Subgroups — C₁₉×#20506 #18026465

Order 4864.

1130 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×SD₁₆×C₁₉	C₂	1	3	yes	maximal
2	C₁₉×#2276	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#2275	C₂	1	1	yes	maximal, characteristic
4	C₄×C₁₉×#106	C₂	1	3	yes	maximal, complemented
5	C₁₉×#1763	C₂	1	3	yes	maximal, complemented
6	C₁₉×#1762	C₂	1	3	yes	maximal
7	C₁₉×#1726	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₁₉	C₂²	1	1	yes	characteristic
9	C₂×C₄×Q₈×C₁₉	C₂²	1	3	yes	—
10	C₂×C₄×D₄×C₁₉	C₂²	1	3	yes	—
11	C₄×SD₁₆×C₁₉	C₄	1	24	yes	complemented
12	C₂×C₁₉×#106	C₂²	1	3	yes	—
13	C₂×C₁₉×#103	C₂²	1	3	yes	—
14	C₂×C₁₉×#102	C₂²	1	3	yes	—
15	C₂×C₁₉×#97	C₂²	1	3	yes	—
16	C₂×C₄×C₈×C₁₉	C₂²	1	3	yes	—
17	C₄×C₄⋊C₄×C₁₉	—	2	3	no	—
18	C₄×C₄⋊C₄×C₁₉	C₂²	1	3	yes	—
19	C₄×C₄⋊C₄×C₁₉	C₂²	1	3	yes	—
20	C₄×C₂²⋊C₄×C₁₉	—	2	3	no	—
21	C₁₉×#376	—	2	1	no	—
22	C₁₉×#376	C₂²	1	1	yes	characteristic
23	C₂×C₁₉×#95	—	2	1	no	—
24	C₁₉×#340	C₂²	1	6	yes	—
25	C₂²×Q₈×C₁₉	C₂³	1	1	yes	characteristic
26	C₂²×D₄×C₁₉	C₂³	1	1	yes	characteristic
27	C₂³×C₄×C₁₉	—	2	3	no	—
28	C₂×SD₁₆×C₁₉	C₂×C₄	1	12	yes	—
29	C₂²×C₈×C₁₉	C₂³	1	3	yes	—
30	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
31	C₄×Q₈×C₁₉	—	2	6	no	—
32	C₄×Q₈×C₁₉	C₂×C₄	1	12	yes	complemented
33	C₄×D₄×C₁₉	—	2	6	no	—
34	C₄×D₄×C₁₉	C₂×C₄	1	12	yes	—
35	C₂×C₄⋊C₄×C₁₉	—	2	3	no	—
36	C₂×C₄⋊C₄×C₁₉	C₂³	1	3	yes	—
37	C₂×C₄⋊C₄×C₁₉	C₂³	1	3	yes	—
38	C₂×C₂²⋊C₄×C₁₉	—	2	3	no	—
39	C₂×C₄²×C₁₉	—	2	3	no	—
40	C₂×C₄²×C₁₉	—	2	3	no	—
41	C₂×C₄²×C₁₉	—	2	3	no	—
42	C₂×C₄²×C₁₉	C₂³	1	3	yes	—
43	C₁₉×#106	C₂×C₄	1	12	yes	complemented
44	C₁₉×#103	C₂×C₄	1	12	yes	complemented
45	C₁₉×#102	C₂×C₄	1	12	yes	—
46	C₁₉×#97	—	2	6	no	—
47	C₄×C₈×C₁₉	C₂×C₄	1	12	yes	complemented
48	C₁₉×#95	—	2	3	no	—
49	C₁₉×#95	—	2	3	no	—
50	C₁₉×#95	D₄	1	1	yes	characteristic
51	C₁₉×#95	D₄	1	3	yes	—
52	C₂⁴×C₁₉	—	2	1	no	—
53	C₂×Q₈×C₁₉	—	2	3	no	—
54	C₂×Q₈×C₁₉	C₂²×C₄	1	6	yes	—
55	C₂×D₄×C₁₉	—	2	3	no	—
56	C₂×D₄×C₁₉	C₂²×C₄	1	6	yes	—
57	C₂²×C₄×C₁₉	—	2	1	no	—
58	C₂²×C₄×C₁₉	—	2	3	no	—
59	C₂²×C₄×C₁₉	—	2	3	no	—
60	C₂²×C₄×C₁₉	—	2	12	no	—
61	C₂²×C₄×C₁₉	—	8	6	no	—
62	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
63	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
64	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
65	SD₁₆×C₁₉	C₄²	1	16	yes	complemented
66	C₂×C₈×C₁₉	—	2	6	no	—
67	C₂×C₈×C₁₉	C₂²×C₄	1	6	yes	—
68	C₂×C₈×C₁₉	C₂²×C₄	1	6	yes	—
69	C₄⋊C₄×C₁₉	—	2	12	no	—
70	C₄⋊C₄×C₁₉	—	2	3	no	—
71	C₄⋊C₄×C₁₉	—	2	3	no	—
72	C₄⋊C₄×C₁₉	C₂²×C₄	1	6	yes	—
73	C₄⋊C₄×C₁₉	C₂²×C₄	1	6	yes	—
74	C₂²⋊C₄×C₁₉	—	2	12	no	—
75	C₄²×C₁₉	—	2	12	no	—
76	C₄²×C₁₉	—	8	3	no	—
77	C₄²×C₁₉	—	8	3	no	—
78	C₄²×C₁₉	C₂²×C₄	1	6	yes	—
79	C₄²×C₁₉	#41	1	6	yes	—
80	#20506	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂³×C₁₉	—	2	6	no	—
82	C₂³×C₁₉	—	4	3	no	—
83	C₂³×C₁₉	—	4	1	no	—
84	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
85	Q₈×C₁₉	—	2	6	no	—
86	Q₈×C₁₉	C₂×C₄²	1	4	yes	—
87	D₄×C₁₉	—	2	6	no	—
88	D₄×C₁₉	C₂×C₄²	1	4	yes	—
89	C₂×C₄×C₁₉	—	2	6	no	—
90	C₂×C₄×C₁₉	—	2	3	no	—
91	C₂×C₄×C₁₉	—	2	3	no	—
92	C₂×C₄×C₁₉	—	2	6	no	—
93	C₂×C₄×C₁₉	—	2	6	no	—
94	C₂×C₄×C₁₉	—	2	6	no	—
95	C₂×C₄×C₁₉	—	4	3	no	—
96	C₂×C₄×C₁₉	—	4	6	no	—
97	C₂×C₄×C₁₉	—	4	3	no	—
98	C₂×C₄×C₁₉	—	8	12	no	—
99	C₂×C₄×C₁₉	C₄×D₄	1	6	yes	—
100	C₂×C₄×C₁₉	C₄×D₄	1	6	yes	—
101	C₂×C₄×C₁₉	C₂³×C₄	1	3	yes	—
102	C₂×C₄×C₁₉	C₂×#41	1	3	yes	—
103	C₈×C₁₉	—	2	6	no	—
104	C₈×C₁₉	C₂×C₄²	1	4	yes	—
105	C₂×C₄×SD₁₆	C₂×C₁₉	1	3	yes	—
106	#2276	C₂×C₁₉	1	1	yes	characteristic, complemented
107	#2275	C₂×C₁₉	1	1	yes	characteristic
108	C₄×#106	C₂×C₁₉	1	3	yes	complemented
109	#1763	C₂×C₁₉	1	3	yes	complemented
110	#1762	C₂×C₁₉	1	3	yes	—
111	#1726	C₂×C₁₉	1	1	yes	characteristic, complemented
112	C₂²×C₁₉	—	2	4	no	—
113	C₂²×C₁₉	—	4	6	no	—
114	C₂²×C₁₉	—	4	6	no	—
115	C₂²×C₁₉	C₂×C₄×D₄	1	3	yes	—
116	C₂²×C₁₉	#577	1	3	yes	—
117	C₂²×C₁₉	#577	1	1	yes	characteristic
118	C₄×C₁₉	—	2	6	no	—
119	C₄×C₁₉	—	2	6	no	—
120	C₄×C₁₉	—	2	4	no	—
121	C₄×C₁₉	—	4	6	no	—
122	C₄×C₁₉	—	4	6	no	—
123	C₄×C₁₉	C₂²×C₄²	1	1	yes	characteristic
124	C₄×C₁₉	C₄×#41	1	3	yes	—
125	C₂²×SD₁₆	C₂²×C₁₉	1	1	yes	characteristic
126	C₂×C₄×Q₈	C₂²×C₁₉	1	3	yes	—
127	C₂×C₄×D₄	C₂²×C₁₉	1	3	yes	—
128	C₄×SD₁₆	C₄×C₁₉	1	24	yes	complemented
129	C₂×#106	C₂²×C₁₉	1	3	yes	—
130	C₂×#103	C₂²×C₁₉	1	3	yes	—
131	C₂×#102	C₂²×C₁₉	1	3	yes	—
132	C₂×#97	C₂²×C₁₉	1	3	yes	—
133	C₂×C₄×C₈	C₂²×C₁₉	1	3	yes	—
134	C₄×C₄⋊C₄	—	2	3	no	—
135	C₄×C₄⋊C₄	C₂²×C₁₉	1	3	yes	—
136	C₄×C₄⋊C₄	C₂²×C₁₉	1	3	yes	—
137	C₄×C₂²⋊C₄	—	2	3	no	—
138	#376	—	2	1	no	—
139	#376	C₂²×C₁₉	1	1	yes	characteristic
140	C₂×#95	—	2	1	no	—
141	#340	C₂²×C₁₉	1	6	yes	—
142	C₂×C₁₉	—	4	4	no	—
143	C₂×C₁₉	C₄²×D₄	1	1	yes	characteristic
144	C₂×C₁₉	#2949	1	3	yes	—
145	C₂×C₁₉	#2952	1	3	yes	—
146	C₂²×Q₈	C₂³×C₁₉	1	1	yes	characteristic
147	C₂²×D₄	C₂³×C₁₉	1	1	yes	characteristic
148	C₂³×C₄	—	2	3	no	—
149	C₂×SD₁₆	C₂×C₄×C₁₉	1	12	yes	—
150	C₂²×C₈	C₂³×C₁₉	1	3	yes	—
151	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
152	C₄×Q₈	—	2	6	no	—
153	C₄×Q₈	C₂×C₄×C₁₉	1	12	yes	complemented
154	C₄×D₄	—	2	6	no	—
155	C₄×D₄	C₂×C₄×C₁₉	1	12	yes	—
156	C₂×C₄⋊C₄	—	2	3	no	—
157	C₂×C₄⋊C₄	C₂³×C₁₉	1	3	yes	—
158	C₂×C₄⋊C₄	C₂³×C₁₉	1	3	yes	—
159	C₂×C₂²⋊C₄	—	2	3	no	—
160	C₂×C₄²	—	2	3	no	—
161	C₂×C₄²	—	2	3	no	—
162	C₂×C₄²	—	2	3	no	—
163	C₂×C₄²	C₂³×C₁₉	1	3	yes	—
164	#106	C₂×C₄×C₁₉	1	12	yes	complemented
165	#103	C₂×C₄×C₁₉	1	12	yes	complemented
166	#102	C₂×C₄×C₁₉	1	12	yes	—
167	#97	—	2	6	no	—
168	C₄×C₈	C₂×C₄×C₁₉	1	12	yes	complemented
169	#95	—	2	3	no	—
170	#95	—	2	3	no	—
171	#95	D₄×C₁₉	1	1	yes	characteristic
172	#95	D₄×C₁₉	1	3	yes	—
173	C₁₉	#20506	1	1	yes	characteristic, direct product factor, Sylow
174	C₂⁴	—	2	1	no	—
175	C₂×Q₈	—	2	3	no	—
176	C₂×Q₈	C₂²×C₄×C₁₉	1	6	yes	—
177	C₂×D₄	—	2	3	no	—
178	C₂×D₄	C₂²×C₄×C₁₉	1	6	yes	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	2	3	no	—
181	C₂²×C₄	—	2	3	no	—
182	C₂²×C₄	—	2	12	no	—
183	C₂²×C₄	—	8	6	no	—
184	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
185	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
186	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
187	SD₁₆	C₄²×C₁₉	1	16	yes	complemented
188	C₂×C₈	—	2	6	no	—
189	C₂×C₈	C₂²×C₄×C₁₉	1	6	yes	—
190	C₂×C₈	C₂²×C₄×C₁₉	1	6	yes	—
191	C₄⋊C₄	—	2	12	no	—
192	C₄⋊C₄	—	2	3	no	—
193	C₄⋊C₄	—	2	3	no	—
194	C₄⋊C₄	C₂²×C₄×C₁₉	1	6	yes	—
195	C₄⋊C₄	C₂²×C₄×C₁₉	1	6	yes	—
196	C₂²⋊C₄	—	2	12	no	—
197	C₄²	—	2	12	no	—
198	C₄²	—	8	3	no	—
199	C₄²	—	8	3	no	—
200	C₄²	C₂²×C₄×C₁₉	1	6	yes	—
201	C₄²	#41×C₁₉	1	6	yes	—
202	C₂³	—	2	6	no	—
203	C₂³	—	4	3	no	—
204	C₂³	—	4	1	no	—
205	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
206	Q₈	—	2	6	no	—
207	Q₈	C₂×C₄²×C₁₉	1	4	yes	—
208	D₄	—	2	6	no	—
209	D₄	C₂×C₄²×C₁₉	1	4	yes	—
210	C₂×C₄	—	2	6	no	—
211	C₂×C₄	—	2	3	no	—
212	C₂×C₄	—	2	3	no	—
213	C₂×C₄	—	2	6	no	—
214	C₂×C₄	—	2	6	no	—
215	C₂×C₄	—	2	6	no	—
216	C₂×C₄	—	4	3	no	—
217	C₂×C₄	—	4	6	no	—
218	C₂×C₄	—	4	3	no	—
219	C₂×C₄	—	8	12	no	—
220	C₂×C₄	C₄×D₄×C₁₉	1	6	yes	—
221	C₂×C₄	C₄×D₄×C₁₉	1	6	yes	—
222	C₂×C₄	C₂³×C₄×C₁₉	1	3	yes	—
223	C₂×C₄	C₂×#41×C₁₉	1	3	yes	—
224	C₈	—	2	6	no	—
225	C₈	C₂×C₄²×C₁₉	1	4	yes	—
226	C₂²	—	2	4	no	—
227	C₂²	—	4	6	no	—
228	C₂²	—	4	6	no	—
229	C₂²	C₂×C₄×D₄×C₁₉	1	3	yes	—
230	C₂²	C₁₉×#577	1	3	yes	—
231	C₂²	C₁₉×#577	1	1	yes	characteristic
232	C₄	—	2	6	no	—
233	C₄	—	2	6	no	—
234	C₄	—	2	4	no	—
235	C₄	—	4	6	no	—
236	C₄	—	4	6	no	—
237	C₄	C₂²×C₄²×C₁₉	1	1	yes	characteristic
238	C₄	C₄×#41×C₁₉	1	3	yes	—
239	C₂	—	4	4	no	—
240	C₂	C₄²×D₄×C₁₉	1	1	yes	characteristic
241	C₂	C₁₉×#2949	1	3	yes	—
242	C₂	C₁₉×#2952	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂²×D₄
commutator	C₄	C₂²×C₄²×C₁₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
prime radical	C₂²×D₄×C₁₉	C₂³
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₁₉×#376 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#2276	C₂	semidirect
4. C₄×C₁₉×#106	C₂	semidirect
5. C₁₉×#1763	C₂	semidirect
7. C₁₉×#1726	C₂	semidirect
11. C₄×SD₁₆×C₁₉	C₄	semidirect
32. C₄×Q₈×C₁₉	C₂×C₄	semidirect
43. C₁₉×#106	C₂×C₄	semidirect
44. C₁₉×#103	C₂×C₄	semidirect
47. C₄×C₈×C₁₉	C₂×C₄	semidirect
65. SD₁₆×C₁₉	C₄²	semidirect
80. #20506	C₁₉	direct (N × H)
106. #2276	C₂×C₁₉	semidirect
108. C₄×#106	C₂×C₁₉	semidirect
109. #1763	C₂×C₁₉	semidirect
111. #1726	C₂×C₁₉	semidirect
128. C₄×SD₁₆	C₄×C₁₉	semidirect
153. C₄×Q₈	C₂×C₄×C₁₉	semidirect
164. #106	C₂×C₄×C₁₉	semidirect
165. #103	C₂×C₄×C₁₉	semidirect
168. C₄×C₈	C₂×C₄×C₁₉	semidirect
173. C₁₉	#20506	direct (N × H)
187. SD₁₆	C₄²×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×SD₁₆×C₁₉	2
C₁₉×#2276	2
C₁₉×#2275	2
C₄×C₁₉×#106	2
C₁₉×#1763	2
C₁₉×#1762	2
C₁₉×#1726	2
#20506	19

Subgroups — C19×#20506 #18026465

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#20506 #18026465