Subgroups — C₁₉×#20710 #18026669

Order 4864.

1276 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3428	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#410	C₂	1	2	yes	maximal
3	C₁₉×#1828	C₂	1	8	yes	maximal, complemented
4	C₁₉×#1740	C₂	1	4	yes	maximal, complemented
5	C₂²×#41×C₁₉	C₂²	1	1	yes	characteristic
6	C₁₉×#520	—	2	4	no	—
7	C₁₉×#519	C₄	1	2	yes	complemented
8	C₁₉×#518	C₄	1	2	yes	complemented
9	C₁₉×#518	C₂²	1	4	yes	—
10	C₄×#41×C₁₉	—	2	4	no	—
11	C₂×C₁₉×#117	C₄	1	4	yes	complemented
12	C₂×C₁₉×#117	C₂²	1	4	yes	—
13	C₂×C₁₉×#117	C₂²	1	2	yes	—
14	C₁₉×#410	—	2	4	no	—
15	C₁₉×#410	—	2	4	no	—
16	C₂×C₁₉×#99	C₂²	1	8	yes	—
17	C₁₉×#342	C₂²	1	16	yes	—
18	C₂×#41×C₁₉	—	2	4	no	—
19	C₂×#41×C₁₉	—	2	4	no	—
20	C₂×#41×C₁₉	D₄	1	4	yes	—
21	C₂×#41×C₁₉	D₄	1	4	yes	—
22	C₂²×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×D₄×C₁₉	C₂³	1	2	yes	—
25	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
27	C₄×Q₈×C₁₉	—	2	8	no	—
28	C₄×D₄×C₁₉	—	2	8	no	—
29	C₄×D₄×C₁₉	—	2	16	no	—
30	C₁₉×#117	—	2	4	no	—
31	C₁₉×#117	—	2	4	no	—
32	C₁₉×#117	—	2	2	no	—
33	C₁₉×#117	—	2	2	no	—
34	C₁₉×#117	—	2	4	no	—
35	C₁₉×#117	—	2	4	no	—
36	C₁₉×#117	—	2	4	no	—
37	C₁₉×#117	—	2	4	no	—
38	C₁₉×#117	C₂×C₄	1	4	yes	—
39	C₁₉×#117	C₂×C₄	1	8	yes	—
40	C₁₉×#117	C₂×C₄	1	4	yes	—
41	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
42	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
44	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	8	yes	—
45	C₂×C₂²⋊C₄×C₁₉	C₂³	1	8	yes	—
46	C₂×C₂²⋊C₄×C₁₉	C₂³	1	4	yes	—
47	C₂×C₄²×C₁₉	—	2	4	no	—
48	C₂×C₄²×C₁₉	—	2	2	no	—
49	C₂×C₄²×C₁₉	—	2	4	no	—
50	C₁₉×#99	—	2	16	no	—
51	C₂⁴×C₁₉	C₂²×C₄	1	2	yes	—
52	C₂⁴×C₁₉	C₂⁴	1	1	yes	characteristic
53	#41×C₁₉	—	2	8	no	—
54	#41×C₁₉	—	2	8	no	—
55	#41×C₁₉	—	2	8	no	—
56	#41×C₁₉	—	2	8	no	—
57	C₂×Q₈×C₁₉	—	2	4	no	—
58	C₂×Q₈×C₁₉	C₂²⋊C₄	1	4	yes	—
59	C₂×D₄×C₁₉	—	2	8	no	—
60	C₂×D₄×C₁₉	—	2	4	no	—
61	C₂×D₄×C₁₉	C₂²⋊C₄	1	4	yes	—
62	C₂×D₄×C₁₉	C₂×D₄	1	8	yes	—
63	C₂²×C₄×C₁₉	—	2	4	no	—
64	C₂²×C₄×C₁₉	—	2	4	no	—
65	C₂²×C₄×C₁₉	—	2	8	no	—
66	C₂²×C₄×C₁₉	—	2	4	no	—
67	C₂²×C₄×C₁₉	—	2	8	no	—
68	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
69	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
70	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₁₉	#41	1	4	yes	—
76	C₂²×C₄×C₁₉	#41	1	4	yes	—
77	C₄⋊C₄×C₁₉	—	2	4	no	—
78	C₄⋊C₄×C₁₉	—	2	8	no	—
79	C₄⋊C₄×C₁₉	—	2	4	no	—
80	C₄⋊C₄×C₁₉	—	4	4	no	—
81	C₄⋊C₄×C₁₉	—	4	4	no	—
82	C₂²⋊C₄×C₁₉	—	2	8	no	—
83	C₂²⋊C₄×C₁₉	—	2	4	no	—
84	C₂²⋊C₄×C₁₉	—	2	8	no	—
85	C₂²⋊C₄×C₁₉	—	2	8	no	—
86	C₂²⋊C₄×C₁₉	C₂²×C₄	1	8	yes	—
87	C₄²×C₁₉	—	2	4	no	—
88	C₄²×C₁₉	—	2	8	no	—
89	C₄²×C₁₉	—	2	4	no	—
90	C₄²×C₁₉	—	4	4	no	—
91	C₄²×C₁₉	—	4	4	no	—
92	#20710	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂³×C₁₉	—	2	4	no	—
94	C₂³×C₁₉	—	8	1	no	—
95	C₂³×C₁₉	—	8	2	no	—
96	C₂³×C₁₉	C₂×C₂²⋊C₄	1	4	yes	—
97	C₂³×C₁₉	C₂³×C₄	1	1	yes	characteristic
98	C₂³×C₁₉	C₂²×D₄	1	2	yes	—
99	C₂³×C₁₉	C₂×#41	1	4	yes	—
100	Q₈×C₁₉	—	2	8	no	—
101	D₄×C₁₉	—	2	16	no	—
102	D₄×C₁₉	—	2	8	no	—
103	C₂×C₄×C₁₉	—	2	4	no	—
104	C₂×C₄×C₁₉	—	2	4	no	—
105	C₂×C₄×C₁₉	—	2	8	no	—
106	C₂×C₄×C₁₉	—	2	4	no	—
107	C₂×C₄×C₁₉	—	2	8	no	—
108	C₂×C₄×C₁₉	—	4	8	no	—
109	C₂×C₄×C₁₉	—	8	4	no	—
110	C₂×C₄×C₁₉	—	8	8	no	—
111	C₂×C₄×C₁₉	—	8	8	no	—
112	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	4	yes	—
113	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
114	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₉	#117	1	2	yes	—
117	C₂×C₄×C₁₉	#117	1	4	yes	—
118	C₂×C₄×C₁₉	#117	1	2	yes	—
119	C₂×C₄×C₁₉	#123	1	4	yes	—
120	C₂×C₄×C₁₉	#123	1	4	yes	—
121	C₂×C₄×C₁₉	#123	1	4	yes	—
122	C₂×C₄×C₁₉	#123	1	4	yes	—
123	#3428	C₂×C₁₉	1	1	yes	characteristic
124	C₂×#410	C₂×C₁₉	1	2	yes	—
125	#1828	C₂×C₁₉	1	8	yes	complemented
126	#1740	C₂×C₁₉	1	4	yes	complemented
127	C₂²×C₁₉	—	2	4	no	—
128	C₂²×C₁₉	—	4	2	no	—
129	C₂²×C₁₉	—	4	4	no	—
130	C₂²×C₁₉	—	4	1	no	—
131	C₂²×C₁₉	—	4	4	no	—
132	C₂²×C₁₉	—	8	4	no	—
133	C₂²×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
134	C₂²×C₁₉	C₂×#117	1	2	yes	—
135	C₂²×C₁₉	C₂×#123	1	4	yes	—
136	C₄×C₁₉	—	2	4	no	—
137	C₄×C₁₉	—	4	8	no	—
138	C₄×C₁₉	—	8	8	no	—
139	C₄×C₁₉	#381	1	2	yes	—
140	C₄×C₁₉	#381	1	1	yes	characteristic
141	C₄×C₁₉	#381	1	4	yes	—
142	C₄×C₁₉	#381	1	1	yes	characteristic
143	C₂²×#41	C₂²×C₁₉	1	1	yes	characteristic
144	#520	—	2	4	no	—
145	#519	C₄×C₁₉	1	2	yes	complemented
146	#518	C₄×C₁₉	1	2	yes	complemented
147	#518	C₂²×C₁₉	1	4	yes	—
148	C₄×#41	—	2	4	no	—
149	C₂×#117	C₄×C₁₉	1	4	yes	complemented
150	C₂×#117	C₂²×C₁₉	1	4	yes	—
151	C₂×#117	C₂²×C₁₉	1	2	yes	—
152	#410	—	2	4	no	—
153	#410	—	2	4	no	—
154	C₂×#99	C₂²×C₁₉	1	8	yes	—
155	#342	C₂²×C₁₉	1	16	yes	—
156	C₂×C₁₉	—	2	1	no	—
157	C₂×C₁₉	—	2	4	no	—
158	C₂×C₁₉	—	2	2	no	—
159	C₂×C₁₉	—	4	4	no	—
160	C₂×C₁₉	C₂×#381	1	1	yes	characteristic
161	C₂×#41	—	2	4	no	—
162	C₂×#41	—	2	4	no	—
163	C₂×#41	D₄×C₁₉	1	4	yes	—
164	C₂×#41	D₄×C₁₉	1	4	yes	—
165	C₂²×Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
166	C₂²×D₄	C₂×C₄×C₁₉	1	1	yes	characteristic
167	C₂²×D₄	C₂³×C₁₉	1	2	yes	—
168	C₂³×C₄	C₂×C₄×C₁₉	1	2	yes	—
169	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
170	C₄×Q₈	—	2	8	no	—
171	C₄×D₄	—	2	8	no	—
172	C₄×D₄	—	2	16	no	—
173	#117	—	2	4	no	—
174	#117	—	2	4	no	—
175	#117	—	2	2	no	—
176	#117	—	2	2	no	—
177	#117	—	2	4	no	—
178	#117	—	2	4	no	—
179	#117	—	2	4	no	—
180	#117	—	2	4	no	—
181	#117	C₂×C₄×C₁₉	1	4	yes	—
182	#117	C₂×C₄×C₁₉	1	8	yes	—
183	#117	C₂×C₄×C₁₉	1	4	yes	—
184	C₂×C₄⋊C₄	—	2	4	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	4	no	—
187	C₂×C₂²⋊C₄	C₂×C₄×C₁₉	1	8	yes	—
188	C₂×C₂²⋊C₄	C₂³×C₁₉	1	8	yes	—
189	C₂×C₂²⋊C₄	C₂³×C₁₉	1	4	yes	—
190	C₂×C₄²	—	2	4	no	—
191	C₂×C₄²	—	2	2	no	—
192	C₂×C₄²	—	2	4	no	—
193	#99	—	2	16	no	—
194	C₁₉	#20710	1	1	yes	characteristic, direct product factor, Sylow
195	C₂⁴	C₂²×C₄×C₁₉	1	2	yes	—
196	C₂⁴	C₂⁴×C₁₉	1	1	yes	characteristic
197	#41	—	2	8	no	—
198	#41	—	2	8	no	—
199	#41	—	2	8	no	—
200	#41	—	2	8	no	—
201	C₂×Q₈	—	2	4	no	—
202	C₂×Q₈	C₂²⋊C₄×C₁₉	1	4	yes	—
203	C₂×D₄	—	2	8	no	—
204	C₂×D₄	—	2	4	no	—
205	C₂×D₄	C₂²⋊C₄×C₁₉	1	4	yes	—
206	C₂×D₄	C₂×D₄×C₁₉	1	8	yes	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	8	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	8	no	—
212	C₂²×C₄	C₂²⋊C₄×C₁₉	1	4	yes	—
213	C₂²×C₄	C₂²⋊C₄×C₁₉	1	4	yes	—
214	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
215	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
216	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
218	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
219	C₂²×C₄	#41×C₁₉	1	4	yes	—
220	C₂²×C₄	#41×C₁₉	1	4	yes	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	2	8	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	4	4	no	—
225	C₄⋊C₄	—	4	4	no	—
226	C₂²⋊C₄	—	2	8	no	—
227	C₂²⋊C₄	—	2	4	no	—
228	C₂²⋊C₄	—	2	8	no	—
229	C₂²⋊C₄	—	2	8	no	—
230	C₂²⋊C₄	C₂²×C₄×C₁₉	1	8	yes	—
231	C₄²	—	2	4	no	—
232	C₄²	—	2	8	no	—
233	C₄²	—	2	4	no	—
234	C₄²	—	4	4	no	—
235	C₄²	—	4	4	no	—
236	C₂³	—	2	4	no	—
237	C₂³	—	8	1	no	—
238	C₂³	—	8	2	no	—
239	C₂³	C₂×C₂²⋊C₄×C₁₉	1	4	yes	—
240	C₂³	C₂³×C₄×C₁₉	1	1	yes	characteristic
241	C₂³	C₂²×D₄×C₁₉	1	2	yes	—
242	C₂³	C₂×#41×C₁₉	1	4	yes	—
243	Q₈	—	2	8	no	—
244	D₄	—	2	16	no	—
245	D₄	—	2	8	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	8	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	4	8	no	—
252	C₂×C₄	—	8	4	no	—
253	C₂×C₄	—	8	8	no	—
254	C₂×C₄	—	8	8	no	—
255	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	4	yes	—
256	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
257	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
258	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
259	C₂×C₄	C₁₉×#117	1	2	yes	—
260	C₂×C₄	C₁₉×#117	1	2	yes	—
261	C₂×C₄	C₁₉×#117	1	4	yes	—
262	C₂×C₄	C₁₉×#123	1	4	yes	—
263	C₂×C₄	C₁₉×#123	1	4	yes	—
264	C₂×C₄	C₁₉×#123	1	4	yes	—
265	C₂×C₄	C₁₉×#123	1	4	yes	—
266	C₂²	—	2	4	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	4	no	—
269	C₂²	—	4	1	no	—
270	C₂²	—	4	4	no	—
271	C₂²	—	8	4	no	—
272	C₂²	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
273	C₂²	C₂×C₁₉×#117	1	2	yes	—
274	C₂²	C₂×C₁₉×#123	1	4	yes	—
275	C₄	—	2	4	no	—
276	C₄	—	4	8	no	—
277	C₄	—	8	8	no	—
278	C₄	C₁₉×#381	1	1	yes	characteristic
279	C₄	C₁₉×#381	1	2	yes	—
280	C₄	C₁₉×#381	1	1	yes	characteristic
281	C₄	C₁₉×#381	1	4	yes	—
282	C₂	—	2	1	no	—
283	C₂	—	2	4	no	—
284	C₂	—	2	2	no	—
285	C₂	—	4	4	no	—
286	C₂	C₂×C₁₉×#381	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₉	#381
commutator	C₂³	C₂³×C₄×C₁₉
Frattini, cosocle	C₂⁴	C₂⁴×C₁₉
socle	C₂×C₁₉	C₂×#381
prime radical	C₂²×#41×C₁₉	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂³×C₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₉×#1828	C₂	semidirect
4. C₁₉×#1740	C₂	semidirect
7. C₁₉×#519	C₄	semidirect
8. C₁₉×#518	C₄	semidirect
11. C₂×C₁₉×#117	C₄	semidirect
92. #20710	C₁₉	direct (N × H)
125. #1828	C₂×C₁₉	semidirect
126. #1740	C₂×C₁₉	semidirect
145. #519	C₄×C₁₉	semidirect
146. #518	C₄×C₁₉	semidirect
149. C₂×#117	C₄×C₁₉	semidirect
194. C₁₉	#20710	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×#3428	2
C₂×C₁₉×#410	2
C₁₉×#1828	2
C₁₉×#1740	2
#20710	19

Subgroups — C19×#20710 #18026669

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#20710 #18026669