Subgroups — C₁₉×#10155 #18016114

Order 4864.

552 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×#3209	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#3165	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#494	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	C₂²×Q₁₆×C₁₉	—	2	2	no	—
9	C₁₉×#533	C₂²	1	1	yes	characteristic
10	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₁₉×#501	—	2	2	no	—
13	C₁₉×#500	—	2	1	no	—
14	C₁₉×#498	—	2	1	no	—
15	C₁₉×#494	—	2	2	no	—
16	C₁₉×#494	—	2	1	no	—
17	C₁₉×#494	—	2	2	no	—
18	C₁₉×#487	—	2	1	no	—
19	C₁₉×#462	—	2	2	no	—
20	C₁₉×#451	—	2	2	no	—
21	C₁₉×#431	C₂²	1	1	yes	characteristic
22	C₁₉×#428	—	2	2	no	—
23	C₁₉×#423	C₂²	1	1	yes	characteristic
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	D₄	1	2	yes	—
31	C₂×Q₁₆×C₁₉	—	2	8	no	—
32	C₂×Q₁₆×C₁₉	—	2	4	no	—
33	C₂×Q₁₆×C₁₉	—	4	8	no	—
34	C₂×M₁₆×C₁₉	—	2	1	no	—
35	C₂×M₁₆×C₁₉	D₄	1	2	yes	—
36	C₂²×C₈×C₁₉	—	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	—	2	1	no	—
42	C₁₉×C₄⋊Q₈	—	2	1	no	—
43	C₁₉×C₄⋊Q₈	—	2	1	no	—
44	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
45	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
46	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
47	C₁₉×#125	—	2	1	no	—
48	C₁₉×C₂²⋊Q₈	—	2	1	no	—
49	C₁₉×C₂²⋊Q₈	—	2	1	no	—
50	C₄×Q₈×C₁₉	—	2	1	no	—
51	C₄×Q₈×C₁₉	—	2	1	no	—
52	C₁₉×#117	—	2	1	no	—
53	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
54	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
55	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
56	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
57	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
58	C₁₉×#107	—	2	4	no	—
59	C₁₉×#106	—	2	4	no	—
60	C₁₉×C₄⋊C₈	—	2	1	no	—
61	C₁₉×C₄⋊C₈	—	2	1	no	—
62	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
63	C₁₉×#103	—	2	4	no	—
64	C₁₉×C₂²⋊C₈	—	2	1	no	—
65	C₁₉×#97	—	2	1	no	—
66	C₄×C₈×C₁₉	—	2	1	no	—
67	C₄×C₈×C₁₉	—	2	1	no	—
68	C₄×C₈×C₁₉	—	2	1	no	—
69	C₄×C₈×C₁₉	D₄	1	2	yes	—
70	C₂×Q₈×C₁₉	—	2	1	no	—
71	C₂×Q₈×C₁₉	—	2	2	no	—
72	C₂×Q₈×C₁₉	—	2	2	no	—
73	C₂×Q₈×C₁₉	—	2	2	no	—
74	C₂×Q₈×C₁₉	—	2	1	no	—
75	C₂×Q₈×C₁₉	—	4	4	no	—
76	C₂×Q₈×C₁₉	—	4	4	no	—
77	C₂×Q₈×C₁₉	—	4	1	no	—
78	C₂²×C₄×C₁₉	—	2	2	no	—
79	C₂²×C₄×C₁₉	—	4	1	no	—
80	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
83	Q₁₆×C₁₉	—	4	16	no	—
84	Q₁₆×C₁₉	—	8	4	no	—
85	M₁₆×C₁₉	—	2	2	no	—
86	M₁₆×C₁₉	—	4	2	no	—
87	M₁₆×C₁₉	Q₁₆	1	4	yes	complemented
88	C₂×C₈×C₁₉	—	2	4	no	—
89	C₂×C₈×C₁₉	—	2	2	no	—
90	C₂×C₈×C₁₉	—	2	2	no	—
91	C₂×C₈×C₁₉	—	2	2	no	—
92	C₂×C₈×C₁₉	—	4	1	no	—
93	C₄⋊C₄×C₁₉	—	2	2	no	—
94	C₄⋊C₄×C₁₉	—	2	1	no	—
95	C₄⋊C₄×C₁₉	—	2	1	no	—
96	C₄⋊C₄×C₁₉	—	2	1	no	—
97	C₄⋊C₄×C₁₉	—	2	2	no	—
98	C₄⋊C₄×C₁₉	—	2	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	C₄⋊C₄×C₁₉	—	4	1	no	—
105	C₄⋊C₄×C₁₉	—	4	1	no	—
106	C₄⋊C₄×C₁₉	—	4	1	no	—
107	C₂²⋊C₄×C₁₉	—	4	1	no	—
108	C₄²×C₁₉	—	4	1	no	—
109	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
113	#10155	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂³×C₁₉	#121	1	1	yes	characteristic
115	Q₈×C₁₉	—	4	4	no	—
116	Q₈×C₁₉	—	4	4	no	—
117	Q₈×C₁₉	—	4	2	no	—
118	Q₈×C₁₉	—	4	4	no	—
119	Q₈×C₁₉	—	4	2	no	—
120	Q₈×C₁₉	—	8	2	no	—
121	C₂×C₄×C₁₉	—	2	2	no	—
122	C₂×C₄×C₁₉	—	2	2	no	—
123	C₂×C₄×C₁₉	—	4	1	no	—
124	C₂×C₄×C₁₉	—	4	1	no	—
125	C₂×C₄×C₁₉	—	4	2	no	—
126	C₂×C₄×C₁₉	—	4	2	no	—
127	C₂×C₄×C₁₉	—	8	2	no	—
128	C₂×C₄×C₁₉	—	8	2	no	—
129	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
130	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
131	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
132	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
133	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
134	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
135	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
136	C₂×C₄×C₁₉	#137	1	2	yes	—
137	C₈×C₁₉	—	2	4	no	—
138	C₈×C₁₉	—	2	4	no	—
139	C₈×C₁₉	—	4	2	no	—
140	C₈×C₁₉	—	8	1	no	—
141	#3209	C₂×C₁₉	1	1	yes	characteristic
142	#3165	C₂×C₁₉	1	1	yes	characteristic
143	C₂×#494	C₂×C₁₉	1	1	yes	characteristic
144	#1997	C₂×C₁₉	1	1	yes	characteristic
145	#1549	C₂×C₁₉	1	1	yes	characteristic
146	#1493	C₂×C₁₉	1	1	yes	characteristic
147	#1278	C₂×C₁₉	1	1	yes	characteristic
148	C₂²×C₁₉	—	2	1	no	—
149	C₂²×C₁₉	—	2	1	no	—
150	C₂²×C₁₉	#392	1	1	yes	characteristic
151	C₂²×C₁₉	#462	1	2	yes	—
152	C₄×C₁₉	—	2	1	no	—
153	C₄×C₁₉	—	2	1	no	—
154	C₄×C₁₉	—	8	2	no	—
155	C₄×C₁₉	—	8	2	no	—
156	C₄×C₁₉	—	8	1	no	—
157	C₄×C₁₉	—	8	1	no	—
158	C₄×C₁₉	#451	1	2	yes	—
159	C₄×C₁₉	#453	1	2	yes	—
160	C₄×C₁₉	#473	1	2	yes	—
161	C₄×C₁₉	#494	1	1	yes	characteristic
162	C₄×C₁₉	#497	1	1	yes	characteristic
163	C₂²×Q₁₆	—	2	2	no	—
164	#533	C₂²×C₁₉	1	1	yes	characteristic
165	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
166	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
167	#501	—	2	2	no	—
168	#500	—	2	1	no	—
169	#498	—	2	1	no	—
170	#494	—	2	2	no	—
171	#494	—	2	1	no	—
172	#494	—	2	2	no	—
173	#487	—	2	1	no	—
174	#462	—	2	2	no	—
175	#451	—	2	2	no	—
176	#431	C₂²×C₁₉	1	1	yes	characteristic
177	#428	—	2	2	no	—
178	#423	C₂²×C₁₉	1	1	yes	characteristic
179	C₄×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
180	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
181	#343	—	2	2	no	—
182	#332	—	2	2	no	—
183	Q₈⋊C₈	—	2	2	no	—
184	C₂×C₁₉	—	2	2	no	—
185	C₂×C₁₉	#2002	1	1	yes	characteristic
186	C₂×C₁₉	#2016	1	1	yes	characteristic
187	C₂×C₁₉	#2020	1	1	yes	characteristic
188	C₂²×Q₈	—	2	1	no	—
189	C₂²×Q₈	D₄×C₁₉	1	2	yes	—
190	C₂×Q₁₆	—	2	8	no	—
191	C₂×Q₁₆	—	2	4	no	—
192	C₂×Q₁₆	—	4	8	no	—
193	C₂×M₁₆	—	2	1	no	—
194	C₂×M₁₆	D₄×C₁₉	1	2	yes	—
195	C₂²×C₈	—	2	1	no	—
196	C₄⋊Q₈	—	2	1	no	—
197	C₄⋊Q₈	—	2	1	no	—
198	C₄⋊Q₈	—	2	1	no	—
199	C₄⋊Q₈	—	2	1	no	—
200	C₄⋊Q₈	—	2	1	no	—
201	C₄⋊Q₈	—	2	1	no	—
202	C₄⋊Q₈	—	2	1	no	—
203	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
204	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
205	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
206	#125	—	2	1	no	—
207	C₂²⋊Q₈	—	2	1	no	—
208	C₂²⋊Q₈	—	2	1	no	—
209	C₄×Q₈	—	2	1	no	—
210	C₄×Q₈	—	2	1	no	—
211	#117	—	2	1	no	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄⋊C₄	—	2	1	no	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
217	#107	—	2	4	no	—
218	#106	—	2	4	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	2	1	no	—
221	C₄⋊C₈	D₄×C₁₉	1	2	yes	—
222	#103	—	2	4	no	—
223	C₂²⋊C₈	—	2	1	no	—
224	#97	—	2	1	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	—	2	1	no	—
227	C₄×C₈	—	2	1	no	—
228	C₄×C₈	D₄×C₁₉	1	2	yes	—
229	C₁₉	#10155	1	1	yes	characteristic, direct product factor, Sylow
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	2	1	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	2	1	no	—
235	C₂×Q₈	—	4	4	no	—
236	C₂×Q₈	—	4	4	no	—
237	C₂×Q₈	—	4	1	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
242	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
243	Q₁₆	—	4	16	no	—
244	Q₁₆	—	8	4	no	—
245	M₁₆	—	2	2	no	—
246	M₁₆	—	4	2	no	—
247	M₁₆	Q₁₆×C₁₉	1	4	yes	complemented
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	4	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₂²⋊C₄	—	4	1	no	—
268	C₄²	—	4	1	no	—
269	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
270	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
271	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
272	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
273	C₂³	C₁₉×#121	1	1	yes	characteristic
274	Q₈	—	4	4	no	—
275	Q₈	—	4	4	no	—
276	Q₈	—	4	2	no	—
277	Q₈	—	4	4	no	—
278	Q₈	—	4	2	no	—
279	Q₈	—	8	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	8	2	no	—
287	C₂×C₄	—	8	2	no	—
288	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
289	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
290	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
291	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
292	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
293	C₂×C₄	C₁₉×#127	1	1	yes	characteristic
294	C₂×C₄	C₂×Q₁₆×C₁₉	1	2	yes	—
295	C₂×C₄	C₁₉×#137	1	2	yes	—
296	C₈	—	2	4	no	—
297	C₈	—	2	4	no	—
298	C₈	—	4	2	no	—
299	C₈	—	8	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	C₁₉×#392	1	1	yes	characteristic
303	C₂²	C₁₉×#462	1	2	yes	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	8	2	no	—
307	C₄	—	8	2	no	—
308	C₄	—	8	1	no	—
309	C₄	—	8	1	no	—
310	C₄	C₁₉×#451	1	2	yes	—
311	C₄	C₁₉×#453	1	2	yes	—
312	C₄	C₁₉×#473	1	2	yes	—
313	C₄	C₁₉×#494	1	1	yes	characteristic
314	C₄	C₁₉×#497	1	1	yes	characteristic
315	C₂	—	2	2	no	—
316	C₂	C₁₉×#2002	1	1	yes	characteristic
317	C₂	C₁₉×#2016	1	1	yes	characteristic
318	C₂	C₁₉×#2020	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
87. M₁₆×C₁₉	Q₁₆	semidirect
113. #10155	C₁₉	direct (N × H)
229. C₁₉	#10155	direct (N × H)
247. M₁₆	Q₁₆×C₁₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₁₉×#3209	2
C₁₉×#3165	2
C₂×C₁₉×#494	2
C₁₉×#1997	2
C₁₉×#1549	2
C₁₉×#1493	2
C₁₉×#1278	2
#10155	19

Subgroups — C19×#10155 #18016114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#10155 #18016114