Subgroups — C₁₉×#13451 #18019410

Order 4864.

382 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#366	C₂	1	2	yes	maximal
3	C₁₉×#1942	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1380	C₂	1	2	yes	maximal
5	C₁₉×#1374	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₉×C₄⋊Q₈	—	2	1	no	—
7	C₂²×C₁₆×C₁₉	C₂²	1	2	yes	—
8	C₁₉×#500	—	2	4	no	—
9	C₂×C₁₉×#107	—	2	1	no	—
10	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
11	C₂×C₁₉×#107	C₂²	1	2	yes	—
12	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
14	C₁₉×#389	—	2	1	no	—
15	C₁₉×#366	—	2	4	no	—
16	C₁₉×#340	—	2	1	no	—
17	C₂²×Q₈×C₁₉	—	4	1	no	—
18	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
20	C₁₉×C₄⋊Q₈	—	2	4	no	—
21	C₁₉×C₄⋊Q₈	—	4	2	no	—
22	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
23	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
24	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
25	C₂×C₄⋊C₄×C₁₉	—	4	1	no	—
26	C₂×C₄⋊C₄×C₁₉	—	4	1	no	—
27	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₁₉	—	2	2	no	—
29	C₂×C₁₆×C₁₉	—	2	2	no	—
30	C₂×C₁₆×C₁₉	Q₈	1	4	yes	complemented
31	C₁₉×#107	—	2	4	no	—
32	C₁₉×#107	—	2	2	no	—
33	C₁₉×#107	—	2	1	no	—
34	C₁₉×#107	—	2	2	no	—
35	C₁₉×#107	D₄	1	4	yes	—
36	C₁₉×#107	Q₈	1	2	yes	complemented
37	C₄×C₈×C₁₉	—	2	2	no	—
38	C₁₉×#95	—	4	1	no	—
39	C₂×Q₈×C₁₉	—	4	4	no	—
40	C₂×Q₈×C₁₉	—	8	2	no	—
41	C₂×Q₈×C₁₉	—	8	2	no	—
42	C₂²×C₄×C₁₉	—	4	2	no	—
43	C₂²×C₄×C₁₉	—	4	1	no	—
44	C₂²×C₄×C₁₉	—	4	1	no	—
45	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
46	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₁₉	—	2	1	no	—
49	C₂×C₈×C₁₉	—	2	1	no	—
50	C₂×C₈×C₁₉	Q₁₆	1	2	yes	—
51	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
52	C₂×C₈×C₁₉	C₂×Q₈	1	2	yes	—
53	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
55	C₄⋊C₄×C₁₉	—	2	4	no	—
56	C₄⋊C₄×C₁₉	—	2	2	no	—
57	C₄⋊C₄×C₁₉	—	4	4	no	—
58	C₄⋊C₄×C₁₉	—	4	2	no	—
59	C₄⋊C₄×C₁₉	—	4	1	no	—
60	C₄⋊C₄×C₁₉	—	4	1	no	—
61	C₄⋊C₄×C₁₉	—	4	1	no	—
62	C₄⋊C₄×C₁₉	—	8	2	no	—
63	C₄²×C₁₉	—	2	1	no	—
64	C₄²×C₁₉	—	2	1	no	—
65	C₁₆×C₁₉	—	2	4	no	—
66	#13451	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
68	Q₈×C₁₉	—	8	8	no	—
69	C₂×C₄×C₁₉	—	2	1	no	—
70	C₂×C₄×C₁₉	—	2	1	no	—
71	C₂×C₄×C₁₉	—	2	1	no	—
72	C₂×C₄×C₁₉	—	2	1	no	—
73	C₂×C₄×C₁₉	—	4	4	no	—
74	C₂×C₄×C₁₉	—	4	2	no	—
75	C₂×C₄×C₁₉	—	8	1	no	—
76	C₂×C₄×C₁₉	—	8	2	no	—
77	C₂×C₄×C₁₉	—	8	2	no	—
78	C₂×C₄×C₁₉	—	8	1	no	—
79	C₂×C₄×C₁₉	—	8	1	no	—
80	C₂×C₄×C₁₉	—	8	1	no	—
81	C₂×C₄×C₁₉	—	8	1	no	—
82	C₂×C₄×C₁₉	D₁₆	1	2	yes	—
83	C₂×C₄×C₁₉	Q₃₂	1	2	yes	—
84	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
85	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
88	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
89	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
90	C₈×C₁₉	—	2	2	no	—
91	C₈×C₁₉	C₂²⋊Q₈	1	2	yes	—
92	C₈×C₁₉	C₄⋊Q₈	1	2	yes	—
93	C₂×#500	C₂×C₁₉	1	1	yes	characteristic
94	C₂×#366	C₂×C₁₉	1	2	yes	—
95	#1942	C₂×C₁₉	1	1	yes	characteristic
96	#1380	C₂×C₁₉	1	2	yes	—
97	#1374	C₂×C₁₉	1	1	yes	characteristic
98	C₂²×C₁₉	#447	1	1	yes	characteristic
99	C₂²×C₁₉	#474	1	1	yes	characteristic
100	C₂²×C₁₉	#500	1	1	yes	characteristic
101	C₂²×C₁₉	C₂×D₁₆	1	1	yes	characteristic
102	C₂²×C₁₉	C₂×Q₃₂	1	1	yes	characteristic
103	C₂²×C₁₉	#509	1	1	yes	characteristic
104	C₂²×C₁₉	#510	1	1	yes	characteristic
105	C₄×C₁₉	—	2	2	no	—
106	C₄×C₁₉	—	2	2	no	—
107	C₄×C₁₉	—	8	4	no	—
108	C₄×C₁₉	—	8	2	no	—
109	C₄×C₁₉	—	16	1	no	—
110	C₄×C₁₉	#395	1	1	yes	characteristic
111	C₄×C₁₉	#451	1	1	yes	characteristic
112	C₄×C₁₉	#477	1	1	yes	characteristic
113	C₄×C₁₉	#501	1	1	yes	characteristic
114	C₂×C₄⋊Q₈	—	2	1	no	—
115	C₂²×C₁₆	C₂²×C₁₉	1	2	yes	—
116	#500	—	2	4	no	—
117	C₂×#107	—	2	1	no	—
118	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
119	C₂×#107	C₂²×C₁₉	1	2	yes	—
120	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
121	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
122	#389	—	2	1	no	—
123	#366	—	2	4	no	—
124	#340	—	2	1	no	—
125	C₂×C₁₉	#2057	1	1	yes	characteristic
126	C₂×C₁₉	#2184	1	1	yes	characteristic
127	C₂×C₁₉	#2188	1	1	yes	characteristic
128	C₂×C₁₉	#2224	1	1	yes	characteristic
129	C₂×C₁₉	#2227	1	1	yes	characteristic
130	C₂×C₁₉	#2252	1	1	yes	characteristic
131	C₂×C₁₉	#2255	1	1	yes	characteristic
132	C₂²×Q₈	—	4	1	no	—
133	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
134	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
135	C₄⋊Q₈	—	2	4	no	—
136	C₄⋊Q₈	—	4	2	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	—	2	1	no	—
139	C₂×C₄⋊C₄	—	2	1	no	—
140	C₂×C₄⋊C₄	—	4	1	no	—
141	C₂×C₄⋊C₄	—	4	1	no	—
142	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
143	C₂×C₁₆	—	2	2	no	—
144	C₂×C₁₆	—	2	2	no	—
145	C₂×C₁₆	Q₈×C₁₉	1	4	yes	complemented
146	#107	—	2	4	no	—
147	#107	—	2	2	no	—
148	#107	—	2	1	no	—
149	#107	—	2	2	no	—
150	#107	D₄×C₁₉	1	4	yes	—
151	#107	Q₈×C₁₉	1	2	yes	complemented
152	C₄×C₈	—	2	2	no	—
153	#95	—	4	1	no	—
154	C₁₉	#13451	1	1	yes	characteristic, direct product factor, Sylow
155	C₂×Q₈	—	4	4	no	—
156	C₂×Q₈	—	8	2	no	—
157	C₂×Q₈	—	8	2	no	—
158	C₂²×C₄	—	4	1	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	—	4	1	no	—
161	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
162	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
163	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	Q₁₆×C₁₉	1	2	yes	—
167	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
168	C₂×C₈	C₂×Q₈×C₁₉	1	2	yes	—
169	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
170	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	4	4	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	4	1	no	—
176	C₄⋊C₄	—	4	1	no	—
177	C₄⋊C₄	—	4	1	no	—
178	C₄⋊C₄	—	8	2	no	—
179	C₄²	—	2	1	no	—
180	C₄²	—	2	1	no	—
181	C₁₆	—	2	4	no	—
182	C₂³	C₂×D₈×C₁₉	1	1	yes	characteristic
183	Q₈	—	8	8	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	4	4	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	8	1	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	8	2	no	—
193	C₂×C₄	—	8	1	no	—
194	C₂×C₄	—	8	1	no	—
195	C₂×C₄	—	8	1	no	—
196	C₂×C₄	—	8	1	no	—
197	C₂×C₄	C₁₉×D₁₆	1	2	yes	—
198	C₂×C₄	C₁₉×Q₃₂	1	2	yes	—
199	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
200	C₂×C₄	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
201	C₂×C₄	C₁₉×C₄⋊Q₈	1	1	yes	characteristic
202	C₂×C₄	C₂×Q₁₆×C₁₉	1	1	yes	characteristic
203	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
204	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	C₁₉×C₂²⋊Q₈	1	2	yes	—
207	C₈	C₁₉×C₄⋊Q₈	1	2	yes	—
208	C₂²	C₁₉×#447	1	1	yes	characteristic
209	C₂²	C₁₉×#474	1	1	yes	characteristic
210	C₂²	C₁₉×#500	1	1	yes	characteristic
211	C₂²	C₂×C₁₉×D₁₆	1	1	yes	characteristic
212	C₂²	C₂×C₁₉×Q₃₂	1	1	yes	characteristic
213	C₂²	C₁₉×#509	1	1	yes	characteristic
214	C₂²	C₁₉×#510	1	1	yes	characteristic
215	C₄	—	2	2	no	—
216	C₄	—	2	2	no	—
217	C₄	—	8	4	no	—
218	C₄	—	8	2	no	—
219	C₄	—	16	1	no	—
220	C₄	C₁₉×#395	1	1	yes	characteristic
221	C₄	C₁₉×#451	1	1	yes	characteristic
222	C₄	C₁₉×#477	1	1	yes	characteristic
223	C₄	C₁₉×#501	1	1	yes	characteristic
224	C₂	C₁₉×#2057	1	1	yes	characteristic
225	C₂	C₁₉×#2184	1	1	yes	characteristic
226	C₂	C₁₉×#2188	1	1	yes	characteristic
227	C₂	C₁₉×#2224	1	1	yes	characteristic
228	C₂	C₁₉×#2227	1	1	yes	characteristic
229	C₂	C₁₉×#2252	1	1	yes	characteristic
230	C₂	C₁₉×#2255	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₉	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
30. C₂×C₁₆×C₁₉	Q₈	semidirect
36. C₁₉×#107	Q₈	semidirect
66. #13451	C₁₉	direct (N × H)
145. C₂×C₁₆	Q₈×C₁₉	semidirect
151. #107	Q₈×C₁₉	semidirect
154. C₁₉	#13451	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×#500	2
C₂×C₁₉×#366	2
C₁₉×#1942	2
C₁₉×#1380	2
C₁₉×#1374	2
#13451	19

Subgroups — C19×#13451 #18019410

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13451 #18019410