Subgroups — C₁₉×#21952 #18027911

Order 4864.

826 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#431	C₂	1	2	yes	maximal
2	C₁₉×#2285	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1939	C₂	1	8	yes	maximal, complemented
4	C₁₉×#1798	C₂	1	4	yes	maximal
5	C₂×C₁₉×#117	C₂²	1	2	yes	—
6	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
7	C₁₉×#431	—	2	8	no	—
8	C₂×C₁₉×C₄⋊C₈	C₂²	1	4	yes	—
9	C₂×C₁₉×C₄⋊C₈	C₂²	1	8	yes	—
10	C₂×C₁₉×C₂²⋊C₈	C₂²	1	4	yes	—
11	C₂×C₄×C₈×C₁₉	C₂²	1	4	yes	—
12	C₁₉×#384	C₄	1	2	yes	—
13	C₁₉×#382	C₄	1	2	yes	—
14	C₁₉×#380	C₄	1	4	yes	—
15	C₄×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
16	C₁₉×#336	C₂²	1	8	yes	—
17	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
18	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
19	C₂²×C₈×C₁₉	C₂³	1	8	yes	—
20	C₁₉×#117	—	4	4	no	—
21	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	8	yes	—
22	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
23	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₉	D₄	1	4	yes	—
27	C₂×C₄²×C₁₉	Q₈	1	4	yes	—
28	C₂×C₄²×C₁₉	C₂³	1	4	yes	—
29	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
30	C₁₉×C₄⋊C₈	—	2	8	no	—
31	C₁₉×C₄⋊C₈	—	4	8	no	—
32	C₁₉×C₂²⋊C₈	—	2	8	no	—
33	C₄×C₈×C₁₉	—	2	8	no	—
34	C₁₉×#95	C₂×C₄	1	8	yes	—
35	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₉	—	2	8	no	—
37	C₂²×C₄×C₁₉	—	2	2	no	—
38	C₂²×C₄×C₁₉	—	2	2	no	—
39	C₂²×C₄×C₁₉	C₄⋊C₄	1	8	yes	—
40	C₂²×C₄×C₁₉	C₂²×C₄	1	8	yes	—
41	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
43	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
44	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
46	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
47	C₂×C₈×C₁₉	—	2	8	no	—
48	C₂×C₈×C₁₉	—	2	8	no	—
49	C₂×C₈×C₁₉	—	2	8	no	—
50	C₄⋊C₄×C₁₉	—	2	16	no	—
51	C₄⋊C₄×C₁₉	—	4	4	no	—
52	C₂²⋊C₄×C₁₉	—	4	4	no	—
53	C₄²×C₁₉	—	2	8	no	—
54	C₄²×C₁₉	C₄⋊C₄	1	4	yes	—
55	C₄²×C₁₉	C₄⋊C₄	1	4	yes	—
56	C₄²×C₁₉	C₂×D₄	1	4	yes	—
57	C₄²×C₁₉	C₂×Q₈	1	4	yes	—
58	#21952	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂³×C₁₉	—	2	1	no	—
60	C₂³×C₁₉	—	2	1	no	—
61	C₂³×C₁₉	—	2	4	no	—
62	C₂³×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
63	C₂³×C₁₉	C₂³×C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₉	—	2	1	no	—
65	C₂×C₄×C₁₉	—	2	1	no	—
66	C₂×C₄×C₁₉	—	2	4	no	—
67	C₂×C₄×C₁₉	—	2	4	no	—
68	C₂×C₄×C₁₉	—	2	4	no	—
69	C₂×C₄×C₁₉	—	2	16	no	—
70	C₂×C₄×C₁₉	—	2	8	no	—
71	C₂×C₄×C₁₉	—	2	16	no	—
72	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
73	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	4	yes	—
74	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	4	yes	—
75	C₂×C₄×C₁₉	#131	1	8	yes	—
76	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
77	C₂×C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
78	C₂×C₄×C₁₉	H(5,𝔽₂)	1	2	yes	—
79	C₂×C₄×C₁₉	2₋¹⁺⁴	1	2	yes	—
80	C₈×C₁₉	—	4	8	no	—
81	C₂×#431	C₂×C₁₉	1	2	yes	—
82	#2285	C₂×C₁₉	1	1	yes	characteristic
83	#1939	C₂×C₁₉	1	8	yes	complemented
84	#1798	C₂×C₁₉	1	4	yes	—
85	C₂²×C₁₉	—	2	2	no	—
86	C₂²×C₁₉	—	2	4	no	—
87	C₂²×C₁₉	—	4	2	no	—
88	C₂²×C₁₉	—	4	2	no	—
89	C₂²×C₁₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₁₉	#518	1	1	yes	characteristic
91	C₂²×C₁₉	#519	1	1	yes	characteristic
92	C₂²×C₁₉	C₂×#131	1	4	yes	—
93	C₄×C₁₉	—	2	2	no	—
94	C₄×C₁₉	—	2	4	no	—
95	C₄×C₁₉	—	4	8	no	—
96	C₄×C₁₉	#424	1	8	yes	—
97	C₄×C₁₉	#537	1	2	yes	—
98	C₄×C₁₉	#544	1	2	yes	—
99	C₂×#117	C₂²×C₁₉	1	2	yes	—
100	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
101	#431	—	2	8	no	—
102	C₂×C₄⋊C₈	C₂²×C₁₉	1	4	yes	—
103	C₂×C₄⋊C₈	C₂²×C₁₉	1	8	yes	—
104	C₂×C₂²⋊C₈	C₂²×C₁₉	1	4	yes	—
105	C₂×C₄×C₈	C₂²×C₁₉	1	4	yes	—
106	#384	C₄×C₁₉	1	2	yes	—
107	#382	C₄×C₁₉	1	2	yes	—
108	#380	C₄×C₁₉	1	4	yes	—
109	C₄×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
110	#336	C₂²×C₁₉	1	8	yes	—
111	C₂×C₁₉	—	4	2	no	—
112	C₂×C₁₉	#2316	1	1	yes	characteristic
113	C₂×C₁₉	C₂×#424	1	2	yes	—
114	C₂×C₁₉	#2979	1	2	yes	—
115	C₂×C₁₉	#2986	1	2	yes	—
116	C₂³×C₄	C₂×C₄×C₁₉	1	2	yes	—
117	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
118	C₂²×C₈	C₂³×C₁₉	1	8	yes	—
119	#117	—	4	4	no	—
120	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	8	yes	—
121	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	4	yes	—
122	C₂×C₂²⋊C₄	C₂×C₄×C₁₉	1	4	yes	—
123	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
124	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
125	C₂×C₄²	D₄×C₁₉	1	4	yes	—
126	C₂×C₄²	Q₈×C₁₉	1	4	yes	—
127	C₂×C₄²	C₂³×C₁₉	1	4	yes	—
128	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
129	C₄⋊C₈	—	2	8	no	—
130	C₄⋊C₈	—	4	8	no	—
131	C₂²⋊C₈	—	2	8	no	—
132	C₄×C₈	—	2	8	no	—
133	#95	C₂×C₄×C₁₉	1	8	yes	—
134	C₁₉	#21952	1	1	yes	characteristic, direct product factor, Sylow
135	C₂⁴	C₂²×C₄×C₁₉	1	1	yes	characteristic
136	C₂²×C₄	—	2	8	no	—
137	C₂²×C₄	—	2	2	no	—
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	C₄⋊C₄×C₁₉	1	8	yes	—
140	C₂²×C₄	C₂²×C₄×C₁₉	1	8	yes	—
141	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
142	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
143	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
144	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
145	C₂²×C₄	C₂×Q₈×C₁₉	1	2	yes	—
146	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
147	C₂×C₈	—	2	8	no	—
148	C₂×C₈	—	2	8	no	—
149	C₂×C₈	—	2	8	no	—
150	C₄⋊C₄	—	2	16	no	—
151	C₄⋊C₄	—	4	4	no	—
152	C₂²⋊C₄	—	4	4	no	—
153	C₄²	—	2	8	no	—
154	C₄²	C₄⋊C₄×C₁₉	1	4	yes	—
155	C₄²	C₄⋊C₄×C₁₉	1	4	yes	—
156	C₄²	C₂×D₄×C₁₉	1	4	yes	—
157	C₄²	C₂×Q₈×C₁₉	1	4	yes	—
158	C₂³	—	2	1	no	—
159	C₂³	—	2	1	no	—
160	C₂³	—	2	4	no	—
161	C₂³	C₂×C₄⋊C₄×C₁₉	1	2	yes	—
162	C₂³	C₂³×C₄×C₁₉	1	1	yes	characteristic
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	16	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	2	8	no	—
170	C₂×C₄	—	2	16	no	—
171	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	2	yes	—
172	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	4	yes	—
173	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	4	yes	—
174	C₂×C₄	C₁₉×#131	1	8	yes	—
175	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
176	C₂×C₄	C₂²×Q₈×C₁₉	1	1	yes	characteristic
177	C₂×C₄	C₁₉×H(5,𝔽₂)	1	2	yes	—
178	C₂×C₄	C₁₉×2₋¹⁺⁴	1	2	yes	—
179	C₈	—	4	8	no	—
180	C₂²	—	2	2	no	—
181	C₂²	—	2	4	no	—
182	C₂²	—	4	2	no	—
183	C₂²	—	4	2	no	—
184	C₂²	C₂²×C₄⋊C₄×C₁₉	1	1	yes	characteristic
185	C₂²	C₁₉×#518	1	1	yes	characteristic
186	C₂²	C₁₉×#519	1	1	yes	characteristic
187	C₂²	C₂×C₁₉×#131	1	4	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	2	4	no	—
190	C₄	—	4	8	no	—
191	C₄	C₁₉×#424	1	8	yes	—
192	C₄	C₁₉×#537	1	2	yes	—
193	C₄	C₁₉×#544	1	2	yes	—
194	C₂	—	4	2	no	—
195	C₂	C₁₉×#2316	1	1	yes	characteristic
196	C₂	C₂×C₁₉×#424	1	2	yes	—
197	C₂	C₁₉×#2979	1	2	yes	—
198	C₂	C₁₉×#2986	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₁₉×#1939	C₂	semidirect
58. #21952	C₁₉	direct (N × H)
83. #1939	C₂×C₁₉	semidirect
134. C₁₉	#21952	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×#431	2
C₁₉×#2285	2
C₁₉×#1939	2
C₁₉×#1798	2
#21952	19

Subgroups — C19×#21952 #18027911

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#21952 #18027911