Subgroups — C₁₉×#9876 #18015835

Order 4864.

486 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	C₄×C₁₉×#125	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1842	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₉×#97	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1279	C₂	1	4	yes	maximal
5	C₂×C₁₉×#125	C₄	1	1	yes	characteristic
6	C₂×C₁₉×C₄⋊C₈	—	2	2	no	—
7	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₁₉×#97	—	2	2	no	—
9	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
10	C₁₉×#389	C₄	1	2	yes	—
11	C₁₉×#387	—	2	2	no	—
12	C₁₉×#382	C₄	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₁₉	—	2	2	no	—
14	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
15	C₄³×C₁₉	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₁₉	—	2	2	no	—
17	C₂²×C₈×C₁₉	—	2	2	no	—
18	C₁₉×#125	—	2	4	no	—
19	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
20	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
21	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
22	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₁₉	—	2	2	no	—
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	2	yes	—
27	C₂×C₄²×C₁₉	D₄	1	2	yes	—
28	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
29	C₁₉×C₄⋊C₈	—	2	4	no	—
30	C₁₉×C₄⋊C₈	—	4	4	no	—
31	C₁₉×#97	—	2	8	no	—
32	C₄×C₈×C₁₉	—	2	2	no	—
33	C₄×C₈×C₁₉	Q₈	1	4	yes	—
34	C₁₉×#95	—	2	2	no	—
35	C₁₉×#95	—	2	2	no	—
36	C₁₉×#95	—	2	2	no	—
37	C₂²×C₄×C₁₉	—	2	2	no	—
38	C₂²×C₄×C₁₉	—	2	2	no	—
39	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₁₉	—	2	8	no	—
45	C₂×C₈×C₁₉	—	2	4	no	—
46	C₂×C₈×C₁₉	—	4	2	no	—
47	C₂×C₈×C₁₉	—	4	4	no	—
48	C₄⋊C₄×C₁₉	—	2	8	no	—
49	C₄⋊C₄×C₁₉	—	2	2	no	—
50	C₄⋊C₄×C₁₉	—	2	4	no	—
51	C₄⋊C₄×C₁₉	—	4	4	no	—
52	C₄⋊C₄×C₁₉	M₁₆	1	4	yes	—
53	C₄²×C₁₉	—	2	4	no	—
54	C₄²×C₁₉	—	2	2	no	—
55	C₄²×C₁₉	—	2	2	no	—
56	C₄²×C₁₉	—	2	2	no	—
57	C₄²×C₁₉	—	2	2	no	—
58	C₄²×C₁₉	—	4	2	no	—
59	C₄²×C₁₉	—	4	2	no	—
60	C₄²×C₁₉	C₂×Q₈	1	2	yes	—
61	C₄²×C₁₉	#41	1	2	yes	—
62	#9876	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₉	—	2	2	no	—
65	C₂×C₄×C₁₉	—	2	2	no	—
66	C₂×C₄×C₁₉	—	2	2	no	—
67	C₂×C₄×C₁₉	—	2	2	no	—
68	C₂×C₄×C₁₉	—	2	4	no	—
69	C₂×C₄×C₁₉	—	2	2	no	—
70	C₂×C₄×C₁₉	—	2	2	no	—
71	C₂×C₄×C₁₉	—	4	2	no	—
72	C₂×C₄×C₁₉	—	4	2	no	—
73	C₂×C₄×C₁₉	—	4	2	no	—
74	C₂×C₄×C₁₉	—	4	2	no	—
75	C₂×C₄×C₁₉	—	4	2	no	—
76	C₂×C₄×C₁₉	#104	1	4	yes	—
77	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
78	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
79	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
80	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
82	C₂×C₄×C₁₉	#131	1	2	yes	—
83	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
84	C₂×C₄×C₁₉	#137	1	2	yes	—
85	C₈×C₁₉	—	2	8	no	—
86	C₈×C₁₉	—	8	2	no	—
87	C₄×#125	C₂×C₁₉	1	1	yes	characteristic
88	#1842	C₂×C₁₉	1	1	yes	characteristic
89	C₄×#97	C₂×C₁₉	1	1	yes	characteristic
90	#1279	C₂×C₁₉	1	4	yes	—
91	C₂²×C₁₉	#391	1	1	yes	characteristic
92	C₂²×C₁₉	#407	1	1	yes	characteristic
93	C₂²×C₁₉	#408	1	1	yes	characteristic
94	C₂²×C₁₉	#417	1	2	yes	—
95	C₂²×C₁₉	C₂×#104	1	2	yes	—
96	C₄×C₁₉	—	2	2	no	—
97	C₄×C₁₉	—	2	4	no	—
98	C₄×C₁₉	—	4	2	no	—
99	C₄×C₁₉	—	4	4	no	—
100	C₄×C₁₉	—	8	2	no	—
101	C₄×C₁₉	#446	1	4	yes	—
102	C₄×C₁₉	#490	1	1	yes	characteristic
103	C₄×C₁₉	#491	1	1	yes	characteristic
104	C₄×C₁₉	#501	1	2	yes	—
105	C₂×#125	C₄×C₁₉	1	1	yes	characteristic
106	C₂×C₄⋊C₈	—	2	2	no	—
107	C₂×C₄⋊C₈	C₂²×C₁₉	1	2	yes	—
108	C₂×#97	—	2	2	no	—
109	C₂×C₄×C₈	C₂²×C₁₉	1	2	yes	—
110	#389	C₄×C₁₉	1	2	yes	—
111	#387	—	2	2	no	—
112	#382	C₄×C₁₉	1	1	yes	characteristic
113	C₄×C₄⋊C₄	—	2	2	no	—
114	C₄×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
115	C₄³	C₂²×C₁₉	1	1	yes	characteristic
116	C₂×C₁₉	#1483	1	2	yes	—
117	C₂×C₁₉	#1489	1	2	yes	—
118	C₂×C₁₉	#1985	1	1	yes	characteristic
119	C₂×C₁₉	#1992	1	1	yes	characteristic
120	C₂×C₁₉	#1994	1	1	yes	characteristic
121	C₂²×C₈	—	2	2	no	—
122	C₂²×C₈	—	2	2	no	—
123	#125	—	2	4	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
127	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
128	C₂×C₄²	—	2	2	no	—
129	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
130	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
131	C₂×C₄²	D₄×C₁₉	1	2	yes	—
132	C₂×C₄²	D₄×C₁₉	1	2	yes	—
133	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
134	C₄⋊C₈	—	2	4	no	—
135	C₄⋊C₈	—	4	4	no	—
136	#97	—	2	8	no	—
137	C₄×C₈	—	2	2	no	—
138	C₄×C₈	Q₈×C₁₉	1	4	yes	—
139	#95	—	2	2	no	—
140	#95	—	2	2	no	—
141	#95	—	2	2	no	—
142	C₁₉	#9876	1	1	yes	characteristic, direct product factor, Sylow
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
146	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
147	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
148	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
149	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
150	C₂×C₈	—	2	8	no	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	4	4	no	—
154	C₄⋊C₄	—	2	8	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄⋊C₄	—	4	4	no	—
158	C₄⋊C₄	M₁₆×C₁₉	1	4	yes	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	4	no	—
164	C₄²	—	4	2	no	—
165	C₄²	—	4	2	no	—
166	C₄²	C₂×Q₈×C₁₉	1	2	yes	—
167	C₄²	#41×C₁₉	1	2	yes	—
168	C₂³	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	C₁₉×#104	1	4	yes	—
182	C₂×C₄	C₄×Q₈×C₁₉	1	2	yes	—
183	C₂×C₄	C₁₉×C₂²⋊Q₈	1	2	yes	—
184	C₂×C₄	C₁₉×#124	1	1	yes	characteristic
185	C₂×C₄	C₁₉×C₄⋊Q₈	1	1	yes	characteristic
186	C₂×C₄	C₂×M₁₆×C₁₉	1	2	yes	—
187	C₂×C₄	C₁₉×#131	1	2	yes	—
188	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	2	yes	—
189	C₂×C₄	C₁₉×#137	1	2	yes	—
190	C₈	—	2	8	no	—
191	C₈	—	8	2	no	—
192	C₂²	C₁₉×#391	1	1	yes	characteristic
193	C₂²	C₁₉×#407	1	1	yes	characteristic
194	C₂²	C₁₉×#408	1	1	yes	characteristic
195	C₂²	C₁₉×#417	1	2	yes	—
196	C₂²	C₂×C₁₉×#104	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	2	4	no	—
199	C₄	—	4	2	no	—
200	C₄	—	4	4	no	—
201	C₄	—	8	2	no	—
202	C₄	C₁₉×#446	1	4	yes	—
203	C₄	C₁₉×#490	1	1	yes	characteristic
204	C₄	C₁₉×#491	1	1	yes	characteristic
205	C₄	C₁₉×#501	1	2	yes	—
206	C₂	C₁₉×#1483	1	2	yes	—
207	C₂	C₁₉×#1489	1	2	yes	—
208	C₂	C₁₉×#1985	1	1	yes	characteristic
209	C₂	C₁₉×#1992	1	1	yes	characteristic
210	C₂	C₁₉×#1994	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₉	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
socle, prime radical	C₂³×C₁₉	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₉ ◁ C₄³×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
62. #9876	C₁₉	direct (N × H)
142. C₁₉	#9876	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₁₉×#125	2
C₁₉×#1842	2
C₄×C₁₉×#97	2
C₁₉×#1279	2
#9876	19

Subgroups — C19×#9876 #18015835

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#9876 #18015835