Subgroups — C₁₉×#13374 #18019333

Order 4864.

360 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×#3163	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₉×#2145	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#360	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1976	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1386	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#1384	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1378	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₉×#135	C₂²	1	1	yes	characteristic
9	C₁₉×#544	—	2	1	no	—
10	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
12	C₁₉×#501	—	2	2	no	—
13	C₁₉×#499	—	2	1	no	—
14	C₁₉×#498	—	2	1	no	—
15	C₁₉×#428	—	2	1	no	—
16	C₁₉×#427	C₂²	1	1	yes	characteristic
17	C₁₉×#427	C₂²	1	1	yes	characteristic
18	C₂×C₁₉×#106	C₂²	1	1	yes	characteristic
19	C₁₉×#421	—	2	1	no	—
20	C₁₉×#410	—	2	1	no	—
21	C₁₉×#405	C₂²	1	1	yes	characteristic
22	C₁₉×#360	—	2	2	no	—
23	C₁₉×#360	—	2	1	no	—
24	C₁₉×#360	—	2	1	no	—
25	C₂×#41×C₁₉	—	2	1	no	—
26	C₁₉×#135	—	2	1	no	—
27	C₁₉×#135	—	2	1	no	—
28	C₁₉×#135	—	2	1	no	—
29	C₁₉×#135	—	2	1	no	—
30	C₂×Q₁₆×C₁₉	Q₈	1	1	yes	characteristic, complemented
31	C₂×SD₁₆×C₁₉	—	2	1	no	—
32	C₂×D₈×C₁₉	Q₈	1	1	yes	characteristic, complemented
33	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
35	C₁₉×C₄⋊Q₈	—	2	2	no	—
36	C₁₉×C₄⋊Q₈	—	4	1	no	—
37	C₁₉×#125	—	2	2	no	—
38	C₁₉×#125	—	4	1	no	—
39	C₁₉×C₂²⋊Q₈	—	4	2	no	—
40	C₁₉×#117	—	2	1	no	—
41	C₁₉×#117	—	2	1	no	—
42	C₁₉×#117	D₄	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
44	C₁₉×M₃₂	—	2	1	no	—
45	C₁₉×M₃₂	—	2	1	no	—
46	C₁₉×M₃₂	—	2	1	no	—
47	C₁₉×M₃₂	—	2	1	no	—
48	C₂×C₁₆×C₁₉	—	2	1	no	—
49	C₂×C₁₆×C₁₉	—	2	1	no	—
50	C₁₉×#107	—	2	2	no	—
51	C₁₉×#107	—	2	1	no	—
52	C₁₉×#107	—	2	1	no	—
53	C₁₉×#106	—	2	2	no	—
54	C₁₉×#106	—	2	1	no	—
55	C₁₉×#106	—	2	1	no	—
56	C₁₉×#106	D₄	1	2	yes	—
57	C₁₉×#106	D₄	1	1	yes	characteristic
58	C₁₉×#106	D₄	1	1	yes	characteristic
59	C₁₉×#104	—	4	2	no	—
60	C₁₉×#99	—	4	2	no	—
61	C₁₉×#97	—	2	1	no	—
62	C₄×C₈×C₁₉	—	2	1	no	—
63	#41×C₁₉	—	4	1	no	—
64	#41×C₁₉	—	4	1	no	—
65	#41×C₁₉	—	4	1	no	—
66	#41×C₁₉	—	4	1	no	—
67	C₂×Q₈×C₁₉	—	2	1	no	—
68	C₂×Q₈×C₁₉	—	4	2	no	—
69	C₂×D₄×C₁₉	—	2	1	no	—
70	C₂×D₄×C₁₉	—	4	1	no	—
71	C₂²×C₄×C₁₉	—	4	1	no	—
72	C₂²×C₄×C₁₉	—	4	1	no	—
73	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
74	Q₁₆×C₁₉	—	2	1	no	—
75	Q₁₆×C₁₉	—	2	1	no	—
76	SD₁₆×C₁₉	—	4	1	no	—
77	SD₁₆×C₁₉	—	4	1	no	—
78	D₈×C₁₉	—	2	1	no	—
79	D₈×C₁₉	—	2	1	no	—
80	M₁₆×C₁₉	—	2	1	no	—
81	M₁₆×C₁₉	SD₁₆	1	2	yes	—
82	C₂×C₈×C₁₉	—	2	1	no	—
83	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
87	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
88	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
89	C₄⋊C₄×C₁₉	—	2	1	no	—
90	C₄⋊C₄×C₁₉	—	2	2	no	—
91	C₄⋊C₄×C₁₉	—	2	1	no	—
92	C₄⋊C₄×C₁₉	—	2	1	no	—
93	C₄⋊C₄×C₁₉	—	4	2	no	—
94	C₄⋊C₄×C₁₉	—	4	2	no	—
95	C₄⋊C₄×C₁₉	—	4	2	no	—
96	C₄⋊C₄×C₁₉	—	4	1	no	—
97	C₄⋊C₄×C₁₉	—	4	1	no	—
98	C₂²⋊C₄×C₁₉	—	4	1	no	—
99	C₂²⋊C₄×C₁₉	—	4	1	no	—
100	C₂²⋊C₄×C₁₉	D₈	1	2	yes	—
101	C₄²×C₁₉	—	2	1	no	—
102	C₄²×C₁₉	—	4	1	no	—
103	C₄²×C₁₉	—	4	1	no	—
104	C₁₆×C₁₉	—	4	1	no	—
105	C₁₆×C₁₉	—	4	1	no	—
106	#13374	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂³×C₁₉	—	4	1	no	—
108	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
109	Q₈×C₁₉	—	4	1	no	—
110	Q₈×C₁₉	—	4	1	no	—
111	Q₈×C₁₉	—	16	2	no	—
112	D₄×C₁₉	—	4	1	no	—
113	D₄×C₁₉	—	4	1	no	—
114	D₄×C₁₉	—	8	1	no	—
115	D₄×C₁₉	—	8	1	no	—
116	C₂×C₄×C₁₉	—	2	2	no	—
117	C₂×C₄×C₁₉	—	4	1	no	—
118	C₂×C₄×C₁₉	—	4	1	no	—
119	C₂×C₄×C₁₉	—	4	2	no	—
120	C₂×C₄×C₁₉	—	4	1	no	—
121	C₂×C₄×C₁₉	—	8	1	no	—
122	C₂×C₄×C₁₉	—	8	1	no	—
123	C₂×C₄×C₁₉	—	8	1	no	—
124	C₂×C₄×C₁₉	—	16	1	no	—
125	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
128	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
131	C₈×C₁₉	—	2	2	no	—
132	C₈×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
133	C₈×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
134	C₈×C₁₉	#124	1	1	yes	characteristic
135	C₈×C₁₉	#124	1	1	yes	characteristic
136	#3163	C₂×C₁₉	1	1	yes	characteristic, complemented
137	#2145	C₂×C₁₉	1	1	yes	characteristic
138	C₂×#360	C₂×C₁₉	1	1	yes	characteristic
139	#1976	C₂×C₁₉	1	1	yes	characteristic
140	#1386	C₂×C₁₉	1	1	yes	characteristic, complemented
141	#1384	C₂×C₁₉	1	1	yes	characteristic, complemented
142	#1378	C₂×C₁₉	1	1	yes	characteristic, complemented
143	C₂²×C₁₉	—	4	1	no	—
144	C₂²×C₁₉	—	8	1	no	—
145	C₂²×C₁₉	—	16	1	no	—
146	C₂²×C₁₉	#447	1	1	yes	characteristic
147	C₂²×C₁₉	#474	1	1	yes	characteristic
148	C₂²×C₁₉	#486	1	1	yes	characteristic
149	C₄×C₁₉	—	4	2	no	—
150	C₄×C₁₉	—	8	1	no	—
151	C₄×C₁₉	—	8	2	no	—
152	C₄×C₁₉	—	8	1	no	—
153	C₄×C₁₉	—	8	1	no	—
154	C₄×C₁₉	—	16	1	no	—
155	C₄×C₁₉	#393	1	1	yes	characteristic
156	C₄×C₁₉	#448	1	1	yes	characteristic
157	C₄×C₁₉	#476	1	1	yes	characteristic
158	C₄×C₁₉	#489	1	1	yes	characteristic
159	C₂×#135	C₂²×C₁₉	1	1	yes	characteristic
160	#544	—	2	1	no	—
161	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
162	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
163	#501	—	2	2	no	—
164	#499	—	2	1	no	—
165	#498	—	2	1	no	—
166	#428	—	2	1	no	—
167	#427	C₂²×C₁₉	1	1	yes	characteristic
168	#427	C₂²×C₁₉	1	1	yes	characteristic
169	C₂×#106	C₂²×C₁₉	1	1	yes	characteristic
170	#421	—	2	1	no	—
171	#410	—	2	1	no	—
172	#405	C₂²×C₁₉	1	1	yes	characteristic
173	#360	—	2	2	no	—
174	#360	—	2	1	no	—
175	#360	—	2	1	no	—
176	C₂×C₁₉	—	2	1	no	—
177	C₂×C₁₉	—	2	1	no	—
178	C₂×C₁₉	—	2	1	no	—
179	C₂×C₁₉	—	16	1	no	—
180	C₂×C₁₉	#2023	1	1	yes	characteristic
181	C₂×#41	—	2	1	no	—
182	#135	—	2	1	no	—
183	#135	—	2	1	no	—
184	#135	—	2	1	no	—
185	#135	—	2	1	no	—
186	C₂×Q₁₆	Q₈×C₁₉	1	1	yes	characteristic, complemented
187	C₂×SD₁₆	—	2	1	no	—
188	C₂×D₈	Q₈×C₁₉	1	1	yes	characteristic, complemented
189	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
190	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
191	C₄⋊Q₈	—	2	2	no	—
192	C₄⋊Q₈	—	4	1	no	—
193	#125	—	2	2	no	—
194	#125	—	4	1	no	—
195	C₂²⋊Q₈	—	4	2	no	—
196	#117	—	2	1	no	—
197	#117	—	2	1	no	—
198	#117	D₄×C₁₉	1	1	yes	characteristic
199	C₂×C₄⋊C₄	—	2	1	no	—
200	M₃₂	—	2	1	no	—
201	M₃₂	—	2	1	no	—
202	M₃₂	—	2	1	no	—
203	M₃₂	—	2	1	no	—
204	C₂×C₁₆	—	2	1	no	—
205	C₂×C₁₆	—	2	1	no	—
206	#107	—	2	2	no	—
207	#107	—	2	1	no	—
208	#107	—	2	1	no	—
209	#106	—	2	2	no	—
210	#106	—	2	1	no	—
211	#106	—	2	1	no	—
212	#106	D₄×C₁₉	1	2	yes	—
213	#106	D₄×C₁₉	1	1	yes	characteristic
214	#106	D₄×C₁₉	1	1	yes	characteristic
215	#104	—	4	2	no	—
216	#99	—	4	2	no	—
217	#97	—	2	1	no	—
218	C₄×C₈	—	2	1	no	—
219	C₁₉	#13374	1	1	yes	characteristic, direct product factor, Sylow
220	#41	—	4	1	no	—
221	#41	—	4	1	no	—
222	#41	—	4	1	no	—
223	#41	—	4	1	no	—
224	C₂×Q₈	—	2	1	no	—
225	C₂×Q₈	—	4	2	no	—
226	C₂×D₄	—	2	1	no	—
227	C₂×D₄	—	4	1	no	—
228	C₂²×C₄	—	4	1	no	—
229	C₂²×C₄	—	4	1	no	—
230	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
231	Q₁₆	—	2	1	no	—
232	Q₁₆	—	2	1	no	—
233	SD₁₆	—	4	1	no	—
234	SD₁₆	—	4	1	no	—
235	D₈	—	2	1	no	—
236	D₈	—	2	1	no	—
237	M₁₆	—	2	1	no	—
238	M₁₆	SD₁₆×C₁₉	1	2	yes	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
241	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
242	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
243	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
244	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
245	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	1	no	—
254	C₄⋊C₄	—	4	1	no	—
255	C₂²⋊C₄	—	4	1	no	—
256	C₂²⋊C₄	—	4	1	no	—
257	C₂²⋊C₄	D₈×C₁₉	1	2	yes	—
258	C₄²	—	2	1	no	—
259	C₄²	—	4	1	no	—
260	C₄²	—	4	1	no	—
261	C₁₆	—	4	1	no	—
262	C₁₆	—	4	1	no	—
263	C₂³	—	4	1	no	—
264	C₂³	C₂×D₈×C₁₉	1	1	yes	characteristic
265	Q₈	—	4	1	no	—
266	Q₈	—	4	1	no	—
267	Q₈	—	16	2	no	—
268	D₄	—	4	1	no	—
269	D₄	—	4	1	no	—
270	D₄	—	8	1	no	—
271	D₄	—	8	1	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	8	1	no	—
278	C₂×C₄	—	8	1	no	—
279	C₂×C₄	—	8	1	no	—
280	C₂×C₄	—	16	1	no	—
281	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
282	C₂×C₄	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
283	C₂×C₄	C₁₉×#124	1	1	yes	characteristic
284	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
285	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
286	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
287	C₈	—	2	2	no	—
288	C₈	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
289	C₈	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
290	C₈	C₁₉×#124	1	1	yes	characteristic
291	C₈	C₁₉×#124	1	1	yes	characteristic
292	C₂²	—	4	1	no	—
293	C₂²	—	8	1	no	—
294	C₂²	—	16	1	no	—
295	C₂²	C₁₉×#447	1	1	yes	characteristic
296	C₂²	C₁₉×#474	1	1	yes	characteristic
297	C₂²	C₁₉×#486	1	1	yes	characteristic
298	C₄	—	4	2	no	—
299	C₄	—	8	2	no	—
300	C₄	—	8	1	no	—
301	C₄	—	8	1	no	—
302	C₄	—	8	1	no	—
303	C₄	—	16	1	no	—
304	C₄	C₁₉×#393	1	1	yes	characteristic
305	C₄	C₁₉×#448	1	1	yes	characteristic
306	C₄	C₁₉×#476	1	1	yes	characteristic
307	C₄	C₁₉×#489	1	1	yes	characteristic
308	C₂	—	2	1	no	—
309	C₂	—	2	1	no	—
310	C₂	—	2	1	no	—
311	C₂	—	16	1	no	—
312	C₂	C₁₉×#2023	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₉	#2023
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₂×C₁₉×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#405 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₁₉×#3163	C₂	semidirect
5. C₁₉×#1386	C₂	semidirect
6. C₁₉×#1384	C₂	semidirect
7. C₁₉×#1378	C₂	semidirect
30. C₂×Q₁₆×C₁₉	Q₈	semidirect
32. C₂×D₈×C₁₉	Q₈	semidirect
106. #13374	C₁₉	direct (N × H)
136. #3163	C₂×C₁₉	semidirect
140. #1386	C₂×C₁₉	semidirect
141. #1384	C₂×C₁₉	semidirect
142. #1378	C₂×C₁₉	semidirect
186. C₂×Q₁₆	Q₈×C₁₉	semidirect
188. C₂×D₈	Q₈×C₁₉	semidirect
219. C₁₉	#13374	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×#3163	2
C₁₉×#2145	2
C₂×C₁₉×#360	2
C₁₉×#1976	2
C₁₉×#1386	2
C₁₉×#1384	2
C₁₉×#1378	2
#13374	19

Subgroups — C19×#13374 #18019333

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13374 #18019333