Subgroups — C₁₉×#34184 #18040143

Order 4864.

3006 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#520	C₂	1	2	yes	maximal
2	C₂×C₄×#41×C₁₉	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2351	C₂	1	4	yes	maximal, complemented
4	C₁₉×#2349	C₂	1	12	yes	maximal, complemented
5	C₁₉×#2317	C₂	1	6	yes	maximal, complemented
6	C₁₉×#2285	C₂	1	3	yes	maximal, complemented
7	C₂×C₁₉×#382	C₂	1	3	yes	maximal, complemented
8	C₂²×#41×C₁₉	C₂²	1	1	yes	characteristic
9	C₁₉×#520	—	2	16	no	—
10	C₄×#41×C₁₉	C₄	1	16	yes	complemented
11	C₂×C₄×Q₈×C₁₉	C₂²	1	4	yes	—
12	C₂×C₄×Q₈×C₁₉	C₂²	1	2	yes	—
13	C₂×C₄×D₄×C₁₉	C₂²	1	12	yes	—
14	C₂×C₄×D₄×C₁₉	C₂²	1	6	yes	—
15	C₂×C₁₉×#117	C₂²	1	6	yes	—
16	C₂×C₁₉×#117	C₂²	1	3	yes	—
17	C₂²×C₄⋊C₄×C₁₉	C₂²	1	6	yes	—
18	C₂²×C₄²×C₁₉	C₂²	1	3	yes	—
19	C₁₉×#389	C₂²	1	24	yes	—
20	C₁₉×#385	C₂²	1	24	yes	—
21	C₁₉×#384	C₂²	1	6	yes	—
22	C₁₉×#382	C₂²	1	3	yes	—
23	C₁₉×#382	C₂²	1	6	yes	—
24	C₁₉×#382	C₂²	1	12	yes	—
25	C₁₉×#382	C₂²	1	1	yes	characteristic
26	C₁₉×#380	C₂²	1	12	yes	—
27	C₄×C₄⋊C₄×C₁₉	C₂²	1	12	yes	—
28	C₂×C₁₉×#95	C₂²	1	12	yes	—
29	C₂×#41×C₁₉	C₂×C₄	1	8	yes	—
30	C₂×#41×C₁₉	D₄	1	8	yes	—
31	C₂×#41×C₁₉	Q₈	1	8	yes	—
32	C₂²×Q₈×C₁₉	C₂³	1	1	yes	characteristic
33	C₂²×D₄×C₁₉	C₂³	1	3	yes	—
34	C₂³×C₄×C₁₉	C₂³	1	12	yes	—
35	C₂³×C₄×C₁₉	C₂³	1	6	yes	—
36	C₂³×C₄×C₁₉	C₂³	1	3	yes	—
37	C₄×Q₈×C₁₉	—	2	16	no	—
38	C₄×Q₈×C₁₉	C₂×C₄	1	16	yes	complemented
39	C₄×D₄×C₁₉	—	2	48	no	—
40	C₄×D₄×C₁₉	C₂×C₄	1	48	yes	complemented
41	C₁₉×#117	—	2	24	no	—
42	C₁₉×#117	C₂×C₄	1	24	yes	complemented
43	C₂×C₄⋊C₄×C₁₉	—	2	24	no	—
44	C₂×C₄⋊C₄×C₁₉	C₂³	1	6	yes	—
45	C₂×C₄⋊C₄×C₁₉	C₂³	1	12	yes	—
46	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₁₉	C₂³	1	24	yes	—
48	C₂×C₄⋊C₄×C₁₉	C₂³	1	6	yes	—
49	C₂×C₂²⋊C₄×C₁₉	C₂³	1	12	yes	—
50	C₂×C₂²⋊C₄×C₁₉	C₂³	1	6	yes	—
51	C₂×C₄²×C₁₉	C₂×C₄	1	24	yes	complemented
52	C₂×C₄²×C₁₉	C₂³	1	12	yes	—
53	C₂×C₄²×C₁₉	C₂³	1	6	yes	—
54	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₁₉	C₂³	1	3	yes	—
56	C₁₉×#95	C₂³	1	12	yes	—
57	C₁₉×#95	C₂³	1	24	yes	—
58	C₁₉×#95	C₂³	1	4	yes	—
59	C₂⁴×C₁₉	C₂⁴	1	3	yes	—
60	#41×C₁₉	C₄⋊C₄	1	64	yes	complemented
61	C₂×Q₈×C₁₉	C₂²×C₄	1	4	yes	—
62	C₂×Q₈×C₁₉	C₂×D₄	1	4	yes	—
63	C₂×Q₈×C₁₉	C₂×Q₈	1	4	yes	—
64	C₂×D₄×C₁₉	C₂²×C₄	1	12	yes	—
65	C₂×D₄×C₁₉	C₂×D₄	1	12	yes	—
66	C₂×D₄×C₁₉	C₂×Q₈	1	12	yes	—
67	C₂²×C₄×C₁₉	—	2	24	no	—
68	C₂²×C₄×C₁₉	—	4	12	no	—
69	C₂²×C₄×C₁₉	—	4	24	no	—
70	C₂²×C₄×C₁₉	C₂²×C₄	1	24	yes	—
71	C₂²×C₄×C₁₉	C₂²×C₄	1	12	yes	—
72	C₂²×C₄×C₁₉	C₂×D₄	1	12	yes	—
73	C₂²×C₄×C₁₉	C₂×Q₈	1	12	yes	—
74	C₂²×C₄×C₁₉	C₂⁴	1	4	yes	—
75	C₂²×C₄×C₁₉	C₂⁴	1	12	yes	—
76	C₂²×C₄×C₁₉	C₂⁴	1	2	yes	—
77	C₂²×C₄×C₁₉	C₂⁴	1	6	yes	—
78	C₂²×C₄×C₁₉	C₂⁴	1	3	yes	—
79	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
80	C₄⋊C₄×C₁₉	—	2	12	no	—
81	C₄⋊C₄×C₁₉	—	2	24	no	—
82	C₄⋊C₄×C₁₉	—	2	4	no	—
83	C₄⋊C₄×C₁₉	—	4	24	no	—
84	C₄⋊C₄×C₁₉	C₂²×C₄	1	24	yes	—
85	C₂²⋊C₄×C₁₉	—	2	24	no	—
86	C₂²⋊C₄×C₁₉	C₂²×C₄	1	24	yes	—
87	C₄²×C₁₉	—	2	24	no	—
88	C₄²×C₁₉	C₂²×C₄	1	4	yes	—
89	C₄²×C₁₉	C₂²×C₄	1	12	yes	—
90	C₄²×C₁₉	C₂²×C₄	1	24	yes	—
91	#34184	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₁₉	—	2	12	no	—
93	C₂³×C₁₉	C₂³×C₄	1	6	yes	—
94	C₂³×C₁₉	C₂²×D₄	1	6	yes	—
95	C₂³×C₁₉	C₂²×Q₈	1	6	yes	—
96	C₂³×C₁₉	C₂⁵	1	1	yes	characteristic
97	Q₈×C₁₉	C₂×C₄⋊C₄	1	16	yes	—
98	D₄×C₁₉	C₂×C₄⋊C₄	1	48	yes	—
99	C₂×C₄×C₁₉	—	2	12	no	—
100	C₂×C₄×C₁₉	—	2	4	no	—
101	C₂×C₄×C₁₉	—	2	4	no	—
102	C₂×C₄×C₁₉	—	2	12	no	—
103	C₂×C₄×C₁₉	—	2	8	no	—
104	C₂×C₄×C₁₉	—	2	24	no	—
105	C₂×C₄×C₁₉	—	4	24	no	—
106	C₂×C₄×C₁₉	—	8	24	no	—
107	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	48	yes	—
108	C₂×C₄×C₁₉	C₂³×C₄	1	4	yes	—
109	C₂×C₄×C₁₉	C₂³×C₄	1	12	yes	—
110	C₂×C₄×C₁₉	C₂³×C₄	1	2	yes	—
111	C₂×C₄×C₁₉	C₂³×C₄	1	6	yes	—
112	C₂×C₄×C₁₉	C₂²×D₄	1	6	yes	—
113	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
114	C₂×C₄×C₁₉	C₂²×Q₈	1	2	yes	—
115	C₂×C₄×C₁₉	C₂²×Q₈	1	6	yes	—
116	C₂×#520	C₂×C₁₉	1	2	yes	—
117	C₂×C₄×#41	C₂×C₁₉	1	1	yes	characteristic
118	#2351	C₂×C₁₉	1	4	yes	complemented
119	#2349	C₂×C₁₉	1	12	yes	complemented
120	#2317	C₂×C₁₉	1	6	yes	complemented
121	#2285	C₂×C₁₉	1	3	yes	complemented
122	C₂×#382	C₂×C₁₉	1	3	yes	complemented
123	C₂²×C₁₉	—	2	12	no	—
124	C₂²×C₁₉	—	2	12	no	—
125	C₂²×C₁₉	—	2	12	no	—
126	C₂²×C₁₉	C₂²×C₄⋊C₄	1	12	yes	—
127	C₂²×C₁₉	C₂⁴×C₄	1	1	yes	characteristic
128	C₂²×C₁₉	C₂³×D₄	1	1	yes	characteristic
129	C₂²×C₁₉	C₂³×Q₈	1	1	yes	characteristic
130	C₂²×C₁₉	#585	1	4	yes	—
131	C₄×C₁₉	—	2	4	no	—
132	C₄×C₁₉	—	2	12	no	—
133	C₄×C₁₉	—	4	4	no	—
134	C₄×C₁₉	—	4	12	no	—
135	C₄×C₁₉	C₂²×C₄⋊C₄	1	4	yes	—
136	C₄×C₁₉	C₂²×C₄⋊C₄	1	12	yes	—
137	C₂²×#41	C₂²×C₁₉	1	1	yes	characteristic
138	#520	—	2	16	no	—
139	C₄×#41	C₄×C₁₉	1	16	yes	complemented
140	C₂×C₄×Q₈	C₂²×C₁₉	1	4	yes	—
141	C₂×C₄×Q₈	C₂²×C₁₉	1	2	yes	—
142	C₂×C₄×D₄	C₂²×C₁₉	1	12	yes	—
143	C₂×C₄×D₄	C₂²×C₁₉	1	6	yes	—
144	C₂×#117	C₂²×C₁₉	1	6	yes	—
145	C₂×#117	C₂²×C₁₉	1	3	yes	—
146	C₂²×C₄⋊C₄	C₂²×C₁₉	1	6	yes	—
147	C₂²×C₄²	C₂²×C₁₉	1	3	yes	—
148	#389	C₂²×C₁₉	1	24	yes	—
149	#385	C₂²×C₁₉	1	24	yes	—
150	#384	C₂²×C₁₉	1	6	yes	—
151	#382	C₂²×C₁₉	1	6	yes	—
152	#382	C₂²×C₁₉	1	12	yes	—
153	#382	C₂²×C₁₉	1	3	yes	—
154	#382	C₂²×C₁₉	1	1	yes	characteristic
155	#380	C₂²×C₁₉	1	12	yes	—
156	C₄×C₄⋊C₄	C₂²×C₁₉	1	12	yes	—
157	C₂×#95	C₂²×C₁₉	1	12	yes	—
158	C₂×C₁₉	—	2	12	no	—
159	C₂×C₁₉	C₂³×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₁₉	#3428	1	2	yes	—
161	C₂×C₁₉	#3500	1	2	yes	—
162	C₂×C₁₉	#3504	1	2	yes	—
163	C₂×#41	C₂×C₄×C₁₉	1	8	yes	—
164	C₂×#41	D₄×C₁₉	1	8	yes	—
165	C₂×#41	Q₈×C₁₉	1	8	yes	—
166	C₂²×Q₈	C₂³×C₁₉	1	1	yes	characteristic
167	C₂²×D₄	C₂³×C₁₉	1	3	yes	—
168	C₂³×C₄	C₂³×C₁₉	1	12	yes	—
169	C₂³×C₄	C₂³×C₁₉	1	6	yes	—
170	C₂³×C₄	C₂³×C₁₉	1	3	yes	—
171	C₄×Q₈	—	2	16	no	—
172	C₄×Q₈	C₂×C₄×C₁₉	1	16	yes	complemented
173	C₄×D₄	—	2	48	no	—
174	C₄×D₄	C₂×C₄×C₁₉	1	48	yes	complemented
175	#117	—	2	24	no	—
176	#117	C₂×C₄×C₁₉	1	24	yes	complemented
177	C₂×C₄⋊C₄	—	2	24	no	—
178	C₂×C₄⋊C₄	C₂³×C₁₉	1	6	yes	—
179	C₂×C₄⋊C₄	C₂³×C₁₉	1	12	yes	—
180	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
181	C₂×C₄⋊C₄	C₂³×C₁₉	1	24	yes	—
182	C₂×C₄⋊C₄	C₂³×C₁₉	1	6	yes	—
183	C₂×C₂²⋊C₄	C₂³×C₁₉	1	12	yes	—
184	C₂×C₂²⋊C₄	C₂³×C₁₉	1	6	yes	—
185	C₂×C₄²	C₂×C₄×C₁₉	1	24	yes	complemented
186	C₂×C₄²	C₂³×C₁₉	1	12	yes	—
187	C₂×C₄²	C₂³×C₁₉	1	3	yes	—
188	C₂×C₄²	C₂³×C₁₉	1	6	yes	—
189	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
190	#95	C₂³×C₁₉	1	12	yes	—
191	#95	C₂³×C₁₉	1	24	yes	—
192	#95	C₂³×C₁₉	1	4	yes	—
193	C₁₉	#34184	1	1	yes	characteristic, direct product factor, Sylow
194	C₂⁴	C₂⁴×C₁₉	1	3	yes	—
195	#41	C₄⋊C₄×C₁₉	1	64	yes	complemented
196	C₂×Q₈	C₂²×C₄×C₁₉	1	4	yes	—
197	C₂×Q₈	C₂×D₄×C₁₉	1	4	yes	—
198	C₂×Q₈	C₂×Q₈×C₁₉	1	4	yes	—
199	C₂×D₄	C₂²×C₄×C₁₉	1	12	yes	—
200	C₂×D₄	C₂×D₄×C₁₉	1	12	yes	—
201	C₂×D₄	C₂×Q₈×C₁₉	1	12	yes	—
202	C₂²×C₄	—	2	24	no	—
203	C₂²×C₄	—	4	12	no	—
204	C₂²×C₄	—	4	24	no	—
205	C₂²×C₄	C₂²×C₄×C₁₉	1	24	yes	—
206	C₂²×C₄	C₂²×C₄×C₁₉	1	12	yes	—
207	C₂²×C₄	C₂×D₄×C₁₉	1	12	yes	—
208	C₂²×C₄	C₂×Q₈×C₁₉	1	12	yes	—
209	C₂²×C₄	C₂⁴×C₁₉	1	12	yes	—
210	C₂²×C₄	C₂⁴×C₁₉	1	4	yes	—
211	C₂²×C₄	C₂⁴×C₁₉	1	2	yes	—
212	C₂²×C₄	C₂⁴×C₁₉	1	6	yes	—
213	C₂²×C₄	C₂⁴×C₁₉	1	3	yes	—
214	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
215	C₄⋊C₄	—	2	12	no	—
216	C₄⋊C₄	—	2	24	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	4	24	no	—
219	C₄⋊C₄	C₂²×C₄×C₁₉	1	24	yes	—
220	C₂²⋊C₄	—	2	24	no	—
221	C₂²⋊C₄	C₂²×C₄×C₁₉	1	24	yes	—
222	C₄²	—	2	24	no	—
223	C₄²	C₂²×C₄×C₁₉	1	4	yes	—
224	C₄²	C₂²×C₄×C₁₉	1	12	yes	—
225	C₄²	C₂²×C₄×C₁₉	1	24	yes	—
226	C₂³	—	2	12	no	—
227	C₂³	C₂³×C₄×C₁₉	1	6	yes	—
228	C₂³	C₂²×D₄×C₁₉	1	6	yes	—
229	C₂³	C₂²×Q₈×C₁₉	1	6	yes	—
230	C₂³	C₂⁵×C₁₉	1	1	yes	characteristic
231	Q₈	C₂×C₄⋊C₄×C₁₉	1	16	yes	—
232	D₄	C₂×C₄⋊C₄×C₁₉	1	48	yes	—
233	C₂×C₄	—	2	12	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	12	no	—
237	C₂×C₄	—	2	8	no	—
238	C₂×C₄	—	2	24	no	—
239	C₂×C₄	—	4	24	no	—
240	C₂×C₄	—	8	24	no	—
241	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	48	yes	—
242	C₂×C₄	C₂³×C₄×C₁₉	1	4	yes	—
243	C₂×C₄	C₂³×C₄×C₁₉	1	12	yes	—
244	C₂×C₄	C₂³×C₄×C₁₉	1	2	yes	—
245	C₂×C₄	C₂³×C₄×C₁₉	1	6	yes	—
246	C₂×C₄	C₂²×D₄×C₁₉	1	6	yes	—
247	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
248	C₂×C₄	C₂²×Q₈×C₁₉	1	2	yes	—
249	C₂×C₄	C₂²×Q₈×C₁₉	1	6	yes	—
250	C₂²	—	2	12	no	—
251	C₂²	—	2	12	no	—
252	C₂²	—	2	12	no	—
253	C₂²	C₂²×C₄⋊C₄×C₁₉	1	12	yes	—
254	C₂²	C₂⁴×C₄×C₁₉	1	1	yes	characteristic
255	C₂²	C₂³×D₄×C₁₉	1	1	yes	characteristic
256	C₂²	C₂³×Q₈×C₁₉	1	1	yes	characteristic
257	C₂²	C₁₉×#585	1	4	yes	—
258	C₄	—	2	4	no	—
259	C₄	—	2	12	no	—
260	C₄	—	4	12	no	—
261	C₄	—	4	4	no	—
262	C₄	C₂²×C₄⋊C₄×C₁₉	1	4	yes	—
263	C₄	C₂²×C₄⋊C₄×C₁₉	1	12	yes	—
264	C₂	—	2	12	no	—
265	C₂	C₂³×C₄⋊C₄×C₁₉	1	1	yes	characteristic
266	C₂	C₁₉×#3428	1	2	yes	—
267	C₂	C₁₉×#3500	1	2	yes	—
268	C₂	C₁₉×#3504	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₁₉
Frattini, cosocle	C₂³	C₂⁵×C₁₉
prime radical	C₂²×#41×C₁₉	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₉×#2351	C₂	semidirect
4. C₁₉×#2349	C₂	semidirect
5. C₁₉×#2317	C₂	semidirect
6. C₁₉×#2285	C₂	semidirect
7. C₂×C₁₉×#382	C₂	semidirect
10. C₄×#41×C₁₉	C₄	semidirect
38. C₄×Q₈×C₁₉	C₂×C₄	semidirect
40. C₄×D₄×C₁₉	C₂×C₄	semidirect
42. C₁₉×#117	C₂×C₄	semidirect
51. C₂×C₄²×C₁₉	C₂×C₄	semidirect
60. #41×C₁₉	C₄⋊C₄	semidirect
91. #34184	C₁₉	direct (N × H)
118. #2351	C₂×C₁₉	semidirect
119. #2349	C₂×C₁₉	semidirect
120. #2317	C₂×C₁₉	semidirect
121. #2285	C₂×C₁₉	semidirect
122. C₂×#382	C₂×C₁₉	semidirect
139. C₄×#41	C₄×C₁₉	semidirect
172. C₄×Q₈	C₂×C₄×C₁₉	semidirect
174. C₄×D₄	C₂×C₄×C₁₉	semidirect
176. #117	C₂×C₄×C₁₉	semidirect
185. C₂×C₄²	C₂×C₄×C₁₉	semidirect
193. C₁₉	#34184	direct (N × H)
195. #41	C₄⋊C₄×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₉×#520	2
C₂×C₄×#41×C₁₉	2
C₁₉×#2351	2
C₁₉×#2349	2
C₁₉×#2317	2
C₁₉×#2285	2
C₂×C₁₉×#382	2
#34184	19

Subgroups — C19×#34184 #18040143

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#34184 #18040143