Subgroups — C₁₉×#3272 #14561129

Order 2432.

546 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×AGL(1,ℤ/8)	C₂	1	2	yes	maximal, complemented
2	C₁₉×#547	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄²×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#468	C₂	1	2	yes	maximal, complemented
6	C₁₉×#461	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#459	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₉×#450	C₂	1	2	yes	maximal, complemented
9	C₁₉×#449	C₂	1	2	yes	maximal, complemented
10	C₄×D₈×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₉×#436	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₁₉	C₂²	1	2	yes	complemented
13	C₂²×D₄×C₁₉	—	2	1	no	—
14	C₂²×D₄×C₁₉	C₂²	1	2	yes	—
15	C₁₉×AGL(1,ℤ/8)	—	2	8	no	—
16	C₂×SD₁₆×C₁₉	C₂²	1	2	yes	—
17	C₂×SD₁₆×C₁₉	C₂²	1	2	yes	complemented
18	C₂×SD₁₆×C₁₉	C₂²	1	2	yes	complemented
19	C₂×D₈×C₁₉	C₂²	1	1	yes	characteristic
20	C₂×D₈×C₁₉	C₂²	1	2	yes	complemented
21	C₂×M₁₆×C₁₉	C₂²	1	2	yes	complemented
22	C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
23	C₁₉×#127	C₂²	1	1	yes	characteristic
24	C₁₉×#123	—	2	2	no	—
25	C₁₉×C₂²⋊Q₈	C₂²	1	2	yes	complemented
26	C₁₉×#121	—	2	1	no	—
27	C₁₉×#121	—	2	1	no	—
28	C₁₉×#121	C₂²	1	2	yes	—
29	C₁₉×#120	—	2	2	no	—
30	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
31	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
32	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic, complemented
33	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
34	C₁₉×#107	C₂²	1	1	yes	characteristic, complemented
35	C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
36	C₁₉×#103	C₂²	1	2	yes	complemented
37	C₁₉×#102	C₂²	1	1	yes	characteristic
38	C₁₉×#102	C₂²	1	1	yes	characteristic, complemented
39	C₁₉×#102	C₂²	1	2	yes	complemented
40	C₁₉×C₂²⋊C₈	C₂²	1	2	yes	complemented
41	C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic, complemented
42	C₂⁴×C₁₉	—	2	2	no	—
43	#41×C₁₉	—	2	4	no	—
44	#41×C₁₉	—	4	2	no	—
45	C₂×Q₈×C₁₉	C₂³	1	2	yes	—
46	C₂×D₄×C₁₉	—	2	4	no	—
47	C₂×D₄×C₁₉	—	2	2	no	—
48	C₂×D₄×C₁₉	—	2	2	no	—
49	C₂×D₄×C₁₉	—	2	2	no	—
50	C₂×D₄×C₁₉	—	2	1	no	—
51	C₂×D₄×C₁₉	—	2	1	no	—
52	C₂×D₄×C₁₉	—	4	2	no	—
53	C₂×D₄×C₁₉	—	4	4	no	—
54	C₂×D₄×C₁₉	D₄	1	1	yes	characteristic
55	C₂×D₄×C₁₉	C₂³	1	2	yes	—
56	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
57	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	—	2	1	no	—
59	C₂²×C₄×C₁₉	—	2	2	no	—
60	C₂²×C₄×C₁₉	—	2	1	no	—
61	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
62	SD₁₆×C₁₉	—	2	4	no	—
63	SD₁₆×C₁₉	—	2	4	no	—
64	SD₁₆×C₁₉	—	4	2	no	—
65	D₈×C₁₉	—	2	4	no	—
66	D₈×C₁₉	D₄	1	4	yes	complemented
67	M₁₆×C₁₉	—	2	4	no	—
68	C₂×C₈×C₁₉	C₂³	1	2	yes	—
69	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₁₉	—	2	1	no	—
72	C₄⋊C₄×C₁₉	—	2	2	no	—
73	C₄⋊C₄×C₁₉	—	2	1	no	—
74	C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
75	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
77	C₂²⋊C₄×C₁₉	—	2	2	no	—
78	C₂²⋊C₄×C₁₉	—	2	1	no	—
79	C₂²⋊C₄×C₁₉	—	2	1	no	—
80	C₂²⋊C₄×C₁₉	—	2	2	no	—
81	C₂²⋊C₄×C₁₉	D₄	1	2	yes	—
82	C₄²×C₁₉	C₂³	1	1	yes	characteristic
83	C₂³×C₁₉	—	2	1	no	—
84	C₂³×C₁₉	—	2	1	no	—
85	C₂³×C₁₉	—	2	2	no	—
86	C₂³×C₁₉	—	4	4	no	—
87	C₂³×C₁₉	—	4	4	no	—
88	C₂³×C₁₉	—	8	2	no	—
89	C₂³×C₁₉	C₂×D₄	1	2	yes	—
90	Q₈×C₁₉	—	2	2	no	—
91	Q₈×C₁₉	—	2	2	no	—
92	D₄×C₁₉	—	2	2	no	—
93	D₄×C₁₉	—	2	2	no	—
94	D₄×C₁₉	—	2	1	no	—
95	D₄×C₁₉	—	2	1	no	—
96	D₄×C₁₉	—	2	2	no	—
97	D₄×C₁₉	—	4	4	no	—
98	D₄×C₁₉	—	4	4	no	—
99	D₄×C₁₉	—	4	4	no	—
100	D₄×C₁₉	—	4	2	no	—
101	D₄×C₁₉	—	8	1	no	—
102	D₄×C₁₉	C₂×D₄	1	2	yes	—
103	D₄×C₁₉	C₂×D₄	1	2	yes	complemented
104	C₂×C₄×C₁₉	—	2	2	no	—
105	C₂×C₄×C₁₉	—	2	2	no	—
106	C₂×C₄×C₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	—	2	1	no	—
108	C₂×C₄×C₁₉	—	2	2	no	—
109	C₂×C₄×C₁₉	—	4	4	no	—
110	C₂×C₄×C₁₉	—	4	2	no	—
111	C₂×C₄×C₁₉	—	4	2	no	—
112	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
114	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
116	C₈×C₁₉	—	2	2	no	—
117	C₈×C₁₉	—	2	1	no	—
118	C₈×C₁₉	C₂×D₄	1	2	yes	complemented
119	#3272	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂²×C₁₉	—	2	2	no	—
121	C₂²×C₁₉	—	2	2	no	—
122	C₂²×C₁₉	—	2	2	no	—
123	C₂²×C₁₉	—	2	2	no	—
124	C₂²×C₁₉	—	2	2	no	—
125	C₂²×C₁₉	—	4	1	no	—
126	C₂²×C₁₉	—	4	1	no	—
127	C₂²×C₁₉	—	4	1	no	—
128	C₂²×C₁₉	—	4	1	no	—
129	C₂²×C₁₉	—	4	2	no	—
130	C₂²×C₁₉	—	4	2	no	—
131	C₂²×C₁₉	—	4	2	no	—
132	C₂²×C₁₉	—	8	8	no	—
133	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
134	C₄×C₁₉	—	2	2	no	—
135	C₄×C₁₉	—	2	1	no	—
136	C₄×C₁₉	—	4	2	no	—
137	C₄×C₁₉	—	4	1	no	—
138	C₄×C₁₉	—	4	1	no	—
139	C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	complemented
140	C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
141	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
142	C₂×AGL(1,ℤ/8)	C₂×C₁₉	1	2	yes	complemented
143	#547	C₂×C₁₉	1	1	yes	characteristic, complemented
144	D₄²	C₂×C₁₉	1	1	yes	characteristic, complemented
145	#492	C₂×C₁₉	1	1	yes	characteristic, complemented
146	#468	C₂×C₁₉	1	2	yes	complemented
147	#461	C₂×C₁₉	1	1	yes	characteristic, complemented
148	#459	C₂×C₁₉	1	1	yes	characteristic, complemented
149	#450	C₂×C₁₉	1	2	yes	complemented
150	#449	C₂×C₁₉	1	2	yes	complemented
151	C₄×D₈	C₂×C₁₉	1	1	yes	characteristic, complemented
152	#436	C₂×C₁₉	1	1	yes	characteristic, complemented
153	C₂×C₁₉	—	4	2	no	—
154	C₂×C₁₉	—	4	2	no	—
155	C₂×C₁₉	—	4	2	no	—
156	C₂×C₁₉	—	8	2	no	—
157	C₂×C₁₉	D₄²	1	1	yes	characteristic
158	C₂×C₁₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
159	C₂×C₁₉	#577	1	1	yes	characteristic
160	C₂×#41	C₂²×C₁₉	1	2	yes	complemented
161	C₂²×D₄	—	2	1	no	—
162	C₂²×D₄	C₂²×C₁₉	1	2	yes	—
163	AGL(1,ℤ/8)	—	2	8	no	—
164	C₂×SD₁₆	C₂²×C₁₉	1	2	yes	—
165	C₂×SD₁₆	C₂²×C₁₉	1	2	yes	complemented
166	C₂×SD₁₆	C₂²×C₁₉	1	2	yes	complemented
167	C₂×D₈	C₂²×C₁₉	1	1	yes	characteristic
168	C₂×D₈	C₂²×C₁₉	1	2	yes	complemented
169	C₂×M₁₆	C₂²×C₁₉	1	2	yes	complemented
170	C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic, complemented
171	#127	C₂²×C₁₉	1	1	yes	characteristic
172	#123	—	2	2	no	—
173	C₂²⋊Q₈	C₂²×C₁₉	1	2	yes	complemented
174	#121	—	2	1	no	—
175	#121	—	2	1	no	—
176	#121	C₂²×C₁₉	1	2	yes	—
177	#120	—	2	2	no	—
178	C₄×D₄	C₂²×C₁₉	1	1	yes	characteristic
179	C₄×D₄	C₂²×C₁₉	1	1	yes	characteristic
180	C₄×D₄	C₂²×C₁₉	1	1	yes	characteristic, complemented
181	C₂×C₄⋊C₄	—	2	1	no	—
182	#107	C₂²×C₁₉	1	1	yes	characteristic, complemented
183	C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic, complemented
184	#103	C₂²×C₁₉	1	2	yes	complemented
185	#102	C₂²×C₁₉	1	1	yes	characteristic
186	#102	C₂²×C₁₉	1	1	yes	characteristic, complemented
187	#102	C₂²×C₁₉	1	2	yes	complemented
188	C₂²⋊C₈	C₂²×C₁₉	1	2	yes	complemented
189	C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic, complemented
190	C₁₉	#3272	1	1	yes	characteristic, direct product factor, Sylow
191	C₂⁴	—	2	2	no	—
192	#41	—	2	4	no	—
193	#41	—	4	2	no	—
194	C₂×Q₈	C₂³×C₁₉	1	2	yes	—
195	C₂×D₄	—	2	4	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	—	2	1	no	—
201	C₂×D₄	—	4	2	no	—
202	C₂×D₄	—	4	4	no	—
203	C₂×D₄	D₄×C₁₉	1	1	yes	characteristic
204	C₂×D₄	C₂³×C₁₉	1	2	yes	—
205	C₂×D₄	C₂³×C₁₉	1	1	yes	characteristic
206	C₂×D₄	C₂³×C₁₉	1	1	yes	characteristic
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	C₂³×C₁₉	1	2	yes	—
211	SD₁₆	—	2	4	no	—
212	SD₁₆	—	2	4	no	—
213	SD₁₆	—	4	2	no	—
214	D₈	—	2	4	no	—
215	D₈	D₄×C₁₉	1	4	yes	complemented
216	M₁₆	—	2	4	no	—
217	C₂×C₈	C₂³×C₁₉	1	1	yes	characteristic
218	C₂×C₈	C₂³×C₁₉	1	2	yes	—
219	C₂×C₈	C₂³×C₁₉	1	1	yes	characteristic
220	C₄⋊C₄	—	2	1	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
224	C₄⋊C₄	C₂³×C₁₉	1	1	yes	characteristic
225	C₄⋊C₄	C₂³×C₁₉	1	1	yes	characteristic
226	C₂²⋊C₄	—	2	2	no	—
227	C₂²⋊C₄	—	2	1	no	—
228	C₂²⋊C₄	—	2	1	no	—
229	C₂²⋊C₄	—	2	2	no	—
230	C₂²⋊C₄	D₄×C₁₉	1	2	yes	—
231	C₄²	C₂³×C₁₉	1	1	yes	characteristic
232	C₂³	—	2	1	no	—
233	C₂³	—	2	1	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	4	4	no	—
236	C₂³	—	4	4	no	—
237	C₂³	—	8	2	no	—
238	C₂³	C₂×D₄×C₁₉	1	2	yes	—
239	Q₈	—	2	2	no	—
240	Q₈	—	2	2	no	—
241	D₄	—	2	2	no	—
242	D₄	—	2	2	no	—
243	D₄	—	2	1	no	—
244	D₄	—	2	1	no	—
245	D₄	—	2	2	no	—
246	D₄	—	4	4	no	—
247	D₄	—	4	4	no	—
248	D₄	—	4	4	no	—
249	D₄	—	4	2	no	—
250	D₄	—	8	1	no	—
251	D₄	C₂×D₄×C₁₉	1	2	yes	—
252	D₄	C₂×D₄×C₁₉	1	2	yes	complemented
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
262	C₂×C₄	C₂×D₄×C₁₉	1	2	yes	—
263	C₂×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
264	C₂×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
265	C₈	—	2	2	no	—
266	C₈	—	2	1	no	—
267	C₈	C₂×D₄×C₁₉	1	2	yes	complemented
268	C₂²	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	4	1	no	—
274	C₂²	—	4	1	no	—
275	C₂²	—	4	1	no	—
276	C₂²	—	4	1	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	8	8	no	—
281	C₂²	C₂²×D₄×C₁₉	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	2	1	no	—
284	C₄	—	4	2	no	—
285	C₄	—	4	1	no	—
286	C₄	—	4	1	no	—
287	C₄	C₁₉×AGL(1,ℤ/8)	1	2	yes	complemented
288	C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
289	C₄	C₁₉×H(5,𝔽₂)	1	1	yes	characteristic
290	C₂	—	4	2	no	—
291	C₂	—	4	2	no	—
292	C₂	—	4	2	no	—
293	C₂	—	8	2	no	—
294	C₂	D₄²×C₁₉	1	1	yes	characteristic
295	C₂	C₂×C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
296	C₂	C₁₉×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₄×D₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₉×AGL(1,ℤ/8)	C₂	semidirect
2. C₁₉×#547	C₂	semidirect
3. D₄²×C₁₉	C₂	semidirect
4. C₁₉×#492	C₂	semidirect
5. C₁₉×#468	C₂	semidirect
6. C₁₉×#461	C₂	semidirect
7. C₁₉×#459	C₂	semidirect
8. C₁₉×#450	C₂	semidirect
9. C₁₉×#449	C₂	semidirect
10. C₄×D₈×C₁₉	C₂	semidirect
11. C₁₉×#436	C₂	semidirect
12. C₂×#41×C₁₉	C₂²	semidirect
17. C₂×SD₁₆×C₁₉	C₂²	semidirect
18. C₂×SD₁₆×C₁₉	C₂²	semidirect
20. C₂×D₈×C₁₉	C₂²	semidirect
21. C₂×M₁₆×C₁₉	C₂²	semidirect
22. C₁₉×C₄⋊Q₈	C₂²	semidirect
25. C₁₉×C₂²⋊Q₈	C₂²	semidirect
32. C₄×D₄×C₁₉	C₂²	semidirect
34. C₁₉×#107	C₂²	semidirect
35. C₁₉×C₄⋊C₈	C₂²	semidirect
36. C₁₉×#103	C₂²	semidirect
38. C₁₉×#102	C₂²	semidirect
39. C₁₉×#102	C₂²	semidirect
40. C₁₉×C₂²⋊C₈	C₂²	semidirect
41. C₄×C₈×C₁₉	C₂²	semidirect
66. D₈×C₁₉	D₄	semidirect
103. D₄×C₁₉	C₂×D₄	semidirect
118. C₈×C₁₉	C₂×D₄	semidirect
119. #3272	C₁₉	direct (N × H)
139. C₄×C₁₉	AGL(1,ℤ/8)	semidirect
142. C₂×AGL(1,ℤ/8)	C₂×C₁₉	semidirect
143. #547	C₂×C₁₉	semidirect
144. D₄²	C₂×C₁₉	semidirect
145. #492	C₂×C₁₉	semidirect
146. #468	C₂×C₁₉	semidirect
147. #461	C₂×C₁₉	semidirect
148. #459	C₂×C₁₉	semidirect
149. #450	C₂×C₁₉	semidirect
150. #449	C₂×C₁₉	semidirect
151. C₄×D₈	C₂×C₁₉	semidirect
152. #436	C₂×C₁₉	semidirect
160. C₂×#41	C₂²×C₁₉	semidirect
165. C₂×SD₁₆	C₂²×C₁₉	semidirect
166. C₂×SD₁₆	C₂²×C₁₉	semidirect
168. C₂×D₈	C₂²×C₁₉	semidirect
169. C₂×M₁₆	C₂²×C₁₉	semidirect
170. C₄⋊Q₈	C₂²×C₁₉	semidirect
173. C₂²⋊Q₈	C₂²×C₁₉	semidirect
180. C₄×D₄	C₂²×C₁₉	semidirect
182. #107	C₂²×C₁₉	semidirect
183. C₄⋊C₈	C₂²×C₁₉	semidirect
184. #103	C₂²×C₁₉	semidirect
186. #102	C₂²×C₁₉	semidirect
187. #102	C₂²×C₁₉	semidirect
188. C₂²⋊C₈	C₂²×C₁₉	semidirect
189. C₄×C₈	C₂²×C₁₉	semidirect
190. C₁₉	#3272	direct (N × H)
215. D₈	D₄×C₁₉	semidirect
252. D₄	C₂×D₄×C₁₉	semidirect
267. C₈	C₂×D₄×C₁₉	semidirect
287. C₄	C₁₉×AGL(1,ℤ/8)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₉×AGL(1,ℤ/8)	2
C₁₉×#547	2
D₄²×C₁₉	2
C₁₉×#492	2
C₁₉×#468	2
C₁₉×#461	2
C₁₉×#459	2
C₁₉×#450	2
C₁₉×#449	2
C₄×D₈×C₁₉	2
C₁₉×#436	2
#3272	19

Subgroups — C19×#3272 #14561129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#3272 #14561129