	4	1	no	—
106	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
110	#10157	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂³×C₁₉	#121	1	1	yes	characteristic
112	Q₈×C₁₉	—	4	4	no	—
113	Q₈×C₁₉	—	4	2	no	—
114	Q₈×C₁₉	—	4	2	no	—
115	Q₈×C₁₉	—	8	2	no	—
116	Q₈×C₁₉	—	8	2	no	—
117	Q₈×C₁₉	—	8	2	no	—
118	C₂×C₄×C₁₉	—	2	2	no	—
119	C₂×C₄×C₁₉	—	2	2	no	—
120	C₂×C₄×C₁₉	—	4	1	no	—
121	C₂×C₄×C₁₉	—	4	1	no	—
122	C₂×C₄×C₁₉	—	4	2	no	—
123	C₂×C₄×C₁₉	—	4	2	no	—
124	C₂×C₄×C₁₉	—	8	2	no	—
125	C₂×C₄×C₁₉	—	8	2	no	—
126	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
127	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
128	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
129	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
130	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
131	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
132	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
133	C₂×C₄×C₁₉	#137	1	2	yes	—
134	C₈×C₁₉	—	2	4	no	—
135	C₈×C₁₉	—	4	2	no	—
136	C₈×C₁₉	—	4	2	no	—
137	C₈×C₁₉	—	8	1	no	—
138	#3230	C₂×C₁₉	1	1	yes	characteristic
139	#3165	C₂×C₁₉	1	1	yes	characteristic
140	C₂×#487	C₂×C₁₉	1	1	yes	characteristic
141	#1997	C₂×C₁₉	1	1	yes	characteristic
142	#1549	C₂×C₁₉	1	1	yes	characteristic
143	#1493	C₂×C₁₉	1	1	yes	characteristic
144	#1278	C₂×C₁₉	1	1	yes	characteristic
145	C₂²×C₁₉	—	2	1	no	—
146	C₂²×C₁₉	—	2	1	no	—
147	C₂²×C₁₉	#392	1	1	yes	characteristic
148	C₂²×C₁₉	#462	1	2	yes	—
149	C₄×C₁₉	—	2	1	no	—
150	C₄×C₁₉	—	2	1	no	—
151	C₄×C₁₉	—	8	2	no	—
152	C₄×C₁₉	—	8	1	no	—
153	C₄×C₁₉	—	8	2	no	—
154	C₄×C₁₉	—	8	1	no	—
155	C₄×C₁₉	#451	1	2	yes	—
156	C₄×C₁₉	#454	1	2	yes	—
157	C₄×C₁₉	#471	1	2	yes	—
158	C₄×C₁₉	#494	1	1	yes	characteristic
159	C₄×C₁₉	#497	1	1	yes	characteristic
160	#533	C₂²×C₁₉	1	1	yes	characteristic
161	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
162	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
163	#501	—	2	2	no	—
164	#500	—	2	1	no	—
165	#500	—	2	1	no	—
166	#498	—	2	1	no	—
167	#487	—	2	2	no	—
168	#487	—	2	1	no	—
169	#487	—	2	2	no	—
170	#484	—	2	2	no	—
171	#477	—	2	2	no	—
172	#431	C₂²×C₁₉	1	1	yes	characteristic
173	#428	—	2	2	no	—
174	#423	C₂²×C₁₉	1	1	yes	characteristic
175	C₂×#103	—	2	2	no	—
176	C₄×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
177	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
178	#343	—	2	2	no	—
179	#332	—	2	2	no	—
180	Q₈⋊C₈	—	2	2	no	—
181	C₂×C₁₉	—	2	2	no	—
182	C₂×C₁₉	#2003	1	1	yes	characteristic
183	C₂×C₁₉	#2016	1	1	yes	characteristic
184	C₂×C₁₉	#2019	1	1	yes	characteristic
185	C₂²×Q₈	—	2	1	no	—
186	C₂²×Q₈	—	2	1	no	—
187	C₂×M₁₆	—	2	1	no	—
188	C₂×M₁₆	D₄×C₁₉	1	2	yes	—
189	C₂²×C₈	—	2	1	no	—
190	C₄⋊Q₈	—	2	1	no	—
191	C₄⋊Q₈	—	2	1	no	—
192	C₄⋊Q₈	—	2	1	no	—
193	C₄⋊Q₈	—	2	1	no	—
194	C₄⋊Q₈	—	2	1	no	—
195	C₄⋊Q₈	—	2	1	no	—
196	C₄⋊Q₈	—	2	1	no	—
197	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
198	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
199	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
200	#125	—	2	1	no	—
201	C₂²⋊Q₈	—	2	1	no	—
202	C₂²⋊Q₈	—	2	1	no	—
203	C₄×Q₈	—	2	1	no	—
204	C₄×Q₈	—	2	1	no	—
205	#117	—	2	1	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	D₄×C₁₉	1	2	yes	—
210	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
211	#107	—	2	4	no	—
212	#107	—	2	4	no	—
213	#106	—	2	4	no	—
214	C₄⋊C₈	—	2	1	no	—
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	D₄×C₁₉	1	2	yes	—
217	#103	—	2	8	no	—
218	#103	—	2	4	no	—
219	C₂²⋊C₈	—	2	1	no	—
220	#97	—	2	1	no	—
221	C₄×C₈	—	2	1	no	—
222	C₄×C₈	—	2	1	no	—
223	C₄×C₈	—	2	1	no	—
224	C₄×C₈	D₄×C₁₉	1	2	yes	—
225	C₁₉	#10157	1	1	yes	characteristic, direct product factor, Sylow
226	C₂×Q₈	—	2	1	no	—
227	C₂×Q₈	—	2	2	no	—
228	C₂×Q₈	—	2	1	no	—
229	C₂×Q₈	—	4	4	no	—
230	C₂×Q₈	—	4	1	no	—
231	C₂×Q₈	—	4	1	no	—
232	C₂×Q₈	—	4	1	no	—
233	C₂×Q₈	—	8	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	4	1	no	—
236	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
238	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
239	M₁₆	—	2	2	no	—
240	M₁₆	—	4	2	no	—
241	M₁₆	Q₁₆×C₁₉	1	4	yes	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	4	1	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₄⋊C₄	—	4	1	no	—
258	C₄⋊C₄	—	4	1	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₂²⋊C₄	—	4	1	no	—
262	C₄²	—	4	1	no	—
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	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
267	C₂³	C₁₉×#121	1	1	yes	characteristic
268	Q₈	—	4	4	no	—
269	Q₈	—	4	2	no	—
270	Q₈	—	4	2	no	—
271	Q₈	—	8	2	no	—
272	Q₈	—	8	2	no	—
273	Q₈	—	8	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	8	2	no	—
281	C₂×C₄	—	8	2	no	—
282	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
283	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
284	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
285	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
286	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
287	C₂×C₄	C₁₉×#127	1	1	yes	characteristic
288	C₂×C₄	C₂×Q₁₆×C₁₉	1	2	yes	—
289	C₂×C₄	C₁₉×#137	1	2	yes	—
290	C₈	—	2	4	no	—
291	C₈	—	4	2	no	—
292	C₈	—	4	2	no	—
293	C₈	—	8	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	C₁₉×#392	1	1	yes	characteristic
297	C₂²	C₁₉×#462	1	2	yes	—
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	8	2	no	—
301	C₄	—	8	1	no	—
302	C₄	—	8	2	no	—
303	C₄	—	8	1	no	—
304	C₄	C₁₉×#451	1	2	yes	—
305	C₄	C₁₉×#454	1	2	yes	—
306	C₄	C₁₉×#471	1	2	yes	—
307	C₄	C₁₉×#494	1	1	yes	characteristic
308	C₄	C₁₉×#497	1	1	yes	characteristic
309	C₂	—	2	2	no	—
310	C₂	C₁₉×#2003	1	1	yes	characteristic
311	C₂	C₁₉×#2016	1	1	yes	characteristic
312	C₂	C₁₉×#2019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
prime radical	C₂³×C₁₉	#121
LCS	G ▷ C₂×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
110. #10157	C₁₉	direct (N × H)
225. C₁₉	#10157	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×#3230	2
C₁₉×#3165	2
C₂×C₁₉×#487	2
C₁₉×#1997	2
C₁₉×#1549	2
C₁₉×#1493	2
C₁₉×#1278	2
#10157	19

Subgroups — C19×#10157 #18016116

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#10157 #18016116