Subgroups — C₁₉×#62374 #18068333

Order 4864.

1588 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3556	C₂	1	2	yes	maximal
2	C₁₉×#3395	C₂	1	4	yes	maximal, complemented
3	C₁₉×#3383	C₂	1	4	yes	maximal, complemented
4	C₁₉×#3374	C₂	1	8	yes	maximal, complemented
5	C₁₉×#3261	C₂	1	4	yes	maximal
6	C₁₉×#3255	C₂	1	2	yes	maximal
7	C₁₉×#3177	C₂	1	4	yes	maximal
8	C₁₉×#3170	C₂	1	2	yes	maximal
9	C₁₉×#2960	C₂	1	1	yes	maximal, characteristic
10	C₁₉×#564	—	2	4	no	—
11	C₁₉×#557	—	2	4	no	—
12	C₁₉×#557	C₂²	1	8	yes	—
13	C₁₉×#557	C₂²	1	8	yes	—
14	C₁₉×#541	—	2	8	no	—
15	C₁₉×#541	C₂²	1	8	yes	—
16	C₁₉×#533	C₂²	1	4	yes	—
17	C₁₉×#533	C₂²	1	1	yes	characteristic
18	C₁₉×#500	C₂²	1	2	yes	—
19	C₁₉×#499	C₂²	1	4	yes	—
20	C₁₉×#494	C₂²	1	2	yes	—
21	C₁₉×#494	C₂²	1	4	yes	—
22	C₁₉×#491	C₂²	1	8	yes	—
23	C₁₉×#487	C₂²	1	4	yes	—
24	C₁₉×#479	C₂²	1	16	yes	—
25	C₁₉×#475	C₂²	1	8	yes	—
26	C₁₉×#470	C₂²	1	8	yes	—
27	C₁₉×#467	C₂²	1	4	yes	—
28	C₁₉×#467	C₂²	1	4	yes	—
29	C₁₉×#462	C₂²	1	8	yes	—
30	C₁₉×#462	C₂²	1	16	yes	—
31	C₁₉×#446	C₂²	1	4	yes	—
32	C₈×Q₈×C₁₉	C₂²	1	4	yes	—
33	C₁₉×#441	C₂²	1	8	yes	—
34	C₄×Q₁₆×C₁₉	C₂²	1	4	yes	—
35	C₄×Q₁₆×C₁₉	C₂²	1	4	yes	—
36	C₄×Q₁₆×C₁₉	C₂²	1	8	yes	—
37	C₁₉×#424	C₂²	1	2	yes	—
38	C₁₉×#423	C₂²	1	1	yes	characteristic
39	C₁₉×#405	C₂²	1	2	yes	—
40	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
41	C₂×Q₁₆×C₁₉	C₂³	1	4	yes	—
42	C₂×Q₁₆×C₁₉	C₂³	1	4	yes	—
43	C₂×Q₁₆×C₁₉	C₂³	1	8	yes	—
44	C₂×M₁₆×C₁₉	C₂³	1	2	yes	—
45	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
47	C₁₉×C₄⋊Q₈	—	2	8	no	—
48	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
49	C₁₉×C₄⋊Q₈	C₂³	1	4	yes	—
50	C₁₉×C₄⋊Q₈	C₂³	1	8	yes	—
51	C₁₉×C₄⋊Q₈	C₂³	1	4	yes	—
52	C₁₉×#126	—	2	16	no	—
53	C₁₉×#125	—	2	8	no	—
54	C₁₉×#125	—	2	16	no	—
55	C₁₉×#125	—	2	16	no	—
56	C₁₉×#125	—	2	16	no	—
57	C₁₉×#125	—	2	4	no	—
58	C₁₉×#125	D₄	1	2	yes	—
59	C₁₉×#125	C₂³	1	8	yes	—
60	C₁₉×C₂²⋊Q₈	—	2	4	no	—
61	C₁₉×C₂²⋊Q₈	D₄	1	4	yes	—
62	C₁₉×C₂²⋊Q₈	C₂³	1	8	yes	—
63	C₄×Q₈×C₁₉	—	2	8	no	—
64	C₄×Q₈×C₁₉	—	2	8	no	—
65	C₄×Q₈×C₁₉	—	2	4	no	—
66	C₄×Q₈×C₁₉	C₂³	1	8	yes	—
67	C₄×Q₈×C₁₉	C₂³	1	16	yes	—
68	C₄×Q₈×C₁₉	C₂³	1	4	yes	—
69	C₁₉×#117	—	2	4	no	—
70	C₁₉×#117	C₂³	1	2	yes	—
71	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
72	C₁₉×#107	C₂³	1	4	yes	—
73	C₁₉×#107	C₂³	1	4	yes	—
74	C₁₉×#106	C₂³	1	8	yes	—
75	C₁₉×C₄⋊C₈	C₂³	1	4	yes	—
76	C₁₉×C₄⋊C₈	C₂³	1	8	yes	—
77	C₁₉×#103	C₂³	1	8	yes	—
78	C₁₉×#103	C₂³	1	8	yes	—
79	C₁₉×#103	C₂³	1	16	yes	—
80	C₁₉×#97	C₂³	1	4	yes	—
81	C₄×C₈×C₁₉	C₂³	1	2	yes	—
82	C₄×C₈×C₁₉	C₂³	1	2	yes	—
83	C₄×C₈×C₁₉	C₂³	1	4	yes	—
84	C₂×Q₈×C₁₉	—	2	4	no	—
85	C₂×Q₈×C₁₉	C₂×D₄	1	4	yes	—
86	C₂×Q₈×C₁₉	C₂⁴	1	8	yes	—
87	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
89	Q₁₆×C₁₉	—	2	8	no	—
90	Q₁₆×C₁₉	—	4	4	no	—
91	Q₁₆×C₁₉	—	4	8	no	—
92	M₁₆×C₁₉	—	4	2	no	—
93	C₂×C₈×C₁₉	—	2	2	no	—
94	C₂×C₈×C₁₉	—	4	1	no	—
95	C₂×C₈×C₁₉	C₂⁴	1	2	yes	—
96	C₂×C₈×C₁₉	C₂⁴	1	1	yes	characteristic
97	C₂×C₈×C₁₉	C₂⁴	1	4	yes	—
98	C₂×C₈×C₁₉	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₁₉	—	2	4	no	—
100	C₄⋊C₄×C₁₉	—	2	4	no	—
101	C₄⋊C₄×C₁₉	—	2	4	no	—
102	C₄⋊C₄×C₁₉	—	2	8	no	—
103	C₄⋊C₄×C₁₉	—	2	16	no	—
104	C₄⋊C₄×C₁₉	—	2	16	no	—
105	C₄⋊C₄×C₁₉	—	2	8	no	—
106	C₄⋊C₄×C₁₉	—	2	16	no	—
107	C₄⋊C₄×C₁₉	C₂×D₄	1	4	yes	—
108	C₄⋊C₄×C₁₉	C₂×D₄	1	4	yes	—
109	C₄⋊C₄×C₁₉	C₂×D₄	1	8	yes	—
110	C₄⋊C₄×C₁₉	C₂⁴	1	8	yes	—
111	C₂²⋊C₄×C₁₉	—	2	8	no	—
112	C₂²⋊C₄×C₁₉	C₂×D₄	1	4	yes	—
113	C₄²×C₁₉	—	2	8	no	—
114	C₄²×C₁₉	—	2	16	no	—
115	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₁₉	C₂⁴	1	2	yes	—
118	C₄²×C₁₉	C₂⁴	1	4	yes	—
119	#62374	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
121	Q₈×C₁₉	—	2	8	no	—
122	Q₈×C₁₉	—	2	8	no	—
123	Q₈×C₁₉	—	2	8	no	—
124	Q₈×C₁₉	—	4	8	no	—
125	C₂×C₄×C₁₉	—	2	4	no	—
126	C₂×C₄×C₁₉	—	2	1	no	—
127	C₂×C₄×C₁₉	—	2	1	no	—
128	C₂×C₄×C₁₉	—	2	8	no	—
129	C₂×C₄×C₁₉	—	2	8	no	—
130	C₂×C₄×C₁₉	C₂²×D₄	1	8	yes	—
131	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
133	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
134	C₂×C₄×C₁₉	C₂⁵	1	1	yes	characteristic
135	C₈×C₁₉	—	2	1	no	—
136	C₈×C₁₉	—	2	2	no	—
137	C₈×C₁₉	—	2	4	no	—
138	C₈×C₁₉	2₋¹⁺⁴	1	2	yes	—
139	#3556	C₂×C₁₉	1	2	yes	—
140	#3395	C₂×C₁₉	1	4	yes	complemented
141	#3383	C₂×C₁₉	1	4	yes	complemented
142	#3374	C₂×C₁₉	1	8	yes	complemented
143	#3261	C₂×C₁₉	1	4	yes	—
144	#3255	C₂×C₁₉	1	2	yes	—
145	#3177	C₂×C₁₉	1	4	yes	—
146	#3170	C₂×C₁₉	1	2	yes	—
147	#2960	C₂×C₁₉	1	1	yes	characteristic
148	C₂²×C₁₉	—	2	1	no	—
149	C₂²×C₁₉	—	2	2	no	—
150	C₂²×C₁₉	C₂³×D₄	1	1	yes	characteristic
151	C₄×C₁₉	—	2	1	no	—
152	C₄×C₁₉	—	2	2	no	—
153	C₄×C₁₉	—	2	8	no	—
154	C₄×C₁₉	—	4	8	no	—
155	C₄×C₁₉	—	4	8	no	—
156	C₄×C₁₉	#578	1	4	yes	—
157	C₄×C₁₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
158	C₄×C₁₉	#585	1	1	yes	characteristic
159	#564	—	2	4	no	—
160	#557	—	2	4	no	—
161	#557	C₂²×C₁₉	1	8	yes	—
162	#557	C₂²×C₁₉	1	8	yes	—
163	#541	—	2	8	no	—
164	#541	C₂²×C₁₉	1	8	yes	—
165	#533	C₂²×C₁₉	1	4	yes	—
166	#533	C₂²×C₁₉	1	1	yes	characteristic
167	#500	C₂²×C₁₉	1	2	yes	—
168	#499	C₂²×C₁₉	1	4	yes	—
169	#494	C₂²×C₁₉	1	2	yes	—
170	#494	C₂²×C₁₉	1	4	yes	—
171	#491	C₂²×C₁₉	1	8	yes	—
172	#487	C₂²×C₁₉	1	4	yes	—
173	#479	C₂²×C₁₉	1	16	yes	—
174	#475	C₂²×C₁₉	1	8	yes	—
175	#470	C₂²×C₁₉	1	8	yes	—
176	#467	C₂²×C₁₉	1	4	yes	—
177	#467	C₂²×C₁₉	1	4	yes	—
178	#462	C₂²×C₁₉	1	8	yes	—
179	#462	C₂²×C₁₉	1	16	yes	—
180	#446	C₂²×C₁₉	1	4	yes	—
181	C₈×Q₈	C₂²×C₁₉	1	4	yes	—
182	#441	C₂²×C₁₉	1	8	yes	—
183	C₄×Q₁₆	C₂²×C₁₉	1	4	yes	—
184	C₄×Q₁₆	C₂²×C₁₉	1	4	yes	—
185	C₄×Q₁₆	C₂²×C₁₉	1	8	yes	—
186	#424	C₂²×C₁₉	1	2	yes	—
187	#423	C₂²×C₁₉	1	1	yes	characteristic
188	#405	C₂²×C₁₉	1	2	yes	—
189	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
190	C₂×C₁₉	—	4	1	no	—
191	C₂×C₁₉	#3497	1	1	yes	characteristic
192	C₂×C₁₉	C₂×#578	1	2	yes	—
193	C₂×Q₁₆	C₂³×C₁₉	1	4	yes	—
194	C₂×Q₁₆	C₂³×C₁₉	1	4	yes	—
195	C₂×Q₁₆	C₂³×C₁₉	1	8	yes	—
196	C₂×M₁₆	C₂³×C₁₉	1	2	yes	—
197	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
199	C₄⋊Q₈	—	2	8	no	—
200	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
201	C₄⋊Q₈	C₂³×C₁₉	1	4	yes	—
202	C₄⋊Q₈	C₂³×C₁₉	1	8	yes	—
203	C₄⋊Q₈	C₂³×C₁₉	1	4	yes	—
204	#126	—	2	16	no	—
205	#125	—	2	8	no	—
206	#125	—	2	16	no	—
207	#125	—	2	16	no	—
208	#125	—	2	16	no	—
209	#125	—	2	4	no	—
210	#125	D₄×C₁₉	1	2	yes	—
211	#125	C₂³×C₁₉	1	8	yes	—
212	C₂²⋊Q₈	—	2	4	no	—
213	C₂²⋊Q₈	D₄×C₁₉	1	4	yes	—
214	C₂²⋊Q₈	C₂³×C₁₉	1	8	yes	—
215	C₄×Q₈	—	2	8	no	—
216	C₄×Q₈	—	2	8	no	—
217	C₄×Q₈	—	2	4	no	—
218	C₄×Q₈	C₂³×C₁₉	1	8	yes	—
219	C₄×Q₈	C₂³×C₁₉	1	16	yes	—
220	C₄×Q₈	C₂³×C₁₉	1	4	yes	—
221	#117	—	2	4	no	—
222	#117	C₂³×C₁₉	1	2	yes	—
223	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
224	#107	C₂³×C₁₉	1	4	yes	—
225	#107	C₂³×C₁₉	1	4	yes	—
226	#106	C₂³×C₁₉	1	8	yes	—
227	C₄⋊C₈	C₂³×C₁₉	1	4	yes	—
228	C₄⋊C₈	C₂³×C₁₉	1	8	yes	—
229	#103	C₂³×C₁₉	1	8	yes	—
230	#103	C₂³×C₁₉	1	8	yes	—
231	#103	C₂³×C₁₉	1	16	yes	—
232	#97	C₂³×C₁₉	1	4	yes	—
233	C₄×C₈	C₂³×C₁₉	1	2	yes	—
234	C₄×C₈	C₂³×C₁₉	1	2	yes	—
235	C₄×C₈	C₂³×C₁₉	1	4	yes	—
236	C₁₉	#62374	1	1	yes	characteristic, direct product factor, Sylow
237	C₂×Q₈	—	2	4	no	—
238	C₂×Q₈	C₂×D₄×C₁₉	1	4	yes	—
239	C₂×Q₈	C₂⁴×C₁₉	1	8	yes	—
240	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
241	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
242	Q₁₆	—	2	8	no	—
243	Q₁₆	—	4	4	no	—
244	Q₁₆	—	4	8	no	—
245	M₁₆	—	4	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	C₂⁴×C₁₉	1	1	yes	characteristic
249	C₂×C₈	C₂⁴×C₁₉	1	2	yes	—
250	C₂×C₈	C₂⁴×C₁₉	1	4	yes	—
251	C₂×C₈	C₂⁴×C₁₉	1	1	yes	characteristic
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	8	no	—
256	C₄⋊C₄	—	2	16	no	—
257	C₄⋊C₄	—	2	16	no	—
258	C₄⋊C₄	—	2	8	no	—
259	C₄⋊C₄	—	2	16	no	—
260	C₄⋊C₄	C₂×D₄×C₁₉	1	4	yes	—
261	C₄⋊C₄	C₂×D₄×C₁₉	1	4	yes	—
262	C₄⋊C₄	C₂×D₄×C₁₉	1	8	yes	—
263	C₄⋊C₄	C₂⁴×C₁₉	1	8	yes	—
264	C₂²⋊C₄	—	2	8	no	—
265	C₂²⋊C₄	C₂×D₄×C₁₉	1	4	yes	—
266	C₄²	—	2	8	no	—
267	C₄²	—	2	16	no	—
268	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
269	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
270	C₄²	C₂⁴×C₁₉	1	2	yes	—
271	C₄²	C₂⁴×C₁₉	1	4	yes	—
272	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
273	Q₈	—	2	8	no	—
274	Q₈	—	2	8	no	—
275	Q₈	—	2	8	no	—
276	Q₈	—	4	8	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	8	no	—
281	C₂×C₄	—	2	8	no	—
282	C₂×C₄	C₂²×D₄×C₁₉	1	8	yes	—
283	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
284	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
285	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
286	C₂×C₄	C₂⁵×C₁₉	1	1	yes	characteristic
287	C₈	—	2	1	no	—
288	C₈	—	2	2	no	—
289	C₈	—	2	4	no	—
290	C₈	C₁₉×2₋¹⁺⁴	1	2	yes	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	2	no	—
293	C₂²	C₂³×D₄×C₁₉	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	8	no	—
297	C₄	—	4	8	no	—
298	C₄	—	4	8	no	—
299	C₄	C₁₉×#578	1	4	yes	—
300	C₄	C₂×C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
301	C₄	C₁₉×#585	1	1	yes	characteristic
302	C₂	—	4	1	no	—
303	C₂	C₁₉×#3497	1	1	yes	characteristic
304	C₂	C₂×C₁₉×#578	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₉
prime radical	C₂³×C₁₉	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₁₉×#533 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#3395	C₂	semidirect
3. C₁₉×#3383	C₂	semidirect
4. C₁₉×#3374	C₂	semidirect
119. #62374	C₁₉	direct (N × H)
140. #3395	C₂×C₁₉	semidirect
141. #3383	C₂×C₁₉	semidirect
142. #3374	C₂×C₁₉	semidirect
236. C₁₉	#62374	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×#3556	2
C₁₉×#3395	2
C₁₉×#3383	2
C₁₉×#3374	2
C₁₉×#3261	2
C₁₉×#3255	2
C₁₉×#3177	2
C₁₉×#3170	2
C₁₉×#2960	2
#62374	19

Subgroups — C19×#62374 #18068333

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#62374 #18068333