Subgroups — C₁₉×#12844 #18018803

Order 4864.

542 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₈×C₁₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×C₂²⋊C₁₆	C₂	1	2	yes	maximal
3	C₁₉×#1374	C₂	1	4	yes	maximal, complemented
4	C₂³×C₈×C₁₉	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₁₉	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₁₉	C₂²	1	4	yes	—
8	C₂×C₄×C₈×C₁₉	—	2	4	no	—
9	C₂×C₄×C₈×C₁₉	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
11	C₁₉×C₂²⋊C₁₆	—	2	4	no	—
12	C₂³×C₄×C₁₉	C₈	1	1	yes	characteristic
13	C₂³×C₄×C₁₉	C₈	1	1	yes	characteristic
14	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₉	—	2	2	no	—
16	C₂²×C₈×C₁₉	—	2	4	no	—
17	C₂²×C₈×C₁₉	—	2	4	no	—
18	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₁₉	D₄	1	4	yes	—
21	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₁₉	—	2	2	no	—
23	C₂×C₄²×C₁₉	—	2	2	no	—
24	C₂×C₄²×C₁₉	C₈	1	2	yes	—
25	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
26	C₂×C₁₆×C₁₉	—	2	4	no	—
27	C₂×C₁₆×C₁₉	—	2	8	no	—
28	C₄×C₈×C₁₉	—	2	4	no	—
29	C₄×C₈×C₁₉	—	2	4	no	—
30	C₂⁴×C₁₉	C₂×C₈	1	1	yes	characteristic
31	C₂²×C₄×C₁₉	—	2	2	no	—
32	C₂²×C₄×C₁₉	—	2	2	no	—
33	C₂²×C₄×C₁₉	—	2	2	no	—
34	C₂²×C₄×C₁₉	—	2	4	no	—
35	C₂²×C₄×C₁₉	—	2	2	no	—
36	C₂²×C₄×C₁₉	—	2	2	no	—
37	C₂²×C₄×C₁₉	C₁₆	1	4	yes	complemented
38	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₁₉	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₁₉	C₂×C₈	1	2	yes	—
41	C₂²×C₄×C₁₉	C₂×C₈	1	2	yes	—
42	C₂²×C₄×C₁₉	M₁₆	1	2	yes	—
43	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₉	—	2	8	no	—
45	C₂×C₈×C₁₉	—	2	2	no	—
46	C₂×C₈×C₁₉	—	2	2	no	—
47	C₂×C₈×C₁₉	—	2	2	no	—
48	C₂×C₈×C₁₉	—	2	8	no	—
49	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
50	C₂×C₈×C₁₉	M₁₆	1	4	yes	—
51	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
52	C₂×C₈×C₁₉	#41	1	4	yes	—
53	C₄²×C₁₉	—	2	4	no	—
54	C₄²×C₁₉	—	2	2	no	—
55	C₄²×C₁₉	—	2	2	no	—
56	C₁₆×C₁₉	—	4	4	no	—
57	#12844	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂³×C₁₉	—	2	2	no	—
59	C₂³×C₁₉	—	2	2	no	—
60	C₂³×C₁₉	C₂²⋊C₈	1	2	yes	—
61	C₂³×C₁₉	C₂×C₁₆	1	2	yes	—
62	C₂³×C₁₉	M₃₂	1	2	yes	—
63	C₂³×C₁₉	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₁₉	—	2	8	no	—
65	C₂×C₄×C₁₉	—	2	8	no	—
66	C₂×C₄×C₁₉	—	2	2	no	—
67	C₂×C₄×C₁₉	—	2	2	no	—
68	C₂×C₄×C₁₉	—	2	2	no	—
69	C₂×C₄×C₁₉	—	2	2	no	—
70	C₂×C₄×C₁₉	—	2	2	no	—
71	C₂×C₄×C₁₉	—	2	2	no	—
72	C₂×C₄×C₁₉	—	2	2	no	—
73	C₂×C₄×C₁₉	—	2	4	no	—
74	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
75	C₂×C₄×C₁₉	C₂×C₁₆	1	4	yes	—
76	C₂×C₄×C₁₉	M₃₂	1	4	yes	—
77	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₉	#117	1	2	yes	—
79	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
80	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
81	C₈×C₁₉	—	2	2	no	—
82	C₈×C₁₉	—	2	4	no	—
83	C₈×C₁₉	#123	1	4	yes	—
84	C₂²×C₄×C₈	C₂×C₁₉	1	1	yes	characteristic
85	C₂×C₂²⋊C₁₆	C₂×C₁₉	1	2	yes	—
86	#1374	C₂×C₁₉	1	4	yes	complemented
87	C₂²×C₁₉	—	2	2	no	—
88	C₂²×C₁₉	—	2	2	no	—
89	C₂²×C₁₉	—	2	4	no	—
90	C₂²×C₁₉	—	2	4	no	—
91	C₂²×C₁₉	C₂²⋊C₁₆	1	4	yes	—
92	C₂²×C₁₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
93	C₂²×C₁₉	#431	1	2	yes	—
94	C₂²×C₁₉	C₂²×C₁₆	1	1	yes	characteristic
95	C₂²×C₁₉	C₂×M₃₂	1	1	yes	characteristic
96	C₂²×C₁₉	C₂×M₃₂	1	2	yes	—
97	C₄×C₁₉	—	2	2	no	—
98	C₄×C₁₉	—	2	4	no	—
99	C₄×C₁₉	—	2	4	no	—
100	C₄×C₁₉	#381	1	1	yes	characteristic
101	C₄×C₁₉	#407	1	1	yes	characteristic
102	C₄×C₁₉	#432	1	2	yes	—
103	C₂³×C₈	C₄×C₁₉	1	1	yes	characteristic
104	C₂³×C₈	C₂²×C₁₉	1	1	yes	characteristic
105	C₂²×C₄²	C₄×C₁₉	1	1	yes	characteristic
106	C₂²×C₁₆	C₂²×C₁₉	1	4	yes	—
107	C₂×C₄×C₈	—	2	4	no	—
108	C₂×C₄×C₈	C₄×C₁₉	1	2	yes	—
109	C₂×C₄×C₈	C₂²×C₁₉	1	2	yes	—
110	C₂²⋊C₁₆	—	2	4	no	—
111	C₂×C₁₉	—	2	4	no	—
112	C₂×C₁₉	#1817	1	1	yes	characteristic
113	C₂×C₁₉	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
114	C₂×C₁₉	#2112	1	1	yes	characteristic
115	C₂×C₁₉	#2162	1	2	yes	—
116	C₂×C₁₉	#2163	1	2	yes	—
117	C₂³×C₄	C₈×C₁₉	1	1	yes	characteristic
118	C₂³×C₄	C₈×C₁₉	1	1	yes	characteristic
119	C₂³×C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
120	C₂²×C₈	—	2	2	no	—
121	C₂²×C₈	—	2	4	no	—
122	C₂²×C₈	—	2	4	no	—
123	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
124	C₂²×C₈	C₂×C₄×C₁₉	1	2	yes	—
125	C₂²×C₈	D₄×C₁₉	1	4	yes	—
126	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
127	C₂×C₄²	—	2	2	no	—
128	C₂×C₄²	—	2	2	no	—
129	C₂×C₄²	C₈×C₁₉	1	2	yes	—
130	C₂×C₄²	C₂×C₄×C₁₉	1	2	yes	—
131	C₂×C₁₆	—	2	4	no	—
132	C₂×C₁₆	—	2	8	no	—
133	C₄×C₈	—	2	4	no	—
134	C₄×C₈	—	2	4	no	—
135	C₁₉	#12844	1	1	yes	characteristic, direct product factor, Sylow
136	C₂⁴	C₂×C₈×C₁₉	1	1	yes	characteristic
137	C₂²×C₄	—	2	2	no	—
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	C₁₆×C₁₉	1	4	yes	complemented
144	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
145	C₂²×C₄	C₂×C₈×C₁₉	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₈×C₁₉	1	2	yes	—
147	C₂²×C₄	C₂×C₈×C₁₉	1	2	yes	—
148	C₂²×C₄	M₁₆×C₁₉	1	2	yes	—
149	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
150	C₂×C₈	—	2	8	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	8	no	—
155	C₂×C₈	C₂²⋊C₄×C₁₉	1	2	yes	—
156	C₂×C₈	M₁₆×C₁₉	1	4	yes	—
157	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
158	C₂×C₈	#41×C₁₉	1	4	yes	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	4	no	—
162	C₁₆	—	4	4	no	—
163	C₂³	—	2	2	no	—
164	C₂³	—	2	2	no	—
165	C₂³	C₁₉×C₂²⋊C₈	1	2	yes	—
166	C₂³	C₂×C₁₆×C₁₉	1	2	yes	—
167	C₂³	C₁₉×M₃₂	1	2	yes	—
168	C₂³	C₂²×C₈×C₁₉	1	1	yes	characteristic
169	C₂×C₄	—	2	8	no	—
170	C₂×C₄	—	2	8	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	C₁₉×C₂²⋊C₈	1	2	yes	—
180	C₂×C₄	C₂×C₁₆×C₁₉	1	4	yes	—
181	C₂×C₄	C₁₉×M₃₂	1	4	yes	—
182	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
183	C₂×C₄	C₁₉×#117	1	2	yes	—
184	C₂×C₄	C₂×M₁₆×C₁₉	1	2	yes	—
185	C₂×C₄	C₂×M₁₆×C₁₉	1	1	yes	characteristic
186	C₈	—	2	2	no	—
187	C₈	—	2	4	no	—
188	C₈	C₁₉×#123	1	4	yes	—
189	C₂²	—	2	4	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	2	2	no	—
192	C₂²	—	2	2	no	—
193	C₂²	C₁₉×C₂²⋊C₁₆	1	4	yes	—
194	C₂²	C₂×C₁₉×C₂²⋊C₈	1	1	yes	characteristic
195	C₂²	C₁₉×#431	1	2	yes	—
196	C₂²	C₂²×C₁₆×C₁₉	1	1	yes	characteristic
197	C₂²	C₂×C₁₉×M₃₂	1	2	yes	—
198	C₂²	C₂×C₁₉×M₃₂	1	1	yes	characteristic
199	C₄	—	2	2	no	—
200	C₄	—	2	4	no	—
201	C₄	—	2	4	no	—
202	C₄	C₁₉×#381	1	1	yes	characteristic
203	C₄	C₁₉×#407	1	1	yes	characteristic
204	C₄	C₁₉×#432	1	2	yes	—
205	C₂	—	2	4	no	—
206	C₂	C₁₉×#1817	1	1	yes	characteristic
207	C₂	C₂×C₁₉×C₂²⋊C₁₆	1	1	yes	characteristic
208	C₂	C₁₉×#2112	1	1	yes	characteristic
209	C₂	C₁₉×#2162	1	2	yes	—
210	C₂	C₁₉×#2163	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈×C₁₉	C₂³
commutator	C₂²	C₂²×C₁₆×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
socle	C₂³×C₁₉	C₂²×C₈
prime radical	C₂⁴×C₁₉	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₉×#1374	C₂	semidirect
37. C₂²×C₄×C₁₉	C₁₆	semidirect
57. #12844	C₁₉	direct (N × H)
86. #1374	C₂×C₁₉	semidirect
135. C₁₉	#12844	direct (N × H)
143. C₂²×C₄	C₁₆×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₈×C₁₉	2
C₂×C₁₉×C₂²⋊C₁₆	2
C₁₉×#1374	2
#12844	19

Subgroups — C19×#12844 #18018803

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12844 #18018803