Subgroups — C₁₉×#60660 #18066619

Order 4864.

1770 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3458	C₂	1	2	yes	maximal
2	C₁₉×#3164	C₂	1	8	yes	maximal, complemented
3	C₁₉×#3163	C₂	1	8	yes	maximal, complemented
4	C₁₉×#3162	C₂	1	4	yes	maximal, complemented
5	C₁₉×#3160	C₂	1	4	yes	maximal, complemented
6	C₁₉×#2911	C₂	1	2	yes	maximal
7	C₂×C₁₉×#427	C₂	1	2	yes	maximal
8	C₂×C₁₉×#405	C₂	1	1	yes	maximal, characteristic
9	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
11	C₁₉×#544	—	2	4	no	—
12	C₁₉×#544	C₂²	1	16	yes	—
13	C₁₉×#543	—	2	4	no	—
14	C₁₉×#533	—	2	8	no	—
15	C₁₉×#533	C₂²	1	8	yes	—
16	C₂×C₁₉×#117	C₂²	1	4	yes	—
17	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
18	C₁₉×#501	C₂²	1	32	yes	complemented
19	C₁₉×#500	C₂²	1	8	yes	complemented
20	C₁₉×#499	C₂²	1	8	yes	complemented
21	C₁₉×#499	C₂²	1	8	yes	complemented
22	C₁₉×#498	C₂²	1	8	yes	complemented
23	C₁₉×#428	C₂²	1	16	yes	—
24	C₁₉×#427	C₂²	1	8	yes	—
25	C₁₉×#427	C₂²	1	4	yes	—
26	C₁₉×#427	C₂²	1	4	yes	—
27	C₁₉×#427	C₂²	1	4	yes	—
28	C₁₉×#427	C₂²	1	4	yes	—
29	C₂×C₁₉×#107	C₂²	1	4	yes	—
30	C₂×C₁₉×#106	C₂²	1	4	yes	—
31	C₁₉×#405	C₂²	1	4	yes	—
32	C₁₉×#405	C₂²	1	4	yes	—
33	C₂×C₁₉×#97	C₂²	1	2	yes	—
34	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
35	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₁₉	Q₈	1	8	yes	complemented
37	C₂×M₁₆×C₁₉	C₂³	1	4	yes	—
38	C₂²×C₈×C₁₉	Q₈	1	8	yes	complemented
39	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
43	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
44	C₁₉×C₄⋊Q₈	—	2	16	no	—
45	C₁₉×C₄⋊Q₈	C₂³	1	16	yes	—
46	C₁₉×#125	—	2	16	no	—
47	C₁₉×#125	C₂³	1	16	yes	—
48	C₁₉×C₂²⋊Q₈	—	2	16	no	—
49	C₁₉×C₂²⋊Q₈	—	2	32	no	—
50	C₄×Q₈×C₁₉	—	2	16	no	—
51	C₁₉×#117	—	2	4	no	—
52	C₁₉×#117	—	2	4	no	—
53	C₁₉×#117	D₄	1	4	yes	—
54	C₁₉×#117	C₂³	1	8	yes	—
55	C₁₉×#117	C₂³	1	8	yes	—
56	C₁₉×#117	C₂³	1	4	yes	—
57	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
58	C₂×C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
59	C₂×C₂²⋊C₄×C₁₉	—	2	4	no	—
60	C₂×C₂²⋊C₄×C₁₉	D₄	1	2	yes	—
61	C₂×C₄²×C₁₉	—	2	4	no	—
62	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
63	C₁₉×#107	C₂³	1	8	yes	—
64	C₁₉×#107	C₂³	1	8	yes	—
65	C₁₉×#107	C₂³	1	16	yes	—
66	C₁₉×#106	C₂³	1	8	yes	—
67	C₁₉×#106	C₂³	1	8	yes	—
68	C₁₉×#106	C₂³	1	16	yes	—
69	C₁₉×#97	C₂³	1	8	yes	—
70	C₄×C₈×C₁₉	C₂³	1	4	yes	—
71	C₄×C₈×C₁₉	C₂³	1	4	yes	—
72	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
73	C₂×Q₈×C₁₉	—	2	16	no	—
74	C₂²×C₄×C₁₉	—	2	8	no	—
75	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
76	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₁₉	C₂×Q₈	1	4	yes	—
79	C₂²×C₄×C₁₉	C₂×Q₈	1	4	yes	—
80	C₂²×C₄×C₁₉	C₂⁴	1	2	yes	—
81	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
82	M₁₆×C₁₉	C₂×Q₈	1	16	yes	—
83	C₂×C₈×C₁₉	C₂×Q₈	1	4	yes	—
84	C₂×C₈×C₁₉	C₂×Q₈	1	4	yes	—
85	C₂×C₈×C₁₉	C₂×Q₈	1	8	yes	—
86	C₂×C₈×C₁₉	C₂×Q₈	1	8	yes	—
87	C₂×C₈×C₁₉	C₂×Q₈	1	8	yes	—
88	C₂×C₈×C₁₉	C₂⁴	1	1	yes	characteristic
89	C₂×C₈×C₁₉	C₂⁴	1	1	yes	characteristic
90	C₂×C₈×C₁₉	C₂⁴	1	2	yes	—
91	C₂×C₈×C₁₉	C₂⁴	1	4	yes	—
92	C₄⋊C₄×C₁₉	—	2	8	no	—
93	C₄⋊C₄×C₁₉	—	2	16	no	—
94	C₄⋊C₄×C₁₉	—	2	32	no	—
95	C₄⋊C₄×C₁₉	C₂×D₄	1	8	yes	—
96	C₄⋊C₄×C₁₉	C₂⁴	1	16	yes	—
97	C₂²⋊C₄×C₁₉	—	2	8	no	—
98	C₂²⋊C₄×C₁₉	—	2	8	no	—
99	C₂²⋊C₄×C₁₉	C₂×D₄	1	8	yes	—
100	C₄²×C₁₉	—	2	8	no	—
101	C₄²×C₁₉	—	2	8	no	—
102	C₄²×C₁₉	C₂×D₄	1	4	yes	—
103	C₄²×C₁₉	C₂⁴	1	4	yes	—
104	#60660	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂³×C₁₉	—	4	2	no	—
106	C₂³×C₁₉	C₄⋊Q₈	1	4	yes	—
107	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
108	C₂³×C₁₉	C₂²×D₄	1	2	yes	—
109	Q₈×C₁₉	—	8	16	no	—
110	C₂×C₄×C₁₉	—	2	16	no	—
111	C₂×C₄×C₁₉	—	4	4	no	—
112	C₂×C₄×C₁₉	—	8	8	no	—
113	C₂×C₄×C₁₉	C₄⋊Q₈	1	4	yes	—
114	C₂×C₄×C₁₉	C₄⋊Q₈	1	8	yes	—
115	C₂×C₄×C₁₉	C₂²×D₄	1	8	yes	—
116	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
117	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₂²×Q₈	1	4	yes	—
119	C₂×C₄×C₁₉	C₂²×Q₈	1	2	yes	—
120	C₂×C₄×C₁₉	C₂²×Q₈	1	2	yes	—
121	C₂×C₄×C₁₉	C₂²×Q₈	1	4	yes	—
122	C₂×C₄×C₁₉	C₂⁵	1	1	yes	characteristic
123	C₈×C₁₉	C₂²×Q₈	1	2	yes	—
124	C₈×C₁₉	C₂²×Q₈	1	2	yes	—
125	C₈×C₁₉	C₂²×Q₈	1	4	yes	—
126	C₈×C₁₉	C₂²×Q₈	1	8	yes	—
127	#3458	C₂×C₁₉	1	2	yes	—
128	#3164	C₂×C₁₉	1	8	yes	complemented
129	#3163	C₂×C₁₉	1	8	yes	complemented
130	#3162	C₂×C₁₉	1	4	yes	complemented
131	#3160	C₂×C₁₉	1	4	yes	complemented
132	#2911	C₂×C₁₉	1	2	yes	—
133	C₂×#427	C₂×C₁₉	1	2	yes	—
134	C₂×#405	C₂×C₁₉	1	1	yes	characteristic
135	C₂²×C₁₉	—	2	2	no	—
136	C₂²×C₁₉	—	2	4	no	—
137	C₂²×C₁₉	—	4	4	no	—
138	C₂²×C₁₉	C₂×C₄⋊Q₈	1	2	yes	—
139	C₂²×C₁₉	C₂×C₄⋊Q₈	1	4	yes	—
140	C₂²×C₁₉	C₂³×D₄	1	1	yes	characteristic
141	C₄×C₁₉	—	2	8	no	—
142	C₄×C₁₉	—	4	16	no	—
143	C₄×C₁₉	C₂×C₄⋊Q₈	1	4	yes	—
144	C₄×C₁₉	C₂×C₄⋊Q₈	1	2	yes	—
145	C₄×C₁₉	C₂³×Q₈	1	1	yes	characteristic
146	C₄×C₁₉	C₂³×Q₈	1	1	yes	characteristic
147	C₂²×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
148	C₂³×C₈	C₂²×C₁₉	1	1	yes	characteristic
149	#544	—	2	4	no	—
150	#544	C₂²×C₁₉	1	16	yes	—
151	#543	—	2	4	no	—
152	#533	—	2	8	no	—
153	#533	C₂²×C₁₉	1	8	yes	—
154	C₂×#117	C₂²×C₁₉	1	4	yes	—
155	C₂×#117	C₂²×C₁₉	1	1	yes	characteristic
156	#501	C₂²×C₁₉	1	32	yes	complemented
157	#500	C₂²×C₁₉	1	8	yes	complemented
158	#499	C₂²×C₁₉	1	8	yes	complemented
159	#499	C₂²×C₁₉	1	8	yes	complemented
160	#498	C₂²×C₁₉	1	8	yes	complemented
161	#428	C₂²×C₁₉	1	16	yes	—
162	#427	C₂²×C₁₉	1	4	yes	—
163	#427	C₂²×C₁₉	1	8	yes	—
164	#427	C₂²×C₁₉	1	4	yes	—
165	#427	C₂²×C₁₉	1	4	yes	—
166	#427	C₂²×C₁₉	1	4	yes	—
167	C₂×#107	C₂²×C₁₉	1	4	yes	—
168	C₂×#106	C₂²×C₁₉	1	4	yes	—
169	#405	C₂²×C₁₉	1	4	yes	—
170	#405	C₂²×C₁₉	1	4	yes	—
171	C₂×#97	C₂²×C₁₉	1	2	yes	—
172	C₂×C₄×C₈	C₂²×C₁₉	1	2	yes	—
173	C₂×C₁₉	—	2	2	no	—
174	C₂×C₁₉	—	2	4	no	—
175	C₂×C₁₉	C₂²×C₄⋊Q₈	1	1	yes	characteristic
176	C₂×C₁₉	#3584	1	2	yes	—
177	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
178	C₂×M₁₆	Q₈×C₁₉	1	8	yes	complemented
179	C₂×M₁₆	C₂³×C₁₉	1	4	yes	—
180	C₂²×C₈	Q₈×C₁₉	1	8	yes	complemented
181	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
182	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
183	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
184	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
185	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
186	C₄⋊Q₈	—	2	16	no	—
187	C₄⋊Q₈	C₂³×C₁₉	1	16	yes	—
188	#125	—	2	16	no	—
189	#125	C₂³×C₁₉	1	16	yes	—
190	C₂²⋊Q₈	—	2	16	no	—
191	C₂²⋊Q₈	—	2	32	no	—
192	C₄×Q₈	—	2	16	no	—
193	#117	—	2	4	no	—
194	#117	—	2	4	no	—
195	#117	D₄×C₁₉	1	4	yes	—
196	#117	C₂³×C₁₉	1	8	yes	—
197	#117	C₂³×C₁₉	1	8	yes	—
198	#117	C₂³×C₁₉	1	4	yes	—
199	C₂×C₄⋊C₄	D₄×C₁₉	1	2	yes	—
200	C₂×C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
201	C₂×C₂²⋊C₄	—	2	4	no	—
202	C₂×C₂²⋊C₄	D₄×C₁₉	1	2	yes	—
203	C₂×C₄²	—	2	4	no	—
204	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
205	#107	C₂³×C₁₉	1	8	yes	—
206	#107	C₂³×C₁₉	1	8	yes	—
207	#107	C₂³×C₁₉	1	16	yes	—
208	#106	C₂³×C₁₉	1	8	yes	—
209	#106	C₂³×C₁₉	1	8	yes	—
210	#106	C₂³×C₁₉	1	16	yes	—
211	#97	C₂³×C₁₉	1	8	yes	—
212	C₄×C₈	C₂³×C₁₉	1	4	yes	—
213	C₄×C₈	C₂³×C₁₉	1	4	yes	—
214	C₁₉	#60660	1	1	yes	characteristic, direct product factor, Sylow
215	C₂⁴	C₂×D₄×C₁₉	1	1	yes	characteristic
216	C₂×Q₈	—	2	16	no	—
217	C₂²×C₄	—	2	8	no	—
218	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
219	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
220	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
221	C₂²×C₄	C₂×Q₈×C₁₉	1	4	yes	—
222	C₂²×C₄	C₂×Q₈×C₁₉	1	4	yes	—
223	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
224	C₂²×C₄	C₂⁴×C₁₉	1	2	yes	—
225	M₁₆	C₂×Q₈×C₁₉	1	16	yes	—
226	C₂×C₈	C₂×Q₈×C₁₉	1	4	yes	—
227	C₂×C₈	C₂×Q₈×C₁₉	1	4	yes	—
228	C₂×C₈	C₂×Q₈×C₁₉	1	8	yes	—
229	C₂×C₈	C₂×Q₈×C₁₉	1	8	yes	—
230	C₂×C₈	C₂×Q₈×C₁₉	1	8	yes	—
231	C₂×C₈	C₂⁴×C₁₉	1	1	yes	characteristic
232	C₂×C₈	C₂⁴×C₁₉	1	1	yes	characteristic
233	C₂×C₈	C₂⁴×C₁₉	1	4	yes	—
234	C₂×C₈	C₂⁴×C₁₉	1	2	yes	—
235	C₄⋊C₄	—	2	8	no	—
236	C₄⋊C₄	—	2	16	no	—
237	C₄⋊C₄	—	2	32	no	—
238	C₄⋊C₄	C₂×D₄×C₁₉	1	8	yes	—
239	C₄⋊C₄	C₂⁴×C₁₉	1	16	yes	—
240	C₂²⋊C₄	—	2	8	no	—
241	C₂²⋊C₄	—	2	8	no	—
242	C₂²⋊C₄	C₂×D₄×C₁₉	1	8	yes	—
243	C₄²	—	2	8	no	—
244	C₄²	—	2	8	no	—
245	C₄²	C₂×D₄×C₁₉	1	4	yes	—
246	C₄²	C₂⁴×C₁₉	1	4	yes	—
247	C₂³	—	4	2	no	—
248	C₂³	C₁₉×C₄⋊Q₈	1	4	yes	—
249	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
250	C₂³	C₂²×D₄×C₁₉	1	2	yes	—
251	Q₈	—	8	16	no	—
252	C₂×C₄	—	2	16	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	8	8	no	—
255	C₂×C₄	C₁₉×C₄⋊Q₈	1	8	yes	—
256	C₂×C₄	C₁₉×C₄⋊Q₈	1	4	yes	—
257	C₂×C₄	C₂²×D₄×C₁₉	1	8	yes	—
258	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
259	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
260	C₂×C₄	C₂²×Q₈×C₁₉	1	2	yes	—
261	C₂×C₄	C₂²×Q₈×C₁₉	1	4	yes	—
262	C₂×C₄	C₂²×Q₈×C₁₉	1	2	yes	—
263	C₂×C₄	C₂²×Q₈×C₁₉	1	4	yes	—
264	C₂×C₄	C₂⁵×C₁₉	1	1	yes	characteristic
265	C₈	C₂²×Q₈×C₁₉	1	2	yes	—
266	C₈	C₂²×Q₈×C₁₉	1	2	yes	—
267	C₈	C₂²×Q₈×C₁₉	1	4	yes	—
268	C₈	C₂²×Q₈×C₁₉	1	8	yes	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	4	4	no	—
272	C₂²	C₂×C₁₉×C₄⋊Q₈	1	2	yes	—
273	C₂²	C₂×C₁₉×C₄⋊Q₈	1	4	yes	—
274	C₂²	C₂³×D₄×C₁₉	1	1	yes	characteristic
275	C₄	—	2	8	no	—
276	C₄	—	4	16	no	—
277	C₄	C₂×C₁₉×C₄⋊Q₈	1	2	yes	—
278	C₄	C₂×C₁₉×C₄⋊Q₈	1	4	yes	—
279	C₄	C₂³×Q₈×C₁₉	1	1	yes	characteristic
280	C₄	C₂³×Q₈×C₁₉	1	1	yes	characteristic
281	C₂	—	2	2	no	—
282	C₂	—	2	4	no	—
283	C₂	C₂²×C₁₉×C₄⋊Q₈	1	1	yes	characteristic
284	C₂	C₁₉×#3584	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₉
socle	C₂²×C₁₉	C₂³×D₄
prime radical	C₂⁴×C₁₉	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂×C₁₉×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#3164	C₂	semidirect
3. C₁₉×#3163	C₂	semidirect
4. C₁₉×#3162	C₂	semidirect
5. C₁₉×#3160	C₂	semidirect
18. C₁₉×#501	C₂²	semidirect
19. C₁₉×#500	C₂²	semidirect
20. C₁₉×#499	C₂²	semidirect
21. C₁₉×#499	C₂²	semidirect
22. C₁₉×#498	C₂²	semidirect
36. C₂×M₁₆×C₁₉	Q₈	semidirect
38. C₂²×C₈×C₁₉	Q₈	semidirect
104. #60660	C₁₉	direct (N × H)
128. #3164	C₂×C₁₉	semidirect
129. #3163	C₂×C₁₉	semidirect
130. #3162	C₂×C₁₉	semidirect
131. #3160	C₂×C₁₉	semidirect
156. #501	C₂²×C₁₉	semidirect
157. #500	C₂²×C₁₉	semidirect
158. #499	C₂²×C₁₉	semidirect
159. #499	C₂²×C₁₉	semidirect
160. #498	C₂²×C₁₉	semidirect
178. C₂×M₁₆	Q₈×C₁₉	semidirect
180. C₂²×C₈	Q₈×C₁₉	semidirect
214. C₁₉	#60660	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×#3458	2
C₁₉×#3164	2
C₁₉×#3163	2
C₁₉×#3162	2
C₁₉×#3160	2
C₁₉×#2911	2
C₂×C₁₉×#427	2
C₂×C₁₉×#405	2
#60660	19

Subgroups — C19×#60660 #18066619

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#60660 #18066619