Subgroups — C₁₉×#33604 #18039563

Order 4864.

690 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#508	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₉×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#3050	C₂	1	2	yes	maximal, complemented
4	C₁₉×#2220	C₂	1	4	yes	maximal, complemented
5	C₁₉×#2219	C₂	1	2	yes	maximal, complemented
6	C₁₉×#2218	C₂	1	2	yes	maximal, complemented
7	C₁₉×#2158	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₉×#2138	C₂	1	2	yes	maximal, complemented
9	C₂×C₁₉×#135	C₂²	1	2	yes	—
10	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
11	C₁₉×#525	—	2	2	no	—
12	C₁₉×#508	—	2	4	no	—
13	C₂×C₁₉×Q₃₂	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₁₉×SD₃₂	C₂²	1	2	yes	complemented
15	C₂×C₁₉×D₁₆	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₁₉×M₃₂	—	2	4	no	—
17	C₂×C₁₉×M₃₂	C₂²	1	2	yes	—
18	C₂×C₁₉×M₃₂	C₂²	1	2	yes	—
19	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
20	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
21	C₁₉×#467	C₂²	1	4	yes	—
22	C₁₉×#466	C₂²	1	4	yes	—
23	C₁₉×#465	—	2	4	no	—
24	C₁₉×#427	C₂²	1	2	yes	—
25	C₁₉×#416	—	2	2	no	—
26	C₁₉×#367	C₂²	1	2	yes	complemented
27	C₁₉×#366	C₂²	1	2	yes	complemented
28	C₁₉×#358	C₂²	1	4	yes	complemented
29	C₁₉×#357	C₂²	1	4	yes	complemented
30	C₂×#41×C₁₉	—	2	2	no	—
31	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
32	C₁₉×#135	—	2	4	no	—
33	C₁₉×#135	—	2	4	no	—
34	C₂×Q₁₆×C₁₉	C₂³	1	2	yes	—
35	C₂×SD₁₆×C₁₉	—	2	2	no	—
36	C₂×D₈×C₁₉	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₉	—	2	2	no	—
38	C₂²×C₈×C₁₉	—	2	2	no	—
39	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₉	D₄	1	2	yes	—
41	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
42	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
43	C₁₉×#123	—	4	2	no	—
44	C₁₉×C₂²⋊Q₈	—	2	4	no	—
45	C₁₉×#121	—	2	4	no	—
46	C₁₉×#120	—	4	2	no	—
47	C₄×D₄×C₁₉	—	4	4	no	—
48	C₁₉×#117	—	2	2	no	—
49	C₁₉×Q₃₂	—	2	2	no	—
50	C₁₉×SD₃₂	—	2	4	no	—
51	C₁₉×D₁₆	—	2	2	no	—
52	C₁₉×M₃₂	—	2	4	no	—
53	C₁₉×M₃₂	—	2	4	no	—
54	C₂×C₁₆×C₁₉	—	2	2	no	—
55	C₂×C₁₆×C₁₉	D₄	1	4	yes	complemented
56	C₂×C₁₆×C₁₉	C₂³	1	2	yes	—
57	C₂×C₁₆×C₁₉	C₂³	1	1	yes	characteristic
58	C₂×C₁₆×C₁₉	C₂³	1	1	yes	characteristic
59	C₁₉×#107	C₂³	1	4	yes	—
60	C₁₉×#106	—	2	2	no	—
61	C₁₉×#103	—	2	4	no	—
62	C₁₉×#102	—	2	4	no	—
63	C₂⁴×C₁₉	D₈	1	1	yes	characteristic
64	#41×C₁₉	—	4	4	no	—
65	#41×C₁₉	—	4	4	no	—
66	C₂×Q₈×C₁₉	—	2	2	no	—
67	C₂×D₄×C₁₉	—	2	2	no	—
68	C₂×D₄×C₁₉	—	4	4	no	—
69	C₂×D₄×C₁₉	—	4	2	no	—
70	C₂²×C₄×C₁₉	—	2	2	no	—
71	C₂²×C₄×C₁₉	—	2	2	no	—
72	C₂²×C₄×C₁₉	—	4	2	no	—
73	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₁₉	D₈	1	2	yes	—
75	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
77	Q₁₆×C₁₉	—	2	2	no	—
78	Q₁₆×C₁₉	—	2	2	no	—
79	SD₁₆×C₁₉	—	4	4	no	—
80	D₈×C₁₉	—	2	2	no	—
81	D₈×C₁₉	—	2	2	no	—
82	C₂×C₈×C₁₉	—	2	2	no	—
83	C₂×C₈×C₁₉	—	2	4	no	—
84	C₂×C₈×C₁₉	—	2	2	no	—
85	C₂×C₈×C₁₉	—	2	2	no	—
86	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₁₉	#41	1	2	yes	—
90	C₂×C₈×C₁₉	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄×C₁₉	—	2	4	no	—
92	C₄⋊C₄×C₁₉	—	4	4	no	—
93	C₂²⋊C₄×C₁₉	—	4	2	no	—
94	C₂²⋊C₄×C₁₉	—	4	4	no	—
95	C₂²⋊C₄×C₁₉	—	4	4	no	—
96	C₄²×C₁₉	—	4	2	no	—
97	C₁₆×C₁₉	—	2	2	no	—
98	C₁₆×C₁₉	C₂×D₄	1	2	yes	—
99	C₁₆×C₁₉	C₂×D₄	1	2	yes	—
100	#33604	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂³×C₁₉	—	2	2	no	—
102	C₂³×C₁₉	—	4	2	no	—
103	C₂³×C₁₉	—	4	2	no	—
104	C₂³×C₁₉	C₂×D₈	1	2	yes	—
105	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
106	Q₈×C₁₉	—	4	2	no	—
107	Q₈×C₁₉	—	4	2	no	—
108	D₄×C₁₉	—	4	2	no	—
109	D₄×C₁₉	—	4	2	no	—
110	D₄×C₁₉	—	8	8	no	—
111	D₄×C₁₉	—	8	4	no	—
112	C₂×C₄×C₁₉	—	2	2	no	—
113	C₂×C₄×C₁₉	—	2	2	no	—
114	C₂×C₄×C₁₉	—	2	2	no	—
115	C₂×C₄×C₁₉	—	2	4	no	—
116	C₂×C₄×C₁₉	—	4	4	no	—
117	C₂×C₄×C₁₉	—	4	2	no	—
118	C₂×C₄×C₁₉	—	8	4	no	—
119	C₂×C₄×C₁₉	#121	1	2	yes	—
120	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
121	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₁₉	#135	1	2	yes	—
123	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
124	C₈×C₁₉	—	2	2	no	—
125	C₈×C₁₉	#121	1	2	yes	—
126	C₈×C₁₉	C₂²×D₄	1	1	yes	characteristic
127	C₈×C₁₉	C₂×#41	1	1	yes	characteristic
128	C₂×#508	C₂×C₁₉	1	1	yes	characteristic, complemented
129	C₂²×M₃₂	C₂×C₁₉	1	1	yes	characteristic, complemented
130	#3050	C₂×C₁₉	1	2	yes	complemented
131	#2220	C₂×C₁₉	1	4	yes	complemented
132	#2219	C₂×C₁₉	1	2	yes	complemented
133	#2218	C₂×C₁₉	1	2	yes	complemented
134	#2158	C₂×C₁₉	1	1	yes	characteristic, complemented
135	#2138	C₂×C₁₉	1	2	yes	complemented
136	C₂²×C₁₉	—	2	2	no	—
137	C₂²×C₁₉	—	2	4	no	—
138	C₂²×C₁₉	—	2	2	no	—
139	C₂²×C₁₉	—	4	4	no	—
140	C₂²×C₁₉	—	8	4	no	—
141	C₂²×C₁₉	—	8	2	no	—
142	C₂²×C₁₉	#466	1	2	yes	—
143	C₂²×C₁₉	C₂²×D₈	1	1	yes	characteristic
144	C₄×C₁₉	—	2	2	no	—
145	C₄×C₁₉	—	8	4	no	—
146	C₄×C₁₉	—	8	2	no	—
147	C₄×C₁₉	#466	1	2	yes	—
148	C₄×C₁₉	C₂×#121	1	1	yes	characteristic
149	C₄×C₁₉	C₂×#135	1	1	yes	characteristic
150	C₂×#135	C₂²×C₁₉	1	2	yes	—
151	C₂³×C₈	C₂²×C₁₉	1	1	yes	characteristic
152	#525	—	2	2	no	—
153	#508	—	2	4	no	—
154	C₂×Q₃₂	C₂²×C₁₉	1	1	yes	characteristic, complemented
155	C₂×SD₃₂	C₂²×C₁₉	1	2	yes	complemented
156	C₂×D₁₆	C₂²×C₁₉	1	1	yes	characteristic, complemented
157	C₂×M₃₂	—	2	4	no	—
158	C₂×M₃₂	C₂²×C₁₉	1	2	yes	—
159	C₂×M₃₂	C₂²×C₁₉	1	2	yes	—
160	C₂²×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
161	C₂²×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
162	#467	C₂²×C₁₉	1	4	yes	—
163	#466	C₂²×C₁₉	1	4	yes	—
164	#465	—	2	4	no	—
165	#427	C₂²×C₁₉	1	2	yes	—
166	#416	—	2	2	no	—
167	#367	C₂²×C₁₉	1	2	yes	complemented
168	#366	C₂²×C₁₉	1	2	yes	complemented
169	#358	C₂²×C₁₉	1	4	yes	complemented
170	#357	C₂²×C₁₉	1	4	yes	complemented
171	C₂×C₁₉	—	2	2	no	—
172	C₂×C₁₉	—	4	2	no	—
173	C₂×C₁₉	—	16	2	no	—
174	C₂×C₁₉	C₂×#466	1	1	yes	characteristic
175	C₂×C₁₉	#3414	1	2	yes	—
176	C₂×#41	—	2	2	no	—
177	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
178	#135	—	2	4	no	—
179	#135	—	2	4	no	—
180	C₂×Q₁₆	C₂³×C₁₉	1	2	yes	—
181	C₂×SD₁₆	—	2	2	no	—
182	C₂×D₈	C₂³×C₁₉	1	2	yes	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
186	C₂²×C₈	D₄×C₁₉	1	2	yes	—
187	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
188	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
189	#123	—	4	2	no	—
190	C₂²⋊Q₈	—	2	4	no	—
191	#121	—	2	4	no	—
192	#120	—	4	2	no	—
193	C₄×D₄	—	4	4	no	—
194	#117	—	2	2	no	—
195	Q₃₂	—	2	2	no	—
196	SD₃₂	—	2	4	no	—
197	D₁₆	—	2	2	no	—
198	M₃₂	—	2	4	no	—
199	M₃₂	—	2	4	no	—
200	C₂×C₁₆	—	2	2	no	—
201	C₂×C₁₆	D₄×C₁₉	1	4	yes	complemented
202	C₂×C₁₆	C₂³×C₁₉	1	2	yes	—
203	C₂×C₁₆	C₂³×C₁₉	1	1	yes	characteristic
204	C₂×C₁₆	C₂³×C₁₉	1	1	yes	characteristic
205	#107	C₂³×C₁₉	1	4	yes	—
206	#106	—	2	2	no	—
207	#103	—	2	4	no	—
208	#102	—	2	4	no	—
209	C₁₉	#33604	1	1	yes	characteristic, direct product factor, Sylow
210	C₂⁴	D₈×C₁₉	1	1	yes	characteristic
211	#41	—	4	4	no	—
212	#41	—	4	4	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	4	4	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	D₈×C₁₉	1	2	yes	—
221	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
223	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
224	Q₁₆	—	2	2	no	—
225	Q₁₆	—	2	2	no	—
226	SD₁₆	—	4	4	no	—
227	D₈	—	2	2	no	—
228	D₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
234	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
235	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
236	C₂×C₈	#41×C₁₉	1	2	yes	—
237	C₂×C₈	C₂⁴×C₁₉	1	1	yes	characteristic
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	4	4	no	—
240	C₂²⋊C₄	—	4	2	no	—
241	C₂²⋊C₄	—	4	4	no	—
242	C₂²⋊C₄	—	4	4	no	—
243	C₄²	—	4	2	no	—
244	C₁₆	—	2	2	no	—
245	C₁₆	C₂×D₄×C₁₉	1	2	yes	—
246	C₁₆	C₂×D₄×C₁₉	1	2	yes	—
247	C₂³	—	2	2	no	—
248	C₂³	—	4	2	no	—
249	C₂³	—	4	2	no	—
250	C₂³	C₂×D₈×C₁₉	1	2	yes	—
251	C₂³	C₂×D₈×C₁₉	1	1	yes	characteristic
252	Q₈	—	4	2	no	—
253	Q₈	—	4	2	no	—
254	D₄	—	4	2	no	—
255	D₄	—	4	2	no	—
256	D₄	—	8	8	no	—
257	D₄	—	8	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	8	4	no	—
265	C₂×C₄	C₁₉×#121	1	2	yes	—
266	C₂×C₄	C₂×D₈×C₁₉	1	2	yes	—
267	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
268	C₂×C₄	C₁₉×#135	1	2	yes	—
269	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	C₁₉×#121	1	2	yes	—
272	C₈	C₂²×D₄×C₁₉	1	1	yes	characteristic
273	C₈	C₂×#41×C₁₉	1	1	yes	characteristic
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	4	no	—
277	C₂²	—	4	4	no	—
278	C₂²	—	8	2	no	—
279	C₂²	—	8	4	no	—
280	C₂²	C₁₉×#466	1	2	yes	—
281	C₂²	C₂²×D₈×C₁₉	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	8	2	no	—
284	C₄	—	8	4	no	—
285	C₄	C₁₉×#466	1	2	yes	—
286	C₄	C₂×C₁₉×#121	1	1	yes	characteristic
287	C₄	C₂×C₁₉×#135	1	1	yes	characteristic
288	C₂	—	2	2	no	—
289	C₂	—	4	2	no	—
290	C₂	—	16	2	no	—
291	C₂	C₂×C₁₉×#466	1	1	yes	characteristic
292	C₂	C₁₉×#3414	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₉	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₉
socle	C₂²×C₁₉	C₂²×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂³×C₄×C₁₉ ◁ C₂³×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₉×#508	C₂	semidirect
2. C₂²×C₁₉×M₃₂	C₂	semidirect
3. C₁₉×#3050	C₂	semidirect
4. C₁₉×#2220	C₂	semidirect
5. C₁₉×#2219	C₂	semidirect
6. C₁₉×#2218	C₂	semidirect
7. C₁₉×#2158	C₂	semidirect
8. C₁₉×#2138	C₂	semidirect
13. C₂×C₁₉×Q₃₂	C₂²	semidirect
14. C₂×C₁₉×SD₃₂	C₂²	semidirect
15. C₂×C₁₉×D₁₆	C₂²	semidirect
26. C₁₉×#367	C₂²	semidirect
27. C₁₉×#366	C₂²	semidirect
28. C₁₉×#358	C₂²	semidirect
29. C₁₉×#357	C₂²	semidirect
55. C₂×C₁₆×C₁₉	D₄	semidirect
100. #33604	C₁₉	direct (N × H)
128. C₂×#508	C₂×C₁₉	semidirect
129. C₂²×M₃₂	C₂×C₁₉	semidirect
130. #3050	C₂×C₁₉	semidirect
131. #2220	C₂×C₁₉	semidirect
132. #2219	C₂×C₁₉	semidirect
133. #2218	C₂×C₁₉	semidirect
134. #2158	C₂×C₁₉	semidirect
135. #2138	C₂×C₁₉	semidirect
154. C₂×Q₃₂	C₂²×C₁₉	semidirect
155. C₂×SD₃₂	C₂²×C₁₉	semidirect
156. C₂×D₁₆	C₂²×C₁₉	semidirect
167. #367	C₂²×C₁₉	semidirect
168. #366	C₂²×C₁₉	semidirect
169. #358	C₂²×C₁₉	semidirect
170. #357	C₂²×C₁₉	semidirect
201. C₂×C₁₆	D₄×C₁₉	semidirect
209. C₁₉	#33604	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₉×#508	2
C₂²×C₁₉×M₃₂	2
C₁₉×#3050	2
C₁₉×#2220	2
C₁₉×#2219	2
C₁₉×#2218	2
C₁₉×#2158	2
C₁₉×#2138	2
#33604	19

Subgroups — C19×#33604 #18039563

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#33604 #18039563