Subgroups — C₁₉×#14651 #18020610

Order 4864.

1810 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#392	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1443	C₂	1	2	yes	maximal, complemented
3	C₁₉×#1440	C₂	1	3	yes	maximal, complemented
4	C₁₉×#1438	C₂	1	6	yes	maximal, complemented
5	C₁₉×#1437	C₂	1	3	yes	maximal, complemented
6	C₂³×D₄×C₁₉	C₂²	1	3	yes	—
7	C₂²×C₂²⋊C₄×C₁₉	C₂²	1	6	yes	—
8	C₂²×C₂²⋊C₄×C₁₉	C₂²	1	3	yes	—
9	C₂²×C₂²⋊C₄×C₁₉	C₂²	1	3	yes	—
10	C₂²×C₂²⋊C₄×C₁₉	C₂²	1	3	yes	—
11	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
12	C₁₉×#392	C₄	1	8	yes	complemented
13	C₂×C₁₉×#95	C₂²	1	6	yes	—
14	C₂×C₁₉×#95	C₂²	1	3	yes	—
15	C₂×C₁₉×#95	C₂²	1	6	yes	—
16	C₂×C₁₉×#95	C₂²	1	1	yes	characteristic
17	C₂⁵×C₁₉	C₂³	1	3	yes	—
18	C₂²×D₄×C₁₉	—	4	3	no	—
19	C₂²×D₄×C₁₉	—	4	3	no	—
20	C₂²×D₄×C₁₉	—	4	6	no	—
21	C₂²×D₄×C₁₉	—	4	6	no	—
22	C₂²×D₄×C₁₉	C₂×C₄	1	12	yes	—
23	C₂³×C₄×C₁₉	C₂³	1	6	yes	—
24	C₂³×C₄×C₁₉	C₂³	1	3	yes	—
25	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
26	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
27	C₂×C₂²⋊C₄×C₁₉	—	2	12	no	—
28	C₂×C₂²⋊C₄×C₁₉	—	4	3	no	—
29	C₂×C₂²⋊C₄×C₁₉	—	4	3	no	—
30	C₂×C₂²⋊C₄×C₁₉	—	4	6	no	—
31	C₂×C₂²⋊C₄×C₁₉	—	4	6	no	—
32	C₂×C₂²⋊C₄×C₁₉	—	4	3	no	—
33	C₂×C₂²⋊C₄×C₁₉	—	4	3	no	—
34	C₂×C₂²⋊C₄×C₁₉	—	4	6	no	—
35	C₂×C₂²⋊C₄×C₁₉	—	4	6	no	—
36	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	12	yes	—
37	C₂×C₄²×C₁₉	—	4	3	no	—
38	C₁₉×#95	—	4	6	no	—
39	C₁₉×#95	—	4	3	no	—
40	C₁₉×#95	—	4	6	no	—
41	C₁₉×#95	C₂×C₄	1	4	yes	—
42	C₂⁴×C₁₉	—	2	6	no	—
43	C₂⁴×C₁₉	—	2	3	no	—
44	C₂⁴×C₁₉	—	2	3	no	—
45	C₂⁴×C₁₉	—	2	6	no	—
46	C₂⁴×C₁₉	—	2	6	no	—
47	C₂⁴×C₁₉	—	2	6	no	—
48	C₂⁴×C₁₉	—	2	6	no	—
49	C₂⁴×C₁₉	—	2	6	no	—
50	C₂⁴×C₁₉	C₂²×C₄	1	6	yes	—
51	C₂⁴×C₁₉	C₂⁴	1	1	yes	characteristic
52	C₂×D₄×C₁₉	—	4	12	no	—
53	C₂×D₄×C₁₉	—	4	24	no	—
54	C₂×D₄×C₁₉	—	4	24	no	—
55	C₂×D₄×C₁₉	—	4	12	no	—
56	C₂×D₄×C₁₉	—	4	12	no	—
57	C₂²×C₄×C₁₉	—	2	3	no	—
58	C₂²×C₄×C₁₉	—	2	6	no	—
59	C₂²×C₄×C₁₉	—	2	3	no	—
60	C₂²×C₄×C₁₉	—	2	3	no	—
61	C₂²×C₄×C₁₉	—	2	3	no	—
62	C₂²×C₄×C₁₉	—	2	6	no	—
63	C₂²×C₄×C₁₉	—	2	6	no	—
64	C₂²×C₄×C₁₉	—	2	6	no	—
65	C₂²×C₄×C₁₉	—	2	6	no	—
66	C₂²×C₄×C₁₉	—	2	6	no	—
67	C₂²×C₄×C₁₉	—	2	6	no	—
68	C₂²×C₄×C₁₉	—	2	6	no	—
69	C₂²×C₄×C₁₉	—	2	6	no	—
70	C₂²×C₄×C₁₉	—	2	6	no	—
71	C₂²×C₄×C₁₉	—	2	6	no	—
72	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₁₉	C₂²×C₄	1	6	yes	—
74	C₂²×C₄×C₁₉	#41	1	4	yes	—
75	C₂²⋊C₄×C₁₉	—	4	12	no	—
76	C₂²⋊C₄×C₁₉	—	4	24	no	—
77	C₂²⋊C₄×C₁₉	—	4	12	no	—
78	C₂²⋊C₄×C₁₉	—	4	12	no	—
79	C₄²×C₁₉	—	4	4	no	—
80	#14651	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂³×C₁₉	—	2	6	no	—
82	C₂³×C₁₉	—	2	6	no	—
83	C₂³×C₁₉	—	2	6	no	—
84	C₂³×C₁₉	—	2	12	no	—
85	C₂³×C₁₉	—	2	12	no	—
86	C₂³×C₁₉	—	4	6	no	—
87	C₂³×C₁₉	—	4	6	no	—
88	C₂³×C₁₉	—	4	6	no	—
89	C₂³×C₁₉	—	4	6	no	—
90	C₂³×C₁₉	—	4	6	no	—
91	C₂³×C₁₉	—	4	6	no	—
92	C₂³×C₁₉	—	4	3	no	—
93	C₂³×C₁₉	—	4	6	no	—
94	C₂³×C₁₉	—	4	6	no	—
95	C₂³×C₁₉	—	4	6	no	—
96	C₂³×C₁₉	—	4	6	no	—
97	C₂³×C₁₉	—	4	3	no	—
98	C₂³×C₁₉	—	4	3	no	—
99	C₂³×C₁₉	—	4	6	no	—
100	C₂³×C₁₉	—	4	3	no	—
101	C₂³×C₁₉	—	4	6	no	—
102	C₂³×C₁₉	C₂³×C₄	1	1	yes	characteristic
103	C₂³×C₁₉	C₂×#41	1	2	yes	—
104	C₂³×C₁₉	H(5,𝔽₂)	1	3	yes	—
105	C₂³×C₁₉	H(5,𝔽₂)	1	6	yes	—
106	C₂³×C₁₉	H(5,𝔽₂)	1	3	yes	—
107	D₄×C₁₉	—	4	48	no	—
108	C₂×C₄×C₁₉	—	2	6	no	—
109	C₂×C₄×C₁₉	—	2	6	no	—
110	C₂×C₄×C₁₉	—	2	12	no	—
111	C₂×C₄×C₁₉	—	2	12	no	—
112	C₂×C₄×C₁₉	—	4	2	no	—
113	C₂×C₄×C₁₉	—	4	1	no	—
114	C₂×C₄×C₁₉	—	4	6	no	—
115	C₂×C₄×C₁₉	—	4	3	no	—
116	C₂×C₄×C₁₉	—	4	6	no	—
117	C₂×C₄×C₁₉	—	4	3	no	—
118	C₂×C₄×C₁₉	—	4	3	no	—
119	C₂×C₄×C₁₉	—	4	6	no	—
120	C₂×C₄×C₁₉	—	4	6	no	—
121	C₂×C₄×C₁₉	—	4	6	no	—
122	C₂×C₄×C₁₉	—	4	6	no	—
123	C₂×C₄×C₁₉	—	4	6	no	—
124	C₂×C₄×C₁₉	—	4	6	no	—
125	C₂×C₄×C₁₉	—	4	6	no	—
126	C₂×#392	C₂×C₁₉	1	1	yes	characteristic
127	#1443	C₂×C₁₉	1	2	yes	complemented
128	#1440	C₂×C₁₉	1	3	yes	complemented
129	#1438	C₂×C₁₉	1	6	yes	complemented
130	#1437	C₂×C₁₉	1	3	yes	complemented
131	C₂²×C₁₉	—	4	6	no	—
132	C₂²×C₁₉	—	4	12	no	—
133	C₂²×C₁₉	—	4	12	no	—
134	C₂²×C₁₉	—	4	6	no	—
135	C₂²×C₁₉	—	4	6	no	—
136	C₂²×C₁₉	—	8	6	no	—
137	C₂²×C₁₉	—	8	6	no	—
138	C₂²×C₁₉	—	8	6	no	—
139	C₂²×C₁₉	—	8	6	no	—
140	C₂²×C₁₉	C₄×#41	1	1	yes	characteristic
141	C₂²×C₁₉	#518	1	3	yes	—
142	C₂²×C₁₉	#518	1	3	yes	—
143	C₂²×C₁₉	#538	1	6	yes	—
144	C₂²×C₁₉	#540	1	6	yes	—
145	C₂²×C₁₉	#560	1	1	yes	characteristic
146	C₂²×C₁₉	#560	1	6	yes	—
147	C₂²×C₁₉	#560	1	3	yes	—
148	C₂²×C₁₉	#560	1	3	yes	—
149	C₂²×C₁₉	H(3,𝔽₄)	1	3	yes	—
150	C₄×C₁₉	—	4	4	no	—
151	C₄×C₁₉	—	4	6	no	—
152	C₄×C₁₉	—	4	2	no	—
153	C₄×C₁₉	—	8	6	no	—
154	C₂³×D₄	C₂²×C₁₉	1	3	yes	—
155	C₂²×C₂²⋊C₄	C₂²×C₁₉	1	6	yes	—
156	C₂²×C₂²⋊C₄	C₂²×C₁₉	1	3	yes	—
157	C₂²×C₂²⋊C₄	C₂²×C₁₉	1	3	yes	—
158	C₂²×C₂²⋊C₄	C₂²×C₁₉	1	3	yes	—
159	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
160	#392	C₄×C₁₉	1	8	yes	complemented
161	C₂×#95	C₂²×C₁₉	1	6	yes	—
162	C₂×#95	C₂²×C₁₉	1	3	yes	—
163	C₂×#95	C₂²×C₁₉	1	6	yes	—
164	C₂×#95	C₂²×C₁₉	1	1	yes	characteristic
165	C₂×C₁₉	—	8	6	no	—
166	C₂×C₁₉	#2333	1	3	yes	—
167	C₂×C₁₉	#2375	1	3	yes	—
168	C₂×C₁₉	#2380	1	1	yes	characteristic
169	C₂×C₁₉	#2825	1	2	yes	—
170	C₂×C₁₉	#2828	1	6	yes	—
171	C₂⁵	C₂³×C₁₉	1	3	yes	—
172	C₂²×D₄	—	4	6	no	—
173	C₂²×D₄	—	4	6	no	—
174	C₂²×D₄	—	4	3	no	—
175	C₂²×D₄	—	4	3	no	—
176	C₂²×D₄	C₂×C₄×C₁₉	1	12	yes	—
177	C₂³×C₄	C₂³×C₁₉	1	6	yes	—
178	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
179	C₂³×C₄	C₂³×C₁₉	1	3	yes	—
180	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
181	C₂×C₂²⋊C₄	—	2	12	no	—
182	C₂×C₂²⋊C₄	—	4	3	no	—
183	C₂×C₂²⋊C₄	—	4	3	no	—
184	C₂×C₂²⋊C₄	—	4	6	no	—
185	C₂×C₂²⋊C₄	—	4	6	no	—
186	C₂×C₂²⋊C₄	—	4	6	no	—
187	C₂×C₂²⋊C₄	—	4	6	no	—
188	C₂×C₂²⋊C₄	—	4	3	no	—
189	C₂×C₂²⋊C₄	—	4	3	no	—
190	C₂×C₂²⋊C₄	C₂×C₄×C₁₉	1	12	yes	—
191	C₂×C₄²	—	4	3	no	—
192	#95	—	4	6	no	—
193	#95	—	4	3	no	—
194	#95	—	4	6	no	—
195	#95	C₂×C₄×C₁₉	1	4	yes	—
196	C₁₉	#14651	1	1	yes	characteristic, direct product factor, Sylow
197	C₂⁴	—	2	6	no	—
198	C₂⁴	—	2	3	no	—
199	C₂⁴	—	2	3	no	—
200	C₂⁴	—	2	6	no	—
201	C₂⁴	—	2	6	no	—
202	C₂⁴	—	2	6	no	—
203	C₂⁴	—	2	6	no	—
204	C₂⁴	—	2	6	no	—
205	C₂⁴	C₂²×C₄×C₁₉	1	6	yes	—
206	C₂⁴	C₂⁴×C₁₉	1	1	yes	characteristic
207	C₂×D₄	—	4	12	no	—
208	C₂×D₄	—	4	24	no	—
209	C₂×D₄	—	4	24	no	—
210	C₂×D₄	—	4	12	no	—
211	C₂×D₄	—	4	12	no	—
212	C₂²×C₄	—	2	3	no	—
213	C₂²×C₄	—	2	6	no	—
214	C₂²×C₄	—	2	6	no	—
215	C₂²×C₄	—	2	6	no	—
216	C₂²×C₄	—	2	3	no	—
217	C₂²×C₄	—	2	6	no	—
218	C₂²×C₄	—	2	3	no	—
219	C₂²×C₄	—	2	3	no	—
220	C₂²×C₄	—	2	6	no	—
221	C₂²×C₄	—	2	6	no	—
222	C₂²×C₄	—	2	6	no	—
223	C₂²×C₄	—	2	6	no	—
224	C₂²×C₄	—	2	6	no	—
225	C₂²×C₄	—	2	6	no	—
226	C₂²×C₄	—	2	6	no	—
227	C₂²×C₄	C₂²×C₄×C₁₉	1	6	yes	—
228	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
229	C₂²×C₄	#41×C₁₉	1	4	yes	—
230	C₂²⋊C₄	—	4	12	no	—
231	C₂²⋊C₄	—	4	24	no	—
232	C₂²⋊C₄	—	4	12	no	—
233	C₂²⋊C₄	—	4	12	no	—
234	C₄²	—	4	4	no	—
235	C₂³	—	2	6	no	—
236	C₂³	—	2	6	no	—
237	C₂³	—	2	6	no	—
238	C₂³	—	2	12	no	—
239	C₂³	—	2	12	no	—
240	C₂³	—	4	6	no	—
241	C₂³	—	4	6	no	—
242	C₂³	—	4	6	no	—
243	C₂³	—	4	6	no	—
244	C₂³	—	4	6	no	—
245	C₂³	—	4	6	no	—
246	C₂³	—	4	3	no	—
247	C₂³	—	4	6	no	—
248	C₂³	—	4	6	no	—
249	C₂³	—	4	6	no	—
250	C₂³	—	4	6	no	—
251	C₂³	—	4	3	no	—
252	C₂³	—	4	3	no	—
253	C₂³	—	4	6	no	—
254	C₂³	—	4	3	no	—
255	C₂³	—	4	6	no	—
256	C₂³	C₂³×C₄×C₁₉	1	1	yes	characteristic
257	C₂³	C₂×#41×C₁₉	1	2	yes	—
258	C₂³	C₁₉×H(5,𝔽₂)	1	3	yes	—
259	C₂³	C₁₉×H(5,𝔽₂)	1	6	yes	—
260	C₂³	C₁₉×H(5,𝔽₂)	1	3	yes	—
261	D₄	—	4	48	no	—
262	C₂×C₄	—	2	6	no	—
263	C₂×C₄	—	2	6	no	—
264	C₂×C₄	—	2	12	no	—
265	C₂×C₄	—	2	12	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	6	no	—
269	C₂×C₄	—	4	3	no	—
270	C₂×C₄	—	4	6	no	—
271	C₂×C₄	—	4	3	no	—
272	C₂×C₄	—	4	3	no	—
273	C₂×C₄	—	4	6	no	—
274	C₂×C₄	—	4	6	no	—
275	C₂×C₄	—	4	6	no	—
276	C₂×C₄	—	4	6	no	—
277	C₂×C₄	—	4	6	no	—
278	C₂×C₄	—	4	6	no	—
279	C₂×C₄	—	4	6	no	—
280	C₂²	—	4	6	no	—
281	C₂²	—	4	12	no	—
282	C₂²	—	4	12	no	—
283	C₂²	—	4	6	no	—
284	C₂²	—	4	6	no	—
285	C₂²	—	8	6	no	—
286	C₂²	—	8	6	no	—
287	C₂²	—	8	6	no	—
288	C₂²	—	8	6	no	—
289	C₂²	C₄×#41×C₁₉	1	1	yes	characteristic
290	C₂²	C₁₉×#518	1	3	yes	—
291	C₂²	C₁₉×#518	1	3	yes	—
292	C₂²	C₁₉×#538	1	6	yes	—
293	C₂²	C₁₉×#540	1	6	yes	—
294	C₂²	C₁₉×#560	1	1	yes	characteristic
295	C₂²	C₁₉×#560	1	6	yes	—
296	C₂²	C₁₉×#560	1	3	yes	—
297	C₂²	C₁₉×#560	1	3	yes	—
298	C₂²	C₁₉×H(3,𝔽₄)	1	3	yes	—
299	C₄	—	4	4	no	—
300	C₄	—	4	6	no	—
301	C₄	—	4	2	no	—
302	C₄	—	8	6	no	—
303	C₂	—	8	6	no	—
304	C₂	C₁₉×#2333	1	3	yes	—
305	C₂	C₁₉×#2375	1	3	yes	—
306	C₂	C₁₉×#2380	1	1	yes	characteristic
307	C₂	C₁₉×#2825	1	2	yes	—
308	C₂	C₁₉×#2828	1	6	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₉	C₂⁴
commutator	C₂³	C₂³×C₄×C₁₉
Frattini, cosocle	C₂⁴	C₂⁴×C₁₉
prime radical	C₂×C₁₉×#392	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#1443	C₂	semidirect
3. C₁₉×#1440	C₂	semidirect
4. C₁₉×#1438	C₂	semidirect
5. C₁₉×#1437	C₂	semidirect
12. C₁₉×#392	C₄	semidirect
80. #14651	C₁₉	direct (N × H)
127. #1443	C₂×C₁₉	semidirect
128. #1440	C₂×C₁₉	semidirect
129. #1438	C₂×C₁₉	semidirect
130. #1437	C₂×C₁₉	semidirect
160. #392	C₄×C₁₉	semidirect
196. C₁₉	#14651	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×#392	2
C₁₉×#1443	2
C₁₉×#1440	2
C₁₉×#1438	2
C₁₉×#1437	2
#14651	19

Subgroups — C19×#14651 #18020610

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#14651 #18020610