Subgroups — C₁₉×#10171 #18016130

Order 4864.

488 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×#3139	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×#1546	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×#494	—	2	1	no	—
12	C₁₉×#491	—	2	2	no	—
13	C₁₉×#487	—	2	2	no	—
14	C₁₉×#487	—	2	1	no	—
15	C₁₉×#487	—	2	2	no	—
16	C₁₉×#487	—	2	1	no	—
17	C₁₉×#487	—	2	1	no	—
18	C₁₉×#462	—	2	2	no	—
19	C₁₉×#451	—	2	2	no	—
20	C₁₉×#431	C₂²	1	1	yes	characteristic
21	C₁₉×#423	C₂²	1	1	yes	characteristic
22	C₁₉×#419	—	2	2	no	—
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	D₄	1	2	yes	—
31	C₂×Q₁₆×C₁₉	—	2	4	no	—
32	C₂×M₁₆×C₁₉	—	2	1	no	—
33	C₂×M₁₆×C₁₉	D₄	1	2	yes	—
34	C₂²×C₈×C₁₉	—	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	—	2	1	no	—
42	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
43	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
44	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
45	C₁₉×#125	—	2	1	no	—
46	C₁₉×C₂²⋊Q₈	—	2	1	no	—
47	C₁₉×C₂²⋊Q₈	—	2	1	no	—
48	C₄×Q₈×C₁₉	—	2	1	no	—
49	C₄×Q₈×C₁₉	—	2	1	no	—
50	C₁₉×#117	—	2	1	no	—
51	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
52	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
53	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
54	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
55	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
56	C₁₉×C₄⋊C₈	—	2	1	no	—
57	C₁₉×C₄⋊C₈	—	2	1	no	—
58	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
59	C₁₉×#103	—	2	8	no	—
60	C₁₉×#103	—	2	4	no	—
61	C₁₉×#103	—	2	4	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	2	no	—
72	C₂×Q₈×C₁₉	—	2	2	no	—
73	C₂×Q₈×C₁₉	—	2	1	no	—
74	C₂×Q₈×C₁₉	—	4	4	no	—
75	C₂×Q₈×C₁₉	—	4	1	no	—
76	C₂×Q₈×C₁₉	—	4	2	no	—
77	C₂×Q₈×C₁₉	—	4	2	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	—	8	4	no	—
84	M₁₆×C₁₉	—	2	2	no	—
85	M₁₆×C₁₉	—	2	2	no	—
86	M₁₆×C₁₉	—	4	2	no	—
87	C₂×C₈×C₁₉	—	2	2	no	—
88	C₂×C₈×C₁₉	—	2	2	no	—
89	C₂×C₈×C₁₉	—	2	2	no	—
90	C₂×C₈×C₁₉	—	4	2	no	—
91	C₂×C₈×C₁₉	—	4	1	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	C₄⋊C₄×C₁₉	—	4	1	no	—
105	C₄⋊C₄×C₁₉	—	4	1	no	—
106	C₂²⋊C₄×C₁₉	—	4	1	no	—
107	C₄²×C₁₉	—	4	1	no	—
108	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
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	#10171	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂³×C₁₉	#121	1	1	yes	characteristic
114	Q₈×C₁₉	—	4	4	no	—
115	Q₈×C₁₉	—	4	4	no	—
116	Q₈×C₁₉	—	4	2	no	—
117	Q₈×C₁₉	—	4	4	no	—
118	Q₈×C₁₉	—	4	2	no	—
119	Q₈×C₁₉	—	8	1	no	—
120	Q₈×C₁₉	—	8	1	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	#135	1	2	yes	—
136	C₂×C₄×C₁₉	#137	1	2	yes	—
137	C₈×C₁₉	—	4	2	no	—
138	C₈×C₁₉	—	4	2	no	—
139	C₈×C₁₉	—	4	2	no	—
140	C₈×C₁₉	—	8	1	no	—
141	#3209	C₂×C₁₉	1	1	yes	characteristic
142	#3139	C₂×C₁₉	1	1	yes	characteristic
143	C₂×#487	C₂×C₁₉	1	1	yes	characteristic
144	#1997	C₂×C₁₉	1	1	yes	characteristic
145	#1546	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	#464	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	1	no	—
156	C₄×C₁₉	—	8	2	no	—
157	C₄×C₁₉	—	8	1	no	—
158	C₄×C₁₉	#452	1	2	yes	—
159	C₄×C₁₉	#455	1	2	yes	—
160	C₄×C₁₉	#473	1	2	yes	—
161	C₄×C₁₉	#495	1	1	yes	characteristic
162	C₄×C₁₉	#497	1	1	yes	characteristic
163	#533	C₂²×C₁₉	1	1	yes	characteristic
164	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
165	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
166	#494	—	2	1	no	—
167	#491	—	2	2	no	—
168	#487	—	2	2	no	—
169	#487	—	2	1	no	—
170	#487	—	2	1	no	—
171	#487	—	2	1	no	—
172	#487	—	2	2	no	—
173	#462	—	2	2	no	—
174	#451	—	2	2	no	—
175	#431	C₂²×C₁₉	1	1	yes	characteristic
176	#423	C₂²×C₁₉	1	1	yes	characteristic
177	#419	—	2	2	no	—
178	C₂×#103	—	2	2	no	—
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	#2004	1	1	yes	characteristic
186	C₂×C₁₉	#2015	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	4	no	—
191	C₂×M₁₆	—	2	1	no	—
192	C₂×M₁₆	D₄×C₁₉	1	2	yes	—
193	C₂²×C₈	—	2	1	no	—
194	C₄⋊Q₈	—	2	1	no	—
195	C₄⋊Q₈	—	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	D₄×C₁₉	1	2	yes	—
202	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
203	C₄⋊Q₈	D₄×C₁₉	1	1	yes	characteristic
204	#125	—	2	1	no	—
205	C₂²⋊Q₈	—	2	1	no	—
206	C₂²⋊Q₈	—	2	1	no	—
207	C₄×Q₈	—	2	1	no	—
208	C₄×Q₈	—	2	1	no	—
209	#117	—	2	1	no	—
210	C₂×C₄⋊C₄	—	2	1	no	—
211	C₂×C₄⋊C₄	—	2	1	no	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	—	2	1	no	—
217	C₄⋊C₈	D₄×C₁₉	1	2	yes	—
218	#103	—	2	8	no	—
219	#103	—	2	4	no	—
220	#103	—	2	4	no	—
221	#103	—	2	4	no	—
222	C₂²⋊C₈	—	2	1	no	—
223	#97	—	2	1	no	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	—	2	1	no	—
227	C₄×C₈	D₄×C₁₉	1	2	yes	—
228	C₁₉	#10171	1	1	yes	characteristic, direct product factor, Sylow
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	—	2	1	no	—
233	C₂×Q₈	—	2	1	no	—
234	C₂×Q₈	—	4	4	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂×Q₈	—	4	2	no	—
237	C₂×Q₈	—	4	2	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	—	8	4	no	—
244	M₁₆	—	2	2	no	—
245	M₁₆	—	2	2	no	—
246	M₁₆	—	4	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	4	2	no	—
251	C₂×C₈	—	4	1	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	4	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	C₂²⋊C₄	—	4	1	no	—
267	C₄²	—	4	1	no	—
268	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
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	C₂³	C₁₉×#121	1	1	yes	characteristic
273	Q₈	—	4	4	no	—
274	Q₈	—	4	4	no	—
275	Q₈	—	4	2	no	—
276	Q₈	—	4	4	no	—
277	Q₈	—	4	2	no	—
278	Q₈	—	8	1	no	—
279	Q₈	—	8	1	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	C₁₉×#135	1	2	yes	—
295	C₂×C₄	C₁₉×#137	1	2	yes	—
296	C₈	—	4	2	no	—
297	C₈	—	4	2	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×#464	1	2	yes	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	8	2	no	—
307	C₄	—	8	1	no	—
308	C₄	—	8	2	no	—
309	C₄	—	8	1	no	—
310	C₄	C₁₉×#452	1	2	yes	—
311	C₄	C₁₉×#455	1	2	yes	—
312	C₄	C₁₉×#473	1	2	yes	—
313	C₄	C₁₉×#495	1	1	yes	characteristic
314	C₄	C₁₉×#497	1	1	yes	characteristic
315	C₂	—	2	2	no	—
316	C₂	C₁₉×#2004	1	1	yes	characteristic
317	C₂	C₁₉×#2015	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
112. #10171	C₁₉	direct (N × H)
228. C₁₉	#10171	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×#3209	2
C₁₉×#3139	2
C₂×C₁₉×#487	2
C₁₉×#1997	2
C₁₉×#1546	2
C₁₉×#1493	2
C₁₉×#1278	2
#10171	19

Subgroups — C19×#10171 #18016130

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#10171 #18016130